Chapter 7Molecular Dynamics

Molecular dynamics

Studies the time evolution of an energy-minimized molecular system and allows the prediction of thermodynamics and transport properties directly from the underlying interactions between the atoms

Molecular dynamics simulations generate information at the microscopic level, including atomic positions and velocities.

The conversion of this microscopic information to macroscopic observables such as pressure, energy, heat capacities, etc., and transport properties requires statistical mechanics.

MD Simulation

Microscopic information

Macroscopic thermodynamics and transport properties

Predictions using Statistical Mechanics

Calculation using Newton’s second law

• Molecular dynamics calculates the “real” dynamics, i.e. behavior of the system, from which the time averages of the system’s properties can be calculated.

• Molecular dynamics is a deterministic method, which means that the state of the system at any future time can be predicted from its current state.

• Treat the atoms as classical particles.

2

2

dt

RdmmaF

F = Force on each atom

DIFF. EQN: SOLVE NUMERICALLY TO DETERMINE THE TRAJECTORIES OF THE ATOMS ON THE PES

PotEF

NEWTON’S EQUATION OF MOTION

Numerical Integration methods

Used for solving the previous equation numerically:

•Verlet algorithm

•Leap-Frog algorithm

•Velocity-Verlet algorithm

And many more….

Verlet algorithmThe most widely used method in molecular dynamics.

It uses the positions for all atoms at time t, and the positions from the previous step at time (t-δt) to calculate the forces at present time and then new positions at time (t+δt)

The velocities do not explicitly appear in the Verlet algorithm but they can be calculated

r

v

F

t-t t t+t

Given current position and position at end of previous time step

r

v

F

t-t t t+t

Compute the force at the current position

r

v

F

t-t t t+t

Compute new position from present and previous positions, and present force

r

v

F

t-2t t-t t t+t

Advance to next time step,repeat

Leap-Frog algorithmThe leap-frog method is the variation of Verlet algorithm.

It uses the positions at time t, and the velocities from the previous half step at time (t-½δt) to calculate the force at present time t and the velocities at next half step at time (t+½δt)

The name of this method comes from its nature, i.e., velocities make ‘leap-frog’ jumps over the positions to give their values at tt

2

1

r

v

F

t-t t t+t

Given current position, and velocity at last half-step

r

v

F

t-t t t+t

Compute current forcer

v

F

t-t t t+t

Compute velocity at next half-step

r

v

F

t-t t t+t

Compute next positionr

v

F

t-2t t-t t t+t

Advance to next time step,repeat

Velocity-Verlet algorithm

The velocity Verlet algorithm gives positions, velocities and accelerations at the same time t to compute this quantities at later time

It does not compromise precision

r

v

F

t-t t t+t

Given current position, velocity, and force

r

v

F

t-t t t+t

Compute new positionr

v

F

t-t t t+t

Compute velocity at half stepr

v

F

t-t t t+t

Compute force at new positionr

v

F

t-t t t+t

Compute velocity at full stepr

v

F

t-2t t-t t t+t

Advance to next time step,repeat

An experiment is usually made on a macroscopic sample that contains an extremely large number of atoms or molecules sampling an enormous number of conformations.

In statistical mechanics, averages corresponding to experimental observables are defined in terms of Ensemble averages;

Meaning of Ensembles

• NVE = number of atoms (N), system volume (V) and potential energy (E) remain constant during the simulation

• NVT = number of atoms (N), system volume (V) and temperature (T) remain constant during the simulation

• NPT = number of atoms (N), pressure (P) and temperature (T) remain constant during the simulation

MD in different thermodynamic ensembles

• MD naturally samples the NVE (microcanonical ensemble): this is not very useful for most applications.

• More useful ensembles include:– Canonical (NVT): requires the particles to

interact with a thermostat– Isobaric-Isothermal (NPT): requires the

particles to interact with a thermostat and a barostat

Thermostats

• Several possible approaches – Berendsen Method: velocities are exponentially scaled

by a factor at each time step so that the system is forced towards the desired temperature

– Nosè and Nosè-Hoover: an additional degree of freedom is added to the system, so that the system is forced towards the desired temperature

– Anderson Method: random collisions take place between the atoms in the system and the heat bath, so that the system is forced towards the desired temperature

Barostats

• They work similarly to thermostats to control pressure. For example:

– Berendsen Method: velocities are exponentially scaled by a factor at each time step so that the system is forced towards the desired pressure

Practical aspects of computer simulations

• Setting up a simulation– Define which energy model (force field) should be used to describe

atomic interactions;

• Choosing initial configurationNeeds careful design – very important phase of the experiment– It may come from the experimental data (x-rays, PDB files)– You can build it from scratch

• Running a simulation– Set up initial configuration– Equilibration phase by doing Molecular Mechanics

• System evolves from initial configuration which may not correspond to the energy minimum

– Defining simulation length and time per step T– Production phase by running the Molecular Dynamics calculation

• Thermodynamic properties of the system are calculated and transport properties predicted. Trajectory file is saved

– Data analysis

What is done internally by the program?

1. Given initial positions, initial velocities are assigned

2. Forces on every atom are computed

3. Advance the system by a time step t

4. Recalculate forces and velocities in the new step

5. Repeat the process in the direction of minimal forces

CAUTION: ONLY if t is small enough, the output of the simulation is reasonable

What T should I use???

Typical ranges for molecular motion:

Description Time scale Amplitude (Å)

Bond stretching, angle bending

Femto to pico seconds10-15 - 10-12s

0.001 - 0.1

Collective motion Pico to nano seconds10-12 - 10-9s

0.1 - 10

Peptide folding Nano to micro seconds10-9 - 10-6s

1 - 100

Protein folding Micro to seconds10-6 - 10s

10 - 100

Choosing a time step t

• Not too short so that conformations are efficiently sampled

• Not too long to prevent wild fluctuations or system ‘blow-up’

• An order of magnitude less than the fastest motion is ideal:

–Usually bond stretching is the fastest motion:

C-H is ~10fs so use time step of 1fs

–Not interested in these motions?

Constrain these bonds and double the time stepSuggested time steps:

• Rigid Translation, 10 fs• Flexible molecules and rigid bonds, 2fs• Both flexible molecules and bonds, 1fs

• An equilibration period is ALWAYS needed– lets system “forget” artificial initial configuration– length of period depends on relaxation time of system

Running molecular dynamics

Relaxation cycles

(run a MM calculation first)

Production cycles

(this is your MD simulation)

Start your MD here

Calculation of Thermodynamics Properties

• Energy• Heat Capacity• Pressure• Temperature

Energy–The internal energy is easily obtained from a simulation as the average of

the energies of the states that are examined during the course of the simulation:

M

iiE

MEU

1

1

Heat capacity–The heat capacity is formally defined as the partial derivative of the internal

energy with the respect to temperature:

Two ways for calculating CV:

• by performing a series of simulations at different temperatures, and then differentiating the energy with respect to the temperature.

• from a single simulation by considering the instantaneous fluctuations in the energy.

(6.7)

VV T

UC

(6.8)

M is the number of steps of the simulation

Pressure– The pressure is usually calculated through fij, the force acting between

atoms i and j, then we have the following expression for the pressure:

N

i

N

ijijijB frTNk

VP

1 13

11(6.13)

Temperature

– In a canonical (NVT) ensemble the total temperature is constant, and it is whatever you set.

– In the microcanonical ensemble (NVE), the temperature fluctuates. The temperature is directly related to the kinetic energy (K) of the system as follows:

N

i

B

i

i NTk

m

pK

1

2

)3(22

According to the “equipartition theorem” energy of each degree of freedom contributes as kBT/2. If there are N particles, each with three degrees of freedom, then the kinetic energy should equal to 3NkBT/2.

(6.14)

Prediction of Time Dependent Properties

• Diffusion Coefficient• Velocity Autocorrelation function• Radial distribution function

Diffusion• consider tagging molecules and watching how they

migrate

(Self-diffusion=Diffusion within a pure substance)

Diffusion equations– Combine mass balance with Fick’s law, which gives how

concentration is changing with r and t

– This can be related to the Mean square displacement (<r2(t)>)

Diffusion

Interpretation of Diffusion

– Asymptotic linear behavior of mean-square displacement gives diffusion constant

– Independent data can be collected for each molecule

2 ( ) 2r t dDt

Einstein equation:

2r

t

dD

curve

the of part

this of Slope

2

d = dimension of the movement (d =3 for 3-D movement)

<r2(t)>= mean square displacement D= diffusion coefficient

Velocity Autocorrelation Function (VAC)

0

1(0) ( )D d

d

v v

Backscattering

Definition: VAC gives how the velocity of an atom remains correlated to its initial velocity as time evolves

•Typical behavior:

Zero slope (soft potentials)

Diffusion coefficient is area under the curve:

Asymptotic behavior (t) is nontrivial (depends on d)

VAC

Time

D= diffusion coefficient<v(0).v(t)>=VAC(t)

Calculating the Diffusion Coefficient (D): Two options

1. Using the mean square displacement (<r2(t)>):

2. Using the velocity autocorrelation function (VAC(t)):

0

)(1

dt tVACd

D

)(2

1 2 trdt

D

d = dimension of the movement (d =3 for 3-D movement)

Radial Distribution Function

• Radial distribution function, g(r)– key quantity in statistical mechanics– quantifies correlation between pair of atoms

• Definition:

( )( )

( )id

r dg r

r d

r

r

Number of atoms at r for ideal gas

Number of atoms at r in actual system

( )id Nr d d

V r r

dr4

3

2

1

0

543210

Hard-sphere g(r) Low density High density

g(r), gives the probability of finding an atom (or molecule, if simulating a molecular fluid) at distance r from another atom (or molecule) compared to the ideal gas distribution