Log-periodic oscillations as real-time signatures of hierarchical dynamics in proteins

Emanuel Dorbath; Adnan Gulzar; Gerhard Stock

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arXiv · 2311.11839

Log-periodic oscillations as real-time signatures of hierarchical dynamics in proteins

Emanuel Dorbath·Adnan Gulzar·Gerhard Stock

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Abstract

The time-dependent relaxation of a dynamical system may exhibit a power-law behavior that is superimposed by log-periodic oscillations. Sornette [Phys. Rep. 297, 239 (1998)] showed that this behavior can be explained by a discrete scale invariance of the system, which is associated with discrete and equidistant timescales on a logarithmic scale. Examples include such diverse fields as financial crashes, random diffusion, and quantum topological materials. Recent time-resolved experiments and molecular dynamics simulations suggest that discrete scale invariance may also apply to hierarchical dynamics in proteins, where several fast local conformational changes are a prerequisite for a slow global transition to occur. Employing entropy-based timescale analysis and Markov state modeling to a simple one-dimensional hierarchical model and biomolecular simulation data, it is found that hierarchical systems quite generally give rise to logarithmically spaced discrete timescales. By introducing a one-dimensional reaction coordinate that collectively accounts for the hierarchically coupled degrees of freedom, the free energy landscape exhibits a characteristic staircase shape with two metastable end states, which causes the log-periodic time evolution of the system. The period of the log-oscillations reflects the effective roughness of the energy landscape, and can in simple cases be interpreted in terms of the barriers of the staircase landscape.

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Emanuel Dorbath, Adnan Gulzar, Gerhard Stock. 2023-11-20. Log-periodic oscillations as real-time signatures of hierarchical dynamics in proteins. https://arxiv.org/abs/2311.11839

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