Illustration of assembly reaction scheme and classification.

a Illustration of the chemical reaction network associated with the formation of assemblies Ai with size i. b Identification of three classes based on assembly dimension: d=1,2,3. c Classification of assemblies based on the scaling of their Flory-Huggins interaction propensity.

Parameters corresponding to the figures in the main text. We made use of the temperature scale T0 = χ/kB.

Phase diagram and assembly size distributions for different classes and assembly strengths.

a Phase diagram as a function of ϕtot and rescaled temperature T/T0 (with T0 = χ/kB) in the regime of high assembly strength, i.e. −eint/χ≫1. The green line is the volume fraction threshold ϕ∗(T) at which intermediate-sized assemblies start to appear, which in this regime precedes the binodal (coloured curve). As indicated by the colour code, the monomer fraction ϕ /ϕtot mildly varies in the two phases. b Size distributions and c pictorial representations corresponding to different regions of the phase diagram, defined by the relative position of the binodal and the assembly threshold. In region “i”, the system is homogeneous and composed of monomers only. Increasing the total volume fraction of assemblies ϕtot beyond the assembly threshold ϕ∗, the system enters region “ii” where intermediate assemblies appear. Here, the sizes corresponding to the maximum and the average of the distribution ϕi scale with , see Appendix C. Finally, once ϕtot exceeds the binodal, the system enters region “v” and demixes in two phases, both rich in intermediate assemblies. In d-f we focus on the low assembly strength regime, i.e. − eint/χ∼1. In phase diagram d, the binodal now precedes in ϕtot the assembly threshold. e In region “iv”, the system phase separates but in both phases monomers dominate the size distribution, while in region “v” the ϕtot-rich phase becomes populated by intermediate-sized assemblies. Progressively lowering the temperature allows switching between these regions, as depicted in f. g,h Behaviour of dilute mixtures as a function of assembly strength, for the two different classes. Notably, assembly below saturation becomes much more accessible for class 2, as can be seen by comparing the green regions “ii” in g and h.

Gelation transition in phase-separating systems.

a Phase diagram for planar (d=2) and three-dimensional (d=3) assemblies in the limit M→ ∞, as a function of ϕtot and the rescaled temperature T/T0 (with T0 = χ/kB). The coloured curve represents the binodal associated with the free energy f, which accounts for the emergence of an infinite assembly. The colour code of the binodal line depicts the monomer fraction ϕ1/ϕtot in the phases, and the black cross ‘x’ indicates the triple point. In the region labelled as “sol-sol”, the system demixes into two phases both populated mainly by monomers, see panel b, with . In the region labelled as “sol-gel”, on the other hand, a phase (the “sol”), obeying , coexists with a phase (the “gel”) that is a macroscopic assembly, containing no solvent . The latter scenario is represented in panel b, right side. c Lowering the temperature allows transitions from the “sol-sol” to the “sol-gel” region, which manifest with a jump in the total volume fraction of the protein-rich phase.

Assembly kinetics at phase equilibrium.

Assuming that the relaxation to phase equilibrium is fast compared to assembly kinetics, we study the slow relaxation to assembly equilibrium in a compartmentalized system. a In the sketch, starting from an initial state composed of monomers and solvent only, assemblies selectively appear in phase I, increasing its volume V I and total volume fraction . b, c For Class 1, as time proceeds, the total macromolecule volume fraction in the two phases, , changes inducing the growth of phase I. In d and e we show the time evolution of the full-size distribution in phase II and I, respectively. f, g For Class 2, as time proceeds, changes in total macromolecule volume fraction in the two phases cause a shrinkage of phase I. This is reminiscent of recent experimental findings that quantify droplet volume changes along with droplet ageing [60]. h, i time evolution of assembly volume fractions ϕi(t) in phase II and I, respectively. Time is measured in units of the discretization time step , where the rate is introduced in Eq. (F5).

Identification of shrinkage and growth regions for different classes.

Here, we study phase-separating systems initially composed of monomers only and we monitor phase volume changes as they relax to thermodynamic equilibrium. a For linear assemblies (d=1) belonging to class 1 the final binodal line (coloured curve) is wider than the initial one (black curve), corresponding to monomers and solvent only (black curve). Areas in orange and light blue correspond to growth and shrinkage of the ϕtot-rich phase (phase I), respectively. b The hehaviour of linear assemblies (d=1) belonging belonging to class 2 is remarkably different. Since, in this class, the interaction with the solvent is screened, the final binodal is shrunk compared to the initial one. As a consequence of the shrinkage, the domain corresponding to phase I growth (light blue area) precedes in ϕtot the shrinkage domain (orange area), for class 2.

A volume fraction threshold separates two assembly regimes in homogeneous systems.

a Illustration of assemblies belonging to Class 1 with different spatial dimension. b Assembly size distribution at low total macromolecular volume fraction: ϕtot = 0.2ϕ∗. Disregarding assembly dimension, d, the macromolecules are mainly in the monomer state, i.e., ϕ1 ≃ϕtot. c For ϕtot = 10ϕ∗, the monomer volume fraction saturates at ϕ1 ≃ ϕ∗ and big assemblys begin to populate the system. For linear assemblies (corresponding to d = 1 in Eq. (8)), the distribution becomes peaked at an intermediate value imax > 1 and then exponentially cut off. For planar and three-dimensional assemblies, d = 2, 3, the distribution becomes bimodal, with peaks at i = 1 and i = M, the maximum assembly size (M = 50). This bimodal behaviour hints at the emergence of a gelation transition in the limit M→ ∞. In the insets, we show the scaling of concentrations ci = ϕi/νi with assembly size. For d = 2, 3 and above the ϕ∗ threshold, deviations from the classical exponential decay are present. Here eint/χ = −1, sint/kB = −1, M = 50, T/T0 = 0.25.

Gel-sol free energies.

a The coloured curved indicates the binodal obtained with the Maxwell construction for f = fsol + fgel, together with the estimate ϕsg(T) (black line, defined in Eq. (D2)) for the transition between homogeneous and gel states. The black cross ‘x’ indicates the triple point. c-e Maxwell construction for three different temperature values, the coloured and black, dashed curves represent convex and concave branches of f, respectively.

The influence of assemblies on the system phase behaviour.

a Focusing on systems with d = 1 belonging to class 1, we compare three binodals corresponding to assembly strength eint/χ = 0.5, −1, −2 (coloured curves) and the reference binary mixture composed of monomers and solvent only (black curve). The latter can be associated with the limit eint/χ→ ∞. The region enclosed by the binodal, corresponding to phase separation, expands even for assemblies with no assembly energy eint/χ = 0. This can be explained by the entropic advantage caused by size polydispersity. b Dependence of the critical volume fraction and critical temperature on the assembly strength eint/χ. The presence of assemblies causes T c and ϕc to deviate from the reference values (black dashed lines) corresponding to a binary mixture with monomers and solvent only (eint/χ→ ∞). In particular, for Class 1, making assemblies more energetically favourable, i.e. decreasing eint/χ, induces an increase in T c and a decrease in ϕc, in turn making phase separation more accessible. Here sint/kB = −2, M =∞. c Comparison between three binodal lines corresponding to systems belonging to class 2 and d = 1, with assembly energies eint/χ = 0, −0.5, −1 (coloured curves) and the reference binary mixture composed of monomers and solvent only (black curve). d For Class 2, decreasing eint/χ, causes T c and ϕc to decrease, overall hindering phase separation. This is caused by the interaction propensity screening in monomers at the bulk of assemblies belonging to class 2, see Eq. (10). Here sint/kB = −2, M = ∞. χ′ = 0.2χ.

The influence of phase separation on assembly size.

a Comparison between the size distribution in a homogeneous system, and in the corresponding phase-separated system (averaged in both compartments). Here, we consider linear assemblies (d = 1), M→ ∞, ϕtot = 0.016 and T/T0 = 0.2. We note that the presence of compartments can favour assembly formation, even when the corresponding homogeneous mixture is populated mainly by monomers. The difference in distributions can be quantified utilizing the functional distance, defined in Eq. (E2). b The magnitude of this distance depends on the droplet size and the temperature chosen. The volume corresponding to the maximum distribution distance shifts towards lower values with decreasing temperature T/T0. The distributions separated by the maximum distance, for T/T0 = 0.2, are the ones displayed in a. eint/χ = −0.5, sint/kB = −2, T/T0 = 0.25.

Kinetic trajectory in the multicomponent phase diagram

Illustration of the assembly kinetics at phase equilibrium, for systems corresponding to M = 3 and initially composed of monomers only.