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

Aug 2022

We study universal aspects of polymer conformations and transversefluctuations for a single swollen chain characterized by a contour length $L$and a persistence length $\ell_p$ in two dimensions (2D) and in threedimensions (3D) in the bulk, as well as in the presence of excluded volume (EV)particles of different sizes occupying different volume fractions. In theabsence of the EV particles we extend the previously established universalscaling 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-enddistance $\langle R_N^2\rangle/(2 L\ell_p)$ and the scaled transversefluctuation $\sqrt{\langle{l_{\perp}^2}\rangle}/{L}$ 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 andtransverse fluctuations. However, unlike in 2D, where the Gaussian regime isabsent due to extreme dominance of the EV interaction, we find the Gaussianregime is present, albeit very narrow in 3D. The scaled transverse fluctuationin the limit $L/\ell_p \ll 1$ is independent of the physical dimension andscales 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 scaledfluctuation scales as $\sqrt{\langle{l_{\perp}^2}\rangle}/{L} \sim(L/\ell_p)^{\nu-1}$, where $\nu$ is Flory exponent for the correspondingspatial dimension ($\nu_{2D}=0.75$, and $\nu_{3D}=0.58$). When EV particles areadded into the system, our results indicate that the crowding density eitherdoes not or only weakly affects the universal scaling relations. We discuss theimplications of these results in living matter by showing the experimentalresult for a dsDNA onto the master plot.

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