In this article, the general formulation of the multidimensional simulated-tempering replica-exchange method is described. In previous works, the one-dimensional replica-exchange simulated-tempering and simulated-tempering replica-exchange methods were developed. For the former method, the weight factor of the one-dimensional simulated tempering is determined by a short replica-exchange simulation and multiple-histogram reweighing techniques. For the latter method, the production run is a replica-exchange simulation with a few replicas not in the canonical ensembles but in the simulated-tempering ensembles which cover wide ranges of temperature. Recently, the general formulation of the multidimensional replica-exchange simulated tempering was presented. In this article, the extension of the simulated-tempering replica-exchange method for the multidimensional version is given. As an example of applications of the algorithm, a two-dimensional replica-exchange simulation and two simulated-tempering replica-exchange simulations have been performed. Here, an α-helical peptide system with a model solvent has been used for the applications.

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