Molecular Dynamics, Monte Carlo and Docking

Lecture 21

Introduction to BioinformaticsMNW2

If you throw up a stone, it is Physics.

If you throw up a stone, it is Physics. If it lands on your head,it is Biophysics.

If you throw up a stone, it is Physics. If it lands on your head,it is Biophysics.

If you write a computer program, it is Informatics.

If you throw up a stone, it is Physics. If it lands on your head,it is Biophysics.

If you write a computer program, it is Informatics. If there is a bug in it, it is Bioinformatics

Allowed phi-psi angles

Red areas are preferred, yellow areas are allowed, and white is avoided

2.3a Hamiltonian equations of motion

Hamiltonian equations (one degree of freedom):

H – Hamiltonian function, Hamiltonian,q, p –Canonical variables: generalized coordinate (q) and

momentum [impulses] (p).

q = coordinates

p = momentum

v(t) = (r(t + ∆t) - r(t - ∆t))/2∆t

Molecular systems strive to be in the lowest energy state

Molecular DynamicsKnowledge of the atomic forces and masses can be used to solve the position of each atom along a series of extremely small time steps (on the order of femtoseconds = 10-15 seconds). The resulting series of snapshots of structural changes over time is called a trajectory. The use of this method to compute trajectories can be more easily seen when Newton's equation is expressed in the following form:

The "leapfrog" method is a common numerical approach to calculating trajectories based on Newton's equation. The steps can be summarized as follows:

Force fieldThe potential energy of a system can be expressed as a sum of valence (or bond), crossterm, and nonbond interactions:

The energy of valence interactions comprises bond stretching (Ebond), valence angle bending (Eangle), dihedral angle torsion (Etorsion), and inversion (also called out-of-plane interactions) (Einversion or Eoop) terms, which are part of nearly all force fields for covalent systems. A Urey-Bradley term (EUB) may be used to account for interactions between atom pairs involved in 1-3 configurations (i.e., atoms bound to a common atom):

Evalence = Ebond + Eangle + Etorsion + Eoop + EUB

Modern (second-generation) forcefields include cross terms to account for such factors as bond or angle distortions caused by nearby atoms. Crossterms can include the following terms: stretch-stretch, stretch-bend-stretch, bend-bend, torsion-stretch, torsion-bend-bend, bend-torsion-bend, stretch-torsion-stretch.

The energy of interactions between nonbonded atoms is accounted for by van derWaals (EvdW), electrostatic (ECoulomb), and (in some older forcefields) hydrogen bond (Ehbond) terms:

Enonbond = EvdW + ECoulomb + Ehbond

Force field

f = a/r12 - b/r6 Van der Waals forcesdistance

ener

gy

The Lennard-Jones potential is mildly attractive as two uncharged molecules or atoms approach one another from a distance, but strongly repulsive when they approach too close. The resulting potential is shown. At equilibrium, the pair of atoms or molecules tend to go toward a separation corresponding to the minimum of the Lennard--Jones potential (a separation of 0.38 nanometers for the case shown in the Figure)

F = kqiqj/r2

Figure: Snapshots of ubiquitin pulling with constant velocity at three different time steps.

ERGODIC(l) of or relating to a process in which a sequence or sizable sample is equally representative of the whole (as in regard to a statistical parameter); (2) involving or relating to the probability that any state will recur, especially having zero probability that any state will never recur. (WEBSTER'S DICTIONARY) A collection of systems forms an ergodic ensemble if the modes of behavior found in any one system from time to time resemble its behavior at other temporal periods and if the behavior of any other system when chosen at random also is like the one system. We do not require identical performance, only quite similar time averages and number averages. (If you cannot tell one youth from another or one adult from another, they belong to an ergodic ensemble.) In an ergodic population, any single individual is representative of the entire population. The salient characteristics of this individual are essentially identical with any other member of the group. (Iberall)

#! /usr/bin/perl#=============================================================================== # # $Id: mcdemo.pl,v 1.1.1.1 2003/03/12 16:13:28 jkleinj Exp $ # # mcdemo: Demo program for MC simulation of the number pi # # (C) 2003 Jens Kleinjung# # Dr Jens Kleinjung, Room P440 | jkleinj@cs.vu.nl# Bioinformatics Unit, Faculty of Sciences | Tel +31-20-444-7783 # Free University Amsterdam | Fax +31-20-444-7653 # De Boelelaan 1081A, 1081 HV Amsterdam | http://www.cs.vu.nl/~jkleinj# #=============================================================================== # preset parameters $hits = 1; $miss = 1;

for ($i=0; $i<100000; $i++) {

# assign random x,y coordinates $x = rand; $y = rand;

# calculate radius $r = sqrt(($x*$x)+($y*$y));

# sum up hits and misses if ($r <= 1)

{ $hits++; } else

{ $miss++; }

# calculate pi $pi = (4*$hits)/($hits +$miss);

# print pi if ($i%100 == 0) { print("$i $pi\n"); }

}

#===============================================================================

In many conformational search methods based on Monte Carlo (MC),after a MC move, the system is energy minimised, i.e. put in the lowest local energy conformation, for example by gradient descent (steepest descent).

antibody HyHEL-63 (cyan) complexed with Hen Egg White Lysozyme

The X-ray structure of the antibody HyHEL-63 (cyan) uncomplexed and complexed with Hen Egg White Lysozyme (yellow) has shown that there are small but significant, local conformational changes in the antibody paratope on binding. The structure also reveals that most of the charged epitope residues face the antibody. Details are in Li YL, Li HM, Smith-Gill SJ and Mariuzza RA (2000) The conformations of the X-ray structure Three-dimensional structures of the free and antigen-bound Fab from monoclonal antilysozyme antibody HyHEL-63. Biochemistry 39: 6296-6309. Salt links and electrostatic interactions provide much of the free energy of binding. Most of the charged residues face in interface in the X-ray structure. The importance of the salt link between Lys97 of HEL and Asp27 of the antibody heavy chain is revealed by molecular dynamics simulations. After 1NSec of MD simulation at 100°C the overall conformation of the complex has changed, but the salt link persists. Details are described in Sinha N and Smith-Gill SJ (2002) Electrostatics in protein binding and function. Current Protein & Peptide Science 3: 601-614.