Universality in conformations and transverse fluctuations of a semi-flexible polymer in a crowded environment

TL;DR

This study reveals universal scaling behaviors of semi-flexible polymers' conformations and fluctuations in 2D and 3D environments, including crowded conditions, with implications for biological systems like DNA.

Contribution

It extends universal scaling relations for polymer conformations to 3D and examines the effects of crowding, showing weak influence on these relations.

Findings

Scaled end-to-end distance and transverse fluctuations collapse onto a master curve.

Gaussian regime is very narrow in 3D, absent in 2D due to EV interactions.

Crowding density has minimal effect on universal scaling relations.

Abstract

We study universal aspects of polymer conformations and transverse fluctuations for a single swollen chain characterized by a contour length $L$ and a persistence length $\ell_p$ in two dimensions (2D) and in three dimensions (3D) in the bulk, as well as in the presence of excluded volume (EV) particles of different sizes occupying different volume fractions. In the absence of the EV particles we extend the previously established universal scaling relations in 2D (A. Huang, A. Bhattacharya, and K. Binder, J. Chem. 140, 214902 (2014)) to include 3D and demonstrate that the scaled end-to-end distance $\langle R_N^2\rangle/(2 L\ell_p)$ and the scaled transverse fluctuation $\sqrt{\langle{l_{\perp}^2}\rangle}/{L}$ as a function of $L/\ell_p$ collapse onto the same master curve, where $\langle R_N^2\rangle$ and $\langle{l_{\perp}^2\rangle}$ are the mean-square end-to-end distance and transverse fluctuations. However, unlike in 2D, where the Gaussian regime is absent due to extreme dominance of the EV interaction, we find the Gaussian regime is present, albeit very narrow in 3D. The scaled transverse fluctuation in the limit $L/\ell_p \ll 1$ is independent of the physical dimension and scales as $\sqrt{\langle{l_{\perp}^2}\rangle}/{L} \sim (L/\ell_p)^{\zeta-1}$, where $\zeta = 1.5$ is the roughening exponent. For $L/\ell_p \gg 1$ the scaled fluctuation scales as $\sqrt{\langle{l_{\perp}^2}\rangle}/{L} \sim (L/\ell_p)^{\nu-1}$, where $\nu$ is Flory exponent for the corresponding spatial dimension ($\nu_{2D}=0.75$, and $\nu_{3D}=0.58$). When EV particles are added into the system, our results indicate that the crowding density either does not or only weakly affects the universal scaling relations. We discuss the implications of these results in living matter by showing the experimental result for a dsDNA onto the master plot.