Minimization v.s. Dyanmics

• A dynamics calculation alters the atomic positions in a step-wise fashion, analogous to energy minimization. However, the steps in molecular dynamics meaningfully represent the changes in atomic positions over time.

• The individual steps in energy minimization are merely directed at establishing a down-hill direction to a minimum.

minimum

Minimization v.s. Dynamics

saddle pointmaxima

Understanding the Mechanisms

The Basics of Molecular Dynamics

Recall Newton’s equation of motion given E(r):

A dynamics trajectory can tell us how a processinvolves over time, i.e. kinetics.

Time Integration Algorithms

• Dynamics simulation is intrinsically chaotic. The finite accuracy of any computer program on any computer hardware will make any trajectory deviate from analytical result on long time scales.

• This is actually very helpful for thermodynamics analysis.

• The important requirement is the conservation of energy and momentum.

How about Thermodynamics?

• Boltzmann’s Law

n = n0 exp(-U/kBT)

U: potential energy; n: number density of molecules; n0 is the reference number density at zero energy.

This can be generalized to (E = U + K):

How about Thermodynamics?

Dynamics Averaging v.s. Thermodynamic Averaging

• Basic principle: a molecular dynamics trajectory visits all possible places (r) and velocities (p) with probability proportional to exp(-E/kT)

if long enough simulation is performed.• Thermodynamic average is over all (p,r) points

at a single time. Dynamic average is over a single (p,r) point at a time but over all times.

• The agreement of the two averages is the ergodic hypothesis.

Kinetic Averaging from Dynamics

• Assumption: the long-time events can be described even if the dynamics algorithm is intrinsically chaotic.

• Given that a single trajectory describes the behavior of a single molecule, kinetics will be obtained by simply running numerous independent trajectories. This is essentially to simulate a measurement in a test tube.

• This can be verified when compared with high-quality kinetics measurements.

Kinetics – A Case Study: Actin Filament Polymerization

• Actin filaments are polymers that drive cell shape changes, cell locomotion.

• Actin filaments also participate in muscle contraction.

Actin Filaments Assembly

• Self-assembly--not catalyzed by enzyme, rate is diffusion limited

• Assembled from identical monomeric actin subunits

The Two Ends Grow at Different Rates

When short myosin S1-decorated filaments are the nuclei for actin polymerization, the resulting elongated filaments have much longerundecorated (+) end, indicating faster monomer addition at that end (5- to 10-fold faster).

Asymmetry in Actin Polymerization

• Method: Simulations of binding of a monomer to the end of a filament over a range of controlling parameters with Brownian dynamics

• Analysis: Computation of rate constants for binding by keeping track of the number of successful binding events at each end of the filament.

Asymmetry in Actin Polymerization

Asymmetry in Actin Polymerization

Practical Aspects of Dynamics

• Starting a simulation • Controlling the system

(temp, press, density)• Equilibrating• Looking at the atoms

Starting a Simulation

• Starting from scratch: Positions are usually from a crystal structure (NMR structures are becoming common) solvated in a water box. Velocities are assigned from a Maxwell distribution at a certain temperature T.

• Continuing a simulation: Make sure both positions and velocities are used.

Controlling the Simulation

• The density is controlled by the choice of the box volume V.

• The temperature can be measured and controlled by rescaling the velocities.

• The pressure can be measured during the run and controlled by rescaling the volume.

Temperature Control

• T(t) is computed from the system kinetic energy

• No longer follow Newtonian trajectory. Energy is no longer conserved.

Reaching Equilibrium

• Indicators of the state are not stationary (that is, fluctuating around a fixed value), but relaxing towards a new value (that is, fluctuating around a value that is slowly drifting with time).

• In all cases, we usually want equilibrium to be reached before performing measurements on the system.

First thing first, you should at least watch your molecules evolve over time!

Initial Analysis

Initial Analysis

• Constant temperature simulations should have stable temperature, i.e. not drifting to a different temperature than specified.

• At constant temperature, the potential energy will only fluctuate around a constant if an equilibrium state is reached. In reality it most likely goes down even after the initial equilibrium. This is because the system has not reached the nearest energy basin, or there exists discrepancy between theory and experiment.

• Root Mean Squared Displacement. How far is your protein drifting away from the crystal structure? Note only consider backbone atoms that are not in contact with other molecules (crystal packing), or are poorly defined.