Fundamentals of Molecular Dynamics Simulations Fundamentals of MD Simulations Fundamentals of Molecular Dynamics Simulations Presented by: Mohamad Raad, Ph.D. Student Dec.10 , 2012 Tutors: Dr. H. Behnejad Faculty of Science, University of Tehran

Fundamentals of MD Simulations

Why MD simulations? Link physics, chemistry and biology Model phenomena that cannot be observed experimentally Understand protein folding… Access to thermodynamics quantities (free energies, binding energies,…)

Introduction Molecular dynamics is a form of simulations in which atoms and molecules are allowed to interact for a period of time by approximations of known physics, giving a view of the motion of the particles.   Main justification of the MD method is that statistical ensemble averages are equal to time averages of the system. MD simulation generates a sequence of points in phase space connected in time, result in trajectory of all particles. Use current positions, velocity and acceleration of atoms and find the position and velocity in next time step. We combine these infinitesimal steps to find the trajectory of the particle.

Steps in Performing an MD Simulation Selection of interaction model Selection of boundary conditions Selection of initial conditions (positions, velocities . . . ) Selection of ensemble (NVE, NVT, NPT . . . ) Selection of target temperature, density/pressure . . . Selection of integrator, thermostat, barostat . . . Perform simulation until equilibration is reached (property dependent) Perform production simulation to collect thermodynamic averages, positions, velocities Analyze the results via post-processing Initialize: Initial particle positions, velocities ... Calculate forces from particle positions Loop over time- steps Solve Newton’s eqns. of motion (integrate) Perform analysis / Write data to disk

Basic MD Simulation : The Idea Follow the exactly same procedure as real experiments Prepare sample prepare N particles solve equation of motions Connect sample to measuring instruments (e.g. thermometer, viscometer,…) after equilibration time, actual measurement begins Measure the property of interest for a certain time interval average properties Example : measurement of temperature

How does MD work Initialize Extract data Approach to equilibrium The algorithm of a standard MD simulation for studying systems in equilibrium is: Initialize Approach to equilibrium Extract data

Initialize Parameters in use: the number of particles, the form of interaction, the boundary condition, the temperature etc. Positions, e.g. the sites of a Bravais-fcc lattice Momenta, e.g. the Maxwell distribution

Approach to equilibrium The Verlet algorithm (original form): h: the time step, typically 10-15 ~ 10-14 seconds

Extract data The expectation value of a static physical quantity is determined as a time average according to The correlation function (in an isotropic system) is defined by n(r): the average number of pairs within [r, r+Δr]

MD as a tool for minimization Energy Molecular dynamics uses thermal energy to explore the energy surface State A State B position Energy minimization stops at local minima

Fundamentals of MD Simulations neighbor lists force calculation is the most time-consuming part in a MD simulation → save CPU time by using neighbor lists Verlet list: cell list: update when a particle has made a displacement > (rskin-rcut)/2 most efficient is a combination of Verlet and cell list 34

Equation of Motion Classical Newton’s equation of motion Three formulation Newtonian Lagrangian Hamiltonian Hamiltonian preferred for many-body systems solution of 2N differential equations Solution methods : Finite Difference Method

Newton‘s equations of motion Fundamentals of MD Simulations Newton‘s equations of motion classical system of N particles at positions Upot : potential function, describes interactions between the particles simplest case: pairwise additive interaction between point particles solution of equations of motion yield trajectories of the particles, i.e. positions and velocities of all the particles as a function of time 36

The Lagrange and Hamilton Forms Fundamentals of MD Simulations The Lagrange and Hamilton Forms These equations also apply to the center of mass motion of a molecule with fi representing the total force on molecule i. The generalized momentum pk conjugate to qk is defined as An alternative formulation uses Hamilton’s equations For Cartesian coordinates Hamilton’s equations become

What the integration algorithm does Fundamentals of MD Simulations What the integration algorithm does Advance the system by a small time step Dt during which forces are considered constant Recalculate forces and velocities Repeat the process If Dt is small enough, solution is a reasonable approximation

Verlet algorithm

A widely-used algorithm: Leap-frog Verlet Fundamentals of MD Simulations A widely-used algorithm: Leap-frog Verlet Using accelerations of the current time step, compute the velocities at half-time step: v(t+Dt/2) = v(t – Dt/2) + a(t)Dt v t-Dt/2 t t+Dt/2 t+Dt t+3Dt/2 t+2Dt

A widely-used algorithm: Leap-frog Verlet Fundamentals of MD Simulations A widely-used algorithm: Leap-frog Verlet Using accelerations of the current time step, compute the velocities at half-time step: v(t+Dt/2) = v(t – Dt/2) + a(t)Dt Then determine positions at the next time step: X(t+Dt) = X(t) + v(t + Dt/2)Dt v X t-Dt/2 t t+Dt/2 t+Dt t+3Dt/2 t+2Dt

A widely-used algorithm: Leap-frog Verlet Fundamentals of MD Simulations A widely-used algorithm: Leap-frog Verlet Using accelerations of the current time step, compute the velocities at half-time step: v(t+Dt/2) = v(t – Dt/2) + a(t)Dt Then determine positions at the next time step: X(t+Dt) = X(t) + v(t + Dt/2)Dt v v X t-Dt/2 t t+Dt/2 t+Dt t+3Dt/2 t+2Dt

Fundamentals of MD Simulations

CONSTANT TEMPERATURE (NVT) METHODS Andersén method Langevin thermostat Dissipative Particle Dynamics thermostat Nosé-Hoover method Nosé-Hoover Chain Method

CONSTANT PRESSURE AND TEMPERATURE (NPT) Andersén Pressure Control Parrinello-Rahman Pressure Control

simulations at constant temperature: a simple thermostat idea: with a frequency ν assign new velocities to randomly selected particles according to a Maxwell-Boltzmann (MB) distribution with the desired temperature simple version: assign periodically new velocities to all the particles (typically every 150 time steps) algorithm: take new velocities from distribution total momentum should be zero scale velocities to desired temperature

Where is the Physics? Potential energy functions Interaction force Cut-off distance Integrators Energy conservation Time reversibility Physical constraints Constant temperature Constant pressure

How do you run a MD simulation? For each time step: Compute the force on each atom: Solve Newton’s 2nd law of motion for each atom, to get new coordinates and velocities Store coordinates Stop X: cartesian vector of the system M diagonal mass matrix .. means second order differentiation with respect to time Newton’s equation cannot be solved analytically: Use stepwise numerical integration

MC vs. MD MC Probabilistic simulation technique Limitations require the knowledge of an equilibrium distribution rigorous sampling of large number of possible phase-space gives only configurational properties (not dynamic properties !) MD Deterministic simulation technique Fully numerical formalism numerical solution of N-body system

Disadvantages: MD simulations compared with MC simulations (II) More limited scale Difficult to be incorporated with quantum mechanics Advantages: Dynamical phenomena Nonequilibrium properties Structural properties

Scales in Simulations Time Scale 10-6 S 10-8 S 10-12 S 10-10 M 10-8 M continuum mesoscale Monte Carlo Time Scale 10-6 S molecular dynamics domain 10-8 S quantum chemistry exp(- D E/kT) 10-12 S F = MA Hy = Ey 10-10 M 10-8 M 10-6 M 10-4 M Length Scale

M. Allen, D. J. Tildesley, Computer Simulation of Liquids (Clarendon Press, Oxford, 1987) D. Rapaport, The Art of Molecular Dynamics (Cambridge University Press, Cambridge, 1995) D. Frenkel, B. Smit, Understanding Molecular Simulation: From Algorithms to Applications (Academic Press, San Diego, 1996) K. Binder, G. Ciccotti (eds.), Monte Carlo and Molecular Dynamics of Condensed Matter Systems (Societa Italiana di Fisica, Bologna, 1996)

Thank you for your attention! J. M. Thijssen, Computational Physics, Cambridge University Press (1999) T. Pang, An Introduction to Computational Physics, Cambridge University Press (1997) and the references therein. Thank you for your attention!