Statistical temperature molecular dynamics

Statistical temperature molecular dynamics (STMD) [Kim et al., 2006, 2007] is a flat-histogram sampling technique which is based upon the Wang-Landau Monte Carlo algorithm [Wang and Landau, 2001b,a]. In Wang-Landau sampling, the cen- tral idea is to arrive at a flat potential energy distribution by weighting the Monte Carlo acceptance probability by w(U) = 1/⌦(U). ⌦(U) is the density of states, connected to the microcanonical entropy as S(U) = kBln⌦(U). This leads to a

uniform random walk in potential energy space.

⌦(U) is not known a priori, so a running estimate ˜⌦(U) is kept. Initially, ˜

⌦(U) = 1 is set. Next, every time an energy Ui is visited, the density of states is updated via the operation ˜⌦(Ui)!f⌦(˜ Ui), wheref is a modification factor and is greater than 1. Each update operation diminishes the probability of a return visit toUi.

A histogram of visits to energy states is kept, and, when it becomes flat within a given tolerance, the modification factor f is decreased (conventionally by the operationf _!pf), the histogram is reset, and the simulation continues. In the limitf !1, ˜⌦(U)!⌦(U).

In STMD, the statistical temperature ˜T(U) is the object of the running estimate, instead of ˜⌦(U). The two are connected as follows:

1 ˜ T(U) = @S˜(U) @U =kB @ln ˜⌦(U) @U . (2.19)

Rewriting as a central finite di↵erence approximation:

1 ˜ T(Ui) ⇡ ˜ S(Ui+1) S˜(Ui 1) 2 , (2.20)

in which is the di↵erence in energy between two adjacent bins in the statistical temperature histogram. Feeding the WL update scheme ˜⌦(Ui) ! f⌦(˜ Ui) into the microcanonical entropy S = kBln⌦(U), we obtain an entropy update scheme

˜

S(Ui) ! S˜(Ui) +kBlnf. Equation (2.20) shows that the statistical temperature

estimate ˜T(Ui) needs to be updated if eitherUi±1are visited. Given that the current

value for ˜T(Ui) is derived from equation (2.20), the entropy update scheme tells us how to update ˜T(Ui) if Ui+1 is visited;

1 ˜ Tnew(Ui) ⇡ ˜ Snew(Ui+1) S˜(Ui 1) 2 = 1 ˜ Told(Ui) +kBlnf 2 , (2.21) or ifUi 1 is visited; 1 ˜ Tnew(Ui) ⇡ S˜(Ui+1) S˜new(Ui 1) 2 = 1 ˜ Told(Ui) kBlnf 2 . (2.22)

Equivalently, if Ui is visited, both ˜T(Ui±1) must be updated as follows:

1 ˜ Tnew(Ui±1) ⇡ _˜ 1 Told(Ui±1) ⌥kBlnf 2 , (2.23)

and this is the STMD update scheme.

An alternative but equivalent formulation due to Allen and Quigley [2013], which highlights the similarity between STMD and Wang-Landau further, is to preserve the Wang-Landau entropy update scheme, ˜S(Ui)!S˜(Ui) +kBlnf, and to

calculate ˜T(Ui) when needed as a central di↵erence approximation, using equation (2.20).

Carlo, altering the acceptance probability to obtain non-Boltzmann sampling. To implement statistical temperature inmolecular dynamics, the generalized ensemble simulation technique is used [Nakajima et al., 1997], with the temperature held at T0. In the canonical ensemble,

P(U) = 1 Z⌦(U)e

U/kBT0_{, (2.24)}

in whichZ is the canonical partition function and T0 is the thermostat-maintained

temperature of the simulation, and not the statistical temperature estimate. A flat distribution is obtained by altering the potentialU to beUMC(U):

⌦(U)e UMC(U)/kBT0 _{=C , (2.25)}

whereC is a constant of choice; let C= 1 for ease. Thus,

UMC(U) =kBT0ln⌦(U) =T0S(U). (2.26)

Implementing this potential gives

fSTMD= r(T0S(U)) = T0

@S(U)

@U rU =

T0

T(U)fTrue, (2.27) wherefSTMD is the scaled force on each atom due to the multicanonical potential,

andfTrueis the force on the respective atom due to the normal (canonical) potential.

After collecting the data on ˜T(U) in a simulation, the estimate for the entropy is then given by ˜ S(U) = Z U Ul 1 ˜ T(U) dU , (2.28)

with an arbitrary lower integration limit Ul. Now the canonical ensemble average of an observable can be calculated for the system.

2.3

Summary

The problem of simulation of intrinsically disordered proteins is a recent one, and it would be of great utility to the intrinsically disordered protein community to develop reliable methods by which simulations of intrinsically disordered proteins can be accelerated. Little is known about the degree to which protein representations can be coarse-grained in the case of IDPs, but it is less likely that models with significant simplifications of the backbone could yield good results. Two models have been selected for experimental study with n16N systems; PLUM and PRIME20, and these both maintain three beads per residue on the backbone, and one on the side- chain.

Accelerated sampling schemes disconnect a molecular dynamics experiment from realistic reproductions of the dynamics by introducing unphysical alterations, leaving the statics of the system available for retrieval. Two very general methods have been recruited here; replica exchange and statistical temperature. The former simulates the system at high temperature in parallel to the reference temperature, and executes Monte Carlo style dice throws to swap the systems’ configurations in keeping with canonical ensemble probability statistics. The high temperature replica explores phase space relatively freely. The latter implements a multicanoni- cal potential which progressively reduces the probability of visiting the most likely configurations of the system through learning the density of states⌦(U). The sys- tem arrives at a uniform sampling of the potential energy space, and data from this uniform sampling regime can be reconstructed into canonical statistics.