We apply moving root-mean-square deviation (mRMSD), which does not require a reference structure, as a method for analyzing protein dynamics. This method can be used to calculate the root-mean-square deviation (RMSD) of structure between two specified time points and to analyze protein dynamics behavior through time series analysis. We applied this method to the Trp-cage trajectory calculated by the Anton supercomputer and found that it shows regions of stable states as well as the conventional RMSD. In addition, we extracted a characteristic structure in which the side chains of Asp1 and Arg16 form hydrogen bonds near the most stable structure of the Trp-cage. We also determined that ≥20 ns is an appropriate time interval to investigate protein dynamics using mRMSD. Applying this method to NuG2 protein, we found that mRMSD can be used to detect regions of metastable states in addition to the stable state. This method can be applied to molecular dynamics simulations of proteins whose stable structures are unknown.

Protein conformational fluctuations are highly complex and exhibit long-term correlations. Here, molecular dynamics simulations of small proteins demonstrate that these conformational fluctuations directly affect the protein’s instantaneous diffusivity $D_I$. We find that the radius of gyration $R_g$ of the proteins exhibits $1/f$ fluctuations that are synchronous with the fluctuations of $D_I$. Our analysis demonstrates the validity of the local Stokes-Einstein–type relation $D_I\propto 1/(R_g+R_0)$, where $R_0\sim 0.3\mathrm{nm}$ is assumed to be a hydration layer around the protein. From the analysis of different protein types with both strong and weak conformational fluctuations, the validity of the Stokes-Einstein–type relation appears to be a general property.

We give general formulations of the multidimensional multicanonical algorithm, simulated tempering, and replica-exchange method. We generalize the original potential energy function E0 by adding any physical quantity V of interest as a new energy term. These multidimensional generalized-ensemble algorithms then perform a random walk not only in E0 space but also in V space. Among the three algorithms, the replica-exchange method is the easiest to perform because the weight factor is just a product of regular Boltzmann-like factors, while the weight factors for the multicanonical algorithm and simulated tempering are not a priori known. We give a simple procedure for obtaining the weight factors for these two latter algorithms, which uses a short replica-exchange simulation and the multiple-histogram reweighting techniques. As an example of applications of these algorithms, we have performed a two-dimensional replica-exchange simulation and a two-dimensional simulated-tempering simulation using an α-helical peptide system. From these simulations, we study the helix-coil transitions of the peptide in gas phase and in aqueous solution.

This article reports the first attempt to apply the relaxation mode analysis method to a simulation of a biomolecular system. In biomolecular systems, the principal component analysis is a well-known method for analyzing the static properties of fluctuations of structures obtained by a simulation and classifying the structures into some groups. On the other hand, the relaxation mode analysis has been used to analyze the dynamic properties of homopolymer systems. In this article, a long Monte Carlo simulation of Met-enkephalin in gas phase has been performed. The results are analyzed by the principal component analysis and relaxation mode analysis methods. We compare the results of both methods and show the effectiveness of the relaxation mode analysis.

We discuss multidimensional generalizations of multicanonical algorithm, simulated tempering, and replica-exchange method. We generalize the original potential-energy function $E_0$ by adding any physical quantity $V$ of interest as a new energy term with a coupling constant $\lambda$. We then perform a multidimensional multicanonical simulation where a random walk in $E_0$ and $V$ spaces is realized. We can alternately perform a multidimensional simulated-tempering simulation where a random walk in temperature $T$ and parameter $\lambda$ is realized. The results of the multidimensional replica-exchange simulations can be used to determine the weight factors for these multidimensional multicanonical and simulated-tempering simulations.

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.