Dynamical mean-field theory of active, self-interacting heteropolymers

Abstract

Analytical theories of polymer dynamics typically focus on the linear response around some steady state obtained via perturbative calculations. The present study derives a dynamical, nonlinear mean-field theory that yields unified access to the evolution of conformations, contact probabilities, and fluctuations of active polymers within a single analytical framework. As in previous mean-field descriptions of passive polymers, the object of interest is the two-point correlator, which is the lowest-order relevant observable in the general Langevin equation of a polymer. In contrast, the present focus is on active polymers driven by heterogeneous monomer-level kicks and spatially correlated active flows, while allowing nonreciprocal couplings, in the presence of hydrodynamic and contact interactions. Using a Gaussian closure to truncate higher-order correlations leads to a self-consistent diffusion equation for the chain correlations. In some cases, this can be further simplified to the dynamics of pairwise squared separations in closed form. In the continuum limit, the theory reveals useful analogies to similar mean-field approximations for reaction-diffusion systems, as well as fractional diffusion in different scale-free regimes. Deriving asymptotic solutions in different parameter regimes of the continuum limit recovers the scalings of coiled, globular, and self-avoiding polymers, thus validating the theory. Applied to hydrodynamically coupled active polymers, such as chromatin, the framework predicts a sequence of crossover scalings between coiled, swollen, and hyper-compacted fractal states.