Replica-exchange in molecular dynamics

Part of 2014 SeSE course in Advanced molecular dynamics

Mark Abraham

Frustration in MD

Frustration in MD

I different motions have different time scales

I bond vibration, angle-vibration, side-chain rotation, diffusion,secondary structure (de)formation, macromolecular events, . . .

I need a short enough time step

I a model that doesn’t get enough fine detail right will strugglewith higher-level things, too

Barriers in MD

x

F

Frustration in MD

I barriers more than a few kT exist, and are hard to crossI need extremely large amount of brute-force sampling to get

over themI makes solving problems like protein folding exceedingly

expensive

Ways to grapple with the problem

I give up on fine detail, and use a coarse-graining approachI accelerate the sampling (work smarter!)I throw more hardware at it (e.g. Folding@Home)I write faster software (hard, very hard)

Accelerating the sampling

I if the problem is that kT is too small..

1. increase T

2. sample widely

3. . . .

4. profit!

I unless the landscape changes. . .

Accelerating the sampling - heating it up

x

F

Simulated tempering

I use Monte Carlo approach to permit system to move incontrol parameter space

I typical control parameter is temperature (but not essential)I typically sample the system only when at temperature of

interestI correct if the (Metropolis) exchange criterion is constructed

correctly

I how? For a state s

P((β, s)→ (β′, s)) = min(1, w(β′,s)w(β,s) )

where β = 1kT and w(β, s) = exp [−βU + g(β)]

Simulated tempering (2)

I correct if the exchange criterion is constructed correctly

I the optimal g(β) is the free energy. . .I so you’re good if you already know the relative likelihood of

each conformation at each temperature

I works great if you already know the answer to a harderproblem than the original

I (but you can use an iterative scheme to converge on theanswer)

Parallel tempering (a.k.a replica exchange)

I side-steps the prior-knowledge problem by running anindependent copy of the simulation at each control parameter

I (note, throwing more hardware at the problem!)I now the exchange is between copies at different control

parameters, each of which is known to be sampled from acorrect ensemble already

I this eliminates g(β) from the generalized exchangecriterion. . .

Parallel tempering - the exchange criterion

P((β, s)↔ (β′, s ′)) = min(1, w(β,s′)w(β′,s)w(β,s)w(β′,s′))

which for Boltzmann weights reduces to

= min(1, exp[(β′ − β)(U ′ − U)])

Parallel tempering - understanding the exchanges

Figure : Potential energy distributions of alanine dipeptide replicas

Parallel tempering - is this real?

I recall that P(βs) ∝ exp[−βU(x)]I any scheme that satisfies detailed balance forms a Markov

chain whose stationary distribution is the target (generalized)ensemble

I so we require only thatP((β, s))P((β, s)→ (β′, s)) = P((β′, s ′))P((β′, s ′)→ (β, s ′))

I . . . which is exactly what the exchange criterion isconstructed to do

Parallel tempering - is this real? (2)

I high temperature replicas hopefully can cross barriersI if the conformations they sample are representative of

lower-temperature behaviour, then they will be able toexchange down

I if not, they won’t

Ensembles used

I natural to use the NVT ensemble with an increasing range ofT and constant V

I there’s a hidden catch - must rescale the velocities to suit thenew ensemble in order to construct the above exchangecriterion

I probably this should use a velocity-Verlet integrator (x and vat same time)

I in principle, can use other ensembles like NPT

Ensembles used (2)

I NVT at constant volume must increase P with TI that seems unphysicalI worse, the force fields are parameterized for a fixed

temperatureI but the method doesn’t require that the ensembles correspond

to physical onesI merely need overlap of energy distributionI how much overlap determines the probability of accepting an

exchange

Problems with replica exchange

I molecular simulations typically need lots of waterI thus lots of degrees of freedomI energy of the system grows linearly with system sizeI width of energy distributions grow as

√size

I need either more replicas or accept lower overlap

Unphysics is liberating

I if there’s no need to be physical, then may as well be explicitabout it

I large number of schemes proposedI example: resolution exchange

I run replicas at different scales of coarse grainingI at exchange attempts, not only rescale velocities, but

reconstruct the coordinates at the higher/lower grain level

Hamiltonian replica exchange

I replicas can be run varying some other control parameter

I e.g. gradually turn on some biasing potential

I can construct higher-dimensional control-parameter schemesalso

I in a free-energy calculation, exchange between both alchemicaltransformation parameter λ and temperature

Replica exchange with solute tempering

I selectively “heat” only a small region of the systemI modify the parameters to scale the energy, rather than heating

I remember P(βs) ∝ exp[−βU(x)]

I advantage that the energy distribution of only part of thesystem increases over control parameter space

I needs many fewer replicas for a given control parameter spaceI implemented in GROMACS with PLUMED plugin

Questions?