Collapse of a single polymer chain: Effects of chain stiffness and attraction range

TL;DR

This work probes how chain stiffness $l_p$ and attraction range $r_c$ shape the collapse of a single polymer chain using Pruned-Enriched Rosenbluth Method (PERM) Monte Carlo on a 3D lattice. The results reveal two regimes: when $l_p lose to r_c$, collapse is sharp and often first-order for stiff chains, while for $l_p arrow r_c$ the transition is rounded and gradual, with the rounding persisting at large $N$. The theta-temperature $T_ heta$ can either increase with $l_p$ at small $r_c$ or decrease with $l_p$ at large $r_c$, illustrating a nontrivial interplay between stiffness and attraction range. These findings provide a coherent framework for understanding DNA/RNA condensation phenomena and guide design principles for responsive polymer systems, with implications for biomolecular folding and nanomaterials.

Abstract

Chain-like macromolecules in solution, whether biological or synthetic, transform from an extended conformation to a compact one when temperature or other system parameters change. This collapse transition is relevant in various phenomena, including DNA condensation, protein folding, and the behavior of polymers in solution. We investigate the interplay of chain stiffness and range of attraction between monomers in the collapse of a single polymer chain. We use Monte Carlo simulations based on the pruned-enriched Rosenbluth method. Two distinct behaviors are found depending on chain stiffness (represented by the persistence length lp) and attraction range rc. When lp is larger than rc, the chain collapses sharply with decreasing temperature, whereas if lp is smaller than rc, it contracts gradually. Notably, in the regime of small lp and large rc, this rounding into a gradual compaction persists upon increasing the chain length and may remain in place in the limit of infinite chain length. Furthermore, for small rc, the transition temperature (theta-temperature) increases with lp, whereas for large rc the theta-temperature decreases with lp. Thus, stiffness promotes collapse for small rc but suppresses it for large rc. Our findings are in agreement with recent experiments on the contraction of single-stranded RNA as compared to double-stranded DNA, and provide valuable insights for understanding polymer collapse and the essential polymer parameters affecting it.

Figures (9)

• Figure 1: (a) Swelling factor $\langle R_g^2\rangle/N$ of the gyration radius as a function of rescaled temperature $T^{*}\equiv k_{\rm B} T/\varepsilon$, plotted on a semi-log plot. Different symbols correspond to different values of the persistence length: $l_p\,{=}\,0$ (circle), $1$ (square), $2$ (triangle), and $3$ (diamond), while different colors denote different values of the attraction range: $r_c\,{=}\,1$ (blue), $2$ (red), and $3$ (green). The curves are fits to a sigmoid-like function [Eq. (\ref{['fit']})]. For $l_p\,{=}\,3$, the fit is less accurate but the derivative peak remains reliable. (b) and (c) Derivative $d(\langle R_g^2\rangle/N)/dT^*$ of the fitted curve as a function of the rescaled temperature $T^*$ on a semi-logarithmic scale, for different values of $l_p$ and $r_c$ (legend in (a)). (b) $l_p \,{=}\,3$; (c) $l_p\,{=}\, 0$, $1$, $2$. The inset in (c) shows an enlarged view of the $r_c\,{=}\,2$ and $3$ curves over a narrow range of small derivative values, highlighting the very weak peaks. The data correspond to chains with $N\,{=}\,500$, and the bond length $b$ is taken as the unit length.
• Figure 2: (a) and (c) Swelling factor $\langle R_g^2\rangle/N$ as a function of the rescaled temperature $T^{*}$ on a semi-log plot. (a) for $l_p\,{=}\,3$, and (c) for $l_p \,{=}\, 0$, $1$, $2$. The three colors correspond to different $r_c$ values, as in Fig. \ref{['fig1']} (see also the legend). Lines are guides to the eye. The inset in (a) shows an enlarged view of the $r_c \,{=}\, 2$ curve in the temperature range $19\le T^{*} \le 22$. (b) and (d) Probability distribution histograms of $R_g$ for different temperatures: $T^{*}\,{=}\,18.0$ (red), $T^{*}\,{=}\,19.0$ (blue), $T^{*}\,{=}\,20.0$ (green), and $T^{*}\,{=}\,21.0$ (black). For all temperatures in (b) $l_p\,{=}\,3$ and $r_c\,{=}\,2$, and in (d) $l_p\,{=}\,0$ and $r_c\,{=}\,2$. Only the histogram outline is shown for clarity. Results are for chains of length $N\,{=}\,500$, and the bond length $b$ is taken as the unit length.
• Figure 3: Swelling factor, $\langle R_g^2\rangle/N$, as a function of chain length, $N$, plotted on a semi-log plot. (a) $l_p\,{=}\,0, r_c\,{=}\,2$ for $T^{*}\equiv k_{\rm B} T/\varepsilon\,{=}\,18.7,\, 18.8,\, 18.9,\, 19.0,$ and $19.1$ (see legend). The transition temperature is estimated as $T^{*}\simeq 18.9 \pm 0.1$. (b) $l_p\,{=}\,3, r_c\,{=}\,2$ for $T^{*}\,{=}\,20.6,\, 20.7,\, 20.8,\, 20.9,\, 21.0,$ and $21.1$ (see legend). The transition temperature is estimated as $T^{*}\simeq 21.0 \pm 0.1$.
• Figure 4: Dependence of $T^{*}_\theta/r_c^3$ on $l_p$ for $r_c\,{=}\,1, 2,$ and $3$. All error bars are smaller than the symbol size. The chain length in all cases is $N\,{=}\,500$, and the bond length $b$ is taken as the unit length.
• Figure 5: (a) A typical chain configuration for $l_p\,{=}\,3$ and $r_c\,{=}\,1$ at $T^{*}_\theta=5.4$, with the detected hairpin segment highlighted in red. A blow-up of the hairpin segment is also shown. (b) A typical chain configuration for $l_p\,{=}\,3$ and $r_c\,{=}\,3$ at $T^{*}_\theta=40.3$ with no hairpins. In both cases, the chain length is $N\,{=}\,500$.