Effective Hamiltonian of topologically stabilized polymer states

TL;DR

This paper demonstrates that a simple quadratic Hamiltonian can produce fractal polymer conformations with variable dimensions, providing a potential microscopic model for topologically stabilized states like mitotic chromosomes.

Contribution

It introduces a Gaussian network model capable of generating fractal conformations with adjustable dimensions, aligning well with numerical data and offering new mathematical representations of fractal Brownian motion.

Findings

Fractal conformations with dimensions between 2 and 3 are achievable with a quadratic Hamiltonian.

Monomer-to-monomer distances follow a Gaussian distribution in these states.

The model maps polymer conformations onto subdiffusive fractal Brownian trajectories.

Abstract

Topologically stabilized polymer conformations observed in melts of nonconcatenated polymer rings and crumpled globules, are considered to be a good candidate for the description of the spatial structure of mitotic chromosomes. Despite significant efforts, the microscopic Hamiltonian capable of describing such systems, remains yet inaccessible. In this paper we consider a Gaussian network - a system with a simple Hamiltonian quadratic in all coordinates - and show that by tuning interactions, one can obtain fractal equilibrium conformations with any fractal dimension between 2 (ideal polymer chain) and 3 (crumpled globule). Monomer-to-monomer distances in topologically stabilized states, according to our analysis of available numerical data, fit very well the Gaussian distribution, giving an additional argument in support of the quadratic Hamiltonian model. Mathematically, the resulting polymer conformations can be mapped onto the trajectories of a subdiffusive fractal Brownian particle. As a by-product of our study, two novel continual integral representations of the fractal Brownian motion are proposed.