Entropic Clustering of Stickers Induces Aging in Biocondensates

TL;DR

A minimal model for the dynamic of stickers and spacers is proposed, and it is shown that entropy maximization of spacers leads to an attractive force between stickers, reminiscent of glassy systems and consistent with the liquid-to-solid transition observed.

Abstract

Biomolecular condensates are cellular phase-separated droplets that usually exhibit a viscoelastic mechanical response. A behavior rationalized by modeling the complex molecules that make up a condensate as stickers and spacers, which assemble into a network-like structure. Condensates usually exhibit a solidification over a long period of time (days), a phenomenon described as aging.The emergence of such a long timescale of evolution from microscopic processes, as well as the associated microscopic reorganization leading to aging, remains mostly an open question. In this article, we explore the connection between the mechanical properties of the condensates and their microscopic structure. We propose a minimal model for the dynamic of stickers and spacers, and show that entropy maximization of spacers leads to an attractive force between stickers. Our system displays a surprisingly slow relaxation toward equilibrium, reminiscent of glassy systems and consistent with the liquid-to-solid transition observed. To explain this behavior, we study the clustering dynamic of stickers and successfully explain the origin of glassy relaxation.

Figures (5)

• Figure 1: Schematic representation of our sticker and spacer model. The disordered background potential is represented as a color coding, while stickers are represented as diffusing particles. We focus on a single polymer dynamic highlighted in bold, while the other effective polymers are in shaded. (a) Represents the initial state in which stickers are homogeneously spread in space. The dynamics of stickers are also represented as small arrows: stickers bound to one polymer can diffuse or bind to another, whereas those bound to two polymers create a cross-link and must unbind before diffusing further. (b) Qualitatively represent a typical final state as predicted by our model.
• Figure 2: (a) Early time evolution of the Percentage of bound stickers in the system. Here, we use $L = 2.10^4$ and $N=100$. (b) Plateau value of the percentage of bound stickers as a function of the binding energy, rescaled by the lineic density of stickers. The scattered points correspond to the measurements obtained from the panel (b).
• Figure 3: (a) sticker distance probability distribution at different time-point. The color coding corresponds to the time, and the analytical solution is provided as dotted line. We simulate systems with fixed lineic density: $N/L = 5.10^{-2}$. Small systems of length $L=100$ successfully converge toward the predicted equilibrium distribution. (b) Large systems, of length $L=2000$, evolves toward the analytic solution without reaching it. (c) We follow the equilibration process by looking at the decay of the free energy per unit length over time, the initial binding regime as been removed for improved visibility. Using a fixed lineic density of stickers, we expect all curves to converge to the same relative value. However, large systems are stuck at a higher value of the free energy, stressing its inability to reach equilibrium. (d) Schematic representation of the partial equilibration of the system into sub-clusters.
• Figure 4: (a) Average cluster size evolution, clusters are built using a hierarchical algorithm that use a minimum distance linkage criteria. The distance used is the average sticker distance computed using the pair correlation function of Eq. \ref{['eq:pcf']}, in our case $\bar{d} = 3.6$. (b) Real part, of the intermediate scattering function for $|\bm{k}| = 1/\bar{r} = 0.37$, where $\bar{r}$ is the average distance between stickers. The stretched exponential fits are in good agreement, and the corresponding $\tau$ values obtained are plotted in the inset. The value of $\alpha$ does not evolve significantly over time, and remains $\approx 0.7$. Notice the abrut decay for small $t_\text{lag}$ that corresponds to the initial binding regime.
• Figure 5: Estimated storage ($G^\prime$) and loss ($G^{\prime\prime}$) modulus, obtained from Eq. \ref{['eq:modulus']}, using ISF measured after different lag-time. Slopes characteristics of Maxwell viscoelastic gel are displayed as reference.