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
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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
Introduction to molecular modeling 3 by Rajmund Kaźmierkiewicz
2. LINUX .....................................................................................................................................................10
3. WHAT IS MODELING? .............................................................................................................................10
7.2. THE EMPIRICAL ENERGY FUNCTION (OR FORCE FIELD) ................................................................................... 16
7.3. BOND STRETCHING .................................................................................................................................. 16
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
10.4. THE TIME STEP ....................................................................................................................................... 35
10.5. THE NEED FOR FASTER COMPUTERS ............................................................................................................. 35
10.6. PHASE SPACE ......................................................................................................................................... 35
10.7. CALCULATION OF AVERAGE PROPERTIES ...................................................................................................... 36
10.9. 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
10.25. 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
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
11.7.4. 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
12.1. 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
12.15. 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
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
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
17. PROTEIN FOLDING ............................................................................................................................ 110
17.1. ENERGY MINIMIZATION ......................................................................................................................... 111
17.2. SOME RELATED METHODS...................................................................................................................... 111
17.3. MONTE CARLO-MINIMIZATION (MCM) ................................................................................................... 111
17.5. PREDICTING PROTEIN SECONDARY STRUCTURE ........................................................................................... 113
17.6. PROTEIN THREADING ............................................................................................................................. 114
17.7. REDUCED OR SIMPLIFIED PROTEIN MODELS ............................................................................................... 114
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:
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:
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 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:
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