Molecular dynamics (1)

Principles and algorithms

Equations of motion

2

00

00

2

2

2

2

)(2

1)()(

,,2,1),(

,,2,1,)(1)(

)(

ttttttt

tt

nitdt

dx

V

dt

xdm

nitVmm

tt

dt

d

dt

d

ii

i

i

ii

ii

iii

avrr

vvrr

vr

rrF

avr

r

Solving the equations of motions results in a microcanonical ensemble (energy is conserved).

Lyapunov instability of trajectoriesFor 3- and more body systems interacting via central forces, an infinitesimably small perturbations of the initial conditions results in FINITE trajectory change after sufficiently long time

Simplistic (Euler) algorithm

2

3

0

0

2

)()(

00

)(1)(

)(

)(

)(2

1)()(

ttEtte

tt

tVmm

tt

ttttt

ttttttt

ii

ii

ii

ii

iii

iiii

i

OO

vvrr

rrF

a

avv

avrr

r

The Verlet algorithm: derivation:

)()()(

...2

)()()(

...)()()(2)(

...)()(2)()(

...)(2

1)()()(

...)(2

1)()()(

3

4

2

2

2

2

ttEtte

t

ttttt

ttttttt

ttttttt

ttttttt

ttttttt

OO

rrv

arrr

arrr

avrr

avrr

The velocity-Verlet algorithmStep 1:

tttt

t

ttttttt

)(2

1)(

2

)(2

1)()( 2

avv

avrr

Step 2:

tttt

ttt

ttUm

tti

ii

)(2

1

2)(

)(1

)(

avv

ra r

Relation to the Verlet algorithm

2

22

)(2

1)(2

)(2

1)()()(

2

1)()(

ttt

tttttttttt

tttt

ar

avravr

rr

The leapfrog algorithm

tt

tttt

ttt

tt

t

2)()(

)(22

vrr

avv

Relation to the Verlet scheme

ttt

tt

tttt

tt

tttt

)()(22

)(2

)(

)()(

ar

vrvr

rr

The three algorithms discussed are variants of the same algorithm.

All three algorithms are reversible in time; if run backward for the same time they restore the starting point.

All these three algorithms have the symplectic property: the total energy oscillates about a value close to the initial total energy (the shadow Hamiltonian). Higher-order algorithms (e.g., the Gear algorithm don’t have this property.

Kinetic energy

Potential energy

Total energy

Total energy

0.0 1.0 2.0 3.0 4.0 5.0

Ene

rgy

[kca

l/mol

]

time [ns]

Time dependence of the potential, kinetic, and total energy of the Ac-Ala10-NHMe (Khalili et al., J. Phys. Chem. B, 2005, 109, 13785-13797)

3,2,1,0 n

dt

trdtrtrtr

n

n

n

The Gear predictor-corrector algorithm (4th order)

c0=3/8, c1=1, c2=3/4, c3=1/6: correction coefficients;

Verlet

Gear4th order

Gear5th order

Gear6th order

Energy error for various integration algorithms

MD simulation procedure

1.Generate a low-energy initial configuration (minimize the potential energy of the system).

2.Generate initial velocities of the atoms.

3.Run simulation; monitor the properties that need to be (approximately) conserved.

zyxa

NiN

m

RTv

iia

,,

,...,2,1,1,0

2

10-15

femto10-12

pico10-9

nano10-6

micro10-3

milli100

secondsbond vibration

loopclosure

helixformation

folding of-hairpins

proteinfolding

all atom MD step

sidechainrotation

MD Package

Explicit Solvent

Implicit Solvent

AMBERa 1 fs(20 fs on ANTON; good symplectic

algorithms)

2 fs

CHARMMb

3 fs 4-5 fs

TINKERc

1 fs 2 fs

Time step t for some standard MD packages

a http://amber.scripps.edu/b http://www.charmm.org/c http:// dasher.wustl.edu/tinker/

Why are the Verlet-like algorithms symplectic?

ttf NN rp ,

We consider an arbitrary function that depends on coordinates and momenta.

p

pr

rrprp

ff

ttfttfdt

d NNNN ,,

pp

rr

Li ˆ

We define the Liouville operator:

Its time derivative is

Fvp

pvr

mm

Thus f(t) can formally be written as:

0,0ˆexp, NNNN ftLittf rprp

If we consider only the first part:

NNNN

p

NN

n

NN

rr

r

tftftf

Li

tm

f

tfft

ftLi

tfLiftf

Li

00,000,0

0ˆ

00,0

00,000exp

...0!2

ˆ0ˆ0

0ˆ

0

2

Fprpprp

p

prp

rrpr

r

rr

For the second part:

However, the two parts of the Liouville operator don’t commute and, consequently

prpr LiLiLiLi ˆexpˆexpˆˆexpexp

pp

rr

n

tt

Lt

iLtiLt

i

LitLitn

prpn

pr

ˆ

2expˆexpˆ

2explim

ˆˆexp

However, we have (the Trotter identity):

prpprr

nn

n

pr

prp

LiLiLiLiLiLic

ct

LitLit

Lt

iLtiLt

i

ˆˆ,ˆ12

1ˆˆ,ˆ24

1

ˆˆexp

ˆ2

expˆexpˆ2

exp

,,3

112

12

Now we apply the approximate Liouville operator to positions and momenta:

ttt

tf

ttt

tfttfLt

i

ttfLt

iLtiLt

i

ttfLitLit

NN

N

NN

NNp

NNprp

NNpr

rrFp

rpprp

rp

rp

,2

,2

,ˆ2

exp

,ˆ2

expˆexpˆ2

exp

,ˆˆexp

Step 1:

Step 2:

N

NN

NN

NN

r

tttt

mtt

ttf

tt

tttt

tf

ttt

tfLti

2

1,

2

2,

2

,2

ˆexp

2

rFprpp

rrpp

rpp

Step 3:

N

N

NNN

N

N

N

N

N

N

N

p

tttt

mt

tttt

t

f

tttt

mt

tttt

t

f

tttt

mt

tt

fLt

i

2

1

,2

2

1

,22

2

1

,2ˆ

2exp

2

2

2

rFpr

rFrFp

rFpr

rFp

rFpr

p

This is exactly the velocity-Verlet algorithm

22

222

t

m

tttttt

ttttttt

t

m

tt

tt

NNN

NNNN

NNN

rFvv

rFvrr

rFvv

By splitting the exponent of the Liouville operator another way we obtain the leapfrog algorithm

ttfL

tiLtiL

ti

ttfLitLit

NNrpr

NNpr

rp

rp

,ˆ2

expˆexpˆ2

exp

,ˆˆexp

Establishing the time step

safe = very small = very small progress

large = flying blind=risk

French Alps „Col de Braus-small” author Ericd. License CC BY-SA 3.0

Crude solution:

In a given time step, we reduce Dt until the change acceleration is sufficiently small

DISADVANTAGE: time reversibility is lost

Refined solution: time-split algorithms

Identify the forces FL that vary „slowly” (e.g., electrostatic forces) and FS that vary „fast” (e.g., the sort-range repulsive forces). Then write the Liouville operator as follows:

SL ppr

SL

LiLiLi

Li

ˆˆˆ

ˆ

p

pp

pr

r

Then we split the Liouville operator in the following way:

forces varying-slow using

t/2at tvelocities

calculate

ˆ2

exp

forcesvarying-fast and t/2mstep with time

stepsVerlet - velocityeconsecutiv

ˆ2

expˆexpˆ2

exp

forces varying-slow using

tat tvelocitiescalculate

ˆ2

exp

ˆexp

LSSL p

m

prpp Lt

i

m

Lm

tiL

m

tiL

m

tiL

ti

Lti

This algorithm is time-reversible if the splitting number m is not changed during the course of the simulation.

References to integration algorithms1. Frenkel, D.; Smit, B. Understanding molecular simulations,

Academic Press, 1996, chapter 4.

2. D.C. Rapaport. The art of molecular dynamics simulation, Cambridge University Press, 1995.

3. Calvo, M. P.; Sanz-Serna, J. M. Numerical Hamiltonian Problems; Chapman & Hall: London, U. K., 1994.

4. Verlet, L. Phys. Rev. 1967, 159, 98.

5. Swope, W. C.; Andersen, H. C.; Berens, P. H.; Wilson, K. R. J. Chem. Phys. 1982, 76, 637.

6. Tuckerman, M.; Berne, B. J.; Martyna, G. J. J. Chem. Phys. 1992, 97, 1990.

7. Ciccotti, G.; Kalibaeva, G. Philos. Trans. R. Soc. London, Ser. A 2004, 362, 1583.

Temperature control (Berendsen thermostat)

n

iziyixiik

k

vvvmE

E

fkTtvv

1

222

2

1

11

f – #degrees of freedom (3n)

t – coupling parameter

Dt – time step

Ek – kinetic energy

: velocities reset to maintain the desired temperature

: microcanonical run

1

Pressure control (Berendsen barostat)

n

i

n

ijijiij

ext

Vp

ppt

vL

1 1

3

1

3

1

1

vrrF

L – the length of the system (e.g., box sizes)

b – isothermal compressibility coefficient

t – coupling parameter

Dt – time step

pext – external pressure