Extracting Slow Fluctuations from Simulation Data

Traditionally, research in physics was conducted in two distinct fields: experimental physics and theoretical physics. In recent years, with the remarkable advancement of computers, the field of computational physics, which uses computers to conduct research in physics, has become recognized as a third field of physics.

An important method in computational physics is computer simulation. This involves conducting simulated experiments on a computer. For example, our group conducts simulations of proteins in water. Specifically, we form molecules by bonding atoms, assume the forces acting between them, and solve Newton's equations of motion to determine the positions and structures of water and protein molecules at short time steps (on the order of femtoseconds). This method is called molecular dynamics simulation and is frequently used in protein research. Currently, even with computers equipped with GPGPUs that can be purchased for a laboratory, it is possible to simulate systems of about 100,000 atoms for microseconds, while specialized computers for protein systems now allow for simulations on the order of milliseconds.

Normal proteins adopt a specific, stable three-dimensional structure in water. Each type of protein performs a specific function within a living organism, and it is known that this function is related not only to the protein's stable average structure but also to the fluctuations of the surrounding structure. In particular, slow (slowly time-varying) structural fluctuations are considered important. Let us consider studying the structural fluctuations of a protein through simulation. In a simulation, the structure of the protein at each moment (the position coordinates of its constituent atoms) is recorded sequentially at certain time intervals. As the scale of computations has increased in recent years, the amount of recorded data has also become enormous. Previously, a method called principal component analysis was used to examine structural fluctuations from this vast amount of data. This is a method that successively seeks the directions in which the projected structural fluctuations are maximized when projected onto a certain axis. The results of this method do not depend on the order in which the studied structures were recorded. In other words, it does not use information from the time axis. In contrast, our group has proposed a method of relaxation mode analysis that uses information from the time axis. In this method, we successively seek the directions in which the projected structural fluctuations are the slowest. It is a method for systematically extracting fluctuations in order of "slowness" rather than "largeness."

Let's look at an application example. Figure (c) below shows the direction of the slowest fluctuation, extracted by relaxation mode analysis from a simulation of a complex formed by one protein molecule and two other protein molecules. The right side of figure (a) shows the time evolution of the fluctuation value projected in that direction. We can see that it changes only once from a small value to a large value during the simulation. The relaxation mode analysis method allows us to extract such "rare events." Figure (b) shows which parts of the protein contribute to this fluctuation, revealing that specific positions contribute significantly.

The relaxation mode analysis method is a versatile method for analyzing the vast amount of time-ordered data obtained from simulations and can be applied to various systems beyond just protein systems.