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Replica-exchange in molecular dynamics
Part of 2014 SeSE course in Advanced molecular dynamics
Mark Abraham
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Frustration in MD
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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
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Barriers in MD
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F
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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
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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)
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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. . .
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Accelerating the sampling - heating it up
x
F
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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)]
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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)
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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. . .
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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)])
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Parallel tempering - understanding the exchanges
Figure : Potential energy distributions of alanine dipeptide replicas
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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
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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
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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
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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
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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
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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
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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
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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
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Questions?