Theory and Simulation of Multiphase Coexistence in Biomolecular Mixtures

Abstract

Biomolecular condensates constitute a newly recognized form of spatial organization in living cells. Although many condensates are believed to form as a result of phase separation, the physicochemical properties that determine the phase behavior of heterogeneous biomolecular mixtures are only beginning to be explored. Theory and simulation provide invaluable tools for probing the relationship between molecular determinants, such as protein and RNA sequences, and the emergence of phase-separated condensates in such complex environments. This review covers recent advances in the prediction and computational design of biomolecular mixtures that phase-separate into many coexisting phases. First, we review efforts to understand the phase behavior of mixtures with hundreds or thousands of species using theoretical models and statistical approaches. We then describe progress in developing analytical theories and coarse-grained simulation models to predict multiphase condensates with the molecular detail required to make contact with biophysical experiments. We conclude by summarizing the challenges ahead for modeling the inhomogeneous spatial organization of biomolecular mixtures in living cells.

Phase Separation in Complex Mixtures with Many Components: Analytical Expressions for Spinodal Manifolds

“…The actual calculation involves a choice for the values of each of the N ( N + 1)/2 virial coefficients B ij . This choice can either be based on the physical properties of the components and determined experimentally or estimated theoretically or alternatively be obtained through a priori assumptions on the statistical distribution of the virial coefficients . The present paper provides a procedure to determine the spinodal and critical points using eqs and , where the calculation is achieved by choosing S c,ij or S sp, ij in the range ⟨−∞,∞⟩ (although the transformation of the parameter space strongly restricts these ranges, as illustrated for N = 3 and positive virial coefficients in the Results section).…”

“…This choice can either be based on the physical properties of the components and determined experimentally or estimated theoretically or alternatively be obtained through a priori assumptions on the statistical distribution of the virial coefficients. 18 The present paper provides a procedure to determine the spinodal and critical points using eqs 20 and 26 , where the calculation is achieved by choosing S c,ij or S sp, ij in the range ⟨−∞,∞⟩ (although the transformation of the parameter space strongly restricts these ranges, as illustrated for N = 3 and positive virial coefficients in the Results section). For physically relevant solutions, the choices of S c,ij or S sp, ij should lead to non-negative coordinates c i,c or c i ,sp —something that can be confirmed easily by substituting the selected S c,ij or S sp, ij in eq 20 .…”

“…The first such methodology was presented two decades ago by Sear and Cuesta 16 and was based on random matrix theory (RMT). 17 This seminal work, valid for mixtures of a very large number of components, prompted various alternative approaches for describing the phase behavior of complex mixtures, culminating in a field that was reviewed recently by Jacobs 18 and Pappu et al 19 As will be shown in the Results section, RMT requires that the number of components fulfills N ≳ 10 3 . This identifies a gap in the numerical approaches for a low number of components and RMT, where the latter is only valid for a very high number of components.…”

Active learning of the thermodynamics-dynamics trade-off in protein condensates

“…Moreover, we have demonstrated that this approach can efficiently converge the Pareto front for a pair of biophysical properties that depend on a complex sequence of computations. We anticipate that this framework could be applied to other combinations of biophysical traits and may also be useful for studying multicomponent and multiphasic systems ( 78 – 80 ). Our work therefore establishes a promising approach for engineering custom biomolecular condensates with tunable thermodynamic and dynamic properties.…”

Phase behavior in multicomponent mixtures

“…The premise of the RMT is that if the random matrix is large enough, the actual values of B ij typically do not matter anymore and only their average and standard deviations are of importance, and for some aspects, even the details of the distribution itself do not play a role anymore. It is not completely clear, however, which simplifications in the calculations are driven by mathematical convenience and which ones are driven by physical relevance, thus yielding some words of caution while applying the RMT approach, as was also mentioned by Jacobs (2023). The exact analytical expressions given in Table 2 allow us to separate the mathematical aspects of the RMT approach from the physical aspects in the calculation.…”