# REPLICA EXCHANGE SIMULATION

Molecular simulation is a way to study the physiological property of proteins, in a traditional molecular simulation, one trajectories was generated under one condition or some conditions that was set, one big problem about such simulation is the lack of efficiency in sampling the canonical distribution due to the trapping into large number of local minima, which slow this sampling of phase space. A number of attempts have been made to overcome this issue. One of them is Replica Exchange Molecular Dynamics.In this method, a number of simulations are performed at different conditions(temperature,reaction coordinates) in parallel, and exchanges of configurations are tried periodically.

Attempt to exchange temperatures with probability

${P}_{accept}=[1,exp[-\frac{1}{k}(\frac{1}{{T}_{i}}-\frac{1}{{T}_{j}})(U({X}_{i})-U({X}_{j})]]$

if ${T}_{i} < {T}_{j}$, and ${E}_{1} > {E}_{2}$, exchange accepted. Exchange probabilities will be zero unless the energy distributions at adjacent temperatures overlap each other.

This is the energy overlap of 1e0q 8 replica at temperature from 300 $^{\circ}{\rm C}$ to 850 $^{\circ}{\rm C}$.

These are trajectories of 8 replicas.

The conditions for each replica could be other constraints rather than temperature, therefore a technique called Hamiltonian Replica Exchange(HREM) was developed. In this method, different replicas correspond to (slightly) different potential functions.

$U(x,\lambda) ={U}_{0}(x)+V(x,\lambda)$

${U}_{0}(x)$ -  "Base Energy"           $V(x,\lambda)$ -  "Perturbation Energy"

The probability of exchange between two replicas in HREM

${U}_{before}=U({x}_{1},{\lambda}_{1})+U({x}_{2},{\lambda}_{2})$    ${U}_{after}=U({x}_{2},{\lambda}_{1})+U({x}_{1},{\lambda}_{2})$

${\alpha}_{12}=exp{-\beta[{U}_{after}-{U}_{before}]}=exp{-\beta[\Delta{U}_{1}+\Delta{U}_{2}]}$

One application of HREM is:

REUS: Replica Exchange Umbrella Sampling (Sugita, Kitao, Okamoto, JCP, 113:6042 (2000))

$V(x,\lambda)=w(x;{d}_{\lambda})$          $w(x)$ = biasing potential

This application was first employed to compute end-to-end distance PMF of a peptide, in which the biasing potentials are harmonic restraining potentials of the end-to-end distance $d$.

To analyze data from multiple replicas, Weighted Histogram Analysis Method(WHAM) is used to merge the data.

REX is an enhanced method, but it is still limited compared to the complexity of conformational space, there is really no way around it.

Temperature REX still appears to be the most applicable in general, Hamiltonian REX(enhance base hopping) and 2D REX(enhance continuum conformation sampling) show more future.