Enhanced sampling via molecular dynamics II: Unbiased approaches

Mark E. Tuckerman Dept. of Chemistry and Courant Institute of

Mathematical Sciences New York University, 100 Washington Square

East, NY 10003

Science, December 22, 2006

Spatial-Warping Transformations

P. Minary, MET, G. J. Martyna SIAM J. Sci. Comput. 30, 2055 (2007) Z. Zhu, MET, S. O. Samuelson, G. J. Martyna, Phys. Rev. Lett. 88 100201 (2002)

Molecular dynamics in a double well

Crossing the barrier is a rare event!

V(x)

time

x

. ( ) heat bath

pxm

dVp F xdx

=

= − = +

Spatial-warping transformations

V(x)

x a= − x a=

Hamiltonian: 2

( , ) ( )2pH x p V xm

= +

Canonical partition function: 2

exp ( )2pQ dp dx V xm

β = − +

∫Change the integration variables:

( )( ) x V y

au f x a dy e β−

−= = − + ∫

1( ( ))V f udx e duβ −

=

Transformed partition function:

( ) ( )2

1 1 exp ( ) ( )2pQ dp du V f u V f um

β − − = − + −

∫

Z. Zhu, MET, S. O. Samuelson, G. J. Martyna, Phys. Rev. Lett. 88 100201 (2002)

Spatial-warping transformations (cont’d)

x a= − x a=

( ) ( )

0 OtherwiseV x a x a

V x− < <

=

Choosing the transformation:

V(x) ( )V x ( ) ( )V x V x−

MD: ( ) ( )1 1 ( ) heat bath ( ) ( ) ( )pu p F um

dF u V f u V f udu

− − = − −= + =

( )V x( )V x

Reference potential spatial warping (REPSWA)

1 1( ( )) ( ( ))V f u V f u− −−

How is the space “warped”?

Barrier: Bk T

Barrier: 5 Bk T

Barrier: 10 Bk T

No Transformation Transformation

5kT

10kT

10kT

V‡

φ

ψ

ψ

φ

How to do biomolecules and polymers

Sampling hindered by barriers in the dihedral angles:

Ramachandran dihedral angles (alanine dipeptide):

( )V φ

P. Minary, MET, G. J. Martyna SIAM J. Sci. Comput. 30, 2055 (2007)

Dihedral angle transformations

Transformed angle: ( )V φ

min

( )min V

u d eφ β χ

φφ φ χ −= + ∫

minφ

tors 1 n.b. 4 2 4nn

( , ) ( ) ( ) (| ( , ) |) (| ( , ) |)i ii

V V S V Sφ φ φ α φ φ∈

= + − −∑r r r r r r r

P. Minary, G. J. Martyna and MET SIAM J. Sci. Comp. 30, 2055 (2008)

6r

A 400-mer alkane chain No nonbonded interactions 397 collective variables!

Nonbonded interactions off

RIS Model

value: 10

No Transformation REPSWA Transformations

β-Barrel Proteins

Honeycutt and Thirumalai PNAS 87, 3526 (1990)

MD

REMC

Transformations PT replicas = 16

Integrators in molecular dynamics

, ( )pq p F qm

= = Equations of motion:

3exp( ) exp ( ) exp exp ( ) ( )2 2t p tiL t F q t F q O t

p m q p ∆ ∂ ∂ ∆ ∂

∆ = ∆ + ∆ ∂ ∂ ∂

( ) (0) (0)( ) (0) (0)

iL tt

q t q qe

p t p pφ∆∆

∆ = = ∆

Time evolution:

where

Pseudocode: 0.5* * ;* ;

GetForce( );0.5* * ;

p p t Fq q t p

qp p t F

← + ∆← + ∆

← + ∆

( )piL F qm q p

∂ ∂= +

∂ ∂

Multiple time scale (r-RESPA) integration

fast slow

pqm

p F F

=

= +

fast slow ref slowpiL F F iL iLm q p p

∂ ∂ ∂= + + = + ∂ ∂ ∂

MET, G. J. Martyna and B. J. Berne, J. Chem. Phys. 97, 1990 (1992)

Multiple Time Scale Integration (cont’d)

3exp( ) = exp( / 2)exp( )exp( / 2) ( )

= exp( / 2) exp( ) exp( / 2)

slow ref slow

n

slow ref slow

iL t iL t iL t iL t O t

iL t iL t iL tδ

∆ ∆ ∆ ∆ + ∆

∆ ∆

Trotter factorized propagator:

exp( ) exp2

exp exp exp2 2

exp2

slow

n

fast fast

slow

tiL t Fp

t p tF t Fp m q p

t Fp

δ δδ

∆ ∂∆ = ∂

∂ ∂ ∂× ∂ ∂ ∂

∆ ∂ ∂

From: G. J. Martyna, MET, D. Tobias, and M. L. Klein Mol. Phys. 87, 1117 (1996).

Illustration of resonance 2 2 fast slowF x F xω= − = −Ω

2

2

(0) (0)2

'( ) (0)cos( ) sin( )

(0)cos( ) (0)sin( )

( ) ( )2

tp p x

px t x t t

p p t x ttp t p x t

ω ωω

ω ω ω

∆′ = − Ω

∆ = ∆ + ∆

′′ = ∆ − ∆∆′′∆ = − Ω ∆

( ) (0)( , , )

( ) (0)x t x

A tp t p

ω∆

= Ω ∆ ∆

A. Sandu and T. S. Schlick, J. Comput. Phys. 151, 74 (1999)

Illustration of resonance (cont’d) 2

2 4 22

1cos( ) sin( ) sin( )2

( , , )sin( ) cos( ) cos( ) sin( )

4 2

tt t tA t

t tt t t t t

ω ω ωω ω

ωω ω ω ω ω

ω ω

∆ Ω∆ − ∆ ∆

Ω ∆ = ∆ Ω ∆ Ω

− ∆ −∆ Ω ∆ ∆ − ∆

Depending on Δt, eigenvalues of A are either complex conjugate pairs

Note: det(A) = 1

2 Tr( ) 2A− < <

or eigenvalues are both real

| Tr( ) | 2A ≥Leads to resonances (|Tr(A)| → 2) at Δt = nπ/ω

Isokinetic dynamics

Constraint the kinetic energy of a system: 2 3 1

2 2i

i i

N kTm

−=∑ p

Introduce constraint via a Lagrange multiplier:

ii i i i

imλ= = −

pr p F p

Derivative of constraint yields multiplier:

[ ]/

0 /

i i ii i i

i i ii ii i i i i

i

m

m m mλ λ= − = ⇒ =

∑∑ ∑ ∑

F pp pp F p

p p

Partition function generated: 2

( )3 1 2 2

N N Ui

i i

Nd kT d em

βδ − −Ω = −

∑∫ ∫ rpp r

Resonance-free but is a poor thermostat.

Resonance-free stochastic isokinetic thermostat

Consider the equations of motion:

1, 1, 2, 1,

1,2, 2,

2

2 11

, 1,...,( )

( )

k k k k

kk k k

dq vdtFdv v dtm

dv v dt v v dt k LG v

dv dt v dt dwQ

G u Q u

λ

λ

γ σ

β −

=

= − = − − =

= − +

= −

Isokinetic constraint: 2 2 11 1, 1,1 1,

1

( , ,...., )1

L

k Lk

Lmv Q v v v v LL

β −

=

+ = Λ =+ ∑

Multiplier:

21 1, 2,

1

1,1 1,

1( , ,..., )

L

k kk

L

LvF Q v vLv v v

λ =

−+=

Λ

∑

P. Minary, MET, G. J. Martyna PRL (2004); B. Leimkuhler, D. Margul, MET Mol. Phys. (in press)

Multiple time-step integrator

Assume forces have fast and slow components: f sF F F= +

( ) ( )f sq v v N OUiL iL iL iL iL iL= + + + +

( ) ( ) ( )1,

1 1,

( ) ( ) ( )1,

1 1,

1,1, 2, 1,

1 1 11, 1, 2 2,

( )

q

Lff f f

v F F kk k

Ls s ss

v F F kk k

L L Lk

N N N k k kk k kk k k

iL vqF

iL v vm v v

FiL v vm v v

G viL v v v v

v v v Q v

λ λ

λ λ

λ λ

=

=

= = =

∂=

∂

∂ ∂= − − ∂ ∂

∂ ∂ = − − ∂ ∂

∂ ∂ ∂ ∂= − − − +

∂ ∂ ∂ ∂

∑

∑

∑ ∑ ∑

21 1, 2,

( ) ( ) 11, ,

L

k kff s s k

F F N

L Q v vvF vF Lλ λ λ =+= = = −Λ Λ Λ

∑

Multiple time-step integrator

2

2, 2,1(0) ( )

2OU

tiL t t

k kee v v e R t

γδδ γδ σ

γ

−− −

= +

( ) ( ) ( ) ( )/2 /2/2 /2 /2 /2 /2 /2s f f sq qN v v OU v v N

niL t iL tiL t iL t iL t iL t iL t iL t iL tiL te e e e e e e e e eδ δδ δ δ∆ ∆ ∆ ∆∆ =

Solution of the Ornstein-Uhlenbeck process:

Numerical illustration of resonance

2 49( ) 0.0252

U q q q= +

Harmonic oscillator with quartic perturbation

3, 100

tL tδ ∆= = [ ]

bins

exact1bins

1( ) ( ; ) ( )N

i ii

t P q t P qN

ς=

= −∑

q

P(q)

Flexible SPC/E water

intra nb intra short long( ) ( ) ( ) ( ) ( ) ( )U U U U U U= + = + +r r r r r r

2 2

max

short LJ

| | /4 2 2long 2

,| |

erfc( )( ) ( )

1 4( ) | ( ) || |

( )

i j ijij

ij ij

ik i

ii

i

q q rU U r

r

U e S qV

S q e

α

α

π απ

−

<

⋅

= +

= −

=

∑

∑ ∑

∑

k

k k

k r

r

r kk

k

( )

( )

(1)long recip res cut

(2)long recip max cut

erf ( )( ) ( ; , ) 1 ( )

erf ( )( ) ( ; , ) 1 1 ( )

i j ijij ij

i j ij

i j ijij ij

i j ij

q q rU U k r r

rq q r

U U k r rr

αα δ θ

αα δ θ

>

>

= + − −

= − − − −

∑

∑

r r

r r

RESPA schemes

F. Paesani et al. J. Chem. Phys. 125, 184507 (2006)

0.5 fs 3 fs ?t t Tδ = ∆ = ∆ =RESPA 1: L = 4, rcut = 6 Å

RESPA 2: L = 4, rcut = 8 Å

RESPA 1: L = 4, rcut = 6 Å

L = 4 (colors) L = 1 (dashed) L = 2 (circles)

Parallel Tempering

MT

1MT −

1T

( )( )( )( )

Z( , , )

KKU

KK

K

efN V T

β−

=r

r (1) ( ) ( )

1

( ,..., ) ( )M

M K

KF f

=

=∏r r r

( )

, 1

( 1) ( )( 1) ( ) ( ) ( 1) 1

( ) ( 1)1

( ) ( 1), 1 1

( ) ( )( , | , ) min 1, min 1,( ) ( )

( ) ( )

K K

K KK K K K K K

K KK K

K KK K K K

f fA ef f

U Uβ β

+

+−∆+ + +

++

++ +

= =

∆ = − −

r rr r r rr r

r r

Swendsen and Wang, Phys. Rev. Lett. 57, 2607 (1986)

Comparison for a 50-mer chain using CHARMM22 REMC/PT replicas = 10; 300 < T < 1000, 10 replicas 20% acceptance Nonbonded interactions off

Comparison for 50-mer using CHARMM22 all interactions

Comparison for 50-mer with all interactions PT replicas = 10; REPSWA α = 0.8; Every 10th dihedral not transformed

Replica exchange with “solute” tempering

A downside of replica-exchange is that U ~ N, therefore, the energy differences between neighboring replica need to shrink as the system size grows, which means more and more replica are needed in order for the acceptance probability to remain approximately fixed.

Replica exchange with solute tempering (REST) was developed to allow just part of the system (the “solute”) to be subject to attempted exchanges. It relies on the idea that different replicas can be subject to different potentials.

( )

( )Z( , , )

K KU

KK

efN V T

β−

=r

r

( ) ( )( ) ( 1) ( 1) ( ), 1 1 1 1( ) ( ) ( ) ( )K K K K

K K K K K K K KU U U Uβ β+ ++ + + +∆ = − + −r r r r

The latest version of REST (called REST2) uses the following ladder of potentials:

0 0

( ) ( ) ( ) ( )K KK s sv vU U U Uβ β

β β= + +r r r r

If the potential energy of a solute-solvent system is:

( ) ( ) ( ) ( )K s sv vU U U U= + +r r r r

P. Liu, B. Kim, R. A. Friesner, B. J. Berne PNAS (2005); L. Wang, R. A. Friesner, B. J. Berne J. Phys. Chem. B (2011)

Replica exchange with “solute” tempering

The ladder of potentials

allows each replica to be run at a single temperature T0 so that the solute probability

0 0( / ) ( ) ( )K s K sU Ue eβ β β β− −=r r

Acceptance probability determined by

( ) ( )( ) ( 1) ( ) ( 1)0, 1 1

1

( ) ( ) ( ) ( )K K K KK K K K s s sv

K K

U U U Uβ

β ββ β

+ ++ +

+

∆ = − − + −

+ r r r r

0 0 0( ) ( ) ( ) ( )K K s K sv vU U U Uβ β β β β= + +r r r r

For full potential ladder:

Solute potential energy

Convergence of alanine dipeptide in solution