[Paper Review] Liquids that form due to dynamics of the molecules that depend on the local density

Richard P. Sear|arXiv (Cornell University)|Mar 24, 2015

RNA Research and Splicing3 citations

TL;DR

This paper proposes that liquids form when molecular dynamics slow down in response to local density, even in out-of-equilibrium systems like mRNA in cells. Using lattice models with density-dependent hopping rates, it shows that such dynamics induce liquid-liquid phase separation, stabilizing droplets by reducing evaporation—offering a generic mechanism for biomolecular condensation in living cells.

ABSTRACT

RNA molecules in living cells form what look like liquid droplets formed by liquid/liquid phase separation. But unlike the molecules in conventional phase separating mixtures, RNA molecules are transported by molecular motors that consume energy and so are out of equilibrium. Motivated by this we consider what sort of simple rules for the dynamics of model mRNA molecules lead to liquid/liquid phase separation. We find that dynamics that slow as the local density of molecules increases, drive the formation of liquids. We also look at the analogous separation of the two blocks of a block copolymer, in which the monomers of one block have dynamics that depend on the local density of monomers of that block. We find that this block condenses and separates from the monomers of the other block. This is a simple model of the out-of-equilibrium domain formation found in the chromatin in the nucleus of cells.

Motivation & Objective

• To understand how out-of-equilibrium dynamics, such as motor-driven transport, can lead to liquid-liquid phase separation in biological systems like mRNA droplets.
• To investigate whether density-dependent dynamics alone—without explicit attractive interactions—can drive condensation into liquid-like droplets.
• To model the separation of chromatin regions in the nucleus, particularly ribosome-producing domains, using a block copolymer framework with density-modulated dynamics.
• To demonstrate that such dynamics, though non-equilibrium in origin, can still satisfy detailed balance and map to equilibrium Ising-like models.
• To provide a minimal, generic mechanism for biomolecular condensation that explains droplet stability in cellular environments without requiring explicit energy-coupled binding.

Proposed method

• Uses a 2D lattice gas model with Kawasaki dynamics, where molecules hop between sites based on local neighbor count and a density-dependent rate.
• Introduces a hopping probability $ p = \exp[-\alpha_D n_n] $, where $ n_n $ is the number of neighbors at the starting site, and $ \alpha_D > 0 $ controls the slowdown at high density.
• Applies the same density-dependent dynamics to block copolymers: one block (type D) has slow hopping in dense regions, while the other (type E) only experiences excluded volume.
• Simulates systems with $ L = 100 $, $ 15\% $ occupancy, and $ r_M = 8 $, using Monte Carlo cycles with random site selection and attempted moves.
• Ensures detailed balance is preserved via the transition probability ratio $ p_{ij}/p_{ji} = \exp[-\alpha_{DP}(n_{MN}(i) - n_{MN}(j))] $, maintaining thermodynamic consistency.
• Uses periodic boundary conditions in the monomeric model but not in the block copolymer model, allowing for phase-separated morphologies to develop naturally.

Experimental results

Research questions

• RQ1 Can density-dependent molecular dynamics alone drive liquid-liquid phase separation without explicit attractive interactions?
• RQ2 How does a slowdown in hopping rate at high local density affect droplet stability and surface tension in non-equilibrium systems?
• RQ3 To what extent can this mechanism explain the formation of biomolecular condensates like mRNA droplets or transcriptionally active chromatin domains?
• RQ4 Does a model with non-equilibrium dynamics but detailed balance still yield qualitatively similar phase behavior to equilibrium systems?
• RQ5 Can this mechanism explain the separation of functional domains in the nucleus, such as nucleolus formation or transcriptional hubs?

Key findings

• Liquids form when the hopping rate of molecules decreases with increasing local density, even in the absence of explicit attractive interactions.
• The model exhibits stable liquid droplets due to suppressed evaporation from dense regions, as molecules are less likely to escape the high-density core.
• In block copolymer simulations, type D monomers (with density-dependent dynamics) form a condensed globule, while type E monomers remain extended, demonstrating microphase separation.
• The system maps to an equilibrium Ising model, satisfying detailed balance despite being driven by non-equilibrium processes, suggesting robustness of the mechanism.
• Effective surface tension is expected to scale with motor work $ w_M $, not $ kT $, indicating a non-equilibrium origin for interfacial properties.
• The mechanism is generic: similar condensation occurs in models with velocity-dependent dynamics, suggesting broad relevance to cellular condensates.