Enhanced conformational sampling via very large time-step

molecular dynamics, novel variable transformations and

adiabatic dynamics

Mark E. Tuckerman

Dept. of Chemistry

and Courant Institute of Mathematical Sciences

New York University, 100 Washington Sq. East

New York, NY 10003

Acknowledgments

• Zhongwei Zhu • Peter Minary• Lula Rosso• Jerry Abrams

• NSF - CAREER• NYU Whitehead Award• NSF – Chemistry, ITR• Camille and Henry Dreyfus

Foundation

Students past and present Postdocs

• Dawn Yarne• Radu Iftimie

Collaborators

• Glenn Martyna• Christopher Mundy

Funding

Talk Outline

• Very large time-step multiple time scale integration that avoids resonance phenomena.

• Novel variable transformations in the partition function for enhancing conformational sampling.

• Adiabatic decoupling along directions with high barriers for direct computation of free energies.

Multiple time scale (r-RESPA) integration

fast slow

pr

mp F F

fast slow ref slow

piL F F iL iL

m r p p

3

x( ) exp( )x(0)

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

slow ref slow

t iL t

iL t iL t iL t O t

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

Resonance Phenomena

• Large time step still limited by frequency of the fast force due to numerical artifacts called resonances.

• Problematic whenever there is high frequency weakly coupled to low frequency motion

Biological Force Fields

Path integrals

Car-Parrinello molecular dynamics

Illustration of resonance

2 2 fast slowF x F x

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 t

tp t p x t

( ) (0)( , , )

( ) (0)

x t xA t

p t p

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 t

A tt t

t t t t t

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

Resonant free multiple time-scale MD

• Resonance means time steps are limited to 5-10 fs for most problems.

• Assign time steps to each force component based on intrinsic time scale.

• Prevent any mode from becoming resonant via a kinetic energy constraint.

• Ensure ergodicity through Nosé-Hoover chain thermostatting techniques.

P. Minary, G. J. Martyna and MET, Phys. Rev. Lett. 93, 150201 (2004).

Review of isokinetic dynamics

Constraint the kinetic energy of a system:

2 3 1

2 2i

i i

NkT

m

p

Introduce constraint via a Lagrange multiplier:

ii i i i

im

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 m

F pp pp F p

p p

Partition function generated:2

( )3 1

2 2N N Ui

i i

Nd kT d e

m

rp

p r

Review of Nosé-Hoover EquationsFor each degree of freeom with coordinate q and velocity v,

1

1

1

2

1,...,

1,..., 1

i

i i i

M

i

i

i

M

i

q v

Fv v v

mv i M

Gv v v i M

Q

Gv

Q

G Qv kT

New equations of motion (Iso-NHC-RESPA)Couple each degree of freedom to the first element of L NHCs of length M

2, 1,

1,

1, 1, 2 , 1,

, , 1,

,

2

1 2

,

,

1,...,

1,..., ; 2,..., 1

j j

j

j j j j

i j i j i j

M j

L M

j i

i j

M j

q v

Fv v

m

Qv vv

kT

v v v v j L

Gv v v j L i M

Q

Gv

Q

1,

1, 2 ,

2,

2

1

1,...,

1

i j

j j

i j

L

j

j L

G Qv kT

LvF Qv v

L

LkT

Ensures the constraint: 1,

2 2

1

( , )1 j

L

j

LK v v mv Qv LkT

L

is satisfied.

Classical non-Hamiltonian statistical mechanics

x (x)General equations of motion:

Consider a solution:

0x x (x )t t

If the equations are non-Hamiltonian.Κ(x) called the compressibility of the equations.

(x) x= (x) 0

In order to generalize Liouville’s theorem, we need to determine:

00

x(x , x )

xt

tJ

Tuckerman, Mundy, Martyna, Europhys. Lett. 45, 149 (1999); Tuckerman, et al. JCP 115, 1678 (2001).

Classical non-Hamiltonian statistical mechanics

Tr(ln M)0(x , x ) det(M)tJ e

Tr(ln M) 1 10 0

M M(x , x ) Tr M (x , x )Tr Mt t

d d dJ e J

dt dt dt

1 0

0 0

x x x

x x x

i i iijt t

ij ijj j jt

dMM M

dt

1 0 0

, , ,0 0

x x x x xMTr M

x x x x x

i j i j kt t t

j i j k ii j i j kt t t

d

dt

10

, , , , ,0

x x x x x xx = (x )

x x x x x x

k i j j j jt t t t t

ki ij kj t ti j k k k ji j k i j k j k jt t t t t

M M

Classical non-Hamiltonian statistical mechanics

0 0(x , x ) (x ) (x , x )t t t

dJ J

dt 0 0(x , x ) 1J

Solution: 0 (x )

0(x , x )t

sds

tJ e

Note that for Hamiltonian systems, κ(x)=0 and J(xt,x0)=1.

Define: (x ) (x , )t t

dw t

dt

Then: 0(x , ) (x ,0)0(x , x ) tw t w

tJ e

Whence:0 0x (x , x ) x t td J d 0(x , ) (x ,0)

0x xtw t wte d e d

Classical non-Hamiltonian statistical mechanics

Define a metric factor:(x , )(x , ) tw t

tg t e

In addition, suppose the dynamical equations have Nc conservationlaws of the form:

(x ) (x ) 0 1,...,k t k t c

dC k N

dt

Then, the dynamical system, assuming ergodicity, will generatea “microcanonical” ensemble whose partition function is:

11

( , , ,..., ) (x) (x) xc

c

N

N k kk

N V C C g C d

Also, assume equilibrium conditions, i.e. that (x , ) (x )t tg t gand the phase space distribution has no explicit time dependence.

Phase space distribution

( ) /(x) U kTg e r

( )(x)

U

kT

r

3

( )3 ( ) ( ) ( )

1

( , N

K vN NLM k k N U

k

d d v e K v v LkT d e

rv r

For the Iso-NHC-RESPA method:

Metric Factor:

For the present system:

Integration of the equations

1,

1,

NHC1

1

s

s

s j

j

N

q vs

q

L

v s s sj

iL iL iL iL

iL vq

iL F v vv v

21

2 1 2

,,1NHC 1 NHC

22

/ 2/ 2/ 2 / 2

1 1 1

T TN Ns s

k

v s sqvk s

nniL t iL tiL t iL t iL tiL t

iL w tiL tiL tiL t iL t iL t

s s s

e e e e e e

e e e e e e

w n w w

Liouville operator decomposition:

Factorized propagator:

Numerical illustration of resonance

2 49( ) 0.025

2U x x x

Harmonic oscillator with quartic perturbation

3 4 100

tL M t

bins

exact1bins

1( ) ( ; ) ( )

N

i ii

t P x t P xN

Flexible TIP3P water2

bond bend 450 kcal mol A 55 kcal molk k

1 2 3 0.5 fs 3 fs ?t t t Intramolecular forces

Short-range forces cutoff = 5Å

Long-range forces10 Å + Ewald

HIV-1 Protease in vacuo

1 2 3 0.5 fs 3 fs ?t t t

1.5 2.5 3.5 4.5

rCH (A)

g(r)

0.9 1.0 1.1 1.2

Conformational sampling in Biophysics

• “Ab initio” protein/nucleic acid structure prediction: Sequence → Folded/active structure.

• Enzyme catalysis.

• Drug docking/Binding free energy.

• Tracking motion water, protons, other ions.

Unfolded State

Native State

Misfolded State

The conformational sampling problem

• Find low free energy structures of complex molecules

• Sampling conformations described by a potential

function: V(r1,…,rN)

• Protein with 100 residues has ~1050 conformations.

• “Rough free energy landscape” in Cartesian space.

• Solution: Find a smoother space in which to work.Z. Zhu, et al. Phys. Rev. Lett. 88, art. No. 100201 (2002)P. Minary, et al. (in preparation)

REPSWA (Reference Potential Spatial Warping Algorithm)

No Transformation Transformation

Barrier Crossing Transformations (cont’d)

‘

‘

‘

‘

Vref(Φ)

A 400-mer alkane chain

RIS Model value: 10

No Transformation Transformation

Model sheet protein No TransformationParallel TemperingDynamic transformation

No TransformationsTransformations

L. Rosso, P. Minary, Z. Zhu and MET, J. Chem. Phys. 116, 4389 (2000)

)0()exp()(

)()(

)()(~

1

11

11

1

xiLttx

TiLTiL

pqF

qm

p

pqF

qm

piL

thth

kk

kk

k

k

Conformational sampling of the solvated alanine dipeptide

ψφ

AFED Tφ,ψ = 5T, Mφ,ψ = 50MC 4.7 nsUmbrella Sampling 50 ns

CHARM22αR

β

[L Rosso, J. B. Abrams and MET (in preparation)]

Conformational sampling of the gas-phase alanine dipeptide

ψφ

AFED Tφ,ψ = 5T, Mφ,ψ = 50MC 3.5 nsUmbrella Sampling 35 ns

CHARM22

β

Conformational sampling of the gas-phase alanine tripeptide

AFED Tφ,ψ = 5T, Mφ,ψ = 50MC 4.7 nsUmbrella Sampling 50 ns

β

Cax7

φ1

ψ1ψ2

φ2

Conformational sampling of the solvated alanine tripeptide

Closed ~ 5Å

Open ~15Å

R

• Protonation state of the active site important in drug binding

RIS Model value: 14

No Transformation Transformation

Water Number Density (Å )

Protease alone: 0.024 Protease + drug: 0.015

Protease alone

Protease + drug

Z. Zhu, D. I. Schuster and MET, Biochemistry 42, 1326 (2003)

Avg. cavity dimensions (Å)HeightWidth

PR alone 20.7 12.3PR + drug 19.2 17.3PR + Saq. 20.2 15.1

Bulk water: 0.033

-3

Conclusions• Isokinetic-NHC-RESPA method allows time steps as large as 100 fs to be used in

typical biophysical problems.

• Variable transformations lead to efficient MD scheme and exactly preserve partition function.

• Speedups of over 106 possible in systems with many backbone dihedral angles.

• Trapped states are largely avoided.

• Future: Combine variable transformations with Iso-NHC-RESPA

• Future: Develop variable transformations for ab initio molecular dynamics, where potential surface is unknown.