Theory of droplet ripening in stiffness gradients

Condensate interfacial forces reposition DNA loci and probe chromatin viscoelasticity

Biomolecular condensates assemble in living cells through phase separation and related phase transitions. An underappreciated feature of these dynamic molecular assemblies is that they form interfaces with other cellular structures, including membranes, cytoskeleton, DNA and RNA, and other membraneless compartments. These interfaces are expected to give rise to capillary forces, but there are few ways of quantifying and harnessing these forces in living cells. Here, we introduce viscoelastic chromatin tethering and organization (VECTOR), which uses light-inducible biomolecular condensates to generate capillary forces at targeted DNA loci. VECTOR can be utilized to programmably reposition genomic loci on a timescale of seconds to minutes, quantitatively revealing local heterogeneity in the viscoelastic material properties of chromatin. These synthetic condensates are built from components that naturally form liquid-like structures in living cells, highlighting the potential role for native condensates to generate forces and do work to reorganize the genome and impact chromatin architecture.

Effective simulations of interacting active droplets

Droplets form a cornerstone of the spatiotemporal organization of biomolecules in cells. These droplets are controlled using physical processes like chemical reactions and imposed gradients, which are costly to simulate using traditional approaches, like solving the Cahn-Hilliard equation. To overcome this challenge, we here present an alternative, efficient method. The main idea is to focus on the relevant degrees of freedom, like droplet positions and sizes. We derive dynamical equations for these quantities using approximate analytical solutions obtained from a sharp interface limit and linearized equations in the bulk phases. We verify our method against fully-resolved simulations and show that it can describe interacting droplets under the influence of chemical reactions and external gradients using only a fraction of the computational costs of traditional methods. Our method can be extended to include other processes in the future and will thus serve as a relevant platform for understanding the dynamics of droplets in cells.

The mechanobiology of nuclear phase separation

The cell nucleus can be thought of as a complex, dynamic, living material, which functions to organize and protect the genome and coordinate gene expression. These functions are achieved via intricate mechanical and biochemical interactions among its myriad components, including the nuclear lamina, nuclear bodies, and the chromatin itself. While the biophysical organization of the nuclear lamina and chromatin have been thoroughly studied, the concept that liquid-liquid phase separation and related phase transitions play a role in establishing nuclear structure has emerged only recently. Phase transitions are likely to be intimately coupled to the mechanobiology of structural elements in the nucleus, but their interplay with one another is still not understood. Here, we review recent developments on the role of phase separation and mechanics in nuclear organization and discuss the functional implications in cell physiology and disease states.

Durotaxis of Passive Nanoparticles on Elastic Membranes

The transport of macromolecules and nanoscopic particles to a target cellular site is a crucial aspect in many physiological processes. This directional motion is generally controlled via active mechanical and chemical processes. Here we show, by means of molecular dynamics simulations and an analytical theory, that completely passive nanoparticles can exhibit directional motion when embedded in nonuniform mechanical environments. Specifically, we study the motion of a passive nanoparticle adhering to a mechanically nonuniform elastic membrane. We observe a nonmonotonic affinity of the particle to the membrane as a function of the membrane's rigidity, which results in the particle transport. This transport can be both up or down the rigidity gradient, depending on the absolute values of the rigidities that the gradient spans across. We conclude that rigidity gradients can be used to direct average motion of passive macromolecules and nanoparticles on deformable membranes, resulting in the preferential accumulation of the macromolecules in regions of certain mechanical properties.

Durotaxis of Passive Nanoparticles on Elastic Membranes

The transport of macromolecules and nanoscopic particles to a target cellular site is a crucial aspect in many physiological processes. This directional motion is generally controlled via active mechanical and chemical processes. Here we show, by means of molecular dynamics simulations and an analytical theory, that completely passive nanoparticles can exhibit directional motion when embedded in nonuniform mechanical environments. Specifically, we study the motion of a passive nanoparticle adhering to a mechanically nonuniform elastic membrane. We observe a nonmonotonic affinity of the particle to the membrane as a function of the membrane's rigidity, which results in the particle transport. This transport can be both up or down the rigidity gradient, depending on the absolute values of the rigidities that the gradient spans across. We conclude that rigidity gradients can be used to direct average motion of passive macromolecules and nanoparticles on deformable membranes, resulting in the preferential accumulation of the macromolecules in regions of certain mechanical properties.

Cavitation controls droplet sizes in elastic media

Biological cells use droplets to separate components and spatially control their interior. Experiments demonstrate that the complex, crowded cellular environment affects the droplet arrangement and their sizes. To understand this behavior, we here construct a theoretical description of droplets growing in an elastic matrix, which is motivated by experiments in synthetic systems where monodisperse emulsions form during a temperature decrease. We show that large droplets only form when they break the surrounding matrix in a cavitation event. The energy barrier associated with cavitation stabilizes small droplets on the order of the mesh size and diminishes the stochastic effects of nucleation. Consequently, the cavitated droplets have similar sizes and highly correlated positions. In particular, we predict the density of cavitated droplets, which increases with faster cooling, as in the experiments. Our model also suggests how adjusting the cooling protocol and the density of nucleation sites affects the droplet size distribution. In summary, our theory explains how elastic matrices affect droplets in the synthetic system, and it provides a framework for understanding the biological case.

A hydro-osmotic coarsening theory of biological cavity formation

Fluid-filled biological cavities are ubiquitous, but their collective dynamics has remained largely unexplored from a physical perspective. Based on experimental observations in early embryos, we propose a model where a cavity forms through the coarsening of myriad of pressurized micrometric lumens, that interact by ion and fluid exchanges through the intercellular space. Performing extensive numerical simulations, we find that hydraulic fluxes lead to a self-similar coarsening of lumens in time, characterized by a robust dynamic scaling exponent. The collective dynamics is primarily controlled by hydraulic fluxes, which stem from lumen pressures differences and are dampened by water permeation through the membrane. Passive osmotic heterogeneities play, on the contrary, a minor role on cavity formation but active ion pumping can largely modify the coarsening dynamics: it prevents the lumen network from a collective collapse and gives rise to a novel coalescence-dominated regime exhibiting a distinct scaling law. Interestingly, we prove numerically that spatially biasing ion pumping may be sufficient to position the cavity, suggesting a novel mode of symmetry breaking to control tissue patterning. Providing generic testable predictions, our model forms a comprehensive theoretical basis for hydro-osmotic interaction between biological cavities, that shall find wide applications in embryo and tissue morphogenesis.

Controlling biomolecular condensates via chemical reactions

Biomolecular condensates are small droplets forming spontaneously in biological cells through phase separation. They play a role in many cellular processes, but it is unclear how cells control them. Cellular regulation often relies on post-translational modifications of proteins. For biomolecular condensates, such chemical modifications could alter the molecular interaction of key condensate components. Here, we test this idea using a theoretical model based on non-equilibrium thermodynamics. In particular, we describe the chemical reactions using transition-state theory, which accounts for the non-ideality of phase separation. We identify that fast control, as in cell signalling, is only possible when external energy input drives the reaction out of equilibrium. If this reaction differs inside and outside the droplet, it is even possible to control droplet sizes. Such an imbalance in the reaction could be created by enzymes localizing to the droplet. Since this situation is typical inside cells, we speculate that our proposed mechanism is used to stabilize multiple droplets with independently controlled size and count. Our model provides a novel and thermodynamically consistent framework for describing droplets subject to non-equilibrium chemical reactions.

Modeling Elastically Mediated Liquid-Liquid Phase Separation

We propose a continuum theory of the liquid-liquid phase separation in an elastic network, where phase-separated microscopic droplets rich in one fluid component can form as an interplay of fluids mixing, droplet nucleation, network deformation, thermodynamic fluctuation, etc. We find that the size of the phase-separated droplets decreases with the shear modulus of the elastic network in the form of ∝[modulus]^{-1/3} and the number density of the droplet increases almost linearly with the shear modulus ∝[modulus], which are verified by the experimental observations. Phase diagrams in the space of (fluid constitution, mixture interaction, network modulus) are provided, which can help to understand similar phase separations in biological cells and also to guide fabrications of synthetic cells with desired phase properties.