introduction to bioinformatics: lecture xv empirical force fields and molecular dynamics

JM - http://folding.chmcc.org 1

Introduction to Bioinformatics: Lecture XVEmpirical Force Fields and Molecular Dynamics

Jarek MellerJarek Meller

Division of Biomedical Informatics, Division of Biomedical Informatics, Children’s Hospital Research Foundation Children’s Hospital Research Foundation & Department of Biomedical Engineering, UC& Department of Biomedical Engineering, UC


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


Molecular systems and interatomic interactions


Molecular systems and interatomic interactions

helix

strand


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


Web watch: folding simulations using MD and distributed computing: Folding@Home

Folding@HomeVijay 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


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.


From quantum models to classical approximations

Born-Oppenheimer approximation, potential energy surface and empiricalforce 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


Force fields for atomistic simulations

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Definition Empirical potential is a certain functional form of the potential energy of a system of interacting atoms with the parametersderived from ab initio calculations and experimental data.

How to get parameters that would have something to do with thephysical reality: experiment and ab initio calculations, also just fitting!


Dispersion interactions and Lennard-Jones potential

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ij rij

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.


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 x2 ; Solution: x(t) = A cos(t+)

Problem Show that 2=k/m

Solving EOM for a system with more than two atoms and Coulomb orLennard-Jones potentials – no analytical solution, numerical integration


Numerical integration of EOM: the Verlet algorithm

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Definition Molecular Dynamics is a technique for atomistic simulationsof 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 firstproposed by Verlet:

Problem Using Taylor’s expansions derive the Verlet formula given above.


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 timescales are required: time limitation and long time dynamics


Long range forces as computational bottleneck

Long range interactions: electrostatic and dispersion interactionslead (in straightforward implementations) to summation over allpairs 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


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.


Ligand diffusion in myoglobin


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

introduction to bioinformatics: lecture xv empirical force fields and molecular dynamics

Documents

folding simulations

md trajectoriesjm http

ucjm http

strandjm http

atomistic simulationsjm

atomistic force fields

molecular mechanics

md simulations molecules