IMA Summer Program Classical and Quantum Approaches in ... · IMA Summer Program Classical and Quantum Approaches in Molecular Modeling Lecture 5: Molecular Sampling Robert D. Skeel

IMA Summer Program

Classical and Quantum Approachesin Molecular Modeling

Lecture 5: Molecular Sampling

Robert D. SkeelDepartment of Computer Science, and of Mathematics

Purdue University

http://bionum.cs.purdue.edu/2007July24.pdf

Acknowledgment: NIH

SamplingCreate a stochastic (or deterministic!) ergodic Markov chain togenerate x1, x2, . . . , xn, . . . having the desired distribution.

• Monte Carlo methods are unbiased in the limit Ntrials→∞but waste information.

• Molecular dynamics with extended Hamiltonians and/orstochastic terms has a bias due to finite ∆t.

Sampling can be accelerated by techniques such as replicaexchange and multicanonical/Wang-Landau sampling.

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Canonical ensembleLet β = 1/kBT .The probability density function factors as

const e−βpTM−1p/2 · ρ(x) where ρ(x) = const e−βU(x).

The constant in ρ(x) is unknown.

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Outline

I. Markov chain Monte Carlo methods

II. Hybrid Monte Carlo methods

III. Molecular dynamics

IV. Replica exchange method

V. Multicanonical/Wang-Landau sampling

VI. Practicalities

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A simple Markov chain Monte Carlo methodGiven x, one step is as follows:

Pick an atom i at random.Generate i.i.d. values ∆x, ∆y, ∆z uniform on [−1

2∆, 12∆]x′ = x with [∆x,∆y,∆z]T added to ~riAccept x′ with probability

min {1, ρ(x′)/ρ(x)}—Metropolis acceptance criterion.

Otherwise, choose x as the new value.Note. A rejected move counts as a step in the chain.

These trial moves have symmetric conditional p.d.f.:ρt(x|x′) = ρt(x′|x).

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MCMC ConvergenceEach configuration xn of the Markov chain has a density ρn(x).Only in the limit n→∞ might ρn(x)→ ρ(x): convergence.

Proposition. convergence⇔ stationarity & ergodicity.

Stationarity means ρn(x) = ρ(x)⇒ ρn+1(x) = ρ(x)

Ergodicity means that all subsets of positive measure will bevisited with probability > 0 in finite time.

The Markov chain given previously is ergodic.

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StationarityProposition. detailed balance⇒ stationarity.

Detailed balance∗ means ρ(x|x′)ρ(x′) = ρ(x′|x)ρ(x)where ρ(x′|x) is the conditional p.d.f. for a complete step.

Proposition. symmetric trial moves & Metropolis criterion⇒detailed balance

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Metropolis-Hastings criterionaccommodates nonsymmetric trial moves by accepting withprobability

min{

1,ρ(x′)ρt(x|x′)ρ(x)ρt(x′|x)

}.

Example is a trial move given by

x′ = x+ ∆tD∇ log ρ(x) +√

2∆tD1/2Z

where D is a constant diagonal matrix and Z is a set ofindependent standard Gaussian random numbers.

I would call this scheme a . . .

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“Brownian dynamics sampler”Basic idea is similar to

force-bias MC (1978)

and almost identical to

smart MC (1978).

Abstracted idea has been called

Metropolis-adjusted Langevin algorithm (1994).

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Outline

I. Markov chain Monte Carlo methods

II. Hybrid Monte Carlo methods

III. Molecular dynamics

IV. Replica exchange method

V. Multicanonical/Wang-Landau sampling

VI. Practicalities

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Hybrid Monte CarloHybrid Monte Carlo uses MD to generate possible moves.Given x :

(1) Generate p from const exp(−12βp

TM−1p).

(2) Obtain x′, p′ from short MD trajectory.

(3) Accept x′ with probability

min{

1,exp(−βH(x′, p′))exp(−βH(x, p))

},

where H(x, p) = 12p

TM−1p+ U(x).

(4) If rejected, choose x.

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Convergence of HMCIt is enough that the integrator

be reversible and volume-preserving.

Counterintuitively, 〈∆H〉 > 0 , ∆H = H(x′, p′)−H(x, p).Indeed, 〈∆H〉 ∝ ∆t2pN .

To get a 50% acceptance rate, we need〈∆H〉 = 0.9099 kBT ,

which implies∆t ∝ N−1/2p

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Outline

I. Markov chain Monte Carlo methods

II. Hybrid Monte Carlo methods

III. Molecular dynamics

IV. Replica exchange method

V. Multicanonical/Wang-Landau sampling

VI. Practicalities

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Deterministic MDInstantaneous temperature T (p) = pTM−1p/kBNd

where Nd = number of DOFs.Nose-Hoover augments Newton’s equations with a thermostat:

ddtx = M−1p,

ddtp = −∇U(x)−ps

Qp,

ddtps = NdkB(T (p)− T ).

where Q = thermal inertia.If T (p) > T , the value of ps will increase,

and eventually ps will be positive,causing p—and T (p)—to decrease.

(And the opposite happens if T (p) < T .)

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Nose-Hoover generates canonical ensemble via

〈A(x, p)〉 = limt→∞

1t

∫ t

0

A(x(t), p(t)) dt.

It is not Hamiltonian but has a conserved quantity

12pTM−1p+ U(x) +

12Q

p2s +NdkBT ln s

whereddts =

1Qpss.

Drift in this “extended energy” may be excessive, however.

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Nose-PoincareIt is defined by the extended Hamiltonian H(x, s, p, ps) =

s

(12s−2pTM−1p+ U(x) +

12Q

p2s +NdkBT ln s− E

)where E is chosen to make H initially zero.

〈A〉 = limt→∞

1t

∫ t

0

A(x(t′), s(t′)−1p(t′)) dt′.

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Stochastic MDRecall Langevin dynamics

Md

dt2x = F (x)− CM d

dtx+ (2kBTCM)1/2

ddtW (t)

where

C is a diagonal matrix of damping constants, e.g., 5 ps−1, and

W (t) is a set of 3N independent canonical Wiener processes.

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Outline

I. Markov chain Monte Carlo methods

II. Hybrid Monte Carlo methods

III. Molecular dynamics

IV. Replica exchange method

V. Multicanonical/Wang-Landau sampling

VI. Practicalities

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Temperature replica exchangeLarge conformational barriers can be surmounted

by raising temperature,and this can be done

while maintaining Boltzmann-Gibbs samplingusing replica-exchange aka parallel tempering (1995).

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MethodSimulate an ensemble of systems at temperaturesT1 < T2 < · · · < Tµ where T1 is the desired temperature.Periodically, choose ν at random from 1, 2, . . . , µ− 1,and consider swapping configurations x(ν) and x(ν+1).Probability of the swapped state relative the unswapped state is

r =ρν(x(ν+1))ρν(x(ν))

ρν+1(x(ν))ρν+1(x(ν+1))

where ρν(x) is the probability density for temperature Tν.Accept the exchange with a probability

min {1, r} .Then continue sampling.

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ShortcomingProbability of rejection increases with system size N .

Number of replicas needed to prevent excessive rejectionsincreases as N1/2.

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Hamiltonian replica exchangeA generalization of replica exchange.

As an example, write U = Upp + Upw + Uww

where the splitting represents protein–protein, protein–water, andwater–water interactions, respectively.

Then considerUν = γνU

pp + Upw + Uww, 1 = γ1 > γ2 > · · · > γµ—hot solute, cold solvent.

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Outline

I. Markov chain Monte Carlo methods

II. Hybrid Monte Carlo methods

III. Molecular dynamics

IV. Replica exchange method

V. Multicanonical/Wang-Landau sampling

VI. Practicalities

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Energy distribution for canonical ensemble

Spread is ∝√NkBT .

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Multicanonical samplingEnergy barriers can be overcome by sampling from a density

for which U(x) has a flat distribution.For suitable ranges Emin ≤ U(x) ≤ Emax, there is such adensity,

ρmulti(x) = const/g(U(x)),

where g(E) is the (configurational) density of states,

g(E) = const∫δ(U(x)− E)dx.

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ReweightingIf we have an approximation to g(E),canonical ensemble averages can be calculated by reweighting

〈A(x)〉 =〈A(x)e−βU(x)/g(U(x))−1〉multi

〈e−βU(x)/g(U(x))−1〉multi,

valid independent of the accuracy of g(E).(If g(E) has good accuracy, there is an alternative.)

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Wang-Landau schemeDiscretize using histograms:

partition Emin ≤ E ≤ Emax into J subintervals,

define basis functions 1j(E) ={

1, in jth subinterval,0, elsewhere.

Construct approximations, g(1)(E), g(2)(E), . . . , of decreasinggranularity:

log g(1)(E) =J∑j=1

N(1)j 1j(E),

log g(2)(E) = log g(1)(E) +12

J∑j=1

N(2)j 1j(E),

et cetera.

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Inner loopInitialize: g0(E) = g(k)(E).Inner loop: for n = 1, 2, . . .

Generate proposal xn from xn−1

Choose xn = xn or xn−1 based onMetropolis criterion for density = const/gn−1(U(x))

# increment histogram valuelog gn(E) = log gn−1(E) +

(12

)k∑Jj=1 1j(U(xn))1j(E),

until well sampled.Set g(k+1)(E) = gn(E).

not a Markov chainScheme tries to create a flat energy distribution,the resistance encountered is a correction to g(k)(E).

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Outline

I. Markov chain Monte Carlo methods

II. Hybrid Monte Carlo methods

III. Molecular dynamics

IV. Replica exchange method

V. Multicanonical/Wang-Landau sampling

VI. Practicalities

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PracticalitiesThe practicalities of doing such calculations involve three steps:

structure building Setting up the input files is best doneinteractively with scripts and visual feedback.visualization programs: RasMol, VMD, PyMOL, . . .

simulation Generating dynamics or sampling trajectories is bestdone in background or remotely.simulation programs: CHARMM, Amber, Gromacs, NAMD,LAMMPS, NWChem, Tinker, . . .

analysis Analyzing trajectory data.

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Simulation specifications

• Specify molecular system & surroundings

• Specify computational tasks

• Select computational model:uncontrolled approximations and error tolerances

- internal forces- external forces, e.g., temperature and pressure control- dynamics (sampling or real)

• (Override defaults for performance parameters)

• Design simulation protocol

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References

• M. P. Allen and D. J. Tildesley, Computer Simulation ofLiquids, 1987,

• D. Frenkel and B. Smit, Understanding Molecular Simulation:From Algorithms to Applications, 2nd edition, 2002.

• A. R. Leach, Molecular Modelling: Principles andApplications, 2nd edition, 2001,

• T. Schlick, Molecular Modeling and Simulation: AnInterdisciplinary Guide, 2002,

• Journal of Chemical Physics

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