Liquid state theory study of the phase behavior and macromolecular scale structure of model biomolecular condensates

Abstract

Biomolecular condensates can form through the liquid–liquid phase separation (LLPS) of proteins and RNAs in cells. However, other states of organization, including mesostructured network microstructures and physical gels, have been observed, the physical mechanism of which are not well understood. We use the Polymer Reference Interaction Site Model liquid state integral equation theory to study the equilibrium behavior of (generally aperiodic in sequence) biomolecular condensates based on a minimal sticker–spacer associating polymer model. The role of polymer packing fraction, sequence, and the strength and range of intermolecular interactions on macromolecular scale spatial organization and phase behavior is studied for typical sticker–spacer sequences. In addition to the prediction of conventional LLPS, a sequence-dependent strongly fluctuating polymeric microemulsion homogeneous state is predicted at high enough concentrations beyond the so-called Lifshitz-like point, which we suggest can be relevant to the dense phase of microstructured biomolecular condensates. New connections between local clustering and the formation of mesoscopic microdomains, the influence of attraction range, compressibility, and the role of spatial correlations across scales, are established. Our results are also germane to understanding the polymer physics of dense solutions of nonperiodic and unique sequence synthetic copolymers and provide a foundation to create new theories for how polymer diffusion and viscosity are modified in globally isotropic and homogeneous dense polymeric microemulsions.

Theory of the center-of-mass diffusion and viscosity of microstructured and variable sequence copolymer liquids

“…Concerning the model employed and theory of structure, we follow our previous work. 17 Briefly, an ideal freely-jointed chain (FJC) model is adopted that consists of spherical interaction sites of type A (spacer) and B (sticker) of identical hard core diameters, σ . the inter-chain interactions between A–B and A–A sites are taken to be purely hard-core, with B–B interactions including an effective short-range attraction of an exponential functional form:Here, α defines the attraction range, and is the attraction energy at contact (Fig.…”

“…The dimensional collective density fluctuation structure factor is, S̃ ( k ) = Ω ( k ) + H ( k ) = [ I − Ω ( k ) C ( k )] −1 Ω ( k )The coupled PRISM equations are closed by the site–site mean spherical approximation (MSA). 43,44 Since in the present study we do not investigate the chain length dependence of phase diagrams which requires more advanced molecular closures, 45 the MSA closure, C αγ ( r ) = − βu αγ ( r ) for r > σ , is chosen for simplicity, because it allows us to make direct connections between PRISM theory and coarse-grained mean field theories that employ the RPA, and because we tested equilibrium predictions based on it against simulations in our prior study 17 that support its reasonableness. Eqn (1) are numerically solved using the Newton–Krylov method built in the pyPRISM package 46 with a choice of 32 768 discretized points in real and reciprocal space following previous studies.…”

“…Our recent biophysically motivated study 17 revealed two distinct regimes in the phase diagram for a sticker–spacer model of biomolecular condensates. At low to intermediate copolymer volume fractions ( ϕ ), the system undergoes macroscopic phase separation, akin to the putative liquid–liquid phase separation observed in biomolecular condensates.…”

“…This is an A/B copolymer model where there are two types of interaction sites that are characterized by strongly attractive sticker–sticker interactions and repulsive or weakly attractive interactions between spacers and/or spacer–stickers. In our previous recent study, 17 we have used liquid state theory (the polymer reference interaction site model, or PRISM, theory 18,19 ) to investigate the thermodynamics, phase behavior and structure of such a copolymer model that is either of a regular ( e.g. , diblock, triblock) or an aperiodic nature germane to biology, in concentrated solutions and melts motivated by the condensate problem.…”

“…Section 2 describes the polymer model employed, briefly reviews PRISM theory, and formulates a non-self-consistent copolymer mode coupling theory and the computation of the collective component of the shear viscosity. Sections 3–5 present our main new results, including for the homopolymer system, copolymer fluids, and results obtained using simplified PRISM-RPA theory 17,40 that ignores the important fluctuation and correlation effects on structure and dynamics. Finally, we summarize our key findings and discuss the limitations and future directions of the present study in Section 6.…”