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Biomolecular dynamics at long timesteps: Bridging the timescale gap between simulation and experimentation

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Annu. Rev. Biophys. Biomol. Struct. 1997. 26:181–222Copyright
c
1997 by Annual Reviews Inc. All rights reserved
BIOMOLECULAR DYNAMICS ATLONG TIMESTEPS: Bridging theTimescale Gap Between Simulationand Experimentation
Tamar Schlick, Eric Barth, and Margaret Mandziuk
The Howard Hughes Medical Institute and New York University, Department of Chemistry and Courant Institute of Mathematical Sciences, 251 Mercer Street,New York, NY 10012
KEY WORDS: molecular and Langevin dynamics, numerical integration, linearized models,normal modes, multiple timesteps
A
BSTRACT
Innovative algorithms have been developed during the past decade for simulatingNewtonian physics for macromolecules. A major goal is alleviation of the severerequirement that the integration timestep be small enough to resolve the fastestcomponents of the motion and thus guarantee numerical stability. This timestepproblemischallengingifstrictlyfastermethodswiththesameall-atomresolutionat small timesteps are sought. Mathematical techniques that have worked well inother multiple-timescale contexts—where the fast motions are rapidly decayingor largely decoupled from others—have not been as successful for biomolecules,where vibrational coupling is strong.This review examines general issues that limit the timestep and describesavailable methods (constrained, reduced-variable, implicit, symplectic, multiple-timestep, and normal-mode-based schemes). A section compares results of se-lected integrators for a model dipeptide, assessing physical and numerical perfor-mance. IncludedisourdualtimestepmethodLN,whichreliesonanapproximatelinearization of the equations of motion every
t
interval (5 fs or less), the so-lution of which is obtained by explicit integration at the inner timestep
τ
(e.g.,0.5 fs). LN is computationally competitive, providing 4–5 speedup factors, andresults are in good agreement, in comparison to 0.5 fs trajectories.These collective algorithmic efforts help ﬁll the gap between the time rangethat can be simulated and the timespans of major biological interest (millisec-onds and longer). Still, only a hierarchy of models and methods, along with
1811056-8700/97/0610-0181$08.00
182
SCHLICK, BARTH & MANDZIUKexperimentational improvements, will ultimately give theoretical modeling thestatus of partner with experiment.
CONTENTS
INTRODUCTION
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
182
The Time Race
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
182
Strong Vibrational Coupling in Biomolecules
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
184
Simulation Classes
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
186
Overview
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
187THE TIMESTEP PROBLEM
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
188
Explicit and Implicit Schemes
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
188
Culprits of the Timestep Limitation and Instability
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
189LARGE-TIMESTEP TECHNIQUES FOR CONTINUOUS DYNAMICS
. . . . . . . . . . . . . . .
193
Constrained Dynamics
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
193
Torsion Dynamics
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
195
Symplectic Schemes
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
196
Implicit Schemes
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
199
Multiple Timestep (MTS) Methods
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
201
Normal-Mode-Based Schemes
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
202
The LN Algorithm
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
204SOME COMPARATIVE NUMERICAL EXPERIMENTS
. . . . . . . . . . . . . . . . . . . . . . . . . . .
208
Langevin vs Newtonian Dynamics
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
211
Constraints
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
211
Performance of LN
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
214
Computational Efﬁciency
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
214PERSPECTIVES
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
215APPENDIX A: OUTLINE OF LIN
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
217
INTRODUCTION
The Time Race
In the world of computation, time is the enemy. Scientists wage a battle of wits to squeezeas many computations as they can into the shortest possible span of time. The larger and more complex the problem, the more cunning their techniques must be (82).
The above battle refers to forecasting the weather, where the developmentof rapidly converging and computationally tractable large-scale optimizationtechniques is key. Yet, the statement rings at least equally true for simulatingbiomolecular motion on modern computers (89), where the crucial numericaltechniques for integrating the equations of motion have ranged from brute-force to ingenious. In molecular dynamics (MD) simulations, insights intomolecularﬂexibilityandactivityaresoughtbynumericallyfollowingmolecularconﬁgurations in time according to Newtonian physics (3, 21, 64). In theory,MD simulations can bridge spatial and temporal resolution and thus capturemolecular motion over a wide range of thermally accessible states. In practice,thenumericaltimestepproblemhaslimitedmostapplicationstostraightforward
BIOMOLECULAR DYNAMICS AT LONG TIMESTEPS
183
Table 1
Some typical vibrational modes
a
Wavelength of Absorptionabsorption [cm
−
1
] frequency [s
−
1
] Period [fs]Vibrational mode (1
/λ
) (
ν
=
c
/λ
) (1
/ν
) Period
/π
[fs]O
−−
H stretch 3200–3600 1.0
×
10
14
9.8 3.1N
−−
H stretchC
−−
H stretch 3000 9.0
×
10
13
11.1 3.5O
−−
C
−−
O asymmetric stretch 2400 7.2
×
10
13
13.9 4.5C
≡≡
C, C
≡≡
N stretch 2100 6.3
×
10
13
15.9 5.1C
==
O (carbonyl) stretch 1700 5.1
×
10
13
19.6 6.2C
==
C stretchH
−−
O
−−
H bend 1600 4.8
×
10
13
20.8 6.4C
−−
N
−−
H bend 1500 4.5
×
10
13
22.2 7.1H
−−
N
−−
H bendC
==
C (aromatic) stretchC
−−
N stretch (amines) 1250 3.8
×
10
13
26.2 8.4Water Libration 800 2.4
×
10
13
41.7 13(rocking)O
−−
C
−−
O bending 700 2.1
×
10
13
47.6 15C
==
C
−−
H bending (alkenes)C
==
C
−−
H bending (aromatic)
a
All values are approximate; a range is associated with each motion depending on the system. The value of
c
=
3.00
×
10
10
cm s
−
1
. The last column indicates the timestep limit for leap-frog stability for a harmonic oscillator:
t
<
2
/ω
=
2
/(
2
πν)
.
integration with very small timesteps compared to the motion of major interest.Consequently, the total length of current trajectories at atomic resolution islimited to the nanosecond timescale.Our battle in the world of biomolecular dynamics is to reliably simulate aslarge a timespan as possible in the smallest amount of computational time. Thereliability issue is a separate topic in its own right, since our force ﬁelds are ap-proximate, quantum effects are ignored, and many other model assumptions orspecial simulation protocols are applied. In addition to these approximations,single-trajectory results must be assessed in the framework of statistical me-chanics. Simulationsarecomputationallytaxingbecauseofthetypicalexpenseof computing the Newtonian forces for a system of thousands of atoms—thesolute and solvent—at each 1-fs timestep. Since conformational changes inmacromolecules occur on a continuum of timescales ranging from 10
−
12
to10
2
s (see Table 1 for the high-frequency end), considerable research has fo-cusedon(
a
)developingalgorithmsthatalleviatetheseverestabilityrequirementdictatedbythehigh-frequencyvibrationalmodes,and(
b
)exploitinghigh-speedparallel computer technology to accelerate MD simulations (16).
184
SCHLICK, BARTH & MANDZIUK
Strong Vibrational Coupling in Biomolecules
A signiﬁcant ﬁnding that emerged from these algorithmic efforts for increas-ing the timestep in biomolecular simulations (36, 41, 69, 89, 97, 102, 116) isthe unexpected difﬁculty of this challenge if one strictly seeks faster methodswith the same all-atom resolution of small-timestep trajectories. Vibrationalmodes are intricately coupled in biomolecules such as proteins and nucleicacids. Therefore, mathematical techniques that have worked well in othermultiple-timescale contexts where the fast motions are decaying rapidly ratherthanoscillatory(e.g.inchemicalreactionswithknownreactantsandproducts),or are largely decoupled from the others (e.g. in fullerenes), have not beendirectly applicable, or as effective, for biomolecules. For example, standardhigh-stability implicit schemes for stiff differential equations, such as implicit-Euler (IE) (43, 73, 95) and implicit-midpoint (IM) (57), are unsatisfactory forproteins and nucleic acids at atomic resolution at large timesteps because of numerical damping (69, 96, 119) and resonance (57) problems, respectively.Implicit methods are also computationally expensive since solution of a non-linear system is required at each timestep (48, 49, 120, 121). Algorithms basedon substructuring (110) require substantial tailoring and perhaps relaxation of goals(i.e. approximateratherthanaccuratereproductionofsmall-timesteptra- jectories)forbiomolecularapplications, andmultiple-timestep(MTS)methodsthatachieveafactorof20ormorespeed-upforfullerenes(74)yieldmuchmoremodest factors (e.g. 4) for macromolecules (45, 116).The intricate vibrational coupling of the multiscale modes associated withglobular systems necessitates good resolution of the high end of the spectruminordertocapturetheslower, large-scalemotions. Constrainingthefastmodesis effective when bond-length stretching is suppressed but not when the bond-angle ﬂexibility is also eliminated (111). For good resolution of the high-fre-quency motion, MTS methods often use a very small timestep (0.25 fs) for thehighest-frequency class (122).Figure 1 illustrates this point. The ﬁgure displays the frequency spectrumfor a blocked alanine residue (N-Acetylalanyl N
-Methylamide, 22 atoms),known commonly as an “alanine dipeptide”, as obtained from MD simulationsusingvariousprotocols. Thesepowerspectrawereobtainedfromvelocitytimeseries (14), with the Fast Fourier Transform routine (Sande-Tukey FFT) (59).All MD simulations were performed in the CHARMM program (19) usingthe Verlet integrator (113) at a timestep of 1 fs over 2
14
steps (
∼
17 ps), butdifferent constraint procedures were enforced via SHAKE (87) in some cases,as follows: (
a
) no constraints, (
b
) constraints on bonds involving hydrogensonly, (
c
) constraints on all bonds, and (
d
) constraints on all bonds and bondangles involving hydrogens. In parts
b
and
c
, bonds were constrained to theirequilibrium values. In part
d
, constraints were made to the starting values
BIOMOLECULAR DYNAMICS AT LONG TIMESTEPS
185
Figure 1
The frequency spectrum for a model of alanine dipeptide (
top
), as obtained from 17-psMD simulations with CHARMM (19) using various constraint protocols via SHAKE (87) with animproved protocol (12), in association with Verlet integration at
t
=
1 fs: (
a
) no constraints; (
b
)constraints only on bonds involving hydrogens; (
c
) constraints on all bonds; and (
d
) constraintson all bonds and on bond angles involving hydrogens. The frequency spectra were obtained fromvelocity time series (14), with the Fast Fourier Transform routine (Sande-Tukey FFT) (59). Thespectral heights are relative.

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