From Protein Structure to Function with Bioinformatics.pdf

9 Protein Dynamics: From Structure to Function 229attention (Fukunishi et al. 2002; Liu et al. 2005; Sugita et al. 2000; Affentrangeret al. 2006; Christen and van Gunsteren 2006; Lyman and Zuckerman 2006).In standard temperature REX MD (Sugita and Okamoto 1999), a generalizedensemble is constructed from M + 1 non-interacting copies, or “replicas”, of thesystem at a range of temperatures {T 0,…,T M} (T m≤ T m + 1; m = 0,…,M), e.g. by distributingthe simulation over M + 1 nodes of a parallel computer (Fig. 9.6 left).A state of this generalized ensemble is characterized by S = {…,s m,…}, where s mrepresents the state of replica m having temperature T m. The algorithm now consistsof two consecutive steps: (a), independent constant-temperature simulations of eachreplica, and (b), exchange of two replicas S = {…,s m,…,s n,…}→ S′= {…,s n′,…,s m,…} according to a Metropolis-like criterion. The exchange acceptance probabilityis thereby given byPS ( → S′ ) = min { 1,exp {( b −b )[ V −V]}}(9.1)m n m nwith V mbeing the potential energy and β m−1= k BT m. Iterating steps a and b, the trajectoriesof the generalized ensemble perform a random walk in temperature space,which in turn induces a random walk in energy space. This facilitates an efficientand statistically correct conformational sampling of the energy landscape of thesystem, even in the presence of multiple local minima.The choice of temperatures is crucial for an optimal performance of the algorithm.Replica temperatures have to be chosen such that (a) the lowest temperatureis small enough to sufficiently sample low-energy states, (b) the highesttemperature is large enough to overcome energy barriers of the system of interest,and (c) the acceptance probability P(S→S¢) is sufficiently high, requiringadequate overlap of potential energy distributions for neighboring replicas. Forlarger systems simulated with explicit solvent the latter condition presents themain bottleneck. A simple estimate (Cheng et al. 2005; Fukunishi et al. 2002)shows that the potential energy difference ΔV~N dfΔT is dominated by the contributionfrom the solvent degrees of freedom N dfsol, constituting the largest frac-t 0 t 1 t 2 t 3 t 0 t 1 t 2 t 3T 2es(T 2 , T 0 )T 0 (T 0 , T 0 )T 1es(T 1 , T 0 )Fig. 9.6 Schematic comparison of standard temperature REX (left panel) and the TEE-REXalgorithm (right panel) for a three-replica simulation. Temperatures are sorted in increasing order,T i + 1> T i. Exchanges («) are attempted (…) with frequency ν ex. Unlike REX, only an essentialsubspace {es} (red boxes) containing a few collective modes is excited within each TEE-REXreplica. Reference replica (T 0, T 0), containing an approximate Boltzmann ensemble, is used foranalysis