Introduction

A class of simulation algorithms closely related to REM (see Chapter 5) are the so-called serial generalized-ensemble (SGE) methods[46]. The basic difference between SGE methods and REM is that in the former no pairs of replicas are necessary to make a trajectory in temperature space and more generally in the generalized ensemble space. In SGE methods only one replica can undergo ensemble transitions which are realized on the basis of a Monte Carlo like criterion. The most known example of SGE algorithm is the simulated tempering (ST) technique[44,47], where weighted sampling is used to produce a random walk in temperature space. An important limitation of SGE approaches is that an evaluation of free energy differences between ensembles is needed as input to ensure equal visitation of the ensembles, and eventually a faster convergence of structural properties[48]. REM was just developed to eliminate the need to know a priori such free energy differences.

ST and temperature-REM yield an extensive exploration of the phase space without configurational restraints. This allows to recover not only the global minimum-energy state, but also any equilibrium thermodynamic quantity as a function of temperature. The potential of mean force (PMF)[51,52] along a chosen collective coordinate can also be computed a posteriori by multiple-histogram reweighting techniques[53,54]. PMF can also be determined by performing generalized-ensemble canonical simulations in the space of the collective coordinate[55] (for example the space of the end-to-end distance of a biopolymer). Comparisons between ST and temperature-REM have been reported[48,49,50]. The overall conclusions of these studies are that ST consistently gives a higher rate of delivering the system between high temperature states and low temperature states, as well as a higher rate of crossing the potential energy space. Moreover ST is well-suited to distributed computing environments because synchronization and communication between replicas/processors can be avoided. On the other side, an effective application of ST and, in general, of SGE methods requires a uniform exploration of the ensemble-space. In order to satisfy this criterion, acceptance rates must be not only high but also symmetric between forward and backward directions of the ensemble-space. This symmetry can be achieved by performing weighted sampling, where weights are correlated with the dimensionless free energies of the ensembles. The knowledge of such free energies is not needed in REM because replica exchanges occur between microstates of the same extended thermodynamic ensemble. To achieve rapid sampling of the ensemble-space through high acceptance rates, we need to choose ensembles appropriately so that neighboring ensembles overlap significantly. As stated above, the most critical aspect in SGE schemes is the determination of weight factors (viz. dimensionless free energy differences between neighboring ensembles). This issue has been the subject of many studies, especially addressed to ST simulations. The first attempts are based on short trial simulations[47,108,109]. The proposed procedures are however quite complicated and computationally expensive for systems with many degrees of freedom. Later, Mitsutake and Okamoto suggested to perform a short REM simulation to estimate ST weight factors[110] via multiple-histogram reweighting[53,54]. A further approximated, but very simple, approach to evaluate weight factors is based on average energies calculated by means of conventional molecular dynamics simulations[111]. The weight factors obtained by the average-energy method of Ref. [111] were later demonstrated to correspond to the first term of a cumulant expansion of free energy differences[49]. Huang et al. used approximated estimates of potential energy distribution functions (from short trial molecular dynamics simulations) to equalize the acceptance rates of forward and backward transitions between neighboring temperatures, ultimately leading to a uniform temperature sampling in ST[112]. The techniques illustrated above have been devised to determine weight factors to be used without further refinement[110] or as an initial guess to be updated during the simulation[112,111]. In the former case, these approximate factors should (hopefully) ensure an almost random walk through the ensemble-space. However, as remarked in Ref. [48], the estimate of accurate weight factors may be very difficult for complex systems. Inaccurate estimates, though unaffecting the basic principles of SGE methods, do affect the sampling performances in terms of simulation time needed to achieve convergence of structural properties[48].

As discussed above, dimensionless free energy differences between ensembles (viz. weight factors) may also be the very aim of the simulation[55] (since they correspond to the PMF along the chosen coordinate). In such cases, accurate determination of weight factors is not simply welcome, but necessary. This can be done a posteriori using multiple-histogram reweighting techniques[53,54], or with more or less efficient updating protocols applied during the simulation[113,112,114,48,115].

In the ORAC program we have implemented SGE simulations, either in a ST-like fashion or in the space of bond, bending and torsional coordinates. These simulations exploit the adaptive method to calculate weight factors developed in Ref. [56]. Such method, called BAR-SGE, is based on a generalized expression[116,117] of the Bennett Acceptance Ratio[118] (BAR) and free energy perturbation[119]. It is asymptotically exact and requires a low computational time per updating step. The algorithm is suited, not only to calculate the free energy on the fly during the simulation, but also as a possible criterion to establish whether equilibration has been reached.