INTRODUCTION TO MOLECULAR MODELING -...
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INTRODUCTION TO MOLECULAR MODELING Rajmund Kaźmierkiewicz, PhD Laboratory of Biomolecular Systems Simulations IFB UG-MUG Course book prepared as part of the project: „Kształcimy najlepszych kompleksowy program rozwoju doktorantów, młodych doktorów oraz akademickiej kadry dydaktycznej Uniwersytetu Gdaoskiego” Project no: UDA-POKL.04.01.01-00-017/10-00 Intercollegiate Faculty of Biotechnology UG-MUG Gdańsk 2011
Transcript of INTRODUCTION TO MOLECULAR MODELING -...
INTRODUCTION TO MOLECULAR MODELING
Rajmund Kaźmierkiewicz, PhD
Laboratory of Biomolecular Systems Simulations
IFB UG-MUG
Course book prepared as part of the project: „Kształcimy najlepszych kompleksowy program rozwoju doktorantów, młodych doktorów oraz akademickiej kadry dydaktycznej Uniwersytetu Gdaoskiego”
Project no: UDA-POKL.04.01.01-00-017/10-00
Intercollegiate Faculty of Biotechnology UG-MUG
Gdańsk 2011
Introduction to molecular modeling 3 by Rajmund Kaźmierkiewicz
Table of contents:
1. INTRODUCTION ....................................................................................................................................... 9
2. LINUX .....................................................................................................................................................10
3. WHAT IS MODELING? .............................................................................................................................10
4. MOLECULAR MECHANICS: LIMITATIONS ................................................................................................11
5. GRAPHICAL REPRESENTATIONS OF MOLECULAR STRUCTURE .................................................................11
6. MOLECULAR STRUCTURE DESCRIPTION ..................................................................................................12
6.1. CARTESIAN COORDINATES ......................................................................................................................... 12
6.2. INTERNAL COORDINATES .......................................................................................................................... 12
6.3. ALTERNATIVE DESCRIPTION OF THE MOLECULAR STRUCTURE ............................................................................ 14
7. ENERGY EXPRESSIONS IN MOLECULAR MECHANICS ...............................................................................15
7.1. POTENTIAL CLASSIFICATION ....................................................................................................................... 15
7.2. THE EMPIRICAL ENERGY FUNCTION (OR FORCE FIELD) ................................................................................... 16
7.3. BOND STRETCHING .................................................................................................................................. 16
7.4. ANGLE BENDING ..................................................................................................................................... 16
7.5. BOND ROTATION (TORSION) ..................................................................................................................... 16
7.6. NON-BONDED INTERACTIONS (VAN DER WAALS) .......................................................................................... 17
7.7. GAY-BERNE POTENTIAL ............................................................................................................................ 17
7.8. NON-BONDED INTERACTIONS (ELECTROSTATIC) ............................................................................................ 17
7.9. THE TOTAL POTENTIAL ENERGY .................................................................................................................. 19
8. EMPIRICAL FORCE FIELD .........................................................................................................................19
8.1. FITTING PARAMETERS .............................................................................................................................. 20
8.2. FITTING CHARGES .................................................................................................................................. 20
8.3. PROBLEMS WITH THE INFINITE RANGE OF NON-BONDED INTERACTIONS ............................................................. 21
8.4. PROBLEMS WITH HIGH VALUES OF ELECTROSTATIC POTENTIAL .......................................................................... 23
8.5. DIELECTRIC PERMITTIVITY ......................................................................................................................... 25
9. OPTIMIZATION OF A STRUCTURE ...........................................................................................................25
9.1. SUCCESSIVE COORDINATE DIRECTION METHOD ............................................................................................ 26
9.2. NEWTON’S METHOD FOR FINDING A MINIMUM ........................................................................................... 27
9.3. STEEPEST DESCENTS ................................................................................................................................ 27
9.3.1. Steepest Descent Method ................................................................................................................ 28
9.4. CONJUGATE GRADIENT METHOD ............................................................................................................... 28
9.4.1. Conjugate gradient method without explicit knowledge of the Hessian matrix ............................. 29
9.5. THE BFGS ALGORITHM FOR UNCONSTRAINED OPTIMIZATION .......................................................................... 30
9.6. TESTING MINIMA ................................................................................................................................... 30
9.7. MINIMIZATION AND MOLECULAR MECHANICS ............................................................................................. 31
10. MOLECULAR DYNAMICS SIMULATIONS ..............................................................................................31
10.1. EQUILIBRATION ...................................................................................................................................... 31
10.2. VELOCITIES IN MD .................................................................................................................................. 32
10.3. DYNAMICS: EQUATIONS OF MOTION .......................................................................................................... 32
4 Introduction to molecular modeling by Rajmund Kaźmierkiewicz
10.3.1. Numerical Solution of the Equations of Motion .......................................................................... 32
10.3.2. The symplectic integration of equations of motion ..................................................................... 32
10.3.3. The Verlet Algorithm ................................................................................................................... 33
10.3.4. Leapfrog Verlet ............................................................................................................................ 33
10.3.5. Velocity Verlet .............................................................................................................................. 34
10.3.6. Gear Predictor-Corrector Method ................................................................................................ 34
10.4. THE TIME STEP ....................................................................................................................................... 35
10.5. THE NEED FOR FASTER COMPUTERS ............................................................................................................. 35
10.6. PHASE SPACE ......................................................................................................................................... 35
10.7. CALCULATION OF AVERAGE PROPERTIES ...................................................................................................... 36
10.8. FLUCTUATIONS ....................................................................................................................................... 37
10.9. OTHER WAYS OF EXPERIMENTAL VERIFICATION OF RESULTS OF MOLECULAR MECHANICS ........................................ 38
10.10. THE PRESSURE ........................................................................................................................................ 39
10.11. THE RADIAL DISTRIBUTION FUNCTION .......................................................................................................... 39
10.12. CALCULATION OF DYNAMIC PROPERTIES FROM MOLECULAR DYNAMICS SIMULATIONS ......................................... 41
10.13. CORRELATIONS AND THE CORRELATION TIME ............................................................................................... 41
10.14. CORRELATION FUNCTIONS AND PROPERTIES ................................................................................................. 41
10.15. THE TIME CORRELATION FUNCTION ............................................................................................................ 41
10.16. VELOCITY AUTOCORRELATION FUNCTION ..................................................................................................... 42
10.17. CALCULATING THE DIFFUSION COEFFICIENT FROM THE MEAN-SQUARE DISPLACEMENT........................................... 43
10.17.1. The Mean Square Displacement .................................................................................................. 43
10.17.2. What is the mean square distance and why is it significant? ...................................................... 44
10.17.3. The Mean Squared Displacement and the Velocity Autocorrelation Function ............................ 45
10.18. MOLECULAR DYNAMICS SIMULATIONS OF LIQUID WATER ................................................................................ 46
10.18.1. Dielectric Relaxation .................................................................................................................... 47
10.18.2. Three-site models for water......................................................................................................... 47
10.18.3. Implicit Treatment of Solvation ................................................................................................... 47
10.19. CONFORMATIONAL SEARCHING, QUENCH DYNAMICS ..................................................................................... 48
10.19.1. Protocol for conformational search: quenched molecular dynamics .......................................... 48
10.20. CONSTRAINTS ......................................................................................................................................... 48
10.20.1. Restrained dynamics as a tool in NMR structure determination ................................................. 48
10.20.2. Use of constraints to increase the integration step ..................................................................... 49
10.20.3. SHAKE and minimization ............................................................................................................. 49
10.21. BOUNDARY CONDITIONS ........................................................................................................................... 50
10.22. BOUNDARY CONDITIONS, „SPECIAL” TREATMENT OF ELECTROSTATIC INTERACTIONS ............................................. 52
10.22.1. Reaction field method .................................................................................................................. 52
10.22.2. Ewald summation ........................................................................................................................ 53
10.22.3. Scaling and other related methods .............................................................................................. 54
10.23. PRACTICAL TIPS FOR SETTING UP MD ......................................................................................................... 54
10.24. MD SIMULATION PROTOCOL ..................................................................................................................... 54
10.24.1. Sample MD simulations work-flow .............................................................................................. 55
10.25. THE „HEATING” DYNAMICS STAGE, THE TEMPERATURE CONTROL ...................................................................... 55
10.25.1. Temperature ................................................................................................................................ 55
10.25.2. MD simulations with a temperature bath. .................................................................................. 55
10.25.3. Barriers, Temperature and Timescales ........................................................................................ 56
10.25.4. Berendsen thermostat ................................................................................................................. 56
10.25.5. Nosé-Hoover thermostat ............................................................................................................. 57
10.25.6. Andersén thermostat ................................................................................................................... 57
Introduction to molecular modeling 5 by Rajmund Kaźmierkiewicz
10.25.7. Trivial Temperature scaling ......................................................................................................... 57
10.25.8. Berendsen barostat ..................................................................................................................... 57
10.25.9. Andersén barostat ....................................................................................................................... 58
10.25.10. Parrinello-Rahman barostat ................................................................................................... 58
10.25.11. So which ones to use? ............................................................................................................. 58
10.26. „PRODUCTION RUN” PROTOCOL, „HEATING” DYNAMICS ................................................................................. 58
10.27. THE REPLICA-EXCHANGE ALGORITHM ......................................................................................................... 59
10.28. SIMULATED ANNEALING ........................................................................................................................... 60
10.29. LANGEVIN EQUATION OF MOTION .............................................................................................................. 60
10.30. BROWNIAN DYNAMICS (BD)..................................................................................................................... 61
11. MONTE CARLO METHOD ....................................................................................................................62
11.1. WHAT IS MONTE CARLO? ........................................................................................................................ 62
11.2. MONTE CARLO (MC) SIMULATION ............................................................................................................ 62
11.2.1. Evolution of Monte Carlo methods so far… ................................................................................. 63
11.2.2. Markov chains ............................................................................................................................. 63
11.2.3. Markov chain Monte Carlo .......................................................................................................... 64
11.3. IMPLEMENTATION OF THE METROPOLIS ALGORITHM (IT IS A KIND OF MARKOV CHAIN) ......................................... 64
11.4. IMPLEMENTATION OF THE METROPOLIS ALGORITHM ..................................................................................... 64
11.5. ADVANTAGES OF METROPOLIS MC SIMULATIONS ......................................................................................... 65
11.6. MONTE CARLO MOVES ............................................................................................................................ 65
11.7. GENETIC ALGORITHMS IN MOLECULAR MODELING ....................................................................................... 65
11.7.1. Genetic algorithms ...................................................................................................................... 66
11.7.2. Guided random search ................................................................................................................ 66
11.7.3. Evolutionary computation ........................................................................................................... 66
11.7.4. Creation of a population of chromosomes .................................................................................. 67
11.7.5. Definition of a fitness function .................................................................................................... 67
11.7.6. Genetic manipulation of the chromosomes ................................................................................ 67
11.7.7. Applications of genetic algorithms in quantitative structure-activity relationships (QSAR) and
drug design ................................................................................................................................................... 69
12. MOLECULAR DOCKING .......................................................................................................................70
12.1. TYPES OF COMPATIBILITIES ........................................................................................................................ 71
12.2. FINDING THE PLACE AND THE ORIENTATION OF THE INTERACTIONS .................................................................... 71
12.3. COMPUTATIONAL TIME ............................................................................................................................ 71
12.4. RIGIDITY VS. FLEXIBILITY ........................................................................................................................... 71
12.5. FLEXIBILITY ............................................................................................................................................ 72
12.6. BOUND AND UNBOUND DOCKING ............................................................................................................... 72
12.7. COMPONENTS OF THE PROBLEM ................................................................................................................ 72
12.8. ASPECTS OF DOCKING .............................................................................................................................. 72
12.9. DOCKING AND DE NOVO DESIGN METHODS .................................................................................................. 72
12.10. ADDITIONAL CHALLENGES IN DOCKING ....................................................................................................... 72
12.11. PROTEIN FLEXIBILITY AND ITS INFLUENCE ON LIGAND BINDING ........................................................................ 73
12.12. CLUSTERING .......................................................................................................................................... 73
12.13. SEARCH ALGORITHMS .............................................................................................................................. 74
12.14. SCORING FUNCTIONS ............................................................................................................................... 74
12.15. RIGID PROTEIN DOCKING ......................................................................................................................... 75
12.16. PARTIAL PROTEIN FLEXIBILITY .................................................................................................................... 75
12.17. FULL PROTEIN FLEXIBILITY ........................................................................................................................ 76
6 Introduction to molecular modeling by Rajmund Kaźmierkiewicz
12.18. EXAMPLES OF DOCKING PROGRAMS ........................................................................................................... 77
12.19. ACTIVATED DYNAMICS ............................................................................................................................. 77
13. HYBRID QM/MM METHOD ................................................................................................................ 78
13.1. BOUNDARY TREATMENT ........................................................................................................................... 78
13.2. IMPROVED BOND TREATMENTS .................................................................................................................. 79
13.3. OTHER APPROACHES ................................................................................................................................ 79
13.4. AVAILABLE SOFTWARE .............................................................................................................................. 80
13.5. AN EXAMPLE - A DIELS-ALDER REACTION .................................................................................................... 80
13.6. GEOMETRY OPTIMIZATION AFTER MM DYNAMICS ........................................................................................ 80
13.7. OSCILLATIONS OF ACTIVE SITES AFTER MM DYNAMICS ................................................................................. 81
13.8. COMPLEX REACTION AFTER MM DYNAMICS ............................................................................................... 81
13.9. ELECTRONIC EXCITATION IN FIXED MM MATRIX .......................................................................................... 82
14. NORMAL MODES AND PRINCIPAL COMPONENT ANALYSIS ................................................................ 82
14.1. ONE MASS AND TWO SPRINGS ................................................................................................................... 83
14.2. TWO MASSES ......................................................................................................................................... 83
14.3. N ATOMS AND POTENTIAL ENERGY FUNCTION ............................................................................................... 85
14.4. FOR THE MULTI-ATOM MOLECULE .............................................................................................................. 86
14.5. HARMONIC APPROXIMATION IN ANHARMONIC SYSTEMS AND IN REAL PROTEINS ................................................. 87
14.6. FORCE CONSTANTS .................................................................................................................................. 88
14.7. NMA USING MOLECULAR MECHANICS, REDUCING THE NUMBER OF VARIABLES. ................................................ 89
14.8. USING NORMAL MODE ANALYSIS TO MODEL PROTEIN DYNAMICS ................................................................... 91
14.9. THE EQUILIBRIUM CORRELATION BETWEEN FLUCTUATIONS .............................................................................. 91
14.10. CALCULATION OF PROTEIN B-FACTORS ........................................................................................................ 92
14.11. EXAMPLES OF APPLICATIONS USING NORMAL MODE ANALYSIS TO MODEL PROTEIN DYNAMICS ............................ 93
14.11.1. Collective dynamics of protofilaments in microtubules: .............................................................. 93
14.11.2. Applications of NMA : ribosome (Application to EM Data) ......................................................... 94
14.11.3. Applications of Normal Mode Analysis to experimental EM maps .............................................. 94
14.12. WHAT ARE THE LIMITATIONS OF NMA ...................................................................................................... 95
14.13. THE PRINCIPAL COMPONENT ANALYSIS (PCA) METHOD ................................................................................. 96
14.13.1. Collective coordinates .................................................................................................................. 96
14.13.2. Building the covariance matrix from your trajectory .................................................................. 96
14.13.3. Visualizing principal components (PC’s) ...................................................................................... 97
14.13.4. Validation of PCA ........................................................................................................................ 98
15. USES OF FREE ENERGY........................................................................................................................ 98
15.1. METHODS AND APPLICATIONS ................................................................................................................... 99
15.2. THERMODYNAMIC INTEGRATION ................................................................................................................ 99
15.3. PERTURBATION METHOD ....................................................................................................................... 100
15.4. THERMODYNAMIC INTEGRATION AND SLOW GROWTH ................................................................................. 101
15.5. THERMODYNAMIC CYCLES ...................................................................................................................... 101
15.6. APPLICATION OF FREE ENERGY SIMULATIONS, PARTITIONING THE FREE ENERGY ................................................. 102
15.7. POTENTIAL OF MEAN FORCE CALCULATIONS .............................................................................................. 103
15.7.1. Potential of Mean Force calculation .......................................................................................... 104
15.8. SIMPLE UMBRELLA SAMPLING ................................................................................................................. 104
15.9. WEIGHTED HISTOGRAM ANALYSIS METHOD (WHAM) ............................................................................... 104
15.9.1. Running a Simulation ................................................................................................................. 104
15.9.2. Reaction Coordinate .................................................................................................................. 105
Introduction to molecular modeling 7 by Rajmund Kaźmierkiewicz
15.9.3. Example: n-Butane .................................................................................................................... 105
15.10. STEERED MOLECULAR DYNAMICS ............................................................................................................ 106
15.11. “RAPID” FREE ENERGY METHODS ............................................................................................................ 106
15.11.1. Linear Interaction Energy (LIE) .................................................................................................. 107
15.11.2. Molecular Mechanics Poisson-Boltzmann Surface Area Method (MM/PBSA) .......................... 107
15.11.3. Example: MM/PBSA .................................................................................................................. 108
15.11.4. Binding free energy of protein-ligand ....................................................................................... 108
15.11.5. Binding free energy of protein-RNA .......................................................................................... 108
16. MOLECULAR DISTANCE GEOMETRY PROBLEM ................................................................................. 109
16.1. CURRENT APPROACHES .......................................................................................................................... 109
16.1.1. Embed Algorithm ....................................................................................................................... 109
16.1.2. Geometric Build-Up ................................................................................................................... 109
17. PROTEIN FOLDING ............................................................................................................................ 110
17.1. ENERGY MINIMIZATION ......................................................................................................................... 111
17.2. SOME RELATED METHODS...................................................................................................................... 111
17.3. MONTE CARLO-MINIMIZATION (MCM) ................................................................................................... 111
17.4. KNOWLEDGE-BASED ENERGIES ................................................................................................................ 112
17.5. PREDICTING PROTEIN SECONDARY STRUCTURE ........................................................................................... 113
17.6. PROTEIN THREADING ............................................................................................................................. 114
17.7. REDUCED OR SIMPLIFIED PROTEIN MODELS ............................................................................................... 114
18. MOLECULAR GRAPHICS SOFTWARE.................................................................................................. 116
19. RECOMMENDED READING ............................................................................................................... 117
Introduction to molecular modeling. 1. Introduction 9 by Rajmund Kaźmierkiewicz
1. Introduction
Given the simplicity and the interpretative power of molecular structural models, apparent
to chemists as early as 1800s, it was only natural that scientists would develop mathematical tools to
aid in understanding molecular structure and the molecular structural changes associated with
chemical reactivity.
Models currently available to scientists for understanding molecular structure are numerous.
They range from simple, plastic, physical molecular representations to sophisticated mathematical
models. Mathematical models include molecular mechanics, the semiempirical quantum methods,
the local density functional approach, and the large-scale computer intensive ab-initio structure
procedures. Each has been usefully applied to (bio)chemical problems and each has practical
limitations. The present text is primarily focused on the use of molecular mechanics models in
(bio)chemistry. Most of these models are more complex than molecular images displayed on a
computer screen but substantially less sophisticated than electronic structure approaches.
Molecular mechanics deals with a simple, empirical “ball-and-spring” model of molecular
structure. Atoms (balls) are connected by springs (bonds) that can be stretched or compressed due
to intra- or intermolecular forces. The sizes of balls and the stiffness of the springs are determined
empirically, that is, they are chosen to reproduce experimental data.
Figure 1. Molecular mechanics is simply the best available method to classically model biomolecules, the building blocks of living systems
Molecular mechanics is often mistaken with bioinformatics. Both of them deal with molecular
structure but bioinformatics focuses mainly on getting the structural information out from the
sequence and molecular mechanics focuses mainly on dealing with the structure once it is obtained.
Both of those methodologies have some common fields of interest but approach them from different
10 Introduction to molecular modeling. 2. Linux by Rajmund Kaźmierkiewicz
directions. The aim of this textbook is to show the Reader that molecular mechanics is not about
thoughtless running program windows and clicking on pretty menu icons. I hope that it will help to
understand basic concepts of molecular mechanics.
2. Linux
The Linux operating system is treated in molecular mechanics as a tool, merely it is a way of
running the computer hardware. The choice of the specific operating system depends on its usability
in that purpose. Linux has some advantages, which makes it a more convenient tool in molecular
mechanics applications than the rest of the operating systems:
It is a true multi-user, multi-tasking operating system for PC hardware
It is flexible and powerful
It is stable; no blue screen of death
It is free; it includes much free software (most of the software is open-source, free and readily
available)
It offers a lot of “flavors”, which are called distributions, to suit most user needs
Currently Linux is a very mature operating system and, according to the top 500 list from
11/2010, about 91.80% supercomputers run on Linux, to put it in perspective, MS Windows share is
about 1%. This means most of the top 500 (www.top500.org) supercomputers run on Linux or other
Unix-like operating systems. If one wishes to run calculations on supercomputers he/she probably
needs to familiarize him-/herself with the basics of the Linux operating system. Some simple tasks in
molecular mechanics do not require running jobs on the supercomputers, but it is advised to learn
Linux using “simple tasks” rather than begin with sending jobs to supercomputers risking a lot of
trouble, out of which the administrator “reprimand” will be probably the least of the problems, if
something goes wrong.
The contemporary Linux distributions are as simple to use as the Other operating systems, in
addition to usual, simple, windows-based tools Linux has an extensive set of commands. They are a
convenient supplement of graphical tools which are especially useful when they are organized in so-
called “shell scripts”. The real power of Linux reveals itself, when One needs to run calculations on
supercomputers remotely. Usually, after being sent through the remote access tools, the “computing
jobs” are submitted to the queue where they await for available computing resources. The results of
the finished jobs are stored on the personal accounts in the supercomputer center. Most of the
computer centers do not enable interactive interpretation of the results, therefore they need to be
downloaded and analyzed locally on the user personal PC.
3. What is Modeling?
The term “computer modeling” has a very broad meaning in (bio)chemistry. It is not exactly
limited to “molecular mechanics”. Usually it involves one or more of the following tasks: a)
construction of a virtual 3-D molecule (a computer model), b) computation of some properties
expected to be associated with a real molecule, of which this is a representation, c) a virtual,
microscopic experiment, d) modeling has been described sometimes also as a “computational
spectrometer”. The computer modeling approaches can be divided into two classes:
Introduction to molecular modeling. 4. Molecular mechanics: limitations 11 by Rajmund Kaźmierkiewicz
Bond-based: bonds between atoms and properties of the bonds are part of the model
methodologies are molecular mechanics, molecular dynamics, docking
Atom-based: positions of the atoms and their electronic structures are input
methodologies are quantum mechanical
Usually the molecular modeling strategy is applied to answer one (or more) specific
questions: 1) what does a molecule look like 2) what do its neighbors look like 3) what does the
neighborhood (the potential energy surface) look like 4) how do we get from one neighborhood to
another (what are the transition states) 5) how does the structure and its neighborhood change with
time 6) how do two or more molecules interact with each other ?
The application of molecular modeling techniques are described throughout the whole
textbook, but the selected (bio)chemical applications can be presented here:
Protein folding landscapes
Interactions, such as: enzyme-substrate, drug-DNA
Interpretation of X-ray diffraction patterns, NOE spectra
Site-directed mutagenesis, the easy way
Homology modeling
Solvation models
4. Molecular mechanics: limitations
Force fields are generally not reactive: bond breaking and formation is not possible in the
simulations. A common suggested solution is to replace the harmonic bond term with a dissociative
Morse potential, but this does not provide the necessary changes in atomic hybridization.
Although long-range interactions (electrostatic, van der Waals) are included in the force
fields, the former rely on the concept of partial charges associated with the nuclei. The latter have
fixed parameters throughout the simulations, so polarization effects are not included.
Transferability of the typical atom parameter sets of different force fields should always be
questioned. However, in practice the construction of different molecules relies on experimental data
and quantum-mechanical calculations.
5. Graphical representations of molecular structure
CH4O chemical formula reflecting only summary of elementary composition of the compound.
CH3OH “condensed” chemical formula, used most often in organic chemistry
Figure 2. Chemical formula depicting full topology (atom names and the organization of chemical
bonds) within the molecule
Figure 3. The projection of the 3D ball-and-stick model representation of the real molecule
H
C
H
H O H
12 Introduction to molecular modeling. 6. Molecular structure description by Rajmund Kaźmierkiewicz
6. Molecular structure description
6.1. Cartesian coordinates
Since Euclidean Space has no preferred origin or direction we need to add a coordinate
system before we can assign numerical values to points and objects in the space. Each point in the
three-dimensional coordinate system can be specified by 3 real numbers, the X, Y and Z coordinates
of that point.
Table 1. The molecular structure described by Cartesian coordinates
Atom number atom name X Y Z
1 C1 3.108 0.653 -8.526
2 C2 4.597 0.674 -8.132
3 1Hl 2.815 -0.349 -8.761
4 2H1 2.517 1.015 -7.711
5 3H1 2.956 1.278 -9.381
6 1H2 4.748 0.049 -7.277
7 2H2 5.187 0.312 -8.947
8 3H2 4.890 1.676 -7.897
They determine an absolute location of each atom in a three-dimensional coordinate system. The
molecular structure described by Cartesian coordinates is rather hard to imagine and usually requires
a computer program to draw the molecule on a computer screen. The structural information
including atom names and Cartesian coordinates can be arranged (formatted) in many possible ways.
One specific arrangement was proposed by authors of the Protein Structural Database and it is called
the PDB format.
6.2. Internal coordinates
Internal coordinates express the relation between atoms in molecules in terms of atom
connectivity, distances, angles and torsional (dihedral) angles. In contrast, Cartesian coordinates
define the molecules in terms of the atomic positions. A complete set of internal coordinates is called
a Z-matrix. Note that only 3N-6 internal coordinates are used in Z-matrix construction, where N
denotes the number of atoms: there are six zero's in the upper right corner of the matrix. The
orientation of the structure in space is not specified. The six “missing” variables correspond to the
three translations and three rotations of the whole structure (with respect to three axes) which do
not change the (internal) energy of system and can therefore be omitted. The orientation in space of
the first three atoms can be defined arbitrarily. Usually, in a Z-matrix, the first atom is the origin. The
second atom is defined by the distance to atom number 1, the third atom by a distance (to atom 1 or
atom 2) and a valence angle between atoms 3-2-1. Starting with the fourth atom the dihedral angle
(4-3-2-1) is introduced. From here every atom is described by a distance, an valence angle and a
dihedral (torsional) angle, with respect to already defined atoms.
Introduction to molecular modeling. 6. Molecular structure description 13 by Rajmund Kaźmierkiewicz
Table 2. The molecular structure described by internal coordinates
Atom name bond length valence angle torsion angle
C 0.000000 000.000000 000.000000 0 0 0
C 1.540000 000.000000 000.000000 1 0 0
H 1.089000 109.471000 000.000000 1 2 0
H 1.089000 109.471000 180.000000 2 1 3
H 1.089000 109.471000 60.000000 1 2 4
H 1.089000 109.471000 -60.000000 2 1 5
H 1.089000 109.471000 180.000000 1 2 6
H 1.089000 109.471000 60.000000 2 1 7
The internal coordinates seem to be more intuitive, usually an experienced user has no
problems imagining very simple molecules just by looking at the set of internal coordinates organized
in the form of a Z-matrix.
Internal coordinates are local, they are determined by positions of already defined atoms.
Molecular mechanics energy is expressed in terms of a combination of internal coordinates of the
system (bonds, valence angles, torsional angles) and interatomic distances (for the non-bonded
interactions). The atomic positions are expressed in terms of Cartesian coordinates.
Internal coordinates can be calculated by the computer from Cartesian coordinates
exploiting vector operations. The bond length, rij, is defined as a distance between two bonded
atoms i and j, and it is the length of the vector between atom i and j:
The valence angle also called the bond angle, , between two consecutive bonds originating on
atom j is calculated by applying the cosine rule:
The valence angle is always positive and not larger than 180°, and it is always the smaller of the two
possible angles.
Figure 4. Internal coordinates: a) bond length, b) valence angle, c) torsional angle
14 Introduction to molecular modeling. 6. Molecular structure description by Rajmund Kaźmierkiewicz
The torsional angle, is a dihedral angle, , between two planes passing through atoms i, j, k and j,
k, l, respectively. It is an angle between vectors normal (i.e., perpendicular) to these planes. The
torsional angle spans the range -180° to 180°. Its absolute value can be calculated as:
where is a unit vector pointing from atom i to j. It is defined as
and
is equal to bond length. Only the absolute value of the torsional angle can be calculated that way.
Additional checking has to be done to obtain the sign of the angle. In molecular mechanics we use
the right-hand screw rule. Some modeling systems may use other conventions.
6.3. Alternative description of the molecular structure
There are also some other “styles” of describing the 3D structures of molecules. Usually they
lead to more complicated mathematical expressions of potential energy functions.
The set of distances between all atoms is an equivalent description of the internal geometry of any
molecule.
Figure 5. An example representation of the molecular structure using distances. Only subset of all distances is shown
Resolution of the structure from mere distances may lead to two solutions, out of which one
represents a model of the real molecule and another that represents a mirror image model of the
real molecule. It is obvious for (bio)chemists that for most molecules only one solution is correct.
Although it may not be immediately apparent but One can reproduce complete molecule geometry
using just significantly detailed contact map representation of the molecular structure.
Introduction to molecular modeling. 7. Energy expressions in molecular mechanics 15 by Rajmund Kaźmierkiewicz
Figure 6. The complete contact map of ubiquitin (PDB code: 1UBQ)
Another style of description of a molecular structure is the so-called “coarse-grain model”. It is
usually a simplified computer model and does not include all atoms. The purpose of introducing such
a model is to speed up the most time consuming tasks in molecular mechanics. The simplified
representation of molecular 3D structure enables for example: computer simulation of protein
folding pathways and simulations of self-assembly of complex cell structures.
Figure 7. An example coarse grained representation of molecular structure using virtual internal variables
7. Energy expressions in molecular mechanics
7.1. Potential classification
A classical potential V can be written in the form
where
V1 is a single-particle term (external fields)
V2 is a pair potential that depends on the interatomic separation (distance, bond length)
V3 is a three body term (angular dependence, bond bending)
V4 is a four-body potential (torsional term)
16 Introduction to molecular modeling. 7. Energy expressions in molecular mechanics by Rajmund Kaźmierkiewicz
7.2. The Empirical Energy Function (or Force Field)
The fundamental interacting unit, in molecular mechanics, is the atom, not individual
electrons. Thousands of atoms can be considered in a calculation. The potential energy of the
collection of atoms can be calculated as a fairly simple function of the atomic coordinates. This
function is called the Potential Energy Function, and is derived empirically by giving good fit to
experimental spectroscopy data. The Potential Energy Function, can be broken down into a sum of
important interaction terms describing contribution of bond stretching, bond angle bending,
torsional angle rotation, non-bonded interactions (Van der Waals interactions and electrostatic
interaction) and the hydrogen bonds contribution.
7.3. Bond stretching
Bond stretching energy term:
the sum is over all covalent bonds.
kri = Hooke’s law spring constant for bond number i
r0i
= equilibrium bond length for bond number i
ri = actual current value of bond length i
In this equation bond stretching is treated as a classical harmonic oscillator term.
7.4. Angle bending
Bond angle bending term:
,
the sum is over all covalent bond angles.
ki
= spring constant for angle deformation
0i
= equilibrium bond angle for angle i
i = current value of bond angle i
7.5. Bond rotation (torsion)
Dihedral (or torsional) angle rotation term:
,
the sum is over dihedral angles
Vi = barrier height
s = 1 for staggered minima
= -1 for eclipsed minima
n = periodicity (n = 3 for ethane, n = 2 for ethene)
= current value of dihedral angle i
Introduction to molecular modeling. 7. Energy expressions in molecular mechanics 17 by Rajmund Kaźmierkiewicz
7.6. Non-bonded interactions (van der Waals)
Non-bonded interaction terms:
where the double sum extends over all possible pairs of atoms separated by more than 2 bonds. The
“combination rules” define εij=(εiεj)1/2 and σij = 1/2(σi+ σj) which are obtained from the single atom
parameters ε and σ.
is the Lennard-Jones form of the Van der Waals energy term, there also exist other
mathematical expressions for this energy contribution term. This is often the most time-consuming
term in simulating large systems.
7.7. Gay-Berne potential
A variant of the Lennard-Jones potential to describe interactions between elongated particles.
Where denotes interparticle unit vector.
This potential is used, for example, in simulations of liquid crystals.
Figure 8. Coarse grained representation of liquid crystals
7.8. Non-bonded interactions (electrostatic)
Electrostatic interaction terms:
qi = partial atomic charges. “The partial atomic charges” are one of a few ways of introducing the
quantum effects into otherwise classical functions. Since electrons are not considered explicitly in
molecular mechanics the only way of taking into account their distribution within the molecule is by
fitting effective “point charges” centered near the atom nucleus to the electrostatic potential
calculated using one of the quantum mechanical ab-initio methods. It is usually done by employing
the so-called restrained electrostatic potential (RESP) algorithm. Such procedures lead to discrete,
centered at the given point(s), charge distributions with partial (non-integer, fractional) values.
= Dielectric constant of medium. The physical origin of this constant is the value of the scalar
dielectric permittivity of the solvent, most often water.
18 Introduction to molecular modeling. 7. Energy expressions in molecular mechanics by Rajmund Kaźmierkiewicz
There are also other terms added by some simulation packages to improve correlation with the
experiment:
“Improper” torsional terms
Out-of-Plane Bending
The potential for moving an atom out of a plane is sometimes treated separately from bending
(although it also involves bending). An out-of-plane coordinate (either χ or d) is displayed below. The
potential is usually taken quadratic in this out-of-plane bend,
Figure 9. Out of plane variable definitions
Hydrogen bonding
Various expressions of stretch/bend cross terms
Cross terms are required to account for some interactions affecting others. For example, a strongly
bent water molecule tends to stretch its O–H bonds. This can be modeled by cross
terms such as
Other cross terms might include stretch-stretch, bend-bend, stretch-torsion and bend-torsion. Force
field models vary in what types of cross terms they use.
Figure 10. Schematic representation of a cholesterol molecule, and definition of the bond distances, bond angles, dihedral angles and Coulomb interactions
Introduction to molecular modeling. 8. Empirical Force Field 19 by Rajmund Kaźmierkiewicz
7.9. The total potential energy
The total potential energy of any molecule is the sum of simple terms allowing for bond stretching,
bond angle bending, bond twisting, van der Waals interactions and electrostatics.
Numerous properties of biomolecules can be simulated with such an empirical energy function.
There are several forms of mathematical expressions of the classical total potential energy of any
molecule. It seems that each author’s ambition is to modify parts of a mathematical function to give
impression of introducing something new. The most common expression is:
It is used together with the database of standard residues (fragments of more complex molecules)
and is accompanied with the set of parameters usually optimized for evaluation of properties of a
given class of chemical compounds. All three (i.e. mathematical expression, the database of standard
residues and the set of parameters) together constitute the Empirical Force Field.
8. Empirical Force Field
Popular Force Fields for Macromolecules, optimized for calculation of properties of proteins and
nucleic acids:
AMBER (Cornell et al. J. Am. Chem. Soc. 1995. 117: 5179)
CHARMM (MacKerell et al. J. Phys. Chem. B. 1998. 102: 3586.)
GROMOS (Schuler et al. J. Comput. Chem. 2001. 22: 1205.)
Parameters of the empirical force fields depend on hybridization and the immediate surroundings of
the given atom. The consequence of this dependence is the high number of “force field atom types”
associated with one atom of the particular chemical element. One may ask: How Many Parameters
are There?
AMBER has 40 atom types.
There are 13 types of carbon:
sp3 carbon
Carbonyl sp2 carbon
Aromatic sp2 carbon
sp2 carbon, double bonded
There are also the bond stretch and angle parameters for each valid combination of atom types.
There are 1 to 3 torsional parameters for many combinations of atoms and there are 30
improper torsions.
There is one set of van der Waals parameters for each atom type, which are combined for each
pairwise interaction.
Atomic charges are set for the atoms in each amino acid/nucleotide residue.
20 Introduction to molecular modeling. 8. Empirical Force Field by Rajmund Kaźmierkiewicz
8.1. Fitting Parameters
Bond stretch and bond angle parameters are fit to IR and RAMAN spectroscopic data from simple
model molecules. Dihedral term parameters are fit to energies derived from ab-initio, usually MP2/6-
31G*, quantum mechanics calculation. Van der Waals parameters are fit to optimize properties of
liquids such as densities and enthalpies of vaporization.
Table 3. The sample force constants and reference bond lengths for selected bonds
Bond r0, (Å) k (kcal mol-1
Å-2
)
Csp3-Csp3 1.523 317
Csp3-Csp2 1.497 317
Csp2 = Csp2 1.337 690
Csp2 = O 1.208 777
Csp3-Nsp3 1.438 367
C-N (amide) 1.345 719
Table 4. The sample force constants and reference angles for selected angles
Angle 0 (deg) k (kcal mol-1
deg-1
)
Csp3-Csp3-Csp3 109.47 0.0099
Csp3-Csp3-H 109.47 0.0079
H-Csp3-H 109.47 0.0070
Csp3-Csp2-Csp3 117.2 0.0099
Csp3-Csp2 = Csp2 121.4 0.0121
Csp3-Csp2 = O 122.5 0.0101
8.2. Fitting Charges
The “atomic charges” are a useful approximation. Quantum mechanics tells us that electrons
are delocalized to probable regions in space, and their charge is shared among nearby atoms. There
is no unique way to assign electrons to particular atoms. For molecular mechanics, we want to
associate charges with atoms. Charges are fit to atomic location using the RESP (Restrained
Electrostatic Potential) method. First, the model molecules are placed in a 3D grid of points, then the
electrostatic potentials are calculated at each point in the grid. Using the potentials, charges are fit to
atomic locations to provide, as closely as possible, the potential at all points of the grid.
bond lengths bond angles charges
Figure 11. Sample non-standard residue with the experimental values of bond lengths and bond angles and the fitted charge values
Introduction to molecular modeling. 8. Empirical Force Field 21 by Rajmund Kaźmierkiewicz
8.3. Problems with the infinite range of non-bonded interactions
Like van der Waals terms, electrostatic terms are typically computed for non-bonded atoms
in a so-called 1-4 relationship, i.e. if atoms are three bonds or are further apart one from each other.
They are also long range interactions and dominate the computation time.
The number of non-bonded interactions grows quadratically with molecule size. The
computation time can be reduced by “cutting off” (excluding) the interactions after a certain
distance. The van der Waals terms decrease relatively quickly (~ R−6) and can be “cut off” around 10
Å. The electrostatic terms decrease slower (~ R−1), and are much harder to be correctly treated with
cutoffs.
Figure 12. Comparison of the „typical” contributions to potential non-bonded energies of interactions (Van der Waals and the electrostatic energies)
The point-charge model has serious deficiencies: (a) electrostatic potentials are not
accurately reproduced; (b) simple models do not allow the charges to change as the molecular
geometry changes, but they should; (this problem is partially overcame by careful parameterization
of the torsional potential) (c) only pairwise interactions are considered, but an electrostatic
interaction can actually change by about 10-20% in the presence of a third body due to induction or
“polarization” effects.
Until recently, the most frequently used method to handle electrostatic and van der Waals
interactions was to ignore all interactions between atoms whose internuclear distance is longer than
a certain cutoff value. Such an approach is usually called the Cut-off Method. In practical
applications, it is convenient to establish a cutoff radius Rc and disregard the interactions between
atoms separated by more than Rc. The same cutoff radius is defined for each atom. This approach
defines a sphere around each atom where all interactions are calculated, beyond this sphere all non-
bonded interactions are ignored. This results in simpler programs and enormous savings of computer
resources, because the number of atomic pairs separated by distance r grows as r2 and becomes
quickly huge. A simple truncation of the potential creates a new problem though: whenever a
r[Å]
E(r)[kcal/mol]
Evdw
Ees(+ )
Ees(+ -)
22 Introduction to molecular modeling. 8. Empirical Force Field by Rajmund Kaźmierkiewicz
particle pair “crosses” the cutoff distance, the energy makes a little jump. The so-called group-based
cutoffs lighten this problem a little bit because all contributions of the entire residue are included (or
omitted) together. In this case all groups should be neutral or almost so and they should be much
smaller than the cut-off radius. Despite these countermeasures a large number of “small energy
jumps” is likely to spoil energy conservation in a simulation. To avoid this problem, the potential is
often shifted in order to vanish at the cutoff radius. Physical quantities are of course affected by this
potential truncation.
A B
Figure 13. The non-bonded cutoffs. A. the interacting atoms without applying cutoff,
B. interacting atoms after applying cutoff
There are several possible choices concerning how the cutoff can be used:
Truncation: the interactions are simply set to zero for interatomic distances greater than the cutoff
distance. This method can lead to large fluctuations in the energy. This method is not often used.
The SHIFT cutoff method: this method modifies the entire potential energy surface such that at the
cutoff distance the interaction potential is zero. The drawback of this method is that equilibrium
distances are slightly decreased.
The SWITCH cutoff method: This method tapers the interaction potential over a predefined range of
distances. The potential takes its usual value up to the first cutoff and is then switched to zero
between the first and last cutoff. This model suffers from strong forces in the switching region which
can slightly perturb the equilibrium structure. The SWITCH function is not recommended when using
short cutoff regions.
An example of the correctly applied switching function.
After applying a correct switching function both energy and gradients are continuous, total energy is
conserved and the thermodynamic properties are not affected.
Introduction to molecular modeling. 8. Empirical Force Field 23 by Rajmund Kaźmierkiewicz
Figure 14. An illustration of various cutoff application methods
Cut-offs also apply to neighbor list updating. In this case only atoms within the neighbor list
need to be considered in calculations of the potential energy. Including “close” atoms avoids
recalculation of the neighbor list on each iteration. The list updating step is carried out using
displacement-based criteria for recalculation of the neighbor list.
Figure 15. The neighbor list updating. Each atom is in the center of its own interaction sphere and there is a list of atoms included within each sphere
8.4. Problems with high values of electrostatic potential
The high values of electrostatic potential around some molecules can be both an advantage
or a disadvantage. It depends what values of electrostatic potential are desired in the given
environment. Even the smallest molecules generate noticeable electrostatic fields.
24 Introduction to molecular modeling. 8. Empirical Force Field by Rajmund Kaźmierkiewicz
Figure 16. Molecular Dipole Moments are the vector sum of the individual bond dipole moments. They depend on the magnitude and direction of the bond dipoles
Molecular Dipole Moments are the vector sum of the individual bond dipole moments. They depend
on the magnitude and direction of the bond dipoles.
The consequence of existence of naturally occurring dipoles is the characteristic behavior of
those molecules which tend to reorient spontaneously (mainly rotate) to accommodate to both the
self-generated and external electric fields. This tendency affects also fragments of molecules if they
possess measurable dipole moments.
The dipole-dipole (or multipole-multipole; multipole is a higher order spatial arrangement of
charges, it takes into account separation of more than two charged interacting sites) interaction can
also be applied in some cases to molecules which are placed (or are observed) from “large”
distances. At larger separations, details of charge distribution are less important. Please keep in
mind, that for molecules a “large distance” term could mean just a couple of nanometers.
Figure 17. An illustration of the dipole-dipole interactions: A means attraction, R means repulsion
NH3 H2O CO2 CH3Cl
OH
H
:
::O=C=O:
.. ..
C
H
ClH
H
D 1.9 D 0.0 D 1.87 D
NH
H
H
:
Introduction to molecular modeling. 9. Optimization of a structure 25 by Rajmund Kaźmierkiewicz
Unfortunately mathematical expressions describing interactions of dipoles and higher
multipoles tend to be more complicated than the simple Coulomb interaction potential.
Figure 18. Sample mathematical expressions relating to the point multipole models based on long-range behavior
Among molecules, interesting for (bio)chemists, are phospholipids and nucleic acids, particularly DNA
molecules, which display high electrostatic field values. Such molecules may display high affinity (i.e.
strong attraction forces) to other charged molecules, especially proteins.
8.5. Dielectric permittivity
Treatment of the dielectric permittivity of the environment of a molecule in molecular mechanics is
interrelated with the treatment of the “solvent model” of the medium surrounding the molecule
computer models. The satisfactory, from the physical point of view, treatment of the dielectric
permittivity is probably not possible. It is a macroscopic, classical entity, whose value comes as a
consequence of resultant cooperative interactions of many molecules, but it is applied to description
of interactions of a single molecule which represents the microscopic world. Most of the classical
force fields do not take into account polarization effects (creation of induced dipoles) of molecules
therefore the single, scalar effective value of dielectric permittivity of the solvent is used.
9. Optimization of a structure
The empirical force field can be represented by the 3N dimensional potential energy hypersurface.
The whole hypersurface is not very interesting. In molecular mechanics only a few points are
important, they are called “the stationary points”. More precisely we need information only about
points where gradient of the potential energy function is equal 0. Is there a reason why we care
about the stationary points (especially minima) on the potential energy hypersurface ?
26 Introduction to molecular modeling. 9. Optimization of a structure by Rajmund Kaźmierkiewicz
saddle point maximum
minimum
Figure 19. Schematic representation of the potential energy hypersurface
The physical meaning of special points on the potential energy hypersurface:
Reactants (substrates of the (bio)chemical reactions, starting material), products and
intermediates (regardless of their lifetime) correspond to energy minima.
Energy minima correspond also to conformations of any compound in its standard, stable state
The most stable conformation (the native conformation) of the molecule corresponds to the
global minimum on the potential energy hypersurface
Energy maxima are (bio)chemically irrelevant.
Saddle points correspond to transition states.
If qi corresponds to one of the normal coordinates of the system,
corresponds to the force constant of this normal vibration.
9.1. Successive Coordinate Direction Method
Starting from a point p we can define a line in the direction specified by a vector n parameterized by
. Along this line, any point is given by
x = p + n
Now f(x) = f(p + n) is a function of one variable which may be minimized using any one-dimensional
method. This process is called the line minimization. The result is a line minimum of f. After one
iteration of this process, the line minimum is then used as the starting point p in the next iteration
for a different choice of the direction vector n.
Introduction to molecular modeling. 9. Optimization of a structure 27 by Rajmund Kaźmierkiewicz
9.2. Newton’s Method for Finding a Minimum
Now we turn to the minimization of a function of n variables using the Newton method,
where and the partial derivatives of are accessible.
Assume that the first and second partial derivatives of exist and are
continuous in a region containing the point , and that there is a minimum at the point . The
quadratic polynomial approximation to is:
A minimum of function occurs where . The expression for can be written
as
If point is close to the point (where a minimum of f occurs), then is invertible and the
above equation can be solved for , and we have
This value of can be used as the next approximation to and is the first step in Newton's method
for finding a minimum
The Newton-Raphson method not only uses the gradient of a function, but also the second order
gradient to determine the search direction. This direction is kept for each new step until a minimum
has been found. Then a new search direction is determined and the process continues. The method
only converges for a positive second order gradient, near the minimum.
Figure 20. Successive minimizations of f(x) along coordinate directions
9.3. Steepest Descents
This minimization method can be summarized in three points:
Step downhill in a direction of local steepest gradient using trial step length
Perform a line minimization to find the optimal step length
Repeat to convergence
28 Introduction to molecular modeling. 9. Optimization of a structure by Rajmund Kaźmierkiewicz
9.3.1. Steepest Descent Method
Here for each iteration of line minimization the direction is chosen to be the local downhill gradient -
f(p). However, though along the downhill gradient to begin with at p, the vector n becomes
perpendicular to the local gradient of f(x) where the current line minimum occurs. Consequently, the
vector n has to make a 90° turn for every iteration. This results in a zigzag path along a "long valley"
to the final minimum of f(x).
Figure 21. Successive minimizations of f(x) using the steepest descent method
Among the Steepest Descents advantages is that it can be easily implemented. It is also very robust
and reliable, it will always get to the minima. Unfortunately it is often very slow to converge.
9.4. Conjugate Gradient Method
Recall that for a scalar quadratic function f, the gradient is given by
f = Hx - b
Along any direction, the variation of this gradient is given by
(f ) = Hx
Suppose that f has been line minimized along the direction u:
uf = 0
say, at p.
Then a successive line minimization along another direction v without spoiling the previous line
minimization should satisfy
u(f ) = 0
where the variation of the gradient is induced by moving along v, hence (f ) = H v.
It follows that we must have
uHv = 0
Any two vectors u and v satisfying the above are said to be conjugate.
For a scalar quadratic function, a sequence of N line minimizations using independent conjugate
directions will lead to the exact minimum.
Introduction to molecular modeling. 9. Optimization of a structure 29 by Rajmund Kaźmierkiewicz
An effective way to find these conjugate directions is via the Fletcher-Reeves algorithm as follows:
Start with an arbitrary initial vector g0 and another vector h0 = g0. The algorithm generates two
sequences of vectors:
g0, g1, g2, …
and
h0, h1, h2, …
using following recurrence: First, calculate
gi+1 = gi - iHhi
where
Then, calculate
hi+1 = gi+1 + ihi
where
9.4.1. Conjugate gradient method without explicit knowledge of the
Hessian matrix
The above algorithm assumes the availability of the Hessian matrix H. If for some reason, e.g. due to
data storage limitation, H is not available, but the gradient of f(x) can still be evaluated, then the
following algorithm for the conjugate gradient method due to Fletcher-Reeves can be employed:
1. Start from some point pi and define gi = - f(pi).
2. Perform line minimization along hi, i.e. minimize f(pi + hi).
3. Use the resulting to assign i = and pi+1 = pi + ihi.
4. This yields gi+1 = -f(pi+1), from which we have hi+1 = gi+1 + ihi, where
as before.
Figure 22. Successive minimizations of f(x) using the conjugate gradient method
30 Introduction to molecular modeling. 9. Optimization of a structure by Rajmund Kaźmierkiewicz
One of the advantages of Conjugate Gradients method is the rapid rate of convergence, in a
quadratic energy landscape, each iteration should converge one degree of freedom. It has also
relatively low storage requirements.
This method has also some disadvantages: it is more complex to code than the steepest descent
algorithm and there is no knowledge of the Hessian explicitly generated.
9.5. The BFGS algorithm for unconstrained optimization
In 1970, an alternative inverse Hessian update formula was suggested independently by Broyden,
Fletcher, Goldfarb and Shanno. Their formula originated a new Quasi—Newton method.
This algorithm can be summarized as follows:
1. Set k := 0, select x(0) and a real positive definite matrix B0.
2. If g(k) = 0, stop. Else dk = -Bkg(k).
3. Compute
4. Update the inverse Hessian approximation Bk+1, set
k := k + 1 and go to step 2.
The inverse Hessian approximation is updated as follows:
where
There exists the (older) Davidon-Fletcher-Powell (DFP) variant, which is mathematically equivalent to
BFGS. It is less tolerant of round-off error or inexact line minimization and it calculates A
(approximates H) rather than H itself.
The BFGS method convergence rate is similar (or better) than the Conjugate Gradient
method and extra physical information is generated from Hessian, but it is still a local minimization
method.
9.6. Testing Minima
Compute the full Hessian (the partial Hessian from an optimization is not accurate enough).
Check the number of negative eigenvalues:
0 required for a minimum.
1 (and only 1) for a transition state
For a minimum, if there are any negative eigenvalues, follow the associated eigenvector to a
lower energy structure.
For a transition state, if there are no negative eigenvalues, follow the lowest eigenvector up hill.
Introduction to molecular modeling. 10. Molecular dynamics simulations 31 by Rajmund Kaźmierkiewicz
9.7. Minimization and Molecular Mechanics
The use of a force field to define structure is often called molecular mechanics.
Use the force field that has been assigned to the atoms in the system.
Find a stable point or a minimum on the potential energy surface in order to begin dynamics.
There will be more than one minimum for a polymer, biopolymer, or a liquid.
There may be a global minimum, but this will not likely be found without a conformational
search.
Molecular dynamics provides information that is complementary to minimization.
Three typical stages: Minimization, Equilibration, Dynamics (The production run)
10. Molecular dynamics simulations
The molecular dynamics technique enables calculation of thermodynamic properties of
molecules (energy, heat capacity) and it provides dynamic information (diffusion coefficient,
dielectric functions, correlated motion). MD allows to study the dynamics of large macromolecules,
including biological systems such as proteins, nucleic acids (DNA, RNA), membranes. Dynamical
events may play a key role in controlling processes which affect functional properties of the
biomolecule. Beyond this “traditional” use, MD is nowadays also used for other purposes, such as
studies of non-equilibrium processes, and as an efficient tool for optimization of structures
overcoming local energy minima (simulated annealing).
In molecular mechanics, the set of many molecules put together is called a system. It
includes typically a macromolecule, sometimes accompanied by a small ligand, water, ions, it may
also include phospholipids or sugars.
The idea of MD is a simple one: calculate the forces acting on the atoms in a molecular
system and analyze their motion. When enough information on the motion of the individual atoms
has been gathered, it is possible to condense it all using the methods of statistical mechanics to
deduce the bulk properties of the material. These properties include the structure (e.g. crystal
structure, predicted x-ray and neutron diffraction patterns), thermodynamics (e.g. enthalpy,
temperature, pressure) and transport properties (e.g. thermal conductivity, viscosity, diffusion). In
addition molecular dynamics can be used to investigate the detailed atomistic mechanisms
underlying these properties and compare them with theory. It is a valuable bridge between
experiment and theory.
10.1. Equilibration
Equilibration is a protocol for bringing the system to equilibrium at the desired temperature
for the simulation. The protocol consists of assigning velocities and then performing molecular
dynamics until the equilibrium has been reached. Every time the state of the system changes, the
system will be “out of equilibrium” for a while, and it is certainly so at the beginning of the computer
simulation. We are referring here to thermodynamic equilibrium.
Once the system is in equilibrium the current velocities are used for production dynamics.
The production run is the phase of the simulation where properties of the system can be
determined.
32 Introduction to molecular modeling. 10. Molecular dynamics simulations by Rajmund Kaźmierkiewicz
10.2. Velocities in MD
The trajectory in a MD simulation consists of both positions and velocities. The velocities are
assigned (created) based on a coordinate file with the atoms in the optimal (minimized) positions.
The initial velocities are assigned taking them from a Maxwell distribution at a certain temperature T.
Initial randomization of velocities is usually the only place where chance enters a molecular
dynamics simulation. The subsequent time evolution is completely deterministic.
The average velocity is related to the temperature according to:
Figure 23. The Maxwell-Boltzmann velocity distribution in the kinetic theory
10.3. Dynamics: Equations of Motion
Molecular dynamics requires a technique for the solution of the equations of motion for
atomic systems. If we consider a system of atoms, with Cartesian coordinates ri then equation of
motion becomes (Newton’s equation of motion):
where mi is the mass of atom i , is an acceleration and Fi is the force on that atom.
10.3.1. Numerical Solution of the Equations of Motion
To simulate molecular motion we need a means of solving the equations of motion for a
system of many particles. Coupled linear differential equations (equations of motion) for the motion
of various masses in a force field can be solved using finite difference methods. The equations are
solved step-by-step in discrete time intervals t. Finite difference methods use calculation of the
velocity (i.e.
) to produce a new set of positions. The new positions are used to reevaluate the
velocities using the equations of motion. This procedure is repeated for each step of the simulation.
There are several different techniques for propagating the motion of the particles in a
simulation: I. Verlet algorithm (A. basic, B. leapfrog, C. velocity Verlet), II. Gear predictor-corrector.
10.3.2. The symplectic integration of equations of motion
In molecular mechanics a great number of phenomena are modeled by ordinary differential
equations (equations of motion). When solved, analytically or numerically, they describe the time
evolution of the quantities used to model the phenomena. Among these systems there are those
Introduction to molecular modeling. 10. Molecular dynamics simulations 33 by Rajmund Kaźmierkiewicz
called conservative or Hamiltonian. We have to use numerical procedures to solve the equations.
Numerical procedures reduce the differential equations to finite difference equations through
algorithms which are now standard. Stability of these algorithms is a research area on its own. There
are two classes of integration algorithms. The first are symplectic algorithms, they are time reversible
and conserve phase space volume, both properties are highly desired. The second class is non-
symplectic, it is bad because it does not recover time reversibility property of Newton’s equations of
motion and it is unstable due to strong energy drift. The non-symplectic algorithm requires also very
small time step to „force“ stability, although nothing can guarantee the stability in long simulations.
Symplecticity is of fundamental importance and it was discussed in many papers (Mitsutake A, Sugita
Y, Okamoto Y., “Generalized-ensemble algorithms for molecular simulations of biopolymers.”,
Biopolymers. 2001;60(2):96-123.; Feig M, Brooks CL 3rd., “Recent advances in the development and
application of implicit solvent models in biomolecule simulations.”, Curr Opin Struct Biol. 2004
Apr;14(2):217-24.; Kamberaj H, Low RJ, Neal MP, “Time reversible and symplectic integrators for
molecular dynamics simulations of rigid molecules.”, J Chem Phys. 2005 Jun 8;122(22):224114.;
Okumura H, Itoh SG, Okamoto Y “Explicit symplectic integrators of molecular dynamics
algorithms for rigid-body molecules in the canonical, isobaric-isothermal, and related ensembles.“, J
Chem Phys. 2007 Feb 28;126(8):084103.; Sugita Y. “Free-energy landscapes of proteins in solution by
generalized-ensemble simulations.“, Front Biosci. 2009 Jan 1;14:1292-303.). Among those algorithms
discussed in next paragraphs the Verlet algorithms are symplectic, the Gear predictor-corrector
algorithm is not symplectic.
10.3.3. The Verlet Algorithm
The Verlet method is a direct solution of the second order differential equations. In the
Verlet method the velocities are eliminated by comparing two Taylor expansions about the position
at time t.
The Taylor series expansion about +t and -t are summed to give the expression:
r(t + t) = r(t) + t v(t) + (1/2)t2 a(t) + …
r(t - t) = r(t) - t v(t) + (1/2)t2 a(t) + …
r(t + t) = 2r(t) - r(t - t) + t2 a(t) + …
This equation is correct except for errors of the order of t4. The computed velocity (used to
estimate the kinetic energy) is subject to errors of the order of t2.
The velocity is computed by v(t) = [r(t + t) – r(t – t)]/t, on the fly, in this method.
10.3.4. Leapfrog Verlet
The Verlet algorithm may introduce numerical imprecision since numbers of the order of t2
are added to numbers of the order t0 ( 1). For this reason the leapfrog Verlet method is used
r(t + t) = r(t) + t v(t + 1/2t) v(t + 1/2t) = v(t - 1/2t) + t a(t)
The velocity equation is executed first and generates a new mid-step velocity. This velocity is then
used to calculate the new position. The velocity is calculated from
v(t) = (1/2)v(t + 1/2t) +(1/2)v(t - 1/2t)
This leapfrog method also has the advantage that temperature scaling by velocity scaling is feasible.
34 Introduction to molecular modeling. 10. Molecular dynamics simulations by Rajmund Kaźmierkiewicz
10.3.5. Velocity Verlet
The handling of the velocity (and therefore the calculation of the kinetic energy) is NOT
„ideal” in either of the above forms of the Verlet algorithm. The velocity Verlet algorithm stores
positions, velocities, and accelerations:
r(t + t) = r(t) + t v(t) + (1/2)t2 a(t)
v(t + t) = v(t) + (1/2)t[a(t) + a(t + t)]
The above velocity Verlet approach can be shown to be equivalent to the basic Verlet algorithm by
eliminating the velocities.
The equations are implemented in two stages. First, the new positions at time t + t are calculated.
Then the velocities at mid-step are calculated using
v(t + 1/2t) = v(t) + (1/2)t a(t)
The forces and acceleration at time t + t are calculated and then the new velocity is calculated.
v(t + t) = v(t + 1/2t) + (1/2)t a(t + t)
10.3.6. Gear Predictor-Corrector Method
The predictor
If the classical trajectory is continuous, then an estimate of the positions, velocities, accelerations
etc. may be obtained by a Taylor series expansion about time t:
rp(t + t) = r(t) + t v(t) + 1/2 (t)2a(t) + 1/6 (t)3b(t) + ...
vp(t + t) = v(t) + t a(t) + 1/2 (t)2b(t) + ...
ap(t + t) = a(t) + t b(t) + ...
bp(t + t) = b(t) + …
The p superscript refers to predicted values. The variables are :
r = position v = velocity (
)
a = acceleration (
) b = third derivative of position with respect to time
The corrector
The equations of motion are introduced by calculating the acceleration, a due to the force, F.
The force is calculated from the potential function V(rp) at the new positions, rp so that the correct
acceleration is:
ac = F/m = (-grad V(rp))/m.
The predicted positions and velocities must be corrected. The correction term is proportional to the
difference between the predicted and correct acceleration,
The corrector step is: a(t + t) = ac(t + t) – ap(t + t)
rc(t + t) = rp(t + t) + c0a(t + t)
vc(t + t) = vp(t + t) + c1a(t + t)
ac(t + t) = ap(t + t) + c2a(t + t)
bc(t + t) = bp(t + t) + c3a(t + t)
Introduction to molecular modeling. 10. Molecular dynamics simulations 35 by Rajmund Kaźmierkiewicz
10.4. The Time Step
The choice of time step t is of critical importance to the success of the method. The time
step must be short in relation to the length of time it takes for a particle to travel its own length.
Time step should be about 10 times shorter than the period of the highest frequency vibration in the
simulation. The configuration space sampled during the simulation will be greater if the time step is
longer, so in the interest of efficiency of calculation it is desirable to make the time step as long as
possible.
Time
Figure 24. Time Scales of Protein Motions and MD
The examples of possible applications and Time Scales of Protein Motions and MD were depicted in
Brooks, Karplus, & Pettit, "Proteins", Wiley, 1988. The time scales needed for all-atom simulations of
a protein folding process are out of reach of contemporary computers. It is still difficult to simulate a
whole process of protein folding using the conventional MD method.
10.5. The need for faster computers
Compared with other applications in today's computational (bio)chemistry, MD simulations
using classical potentials are less demanding than electronic structure programs. Using a parallel
computer a single job is divided into several smaller ones and they are calculated on multi CPUs
simultaneously. Today, almost all MD programs for biomolecular simulations (like AMBER, CHARMm,
GROMOS, NAMD) can run on parallel computers.
10.6. Phase Space
Phase Space is a concept common for theory (molecular mechanics, statistical mechanics)
and experiment (thermodynamics). Computer simulations generate information at the microscopic
level (atomic and molecular positions and velocities) and statistical mechanics converts this
information into macroscopic terms (for example: pressure and internal energy). The positions and
momenta of the particles can be thought of as coordinates in multidimensional space: phase space.
For a system of N atoms this space has 6N dimensions (3N positions and 3N momenta). represents
the particular point in phase space.
10-15 10-610-910-12 10-3 100
(s)(fs) (ps) (μs)(ns) (ms)
Bond stretching
α-Helix folding
β-Hairpin folding
Protein folding
36 Introduction to molecular modeling. 10. Molecular dynamics simulations by Rajmund Kaźmierkiewicz
10.7. Calculation of Average Properties
We can represent the instantaneous value of some property A as Aobs. The experimental
observable macroscopic property A is given by a time average. The equations governing the time
evolution are none other than Newton’s equations of motion. In a molecular dynamics simulation the
solutions are not performed continuously but in time steps, t.
The ensemble is a central concept in statistical mechanics. Imagine that a given molecular
system is replicated many times over, so that we have an enormous number of copies, each
possessing the same physical characteristics of temperature, density, number of atoms and so on.
Since we are interested in the macroscopic properties of the system, it is not necessary for these
replicas to have exactly the same atomic positions and velocities. In other words the replicas are
allowed to differ microscopically, while retaining the same general properties. Such a collection of
replicated systems is called an ensemble.
Because of the way the ensemble is constructed, if a snapshot of all the replicas is taken at
the same instant, we will find that they differ in the instantaneous values of their bulk properties.
This phenomenon is called fluctuation. Thus the true value of any particular bulk property must be
calculated as an average over all the replicas. This is what is meant by an ensemble average, and the
instantaneous values are said to fluctuate about the mean value.
Molecular dynamics proceeds by a numerical integration of the equations of motion. Each
time step generates a new arrangement of the atoms (called a configuration) and new instantaneous
values for bulk properties such as temperature, pressure, configuration energy etc. To determine the
true or thermodynamic values of these variables requires an ensemble average. In molecular
dynamics this is achieved be performing the average over successive configurations generated by the
simulation. In doing this we are making an implicit assumption that an ensemble average (which
relates to many replicas of the system) is the same as an average over time of one replica (the
system we are simulating). This assumption is known as the Ergodic Hypothesis. Fortunately it seems
to be generally true, provided a long enough time is taken in the average. However it has not yet
been rigorously proved mathematically.
Examples of thermodynamic properties that can be calculated from computer simulations as
ensemble averages include:
Temperature;
Pressure;
Density;
Configuration energy;
Enthalpy;
Structural correlations;
Time correlations;
Elastic properties.
Introduction to molecular modeling. 10. Molecular dynamics simulations 37 by Rajmund Kaźmierkiewicz
10.8. Fluctuations
Most of the properties that we calculate for a molecular system are averages. Well known
properties like temperature, pressure and density are calculated as ensemble averages, and in the
real world they are treated as fixed, measurable quantities, which they generally appear to be.
However all averages are obtained by summing over many numbers, and it would be very unusual
(even pointless) if all the individual numbers summed had exactly the same value. Thus in practice
we expect the average to show some dispersion - individual contributions are scattered about the
mean value. In statistical thermodynamics this dispersion about the average value is known as
fluctuation and it is both a subtle and important property of all physical systems.
When calculating an ensemble average (of say, pressure at a fixed temperature and density),
we take an instantaneous snapshot of a very large set of replicas of the system concerned and
compute the average from the sum of the individual values taken from each replica. Even though
each replica represents the same system at the same pressure, their individual, instantaneous values
differ slightly, because the molecules that bombard the vessel surfaces to create the pressure are not
in synchronization between each replica and cannot possibly give rise to precisely the same surface
forces at the same instant. Thus, with pressure, we expect some fluctuation about the mean value
and indeed, similar arguments can be made for all the bulk properties of the system.
Fluctuations are of fundamental importance in statistical mechanics because they provide
the means by which many physical properties of a molecular system can happen. For instance, the
density of a liquid at equilibrium is a fixed, uniform quantity and we feel justified in considering the
system to be isotropic - the same at all points within its bulk. Yet we know that the molecules in the
system are undergoing diffusion and can easily travel throughout the bulk of the liquid. It is difficult
to imagine how this diffusion can take place if the environment each molecule is in is completely
isotropic. If however we consider the density to be fluctuating minutely from the mean value at
different points in the bulk, we can readily see that such fluctuations would provide a means by
which the diffusion may take place. It is a surprising fact, but most of the physical properties of a bulk
system are driven by fluctuations, and indeed can be calculated directly from them. For this reason it
is possible to view fluctuations as even more fundamental than the average value.
A good example of the importance of fluctuation is provided by the Fluctuation-Dissipation
theorem, which is a theorem of great power in statistical mechanics. This theorem proposes that the
mechanism underpinning the response of a system to an external perturbation, is precisely the same
mechanism by which equilibrium fluctuations are held close to the average bulk value. Thus for
example, a molecule vibrationally excited by an infrared photon, will lose (i.e. dissipate) that energy
to the rest of the system by the same mechanism by which normal vibrational energies are
exchanged (i.e. fluctuate) between molecules at equilibrium. This insight is the basis of a theoretical
description of solution spectroscopy.
Although the fluctuations are extraordinarily small for large systems we must confront the
fact that any real simulation has a limited number of atoms and is carried out for a relatively small
number of steps compared to the systems considered in statistical mechanics. The fluctuation in the
mean-squared energy is:
38 Introduction to molecular modeling. 10. Molecular dynamics simulations by Rajmund Kaźmierkiewicz
This result can be calculated in terms of familiar thermodynamic quantities. For an ideal gas
there is no potential energy contribution and so the energy is <E> = 3/2NkT yielding the ideal gas
specific heat Cv = 3/2 Nk. In general the result is related to the size of system since:
. Practically, we can increase the sampled energy configurations by averaging over a longer
time:
Using fluctuation theory we have the mathematical expression for heat capacity:
10.9. Other ways of experimental verification of results of molecular
mechanics
The potential energy of the molecule calculated from a well-designed empirical force field
represents a strain in the molecule. Augmented with bond/group equivalents and statistical
mechanical corrections, it can be used to estimate the heat of the formation of a compound (which
can be directly compared with the experimental value). This quantity can be used also to compare
the relative stability of different compounds. Unfortunately, in most cases, the calculated potential
energy incorporates some arbitrary component which depends upon the types of atoms and
covalent bonds in the molecule, therefore comparison of the energies calculated for different
molecules cannot be rigorous. For this reason, potential energy will, in most cases, reliably evaluate
the difference in energy between conformers of the same molecule, but will fail if One will attempt
to calculate the change in energy after adding a new fragment into the molecule. Molecular
mechanics can also provide the interaction energy, , of two molecules A and B as:
Where , , and are potential energies of the optimized complex, the optimized molecule A,
and the optimized molecule B; respectively. Note that the type and number of atoms and covalent
bonds in the complex AB is equal to their sum in isolated molecules A and B, and the arbitrary
“energy zero” should cancel out in this case. For this reason, the difference between interaction
energies calculated for different complexes, (equal to ) is the preferred method over
direct comparison of the energies of different complexes (equal to ).
Potential energy functions can also be used to estimate contributions from intramolecular
vibrations to the so-called vibrational free energy and vibrational entropy. These quantities, and
contributions from translation and rotation of the molecule as a whole, vary with temperature and
are the main contributors to the thermodynamic functions such as enthalpy, free energy, specific
heat. One approach is to use the frequencies, , corresponding to normal modes within harmonic
approximation, that is, to calculate them from a mass scaled Hessian matrix at energy minimum. The
expressions for relating classical vibrational contributions to Helmholtz free energy, Fvib, internal
energy, Evib, heat capacity at constant volume
, and entropy Svib of the nonlinear
molecule, are derived in many standard textbooks for statistical mechanics:
Introduction to molecular modeling. 10. Molecular dynamics simulations 39 by Rajmund Kaźmierkiewicz
where R, T, and h are the gas constant, the absolute temperature, and Planck's constant respectively;
and N denotes the number of atoms in the molecule. Frequently, these values are augmented with a
correction to account for vibrations at T = 0 K, which is of quantum origin, by adding energy value at
zero degrees of Kelvin (E0) to the free energy Fvib and the internal energy Evib :
The harmonic approximation is quite accurate for isolated molecules. For complexes of two or more
molecules or systems containing water, the harmonic approximation breaks down. In this case,
molecular dynamics or Monte Carlo approaches are more reliable for estimation of thermodynamic
functions.
10.10. The Pressure
Fluctuations in the pressure are related to the isothermal compressibility, which is very small
for a dense fluid. For this reason calculation of the isothermal compressibility by the method of
fluctuations is a challenging task. In general, pressure is difficult to calculate accurately. The
agreement between the MD and Monte-Carlo (MC) methods is poor compared to the energy and the
statistics are significantly worse than for the energy.
10.11. The radial distribution function
The radial distribution function is an example of a pair correlation function, which describes
how, on average, the atoms in a system are radially packed around each other. This proves to be a
particularly effective way of describing the average structure of disordered molecular systems such
as liquids. Also in systems like liquids, where there is continual movement of the atoms and a single
snapshot of the system shows only the instantaneous disorder, it is extremely useful to be able to
deal with the average structure.
The radial distribution function is useful in other ways. For example, it is something that can
be deduced experimentally from x-ray or neutron diffraction studies, thus providing a direct
comparison between experiment and simulation. It can also be used in conjunction with the
interatomic pair potential function to calculate the internal energy of the system, usually quite
accurately.
40 Introduction to molecular modeling. 10. Molecular dynamics simulations by Rajmund Kaźmierkiewicz
Figure 25. Construction of a radial distribution function
To construct a radial distribution function is simple. Choose an atom in the system and draw
around it a series of concentric spheres, set at a small fixed distance (r) apart (see figure above). At
regular intervals a snapshot of the system is taken and the number of atoms found in each shell is
counted and stored. At the end of the simulation, the average number of atoms in each shell is
calculated. This is then divided by the volume of each shell and the average density of atoms in the
system. The result is the radial distribution function. Mathematically the formula is:
g(r)=n(r)/ r2r)
In which g(r) is the radial distribution function, n(r) is the mean number of atoms in a shell of width
r at distance r, is the mean atom density. The method need not be restricted to one atom. All the
atoms in the system can be treated this way, leading to an improved determination of the radial
distribution function as an average over many atoms.
The radial distribution function is usually plotted as a function of the interatomic separation
r. A typical radial distribution function plot (below) shows a number of important features. Firstly, at
short separations (small r) the radial distribution function is zero. This indicates the effective width of
the atoms, since they cannot approach any more closely. Secondly, a number of obvious peaks
appear, which indicate that the atoms pack around each other in “shells” of neighbors. The
occurrence of peaks at long range indicates a high degree of ordering. Usually, at high temperature
the peaks are broad, indicating thermal motion, while at low temperature they are sharp. They are
particularly sharp in crystalline materials, where atoms are strongly confined in their positions. At
very long range every radial distribution function tends to a value of 1, which happens because the
radial distribution function describes the average density at this range.
Figure 26. Both, MD and MC give similar forms for radial distribution function g(r)
Introduction to molecular modeling. 10. Molecular dynamics simulations 41 by Rajmund Kaźmierkiewicz
10.12. Calculation of Dynamic Properties from Molecular Dynamics
Simulations
In the paper by Jianshu Cao and Gregory A. Voth (J. Chem. Phys. 103(10), 8 September 1995)
a theory for time correlation functions in liquids is developed. It is based on the optimized quadratic
approximation for liquid state potential energy functions.
10.13. Correlations and the Correlation Time
Correlations between two different quantities X and Y are determined via the correlation
coefficient:
Where means covariance, and (X), (Y) are standard deviations. The value of cXY lies
between 0 and 1, with values close to 1 indicating high correlation. If the coefficient cXY is evaluated
at different times, it becomes a time correlation function cXY(t). For identical variables X=Y, cXX(t) is
called an autocorrelation function and its integral from 0 to ∞ is a correlation time.
10.14. Correlation Functions and Properties
The meaning of the coefficient in a simulation is represented by
where represents a point in phase space, that is a set of positions and momenta at a given time
step in the computer MD simulation. Time correlation functions are useful in molecular dynamics
simulations because their time integrals can be related to transport coefficients or other properties:
diffusion viscosity
dielectric constant
thermal conductivity
The Fourier transform of time correlation functions can be related to experimental spectra.
10.15. The Time Correlation Function
In MD, the system is moved in discrete time intervals following Newton's equations of
motion. At any time t we can calculate a property A(t). The time correlation function is the product of
the property at t and at a time t+.
The angle brackets represent statistical averaging. It is defined in statistical mechanics as averaging
over many similar systems (the ensemble). We can use many separate time frames of molecular
dynamics instead of many systems in the ensemble to obtain useful time decays that can be
analyzed.
42 Introduction to molecular modeling. 10. Molecular dynamics simulations by Rajmund Kaźmierkiewicz
10.16. Velocity Autocorrelation Function
The velocity autocorrelation function is a prime example of a time dependent correlation
function, and is important because it reveals the underlying nature of the dynamical processes
operating in a molecular system. It is constructed as follows. At a chosen origin in time (i.e. some
moment when we chose to start the calculation) we store all three components of the velocity vi,
where
vi=[vx(t0),vy(t0),vz(t0)]i
for every atom (i) in the system. We can already calculate the first contribution to the velocity
autocorrelation function, corresponding to time zero (i.e. t=0). This is simply the average of the scalar
products vi . vi for all atoms:
At the next time step in the simulation t = t0 + t; and the corresponding velocity for each atom is
Vi = [Vx(t0 + t), Vy(t0 + t), Vz(t0 + t)]i
and we can calculate the next point of the velocity autocorrelation function as
We can repeat this procedure at each subsequent time step and so obtain a sequence of points in
the velocity autocorrelation function, as follows:
or (for short)
.
Though this can be continued forever, we generally stop after a fixed value of n, and start
again to calculate another velocity autocorrelation function, beginning at a new time origin. The final
velocity autocorrelation function can then be an average of all the velocity autocorrelation function's
we have calculated in the course of our simulation. What could such a function tell us about the
molecular system?
Consider a single atom at time zero. At that instant the atom (i) will have a specific velocity vi.
If the atoms in the system did not interact with each other, the Newton's Laws of motion tell us that
the atom would retain this velocity for all time. This of course means that all our points Cv(t) would
have the same value, and if all the atoms behaved like this, the plot would be a horizontal line. It
follows that a velocity autocorrelation function plot that is almost horizontal, implies very weak
forces are acting in the system.
On the other hand, what happens to the velocity if the forces are small but not negligible?
Then we would expect both its magnitude and direction to change gradually under the influence of
these weak forces. In this case we expect the scalar product of Vi(t=t0) with Vi(t=t0+nt) to decrease
on average, as the velocity is changed. (In statistical mechanics we say that the velocity decorrelates
with time, which is the same as saying the atom 'forgets' what its initial velocity was.) In such a
system, the velocity autocorrelation function plot is a simple exponential decay, revealing the
Introduction to molecular modeling. 10. Molecular dynamics simulations 43 by Rajmund Kaźmierkiewicz
presence of weak forces slowly destroying the velocity correlation. Such a result is typical of the
molecules in a gas.
What happens when the interatomic forces are strong? Strong forces are most evident in
high density systems, such as solids and liquids, where atoms are packed closely together. In these
circumstances the atoms tend to seek out locations where there is a near balance between repulsive
forces and attractive forces, since this is where the atoms are most energetically stable. In solids
these locations are extremely stable, and the atoms cannot escape easily from their positions. Their
motion is therefore an oscillation; the atoms vibrate backwards and forwards, reversing their velocity
at the end of each oscillation. If we now calculate the velocity autocorrelation function, we will
obtain a function that oscillates strongly from positive to negative values and back again. The
oscillations will not be of equal magnitude however, but will decay in time, because there are still
disrupting forces acting on the atoms to change their oscillatory motion. So what we see is a function
resembling a damped harmonic motion.
Liquids behave similarly to solids, but now the atoms do not have fixed regular positions. A
diffusive motion is present to destroy rapidly any oscillatory motion. The velocity autocorrelation
function therefore may perhaps show one very damped oscillation (a function with only one
minimum) before decaying to zero. In simple terms this may be considered a collision between two
atoms before they rebound from one another and diffuse away.
As well as revealing the dynamical processes in a system, the velocity autocorrelation
function has other interesting properties. Firstly, it may be Fourier transformed to project out the
underlying frequencies of the molecular processes. This is closely related to the infra-red spectrum
of the system, which is also concerned with vibration on the molecular scale. Secondly, provided the
velocity autocorrelation function decays to zero at long time, the function may be integrated
mathematically to calculate the diffusion coefficient D0, as in:
This is a special case of a more general relationship between the velocity autocorrelation
function and the mean square displacement, and are known as the Green-Kubo relations, which
relate correlation functions to so-called transport coefficients.
10.17. Calculating the diffusion coefficient from the mean-square
displacement
10.17.1. The Mean Square Displacement
Molecules in liquids and gases do not stay in the same place, but move about constantly. It is
in fact essential that they do so, otherwise they would not possess the property of fluidity. The
phenomenon is apparent if you place a drop of ink into water - after a while the color is evenly
distributed through the liquid. It is obvious that the molecules of the ink have moved through the
bulk of the water. This process is called diffusion and it happens quite naturally in fluids at
equilibrium. (The water molecules themselves are also undergoing diffusion, though this is not so
obvious.)
44 Introduction to molecular modeling. 10. Molecular dynamics simulations by Rajmund Kaźmierkiewicz
The motion of an individual molecule in a dense fluid does not follow a simple path. As it
travels, the molecule is jostled by collisions with other molecules which prevent it from following a
straight line. If the path is examined in close detail, it will be seen to be a good approximation to a
random walk. Mathematically, a random walk is a series of steps, one after another, where each step
is taken in a completely random direction from the one before. This kind of path was famously
analyzed by Albert Einstein in a study of Brownian motion and he showed that the mean square of
the distance travelled by a particle following a random walk is proportional to the time elapsed. This
relationship can be written as
where is the mean square distance and t is time. D and C are constants. The constant D is the
most important of these and defines the diffusion rate. It is called the diffusion coefficient.
10.17.2. What is the mean square distance and why is it significant?
Imagine a single particle undertaking a random walk. For simplicity assume this is a walk in
one dimension (along a straight line). Each consecutive step may be either forward or back, we
cannot predict which, though we can say we are equally likely to step forward as to step back. (A
drunk man comes to mind!) From a given starting position, what distance are we likely to travel after
many steps? This can be determined simply by adding together the steps, taking into account the
fact that steps backwards subtract from the total, while steps forward add to the total. Since both
forward and backward steps are equally probable, we come to the surprising conclusion that the
probable distance travelled sums up to zero!
If however, instead of adding the distance of each step we added the square of the distance,
we realize that we will always be adding positive quantities to the total. In this case the sum will be
some positive number, which grows larger with every step. This obviously gives a better idea about
the distance (squared in this case) that a particle moves. If we assume each step happens at regular
time intervals, we can easily see how the square distance grows with time, and Einstein showed that
it grows linearly with time.
In a molecular system a molecule moves in three dimensions, but the same principle applies.
Also, since we have many molecules to consider we can calculate a square displacement for all of
them. The average square distance, taken over all molecules, gives us the mean square
displacement. This is what makes the mean square displacement significant in science: through its
relation to diffusion it is a measurable quantity, one which relates directly to the underlying motion
of the molecules.
In molecular dynamics the mean square displacement is easily calculated by adding the
squares of the distance. Typical results (for a liquid) resemble the following plot.
Introduction to molecular modeling. 10. Molecular dynamics simulations 45 by Rajmund Kaźmierkiewicz
Figure 27. The linear dependence of the mean square displacement plot is apparent. If the slope of this plot is taken, the diffusion coefficient D may be readily obtained
At very short times however, the plot is not linear. This is because the path a molecule takes
will be an approximate straight line until it collides with its neighbor. Only when it starts the collision
process will its path start to resemble a random walk. Until it makes that first collision, we may say it
moves with approximately constant velocity, which means the distance it travels is proportional to
time, and its mean square displacement is therefore proportional to the time squared. Thus at very
short time, the mean square displacement resembles a parabola. This is of course a simplification -
the collision between molecules is not like the collision between two pebbles, it is not instantaneous
in space or time, but is `spread out' a little in both. This means that the behavior of the mean square
displacement at short time is sometimes more complicated than this mean square displacement plot
shows.
10.17.3. The Mean Squared Displacement and the Velocity
Autocorrelation Function
The mean square displacement and the velocity autocorrelation function seem to be two
very different functions. The mean square displacement is (for the most part) a linear function of
time, while the velocity autocorrelation function displays a complicated dependence on time. But a
little thought will suggest that they must have something in common. Both, in an average sense,
describe the motion of a molecule with time and must therefore be related somehow. The
mathematical relationship is revealing, as the following shows.
We can describe the distance r(t) a molecule moves in time as an integral of its velocity v(t):
The square of this distance is thus
defining u'=u+s and integrating over u, results in the following form where the ensemble average has
also been taken:
46 Introduction to molecular modeling. 10. Molecular dynamics simulations by Rajmund Kaźmierkiewicz
In this equation <v(0) v(s)> is the velocity autocorrelation function, so the relationship between mean
square displacement and velocity autocorrelation function is now apparent. This can also be written
as
What this integral shows is that the mean square displacement is comprised of two parts. The first
term on the right includes the time t explicitly and if we assume that when t is large, the velocity
autocorrelation function decays to zero (as it usually does) then the integral here will have a fixed
value. Since the second term also integrates to a fixed value for large t, we can see that this equation
is equivalent to Einstein's, provided we assume that
and
when t is large. This is a very important result, as it shows how the diffusion coefficient can be
obtained from both the velocity autocorrelation function and the mean square displacement.
Another thing we can see is that when t is small, the time dependence of the velocity
autocorrelation function cannot be ignored (it is no longer constant). So from the above integral, it
follows that the mean square displacement must depend on the behavior of the velocity
autocorrelation function at short time. This means the short time behavior of the mean square
displacement cannot be linear. Molecular motion only becomes random after the velocity
autocorrelation function becomes zero and the molecules have “forgotten” what speed and direction
they began with at t=0.
10.18. Molecular dynamics simulations of liquid water
The water computer model is probably the most frequently used compound in molecular
mechanics. The early attempts to model water molecules originate in 1970 years of XX century:
Rahman and Stillinger - Original four site model 1971
Stillinger and Rahman - Revised model 1973
Jorgensen - TIPS3 model, three site model 1981
Berendsen - Optimization of parameters 1981
Jorgensen - Comparison of models 1983
Review of properties of selected, so-called, three-point water models: M. Pekka, L. Nilsson, J. Phys.
Chem. A 2001, 105: (9954-9960).
In Rahman and Stillinger’s model water molecules were treated as asymmetric rigid rotors. They
defined effective pair potentials to replace higher order potential terms and …used neon parameters
Introduction to molecular modeling. 10. Molecular dynamics simulations 47 by Rajmund Kaźmierkiewicz
for oxygen (that is currently rather unusual). They define also a switching function that allows the
potential to vary smoothly to zero and assigned charges to lone pairs and hydrogens, this is the
reason why it is called the four-site model for H2O.
10.18.1. Dielectric Relaxation
Our lack of knowledge of the true dipole moment in liquid water limits our ability to predict
the static dielectric constant . The four-site model guarantees a tetrahedral hydrogen bond
arrangement, the hydrogen bonds are too short and too directional. To improve the original model
the lone pairs were shortened to make the ST2 model (d = 0.8 Å).
Figure 28. The Stillinger and Rahman (JCP 1974, 60, 1545) model of water molecule
10.18.2. Three-site models for water
There are a few three-site water models:
The original TIPS3 model has positive charges on the hydrogen atoms and a negative charge on
oxygen atom (qO = -2qH). (Jorgensen JACS 1981, 103, 335)
Berendsen parameterized a three-site water model (SPC) and got better agreement with the
experiment. (Berendsen et al. in Intermolecular Forces 1981 p.331)
Comparison of those models is made in Jorgensen et al. (JCP 1983, 79, 926)
The TIP3P model is now frequently used. The second peak of the radial O-O pair distribution
function gO-O tends to disappear for this model. It is a good overall model and it is much less
expensive than TIP4P or other four site models.
10.18.3. Implicit Treatment of Solvation
Figure 29. Mean influence of water captured by the solvation free energy
O
H
H
q = -0.23 e
q = -0.23 e
q = +0.19 e
q = +0.19 e
d
48 Introduction to molecular modeling. 10. Molecular dynamics simulations by Rajmund Kaźmierkiewicz
The Implicit Treatment of Solvation can be very efficient especially with the generalized Born
(GB) approach. For example the generalized Born with a smooth switching (Michael S. Lee, Freddie R.
Salsbury, and Charles L. Brooks, J. Chem. Phys. 116, 10606 (2002); Im W, Feig M, Brooks CL 3rd.,
Biophys J. 2003 Nov;85(5):2900-18.) is about 30 times faster than comparable explicit solvent
simulations (W. Im, M.S. Lee, and C.L. Brooks III, J. Comput. Chem. 24:1691-1702 (2003).). It offers a
good balance (compromise) between accuracy and efficiency.
10.19. Conformational searching, Quench Dynamics
Quench (or quenched) molecular dynamics was historically one of the first conformational
“search” methods. Currently there exist better tools like Monte-Carlo method or Replica Exchange
Molecular Dynamics.
10.19.1. Protocol for conformational search: quenched molecular
dynamics
1. Energy minimization.
2. Equilibration at high temperature for production dynamics. Run production dynamics and save
structures at periodic intervals (for example after every 1 ps).
3. Slowly cool the structures (annealing) and minimize energy of resulting conformations.
4. Save minimized structures for structural studies.
10.20. Constraints
Constraints (restrictions on the conformational “freedom” of the molecule) may be imposed
during minimization, as well as during dynamics. These constraints may be based on experimental
data such as NOEs from an NMR experiment or they may be imposed by a template such that One
forces a ligand to find the minimum closest in structure to a target molecule. Template forcing is also
important for homology modeling. Since it is not possible, at present, to fold a protein by single
energy minimization, one can approach the question of determining the fold of a protein by
comparing it with a structure that has significant amino acid sequence homology.
10.20.1. Restrained dynamics as a tool in NMR structure determination
Distance restraints force two atoms toward a given value
E = k(rij – rtarget)2
where k is the force constant and rtarget is the target distance. An “energy penalty” is paid for
deviation from the target distance. In a typical NOE experiment, usually only the upper bound
distance is known, for example r < 5Å, for that reason an experimental data can be
Figure 30. An illustration of a so-called „flat-bottomed” potential
Introduction to molecular modeling. 10. Molecular dynamics simulations 49 by Rajmund Kaźmierkiewicz
incorporated into a simulation using a so-called „flat-bottomed” potential. A flat-bottomed restraint
function allows flexibility to accommodate typical data where the minimum distance between nuclei
is determined from van der Waal’s radii and the data impose an upper bound.
10.20.2. Use of constraints to increase the integration step
The SHAKE algorithm constrains motions so bond lengths do not exceed preset thresholds. It
uses iterative adjustments in atom positions (one-by-one). The SHAKE algorithm typically improves
(shortens) computational time by about 3 times.
Figure 31. An illustration of the effect of the SHAKE algorithm on molecular structures.
Application of the „SHAKE“ algorithm enables the increase of the integration step from t =
1fs (fs = femtosecond) to t =2 fs.
Figure 32. "Shaking" water
The special “three-point” algorithm (SETTLE) is used for restraining deformations of water models.
Instead of restraining the bond angle there is an artificial H - H bond introduced (and constrained).
10.20.3. SHAKE and minimization
Since SHAKE is an algorithm based on dynamics, the minimization algorithm is not aware of
what SHAKE is doing; for this reason, minimizations generally should be carried out without SHAKE.
One exception is short minimization whose purpose is to remove close contacts between atoms
before molecular dynamics simulations can begin. Even in this case SHAKE can be avoided by
artificial, substantial increase of bond and bond angle force constants values during the short initial
minimization.
50 Introduction to molecular modeling. 10. Molecular dynamics simulations by Rajmund Kaźmierkiewicz
10.21. Boundary conditions
Many current simulations are performed using periodic boundary conditions, so that surface
effects can be avoided and configurations, typically encountered at the macroscopic level of the
system, can be obtained. In this case, a particle interacts not only with all the particles in the
systems, but also with their periodic images. The boundary conditions can be divided into two
classes:
Spatial boundary conditions:
MD simulations of biomolecules can be performed in:
Vacuum: it is of little interest
Condensed phase: the system must be finite
Spherical droplet (finite boundary): boundary artifacts
Periodic box (there is no boundary at all): periodicity artifacts
Thermodynamic boundary conditions:
MD simulations can be performed at different ensembles, according to statistical mechanics they can
be divided into four groups:
Constant NVE: micro-canonical ensemble
Constant NVT: canonical ensemble
Constant μVT: grand-canonical ensemble
Constant NPT: isothermal-isobaric ensemble
Figure 33. Boundary conditions box or droplet?
We cannot simulate infinite systems, but finite systems lead to boundary effects. The
solution is to use periodic boundary conditions (PBC). How to make sure a particle does not interact
with itself ? Use the minimum image convention and cut-off interactions beyond a specified
distance. After applying periodic boundary conditions the electrostatic interactions need “special
treatment” as they are long range.
Figure 34. An illustration of a periodic boundary conditions
Introduction to molecular modeling. 10. Molecular dynamics simulations 51 by Rajmund Kaźmierkiewicz
After applying boundary conditions the finite system is converted into an infinite system
without increasing computational cost. As a sideline result the new “features” are introduced:
unwanted surface effects are eliminated and an artificial periodicity is imposed.
Figure 35. An illustration of an artificial periodicity
Motion of atoms in a “box replicas” mirrors the motion of atoms in the central box. If an
atom leaves the central box, it's replica enters the central box from the other side this implies that
the number of atoms in a central box is conserved. It is worth noting that not only a rectangular box
can be replicated using the periodic boundary conditions.
A B C
Figure 36. A. Example: truncated octahedron; B. Peptide in aqueous solution in a periodic truncated octahedron. C. An illustration of how the cut-off value Rc can be applied to the extended system using periodic
boundary conditions
Figure 37. Nothing can stop particles from interacting with other particles from the neighboring boxes
52 Introduction to molecular modeling. 10. Molecular dynamics simulations by Rajmund Kaźmierkiewicz
Figure 38. An illustration of the minimum image convention
The “minimum image convention”: a particle doesn't interact with all of the other particles,
only with the nearest non-equivalent neighbors, and each atom interacts with at most one image of
every atom (each individual particle in the simulation interacts with the closest image of the
remaining particles in the system).
The periodic boundary conditions (usually) work in three dimensions. In other words, each
system “sees infinite number of its images” along X, Y and Z axes. In order to simulate surface one
needs to apply two-dimensional periodic boundary conditions.
Figure 39. Two-dimensional periodic boundary conditions
10.22. Boundary conditions, „special” treatment of electrostatic
interactions
10.22.1. Reaction field method
Interactions are restricted inside a given radius, everything else outside is taken as a
homogeneous medium with a dielectric constant s
Introduction to molecular modeling. 10. Molecular dynamics simulations 53 by Rajmund Kaźmierkiewicz
Figure 40. An illustration of the reaction field method
System energy is now defined as
For atomistic systems the dipoles can be taken as the charge groups. This approach is
computationally quite fast and it has become a popular approach in the modeling of very large
biostructures. However, the (still unsolved) problem is what value assigns to s?
10.22.2 Ewald summation
The Ewald summation method takes into account all interactions to an atom inside the MD
“central box” as well as from the periodic images (Ann. Phys. 64 (1921) 253):
This sum does not necessarily converge. The idea is to do the summation in such an order that it
does. (And preferably as fast as possible).
54 Introduction to molecular modeling. 10. Molecular dynamics simulations by Rajmund Kaźmierkiewicz
10.22.3. Scaling and other related methods
Standard Ewald summation is O(N2). A full optimization gives a slight improvement, O(N3/2).
Particle-particle particle-Mesh (PPPM, P3M) The method relies on expressing the long-range
interparticle force as the sum of two components: the short-range force, which is only nonzero
within some cut-off radius, and the ”reference” force, that is long-ranged and smooth and can be
approximated on a grid. P3M scales as O(N log N). (Hockney and Eastwood, Computer simulation
using particles (McGraw-Hill, New York. 1981))
Fast multipole method (FMM) looks at different regions in space at different resolutions; the
contributions of regions far away are described as electric multipole expansions. FMM scales as
O(N). However, in practice these methods become more efficient than an optimized Ewald
routine only for particle numbers ~ 105. (J. Comput. Phys. 73 (1987) 325)
For intermediate system sizes, N = 103 – 104, the Particle Mesh Ewald method (PME), that scales
as O(N log N), is a good alternative.
The reciprocal sum in the Ewald summation is approximated using Fast Fourier Transform with
convolutions on a grid where charges are interpolated to the grid points. PME does not
interpolate but evaluates the forces by analytically differentiating the energies. (J. Chem. Phys.
98 (1993) 10089)
For systems that are not fully periodic, in addition to using FMM there is also a method called
Lekner summation (Physica A 176 (1991) 485-498).
10.23. Practical Tips for Setting up MD
1. Decide what you want to simulate (protein, DNA, sugars, water, ions, lipids)
2. Build Individual Components
add missing atoms
add hydrogens
modify ionization states
add functional groups into residues
compute missing energy parameters with quantum mechanics (QM)
3. Solvate Structure
4. Combine Molecular Components (lipid bilayer, water, ions, polymeric chains)
5. Minimize Energy / Equilibrate
10.24. MD Simulation protocol
Depends on the purpose of the simulation:
1. following a process in time implies the dynamical simulation
time is important
running many trajectories
averaging over set of trajectories
2. getting equilibrium properties of the system implies thermodynamical simulation
time is not important
running one trajectory
averaging over time ( vs. average over ensemble in 1.)
Introduction to molecular modeling. 10. Molecular dynamics simulations 55 by Rajmund Kaźmierkiewicz
10.24.1. Sample MD simulations work-flow
Figure 41. Sample MD simulations work-flow
10.25. The „heating” dynamics stage, the temperature control
Currently, a typical molecular dynamics simulation has 102 – 108 particles. It is still very far from
Avogadro’s number, 6 x 1023. So what does it mean temperature and pressure in such a small system?
10.25.1. Temperature
The result from the statistical mechanics - the equipartition theorem of energy
states that for Nf degrees of freedom in the kinetic energy, the temperature function may be given as
10.25.2. MD simulations with a temperature bath.
Figure 42. MD simulations with a temperature bath
56 Introduction to molecular modeling. 10. Molecular dynamics simulations by Rajmund Kaźmierkiewicz
A temperature bath tries to keep the protein at constant temperature or kinetic energy (Ek).
E is not conserved. Now, as the system falls down into a more stable state (lower potential
energy, Ep), the temperature bath “steals” the extra Ek .
On the other hand, if the system moves into a higher (less favorable) potential energy
(exchanging Ek for Ep and thus cooling off), the temperature bath gives a heat “boost” to restore
Ek . Ep is not conserved and is time dependent.
The system samples potential energy Ep with probability proportional to .
The protein now samples a much larger set of phase space (q, p). The (q, p) set is called the
canonical ensemble.
The fluctuations in the E distribution are critical, because large E’s and Ep’s get you OVER energy
barriers.
10.25.3. Barriers, Temperature and Timescales
Figure 43.The protein energy landscape, barriers, temperature and timescales
Protein energy landscape is highly complex and rugged with numerous local minima. Increase of the
temperature enhances barrier crossing. The “enhancement factor” depends exponentially on the
barrier height. After increasing the temperature the typical time-to-live in one high-energetic
conformational state shortens significantly, and also enhances the ability to cross barriers.
10.25.4. Berendsen thermostat
The weak-coupling thermostat, commonly known as the Berendsen thermostat or
temperature control, is a common way for controlling the temperature in MD simulations. (J. Chem.
Phys. 81 (1984) 3684). The basic idea is that One adds a frictional term to the equations of motion,
which then drives the system (exponentially) towards the desired temperature. In practice, the
velocities of the particles are scaled with a factor
The Berendsen algorithm is simple to implement and it is very efficient for reaching the desired
temperature from far-from-equilibrium configurations. The downside is that it has been argued that
this method does not produce the correct statistical ensemble.
Introduction to molecular modeling. 10. Molecular dynamics simulations 57 by Rajmund Kaźmierkiewicz
10.25.5. Nosé-Hoover thermostat
Another popular thermostat is the Nosé-Hoover thermostat, which retains a Boltzmann
equilibrium distribution. (Hoover, Phys. Rev. A 31 (1985) 1695; S. Nosé, J. Chem. Phys. 81 (1984) 511;
Mol. Phys. 52 (1984) 255.)
Again, One can add a frictional term to the equations of motion
with the dynamics of the friction coefficient by
Q = fictional “heat bath mass”. Large Q means weak coupling. Nosé suggested Q ~ gkBT, where g is
the number of degrees of freedom in the
system (~ 6N).
10.25.6. Andersén thermostat
Another straightforward way to control the temperature (with a proper statistical
distribution) is the method by Andersén (J. Chem. Phys. 72 (1980) 2384). After each time step, each
atom is assigned, with some probability, a new velocity corresponding to the desired temperature T0.
This can be thought as coupling all the atoms in the system to an external heat bath. The method
enables one to calculate thermodynamic averages. However, due to the perturbation of the atom
velocities, it is not possible to precisely study atomistic processes in detail.
10.25.7. Trivial Temperature scaling
The simplest possible way to control the temperature in the system would be simply to set
the temperature of the system at every time step exactly to T (this means to scale atom velocities
with a suitable factor). However, for small systems this may cause significant perturbations of the
atom trajectories and the overall dynamics of the system. In addition, this totally suppresses any
possible (natural) fluctuations in system temperature and does not provide the correct statistical
ensemble (NVT, NpT). Proper temperature is characterized by distributions, not averages.
10.25.8. Berendsen barostat
The system pressure is set toward a desired value by changing the dimensions of the
simulation cell size during the simulation. The scaling factor (for each dimension) is
Where P0 is the desired value of pressure, τp the coupling time constant for the pressure scaling and
β the isothermal compressibility of the system. The scaling is done for all components of the atom
positions as well as the simulation cell dimensions. Note that since β appears in the scaling factor as a
product with δt/τp, provided the time step and the coupling constant are chosen wisely (usually
τp>100 δt) , one does not need to know the exact isothermal compressibility of the system in
58 Introduction to molecular modeling. 10. Molecular dynamics simulations by Rajmund Kaźmierkiewicz
question. If used with the Berendsen thermostat one obtains realistic fluctuations in T and P (J.
Chem. Phys. 81 (1984) 3684).
10.25.9. Andersén barostat
The Andersén method for pressure control uses a fictional piston, with a ”mass” Q, to control
the volume of the simulation cell. (J. Chem. Phys. 72 (1980) 2384)
The kinetic and potential energy of the piston are:
and
If the atom positions r and velocities v are written in reduced units s so that r = V1/3s and v =
V1/3(ds/dt), the equations of motion are
Where f are the forces on the atoms, P(t) is the instantaneous pressure and P0 the desired pressure.
10.25.10. Parrinello-Rahman barostat
A method for pressure control, by Parrinello and Rahman was first introduced in J. Appl.
Phys. (52 (1981) 7182). This method has the particular advantage that it allows a variable simulation
cell shape. Basically, the box vectors are set to follow an equation of motion, and the equations of
motion of the particles are also changed as in the Nosé-Hoover thermostat. No instantaneous change
of atom positions takes place. It is advised that for an exhaustive derivation of the Parrinello-Rahman
barostat and the related equations of motion, look at the original paper cited above.
Nosé-Hoover thermostat and Parrinello-Rahman barostat can be used to provide most realistic
fluctuations in temperature and pressure when one is interested in the thermodynamic properties of
a system.
10.25.11. So which ones to use?
Do not use any trivial quenching scaling methods. They suppress fluctuations, and do not
provide the correct statistical ensembles. Berendsen T and p control are simple to implement and
use. In addition, they can steadily drive the system state far from equilibrium toward equilibrated
state. This is very handy at the start of the simulation, where significant fluctuations may take place.
However, if you need to produce the correct statistical ensemble you will need to use other
methods. Nose-Hoover thermostat with Rahman-Parrinello barostat (J. Appl. Phys. 52 (1981) 7182)
is possibly the best option there.
10.26. „Production run” protocol, „heating” dynamics
After an initial minimization stage, typically using the robust steepest descent (SD) and (CG)
algorithms to resolve any initial poor contacts within the system without creating large distortions in
the overall structure, the heating dynamics is carried out. It is the first stage of “production run” MD
simulation protocol, which may take place just after the initial (optimized) structure of the system
becomes available. A molecular dynamics simulation heating stage is employed to add thermal
energy to the system to reach a target temperature. A standard molecular dynamics simulation stage
is then employed to equilibrate the system at a target temperature. The purpose of the equilibration
Introduction to molecular modeling. 10. Molecular dynamics simulations 59 by Rajmund Kaźmierkiewicz
stage is to ensure that the energy in the system is distributed appropriately among all degrees of
freedom. This allows the system to achieve thermal equilibration at the target temperature. The last
part of the standard dynamics cascade is typically the “production stage” of molecular dynamics in
an NVT or NVE statistical ensemble using a velocity or a leap-frog Verlet integration algorithm. The
results of the production stage are stored in the simulation trajectory (this is usually a text file), from
which structural and energetics properties can be calculated and subsequently analyzed.
10.27. The Replica-Exchange algorithm
Figure 44. The Replica-Exchange algorithm
Schematic representation of the exchange of structures between replicas (parallel MD
simulations, starting from the common initial structure, which are carried out at different
temperatures) of the system.
500 K
420 K
355 K
300 K
280 K
Figure 45. Schematic representation of the exchange of structures between replicas
60 Introduction to molecular modeling. 10. Molecular dynamics simulations by Rajmund Kaźmierkiewicz
A drawback of temperature replica-exchange (Chemical Physics Letters 1999, 314(1-2): 141-
151) simulations is that many replicas are needed for simulations in an explicit solvent. An effective
simulation of temperature replica-exchange of a system consisting of small peptides in water, Tmax –
Tmin ~ 300K, may need even 50-80 replicas.
Since the replicas are non-interacting and independent their Hamiltonians (energy functions)
need not be identical. One may use different Hamiltonians and exchange those (R. Affentranger, I.
Tavernelli, E. E. Di Iorio. A Novel Hamiltonian Replica Exchange MD Protocol to Enhance Protein
Conformational Space Sampling, J. Chem. Theory Comput. 2006, 2(2): 217-228) instead of
temperature or restrict the changes in the Hamiltonians to only a subset of the degrees of freedom
to reduce the number of replicas needed.
10.28. Simulated Annealing
The simulated annealing (S. Kirkpatrick, C. D. Gelatt, Jr. and M. P. Vecchi, Science 13
May 1983: Vol. 220 no. 4598 pp. 671-680) is a popular global optimization algorithm. It starts
near 0 K then the temperature rises rapidly (in ~ 1-5 ps) to T ~ 1000K, the system stays at that
temperature for a while (in ~50 ps) and, after that stage, it decreases T in small steps (cooling
schedule, in ~100-500 ps). It is easy to understand and implement but it might be easily trapped in
local minima.
Figure 46. Examples of simulated annealing cooling schedules
10.29. Langevin equation of motion
Langevin equation of motion
where is an acceleration multiplied by mass of the moving particle, represents a friction
force, is a conservative force (analogous to the force from Newton’s equation of motion), is a
random force. According to Langevin, at each time step all particles receive a random force and have
Introduction to molecular modeling. 10. Molecular dynamics simulations 61 by Rajmund Kaźmierkiewicz
their velocities lowered using a constant friction. The friction force is proportional to velocity (f > 0)
and, because of the “-“ sign, it lowers the velocities what implies a decrease in kinetic energy and
temperature. The random force, on the contrary, adds kinetic energy to the particle and it causes the
increase of temperature. Balance of those two forces maintains the system temperature at the set
value. This works like a thermostat, and in fact it is called the Langevin thermostat, and it generates
the correct canonical ensemble.
According to the statistical mechanics the random force can be written as .
Everything under the square root sign depends on an environment of the molecule and is constant
for a given system (only temperature, T may fluctuate). The is a so-called stationary Gaussian
process with zero mean. It means that the random force vanishes after averaging over time. Such a
random force mimics the random collisions of the modeled molecule with the solvent molecules but
they (the solvent molecules) are not included explicitly in the simulations. This method saves a lot of
computational time and it is used sometimes instead of the standard MD with other solvent models.
10.30. Brownian Dynamics (BD)
The dynamic contributions of the solvent are incorporated into the Brownian dynamics
equations as a dissipative random force (Einstein’s derivation on 1905). Therefore, water molecules
are not treated explicitly. Since BD algorithm is derived under the conditions that solvent damping is
large and the inertial memory is lost in a very short time, longer time-steps can be used. In other
words, BD method is suitable for long time simulation.
If the Langevin equation can be expressed as
Here, and represent the position and mass of atom i, respectively. is a frictional coefficient
and is determined by the Stokes’ law, that is, in which
is a Stokes radius of
atom i and is the viscosity of water. is the systematic force on atom i. is a random force on
atom i having a zero mean and a variance ; this derives from
the effects of solvent. For the overdamped (overdamped, in this context, means that “something”
decreases gradually, the system returns - exponentially decays - to equilibrium without oscillating)
limit, we set the left side of Langevin equation to zero,
This equation can be “integrated” using the Verlet-like procedure and it leads to the Brownian
dynamics equation:
where t is a time step and a random noise vector obtained from Gaussian distribution.
62 Introduction to molecular modeling. 11. Monte Carlo Method by Rajmund Kaźmierkiewicz
11. Monte Carlo Method
The Monte Carlo simulation technique has formally existed since the early 1940s, where it had
applications in research into nuclear fusion. There are a number of isolated and undeveloped
instances on much earlier occasions. Buffon's [Georges Louis Leclerc Comte de Buffon (07.09.1707.-
16.04.1788.)] had an idea to drop a needle of length L at random on grid of parallel lines of spacing D.
For L less than or equal D one can obtain probability that the needle intersects the grid P = 2·L/·D.
After dropping the needle N times and count R intersections one obtains P = R / N, and the
approximate value of π, π = 2·L·N/R·D.
11.1. What is Monte Carlo?
Monte Carlo (MC) methods are stochastic techniques, this means they are based on the use
of random numbers and probability statistics to investigate problems. The MC methods are used in
many areas from economics to nuclear physics to regulating the flow of traffic. The way they are
applied varies from field to field, and there are a lot of variants of MC even within (bio)chemistry. MC
methods derive their collective name from the fact that Monte Carlo, the capital of Monaco, has
many casinos and casino roulette wheels are a good example of a random number generator.
Note that these methods only provide an approximation of the answer. The attempt to
minimize errors is the reason there are so many different Monte Carlo methods. Major Components
of MC methods are: the probability distribution function, random number generator, sampling rule,
scoring (function).
11.2. Monte Carlo (MC) Simulation
Instead of evaluating forces to determine incremental atomic motions, Monte Carlo
simulation simply imposes relatively large motions on the system and determines whether or not the
altered structure is energetically feasible at the temperature simulated. The system jumps abruptly
from conformation to conformation, rather than evolving smoothly through time. It can traverse
barriers without feeling them; all that matters is the relative energy of the conformations before and
after the jump. Because MC simulation samples conformation space without a true `time' variable or
a realistic dynamics trajectory, it cannot provide time-dependent quantities. However, it may be
much better than MD in estimating average thermodynamic properties for which the sampling of
many system configurations is important. One of the first applications of the MC method was
calculation of complicated integrals in 1940’s. MC can also be applied to calculate average
macroscopic properties of modeled biomolecular systems, because they are “conveniently” defined
by (rather complicated) integrals especially when an average value of a quantity of interest is defined
by the typical integral:
, where
Introduction to molecular modeling. 11. Monte Carlo Method 63 by Rajmund Kaźmierkiewicz
Figure 47. The interplay between the potential energy and population distribution in MC method
In molecular mechanics one usually looks for the lowest energy conformation. We are in a
very fortunate situation, because from statistical mechanics one can conclude that the system will
preferably populate the lowest energy states. Such a physical low favors sampling of low-energy
structures.
Consider the game of solitaire: what’s the chance of winning with a properly shuffled deck? It
is hard to compute analytically because winning or losing depends on a complex procedure of
reorganizing cards. Why not just play a few hands, and see empirically how many do in fact win? The
Monte Carlo principle states that one can approximate a probability density function (in this
example the “frequency” of winning hands) using only samples from that density (one can estimate
the chance of winning from a couple of card deals).
11.2.1. Evolution of Monte Carlo methods so far…
If we take a look at the short Monte-Carlo history we could see:
1. Uniform points and original integrand…
but this had very poor efficiency
2. Uniform points and transformed integrand…
but this only worked for certain integrands
3. Non-uniform points and scaled integrand…
but this is very cumbersome for complicated integrands…
4. Now, we try Markov chain approaches…
11.2.2. Markov chains
Properties of Markov chain are: it is a sequence of randomly-chosen states, the probability of
transitions between states is independent of history, the entire chain represents a stationary
probability distribution. There are some examples of Markov chains applicable to molecular
mechanics: computer random number generators, Brownian motion, Hidden Markov Models,
(perfectly) encrypted data.
A Markov chain is a sequence of random variables X1, X2, X3 ... with the Markov property,
namely that, given the present state, the future and past states are independent. The possible values
of Xi form a countable set S called the state space of the chain. Markov chains are often described by
a directed graph, where the edges are labeled by the probabilities of going from one state to the
other states.
64 Introduction to molecular modeling. 11. Monte Carlo Method by Rajmund Kaźmierkiewicz
11.2.3. Markov chain Monte Carlo
Assembling the entire distribution for MC is usually hard due to:
Complicated energy landscapes
High-dimensional systems
Extraordinarily difficult normalization
Solutions are:
Build up distribution from Markov chain
Choose local transition probabilities which generate distribution of interest
Each random variable is chosen based on the previous variable in the chain
“Walk” along the Markov chain until convergence reached
Result: Normalization not required, calculations are local
11.3. Implementation of the Metropolis algorithm (it is a kind of
Markov chain)
Figure 48. The Metropolis algorithm
11.4. Implementation of the Metropolis algorithm
1. Metropolis did not suggest any strategy for generation of trial moves
2. “E” does not need to be an energy value
3. There is no requirement for “T” to be constant ! (the consequence is a possibility to “drive” T into
any desired direction). A lot of Metropolis algorithm implementations use Simulated-Annealing
inspired changes of temperature values.
4. b=1/(k*T), if T->0 then exp(-b*DE)->0. (in this case exp(-b*DE)>=rand() is always (except situations
when rand() is also 0) false and trial moves are accepted only if DE<0). This leads to slow (and
random) minimization of energy.
Introduction to molecular modeling. 11. Monte Carlo Method 65 by Rajmund Kaźmierkiewicz
11.5. Advantages of Metropolis MC simulations
It does not require forces
Rapidly-changing energy functions
No differentiation required
Amenable to complex move sets:
Torsions – Rotamers – Tautomers
Monte Carlo “machinery”
Boundary conditions
Finite – Periodic
Interactions
Complete – Truncated – Periodic
How do we choose “moves”?
11.6. Monte Carlo moves
Example trial moves that may be used in the Monte-Carlo procedure:
Rigid body translation
Rigid body rotation
Internal conformational changes (soft vs. stiff modes)
Changes of titration/electronic states
Some “open” questions:
How “big” a move should we take?
Move one particle or many? How “big” a move should we take?
The smaller moves the better acceptance rate and slower sampling
Bigger moves: faster sampling, poorer acceptance rate
Move one particle or many?
Possible to achieve more efficient sampling with correct multi-particle moves
One-particle moves must choose particles at random
11.7. Genetic Algorithms in Molecular Modeling
There are three main types of search methods: calculus-based techniques are local in scope
and depend upon the existence of derivatives, enumerative methods search every point related to an
objective function's domain space, one point at a time. (They are very simple to implement, but may
require significant computation and therefore suffer from a lack of efficiency), guided random search
is based on enumerative approaches. It uses supplementary information to guide the search. Two
major subclasses are simulated annealing and evolutionary computation.
66 Introduction to molecular modeling. 11. Monte Carlo Method by Rajmund Kaźmierkiewicz
11.7.1. Genetic algorithms
Table 5. The Genetic Algorithms take inspiration from Nature
GA Nature Description
Population Population The set of individuals in a given time
point
Chromosome/solution. Information
is hold in strings
Individual (information is in the
DNA, one or more chromosomes)
Individual of the population and the
information it carries
Parameter Gene Basic information unit
Value assign to the parameter Allele The content of information in the
basic unit
All parameter values Genotype Entire coded information
Fitting value Phenotype How information is expressed
Scoring function Environment Selector
11.7.2. Guided random search
Two major subclasses of Guided random search techniques are simulated annealing and
evolutionary computation: simulated annealing is based on thermodynamic considerations (with
annealing interpreted as an optimization procedure). The method probabilistically generates a
sequence of states based on a cooling schedule to converge ultimately to the global optimum
(Metropolis, N ., Rosenbluth, A.W., Rosenbluth, M .N., Teller, A .H ., and Teller, E . (1953), J. Chem.
Phys. 21, 1087-1092 .; Kirkpatrick, S ., Gelatt, C .D., and Vecchi, M .P. (1983), Science 220, 671-680.).
The main goal of evolutionary computation (de Jong, K . and Spears, W. (1993), In , Proceedings of
the Fifth International Conference on Genetic Algorithms (S . Forrest, Ed.). Morgan Kaufmann
Publishers, San Mateo, California, pp . 618–623.) is the application of the concepts of natural
selection to a population of structures in the memory of a computer (Kinnear, K .E .Jr (1994) .
Advances in Genetic Programming. The MIT Press, Cambridge, p. 518 .).
11.7.3. Evolutionary computation
Evolutionary computation can be subdivided into evolution strategies, evolutionary
programming and genetic algorithms. Evolution strategies were proposed in the early 1970s. They
use a real encoding of the problem parameters. Evolution strategies are frequently associated with
engineering optimization problems. They promote mutations rather than recombinations.
Evolutionary programming (also) does not use recombinations but allows any type of encoding. With
genetic algorithms, a population of individuals is created and the population is then evolved by
means of the principles of variation, selection, and inheritance. The genetic algorithms differ from
evolution strategies and evolutionary programming in that this approach emphasizes the use of
specific operators, in particular crossover, that mimic the form of genetic transfer in living organisms
(1). Genetic programming (2) is an extension of genetic algorithms in which members of the
population are parse trees of computer programs. Genetic programming is most easily implemented
where the computer language is tree structured and therefore LISP is often used.
1. Porto, V.W., Fogel, D.B., and Fogel, L.J. (1995) . Alternative neural network training methods. IEEE
Expert Intell. Syst. Appl. 10 (June), 16–22.
Introduction to molecular modeling. 11. Monte Carlo Method 67 by Rajmund Kaźmierkiewicz
2. Koza, J.R . (1992) . Genetic Programming. On the Programming of Computers by Means of Natural
Selection . The MIT Press, Cambridge, p . 819.
Genetic algorithms are based on the Darwinian (Lamarckian principles also can be used)
principles of natural selection and evolution. They manipulate a population of potential solutions to
an optimization (or search) problem. Specifically, they operate on encoded representations of the
GA in Computer-Aided Molecular Design solutions (equivalent to the chromosomes of individuals in
nature). Each solution is associated with a fitness value which reflects how good it is compared to
other solutions in the population. The selection strategy is ultimately responsible for ensuring
survival of the best fitted individuals. Manipulation of “genetic material” is performed through
crossover and mutation operators. How do genetic algorithms work ? A genetic algorithm operates
through a simple cycle including the following stages: encoding mechanism; creation of a population
of chromosomes; definition of a fitness function; genetic manipulation of the chromosomes.
Traditionally a binary encoding is used. This is particularly suitable when the variables are
Boolean (1 or 0 encode the presence or absence of an atom in a molecule). A chromosome consists
of a string of binary digits (bits). When continuous variables are used (like physicochemical
descriptors), a common method of encoding them uses their integer representation. Each variable is
first encoded. The binary codes of all the variables are then concatenated to obtain the binary string
constituting the chromosome.
11.7.4. Creation of a population of chromosomes
The initial population of individuals is usually generated randomly. When designing a genetic
algorithm model, one has to decide what the population size must be. Increasing the population size
increases its diversity and reduces the probability of a premature convergence to a local optimum.
However, this strategy also increases the time required for the population to converge to the optimal
regions in the search space. The most effective population size is dependent on the problem being
solved, the representation used, and the choice of the operators.
11.7.5. Definition of a fitness function
The individuals of the population are exposed to an evaluation function that plays the role of
the environmental pressure in the Darwinian evolution. This function is called fitness. Based on each
individual’s fitness, a mechanism of selection determines mates for the genetic manipulation
process. Low fitness individuals are less likely to be selected than high-fitness individuals as parents
for the next generation. A fitness scaling is often used to prevent the early domination of super-
individuals in the selection process and to promote healthy competition among near equals when the
population has largely converged. Different scaling procedures can be used. Among them one can
use the linear scaling, the sigma truncation, the power law scaling, the sigmoidal scaling and the
Gaussian scaling. Many practical problems contain constraints that must be satisfied in a modeling
process. They can be handled directly by the fitness function.
11.7.6. Genetic manipulation of the chromosomes
Parent selection in a genetic algorithm is to provide more reproductive chances for the most
fit individuals. There are many ways to do this. The classical genetic algorithm uses the roulette
wheel selection scheme. The roulette wheel selection causes a problem when the presence of a few
68 Introduction to molecular modeling. 11. Monte Carlo Method by Rajmund Kaźmierkiewicz
individuals with relatively high fitness values induces the allocation of a large number of offspring to
these individuals, and can cause a premature convergence. The solution consists of using alternate
selection schemes. Among these, the most commonly employed is the tournament selection in
which an individual must win a competition with a randomly selected set of individuals. The winner
of the tournament is the individual with the highest fitness of the tournament competitors. The
winner is then incorporated in the mating pool. The fitness difference provides the selection
pressure, which drives the genetic algorithm to improve the fitness of each succeeding generation.
Crossovers and mutations are genetic operators allowing the creation of new chromosomes
during the reproduction process. Crossover occurs when two parents exchange parts of their
corresponding chromosome. The one-point crossover is the simplest form. It occurs when parts of
two parent chromosomes are swapped after a randomly selected point, creating two children.
Figure 49. The one-point crossover scheme
In a two-point crossover scheme, two crossover points are randomly chosen and segments of the
strings between them are exchanged
Figure 50. The two-point crossover scheme
Introduction to molecular modeling. 11. Monte Carlo Method 69 by Rajmund Kaźmierkiewicz
Multi-point crossover is an extension of the two-point crossover. It consists of an increase in
the number of crossover points. The segmented crossover is a variant of multi-point crossover which
allows the number of crossover points to vary.
Mutations induce sporadic and random alterations of bit strings. Mutation plays a secondary
role in genetic algorithms in restoring lost genetic material. The probability of mutation can be held
constant or can vary throughout a run of the genetic algorithm.
Figure 51. The multi-point crossover scheme
The offspring created by the genetic manipulation process constitute the next population to
be evaluated. The genetic algorithm can replace the whole population or only its less fitted
members. The former strategy is termed the generational approach and the latter the steady-state
approach. The genetic algorithm cycle is repeated until a satisfactory solution to the problem is
found or some other termination criteria are met.
11.7.7. Applications of genetic algorithms in quantitative structure-
activity relationships (QSAR) and drug design
In QSAR studies of large data sets, variable selection and model building are difficult and
time-consuming tasks. The construction of QSAR models requires, in a first step, the design of
representative training and testing sets. Genetic algorithms are used as an attractive alternative for
selecting valuable test series, the selection uses constants encoding of hydrophobic, steric and
electronic effects. Genetic algorithms were also used to propose sets of aromatic substituents which
presented a high information content. In the activity space, genetic algorithms can be used for their
ability to detect outliers. In computer-aided molecular design, genetic algorithms can also be
employed for the identification of appropriate molecular structures given the desired
physicochemical property or biological activity.
Genetic algorithms are linked with a backpropagation neural network in order to produce an
intercommunicating hybrid system, working into a cooperative environment for generating an
optimal modeling solution. Venkatasubramanian (Venkatasubramanian, V., Sundaram, A ., Chan, K.,
and Caruthers, J.M. (1996). Computer-aided molecular design using neural networks and genetic
algorithms. In, Genetic Algorithms in Molecular Modeling (J. Devillers, Ed .). Academic Press, London,
pp. 271-302.) and coworkers proposed a backpropagation neural network-based approach for
solving the forward problem of property prediction of polymers based on the structural
characteristics of their molecular subunits, and a genetic algorithm-based approach for the inverse
problem of constructing a molecular structure given a set of desired macroscopic properties.
70 Introduction to molecular modeling. 12. Molecular docking by Rajmund Kaźmierkiewicz
Molecular docking is the process which allows recognition between molecules through
complementarity of molecular surface structures and energetics. It includes not only the size and
shape of molecular surfaces, but also charge-charge interaction, hydrogen bonding and van der
Waals interaction. The genetic algorithm was used for the first time (Payne, A.W.R. and Glen, R .C.
(1993), J. Mol. Graphics 11, 74-91.) in molecular docking to fit a series of N-methyl-D-aspartate
(NMDA) antagonists to a putative NMDA pharmacophore composed of the distance from the amine
nitrogen to a phosphonate sp2 oxygen and the distance from the carboxylic acid oxygen to the same
phosphonate sp2 oxygen.
The structure–generation genetic algorithm can produce (Jones, G., Willett, P., and Glen, R.C.
(1996), In, Genetic Algorithms in Molecular Modeling (J. Devillers, Ed .). Academic Press, London, pp.
211-242.) very large and diverse sets of reasonable chemical structures for searching by 3D database
programs. The genetic algorithm was used at the conformational analysis stage in the searching of
databases of flexible 3D molecules to find candidate drug structures that fit a query pharmacophore.
The side effect was the estimation of the binding affinities and the most energetically favorable
combination of interactions between a receptor and a flexible ligand, and the superimposition of
flexible molecules with the use of the resulting overlays to suggest possible pharmacophoric
patterns.
Walters and coworkers (Walters, D.E. and Muhammad, T.D. (1996). In, Genetic Algorithms in
Molecular Modeling (J. Devillers, Ed .). Academic Press, London, pp. 193-210.) proposed a program
called Genetically Evolved Receptor Models (GERM) for producing atomic-level models of receptor
binding sites, based on a set of known structure-activity relationships. The generation of these
models requires no a priori information, about a real receptor, other than a reasonable assurance
regarding the SAR data. The receptors generated by the genetic algorithm can be used for:
correlating calculated binding with measured bioactivity, predicting the activity of new compounds
by docking the chemicals, calculating their binding energies and calculating their biological activity.
12. Molecular docking
Molecular docking means finding the binding orientation of two molecules with known
structures. The binding of small molecule ligands to large protein targets is central to numerous
biological processes. The accurate prediction of the binding modes between the ligand and protein,
(the docking problem) is of fundamental importance in modern structure-based drug design. The
task of molecular docking can be divided according to the molecules being involved:
Protein-Ligand docking
Protein-Protein docking
Specific docking algorithms are usually designed to deal with one of these problems but not with
both (different contact area, flexibility, level of representation). Assuming the receptor structure is
available, a primary challenge in lead discovery and optimization is to predict both ligand orientation
and binding affinity.
The major techniques currently available are: molecular dynamics, Monte Carlo methods,
genetic algorithms, fragment-based methods, point complementarity methods, distance geometry
methods, tabu searches and systematic searches.
Introduction to molecular modeling. 12. Molecular docking 71 by Rajmund Kaźmierkiewicz
Docking protocols can be described as a combination of two components; a search strategy
and a scoring function. The search algorithm should generate an optimum number of configurations
that include the experimentally determined binding mode.
12.1. Types of compatibilities
Geometrical compatibility prevents overlap between atoms and maximizes shape
compatibility between the ligand and the binding site. Chemical compatibility maximizes the number
of chemically favorable interactions and minimizes the number of unfavorable interactions.
12.2. Finding the place and the orientation of the interactions
The general problem includes a search for the location of the binding site and a search to
figure out the exact orientation of the ligand in the binding site. A program that does both makes a
Global docking. Sometimes the location of the binding site is known. In this case we only need to
orient the ligand in the binding site. In this case the problem is called Local docking. Global docking is
more demanding in terms of computational time and the results are less accurate. When the
location of the binding site is unknown we have several possibilities: a) Looking for the binding sites
by separate programs (such as detection of cavities, conservation etc.) and then applying a local
docking program, b) Global docking.
Protein-protein docking methods.
The appoaches to protein-protein docking have a lot in common with small molecule docking. The
methods are still based on the combination of search algorithm coupled to scoring function. The
scoring functions are essentially the same (since we are still dealing with atomic interactions),
however the major problem is that the conformational space we now need to search is massive.
12.3. Computational time
Docking programs are often restricted by the computational time, due to the enormous
number of possibilities that should be examined. Docking algorithms should consider the
computational time and often the price is the accuracy of the scoring function. Efficiency is especially
required for drug design.
12.4. Rigidity vs. flexibility
Most of the early algorithms assumed that the docked molecules do not change
conformations. This assumption allows to treat the molecules as rigid bodies, making the algorithm
simpler and faster. This assumption is obviously problematic and was proven to be wrong in some
cases. Newer algorithms try to face the flexibility problems with variety of ways. Other methods try
to handle the flexibility problem indirectly or at least to “minimize the damage” of not incorporating
flexibility. Docking procedures that perform rigid body search are termed rigid docking. Docking
procedures that consider possible conformational changes are termed flexible docking.
72 Introduction to molecular modeling. 12. Molecular docking by Rajmund Kaźmierkiewicz
12.5. Flexibility
Some algorithms break the ligand up into pieces, dock the individual pieces, and try and
reconnect the bound conformations. For example FlexX uses a library of precomputed, minimized
geometries from the Cambridge database with up to 12 minima per bond. Sets of alternative
fragments are selected by choosing simple or multiple pieces in combination. Flexible docking via
molecular dynamics with minimization can handle arbitrary flexibility, however it is extremely slow
compared to other docking methods.
12.6. Bound and unbound docking
In bound docking the goal is to reproduce a known complex where the starting coordinates of the
individual molecules are taken from the crystal of the complex. In the unbound docking, which is a
significantly more difficult problem, the starting coordinates are taken from the unbound molecules.
12.7. Components of the problem
Algorithms to dock molecules need: A. System representation, B. Searching procedure, C.
Scoring function, D. Clustering procedure. The parameters of the problem for docking of 2 rigid
bodies are 3 angles (rotations) and 3 distances (translations). Usually the ligand is not rigid and few
other parameters are required Np = 3 + 3 + Nfb. Where Np is a number of the number of parameters
needed to fully describe ligand position, 3+3 is a Position and Orientation, Nfb is a Number of flexible
bonds. If there are many parameters the problem is more complicated. GA have advantages over
other methods in these situations.
12.8. Aspects of Docking
Docking algorithms need to generate poses (configurations) of ligands in binding site then
score the poses using simple energy function or rank the poses using more complex thermodynamic
functions like ΔG. They need to take into account desolvation and entropic contribution.
12.9. Docking and de novo design methods
There is a difference between docking algorithms and de novo design methods. This is
subjective and in many cases significant overlap in methodology occurs between the two strategies.
Examples of de novo design tools are BUILDER, CONCEPTS, CONCERTS, DLD/MCSS, Genstar, Group-
Build, Grow, HOOK, Legend, LUDI, MCDNLG, SMOG and SPROUT. LUDI is given as an example of a de
novo design tool applied to the docking problem.
12.10. Additional Challenges in Docking
Limited resolution of receptor structures
Flexibility of receptor
Conformational changes in ligand and/or receptor upon binding
Role of water molecules in binding (desolvation of ligand and binding site, waters bridging
protein and ligand)
Introduction to molecular modeling. 12. Molecular docking 73 by Rajmund Kaźmierkiewicz
12.11. Protein Flexibility and Its Influence on Ligand Binding
For a single, fixed conformation to be an adequate representation of the protein, the system
would have to be very rigid. Such a system would correspond with the “lock-and-key” theory of
binding in which the protein exists in a single well-defined state with only one optimal
complementary ligand. However, the energy landscape of most proteins is frequently described in
terms of a folding funnel in which there are many highly unfavorable states that collapse via multiple
routes into possibly several favorable folded states. It is implied by the width of the minima on the
protein potential energy hypersurface that the folded state of the flexible system has higher entropy
and, as such, will have a larger ensemble of occupied states. This set of conformations is a collection
of structurally similar and nearly energetically equivalent conformations of the protein that,
together, make up the folded state.
It is important to note that a conformational potential energy hypersurface and the states
that it represents are condition-dependent. By altering the conditions (ionic strength, pH,
temperature), the minima can shift, changing which state is most populated. The most important
point to consider is that introducing a ligand into the system also changes the environment. It too
may affect the most populated state of the protein; such a case would correspond to an “induced-fit”
system. The protein most likely exists in a full complement of conformations out of which most are in
the native state, some in the induced-fit state, and some in other states. If the ligand binds
preferentially to the induced-fit state with sufficiently favorable free energy (meaning greater than
the free energy difference between the two protein states), the average structure of the protein will
change.
The implications for drug discovery are clear; a single protein structure is only useful to
identify ligands for that particular narrow state of the ensemble of lowest energy protein
conformations. To obtain new leads and properly predict activity of existing inhibitors, multiple
structures are the best option. The available computational methods range in the degree of protein
flexibility that they accommodate (focusing on a narrow set of states, a broader ensemble, or the
substantial fragment of the protein potential energy hypersurface). Keep in mind that most of the
advancements have only appeared in literature over the last five years, and there are only a few
systems to compare experimental data.
Searching procedures applied for docking: Greed search, Monte Carlo search/Random,
Genetic Algorithms. Scoring functions procedures applied for docking: Force fields, Geometric
features, Knowledge based parameters.
12.12. Clustering
No docking algorithm can produce a single, trustworthy structure for the bound complex, but
instead they produce an ensemble of predictions. Each predicted structure has an associated energy
(or enthalpy as the case may be) as well as the relative population. By clustering our data based on
some “distance” criteria, we can gain some sense of similarity between predictions. The distance can
be any of a number of similarity measures, but for 3D structures, RMSD is the standard choice.
74 Introduction to molecular modeling. 12. Molecular docking by Rajmund Kaźmierkiewicz
12.13. Search algorithms
Docking protocols can be described as a combination of two components; a search strategy
and a scoring function. The search algorithm should generate an optimum number of configurations
that include the experimentally determined binding mode. A rigorous search algorithm would
exhaustively check all possible binding modes between the ligand and receptor. All six degrees of
translational and rotational freedom of the ligand would be explored along with the internal
conformational degrees of freedom of both the ligand and protein. However, this is impractical due
to the size of the search space. The practical application of such an extensive search involves the
sampling of many high energy unfavorable states which can restrict the success of an optimization
algorithm.
For a simple system comprising a ligand with four rotatable bonds and six rigid-body
alignment parameters, the search space has been estimated as follows. If the angles are considered
in 10 degree increments and translational parameters on a 0.5 Å grid there are approximately 4×108
rigid body degrees of freedom to sample, corresponding to 6×1014 configurations (including the four
rotatable torsions) to be searched. This would require approximately 2000000 years of
computational time at a rate of 10 configurations per second. As a consequence only a small amount
of the total conformational space can be sampled, and so a balance must be reached between the
computational expense and the amount of the search space examined.
In practice therefore, to sample such a large search space the computational expense is
limited by applying constraints, restraints and approximations to reduce the dimensionality of the
problem in an attempt to locate the global minimum as efficiently as possible. A common
approximation in early docking algorithms was to treat both the ligand and target as rigid bodies and
only the six degrees of translational and rotational freedom were explored.
Although these methods have been successful in certain cases, there is a limitation to the
rigid body docking paradigm in that the ligand conformation must be close to the experimentally
observed conformation when bound to the protein. Numerous examples of conformational change
of the ligand upon binding, for example the binding of cyclosporin A to cyclophilin, have led the drive
to incorporate conformational flexibility into the search algorithm.
A common approach in modeling molecular flexibility is to consider only the conformational
space of the ligand, assuming a rigid receptor throughout the docking protocol. The techniques used
to incorporate conformational flexibility into a docking protocol will be discussed in some detail.
However, the searching algorithm is only half the docking problem; the other factor to be
incorporated into a docking protocol is the scoring function.
12.14. Scoring functions
A large number of current scoring functions are based on force fields that were initially
designed to simulate the function of proteins. A force field is an empirical fit to the potential energy
surface in which the protein exists and is obtained by establishing a model with a combination of
bonded terms (bond distances, bond angles, torsional angles) and non-bonded terms (van der Waals
and electrostatic). Some scoring functions used in molecular docking have been adapted to include
terms such as solvation and entropy. A separate approach is to use pure empirical scoring functions
that are derived using multivariate regression methods of experimental data.
Introduction to molecular modeling. 12. Molecular docking 75 by Rajmund Kaźmierkiewicz
12.15. Rigid Protein Docking
Most of the docking methods used at the present moment in academic and industrial
research all assume a rigid protein. To illustrate the methodology used by these methods I will briefly
discuss three of the most common programs used for docking: Autodock, Dock and FlexX.
Autodock uses a kinematic model for the ligand, it assumes the rigid valence geometry of
ligand and only torsional angles are allowed to change. The ligand begins the search process
randomly outside the binding site and by exploring the values for translations, rotations and its
internal degrees of freedom, it will eventually reach the bound conformation. Distinction between
good and bad docked conformations is carried out by the scoring function. Autodock is able to use
Monte Carlo methods or simulated annealing (SA) in the search process and in its last version
introduced the ability to use genetic algorithms (GA). The routine implemented in the recent release
is a Lamarkian genetic algorithm (LGA), in which a traditional GA is used for global search and is
combined with a Solis and Wets local search procedure. Authors of Autodock show that the new LGA
is able to handle ligands with a larger number of degrees of freedom than SA or traditional GA. FlexX
and Dock both use an incremental construction algorithm which attempts to reconstruct the bound
ligand by first placing a rigid anchor in the binding site and later using a greedy algorithm to add
fragments and complete the ligand structure. Although these programs are more efficient than
Autodock in the sense that they require fewer energy evaluations they also have some tradeoffs.
One of the main problems is that it is not trivial to choose the anchor fragment and its choice will
determine what solutions can be obtained. Also the greedy algorithm propagates errors resulting
from initial bad choices that lead to missing final conformations of lower energy. Recently other
docking programs have also been reported such as DREAM++, QSDock, and Darwin.
12.16. Partial Protein Flexibility
The first approximation used in modeling partial protein flexibility was the soft-docking
method first described by Jiang and Kim (Jiang, F. & Kim, S.H. "Soft docking": matching of molecular
surface cubes. J Mol Biol 219, 79-102 (1991).). The principle underlying this method consists of
decreasing the van der Waals repulsion energy term between the atoms in the binding site and those
in the ligand. This method could result in final solutions that include physically impossible atom
collisions. Due to the mobility available to the protein atoms in the binding site, it is possible that
there is a low energy rearrangement of these that would eliminate collisions while maintaining the
conformation of the ligand returned during its conformational search. This method has the
advantage of being computationally efficient as it still describes the protein using fixed coordinates
and it is easy to implement since it does not require changes to the energy evaluation function
besides changing van der Waals parameters.
The most common approximation used to incorporate partial protein flexibility in modeling
the binding process is to select a few degrees of freedom in the protein binding site and do a
simultaneous search of the combined ligand/protein conformational space. Incorporating select
degrees of freedom from the binding site in the conformational search process is based on the
assumption that these degrees of freedom are the ones playing a major role in determining the
conformational changes during the binding process. This choice requires deep chemical
understanding of the system under study and is therefore difficult to automate.
76 Introduction to molecular modeling. 12. Molecular docking by Rajmund Kaźmierkiewicz
One of the earliest reports of using select degrees of freedom from the protein was
described by Jones et. al. (Jones, G., Willett, P., Glen, R.C., Leach, A.R. & Taylor, R. Development and
validation of a genetic algorithm for flexible docking. J Mol Biol 267, 727- 48 (1997).) and was
implemented in the program Genetic Optimization for Ligand Docking (GOLD). This program
improves on the rigid protein model by performing a conformational search on the binding site with
the aim of improving the hydrogen bonding network between the protein and the ligand. Hydrogen
bonds are local electrostatic interactions between pairs of atoms which play an important energetic
role in ligand recognition and binding. GOLD selects the degrees of freedom in the binding site that
correspond to reorientations of hydrogen bond donor and acceptor groups. These degrees of
freedom represent only a very small fraction of the total conformational space that is available but
should account for a significant difference in binding energy values.
More recent studies have been reported in which other degrees of freedom from aminoacid
sidechains are also used in the conformational search. These are searched using stochastic methods
with arbitrary step sizes or using rotamer libraries. Rotamer libraries consist of discrete sidechain
conformations of low energy which are usually determined from statistical analysis of structural data
derived experimentally.
12.17. Full Protein Flexibility
Ideally ligand docking to a protein could be simulated using Molecular Dynamics (MD). This
has the advantage that not only it takes into account all the degrees of freedom available to the
protein but also enables an explicit modeling of the solvent. Furthermore, accurate energy
calculations can also be carried out using the free energy perturbation method. Unfortunately,
modeling proteins using MD is computationally expensive, and the computational power necessary
to simulate the full process of diffusion and ligand binding without any approximations will be out of
our reach for many years to come.
Mangoni et. al.( Mangoni, M., Roccatano, D. & Di Nola, A. Docking of flexible ligands to
flexible receptors in solution by molecular dynamics simulation. Proteins 35, 153-62 (1999).) reported
a modification to the standard MD protocol which reduces the computational time required for the
docking simulation. The protocol consists of separating the center of mass motion of the ligand from
its internal and rotational motions by coupling the different degrees of freedom to separate thermal
baths. This optimization allows the ligand to sample the space surrounding the binding site faster
while maintaining correct interactions with both protein and solvent.
An alternative approach to model full protein flexibility is to generate a set of rigid protein
conformations that together represent the conformational diversity available to the protein. These
conformations can later be docked to database of ligands using traditional rigid-protein/flexible-
ligand methods. There are several possible methods to generate the ensembles, but unfortunately
their accuracy is proportional to the difficulty in obtaining them. The most accurate ensemble is the
one determined exclusively from experimental data. An example is the case where several structures
of protein/ligand complexes are determined using X-ray crystallography bound to different candidate
drugs. Under these circumstances it is usually possible to observe alternative binding modes directly
(Munshi, S. et al. An alternate binding site for the P1-P3 group of a class of potent HIV- 1 protease
inhibitors as a result of concerted structural change in the 80s loop of the protease. Acta Crystallogr
D Biol Crystallogr 56, 381-8 (2000).).
Introduction to molecular modeling. 12. Molecular docking 77 by Rajmund Kaźmierkiewicz
Another less accurate option is to use the ensemble of structures that results from an
experimental protein structure determination using the NMR (Nuclear Magnetic Resonance)
technique. This docking methodology was first reported by Knegtel et al (Knegtel, R.M., Kuntz, I.D. &
Oshiro, C.M. Molecular docking to ensembles of protein structures. J Mol Biol 266, 424-40 (1997).).
Finally, one can generate an ensemble using computational methods such as Monte Carlo (MC) or
MD sampling. The accuracy of these alternatives is closely related to the accuracy of the force field
used and is limited by the ability of these computational techniques to effectively sample the
conformational space.
A different representation for full protein flexibility is to divide the protein in tightly coupled
domains whose constituent atoms move collectively as one. Hinges connect the domains and the
motion of the protein is simulated similarly to an articulated robot. Required conformational changes
inside domains can be handled using minimization. An application of this model to the docking
problem was reported by Sandak et al (Sandak, B., Wolfson, H.J. & Nussinov, R. Flexible docking
allowing induced fit in proteins: insights from an open to closed conformational isomers. Proteins 32,
159-74 (1998).).
12.18. Examples of Docking Programs
Table 6. The examples of docking programs
Docking approach Examples
Matching of descriptors DOCK, QSDOCK, SLIDE
Incremental construction FlexX, Hammerhead
Monte Carlo Simulated Annealing AutoDock, MCDOCK
Monte Carlo Minimization ICM, QXP
Molecular Dynamics MDD
Genetic Algorithms GOLD, AutoDock3
12.19. Activated Dynamics
There are many biochemical processes involved in proteins and DNA, which occur only
infrequently. For example, docking a ligand to a particular space of a protein takes place on a
millisecond time scale. Therefore, these processes are not adequately sampled in conventional
simulation approaches, because they require a very long dynamic simulation, which is not feasible
even using the fastest computers.
Activated dynamics provides an alternative approach to this problem. In this approach one
performs several molecular dynamics simulations, and each of them has a constrained value of a
reaction coordinate describing the investigated process. In the case of docking of a ligand to the
protein, the reaction coordinate can by represented by a relative distance between the ligand center
of mass, and the center of mass of the protein pocket. Then one performs a series of molecular
dynamics simulations, and each of them has a constrained value of the distance between the ligand
and the protein. The final structure of the molecular dynamics with a bigger reaction coordinate, is
the initial structure of the dynamics using the smaller value of this coordinate. In this series of
simulations, the entire protein relaxes according to the potential of the mean force, representing the
move of the ligand in a direction of the protein pocket.
78 Introduction to molecular modeling. 13. Hybrid QM/MM method by Rajmund Kaźmierkiewicz
13. Hybrid QM/MM method
Hybrid QM/MM method combines quantum mechanical and molecular mechanical method.
It treats just the reacting part of the system quantum mechanically, and uses MM for the
surroundings. It uses a combined Hamiltonian (The total mechanical energy function) for the system:
Figure 52. The hybrid QM/MM method
13.1. Boundary treatment
The boundary term is given by
where i is summed over all MM partial charges, m overall QM nuclei, and e overall QM electrons. The
first term is 1-electron interaction between QM electron density and MM partial charges. The second
term is a standard Coulomb interaction between QM nuclei and MM charges. The final term is
required because electron density (and hence dispersion) is explicitly treated in the QM region, but
not in the MM region.
The valence of the QM region must be satisfied. The MM bond, angle, dihedral terms need a
partner atom to act on, in order to maintain the geometry of the system. The QM/MM is often used
to simulate a solute quantum mechanically, with explicit solvent treated with MM — in this instance,
the problem of QM-MM bonds is avoided.
Conventional solution for boundary treatment is a “link atoms” approach (usually hydrogen
atoms, but sometimes halogens or even methyl groups) are added along the bond (Singh, U. and
Kollman, P. J. Comput. Chem. 1986, 7, 718). The link atom satisfies the valence of the QM region. The
QM atom is used for calculation of all MM bond terms. For non-bonded interactions (electrostatic
terms), originally the link atom did not interact with any MM atom. Better properties are usually
obtained if the link atom interacts with the entire MM region. It leads to poor handling of electron
density.
Introduction to molecular modeling. 13. Hybrid QM/MM method 79 by Rajmund Kaźmierkiewicz
13.2. Improved bond treatments
Local Self-Consistent Field (LSCF) (Warshel, A. and Levitt, M. J. Mol. Biol. 1976, 103, 227) uses
a parameterized frozen orbital along the QM-MM bond, which is not optimized in the SCF.
Generalized Hybrid Orbital (GHO) (Gao, J. et. al. J. Phys. Chem. A 1998, 102, 4714) includes the QM-
MM orbitals in the SCF.
Figure 53. The "frozen orbital" scheme of the QM-MM method
Dynamics of a QM/MM system is almost identical to those of an MM system: Forces are
calculated from first derivatives on each atom. The QM nuclei are treated identically to the MM
partial charges. The system is propagated by standard Newtonian dynamics. QM/MM can also be
used in conjunction with Monte Carlo methods. A complication: the MM atoms affect the QM
electron density, so an SCF is required for every Monte Carlo move. Workaround: approximate the
energy change of the QM region by first-order perturbation theory (it is called the “Perturbative
QM/MC”) as long as moves are far enough away from the QM region (Truong, T. and Stefanovich, E.
Chem. Phys. Lett. 1996, 256, 348).
There are some Drawbacks of QM/MM. Some parameterization is still required for the
boundary treatment. The choice of the size of the QM region is still something of an art. Although the
QM region polarizes in response to the MM partial charges, the reverse is not also true (although
fully polarizable QM/MM methods are being developed). The free energy of a QM system can be
determined via frequency calculation; however, this is rather inaccurate when applied to QM/MM
systems (second derivatives are poorly determined, due to the harmonic approximation).
13.3. Other approaches
ONIOM approach (Svensson, M. et. al. J. Phys. Chem. 1996, 100, 19357) divides the system
into the “real” (full) system and the “model” (subset) and treats the model at high level, and the real
at low level, giving the total energy as
which relies on the approximation
The “model” system still has to be properly terminated. Extension to three level systems is relatively
straightforward (for example, it is also possible to incorporate other methods ab-initio core, semi-
empirical boundary, MM surroundings).
80 Introduction to molecular modeling. 13. Hybrid QM/MM method by Rajmund Kaźmierkiewicz
Empirical Valence Bond method (Warshel, A. and Weiss, M. J. Am. Chem. Soc. 1980, 102,
6218) treats any point on a reaction surface as a combination of two or more valence bond
structures. Parameterization is made from QM or experimental data. It is an effective method, but
must be carefully set up for each system.
Effective Fragment Potential (Webb, P. and Gordon, M. J. Phys. Chem. 1999, 103, 1265) adds
“fragments” to a standard QM treatment, which are fully polarizable and are “parameterized” from
separate ab initio calculations. Treatment of bonds between the ‘true’ QM region and the fragments
is still problematic.
13.4. Available software
CHARMM has been interfaced with MOPAC, GAMESS-US, GAMESS-UK, CADPAC, DeFT.
AMBER works with ROAR and with Gaussian. DYNAMO implements semi-empirical QM/MM. QSite is
a commercial package from Schrödinger, Inc.
13.5. An Example - a Diels-Alder Reaction
Figure 54. The Diels-Alder cyclo-addition reaction catalyzed by 1E9, a catalytic antibody that was raised against a transition-state analogue compound
13.6. Geometry Optimization after MM Dynamics
The combined QM/MM calculations can also be performed for protein systems, for which
experimental geometry is unknown. In this approach we usually start from the protein structure,
which is very similar to the target protein structure, and modify the initial structure manually to get
the final protein geometry. However any modifications of the initial protein structure change
interactions in the entire protein system, therefore we need to relax and equilibrate the final
protein geometry in a process of molecular mechanical dynamics. However, the molecular
mechanical dynamics generates a trajectory, which is a set of several protein structures (snapshots),
rather than just one protein structure. Therefore, the combined QM/MM calculations which are
based on the MM dynamics, should be performed for several protein structures, representing several
Introduction to molecular modeling. 13. Hybrid QM/MM method 81 by Rajmund Kaźmierkiewicz
protein geometries in a process of protein thermal motions. Final results of those QM/MM
calculations will be obtained as average values of individual QM/MM calculations, where each of
them (snapshot) represents a temporary protein structure in a dynamic process. The QM/MM
calculations of an individual protein structure (snapshot) are performed in the same approach, as the
method used in the combined QM/MM calculations in the fixed MM protein matrix.
13.7. Oscillations of Active Sites after MM Dynamics.
When optimal geometries of an active site are already obtained using a series of QM/MM
calculations after MM dynamics, one can calculate oscillation hessian of the active site of each
individual protein snapshot. The individual QM/MM calculations can be performed according to the
same procedure as QM/MM oscillation calculations in the fixed protein matrix. After QM/MM
calculations, the final results will be obtained as average values based on all individual QM/MM
calculations performed for the fixed MM protein matrix. This QM/MM method allows us to calculate
molecular oscillations of active sites in protein systems for which experimental structures are
unknown. This is particularly important for any protein modifications, where some amino acids of the
active site pocket have been replaced by other amino acids. This type of protein modification is very
expensive to perform experimentally, therefore we can use the QM/MM method for prediction
which protein modification will have the biggest impact on the electronic and geometric structure of
the protein active site. By calculating molecular oscillations of the active site for different protein
mutations, we can show what type of protein modification will change the oscillations of the active
site the most. Then this theoretical prediction can be verified experimentally.
13.8. Complex Reaction after MM Dynamics
In most cases, chemical reactions generate products, whose geometrical structures are much
different from geometrical structures of reactants. If these reactions take place in a protein
environment, the entire protein relaxes in response to newly obtained products. Therefore we
cannot use a simple QM/MM approach in the fixed MM protein matrix for calculating more complex
chemical reactions in a protein. The best method is a combination of individual QM/MM calculations
in the fixed MM protein matrix, which is applied to several protein structures obtained during MM
dynamics. Therefore, we usually do a separate MM dynamics for a protein structure representing
reactants, and a similar separate MM dynamics representing products of the reaction. Then for each
MM dynamics, we do a series of combined QM/MM calculations, and calculate the total energy of
the reacting molecules, as average values. A more complex situation is for chemical reactions, which
have one or several intermediate states on a reaction profile between reactants and products. In this
case, we need to perform a separate MM dynamics for each intermediate state, in a similar way as
for reactants and products. For example a methylation reaction, which takes place between a
cofactor, which is part of the protein, and a DNA base, which is completely flipped out the DNA helix.
There is experimental evidence indicating that this chemical reaction involves cysteine from the
protein, and there are two intermediate states between the reactants and the products. Therefore in
order to calculate this reaction, first we need to perform four different MM dynamics. Then for each
MM dynamics we can perform a series of QM/MM calculations, and calculate the energy of reacting
molecules as average values over selected protein snapshots. The same procedure need to be
applied to the other MM dynamics.
82 Introduction to molecular modeling. 14. Normal Modes and Principal Component Analysis by Rajmund Kaźmierkiewicz
13.9. Electronic Excitation in Fixed MM Matrix
The QM/MM calculations of electronic excitations of an active site can be also performed
after MM dynamics. We apply this approach to the protein system which does not have an
experimental structure, or to any protein modification. Each protein modification changes
interactions between protein atoms, therefore, after protein modification we need to perform MM
dynamics in order to relax and equilibrate the protein system. After MM dynamics, we have a series
of protein structures and for randomly selected snapshots, we perform the combined QM/MM
calculations, optimizing the geometry of the active site quantum mechanically. Those calculations
can be performed using the same approach as the QM/MM geometry optimization in the fixed MM
protein matrix. Then for each protein snapshot we perform QM/MM calculations of electronic
excitations, based on the optimal geometry obtained from the QM/MM calculations. Those
calculations are based on the geometry of the active site in its ground electronic state, therefore as
results of those calculations, we have electronic excitations which can be compared with
experimental electronic absorption spectra. We can also perform geometry optimization of the
active site in its excited electronic state, and then we can calculate electronic excitations from the
excited to the ground electronic state. Results of those calculations can be compared with
experimental electronic emission spectra, such as fluorescence or phosphorescence. There are
usually small differences in the geometry of the same molecule in the ground and in the excited
electronic states. Therefore we can use the same protein matrix for the QM/MM calculations in the
ground and in the excited electronic states. However, there are molecular systems where after
electronic excitation the active site of a protein changes its geometry considerably. The best example
of that protein system is bacteriorhodopsin, where electronic excitation changes the conformation of
the active site from cis to trans. In this case we need to perform a separate MM dynamics for both
conformers of the active site, and separate series of QM/MM calculations. Similar as in the QM/MM
calculations of optimal geometries of an active site after MM dynamics, final results of the QM/MM
electronic excitations will be calculated as average values over a series of randomly selected protein
snapshots.
14. Normal Modes and Principal Component Analysis
The vibrations of a molecule are given by its normal modes. Each absorption in a vibrational
spectrum corresponds to a normal mode. For example, the four normal modes of carbon dioxide, are
the symmetric stretch, the asymmetric stretch and two bending modes. The two bending modes
have the same energy and differ only in the direction of the bending motion. Modes that have the
same energy are called degenerate. In the classical treatment of molecular vibrations, each normal
mode is treated as a simple harmonic oscillator.
Figure 55. Normal Modes for a linear triatomic molecule. In the last bending vibration the motion of the atoms is in-and-out of the plane of the paper
Introduction to molecular modeling. 14. Normal Modes and Principal Component Analysis 83 by Rajmund Kaźmierkiewicz
In general linear molecules have 3N-5 normal modes, where N is the number of atoms. The five
remaining degrees of freedom for a linear molecule are three coordinates for the motion of the
center of mass (x, y, z) and two rotational angles. Non-linear molecules have three rotational angles,
hence 3N-6 normal modes. The characteristics of normal modes is summarized below.
Characteristics of Normal Modes:
1. Each normal mode acts like a simple harmonic oscillator.
2. A normal mode is a concerted motion of many atoms.
3. The center of mass doesn’t move.
4. All atoms pass through their equilibrium positions at the same time.
5. Normal modes are independent; they don’t interact.
In the asymmetric stretch and the two bending vibrations for CO2, all the atoms move. The
concerted motion of many of the atoms is a common characteristic of normal modes. However, in
the symmetric stretch, to keep the center of mass constant, the center atom is stationary. In small
molecules all or most of the atoms move in a given normal mode; however, symmetry may require
that a few atoms remain stationary for some normal modes. The last characteristic, that normal
modes are independent, means that normal modes don’t exchange energy. For example, if the
symmetric stretch is excited, the energy stays in the symmetric stretch.
14.1. One mass and two springs
Figure 56. Applying of normal mode analysis to one mass and two springs model
14.2. Two masses
For a single mass on a spring, there is one natural frequency, namely . Let's see what happens
if one has two equal masses and three springs. The two outside spring constants
Figure 57. Applying of normal mode analysis to two masses and three springs model
84 Introduction to molecular modeling. 14. Normal Modes and Principal Component Analysis by Rajmund Kaźmierkiewicz
are the same, but we'll allow the middle one to be different. In general, all three spring constants
could be different, but the math gets messy in that case.
Let x1 and x2 measure the displacements of the left and right masses from their respective
equilibrium positions. We can assume that all of the springs are unstretched at equilibrium, but we
don't actually have to, because the spring force is linear. The middle spring is stretched (or
compressed) by x2-x1, so the F = ma equations on the two masses are
,
.
Concerning the signs of the terms here, they are equal and opposite, as dictated by Newton's third
law, so they are either both right or both wrong. They are indeed both right, as can be seen by taking
the limit of, say, large x2. The force on the left mass is then in the positive direction, which is correct.
These two F = ma equations are “coupled”, in the sense that both x1 and x2 appear in both equations.
How do we go about solving for and ?
Let's guess solutions of the form and
. It is convenient to
write these solutions in vector form:
Substituting these guesses into the equation F = ma , and canceling the factor of , yields
In matrix form, this can be written as
At this point, it seems like we can multiply both sides of this equation by the inverse of the matrix.
This leads to (A1,A2) = (0, 0). This is obviously a solution (the masses just sit there), but we're looking
for a nontrivial solution that actually contains some motion. The only way to escape the preceding
conclusion that A1 and A2 must both be zero is if the inverse of the matrix doesn't exist. For the
present purposes, the only fact we need to know about matrix inverses is that they involve dividing
by the determinant. So if the determinant is zero, then the inverse doesn't exist. This is therefore
what we want. Setting the determinant equal to zero gives the quartic equation,
We now perform the usual step of invoking the fact that the positions x1(t) and x2(t) must be real for
all t, and we obtain
Introduction to molecular modeling. 14. Normal Modes and Principal Component Analysis 85 by Rajmund Kaźmierkiewicz
So what we did above was solve the eigenvectors and eigenvalues of this matrix. The eigenvectors of
a matrix are the special vectors that get carried into a multiple of themselves when acted on by the
matrix. And the multiple (which is m2 here) is called the eigenvalue. Such vectors are indeed
special, because in general a vector gets both stretched (or shrunk) and rotated when acted on by a
matrix. Eigenvectors don't get rotated at all.
14.3. N atoms and potential energy function
Let H be the Hessian matrix for a system with N atoms and potential energy function U:
Motions will depend on the masses, so with M being a 3Nx3N diagonal matrix:
M11=m1, M22=m1, M33=m1, M44=m2 ...
F=M-1/2HM-1/2
Diagonalize F, this is to solve secular equation |F-I|=0, equivalent to solving the eigenvalue problem:
Fx=x)
The eigenvalues correspond to frequencies of motions described by the
eigenvectors xi. It assumes that the system is described by harmonic potentials, which is usually not
true for real systems, but may be approximately true near the minimum of the potential energy; less
good for liquids and very floppy molecules.
For general 3D systems we expect 3N-6 non-zero frequencies: six should be zero,
corresponding to the six overall translation rotation degrees of freedom
Two-Atomic Molecule Three-Atomic Molecule
Figure 58. Natural Modes of Vibrations for Some Simple Systems
86 Introduction to molecular modeling. 14. Normal Modes and Principal Component Analysis by Rajmund Kaźmierkiewicz
Figure 59. Low frequencies and Large collective motions for the multi-atom molecule.
14.4. For the multi-atom molecule
One mode can represent 70-90% of functionally relevant motion.
For many observed movements, the first 12 normal modes contain the relevant degrees of
freedom
Figure 60. The sample Frequency spectrum of a protein
Introduction to molecular modeling. 14. Normal Modes and Principal Component Analysis 87 by Rajmund Kaźmierkiewicz
14.5. Harmonic Approximation in Anharmonic Systems and in real
Proteins
For a theoretical system with a purely harmonic potential V(R), there exists a superposition of
normal modes that exactly expresses any given motion.
For an anharmonic potential V(R), the real world systems, harmonic potential still gives an
excellent approximation near a potential energy minimum, and any small-amplitude motion
around such a minimum can still be well described by a sum of normal modes.
Any classical system can be said to behave harmonically at a sufficiently low temperature.
A typical normal mode analysis computes the characteristic vibrations and the corresponding
frequencies assuming V(R) is harmonic in all degrees of freedom.
Only systems near a potential energy minimum exhibit harmonic behavior, so normally, the
system energy is first minimized to assist with the harmonic assumption.
Around 200 K, the fluctuations of the atoms of a globular protein begin to deviate considerably
from harmonic behavior. As the protein is heated to 300 K, its motion becomes considerably
anharmonic.
These facts should be kept in mind when attempting to interpret in vivo and in
vitro protein behavior using normal modes.
Normal Mode Analysis (NMA), also known as “Harmonic Analysis” is a
classical technique for studying the vibrational and thermal properties of various
molecular structures at the atomic level. It was originally developed for interpretation of the
optical spectra (for example the Fourier Transform Infrared (FTIR) and Raman spectra) of
small molecules. Normal modes of vibration are simple harmonic oscillations about a local
energy minimum, characteristic of a system’s structure and its energy function V(R).
Normal mode analysis is an alternative method to study dynamics of molecules. It
does not require trajectory, because it works with single structure. Conformational fluctuation
is given by a superposition of normal modes. One can use normal mode analysis to refine
small-angle X-ray scattering profiles.
There are two classes of applications of the Normal Mode analysis method:
Separation of frequencies
Simple analytic description of the potential
Figure 61. An illustration to the simple analytic description of the potential
88 Introduction to molecular modeling. 14. Normal Modes and Principal Component Analysis by Rajmund Kaźmierkiewicz
Normal modes are collective motions of coupled atoms, example:
Symmetric stretch, translation, and asymmetric stretch
Figure 62. Linear motions of three masses attached by springs (coupled 1-D harmonic oscillators)
For proteins one has: N masses = N normal modes
Harmonic Potential :
V=∑i=1
3N
∑j=1
3N
{k ij (x i− x j)2}
The equations for non-bonding interactions for example the Van der Waals potential, are very
different. Can one express the equation in harmonic form? Yes, by using Taylor expansion of the
energy function:
14.6. Force Constants
The force constants are equal to the second derivatives of the actual energy function:
V=∑i=1
3N
∑j=1
3N
{k ij (x i− x j)2}
k ij=∂
2E
∂ x i∂ x j
The Normal Modes Analysis (NMA) using Molecular Mechanics is a little bit more
complicated, because the full atomic representation and MM interactions require:
energy minimization
diagonalization of the second derivative of the potential energy (3N x 3N Hessian matrix).
Introduction to molecular modeling. 14. Normal Modes and Principal Component Analysis 89 by Rajmund Kaźmierkiewicz
Calculation and diagonalization of the 3N x 3N Hessian matrix is usually not a trivial task. Some
authors suggest approximations to speed-up these procedures:
Figure 63.NMA using Molecular Mechanics, Memory-Efficient Diagonalization
14.7. NMA using Molecular Mechanics, Reducing the Number of
Variables.
Figure 64. NMA using Molecular Mechanics, Reducing the Number of Variables
One of the most popular approximation is to perform an NMA using Molecular Mechanics and
reducing the number of variables by introduction of the Elastic Network Model approximation
(Monique M Tirion (1996) Phys Rev Lett. T7, 1905-1908):
Figure 65. An Elastic Network Model approximation in NMA method
90 Introduction to molecular modeling. 14. Normal Modes and Principal Component Analysis by Rajmund Kaźmierkiewicz
Vector Quantization:
Encode data (in R3 space) using a finite set {wj} (j=1,…,k) of codebook vectors. Delaunay triangulation
divides R3 into k Voronoi polyhedra (so-called “receptive fields”):
Figure 66. The Voronoi polyhedra
The reduction of the number of variables by vector quantization can help explain the large
scale motions (Bahar, Curr Opin. Struct. Biol., 2005, 15:1-7)
Figure 67. An illustration of reduction of the number of variables by vector quantization
Introduction to molecular modeling. 14. Normal Modes and Principal Component Analysis 91 by Rajmund Kaźmierkiewicz
14.8. Using Normal Mode Analysis to Model Protein Dynamics
Figure 68. Using Normal Mode Analysis to Model Protein Dynamics (Tirion M., Large Amplitude Elastic Motions in Proteins from a Single Parameter, Atomic Analysis. Physical Review Letters 1996, 77:9)
14.9. The equilibrium correlation between fluctuations
The equilibrium correlation (Tirion, M. Large Amplitude Elastic Motions in Proteins from a
Single Parameter, Atomic Analysis. Physical Review Letters. 1996. 77:9) between fluctuations Ri and
Rj of two C carbons i and j is given by:
is a symmetric Kirchhoff matrix (connectivity matrix):
RMS deviation of backbone C atoms per mode (Tirion, M. Large Amplitude Elastic Motions in
Proteins from a Single Parameter, Atomic Analysis. Physical Review Letters. 1996. 77:9).
92 Introduction to molecular modeling. 14. Normal Modes and Principal Component Analysis by Rajmund Kaźmierkiewicz
Figure 69. An illustration of RMS deviation of backbone C atoms per mode
Only a small number of modes contribute to overall motion
Tirion’s “geometric” modes match “energy-based” modes
14.10. Calculation of protein B-factors
Table 7. The B-factor value is located in the last column of the PDB formatted protein structure description
B-factor
ATOM 4 N GLY O 1 26.266 -12.458 5.676 1.00 40.85
ATOM 5 CA GLY O 1 26.236 -11.169 6.450 1.00 33.10
ATOM 6 C GLY O 1 27.338 -10.107 6.224 1.00 28.33
ATOM 7 O GLY O 1 28.478 -10.258 6.644 1.00 33.77
ATOM 8 N ASP O 2 27.085 -9.047 5.480 1.00 24.61
ATOM 9 CA ASP O 2 28.167 -8.101 5.107 1.00 22.56
ATOM 10 C ASP O 2 28.316 -6.857 5.988 1.00 21.47
ATOM 11 O ASP O 2 27.527 -5.948 5.802 1.00 14.42
The numbers in the last column in the protein PDB file designate the temperature factors, or B-
factor, for each atom in the structure.
The B-factor describes the displacement of the atomic positions from an average (mean) value.
For example, the more flexible an atom is the larger the displacement from the mean position
will be (mean-squares displacement).
In graphics programs we can often color a protein according to B-factor value.
Introduction to molecular modeling. 14. Normal Modes and Principal Component Analysis 93 by Rajmund Kaźmierkiewicz
Figure 70. Sample protein structure colored according to B-factor value
14.11. Examples of Applications Using Normal Mode Analysis to
Model Protein Dynamics
14.11.1. Collective dynamics of protofilaments in microtubules:
Figure 71. Mechanisms of muscle action
94 Introduction to molecular modeling. 14. Normal Modes and Principal Component Analysis by Rajmund Kaźmierkiewicz
Figure 72. More details of mechanisms of muscle action
14.11.2. Applications of NMA : ribosome (Application to EM Data)
Figure 73. Rotation of the 30S relative to the 50S: Ratchet-like motion. It is a key mechanical step in the translocation (Frank J., Agrawal R.K. Nature 2000, 318)
14.11.3. Applications of Normal Mode Analysis to experimental EM maps
The application of the Normal Mode Analysis method for the flexible fitting of high-
resolution structures into low-resolution maps of macromolecular complexes from electron
microscopy has been recently described in applications to simulated electron density maps. This
method uses a linear combination of low-frequency normal modes in an iterative manner to deform
the structure optimally to conform to the lower solution electron density map. Gradient-following
techniques in the coordinate space of collective normal modes are used to optimize the overall
correlation coefficient between computed and measured electron densities. With this approach,
multi-scale flexible fitting can be performed using all-atoms or C atoms. (Seth A Darst, Bacterial RNA
polymerase, Current Opinion in Structural Biology, Volume 11, Issue 2, 1 April 2001, Pages 155-162)
Introduction to molecular modeling. 14. Normal Modes and Principal Component Analysis 95 by Rajmund Kaźmierkiewicz
Figure 74. An applications of NMA to experimental EM maps
14.12. What are the Limitations of NMA:
We do not know a priori which is the relevant mode, but the first 12 low-frequency modes are
probable candidates.
The amplitude of the motion is unknown.
NMA requires additional standards for parameterization, i.e. a screening against complementary
experimental data to select the relevant modes and amplitude.
Expert (user) input / evaluation is required
This method is not based on first principles of physics (like MD).
Normal mode analysis is less (computationally) expensive than Molecular Dynamics (MD)
simulation, but because the computer must invert large matrices, it requires much more
memory when dealing with large molecules.
This problem can be overcome somewhat by clumping regions, such as amino acid residues, and
treating them as if they were a single atom, effectively reducing the number of atoms, and
hence the size of the matrices the computer must invert.
Normal modes may break the symmetry of structures due to forced orthogonalization.
96 Introduction to molecular modeling. 14. Normal Modes and Principal Component Analysis by Rajmund Kaźmierkiewicz
14.13. The Principal Component Analysis (PCA) method
Normal mode analysis and principal component analysis are powerful theoretical tools for
studying collective motions in proteins. The former is based on the assumption of harmonicity of the
dynamics, while the latter is valid even when the dynamics is highly anharmonic. The results of the
latter analysis indicate that most important conformational events are taking place in the
conformational subspace spanned by a rather small number of principal modes, and this important
subspace is also spanned by a number of normal modes.
14.13.1. Collective coordinates
Collective variables are projections on eigenvectors obtained either by diagonalization of a
covariance matrix as in PCA or diagonalization of the second derivatives, Hessian matrix as in the
NMA. Diagonalize Hessian matrix:
Principal Component Analysis from MD
Normal Mode Analysis
Functional motions of a protein may be represented by only a few low-frequency modes.
Principal Component Analysis is a mathematical technique, used to find patterns in high-
dimensional datasets, such as protein structures. It allows to find relationships/patterns, which
would be invisible from a pure visual examination. PCA can be applied to MD simulation trajectories
to detect the global, correlated motions of the system (the principal components). One can separate
the configurational space into 2 sub-spaces:
1. The Essential subspace: correlated motions comprising only a few of the degrees of freedom
available to the protein, they are FUNCTIONALLY IMPORTANT
2. The “Irrelevant” subspace: independent, Gaussian fluctuations, which are constrained and of
no/little functional relevance – act locally
Example: a 500 frame trajectory of a 300 residue protein.
14.13.2. Building the covariance matrix from your trajectory
Populate the 900 x 900 matrix (x, y and z Cartesian coordinates of each Cα atom):
where is a time-averaged position
The covariance matrix is then diagonalized, after that procedure the columns of the
transformation matrix become the eigenvectors, each associated with an eigenvalue. Eigenvectors
are then sorted by eigenvalue, the highest eigenvalues represent the most significant relationship
between the dimensions: these are the principal components. Eigenvectors represent a correlated
displacement of groups of atoms through space. Eigenvalues represent the magnitude of this
displacement (nm2).
Introduction to molecular modeling. 14. Normal Modes and Principal Component Analysis 97 by Rajmund Kaźmierkiewicz
Figure 75. First 2 eigenvectors account for about 60% of total positional fluctuations
14.13.3. Visualizing principal components (PC’s)
The motion described by an eigenvector can be visualized by projecting the trajectory onto the
eigenvector and taking the 2 extreme projections and interpolating between them to create an
animation.
Figure 76. Projection of atom from a trajectory onto eigenvector
Figure 77. The sample porcupine plot
98 Introduction to molecular modeling. 15. Uses of Free Energy by Rajmund Kaźmierkiewicz
Porcupine plots can be used to display the motion described by an eigenvector in a static image. A
cone extending from the C position shows the direction of the atom along the
eigenvector.
Figure 78. The sample covariance plot
Covariance plots are a tool to visualize atoms which have a high correlation coefficient from the
covariance matrix. Correlation coefficient measures the “degree of synchronization” of motion of
two atoms.
14.13.4.....Validation of PCA
One may ask: How relevant are the PCs we have calculated and visualized?
1. Divide simulations into two or more parts and compare the eigenvectors for each part, to
measure subspace overlap: higher overlap indicates sampling of only a single energy minimum;
lower overlap indicates more complete sampling.
2. PCs can also measure cosine content of eigenvectors. Hess et al. (Hess, ”Similarities between
principal components of protein dynamics and random diffusion”, Phys. Rev. E 62(6):8438-8448
(2000)) showed that the first few PCs of high-dimensional random diffusion are cosines and that
several protein simulation PCs resemble these cosines. So high cosine content may mean that the
fluctuations in your simulation are due to random diffusion: typically seen when simulation
timescales are too short to reach energy barriers.
15. Uses of Free Energy
Free energy is one of the most important thermodynamic quantities (reaction equilibrium, solvation,
stability, and kinetics). It is used in:
Evaluating Protein-protein and protein-ligand interactions (binding constants,
association and disassociation)
Mutation analysis
Rational drug design
Protein folding unfolding
Introduction to molecular modeling. 15. Uses of Free Energy 99 by Rajmund Kaźmierkiewicz
15.1. Methods and Applications
Most of the free-energy methods are based on calculation of free-energy differences, which
may be the quantity of interest anyway. If reference is simple (such as ideal gas or harmonic crystal),
its absolute free energy can be evaluated analytically. The free-energy evaluation methods can be
divided into three classes:
Free energy perturbation and thermodynamic interaction
Potential of mean force calculations
“Rapid” (and not very precise) free energy methods (Beveridge, D.L. and DiCapua, F.M. (1989)
Free Energy Via Molecular Simulation: Applications to Chemical and Biomolecular Systems, Annu.
Rev. Biophys. Biophys. Chem. 18: 431-492; Brooks, C.L. and Case, D.A. (1993) Simulations of
Peptide Conformational Dynamics and Thermodynamics, Chem. Rev. 93:2487-2502; Kollman, P.
(1993) Free Energy Calculations: Applications to Chemical and Biochemical Phenomena, Chem.
Rev. 93: 2395-2417; Lybrand, T.P. (1990) Computer Simulation of Biomolecular Systems Using
Molecular Dynamics and Free Energy Perturbation Methods, in, Reviews in Computational
Chemistry, Vol.1, Lipkowitz, K.B. and Boyd, D.B., eds. VCH Publishers, New York, pp. 295-320;
Reynolds, C.A., King, P.M., and Richards, W.G. (1992) Free Energy Calculations in Molecular
Biophysics, Mol. Phys. 76, 251-275)
Calculation of thermodynamic quantities from molecular simulation is based on the
principles of statistical mechanics. We need to extend our previous discussions of that topic to
describe application of free energy simulations to biomolecular systems.
15.2. Thermodynamic Integration
For the free energy function, , on the interval to , the free energy difference is
defined by:
Since
then
From statistical mechanics
So (after quite a few substitutions of mathematical expressions) we can write
where the brackets denote an ensemble average over the probability function of . Thus, one can
write
100 Introduction to molecular modeling. 15. Uses of Free Energy by Rajmund Kaźmierkiewicz
In practice the integral is approximated by a summation over discrete intervals in λ. That is,
simulations are run at different values of over the interval 0 to 1, with ensemble averages being
determined at each . In many cases, simulations will be run in the forward direction (0 1) and the
reverse direction (1 0), with the amount of hysteresis between the forward and reverse simulations
being a measure of the statistical uncertainty in the integration. Another approach to obtaining
statistical information is to begin the simulation from a different equilibrated starting structure.
Estimates from all of the starting structures are independent estimates of the true mean and they
should be normally distributed.
15.3. Perturbation Method
The perturbation method (free energy perturbation method, FEP) is an alternative approach to
calculating the free energy. We begin again with the relationship
and we employ the coupling parameter, ,
We now write
We then multiply the numerator by the unity factor. That is,
So (again, after quite a few substitutions of mathematical expressions) we can write
where the subscript 0 indicates configurational averaging over the ensemble of configurations
representative of the initial state of the system. Thus,
We also can show
where configurational averaging is over the ensemble of configurations representative of the final
configuration.
The thermodynamic perturbation method is implemented by first performing Monte Carlo or
molecular dynamics simulations for state 0 and generating the ensemble average for the energy
difference described above (the forward calculation). Then simulations for state 1 are performed to
obtain the corresponding ensemble average (the reverse calculation). The difference in between
the forward and backward calculations is a measure of the statistical uncertainty of the calculations.
The perturbation approach will be accurate only when states 0 and 1 differ by only a small amount,
that is, when they are only perturbations of one another. However, additional methods can be
applied to extend the applicability and accuracy of these perturbation methods. If the states 0 and 1
are not sufficiently similar, the calculation can be divided into a series of steps along the
coordinate. It is recommended that the free energy changes for each step be no more than 2kT (ca.
Introduction to molecular modeling. 15. Uses of Free Energy 101 by Rajmund Kaźmierkiewicz
1.5 kcal/mol). The overall free energy change is then obtained by summing the change from each of
the steps. That is,
where the interval 0 to 1 has been divided into n subintervals.
15.4. Thermodynamic Integration and Slow Growth
Figure 79. An Alternative Approach to Potential of Mean Force Calculations (C. Chipot, P. A. Kollman, D. A. Pearlman, Alternative Approaches to Potential of Mean Force Calculations: Free Energy Perturbation versus
Thermodynamic Integration. Case Study of Some Representative Nonpolar Interactions, Journal of Computational Chemistry 1996, 17(9): 11 12-131)
15.5. Thermodynamic Cycles
This approach is an extension of the free energy methods described above. It is often applied in
studying the relative strength of ligand-receptor interactions and the relative stability of proteins
differing in one or a few amino acids.
Thermodynamic cycle methods were developed because relatively large, complicated changes need
to be taken into account when considering the physical phenomena that occur in ligand-receptor
binding or the effect of a mutation on protein stability. That is, binding of a drug to a receptor will
produce relatively large conformational changes (this is, the protein will favor a particular set of
conformational substates). Binding of a very similar drug to the same site should produce most of the
same changes. The thermodynamic cycle is designed to cancel out the large changes that are
common to binding of either drug to the receptor.
Consider ligands A and A’ and a receptor B. We can write the equilibriums:
102 Introduction to molecular modeling. 15. Uses of Free Energy by Rajmund Kaźmierkiewicz
represent the binding processes in which the large conformational changes
occur. We desire to calculate the quantity
also the nonphysical processes
These processes are part of the overall thermodynamic cycle
Because , a thermodynamic function, is a state property, it is dependent only on the initial and
final states and not on the path between them. Thus,
and are calculated by one of the methods described above. The changes in these processes
are usually relatively small and localized, though it is necessary to apply the coupling parameter
approach.
15.6. Application of free energy simulations, Partitioning the free
energy
Which interactions contribute to the most of the overall free energy? In Thermodynamic Integration:
In FEP, this can be achieved by first perturbing the electrostatic and then the van-der-Waals
parameters. Note: only the sum of the contributions is truly meaningful, the individual contributions
are not state functions (Boresch S. Karplus M: The meaning of component analysis: decomposition of
the free energy in terms of specific interactions. J Mol Biol 1995. 254:801-807).
Introduction to molecular modeling. 15. Uses of Free Energy 103 by Rajmund Kaźmierkiewicz
15.7. Potential of Mean Force Calculations
Figure 80. The Potential of Mean Force (PMF)
We can identify or hypothesize one biological process to take place along some inter- or intra-
molecular coordinates, called reaction coordinates (RC).
PMF is basically the free energy profile alone the reaction coordinates, and all the other degrees
of freedom will be averaged out.
A simple example. We select the distance between two atoms as RC, the PMF is the free
energy change as the separation (r) between the atoms is changed. The distribution of r can be
described by the radial distribution function g(r), so:
For a general RC q:
For multi-dimension cases, (q, s) :
It is often difficult to find suitable RC for detailed biological processes (Jensen M, Park S.
Tajkhorshid E. Schulten K: Energetics of glycerol conduction through aquaglyceroporin GlpF. Proc
Natl Acad Sci USA 2003, 99:6731-6736.).
The logarithmic relationship between the PMF and g(q) means that a small change in the free
energy may correspond to g(q) changing by an order of magnitude or more from its most likely value.
Standard MC or MD methods do not adequately sample regions where g(q) differs drastically from
the most likely value, leading to inaccurate values for the PMF (Johannes Kästner, Hans Martin Senn,
Stephan Thiel, Nikolaj Otte, and Walter Thiel, QM/MM Free-Energy Perturbation Compared to
Thermodynamic Integration and Umbrella Sampling: Application to an Enzymatic Reaction, J. Chem.
Theory Comput., 2006, 2 (2), pp 452–461).
One can calculate the PMF using the FEP method. But FEP is commonly used to study “mutations”,
which are often along non-physical pathways. One usually wants to calculate PMF for a physically
achievable process, so one can get the transition states and derive kinetic quantities such as rate
constants. The traditional way to avoid the sampling problem is Umbrella Sampling.
104 Introduction to molecular modeling. 15. Uses of Free Energy by Rajmund Kaźmierkiewicz
15.7.1. Potential of Mean Force calculation
The goal is to extract degree of freedom from partition function and free energy. The free energy is
related to probability:
For this reason one can use relatively simple approach for PMF calculation:
Run canonical MD or Monte Carlo
Compute probability distribution P(x)
P(x) determines F(x) up to a constant
15.8. Simple Umbrella Sampling
Problem: P(x) converges slowly due to barriers along x.
Solution: Add an additional potential term to the energy to encourage barrier crossing.
Sample with umbrella potential U’(x)
Compute biased probability P’(x)
Estimate unbiased free energy
in this equation F0 is undetermined but it is irrelevant.
15.9. Weighted Histogram Analysis Method (WHAM)
Weighted Histogram Analysis Method determines optimal F values for combining simulations
(Kumar, et al., J Comput Chem, 13, 1011-1021, 1992). Some generalizations are possible:
multidimensional reaction coordinates, multiple temperatures. WHAM equations:
where
Nsims = number of simulations
ni(x)= number of counts in histogram bin associated with x
Ubias,i , Fi = biasing potential and free energy shift from simulation i
P(x) = best estimate of unbiased probability distribution
Fi and P(x) are unknowns
Solve by iteration to self-consistency
15.9.1. Running a Simulation
Choose the reaction coordinate
Choose the number of simulations and the biasing potential
Run the simulations
Compute time series for the value of the reaction coordinate
Apply the WHAM equations
Introduction to molecular modeling. 15. Uses of Free Energy 105 by Rajmund Kaźmierkiewicz
15.9.2. Reaction Coordinate
The choice of the reaction coordinate is sometimes obvious. It may be a dihedral angle like
for butane, the backbone dihedrals like for the alanine dipeptide. Some care is required, because
PMF depends on the choice of coordinate and the volume element may not be constant along
reaction coordinates.
15.9.3. Example: n-Butane
Compute PMF for rotating dihedral of united atom n-butane
PMF integrates out the effects of flexible bonds and angles
Protocol:
18 independent simulations:
500 ps each
Restraint spring constant = 0.02 kcal/mol-deg2
T=300K (stochastic dynamics)
WHAM:
90 bins (4°/bin)
Enforced periodicity
Figure 81. Histograms from Individual Trajectories
Figure 82. Histogram of Combined Trajectories
106 Introduction to molecular modeling. 15. Uses of Free Energy by Rajmund Kaźmierkiewicz
Figure 83.The n-butane PMF
15.10. Steered Molecular Dynamics
In Steered Molecular Dynamics (SMD), time-dependent external forces are applied to a
system, which induce unbinding of ligands and conformational changes in biomolecules on time
scales accessible to MD simulations. Assuming a reaction coordinate x, we add an external force
along the path, a simple way is by a harmonic spring:
Figure 84. The schematic illustration of Steered Molecular Dynamics
The Steered Molecular Dynamics method is similar to experiments by Atomic Force Microscopy, a
“spring” of stiffness k is attached to the ligand and a constant pulling rate is applied to measure the
adhesion forces while the ligand detaches from the protein.
15.11. “Rapid” Free Energy Methods
Free energy calculations are very important in computer-aided drug design. However, if the
calculations take longer to perform than a candidate drug molecule can be synthesized and tested,
then there is little practical benefit from attempting the calculation.
Free energy calculations are time-consuming. It is necessary to develop some alternative
methods, which still being based upon 'exact' statistical mechanics, are intended to provide free
energy with less computational effort than a full free energy calculation.
Introduction to molecular modeling. 15. Uses of Free Energy 107 by Rajmund Kaźmierkiewicz
15.11.1. Linear Interaction Energy (LIE)
The Linear Interaction Energy is a semi-empirical method for estimating absolute binding free
energies of ligands binding to proteins. The interaction between the ligand and protein or solvent is
broken down into the electrostatic and van der Waals contributions.
To determine AF one thus needs to perform just two simulations, one of the ligand in the solvent and
the other of the ligand bound to the protein.
What remains is to determine values of the parameters and . By some analytical theories,
the parameter related to the electrostatic contribution is around 1/2.
For the Van der Waals component no such analytical theory exists. depends on a different
force field, and the nature of the binding sites, different distributions of polar and non-polar groups
in different binding sites. In other words needs to be evaluated for each protein separately (Wang
W. Wang J. Kollman PA: What determines the van der Waals coefficient in the LIE (Linear
Interaction Energy) method to estimate binding free energies using molecular dynamics simulations?
Proteins Struct Funct Genet 1999, 34:395-402.).
15.11.2. Molecular Mechanics Poisson-Boltzmann Surface Area Method
(MM/PBSA)
The MM/PBSA approach represents the post-processing method to evaluate free energies of
binding or to calculate absolute free energies of molecules in solutions, which combines the
molecular mechanical energies with the continuum solvent approaches. In this method, one usually
carries out a MD simulation with explicit water and counterions. Then one post-processes these
structures, removing any solvent and counterions, and calculates the Gibbs free energy (Kollman PA,
Massova L, Reyes C, Kuhn B., Huo S, Chong L. Lee M. Lee T, Duan Y. Wang W, Donini O. Cieplak P.
Srinivasan J. Case D: Cheatham TE. Ill: Calculating structures and free energies of complex molecules:
combining molecular mechanics and continuum models. Acc Chem Res 2000, 33:889-897.):
Calculated average Gibbs free energy:
The components in MM/PBSA equation:
are as follows:
average molecular mechanical
energy
Solvation free energy
Numerical solution of Poisson-Boltzmann equation or
Generalized Born model
Solvent-accessible surface area
Solute entropy, which is likely to be much smaller than other terms. It can be
estimated by harmonic analysis or normal mode analysis,
108 Introduction to molecular modeling. 15. Uses of Free Energy by Rajmund Kaźmierkiewicz
15.11.3. Example: MM/PBSA
15.11.4. Binding free energy of protein-ligand
There are two methods of G evaluation:
1. separate simulations of complex, protein, and ligand or
2. evaluation of all three terms using just the snapshots from complex simulations.
Figure 85. Sample correlation between calculated and experimental protein-ligand binding free energies
The second method is a good approximation in cases that, there are no large conformational changes
of protein and ligand before and after their association (Kuhn B. Kollman PA: Binding of a diverse set
of ligands to avidin and streptavidin: an accurate quantitative prediction of their relative affinities by
a combination of molecular mechanics and continuum solvent models. J Med Chem 2000. 43:3786-
3791).
15.11.5. Binding free energy of protein-RNA
Figure 86. The MM-PBSA free energy differences between free and bound protein and RNA
Introduction to molecular modeling. 16. Molecular Distance Geometry Problem 109 by Rajmund Kaźmierkiewicz
Conformational change upon binding of U1A protein and internal loop (IL) RNA, and are the
MM-PBSA free energy differences between free and bound protein and RNA, respectively. is the
free energy of association of protein and RNA in their bound structures. (Reyes C., Kollmann PA:
Structure and thermodynamics of RNA-protein binding: using molecular dynamics and free energy
analysis to calculating both the free energies of binding and conformational change. J Mol Biol 2000,
297:1145-1158.)
16. Molecular Distance Geometry Problem
Given n atoms a1, …, an and a set of distances di,j between ai and aj,
find the coordinates x1, ..., x3n or a1, ..., an such that
Where S is a set of integer pairs from 1 to 3n.
16.1. Current Approaches
Embed Algorithm by Crippen and Havel
Geometric Build-Up by Blumenthal 1953
CNS Partial Metrization by Brünger et al
Graph Reduction by Hendrickson
Alternating Projection by Glunt and Hayden
Global Optimization by Moré and Wu
Multidimensional Scaling by Trosset, et al
Currently, the first two approaches are most commonly used.
16.1.1. Embed Algorithm
1. bound smooth; keep distances consistent
2. distance metrization; estimate the missing distances
3. repeat (say 1000 times):
a. randomly generate D in between L and U
b. find X using SVD with D
c. if X is found, stop
4. select the best approximation X
5. refine X with simulated annealing
6. final optimization
(Crippen and Havel 1988 (DGII, DGEOM); Brünger et al 1992, 1998 (XPLOR, CNS))
16.1.2. Geometric Build-Up
Geometric Build-Up is a (rather advanced) mathematical procedure used to speed up the
reconstruction of atom locations from the matrix of distances. It uses unique mathematical tools and
concepts (for example, ):
Independent Points: A set of k+1 points in k dimensional space Rk is called independent if it is not
a set of points in Rk-1.
110 Introduction to molecular modeling. 17. Protein Folding by Rajmund Kaźmierkiewicz
Metric Basis: A set of points B in a space S is a metric basis of S provided each point of S is
uniquely determined by its distances from the points in B.
Fundamental Theorem: Any k+1 independent points in k dimensional space Rk form a metric basis
for Rk. (Blumenthal 1953: Theory and Applications of Distance Geometry)
Besides being rather cryptic for non-mathematicians it considerably speeds up calculations. The
geometric build-up algorithm solves a molecular distance geometry problem in O(n) when distances
between all pairs of atoms are given, while the singular value decomposition algorithm requires
O(n2~n3) computing time!
Build up procedures example application:
Figure 87. The X-ray crystallography structure (left) of the HIV-1 RT p 66 protein (4200 atoms) and the structure (right) determined by the geometric build-up algorithm using the distances for all pairs of atoms in the protein.
The algorithm took only 188,859 floating-point operations to obtain the structure, while a conventional singular-value decomposition algorithm required 1,268,200,000 floating-point operations. The RMSD of the
two structures is ~10-4
Å
17. Protein Folding
Proteins are created linearly and then assume their tertiary structure by “folding”. The exact
mechanism is still unknown, however molecular mechanics simulations can be informative. Proteins
assume the lowest energy structure, or sometimes an ensemble of low energy structures. It is most
likely that the hydrophobic collapse is an important “driving” force of the folding process. The local
(secondary) structure tendencies also play a significant role. The folded structure is stabilized
internally by a network of hydrogen bonds, disulphide bonds, electrostatic interactions and salt
bridges. There are three major classes of methods for the tertiary (folded) protein structure
prediction: Homology Modeling/Comparative Modeling, The probe and template sequences are
evolutionarily related
Fold Recognition/Threading, For the query sequence, determine the closest matching structure
from a library of known folds by scoring function
First Principles with Database Information, Secondary and/or tertiary information from
databases/statistical methods; First Principles/Ab-initio without Database Information,
Physiochemical models with most general application
The X-ray crystallography structure (left) of the HIV-1 RT p66 protein (4200
atoms) and the structure (right) determined by the geometric build-up algorithm
using the distances for all pairs of atoms in the protein. The algorithm took only
188,859 floating-point operations to obtain the structure, while a conventional
singular-value decomposition algorithm required 1,268,200,000 floating-point
operations. The RMSD of the two structures is ~10-4 Å.
Introduction to molecular modeling. 17. Protein Folding 111 by Rajmund Kaźmierkiewicz
Kinds of Structure Prediction: Comparative modeling, where the homolog has known structure,
which is adjusted for sequence differences. It uses energy minimization and molecular dynamics.
According to the fold recognition method proteins fall into broad fold classes. Models of folds that
recognize compatible sequences. The method treats structures and sequences separately and it aims
to find connections between (known and unknown) structures and sequences. To some degree it is
an “inverse” problem if compared to folding. Usually it is able to predict more than fold class. An Ab
initio or “new fold” prediction method seeks the structures of proteins having no homologs, and not
recognized by any fold model.
17.1. Energy Minimization
Many forces act on a protein: hydrophobic, the inside of a protein wants to avoid water;
packing, atoms can’t be too close, nor too far away; bond angle/length constraints; long distance,
these are electrostatics interactions and hydrogen bonds, disulphide bonds, salt bridges. Energy
minimization procedure can calculate all of these forces, and minimize. The most important is the
global optimization, which is intractable in a general case, but can be useful. Some related methods
(all of them serve the same purpose, finding the global minimum on the protein energy
hypersurface and that’s why they are “related”): stochastic search methods use random
perturbations; Monte-Carlo simulated annealing uses perturbations, acceptance criterion, schedule;
evolutionary algorithms use ensembles/populations; smoothing methods use deformation of
function being minimized; homotopy/continuation methods use nonlinear equations.
17.2. Some Related Methods
Example of Global optimization algorithms “Optimization” refers to trying to find the global energy
minimum of a potential surface:
Genetic Algorithm (GA)
Simulated Annealing (SA)
Tabu Search (TS)
Ant Colony Optimization (ACO)
A model system: Lennard Jones clusters
17.3. Monte Carlo-Minimization (MCM)
Generate at random a set of structures and locally minimize them.
1. Select the lowest-energy one as the "generative" structure.
2. Carry out a random change (perturbation) of "generative" structure to produce a new
3. conformation.
4. Minimize the energy of the new conformation.
5. Compare its energy to the energy of the "generative" structure by means of the Metropolis
6. criterion [accepted with probability of exp(-E/kT)].
7. If accepted in Metropolis criterion new (minimized) structure becomes the
8. “generative” structure, otherwise the "generative" structure remains unchanged.
9. Iterate into point 3.
112 Introduction to molecular modeling. 17. Protein Folding by Rajmund Kaźmierkiewicz
The Conformation-Family Monte Carlo (CFMC) seems to be an extension of the MCM method. It
uses the Metropolis criterion to move between families. CFMC uses the Boltzmann distribution to
choose conformation from a family. It does not move between structures, but between families.
MD as a tool for minimization
Figure 88. MD as a tool for minimization
Crossing energy barriers
Figure 89. Using MD as a tool for minimization
The actual transition time from A to B is very quick (a few picoseconds). What takes time is waiting.
The average waiting time for going from A to B can be expressed as:
17.4. Knowledge-based Energies
The simplest form of potential energy function for investigation of the protein folding problem is the
Knowledge-based (where “knowledge” usually comes from the protein database PDB) Energy
function.
Native State Randomized State
Figure 90. An illustration of the method for obtaining Knowledge-based Energy function
Energy
positionEnergy minimization
stops at local minima
Molecular dynamics
uses thermal energy
to explore the energy
surfaceState A
State B
A
B
I
G
Position
En
erg
y
time
Po
sitio
n
State A
State B
τ A→B=Ce
ΔG
kT
many
clusters
no
clusters
Introduction to molecular modeling. 17. Protein Folding 113 by Rajmund Kaźmierkiewicz
The “recipe” for the function is as follows:
1. Count pairs of centers of each type at different separations, r, to give Nij(r).
2. Normalize by the expected count for a random arrangement given by Mij(r).
3. Convert to additive score: Eij(r)=log(Nij(r)/Mij(r)).
4. After applying this procedure one can obtain the Pair-wise Energies.
Figure 91. The correlation between pair-wise energies and amino-acid residues centers distributions
Get distribution of distances between pairs of atom centers of a particular type, for example
D-OD1...F-CD2. Normalize and take log to get Energy score: Eij(r)=log(Nij(r)/Mij(r))
The Knowledge-based Energies are closely connected with the Knowledge-based Geometry
they can be created by applying a simple procedure: Cut a protein into overlapping pieces for re-use.
May cluster to have less redundancy. (Jones & Thirup, EMBO J. 5, 819 (1986) Levitt, J. Mol. Biol. 226:
507-533 (1992).) This procedure leads to Fragment Libraries. One may build any structure from a
library of fragments. For 5-residue 20mer library, a protein of length N will consist of (N-3)/2
fragments. (Kolodny et al, J. Mol. Biol., 323: 297(2002).)
17.5. Predicting Protein Secondary Structure
Assign the secondary structure of every residue from a protein structure in the database
Now predict this secondary structure from the amino acid sequence based on the amino acid
residues conformational preferences.
Two general schemes relate secondary structure to sequence:
Statistical: Count how often each type of residue occurs in each type of secondary structure.
Patterns: Look for characteristic sequence patterns that define the ends of -helices, -
turns.
114 Introduction to molecular modeling. 17. Protein Folding by Rajmund Kaźmierkiewicz
17.6. Protein Threading
Make a structure prediction through finding an optimal placement (threading) of a protein
sequence onto each known structure (structural template). The “placement” quality is measured by
some statistics-based energy function. The best overall “placement” among all templates may give a
structure prediction. Protein Threading Needs: construction of a template library, design of energy
function, sequence-structure alignment, template selection and model construction methods.
17.7. Reduced or Simplified Protein Models
Figure 92. One of the earliest simplified protein models created for investigation of protein folding
Each residue in this simplified protein model is represented by 1 atom/residue, it is able to fold
protein with 1000 steps of minimization. Escape from local minima is achieved with normal mode
„jumps” (Levitt & Warshel Nature, 1975)
Simplification of the protein structure representation sometimes does not mean the simplification of
the potential energy function as is apparent in the case of the UNRES force field.
Unres force field potential energy function.
Unres is a united-residue force field for off-lattice protein-structure simulations. United-
residue representation of the polypeptide chain consists of ellipsoids equivalent to side-chain
residues connected with the point dipoles representing peptide bonds.
Some simplified protein models use two dimensional lattice or three-dimensional face-
centered cubic lattice.
Figure 93. Random walk on lattice. Self-avoiding and bounded. Can extend the chain in 4 ways all with a bond angle of 120
o
Introduction to molecular modeling. 17. Protein Folding 115 by Rajmund Kaźmierkiewicz
Figure 94. Random walk generation on three-dimensional face-centered cubic lattice may give also “native-like” folds
One of the best lattice based simplified protein models was created by Kolioski and Skolnick
(Kolinski & Skolnick. Assembly of Protein Structure From Sparse Experimental Data: An Efficient
Monte Carlo Model. Proteins, 32: 475 (1998). )
Figure 95. They use a complicated lattice model for the side chain centroids and Multi-Replica Monte Carlo for protein structure prediction
Even the “Simple Toy Protein Model” consisting of Two kinds of amino acid residues: Hydrophobic
(H) and Polar (P), which represents the Two-Dimensional Self-Avoiding Chain of N residues on a
square lattice can be very useful.
Figure 96. The simplified 24 residues protein model of PPHPPPHPPHHPPHPHHPPHPHHH on a square lattice
The model has to be simple enough for all sequences and conformations to be explored. For such a
short protein there are 224=16777216 possible different sequences, and there are 2158326727
different chain shapes. All of the shapes can be effectively explored. This model enables testing of
our understanding of interplay between hydrophilic/hydrophobic interactions and their influence on
protein folding.
116 Introduction to molecular modeling. 18. Molecular graphics software by Rajmund Kaźmierkiewicz
18. Molecular graphics software
The selected, frequently used, molecular graphics presentation programs are collected in the table:
Software Web page L* W
** Short Description
Abalone http://www.biomolecular-
modeling.com/Abalone/ L W
Simple program for molecular structure
visualization and Biomolecular (rather short)
dynamics simulations of proteins, DNA, ligands.
AutoDock http://autodock.scripps.ed
u/ L W
Automated docking tools designed to predict how
small molecules, such as substrates or drug
candidates, bind to a receptor of known 3D
structure.
Cn3D
http://www.ncbi.nlm.nih.g
ov/Structure/CN3D/cn3d.s
html
L W
Simultaneously displays structure, sequence and
alignment. Based on NCBI MMDB database. Does
not read PDB files.
DINO http://www.dino3d.org/in
tro.php L
DINO is a real-time 3D visualization program for
structural biology data.
DeepView http://spdbv.vital-it.ch/ L W
Molecular graphics with interface allowing analysis
of several proteins at the same time. Can also read
electron density maps, and provides tools to build
into the density.
Discovery
Studio (DS)
Visualizer
http://accelrys.com/produ
cts/discovery-
studio/visualization-
download.php
L W Provides publication quality images with
hierarchical presentation of chemical structure.
iMol http://www.pirx.com/iMol
/
Molecular visualization application for Mac OS X
operating system.
Jmol http://jmol.sourceforge.ne
t/ L W
Open-source Java viewer for chemical structures in
3D.
MarvinSpace
http://www.chemaxon.co
m/products/marvin/marvi
nspace/
L W Web enabled 3D molecule visualization tool.
Molegro http://www.molegro.com/ L W Molecular Viewer for visualization of molecules and
Molegro Virtual Docker results.
MOLMOL http://www.marcsaric.de/i
ndex.php/Molmol L W
Molecular graphics program for displaying,
analyzing, and manipulating the three-dimensional
structure of biological macromolecules, especially
protein and DNA.
NOC http://noch.sourceforge.n
et/ L W
Molecular explorer for protein structure
visualization, crystallographic mapping, modeling
and refinement.
Protein
Explorer
(RasMol)
http://www.umass.edu/mi
crobio/rasmol/ L W
Molecular visualization software for looking at
macromolecular structure and its relation to
function.
PyMOL http://www.pymol.org/ L W
Molecular graphics system with an embedded
Python interpreter designed for real-time
visualization and rapid generation of high-quality
molecular graphics images and animations.
Introduction to molecular modeling. 19. Recommended reading 117 by Rajmund Kaźmierkiewicz
Software Web page L* W
** Short Description
QuteMol http://qutemol.sourceforg
e.net/ W
Generates high resolution antialiased snapshots for
publication quality renderings. Requires a graphics
card.
Ramachandra
n Plot
Explorer
http://boscoh.com/ramapl
ot/ W
Outstanding software to visualize and manipulate
phi and psi angles of peptide bonds.
VMD http://www.ks.uiuc.edu/R
esearch/vmd/ L W
Molecular visualization program for displaying,
animating, and analyzing large biomolecular
systems using 3-D graphics and built-in scripting.
Chimera
http://www.csb.yale.edu/
userguides/graphics/chim
era/chimera_descrip.html
L W UCSF modeling package
LIGPLOT
http://www.csb.yale.edu/
userguides/graphics/ligplo
t/ligplot_descrip.html
L W
Ligand plotting software. Automatically generates
schematic diagrams of protein-ligand interactions
for a given PDB file.
Molscript
http://www.csb.yale.edu/
userguides/graphics/molsc
ript/molscript_descrip.ht
ml
L
MolScript is a program for displaying molecular 3D
structures, such as proteins, in both schematic and
detailed representations.
Raster-3D
http://skuld.bmsc.washing
ton.edu/raster3d/raster3d
.html
L Raster3D is a set of tools for generating high quality
raster images of proteins or other molecules.
ISIS/DRAW http://www.symyx.com/d
ownloads/index.jsp W
It is commercial molecular graphics software. (but
it is free after registration). *Software available under Linux
**Software available under MS Windows
19. Recommended reading
1. Daan Frenkel. Understanding Molecular Simulation, Second Edition: From Algorithms to
Applications.
2. M.P. Allen, D.J. Tildesley. Computer Simulation of Liquids, Oxford University Press, New York
1987.
3. J.M. Haile. Molecular Dynamics Simulations: Elementary Methods, Wiley, New York 1992.
4. Molecular Modelling. Principles and Applications, Eds. A. R. Leach, Addison Wesley Longman,
Essex, England 1996.
