Compounding formula approach to chromatin and active polymer dynamics

TL;DR

The paper develops a scaling framework for active polymers, notably an active Rouse model driven by persistent, non-Markovian noise, and applies it to chromatin-like systems. It introduces a compounding formula that links the mean-squared displacement of a monomer in a long chain to that of an isolated monomer through a tension-propagation–defined effective connectivity $m(\tau)$, capturing how dynamical correlations spread along the chain. By deriving explicit transient and steady-state MSD expressions for various noise correlators $g(u)$, the work reveals distinct scaling regimes, including short-time ballistic-like $\tau^2$ and tension-propagation–controlled $\tau^{3/2}$ or $\tau^{1/2}$ behavior, depending on noise memory. The framework reconciles prior conflicting results in active polymer dynamics, generalizes to other spatially extended systems (e.g., crumpled globules, semiflexible polymers, single-file active particles), and provides a tractable route to incorporate non-Markovian active fluctuations into scaling analyses of complex biomolecular and soft-matter dynamics.

Abstract

Active polymers are ubiquitous in nature, and often kicked by persistent noises that break detailed balance. In order to capture the out-of-equilibrium dynamics of such active polymers, we propose a simple yet reliable analytical framework based on a compounding formula. Connecting polymeric dynamics to the isolated monomeric behavior via the notion of tension propagation, the formula allows us to clarify rich scaling scenarios alongside corresponding intuitive physical pictures. We demonstrate distinctive transient and steady-state scalings due to the non-Markovian nature of the active noise. Aside from a paradigmatic example of an active Rouse polymer, we expect the framework to be applicable to wide variety of spatially extended systems including more general polymers (crumpled globule, semiflexible polymers etc), fluctuation of growing interface, and an array of particles in single-file configuration.

Figures (6)

• Figure 1: The compounding formula \ref{['CF2']} connects the MSD of a monomer of the chain (left) to that of an isolated monomer with an effective friction $m(\tau) \gamma$, with $m$ the number of "dynamically connected monomers" (right).
• Figure 2: The two cases considered. The red dashed area denotes the time interval during which the active noise is applied. In the transient case passive thermal white noise acts up to time $t=0$. In the steady state case the initial condition is set at a time $-T_\infty$, with $T_\infty$ longer than any characteristic times so that at time $t=0$ an active steady state is reached. In both cases the MSD is calculated in the time interval $[0,\tau]$.
• Figure 3: Test of the validity of the compounding formula \ref{['CF2']} for the transient case and three different noises: (a) $g(u) = e^{-u/\tau_A}$, (b) $g(u) = [100 \, e^{-10u/\tau_A} + e^{-u/(10\tau_A)}]/101$ and (c) $g(u) = 1/[1+(u/\tau_A)^{0.7}]$. These are all normalized so that $g(0)=1$. The solid lines are plots of Eq. \ref{['msd-tr']}, the tagged monomer MSD of a connected polymer. The dashed lines are plots of the ratio between Eqs. \ref{['msd-i']} and \ref{['tension_prop']}. The short time MSD scales as $\sim \tau^{3/2}$, while it asymptotically converges at long times to a $\sim \tau^{1/2}$ scaling in the exponentially correlated noise (a) and (b). The power-law correlated noise (c) generates a MSD with a different power-law behavior, see text.
• Figure 4: Test of the compounding formula for the steady state case for the same three noises (a,b,c) as for the transient case of Fig. \ref{['fig:compound']}. The long time scale behavior is the same for steady state and transient case, but the short time scale is characterized by a MSD $\sim \tau^2$ in the former. See text for an explanation. Inset: plot of $H(\tau,n)$ vs. $\tau/\tau_A$ and fixed $n$ for equilibrium dynamics (red circles) and active steady state with exponentially correlated noise (green circles). The latter shows a short time power law ($\sim \tau^2$) behavior.
• Figure 5: Summary of the scaling prediction of the tagged monomer MSD in active Rouse polymer kicked by persistent noise (exponential correlation in this example). Time evolution follows a sequence of regimes $1 \rightarrow 2 \rightarrow 3$ in the transient case, and $4 \rightarrow 3$ in the steady-state case.