Replica exchange

Replica exchange molecular dynamics (REMD) is a method that can be used to speed up the sampling of any type of simulation, especially if conformations are separated by relatively high energy barriers. It involves simulating multiple replicas of the same system at different temperatures and randomly exchanging the complete state of two replicas at regular intervals with the probability:

\[P(1 \leftrightarrow 2)=\min\left(1,\exp\left[ \left(\frac{1}{k_B T_1} - \frac{1}{k_B T_2}\right)(U_1 - U_2) \right] \right)\]

where \(T_1\) and \(T_2\) are the reference temperatures and \(U_1\) and \(U_2\) are the instantaneous potential energies of replicas 1 and 2 respectively. After exchange the velocities are scaled by \((T_1/T_2)^{\pm0.5}\) and a neighbor search is performed the next step. This combines the fast sampling and frequent barrier-crossing of the highest temperature with correct Boltzmann sampling at all the different temperatures 60, 61. We only attempt exchanges for neighboring temperatures as the probability decreases very rapidly with the temperature difference. One should not attempt exchanges for all possible pairs in one step. If, for instance, replicas 1 and 2 would exchange, the chance of exchange for replicas 2 and 3 not only depends on the energies of replicas 2 and 3, but also on the energy of replica 1. In GROMACS this is solved by attempting exchange for all odd pairs on odd attempts and for all even pairs on even attempts. If we have four replicas: 0, 1, 2 and 3, ordered in temperature and we attempt exchange every 1000 steps, pairs 0-1 and 2-3 will be tried at steps 1000, 3000 etc. and pair 1-2 at steps 2000, 4000 etc.

How should one choose the temperatures? The energy difference can be written as:

\[U_1 - U_2 = N_{df} \frac{c}{2} k_B (T_1 - T_2)\]

where \(N_{df}\) is the total number of degrees of freedom of one replica and \(c\) is 1 for harmonic potentials and around 2 for protein/water systems. If \(T_2 = (1+\epsilon) T_1\) the probability becomes:

\[P(1 \leftrightarrow 2) = \exp\left( -\frac{\epsilon^2 c\,N_{df}}{2 (1+\epsilon)} \right) \approx \exp\left(-\epsilon^2 \frac{c}{2} N_{df} \right)\]

Thus for a probability of \(e^{-2}\approx 0.135\) one obtains \(\epsilon \approx 2/\sqrt{c\,N_{df}}\). With all bonds constrained one has \(N_{df} \approx 2\, N_{atoms}\) and thus for \(c\) = 2 one should choose \(\epsilon\) as \(1/\sqrt{N_{atoms}}\). However there is one problem when using pressure coupling. The density at higher temperatures will decrease, leading to higher energy 62, which should be taken into account. The GROMACS website features a so-called REMD calculator, that lets you type in the temperature range and the number of atoms, and based on that proposes a set of temperatures.

An extension to the REMD for the isobaric-isothermal ensemble was proposed by Okabe et al. 63. In this work the exchange probability is modified to:

\[P(1 \leftrightarrow 2)=\min\left(1,\exp\left[ \left(\frac{1}{k_B T_1} - \frac{1}{k_B T_2}\right)(U_1 - U_2) + \left(\frac{P_1}{k_B T_1} - \frac{P_2}{k_B T_2}\right)\left(V_1-V_2\right) \right] \right)\]

where \(P_1\) and \(P_2\) are the respective reference pressures and \(V_1\) and \(V_2\) are the respective instantaneous volumes in the simulations. In most cases the differences in volume are so small that the second term is negligible. It only plays a role when the difference between \(P_1\) and \(P_2\) is large or in phase transitions.

Hamiltonian replica exchange is also supported in GROMACS. In Hamiltonian replica exchange, each replica has a different Hamiltonian, defined by the free energy pathway specified for the simulation. The exchange probability to maintain the correct ensemble probabilities is:

\[P(1 \leftrightarrow 2)=\min\left(1,\exp\left[ \left(\frac{1}{k_B T} - \frac{1}{k_B T}\right)((U_1(x_2) - U_1(x_1)) + (U_2(x_1) - U_2(x_2))) \right] \right)\]

The separate Hamiltonians are defined by the free energy functionality of GROMACS, with swaps made between the different values of \(\lambda\) defined in the mdp file.

Hamiltonian and temperature replica exchange can also be performed simultaneously, using the acceptance criteria:

\[P(1 \leftrightarrow 2)=\min\left(1,\exp\left[ \left(\frac{1}{k_B T} - \right)(\frac{U_1(x_2) - U_1(x_1)}{k_B T_1} + \frac{U_2(x_1) - U_2(x_2)}{k_B T_2}) \right] \right)\]

Gibbs sampling replica exchange has also been implemented in GROMACS 64. In Gibbs sampling replica exchange, all possible pairs are tested for exchange, allowing swaps between replicas that are not neighbors.

Gibbs sampling replica exchange requires no additional potential energy calculations. However there is an additional communication cost in Gibbs sampling replica exchange, as for some permutations, more than one round of swaps must take place. In some cases, this extra communication cost might affect the efficiency.

All replica exchange variants are options of the mdrun program. It will only work when MPI is installed, due to the inherent parallelism in the algorithm. For efficiency each replica can run on a separate rank. See the manual page of mdrun on how to use these multinode features.