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. 140, 1 (1998)
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) → ∞) 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. (submitted)
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)For each degree of freeom with coordinate q and velocity v,
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.
Phase space distribution
x (x) (x)k kk
d g C
x= (x)
x (x)d g
gdt
3
( )3 ( ) ( ) ( )
1
( , N
K vN NLM k k N U
k
d d v e K v v LkT d e
rv r
General non-dissipative non-Hamiltonian dynamics:
General “microcanonical” partition function:
Phase space metric:
For the present system:
MET, C. J. Mundy, G. J. Martyna, Europhys. Lett. (2000)
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
Flexible TIP3P water2
bond bend 450 kcal mol A 55 kcal molk k
1 2 3 0.5 fs 3 fs ?t t t
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
[J. Abrams, L. Rosso 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
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.
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