Analytical Solution to the Flory–Huggins Model

Abstract

A self-consistent analytical solution for binodal concentrations of the two-component Flory–Huggins phase separation model is derived. We show that this form extends the validity of the Ginzburg–Landau expansion away from the critical point to cover the whole phase space. Furthermore, this analytical solution reveals an exponential scaling law of the dilute phase binodal concentration as a function of the interaction strength and chain length. We demonstrate explicitly the power of this approach by fitting experimental protein liquid–liquid phase separation boundaries to determine the effective chain length and solute–solvent interaction energies. Moreover, we demonstrate that this strategy allows us to resolve differences in interaction energy contributions of individual amino acids. This analytical framework can serve as a new way to decode the protein sequence grammar for liquid–liquid phase separation.

Analytical Formulation and Field-Theoretic Simulation of Sequence-Specific Phase Separation of Protein-Like Heteropolymers with Short- and Long-Spatial-Range Interactions

“…Transfer matrix, restricted primitive model simulations, and other techniques have been applied to study complex coacervation of polyampholytes and polyelectrolytes; − and LLPS of single or multiple biomolecular species, especially those involving folded proteins and folded domains, have also been modeled by patchy particles − using simulations , as well as analytical formalisms based on Wertheim’s thermodynamic perturbation theory. , As expected, the required computation increases with structural and energetic details that a model seeks to capture. In this context, it is noteworthy that even basic FH theory, which requires minimal numerical effort yet has recently been made even more tractable by a novel self-consistent solution, can be very useful in advancing knowledge about biomolecular LLPS. This is exemplified by the applications of FH to delineate scenarios of tie-line patterns and their ramifications for homeostasis, ,, to ascertain the extent of void-volume contributions to the hydrostatic pressure dependence of LLPS, and to rationalize experimental data on the impact of aromatic valence on IDP LLPS …”

Theory and Simulation of Multiphase Coexistence in Biomolecular Mixtures

“…The Flory–Huggins model is commonly used to fit experimental data on biomolecular LLPS . Assuming an incompressible fluid with N nonsolvent species, the Flory–Huggins free-energy density is β f v 0 = ∑ i = 1 N ϕ i L i log nobreak0em.25em⁡ ϕ i + ϕ 0 log nobreak0em.25em⁡ ϕ 0 + 1 2 ∑ i = 1 N ∑ j = 1 N ϵ i j ϕ i ϕ j where the volume fraction occupied by species i is ϕ i = L i v 0 ρ i , the degree of polymerization of species i is L i , the size of a monomer is represented by v 0 , and the solvent-occupied volume fraction, ϕ 0 , is determined by the incompressibility constraint, ∑ i = 0 N ϕ i = 1.…”

LLPSvs.LLCPS: analogies and differences

“…For a more in-depth analysis see the references. [41][42][43] The spinodal and binodal lines for varying w, have been plotted in Fig. 1(b).…”