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Introduction to Bioinformatics: Lecture XV
Empirical Force Fields and Molecular Dynamics
Jarek Meller
Division of Biomedical Informatics,
Children’s Hospital Research Foundation
& Department of Biomedical Engineering, UC
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Outline of the lecture
Motivation: atomistic models of molecular systems
Empirical force fields as effective interaction models
for atomistic simulations
Molecular Dynamics algorithm
Kinetics, thermodynamics, conformational search and
docking using MD
Limitations of MD: force fields inaccuracy, long range
interactions, integration stability and time limitations,
ergodicity and sampling problem
Beyond MD: other protocols for atomistic simulations
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Molecular systems and interatomic interactions
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Molecular systems and interatomic interactions
a

helix
b

strand
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Molecular Dynamics as a way to study molecular motion
What is wrong with the previous pictures?
Real molecules “breathe”: molecular motion is
inherent to all chemical processes, “structure” and
function of molecular systems
For example, ligand binding (oxygen to hemoglobin,
hormone to receptor etc.) require inter

and intra

molecular motions
Another example is protein folding
–
check out some
MD trajectories
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Web watch: folding simulations using MD and
distributed computing: Folding@Home
Folding@Home
Vijay S Pande and colleagues, Stanford Univ.
For example, folding simulations of the villin headpiece …
http://www.stanford.edu/group/pandegroup/folding/papers.html
Some more MD movies from Ron Elber’s group:
http://www.cs.cornell.edu/ron/movies.htm
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Two approximations and two families of MD methods
The
quantum
or
first

principles
MD simulations (Car and
Parinello), take explicitly into account the quantum nature of the
chemical bond. The electron density functional for the valence
electrons that determine bonding in the system is computed
using quantum equations, whereas the dynamics of ions (nuclei
with their inner electrons) is followed classically.
In the
classical
mechanics approach to MD simulations
molecules are treated as classical objects, resembling very much
the “ball and stick” model. Atoms correspond to soft balls and
elastic sticks correspond to bonds. The laws of classical
mechanics define the dynamics of the system.
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From quantum models to classical approximations
Born

Oppenheimer approximation, potential energy surface and empirical
force fields, parametrizing atomistic force fields by combination of ab initio,
experiment and fitting …
Ab initio
methods: computational methods of physics and chemistry
that are based on fundamental physical models and, contrary to
empirical methods, do not use experimentally derived parameters
except for fundamental physical constants such as speed of light
c
or Planck constant
h
.
The NIH guide to molecular mechanics:
http://cmm.info.nih.gov/modeling/guide_documents/molecular_mechanics_document.html
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Force fields for atomistic simulations
Definition
Empirical potential
is a certain functional form of the
potential energy of a system of interacting atoms with the parameters
derived from
ab initio
calculations and experimental data.
How to get parameters that would have something to do with the
physical reality: experiment and ab initio calculations, also just fitting!
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Dispersion interactions and Lennard

Jones potential

ij
ij
r
ij
Problem
Find that the minimum of van der Waals
(Lennard

Jones) potential
Dispersion (van der Waals) interactions result from polarization of
electron clouds and their range is significantly shorter than that of
Coulomb interactions.
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Time evolution of the system:
Newton’s equations of motion
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Solving EOM: Coulomb interactions and N

body problem
Solving EOM for a harmonic oscillator
–
simple …
Potential: U(x)=1/2
k
x
2
; Solution: x(t) = A cos(
t+
)
Problem
Show that
2
=k/m
Solving EOM for a system with more than two atoms and Coulomb or
Lennard

Jones potentials
–
no analytical solution, numerical integration
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Numerical integration of EOM: the Verlet algorithm
Definition
Molecular Dynamics
is a technique for atomistic simulations
of complex systems in which the time evolution of the system is followed
using numerical integration of the equations of motion.
One commonly used method of numerical integration of motion was first
proposed by Verlet:
Problem
Using Taylor’s expansions derive the Verlet formula given above.
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Fast motions and the integration time step
For example, O

H bonds vibrate with a period of about 17 fs
To preserve stability of the integration,
t needs to very short

of the order of femtoseconds (even if fastest vibrations are
filtered out)
Except for very fast processes, nano

and micro

seconds time
scales are required: time limitation and long time dynamics
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Long range forces as computational bottleneck
Long range interactions: electrostatic and dispersion interactions
lead (in straightforward implementations) to summation over all
pairs of atoms in the system to compute the forces
Environment, e.g. solvent, membranes, complexes
Implicit solvent models: from effective pair energies to PB models
Explicit solvent models: multiple expansion, periodic boundary
conditions (lattice symmetry), PME
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Examples of problems and MD trajectories
Thermodynamics
: what states are possible, what states are
“visited”, statistics and averages for observables, chemical
processes as driven by free energy differences between states, MD
as a sampling method (different ensembles and the corresponding
MD protocols)
Kinetics
: how fast (and along what trajectory) the system
interconverts between states, rates of processes, mechanistic
insights, MD provides “real” trajectories and intermediate states,
often inaccessible experimentally
Specific applications
: sampling for energy minimization and
structure prediction, homology modeling, sampling for free energy
of ligand binding, folding rates and folding intermediates etc.
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Ligand diffusion in myoglobin
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Ligand diffusion in myoglobin
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Molecular Dynamics as a way to study molecular motion
Quantum (first principles) MD is computationally
expensive
Empirical force fields as a more effective alternative
No chemical change though, problem with
parametrization and numerous approximations (read
inherent limitations of empirical force fields)
Commonly used force fields and MD packages:
Charmm, AMBER, MOIL, GROMOS, Tinker
Other limitations of MD: long range interactions,
integration stability and time limitations, ergodicity and
sampling problem
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