Condensation dynamics of sticky and anchored flexible biopolymers

TL;DR

The paper investigates how end-anchored flexible biopolymers condense under transient crosslinking, linking microscopic binding kinetics to mesoscopic condensation dynamics. Using coarse-grained 3D Brownian dynamics with a kinetic Monte Carlo crosslinking scheme (aLENS) and a DBSCAN-based cluster tracking, it identifies two dynamical pathways—merging and ripening—that govern the approach to a single condensate and derives a minimal free-energy framework that captures the observed scaling. The results connect microscopic parameters like $K_e$ and end-to-end separation $L_{sep}$ to macroscopic condensate properties, offering insights into chromatin reorganization timescales relevant to gene regulation and suggesting extensions to include hydrodynamic and active processes. Collectively, the work unifies previous equilibrium descriptions with non-equilibrium cluster dynamics and provides a mechanistic view of how transient protein–DNA interactions shape nuclear organization.

Abstract

Cells regulate gene expression in part by forming DNA-protein condensates in the nucleus. While existing theories describe the equilibrium size and stability of such condensates, their dynamics remain less understood. Here, we use coarse-grained 3D Brownian-dynamics simulations to study how long, end-anchored biopolymers condense over time due to transient crosslinking. By tracking how clusters nucleate, merge, and disappear, we identify two dominant dynamical pathways, ripening and merging, that govern the progression from an uncompacted chain to a single condensate. We show how microscopic kinetic parameters, protein density, and mechanical constraints shape these pathways. Using insights from the simulations, we construct a minimal mechanistic free-energy model that captures the observed scaling behavior. Together, these results clarify the dynamical determinants of DNA and chromatin reorganization on timescales relevant to gene regulation.

Figures (14)

• Figure 1: (A) Schematic representation of a polymer chain with sticky tails and fixed ends. The color of the beads indicates their index, with black representing the two fixed ends. Sticky tails are depicted as purple bars attached to the beads. Under the influence of kinetic binding (indicated by purple and green arrows), sections of the chain form dense clusters (light blue circle). (B) Simulation snapshots at $t=0\, \mathrm{s}$ (top) and after undergoing condensation at $t=600\, \mathrm{s}$ (bottom). An example cluster is circled in light blue. (C) Bead distance maps from the snapshots shown in (B). The left (right) plot is calculated from bead positions at $t=0\, \mathrm{s}$ ($t=600\, \mathrm{s}$). The circled cluster in (B) corresponds to the region marked in light blue in the right plot.
• Figure 2: (A) Average values of total clustered chain length $\ell_{tot}$ normalized by $L_{tot}$. Curves represent averages over 12 independent realizations. Curve colors correspond to different values of $K_e$ and $L_{sep}$, as indicated at the top of the figure. Shaded areas are $95\%$ confidence intervals of the mean. (B) Equilibrium values of $\langle \ell_{tot}/L_{tot} \rangle$ for different values of $K_e$ and $L_{sep}$. The and the colored boxes around data points correspond to the curves in (A). (C) Comparison of the total clustered chain length from our $K_e=30\per µ\molar$ simulations (dark blue) to previous equilibrium theories with experimentally inferred constants (cyan) Renger2022Quail2021.
• Figure 3: (A,D) Number of clusters $N_c$, (B,E) average cluster size $\bar{\ell}$, and (C,F) variance of clusters $\sigma_\ell^2$ scaled by $\bar{\ell}^{-2}$ as functions of time. Curves are averaged over 12 realizations using different random number generator seeds. Shaded areas are 95% confidence intervals of the mean. (A-C) Parameter sets with $L_{sep}=5µm$ and varied binding affinities ($K_e = 3, 10, 30, 100 \mu M^{-1}$). (D-F) Parameter sets with $K_{e}=30 \mu M^{-1}$ and varied end separations ($L_{sep} = 3,5,7,9 \mu\text{m}$).
• Figure 4: (A,C) Total chain in clusters and (B,D) number of clusters as functions of time with $K_e = 30 \mu M^{-1}$, $L_{sep} = 5\mu m$, and varying crosslinker turnover rates ($k_o=.1,.3,1,3,10,30,100,300 s^{-1}$). Curves are averaged over 12 independent realizations using different random number generator seeds. Shaded areas are 95% confidence intervals of the mean. In (A,B), time is in units of seconds whereas in (C,D), time is rescaled by the inverse of the crosslinker binding rate.
• Figure 5: Visualization of cluster dynamics in a single simulation. (A) Clusters shown in bead index space over time. Filled regions of the same color indicate bead indices belonging to the same cluster, with darker shades representing earlier time points in the cluster's genealogy. (B) Cluster sizes from (A) as a function of time, with merging events marked by circles. (Inset) Zoomed-in view of the first 100 seconds of (B). (C) 3D snapshot of the system at the time indicated by the dotted lines in (A) and (B).