Stretched exponential relaxation in polymer dynamics

Internal friction as a factor in the anomalous chain length dependence of DNA transcriptional dynamics

Recent experiments by Brückner et al. [Science 380, 1357 (2023)] have observed an anomalous chain length dependence of the time of near approach of widely separated pairs of genomic elements on transcriptionally active chromosomal DNA. In this paper, I suggest that the anomaly may have its roots in internal friction between neighboring segments on the DNA backbone. The basis for this proposal is a model of chain dynamics formulated in terms of a continuum scaled Brownian walk (sBw) of polymerization index N. The sBw is an extension of the simple Brownian walk model widely used in path integral calculations of polymer properties, differing from it in containing an additional parameter H (the Hurst index) that can be tuned to produce varying degrees of correlation between adjacent monomers. A calculation using the sBw of the mean time τc for chain closure predicts-under the Wilemski-Fixman approximation for diffusion-controlled reactions-that at early times, τc varies as the 2/3 power of N, in close agreement with the findings of the Brückner et al. study. Other scaling relations of that study, including those related to the probability of loop formation and the mean square displacements of terminal monomers, are also satisfactorily accounted for by the model.

Theory of structural relaxation in glass from the thermodynamics of irreversible processes

This work proposes a fundamental thermodynamic description of structural relaxation in glasses by establishing a link between the Prony series solution to volume relaxation derived from the principles of irreversible thermodynamics and asymmetric Lévy stable distribution of relaxation rates. Additionally, it is shown that the bulk viscosity of glass, and not the shear viscosity, is the transport coefficient governing structural relaxation. We also report the distribution of relaxation times and energy barrier heights underpinning stretched exponential relaxation. It is proposed that this framework may be used for qualitative and quantitative descriptions of the relaxation kinetics in glass.

Pattern formation in reaction-diffusion systems in the presence of non-Markovian diffusion

We study reaction-diffusion systems beyond the Markovian approximation to take into account the effect of memory on the formation of spatiotemporal patterns. Using a non-Markovian Brusselator model as a paradigmatic example, we show how to use reductive perturbation to investigate the formation and stability of patterns. Focusing in detail on the Hopf instability and short-term memory, we derive the corresponding complex Ginzburg-Landau equation that governs the amplitude of the critical mode and we establish the explicit dependence of its parameters on the memory properties. Numerical solution of this memory-dependent complex Ginzburg-Landau equation as well as direct numerical simulation of the non-Markovian Brusselator model illustrates that memory changes the properties of the spatiotemporal patterns. Our results indicate that going beyond the Markovian approximation might be necessary to study the formation of spatiotemporal patterns even in systems with short-term memory. At the same time, our work opens up a new window into the control of these patterns using memory.

Role of non-equilibrium conformations on driven polymer translocation

One of the major theoretical methods in understanding polymer translocation through a nanopore is the Fokker-Planck formalism based on the assumption of quasi-equilibrium of polymer conformations. The criterion for applicability of the quasi-equilibrium approximation for polymer translocation is that the average translocation time per Kuhn segment, ⟨τ⟩/NK, is longer than the relaxation time τ0 of the polymer. Toward an understanding of conditions that would satisfy this criterion, we have performed coarse-grained three dimensional Langevin dynamics and multi-particle collision dynamics simulations. We have studied the role of initial conformations of a polyelectrolyte chain (which were artificially generated with a flow field) on the kinetics of its translocation across a nanopore under the action of an externally applied transmembrane voltage V (in the absence of the initial flow field). Stretched (out-of-equilibrium) polyelectrolyte chain conformations are deliberately and systematically generated and used as initial conformations in translocation simulations. Independent simulations are performed to study the relaxation behavior of these stretched chains, and a comparison is made between the relaxation time scale and the mean translocation time (⟨τ⟩). For such artificially stretched initial states, ⟨τ⟩/NK < τ0, demonstrating the inapplicability of the quasi-equilibrium approximation. Nevertheless, we observe a scaling of ⟨τ⟩ ∼ 1/V over the entire range of chain stretching studied, in agreement with the predictions of the Fokker-Planck model. On the other hand, for realistic situations where the initial artificially imposed flow field is absent, a comparison of experimental data reported in the literature with the theory of polyelectrolyte dynamics reveals that the Zimm relaxation time (τZimm) is shorter than the mean translocation time for several polymers including single stranded DNA (ssDNA), double stranded DNA (dsDNA), and synthetic polymers. Even when these data are rescaled assuming a constant effective velocity of translocation, it is found that for flexible (ssDNA and synthetic) polymers with NK Kuhn segments, the condition ⟨τ⟩/NK < τZimm is satisfied. We predict that for flexible polymers such as ssDNA, a crossover from quasi-equilibrium to non-equilibrium behavior would occur at NK ∼ O(1000).

Relaxation of ultralarge VWF bundles in a microfluidic-AFM hybrid reactor

The crucial role of the biopolymer "Von Willebrand factor" (VWF) in blood platelet binding is tightly regulated by the shear forces to which the protein is exposed in the blood flow. Under high-shear conditions, VWFs ability to immobilize blood platelets is strongly increased due to a change in conformation which at sufficient concentration is accompanied by the formation of ultra large VWF bundles (ULVWF). However, little is known about the dynamic and mechanical properties of such bundles. Combining a surface acoustic wave (SAW) based microfluidic reactor with an atomic force microscope (AFM) we were able to study the relaxation of stretched VWF bundles formed by hydrodynamic stress. We found that the dynamical response of the network is well characterized by stretched exponentials, indicating that the relaxation process proceeds through hopping events between a multitude of minima. This finding is in accordance with current ideas of VWF self-association. The longest relaxation time does not show a clear dependence on the length of the bundle, and is dominated by the internal conformations and effective friction within the bundle.

Non-Markovian Lévy diffusion in nonhomogeneous media

We study the diffusion equation with a position-dependent, power-law diffusion coefficient. The equation possesses the Riesz-Weyl fractional operator and includes a memory kernel. It is solved in the diffusion limit of small wave numbers. Two kernels are considered in detail: the exponential kernel, for which the problem resolves itself to the telegrapher's equation, and the power-law one. The resulting distributions have the form of the Lévy process for any kernel. The renormalized fractional moment is introduced to compare different cases with respect to the diffusion properties of the system.

Solutions of a class of non-Markovian Fokker-Planck equations

We show that a formal solution of a rather general non-Markovian Fokker-Planck equation can be represented in a form of an integral decomposition and thus can be expressed through the solution of the Markovian equation with the same Fokker-Planck operator. This allows us to classify memory kernels into safe ones, for which the solution is always a probability density, and dangerous ones, when this is not guaranteed. The first situation describes random processes subordinated to a Wiener process, while the second one typically corresponds to random processes showing a strong ballistic component. In this case the non-Markovian Fokker-Planck equation is only valid in a restricted range of parameters, initial and boundary conditions.