AbstractBiomolecular condensates form via phase transitions that combine phase separation or demixing and networking of key protein and RNA molecules. Proteins that drive condensate formation are either linear or branched multivalent proteins where multivalence refers to the presence of multiple protein-protein or protein-nucleic acid interaction domains or motifs within a protein. Recent work has shown that multivalent protein drivers of phase transitions are in fact biological instantiations of associative polymers. Such systems can be characterized by stickers-and-spacers architectures where stickers contribute to system-specific spatial hierarchies of directional interactions and spacers control the concentration-dependent inhomogeneities in densities of stickers around one another. The collective effects of interactions among stickers and spacers lead to the emergence of dense droplet phases wherein the stickers form percolated networks of polymers. To enable the calculation of system-specific phase diagrams of multivalent proteins, we have developed LASSI (LAttice simulations of Sticker and Spacer Interactions), which is an efficient open source computational engine for lattice-based polymer simulations built on the stickers and spacers framework. In LASSI, a specific multivalent protein architecture is mapped into a set of beads on the 3-dimensional lattice space with proper coarse-graining, and specific sticker-sticker interactions are modeled as pairwise anisotropic interactions. For efficient and broad search of the conformational ensemble, LASSI uses Monte Carlo methods, and we optimized the move set so that LASSI can handle both dilute and dense systems. Also, we developed quantitative measures to extract phase boundaries from LASSI simulations, using known and hidden collective parameters. We demonstrate the application of LASSI to two known archetypes of linear and branched multivalent proteins. The simulations recapitulate observations from experiments and importantly, they generate novel quantitative insights that augment what can be gleaned from experiments alone. We conclude with a discussion of the advantages of lattice-based approaches such as LASSI and highlight the types of systems across which this engine can be deployed, either to make predictions or to enable the design of novel condensates.Author SummarySpatial and temporal organization of molecular matter is a defining hallmark of cellular ultrastructure and recent attention has focused intensely on organization afforded by membraneless organelles, which are referred to as biomolecular condensates. These condensates form via phase transitions that combine phase separation and networking of condensate-specific protein and nucleic acid molecules. Several questions remain unanswered regarding the driving forces for condensate formation encoded in the architectures of multivalent proteins, the molecular determinants of material properties of condensates, and the determinants of compositional specificity of condensates. Building on recently recognized analogies between associative polymers and multivalent proteins, we have developed and deployed LASSI, an open source computational engine that enables the calculation of system-specific phase diagrams for multivalent proteins. LASSI relies on a priori identification of stickers and spacers within a multivalent protein and mapping the stickers onto a 3-dimensional lattice. A Monte Carlo engine that incorporates a suite of novel and established move sets enables simulations that track density inhomogeneities and changes to the extent of networking among stickers as a function of protein concentration and interaction strengths. Calculation of distribution functions and other nonconserved order parameters allow us to compute full phase diagrams for multivalent proteins modeled using a stickers-and-spacers representation on simple cubic lattices. These predictions are shown to be system-specific and allow us to rationalize experimental observations while also enabling the design of systems with bespoke phase behavior. LASSI can be deployed to study the phase behavior of multicomponent systems, which allows us to make direct contact with cellular biomolecular condensates that are in fact multicomponent systems.
Infinite Orders and Non-D-finite Property of 3-Dimensional Lattice Walks
