# Interactions between Phase-Separated Liquids and Membrane Surfaces
## 2 Physical Properties of Phase-Separating Molecules and Biomolecular Condensates
### Biophysical Drivers of Liquid-Liquid Phase Separation  
The basic principles of biomolecules condensing into a distinct liquid phase can be understood from elementary polymer physics. The Flory-Huggins model calculates the free energy of a solution of homopolymers in a lattice model and assumes a simple parameter \(\chi\), the Flory-Huggins parameter, written as follows:  
\[\chi\ =\ \frac{z}{k_{B}T}\bigg{[}u_{ps}-\frac{1}{2}\big{(}u_{pp}+u_{ss} \big{)}\bigg{]} \tag{1}\]  
(see e.g., [PERSON], [PERSON] [15]). \(\chi\) is dependent on the mean-field energies of solvent-solvent (\(u_{ss}\)), polymer-solvent (\(u_{ps}\)), and polymer-polymer (\(u_{pp}\)) interactions, and temperature (\(z\) is a parameter of the lattice model); it is a part of the enthalpic term of the Flory-Huggins model and has been used to predict polymer phase separation at high \(\chi\)[15; 16; 17]. [PERSON] and [PERSON] provide a model that predicts phase separation between polyions dependent on polyion length and linear charge density [15; 18]. Both of these models, however, rely on polymers of homogeneous composition. More realistic models of biomolecule phase separation explore the effects of sequence heterogeneity and specific interaction domains.  
A wide variety of structural factors may promote or inhibit the ability of a protein to phase separate. For example, many phase-separating proteins contain intrinsically disordered regions (IDRs) that often drive phase separation. The proteins Ddx4 and LAF-1, both associated with P-granules--membraeless organelles implicated in germ cell lineage maintenance [6; 8; 13]--contain such IDRs. Archetypes of IDRs that phase separate include IDRs like that of FUS [19] enriched in polar residues, those like Ddx4 [8] containing oppositely-charged \"blocks\" of charged residues, and those like NICD that carry a net charge that drives condensation with another, oppositely-charged, molecule [13; 20]. In some proteins, IDRs are sufficient to drive phase separation [6], but in others folded domains are necessary alongside IDRs [21]. Folded, interactive domains can also lead to the formation of biomolecular condensates alone. In these systems, multivalency--that is, repeats of specific binding motifs or domains in a single molecule--is known to be an important factor promoting phase separation [2; 13]. The linkers between these domains also affect a protein's ability to phase separate: Weak interactions involving disordered linkers can be a necessary contributor to phase separation [22] and the length and solvent affinity of linkers can impact whether a protein experiences phase separation with increasing concentration before undergoing gelation [23]. The ways in which domains and linkers impact the physics of biomolecular condensate formation and their physical properties have been investigated thoroughly through a \"stickers and spacers\" model. This model builds on [PERSON]'s mean-field theory for associative polymers [24], defining \"sticker\" regions as groups that engage in attractive interactions and \"spacers\" as parts of the polymer chain that interconnect stickers [14]. [PERSON] et al. have recently reviewed this model [14].