Gaussian closure and dynamical mean-field theory for self-avoiding heteropolymers

Abstract

Analytical treatments of polymer dynamics have mostly been restricted to linear response theory around some steady state obtained via perturbative field theory. Here, I derive an analytical framework that yields unified access to the evolution of conformations, contact probabilities, and fluctuations within a dynamical mean-field theory. Starting with the Langevin equation of a hydrodynamically coupled and self-avoiding heteropolymer, the key idea is to focus on the two-point correlator as the lowest-order relevant observable. Truncating higher-order correlations via a Gaussian closure leads to a self-consistent diffusion equation for the chain correlations. The theory is validated by contrasting coiled, globular, and self-avoiding polymers within a single dynamical framework, and predicts hyper-compacted fractal states in hydrodynamically coupled active polymers such as chromatin.

Figures (2)

• Figure 1: Effective diffusion of pairwise correlations. The dynamical mean-field theory, Eq. \ref{['eq:dynamic_mean_field_full']}, can be interpreted as a nonlinear, non-local diffusion-reaction process at the level of pairwise correlations and contact maps. Admissible dynamics in absence of hydrodynamic interactions are shown. In nonreciprocal generalizations, the dashed lines can vanish.
• Figure 2: Asymptotic behavior of a polymer in thermal equilibrium. Classical scalings emerge from a competition between different asymptotic terms in the dynamical mean-field theory, Eq. \ref{['eq:dynamic_mean_field_full']}. Hydrodynamic interactions were neglected because they are irrelevant for equilibrium structure.