Phase Behavior and Cluster Size Distribution in Polymer Solutions: From Homopolymers to Heteropolymers

Biomolecular condensates are dynamic structures which result from the membraneless aggregation of biomolecules. Protein condensation, in particular, has attracted considerable attention since this pathway has been proposed as a possible intermediate and metastable state between the native and the amyloid state, which is well known to be a critical player in neuro-degenerative diseases such as Alzheimer and ALS. These phase transitions have come to the attention of biomedicine, since the reversible progression between the native form and the liquid-like structure is exploited in the cellular environment to carry out a wide variety of functions. The dysregulation of this dynamic equilibrium may, however, propel the condensate to evolve into a pathological amyloid. The physical understanding of the phase behaviour and of the interactions which lead to such outcomes is still incomplete and needs further development. To address this gap in our understanding, the present work develops a theoretical framework aimed at capturing the phase behavior of polymer solutions, starting from classical models and extending them through modern analytical and computational techniques. Chapter 1: homopolymer systems Introduction and the Freed Model The Flory-Huggins theory is a lattice model for a solution of polymers in a solvent. Any site of the lattice can be occupied by either a polymer segment or a solvent molecule. Between nearest neighbours there are nonbonded interactions with attractive interaction energies ϵij. Opting for a mean-field solution streamlines the calculations and gives rise to the well-known Flory-Huggins free energy. The model, however, suffers from some deficiencies (e.g. it is unsuitable for highly concentrated solutions) which can be overcome by an exact treatment. This approximation can be systematically improved by incorporating higher-order energetic and entropic corrections. The resulting model accurately describes a solution of polydisperse polymers at various concentration regimes Phase diagrams The phase diagram for a polymer solution can be traced using analytical methods and computational simulations. The paper elaborates on a self-consistent algorithm to calculate the binodal (and therefore the phase diagram of the system) starting from the Flory-Huggins free energy. Its accuracy is corroborated by simulations in which the condensed phase emerges clearly. A generalised framework for a homopolymer solution Using the exact free energy instead of its mean-field approximation significantly bolsters the potency of the algorithm. The thesis will explore how such an approach can be implemented to retrieve phase diagrams even more precisely for polydisperse, compressible systems across a wide range of concentration regimes. This analytical framework is valid for a solution of polymers with the same sequence and therefore equal disposition of sticker (interacting) regions. Building on this foundation, the next chapter extends the framework to heteropolymers, where the interactions become sequence-dependent and stickers disposition plays a central role in driving gelation and phase separation.3 Relevance and possible outcomes Chapter 2: Generalisation to heteropolymers The work of Semenov and Rubenstein describes the thermodynamic equilibrium of a solution of associative heteropolymers by deriving the free energy of the system with a mean field approach. The mutual polymer interactions are attractive and form reversible bonds. Depending on the interaction strength and concentration, this may lead to gelation with or without concomitant phase separation. The analytical treatment expounded in the first chapter will be extended to such systems. This will allow to capture the phase landscape of a solution of interacting polymers in a general setting, irrespective of concentration, describing condensation and gelation dynamics.