NUMEV Seminar: “Data-Driven Modeling and Time Series Forecasting Using the Generalized Langevin Equation”

“Data-driven modeling and time series forecasting using the generalized Langevin equation”

Roland Netz, Department of Physics, Free University of Berlin

Abstract

Most systems of scientific interest are interacting many-body systems. Their kinetics are typically described in terms of a low-dimensional reaction coordinate, which is generally influenced by the entire system. The dynamics of such a reaction coordinate are governed by the generalized Langevin equation (GLE), an integro-differential stochastic equation, and involve a memory function that describes how the dynamics depend on previous values of the reaction coordinate. The GLE is thus a systematic non-Markovian description of a system’s dynamics in terms of coarse-grained variables. We have recently introduced a novel hybrid projection scheme that allows us to extract the GLE parameters from time series data in a form suitable for both analytical and numerical treatments. In this talk, I will discuss several examples where the GLE can be used to interpret and model data across various scientific fields.

Protein-folding kinetics is typically described as Markovian (i.e., memoryless) diffusion in a one-dimensional free-energy landscape, governed by an instantaneous friction coefficient that is fitted to reproduce experimental or simulated folding times. According to this view, the folding time is dominated by the exponential term of the folding free-energy barrier, the Arrhenius factor, where the friction coefficient only determines the pre-exponential time scale and plays a subordinate role. Analysis of large-scale molecular dynamics simulation trajectories of fast-folding proteins from the Shaw group, performed using the special-purpose computer ANTON, demonstrates that the friction characterizing protein folding exhibits significant memory with a decay time of the same order of magnitude as the folding and unfolding times. Non-Markovian modeling not only accurately reproduces simulations but also demonstrates that memory friction effects lead to anomalous and drastically altered protein kinetics. For the set of proteins for which simulations are available, it is shown that the folding and unfolding times are not dominated by the free-energy barrier but rather by non-Markovian friction.

Memory effects are also present in non-equilibrium systems. Using an appropriate non-equilibrium formulation of the GLE, it is shown that the motion of living organisms is characterized by memory friction, which makes it possible to characterize the internal feedback loops of such organisms and to classify and sort individual organisms. The GLE can even be used to predict complex phenomena such as weather patterns.

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