Michael F. Hagan,Farzaneh Mohajerani
Self-assembly coupled to liquid-liquid phase separation
Liquid condensate droplets with distinct compositions of proteins and nucleic acids are widespread in biological cells. While it is known that such droplets, or compartments, can regulate irreversible protein aggregation, their effect on reversible self-assembly remains largely unexplored. In this article, we use kinetic theory and solution thermodynamics to investigate the effect of liquid-liquid phase separation on the reversible self-assembly of structures with well-defined sizes and architectures. We find that, when assembling subunits preferentially partition into liquid compartments, robustness against kinetic traps and maximum achievable assembly rates can be significantly increased. In particular, both the range of solution conditions leading to productive assembly and the corresponding assembly rates can increase by orders of magnitude. We analyze the rate equation predictions using simple scaling estimates to identify effects of liquid-liquid phase separation as a function of relevant control parameters. These results may elucidate self-assembly processes that underlie normal cellular functions or pathogenesis, and suggest strategies for designing efficient bottom-up assembly for nanomaterials applications.
Self-assembly coupled to liquid-liquid phase separation

Liquid condensate droplets with distinct compositions of proteins and nucleic acids are widespread in biological cells. While it is known that such droplets, or compartments, can regulate irreversible protein aggregation, their effect on reversible self-assembly remains largely unexplored. In this article, we use kinetic theory and solution thermodynamics to investigate the effect of liquid-liquid phase separation on the reversible self-assembly of structures with well-defined sizes and architectures. We find that, when assembling subunits preferentially partition into liquid compartments, robustness against kinetic traps and maximum achievable assembly rates can be significantly increased. In particular, both the range of solution conditions leading to productive assembly and the corresponding assembly rates can increase by orders of magnitude. We analyze the rate equation predictions using simple scaling estimates to identify effects of liquid-liquid phase separation as a function of relevant control parameters. These results may elucidate self-assembly processes that underlie normal cellular functions or pathogenesis, and suggest strategies for designing efficient bottom-up assembly for nanomaterials applications.

The self-assembly of basic subunits into larger structures with well-defined architectures underlies essential functions in biological organisms, where examples of assembled structures include multi-protein filaments such as microtubules or actin [1, 2], scaffolds for vesicular budding [3–9], the outer shells or ‘capsids’ of viruses [10–16], and bacterial microcompartments [17–22] or other proteinaceous organelles [23–27]. However, achieving efficient and high fidelity assembly into target architectures requires precisely tuned subunit interaction strengths and concentrations due to competing thermodynamic and kinetic effects (e.g. [28–47]). The need for such precision could severely constrain the use of assembly for biological function or human engineered applications. Biological organisms employ multiple modes of biochemical and physical regulation to overcome this limitation. In this article, we investigate one such mode—how spatial heterogeneity due to formation of biomolecular condensates, or liquid-liquid phase-separation (LLPS), can dramatically enhance the speed and robustness of self-assembly. While membranous organelles play a prominent role in compartmentalizing eukaryotic cells, it is now clear that condensates act as ‘membrane-less compartments’ to spatially organize cellular interiors within all kingdoms of life (e.g. [48–69]). These compartments are implicated in diverse cellular functions, including transcriptional regulation [53, 70–73], formation of neuronal synapses [74–76], enrichment of specific proteins and nucleic acids [77–82], cellular stress responses [83–86], and cell division [79, 87]. In addition to the roles of condensates in normal cellular function, pathogenic viruses generate or exploit LLPS during various stages of their life cycles [88–93]. Most relevant to this article, many viruses undergo assembly and/or genome packaging within phase-separated compartments known as virus factories, replication sites, Negri bodies, inclusion bodies, or viroplasms [88–108]. In vitro studies show that viral nucleocapsid proteins and RNA molecules undergo LLPS (e.g. [99, 106, 109–111]), and that LLPS accelerates assembly of nucleocapsid-like particles [99]. It is hypothesized that viruses exploit LLPS to avoid host immune responses and coordinate events such as RNA replication, capsid protein translation, assembly, and genome packaging. However, the mechanisms underlying these events are poorly understood. In addition to viruses, other examples of biological self-assembly coupled to LLPS include the formation of clathrin cages to mediate endocytosis [112]; post-synaptic densities [113] and pre-synaptic vesicles release sites (active zones) [114, 115] at neuronal synapses; observations that condensates can both accelerate and suppress aggregation of α-synuclein [116], and actin assembly in polypeptide coacervates [117]. Multiple lines of evidence suggest that condensate formation is driven by favorable interactions among their constituents combined with unfavorable interactions with the bulk exterior cytoplasm or nucleoplasm. Although condensation may be driven, destabilized, or regulated by diverse nonequilibrium effects (e.g. [54, 70–73, 118–121]), equilibrium thermodynamics provides a starting point to model their stability, and their formation is frequently described as LLPS [48, 49, 52, 52–60, 69, 74, 85, 118–131]. Henceforth, we will use the term LLPS, keeping in mind that nonequilibrium effects may also be present. Consistent with equilibrium phase coexistence, the composition of the compartment interior can significantly differ from that of the cytoplasm. Thus, LLPS can provide significant spatiotemporal control over reaction processes by concentrating and colocalizing specific sets of subunit species that preferentially partition into the compartment. These capabilities potentially enable LLPS to strongly regulate self-assembly. Yet, despite recent intense investigations into LLPS, its coupling to assembly has yet to be fully explored. Previous simulations showed that the condensed enzyme complex that forms the interior cargo of bacterial microcompartments can promote nucleation and control the size of the exterior protein shell [43, 45, 132–134]. Most closely related to our work, Refs. [116, 135–137] recently showed that the presence of a compartment can significantly accelerate irreversible protein aggregation into linear fibrils. Here, we investigate the effects of LLPS on reversible self-limited assembly into target structures with finite sizes and well-defined architectures. Self-limited assembly from bulk solution is constrained by competing thermodynamic and kinetic effects—subunit interactions must be sufficiently strong and geometrically precise to stabilize the target structure, but overly high interaction strengths or subunit concentrations lead to kinetic traps (e.g. [28–47]). Avoiding such kinetic traps imposes a ‘speed limit’ on assembly from bulk solution [41, 42, 138]. Using a master equation description of assembly, we show that these thermodynamic and kinetic constraints can be simultaneously satisfied by spatial heterogeneity due to phase-separated compartments. We find that LLPS can significantly accelerate assembly nucleation, consistent with previous studies of irreversible assembly [116, 135–137], but also induces kinetic traps that slow assembly in certain parameter regimes. Crucially though, by enhancing nucleation only within spatially localized regions, LLPS significantly expands the range of subunit concentrations and interaction strengths over which such kinetic traps are avoided, thus promoting assembly robustness. This effect can increase by orders of magnitude the maximum rate of productive assembly into the ordered target structure. The extent of assembly acceleration and robustness enhancement are nonmonotonic functions of the key control parameters: the compartment size, and the partition coefficient of subunits between the compartment and the bulk cytoplasm. We present simple scaling estimates that capture the effect of LLPS on assembly, and reveal the underlying mechanisms that enable regulation. For example, the bulk solution acts as a “buffer” that steadily supplies free subunits to the compartment to enable rapid assembly without kinetic traps. Although we particularly focus on self-limited assembly processes that lead to finite-sized structures, our models are general and many results also apply to unlimited assembly or crystallization.

We have developed a minimal model to describe assembly in the presence of one or more liquid droplets coexisting with a background solution of different composition (Fig 1). We are motivated by processes such as virus assembly, in which viral proteins, nucleic acids and other viral components, and possibly some host proteins phase separate to form liquid compartments within the cellular cytoplasm. For this initial study, we consider only one assembling species in the limit that the assembly subunits comprise a small fraction of the compartment mass, and thus the size and composition of the compartment can be treated as independent of the subunit concentration. We consider a system of subunits that form self-limited assemblies with optimal size N. The subunits are immersed in a multicomponent solvent which is in a state of phase coexistence, with stoichiometry such that there are one or more small compartments rich in one (or more) solvent species coexisting with a much larger background rich in the other species. While we consider protein subunits undergoing assembly, the solvent could be comprised of proteins, nucleic acids, or other macromolecules. The distinction is that ‘subunits’ form ordered (para)crystalline structures such as a capsid, while the primary compartment constituents remain amorphous and (for the regime we consider) in the liquid phase. Our model can be viewed as a minimal starting point motivated by biological assembly coupled to LLPS. For example, in the model proposed for rotavirus assembly, two nonstructural rotavirus proteins (NSP2 and NSP5) undergo LLPS to form a viroplasm. The capsid proteins (VP2 and VP6) partition favorably into the viroplasm due to weak multivalent interactions with NSP2 and NSP5 [94, 98, 104, 107]. Here, NSP2 and NSP5 would correspond to the primary compartment constituents, and the capsid proteins (VP2 and VP6) correspond to the assembly subunits. We note that the rotavirus genomic RNA is also driven to partition into the viroplasm through interactions between RNA binding proteins and NSP2/5, but we do not explicitly consider assembly around RNA in this work to simplify the model. Our assumption of low concentration of assembly subunits in the compartment noted above is consistent with concentrations of VP2/6 in the viroplasm that are small compared to those of NSP2/5 [94, 98, 104, 107]. The driving force for subunits to enter the compartment phase is characterized by the partition coefficient Kc, which at equilibrium satisfies with and the subunit concentrations in the compartment and background. The partition coefficient is related to the change in solvation free energy gc for a subunit that transitions from the background to the compartment as with β = 1/kBT with kBT the thermal energy. Applying standard dilute solution thermodynamics will result in a subunit solvation free energy difference with the form (see Weber et al. [120]) gc ∝ ns(Δχ)(Δϕ), where Δϕ is the difference in solvent composition between the background and compartment, Δχ is the difference in interaction strength (parameterized by the Flory χ parameter) for an interaction site on the subunit between the background and compartment, and ns is the number of interaction sites per subunit. The key point is that even for relatively weak interactions, a subunit with multivalency of ns ≳ 10 could have a partition coefficient as large as Kc ∼ 104 − 105, although Kc ∼ 100 may be a typical value [54]. We denote the volumes of the compartment and background as Vc and Vbg, which are related to the total system volume by Vtot = Vc + Vbg. We will present results in terms of the compartment size ratio, Vr ≡ Vc/Vbg. In most biological systems or in vitro experiments, the compartment volume will be small compared to the background, Vr ≪ 1. For this work we assume a fixed total subunit concentration ρT. For simplicity we will typically consider a single compartment, but we also discuss the case of multiple compartments, which might arise due to microphase separation or arrested phase separation.

Let us consider a single compartment in a eukaryotic cell with radius Rcell = 10 μm. At our default compartment volume ratio of Vr = 10−3, the compartment radius is Rc = 1 μm. Assuming a typical protein subunit with mass 30 kDa and a volume of about 50 nm3, requiring a volume fraction of subunits ≤0.01 results in the total number of proteins in the compartment Nc ≲ 106, which is large compared to a typical assembly size of 100–1000 subunits, and sufficiently large that finite number fluctuations can be neglected, at least to a first approximation. Along these lines, defining a ‘minimum’ compartment size as the smallest compartment containing Nmin ∼ 1000 subunits gives Rmin ≈ 0.1 μM. Since compartment radii scale with subunit number ∝ N(1/3), these estimates are insensitive to the assembly size.

To simulate the assembly kinetics, we adapt the rate equation description originally developed by Zlotnick and coworkers [28, 29, 139] and used by others [138, 140] to describe the self-assembly of 2D polymers (capsids) in bulk solution. Denoting the concentration of an intermediate with n subunits in either phase as with α = c, bg, the time evolution of intermediate concentrations is given by: with the diffusive exchange between the phases given by (see Section A in S1 Text and Refs. [135, 137]) and fn and bn as the association and dissociation rate constants for intermediates of size n. We set the initial condition as , , and . We have made several assumptions to simplify the models, based on previous work (e.g [28, 29, 39, 41, 138–140]). First, we assume that there is only one ‘average’ intermediate structure for each size n. Second, we assume that only individual subunits can associate to or dissociate from an intermediate. This assumption is based on the fact that particle-based computer simulations show that, at the dilute conditions typical of productive assembly reactions, most assembly events involve association of individual subunits [138], and that extending Eq (2) to allow for binding of higher-order oligomers does not qualitatively change the results (see the supplemental material of Ref. [138]). Third, we assume that the domain composition is independent of subunit concentration and assembly. Similarly, we assume that the diffusion coefficient is independent of subunit concentration. These simplifications are based on the assumption of low concentrations of subunits in the compartment and that the other compartment constituents are macromolecules that typically have equal or larger molecular weights as the subunits. To focus on effects of competing reactions on assembly, we also neglect the possible dependence of diffusion coefficients on intermediate size or gc, considered in Refs. [136, 137] respectively. The model can be readily extended to account for these effects. The most important simplification is that we neglect the possibility of malformed (off-pathway) structures. While this is a good assumption under productive assembly conditions, particularly when subunit-subunit interactions have high orientational specificity, malformed structures can lead to kinetic traps at high concentrations or binding affinities [32–34, 36–42, 141–143]. This effect will be considered by performing particle-based simulations in a future work, and we discuss its implications in the Conclusions section. To complete the Master equation description we must specify the transition rates between intermediates. We consider two models, which consider different dependencies of rates on the partial capsid size [138].

We start with a simple generic model for a nucleated self-assembly process denoted as the ‘nucleation and growth (NG) model’ [138]. This can describe linear assembly with nucleation (e.g. assembly of a helical viral capsid or the equilibrium assembly of an actin filament), or polyhedral shell assembly with an initial nucleation step, followed by assembly along a single growth front until the shell closes on itself [29, 138, 140] (see Fig 2A). We consider a system of capsid protein subunits with total concentration ρT that start assembling at the time t = 0. We assume that the rate constants are the same in the compartment and background, so we simplify the presentation by omitting the specification of phase in this subsection. Our reaction is given by: where bi is the dissociation rate constant (with i = {nuc, elong, N}), which is related to the forward rate constant by detailed balance as v0bi = f exp(βgi), with gi the change in interaction free energy upon subunit association to a partial capsid and v0 the standard state volume. The nucleation and elongation phases are distinguished by the fact that association in the nucleation phase has an unfavorable free energy change, gnuc − kBT ln(ρ1v0) > 0, while association in the elongation phase is favorable, gelong − kBT ln(ρ1v0) < 0. For the moment, we assume that there is a single critical nucleus size nnuc. For most results in this article, we will set gnuc = −4kBT and gelong = −17kBT and gN = 2gelong. The small value of gnuc relative to gelong accounts for the fact that the first few subunits to associate make fewer and/or less favorable contacts than subunits in larger intermediates, giving rise to a nucleation barrier, while the large value of gN accounts for the fact that in many assembly geometries the last subunit makes the largest number of contacts upon associating. We have chosen values of gnuc and gelong to be roughly consistent with binding affinity values estimated for virus capsid assembly [28, 30, 31, 144], but the results do not qualitatively change for other affinity values within a given assembly regime.

To test whether our conclusions depend qualitatively on the model geometry, we also consider transition rates based on the ‘classical nucleation theory (CNT)’ model for icosahedral capsids suggested by Zandi et al. [145]. In this model, assembly intermediates are represented as partial spheres that are missing a spherical cap. With the parameterization shown in Fig 2B, the total capsid size is N = 4πR2/a0 with R the capsid radius and a0 the subunit area, and intermediate sizes are given by n = N(1 − cos θ)/2. Subunits along the perimeter of the missing cap have fewer interactions than those in the shell interior, leading to a line tension σ, and the total binding free energy for an intermiate with n subunits is with the perimeter of the missing spherical cap given by with gsub the binding free energy per subunit in a complete capsid and l0 the diameter of a subunit. Following previous work [138, 145], we set the line tension to σ = −gsub/2l0, so that a subunit adding to the perimeter of the capsid satisfies half of its contacts on average. We assume that the forward rate constant is proportional to the number of subunits on the perimeter, fn = f0ln/l0, with f0 the association rate constant for a single binding site, and we set and . The key difference between the CNT and NG models is the dependence of the critical nucleus size on solution conditions. For the NG model the critical nucleus size nnuc is constant, provided exp(gnuc/kBT) < ρ1v0 < exp(gelong/kBT). For the CNT model, the critical nucleus size varies with subunit concentration and interaction strengths, and is given by the maximum in kBT log(ρ1v0)n + Gn, or [145] with Γ = [gsub − ln(ρ1v0)]/σl0. Thus, the critical nucleus size continually changes over time during an assembly process for the CNT model as subunits are depleted, whereas it is constant until the very late stages of a NG assembly process.

We begin by calculating how the equilibrium yield of self-assembled structures depends on subunit concentrations and interaction strengths, as well as the two key control parameters for subunit partitioning into the compartment: the partition coefficient Kc and the compartment size ratio Vr. At equilibrium the subunit concentrations in the compartment and background are related to each other by Kc, and to the total subunit concentration ρT by mass conservation, giving with

We now calculate the effect of the compartment on assembly yields, using the well-justified approximation that intermediates have very low concentrations at equilibrium for self-limited assembly [41, 146]. Thus we consider a two-state system, with finite concentrations of only free subunits and complete assemblies with N. Mass conservation then gives where and are the concentrations of assemblies in the compartment and background. At equilibrium these are related to the free subunit concentration by the law of mass action [41, 147] with α = dom, bg and gsub as the per-subunit interaction energy within a complete assembly (which we assume is the same in the compartment and background). Eqs (10) and (11) can be easily solved numerically. However, we can make the results more transparent by following previous analysis for homogeneous assembly [41, 148] and writing the fraction of subunits in assemblies as This simplifies to In the limit of large optimal assembly size N ≫ 1, Eq (13) satisfies the following asymptotic limits: with ρCAC the critical assembly concentration (CAC) given by (assuming , so that all assembly occurs in the compartment) with as the CAC in a system without coupling to LLPS (i.e. Vr = 0 or Kc = 1). In all subsequent expressions, we will write the limit of no LLPS or Kc → 1 with a superscript ‘0’. The last expression in Eq (16) assumes Vr ≪ 1 and shows that LLPS reduces the CAC by a factor KcKeff. Then using Eq (8) we arrive at the simple result that significant assembly occurs when the total subunit concentration ρT exceeds the local CAC within the compartment. To obtain further insight, we note that KcKeff ≈ (Vr + 1/Kc)−1, giving the asymptotic limits and that maximal enhancement of equilibrium assembly is achieved when Kc ≳ Vr.

We can draw two important conclusions from Eq (16). First, the presence of a compartment allows assembly under conditions where there is no bulk assembly (Fig 3). Second, there is a range of total subunit concentrations ∝ KcKeff over which assembly occurs only in the compartment, thus allowing for spatial control over assembly. As a measure of the extent to which LLPS can spatially control assembly, we define selectivity as . The equilibrium selectivity is then given by We thus see that even a very small partition coefficient leads to strong equilibrium selectivity due to the high-valence nature of an assembled capsid. In particular, an assembled capsid has ∼N interactions with compartment components, but only has three translational degrees of freedom suppressed by partitioning into the compartment volume. However, if assembled capsids and large intermediates are not able to rapidly exchange between the compartment and background [137], the selectivity at finite times may be under kinetic control.

Figs 3 and 4 show the effect of LLPS on assembly kinetics, as measured by the fraction of subunits in complete capsids (xN), obtained by numerically integrating the Master equation (Eq 2) with the NG model (Eq 4). Fig 4A shows xN as a function of time for several initial subunit concentrations ρT in the absence of LLPS. There is an initial lag phase during which intermediate populations build up to a quasi-steady-state, followed by rapid appearance of complete capsids, and then eventually saturation as free subunits are depleted. The duration of the lag phase decreases as 1/ρT. Importantly, the rate of capsid production is nonmonotonic with respect to ρT—yields of complete capsids are suppressed for the highest concentration shown (ρT = 4M) by the monomer starvation kinetic trap arising from depletion of free subunits before capsids finish assembling. These results are discussed further in section Assembly timescales without LLPS and Refs. [41, 138]. This kinetic trapping effect is responsible for the low values of xN at high concentrations for finite-time results in Fig 3. Figs 4B and 3A show how the assembly kinetics is changed by LLPS. With the lowest concentration shown in Fig 4A (ρT = 0.2 μM), xN is shown as a function of time for increasing values of the partition coefficient Kc. We see that the yields and assembly rates increase dramatically, with the duration of the lag phase decreasing and the maximum rate of capsid production (corresponding to the nucleation rate) increasing with Kc. To give a more comprehensive picture, Fig 3A shows xN as a function of both Kc and ρT. We see that assembly occurs at lower concentrations as Kc increases, and that LLPS increases the range of concentrations over which productive assembly occurs, particularly for Kc of . Similarly, Fig 3B and 3C respectively show xN as a function of concentration measured at 1 day, 107 seconds, and equilibrium. With or without LLPS, productive assembly at one day occurs over a much narrower range of concentrations than predicted by equilibrium, due to nucleation barriers at small concentrations and kinetic traps at high concentrations. Even at extremely long times the results have not reached full equilibrium due to kinetic traps at high concentrations. However, LLPS significantly broadens the range of concentrations leading to productive assembly at all timescales. The solid lines in Fig 3A and the dashed lines in Fig 3B and 3C indicate the CAC (ρCAC, Eq (16)), and scaling estimates for threshold concentrations below/above which productive assembly is impeded by large nucleation barriers or kinetic traps respectively (see section Assembly timescales without LLPS). Notice that in addition to making assembly more robust, LLPS also increases the maximum yield achievable at finite times. This increase arises because both nucleation and elongation rates can locally increase within the droplet due to the high local concentration while avoiding free subunit depletion. The ability of LLPS to avoid kinetic traps arises because, for Vr ≪ 1, the background acts like a buffer that steadily supplies free subunit to the compartment even when the nucleation rate is large. As a measure of this behavior, Fig 5 shows the concentration of subunits in the background normalized by the total concentration, as a function of the maximum capsid formation rate (which occurs shortly after the end of the lag phase, before significant free subunit depletion has occurred). Here we have measured at the time point corresponding to the maximum rate. Results are shown for LLPS assembly for the same parameters as in Fig 4, with increasing rates corresponding to increasing values of Kc. For the case without LLPS, we achieve faster rates by increasing the subunit-subunit affinities from (gnuc, gelong) = (−4, −17)kBT to (gnuc, gelong) = (−6, −25.5)kBT. We have increased affinities rather than total concentration (as we do for other results) to simplify comparison of between the two cases. The results stop at gnuc = −6kBT because stronger affinities lead to low yields and decreasing rates due to the monomer starvation trap. We see that with LLPS the subunit concentration remains near ρT even for extremely high assembly rates, whereas subunits are rapidly depleted without LLPS. For higher subunit affinities |gnuc| > 6kBT depletion is so rapid that the monomer starvation trap sets in. Note that, in our Master equation description, subunits will eventually be depleted as xN → 1 even with LLPS, but in reality excluded volume effects (which are neglected in our model) would suppress assembly rates before this point unless complete capsids are expelled from the compartment.

Fig 6 shows the selectivity (xselec) measured from Master equation solutions and the equilibrium result (Eq (19)) as a function of Kc for several values of target capsid size N. We see that finite-time selectivity values closely match the equilibrium results, and that even an extremely small partition coefficient is sufficient to drive highly selective assembly in the compartment for large N.

To gain insights into how LLPS can affect assembly, in this section we derive simple scaling estimates for the timescales associated with the nucleation and growth mechanism of Eq (4). We closely follow Refs. [138, 149], but we extend the analysis to include the effect of a compartment. Although we introduce a number of simplifications, in the next section we show that the resulting scaling estimates provide good approximations when these simplifications are relaxed by numerically solving the Master equation models.

Let us begin by summarizing the analysis of Ref [138] for assembly timescales in the absence of LLPS. As above, we consider a system of subunits with total concentration ρT that form assemblies with optimal size N subunits, and we break the assembly process into ‘nucleation’ and ‘elongation’ phases. For simplicity we assume that the association rate constant f is independent of intermediate size (except where mentioned otherwise), so that the rates of association to each intermediate are fρ1. We now write the time required for an individual assemblage to form as τ = τnuc + τelong with τnuc and τelong the average times for nucleation and elongation, respectively. The elongation timescale can generally be estimated as [138, 146] where we have assumed N ≫ nnuc so that N − nnuc ≈ N. The factor in the numerator indicates that the elongation timescale increases with optimal assembly size (i.e. α > 0) since independent subunit additions must occur. The value of the exponent α will depend on factors such as the dimensionality, the aggregate geometry, and the relative stability of intermediates, but we expect 1/2 ≤ α ≤ 2. For strongly forward-biased assembly during the elongation phase, α = 1 for the NG model and α = 1/2 for the CNT model (see [138] and Section B in S1 Text). Except where specified otherwise, for the scaling estimates in the rest of this article we set α = 1, but the results are easily extendable to other exponent values. The mean nucleation time at the beginning of the reaction can be estimated from the statistics of a random walk biased toward disassembly [29, 138], and can be approximately written as where so that is the interaction free energy of the structure just below the critical nucleus size. The form of Eq (21) can be understood by noting that the pre-critical nucleus is present with concentration , and subunits associate to the precritical nucleus with rate fρT. Note that the special case of nnuc = 2 corresponds to no nucleation barrier (since two subunits must associate to begin assembly), in which case and . We consider this case in Section D in S1 Text. While Eq 21 gives the initial nucleation rate, the nucleation rate decreases over time due to subunit depletion, and asymptotically approaches zero as the concentration of complete capsids approaches its equilibrium value. Thus, we estimate the median assembly time τ1/2 (the time at which the reaction is 50% complete) by treating the system as a two-state reaction with nnuc-th order kinetics, which yields [138] with , and xN as the equilibrium fraction of subunits in complete capsids. The factor of N−1 in Eq 22 accounts for the fact that N subunits are depleted by each assembled capsid. Analogous to crystallization or phase separation, there is a range of subunit concentrations and interaction strengths for which the unassembled state is metastable; i.e., the system is beyond the CAC so assembly is thermodynamically favorable, but the nucleation timescale exceeds experimentally relevant timescales. The boundary of this regime can be estimated by inverting Eq (22). Denoting the ‘relevant’ observation timescale as τobs, we can estimate the threshold subunit concentration below which nucleation will not be observed as When elongation is fast compared to nucleation, the expressions Eq. S5 (in Section B in S1 Text) and Eq 22 respectively predict the duration of the lag phase and the median assembly time. However, these relations begin to fail above threshold values of the subunit concentration or subunit-subunit binding affinity, when nucleation and elongation timescales become comparable. Upon further increasing these parameters, nucleation becomes sufficiently fast that a significant fraction of monomers are depleted before elongation of most structures can complete. Subsequent evolution into complete assemblages then requires exchange of subunits between different intermediates (Ostwald ripening), which is an activated process and thus slow compared to assembly timescales. We describe this condition as the monomer-starvation kinetic trap. The threshold subunit concentration ρ* and interaction energies beyond which the system begins to enter the trap can be estimated by the locus of parameter values at which the median assembly time and elongation time are equal, i.e., τ1/2(ρ*) = τelong(ρ*): and a corresponding assembly timescale Note that corresponds to approximately the minimal timescale or maximal assembly rate (over all subunit concentrations) since both τ1/2 and τelong monotonically decrease with subunit concentration before the onset of kinetic trapping.

We now extend the scaling analysis to account for the presence of a compartment. Based on the conclusion of Section A in S1 Text that exchange of subunits between the background and compartment is typically much faster than assembly rates, we will make a quasi-equilibrium approximation for the relationship between subunit concentrations in the compartment and backround: and . As shown previously for irreversible aggregation [120], the compartment can dramatically amplify the nucleation rate by locally concentrating subunits. The total initial nucleation rate (in both the compartment and background at the beginning of the assembly process) is given by with the nucleation rate in the absence of a compartment, and snuc the acceleration factor for the initial nucleation rate. Eq (26) shows that for nucleation will occur exclusively in the compartment, and the nucleation acceleration factor simplifies to . To estimate the parameters that maximize the initial nucleation rate, we optimize Eq (26) with respect to Vr to obtain and thus a maximum nucleation acceleration of: As in the equilibrium analysis in section Effects of LLPS on self-assembly equilibrium, under optimal conditions nucleation proceeds nearly as if the total subunit density were amplified by the partition coefficient Kc.

We now evaluate the effect of the compartment on the propensity of the system to undergo the monomer-starvation kinetic trap, by evaluating the dependence of the elongation and median assembly timescales on the phase-separation parameters. The median assembly timescale can be computed by the same analysis used above for the nucleation time, leading to and similarly the threshold concentration below which nucleation does not occur on relevant timescales is with given by Eq (23). The elongation time within the compartment, τelong,D or background τelong,B is given by Eq. S5 in Section B in S1 Text with the appropriate local concentration or . To estimate the onset of the kinetic trap, we must account for the numbers of capsids that are forming by both reaction channels (in the compartment or background), so we compute an average elongation time weighted by the relative number of assemblies that form the compartment or background In the limit of strongly forward-biased elongation and so all nucleation occurs in the compartment, the elongation timescale is approximately As discussed in section Assembly timescales without LLPS, the minimum assembly timescale occurs when the nucleation and elongation timescales are equal, τelong(Kc, Vr, ρ*) = τ1/2(Kc, Vr, ρ*); the monomer-starvation kinetic trap begins beyond this point. Using Eqs (26), (28), and (31) results in Finally, we can approximately extend the scaling estimates of this section to the CNT model by substituting Eq (7) for nnuc. Eq (32) shows that a key feature of preferential partitioning into the compartment is the ability of the system to buffer itself against the monomer-starvation kinetic trap while maintaining fast localized assembly in the compartment, as shown in Fig 5. We can further assess this feature in several ways as follows. Fig 7A and 7B compare Eqs (28) and (30) to the median assembly and elongation times computed from the rate equations as a function of ρT and Kc respectively. We see that the scaling estimates and numerical results closely agree until the nucleation and elongation timescales become comparable; the threshold concentration ρ* (Eq (32)) and partition coefficient (Eq. S8 in Section C in S1 Text) are shown as vertical dashed lines in Fig 7A and 7B respectively. Fig 7C and 7D show the median assembly times computed from the rate equations as a function of ρT and Kc and Vr respectively, with the locus of parameter values leading to minimum assembly predicted by Eq (32) shown as white ∘ symbols. Fig 8 shows analogous results for the CNT model. The prediction closely tracks the minimum assembly timescale observed in the numerical results. Below this threshold the median assembly time is closely predicted by Eq (28), with the assembly timescale sped up according to snuc in Eq (26). Above this threshold the numerically computed assembly timescales rapidly increase due to overly fast nucleation and thus onset of the monomer-starvation trap. We also show the CAC and the threshold for achieving assembly within an observation time of 1 day on these plots. Notice that, at a given value of Vr, there is an optimal value of Kc (estimated below) which maximizes robustness of assembly to variations in concentration. In contrast, robustness monotonically increases with decreasing Vr.

Given that the compartment both shifts and broadens the range of parameter values over which productive assembly can occur, it is of interest to determine parameters for which LLPS has the strongest effect on assembly times. To this end, we define the assembly ‘speedup’ as the factor by which the median assembly time decreases with LLPS relative to bulk solution: sLLPS(Kc, Vr) ≡ τ0/τ(Kc, Vr). Recalling that the minimum assembly timescale occurs at ρ* when elongation and nucleation times are equal, we can then maximize the speedup with respect to the partition coefficient to obtain (Section C in S1 Text) Thus, for an optimal compartment partition coefficient, assembly can be sped up (i.e. τ1/2 reduced) by orders of magnitude for small Vr. The degree of speedup increases with: decreasing Vr, increasing critical nucleus size, and decreasing total subunit concentration. These trends can be understood as follows. Decreasing Vr means that subunits are not depleted as quickly within the compartment, thus allowing larger values of Kc and correspondingly higher local concentrations of subunits within the compartment without depleting subunits quickly enough to cause over-nucleation and monomer starvation. A larger critical nucleus size provides a larger separation between nucleation and growth timescales, thus enabling further concentration of subunits in the compartment without over-nucleation. The decreasing dependence on concentration arises because as the system approaches , the assembly timescale without LLPS decreases and thus so does the extent of possible speedup before over-nucleation sets in. However, note in Fig 8 that the maximum optimal concentration in the presence of LLPS exceeds the intrinsic value, , due to the extra regulation of nucleation and growth timescales allowed by a compartment. Also note that LLPS provides speedup even after monomer starvation begins to set in. Fig 9 compares the scaling estimate for speedup (see Eq. S9 in Section C in S1 Text) to the value computed numerically from the rate equations. For the numerical value, we computed for fixed ρT and interaction parameters by numerically integrating the rate equations without LLPS, and then performed numerical minimization over Kc to obtain with respect to Kc for the same ρT and interaction parameters. Then the speedup is given by . We have presented the speedup as a function of concentration normalized by the optimal value in the absence of LLPS so that the results can be shown on the same plot. As shown in Fig 9A, the scaling estimate closely matches the numerical result until ρT exceeds the maximum value of ρ* at which point monomer starvation begins to set in. The results for the CNT model are obtained by substituting Eq (7) into Eqs. S7-S9. The agreement is reasonable but not as close as the NG estimate because Eq (7) is based on the critical nucleus size in the absence of LLPS. As noted above, we see that LLPS continues to speedup the assembly time even in the monomer-starvation regime. Fig 9B shows the speedup as a function of Vr for fixed ρT. Here we see good agreement between the scaling estimate and numerical results, except the numerical speedup diminishes at small Vr for nnuc = 5. This occurs because the minimum assembly timescale has decreased below the diffusion limited timescale (Eq. S4 in Section A in S1 Text) in this regime, which is not accounted for in the scaling estimate.

Notice that decreases with increasing subunit concentration (see Eq. S7 in Section C in S1 Text). For large subunit concentrations (i.e. ), the assumption that assembly occurs primarily in the compartment breaks down and we must consider the full form of snuc (Eq (26)). If we substitute this into the expression for Eq (32), we see that there is a maximum in ρ* at which results in Finally, using Eq 16 shows that the range of subunit concentrations over which assembly is favorable increases as For small Vr, the width of this range, and thus the robustness of assembly to variations in subunit concentration or subunit-subunit binding affinities, increases by orders of magnitude (Fig 7D). We can alternatively specify robustness by defining the region of productive assembly as the set of parameter values for which nucleation occurs within experimentally relevant timescales (e.g. 1 day) and avoids the monomer-starvation trap. To maximize the breadth of this range, we define as the partition coefficient that maximizes the ratio of the monomer-starvation threshold to the nucleation timescale threshold (Eq (29)): .

It is well-established that efficient self-assembly in homogeneous solution is constrained to a narrow window of moderate subunit concentrations and interaction strengths, due to the competing constraints of minimizing nucleation timescales while avoiding kinetic traps [28–47]. Here, we find that when subunits preferentially partition into nano- or microscale compartments, the range of parameters leading to productive assembly can be broadened by more than an order of magnitude, and the corresponding assembly timescales can be reduced by multiple orders of magnitude. Moreover, in part of this parameter range, almost all assembly occurs within the compartment interior, thus allowing spatial control over assembly. These behaviors depend sensitively on two parameters that control phase coexistence: the partition coefficient of subunits into phase separated compartments and the size ratio between the compartments and the cell. In addition, we find that the maximum degree of speedup due to LLPS increases with: decreasing compartment/cell size ratio or subunit concentration and increasing assembly critical nucleus size. These effects arise because the compartment (or compartments) drive high local concentrations of subunits, thus minimizing the local nucleation timescales, but the small size of the compartment limits the total nucleation rate (averaged over the whole system volume). In effect, the bulk exterior acts as a subunit ‘buffer’ that, early in the reaction, steadily supplies subunits to the compartment and thereby suppresses the monomer starvation kinetic trap (see Fig 5). This mechanism has the strongest effect on robustness of assembly to variations in parameter values for small critical nucleus sizes or non-nucleated reactions, for which the homogeneous system lacks an intrinsic difference between nucleation and growth timescales and thus is most sensitive to subunit depletion. However, the decrease in assembly timescales is most dramatic for larger critical nucleus sizes, due to the high-order dependence of assembly timescales on local subunit concentration.

Since these mechanisms depend on localization of subunits, the ability of LLPS to control assembly increases with decreasing compartment size (relative to the total system size). To estimate the relevance of this effect in biological systems, consider that typical compartments in eukaryotic cells range in size from ∼50nm to 10 μm [54, 123, 124, 130]. For a compartment with diameter 1 μm in a cell with diameter 20 μm, the volume ratio of the compartment relative to bulk is Vr ∼ 10−4. From Eqs (35) and (36) and Fig 7, we see that the range of subunit concentrations leading to productive assembly could increase by up to two orders of magnitude, with increases in assembly rates exceeding five orders of magnitude (Eq. S9 in Section C in S1 Text and Fig 9). These increases reflect the ability of compartmentalization to enable fast localized assembly while minimizing the rate of global depletion of subunits.

Since our models are general, the quantitative predictions and scaling formulae described here can apply to a broad range of experimental systems in which there is phase coexistence and the assembly subunits preferentially partition into one phase. Such phase-separated compartments appear to be ubiquitous in cells, and as noted in the introduction, assembly of diverse structures such as clathrin cages, actin filaments, and neuronal synapses can occur within compartments. The systems which most directly inspired this work are the phase-separated compartments generated during viral infections (e.g. virus factories, replication sites, Negri bodies, inclusion bodies, or viroplasms [88–108]), within which viral particles undergo assembly. However, directly testing our theoretical predictions may be easier in in vitro experiments, since there is a greater ability to control the size and composition of compartments [52, 54, 76, 99]. Compartment sizes can be controlled in bulk systems by varying the total density of the phase-separating components, while microfluidic arrays enable precise control over droplet sizes and compositions.

We have focused on a minimal model for this first study of the effects of LLPS on assembly robustness. There are a number of additional physical ingredients that merit further exploration. For parameter regimes that lead to high subunit concentrations within the compartment, the assumption that the subunits do not affect the equilibrium compartment size and composition will break down. Importantly, the rate equation models and scaling estimates considered here do not account for kinetic traps resulting from malformed assemblies, which can arise when subunits bound with incorrect geometries do not have time to anneal before becoming locked into place by association of additional subunits (e.g. [11, 32, 39, 41, 42, 141, 142, 150, 151]). Since association rates increase with concentration, we anticipate that malformed assemblies will occur above a threshold local concentration within the compartment, thus limiting the maximum speed up provided by LLPS. This threshold concentration increases with the geometric specificity of the subunit-subunit interactions. Thus, for sufficiently specific interactions the results described here will not qualitatively change when accounting for malformed structures—there will be a significant range of local concentrations, and thus assembly speed up, before either the malformed structure for monomer starvation kinetic trap set in. We will explore the effects of malformed assemblies on LLPS-coupled assembly in a future work. Further, Schmit and Michaels showed that, if subunit diffusion slows with increasing subunit-compartment attraction strength (gc and Kc in our model), then there is an optimum Kc beyond which assembly slows. The results in our work arising from competing interactions are distinct from this effect. Other important effects to be incorporated include: slow diffusion into/out of the compartment [137], accounting for spatial structure and stochasticity of assembly [32, 152–155], nonequilibrium effects such as synthesis of new subunits or phosphorylization-driven changes in assembly activity, selective partitioning of different species in a multicomponent assembly reaction, and the ability of the compartment to template the size and shape of assemblies, such as occurs in bacterial microcompartments [43, 45, 132, 133]. Ultimately, understanding how different combinations of these physical mechanisms enable phase separation processes to control the time, place, and rate of assembly will engender a more complete understanding of biological self-assembly, and can advance strategies for designing human-engineered nanostructured materials.

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