Entropic alignment of topologically modified ring polymers in cylindrical confinement

TL;DR

This study demonstrates that entropic interactions alone, engineered through internal loops in topologically modified ring polymers, can drive spatial segregation and orientational order in cylindrical confinement. Using a bead-spring model with purely repulsive monomer interactions and targeted cross-links (including rotated-8, Arc-1-2, and Arc-1-10 topologies), the authors show robust demixing along the cylinder axis and emergent Ising-like anti-parallel alignment of polymer segments. Free energy landscapes $F[X_i] = -k_B T \ln p[X_i]$ reveal minima corresponding to specific loop-overlap configurations, explaining the prevalence of anti-parallel arrangements especially under stronger confinement or longer chains. The findings provide a mechanistic, entropy-driven framework for understanding chromosome organization in bacteria and offer insights for designing synthetic polymers with controllable spatial architecture in confined geometries.

Abstract

Under high cylindrical confinement, segments of ring polymers can be localized along the long axis of the cylinder by introducing internal loops within the ring polymer. The emergent organization of the polymer segments occurs because of the entropic repulsion between internal loops. These principles were used to identify the underlying mechanism of bacterial chromosome organization. Here, we outline functional principles associated with entropic interactions, leading to specific orientations of the ring polymers relative to their neighbors in the cylindrical confinement. We achieve this by modifying the ring polymer topology by creating internal loops of two different sizes within the polymer, and thus create an asymmetry. This allows us to strategically manipulate polymer topology such that segments of a polymer face certain other segments of a neighboring polymer. The polymers therefore behave as if they are subjected to an `effective' entropic interaction reminiscent of interactions between Ising spins. But this emergent spatial and orientational organization is not enthalpy-driven. We consider a bead spring model of flexible polymers with only repulsive excluded volume interactions between the monomers. The polymers entropically repel each other and occupy different halves of the cylinder, and moreover, the adjacent polymers preferentially re-orient themselves along the axis of the cylinder. We further substantiate our observations by free energy calculations. To the best of our knowledge, this is the first study of the emergence of effective orientational interactions by harnessing entropic interactions in flexible polymers. The principles elucidated here could be relevant to understand the interactions between different sized loops within a large chromosome.

Figures (13)

• Figure 1: (a) A schematic of a ring polymer that is topologically modified to a 'rotated-8' with 200 monomers in each ring, (b) Different subloops of the two polymers along the long axis of the cylinder. The snapshot from the simulations shows the different subloops in different colors. (c) Probability density distribution $p(z)$ of the center of mass (COMs) of the different loops, denoted by subloop-1, 2, 3, 4. The coordinate $z$ is along the long axis of the cylinder. (d) Position of COM of each subloop as a function of number of iterations (simulation time). The data re-iterates that the loop-COMs stay well separated along $z$, though the subloops of a polymer interchange positions along $z$. (e) Statistical average of the dot product of the two vectors joining the COMs of the subloops of the two polymers as well as plot the probability density of $\langle \cos(\theta) \rangle$.
• Figure 2: (a) Mean density of monomers belonging to different subloops of two polymers of 'rotated 8' architecture as a function of $z$. Each polymer has $N=200$ monomers. Data is normalized by the number of monomers ($100$) in each loop. The legend 'Total' refers to the mean monomer density of the two polymers, normalized by the total number of monomers in the cylinder ($400$). (b) Probability distribution of the distance between the COMs of subLoop-1 and subLoop-3 with varying number of monomers $N$ of the polymer. The cylinder diameters, chosen for 'rotated 8' polymers with $N=200$, 300, 400, and 500 monomers, are $D_c= 5a$, $5.9a$, $6.48a$, and $7a$, respectively. The length of the cylinder, $L$, is chosen such that $D_c/L=5$.
• Figure 3: (a) Schematic of a ring polymer with $N=200$. The tip of the arrows show the position of the monomers on the contour which are cross-linked such that an Arc-1-2 topology is obtained. The polymer has a big subloop (red beads) and two small subloops (black beads). (b) Probability density distribution $p(z)$ of the center of mass (COM) of the monomers of different loops, from a pair of polymers, referred to as $P1$ and $P2$, respectively. The confining cylinder has length $L=25a$ and diameter $D_c=5a$. (c) Mean density of monomers from different loops along the $z$ axis. (d) Schematic of idealized configurations, classified as C1, C2, C3, and C4, that are predominantly attained by a pair of polymers having the Arc-1-2 topology. The big subloop and the two Small subloops are shown as red and black lines, respectively. The blue vector joins the COM of the bigger loop to the crossing point of the two smaller subloops for each polymer. (e) Representative snapshot of two Arc-1-2 polymers from the simulation. Here, the monomers of smaller and bigger loops are shown in different colors.
• Figure 4: We define the angle $\theta$ between the two vectors, one for each polymer, which joins the COM of the monomers of the big loop to the COM of the monomers of the two small loops (combined) of the same polymer. (a) Probability distribution $P(\cos \theta)$. The quantity $\cos(\theta)$ is the dot product of the two vectors for a particular configuration. The small asymmetry in $P(\cos(\theta))$ becomes more prominent as we decrease from $L=25a$ to $L=20a$ and $L=15a$. (b) Relative contributions (per monomer) of the spring potential between neighboring monomers along the chain contour, the excluded volume interactions, and the kinetic energy as a function of time for a pair of polymer in a cylinder of length $L=25a$ and diameter $D_c=5a$. The contribution of excluded volume interactions is minimal.
• Figure 5: The histograms show the probabilities of different configurations in panels (a), (b), (c), respectively for cylinders of lengths $L=25a, L=20a$, $L=15a$ and in all case for a fixed diameter $D_c=5a$. Two Arc-1-2 polymers confined in a cylinder prefers to entropically organize themselves in the anti-parallel C4 configuration. The error bars denote a standard deviation (SD), as obtained from a calculation over 40 independent runs. In panel (d), we show the probability density distributions of COM of the big subloops and the two small subloops of two polymers P1, P2, confined in a cylinder of length $L=15a$.