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2 Οκτ 2013 (πριν από 4 χρόνια και 9 μέρες)

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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

JM
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http://folding.chmcc.org

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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

JM
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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

JM
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http://folding.chmcc.org

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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


JM
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http://folding.chmcc.org

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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

JM
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http://folding.chmcc.org

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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.

JM
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http://folding.chmcc.org

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Ligand diffusion in myoglobin

JM
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http://folding.chmcc.org

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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