To capture the interplay between liquid phase separation and protein aggregation kinetics we start with a model of two coexisting phases. One phase could be rich in proteins for example and coexist with a phase rich in another protein component, lipid, or water. Monomers that are prone to aggregate can partition differently into these phases. This partitioning is determined by the relative interactions between the majority components of each phase with the monomers. We consider the case where the partitioning of monomers is close to equilibrium during the kinetics of aggregation. This assumption is well justified since small, weakly interacting molecules such as the aggregating monomers diffuse between seconds and minutes through a cell of size in the order of tens of $\mu m$ (Brangwynne et al., 2009; Griffin et al., 2011), while typical time scales of aggregation in vitro are in the order of hours (see, for example, Cohen et al., 2013). Furthermore, the diffusion of aggregates is highly hindered as long fibrillar aggregates experience a much larger hydrodynamic drag force and can get entangled with cytoskeletal filaments and other assembled fibrils (de Gennes, 1971; Rubinstein, 1987). Finally, at large enough density and size, fibrils may even form solid-like gels (Mateju et al., 2017) further slowing down their mobility. All these effects imply that we may safely neglect diffusion of large aggregates and consider the typical case that monomers diffuse quickly relative to their aggregation kinetics.

We also consider the case where monomers and aggregates are dilute enough to neglect their influence on the composition of the two coexisting protein phases. Typical values of volume fractions for monomers of Amyloid-β, $c_{\mathrm{m}}{\nu }_{\mathrm{m}}$ (radius of gyration in the range 1–2 nm [Sajfutdinow et al., 2018]), at physiological concentrations between 100 pM to 1 nM are in the range of 10^-9 to 10^-8. Time scale separation and dilute monomers together ensure that the compartment can coexist at thermodynamic equilibrium while the partitioning kinetics of monomers may weakly deviate from the partitioning equilibrium. Thus, we first discuss the partitioning of monomers into phase separated compartments at equilibrium and then consider small deviations from this equilibrium to understand its consequences for protein aggregation.

We consider a system of total volume $V$ hosting a single liquid compartment (a droplet for example) of a condensed phase I of volume ${\displaystyle V^{\mathrm{I}}}$. The compartment itself forms by liquid-liquid phase separation between the two components $A$ and $B$. Compartment I is composed of the component $A$ and a small fraction of component $B$, while compartment II has a small amount of $A$ and a large amount of $B$, as depicted in Figure 1. Each compartment creates a distinct environment for the aggregating monomers.

For simplicity, we discuss the case of an incompressible system where the aggregating monomers ‘m’ and aggregates ‘a’ are dilute, that is $c_{\mathrm{m}}{\nu }_{\text{m}}\ll 1$ and $c_{\mathrm{a}}{\nu }_{\text{a}}\ll 1$, with $c_{\text{m}}$ and $c_{\text{a}}$ denoting the concentrations of monomers and aggregates and ${\nu }_{\text{m}}$ and ${\nu }_{\text{a}}$ are the respective molecular volumes. The assumption of dilute monomers and aggregates imply that for an incompressible system, the volume fractions ${\varphi }_A$ and ${\varphi }_B$ of the protein components $A$ and $B$ obey, ${\varphi }_A+{\varphi }_B=1-c_{\mathrm{m}}{\nu }_{\mathrm{m}}-c_{\text{a}}{\nu }_{\text{a}}\simeq 1$, where we abbreviate ${\varphi }_A=\varphi$ in the following. As a result, the monomers may partition differently into the respective minority and majority phases, but, due to their dilute concentrations, they do not affect the degree of phase separation. Under these circumstances and in the absence of binding processes, the partitioning of monomers in the two phases is governed by the relative interaction strength $\mathrm{\Delta }\chi$ between the monomers with the $A$ and the $B$ components, respectively. If $\mathrm{\Delta }\chi$ is large and positive, monomers favor the presence of the majority component $A$ in compartment I. In this case, we expect a more pronounced partitioning of monomers into compartment I. Contrariwise, when $\mathrm{\Delta }\chi$ is large and negative, monomers favorably partition into compartment II. The degree of monomer partitioning at equilibrium can be calculated using the condition that the chemical potentials of monomers associated with compartment I and II are balanced (see Figure 2(a), and Appendix 1 for the derivation), and allows us to define the monomer partitioning

where ${\displaystyle {\mathrm{c}}_{\mathrm{m}}^{\mathrm{I}}}$, ${\displaystyle {\mathrm{c}}_{\mathrm{m}}^{\mathrm{I}}}$ are the monomer concentrations in phases I and II, respectively, $\nu$ denotes the molecular volume of $A$ and $B$ molecules, and ${\displaystyle {\varphi }^{\mathrm{I}}-{\varphi }^{\mathrm{I}\mathrm{I}}\in [0,1]}$ is the degree of phase separation of the $A$-component. Then the relative partitioning of the total monomer concentration, ${\displaystyle c_{\mathrm{m}}^{\mathrm{t}\mathrm{o}\mathrm{t}}=\left(c_{\mathrm{m}}^{\mathrm{I}}{V}^{\mathrm{I}}+c_{\mathrm{m}}^{\mathrm{I}\mathrm{I}}{V}^{\mathrm{I}\mathrm{I}}\right)/V}$, is given by the expressions ${\displaystyle c_{\mathrm{m}}^{\mathrm{I}}=\xi (\overline{\varphi })\mathrm{\Gamma }{c}_{\mathrm{m}}^{\mathrm{t}\mathrm{o}\mathrm{t}}}$ and ${\displaystyle c_{\mathrm{m}}^{\mathrm{I}\mathrm{I}}=\xi (\overline{\varphi })c_{\mathrm{m}}^{\mathrm{t}\mathrm{o}\mathrm{t}}}$, where the partition degree

captures the impact of the relative size of the compartment volume ${\displaystyle V^{\mathrm{I}}(\overline{\varphi })/V}$. The volume of the compartment I is in turn controlled by the mean volume fraction $\overline{\varphi }$ of $A$ molecules in the system in terms of the relationship ${\displaystyle V^{\mathrm{I}}(\overline{\varphi })=V(\overline{\varphi }-{\varphi }^{\mathrm{I}\mathrm{I}})/({\varphi }^{\mathrm{I}}-{\varphi }^{\mathrm{I}\mathrm{I}})}$, where we neglected the volume contribution of monomers and aggregates due to the considered dilute conditions. For finite sized compartments, the equilibrium volume fractions, ${\displaystyle {\varphi }^{\mathrm{I}}}$ and ${\displaystyle {\varphi }^{\mathrm{I}\mathrm{I}}}$, are slightly increased due to the Laplace pressure. However, for compartments significantly exceeding the size of the molecules the relative increase is weak and is thus neglected in the following (see Appendix 1).

Figure 2

Download asset Open asset

Due to the separation of time scales of monomer diffusion and monomer aggregation, the partitioning of monomers into the compartment is close to equilibrium at all times of the aggregation kinetics and thus the relative fraction of monomers is approximately governed by the monomer partitioning $\mathrm{\Gamma }$, Equation 1. However, as the aggregation kinetics decreases the amount of monomers inside each phase, aggregation couples to the partitioning. This coupling is represented by a diffusive flux of monomer with a rate $J^{\alpha }$ in each phase, that attempts to maintain the monomer partitioning close to equilibrium. In the limit of a sharp interface separating the liquid compartment from the bulk, there is no aggregation at the interface, ${\displaystyle J^{\mathrm{I}}=-J^{\mathrm{I}\mathrm{I}}\equiv J}$. Furthermore, to linear order, the flux ${\displaystyle J}$ between the phases is proportional to the difference of monomer partitioning with respect the equilibrium value $\mathrm{\Gamma }$ (see Appendix 2 for the derivation) and is of the form:

where ${\displaystyle M_{\mathrm{m}}^{\alpha }=c_{\mathrm{m}}^{\alpha }{m}_{\mathrm{m}}}$ (with ${\displaystyle m_{\mathrm{m}}}$ as monomer mass) is the monomer mass concentration in compartment ${\displaystyle \alpha =\mathrm{I},\mathrm{I}\mathrm{I}}$, and $k$ denotes the rate at which monomer partitioning relaxes back to the equilibrium given by Equation 1. For simplicity, we consider the case where diffusion of monomers is constant and equal in each phase, and not affected by the aggregates.

Very generally, in a homogeneous solution, irreversible protein aggregation results from the combined action of several microscopic events, including (i) primary nucleation, whereby monomers spontaneously interact to form the smallest stable aggregate structures, (ii) fibril elongation, and (iii) secondary (i.e. aggregate-dependent) nucleation processes (Michaels and Knowles, 2014; Michaels et al., 2016; Arosio et al., 2016; Michaels et al., 2018; Törnquist et al., 2018). Secondary nucleation mechanisms (Törnquist et al., 2018) have been found to be active in many aggregating protein systems, ranging from prions to amyloidogenic proteins (Zhu et al., 2003; Kundel et al., 2018; Ruschak and Miranker, 2007; Meisl et al., 2014; Cohen et al., 2013); key examples of such secondary nucleation processes include fibril fragmentation, lateral branching and surface-catalyzed secondary nucleation.

In the presence of a liquid compartment, irreversible protein aggregation of fibrillar structures occurs within each phase as a consequence of both primary and secondary nucleation, and growth of aggregates via their ends, each event occurring with rate constants $k_1$, $k_2$, and $k_{+}$(Michaels and Knowles, 2014; Michaels et al., 2016; Arosio et al., 2016; Michaels et al., 2018). We have seen that the key term in our model is the difference between the monomer concentration inside and outside of the compartment which leads to the diffusive flux of monomers $J^{\alpha }$ between the phases (Equation 3a), which connects the effects of phase separation and protein aggregation. The coupled equations describing protein aggregation kinetics in both phases can be written as

Here, Equation (3b) describes the rate of formation of new fibrils in each compartment (${\displaystyle \alpha =\mathrm{I},\mathrm{I}\mathrm{I}}$) through primary nucleation, fragmentation or surface catalyzed secondary nucleation. In the case of primary nucleation, the rate of formation of new aggregates depends solely on the concentration of monomers, where the reaction order $n_1$ describes the concentration dependence of nucleation. For secondary processes, including fragmentation and surface-catalyzed secondary nucleation, the rate of formation of new aggregates is proportional to the aggregate mass concentration; the dependence of the rate on the monomer concentration is described by the reaction order $n_2$ (the case $n_2=0$ corresponding to fragmentation). Note that both primary and secondary nucleation of aggregates are non-classical, multi-step nucleation processes; hence, the reaction orders $n_1$ and $n_2$ do not necessarily correlate to the physical size of nuclei (Šarić et al., 2016). Equation (3c) captures the build-up of aggregate mass within each compartment due to elongation of existing aggregates, which occurs by monomer addition at their ends. Finally, Equation (3d) models the population balance of monomers in each compartment as a result of two effects: (i) monomer depletion due to aggregate growth (see Equation (3c)) and (ii) the monomer flux between compartments I and II; this flux is given by Equation (3a) and ensures that partitioning is maintained close to the monomer partitioning factor $\mathrm{\Gamma }$.

While the monomer partitioning factor $\mathrm{\Gamma }$ (Equation 1) governs the constant ratio of the time dependent concentrations in compartment I and II, the partitioning degree $\xi$ (Equation 2) determines how the total monomer concentration, which decays over time as a result of aggregation, is split between the two compartments at any time point during the kinetics of aggregation. As we will see, both parameters will be crucial in controlling the degree of aggregate partitioning into the compartments.

To understand how protein aggregation kinetics couples to the two phase separated compartments in terms of the physical parameters $\mathrm{\Gamma }$ and $\xi$, we constructed explicit analytical solutions to the set of non-linear kinetic Equation (3) by exploiting an analogy to classical mechanics (Michaels et al., 2016 and Appendix B for details of the calculations), and compared these with numerical solutions of (Equation 3).

In the limit of fast monomer diffusion, the aggregation kinetics in each compartment is controlled by a set of effective rate parameters. The relative magnitude of these effective rates between compartment I and II at early times scales with the monomer partitioning as ${\mathrm{\Gamma }}^{n_1}$, while at late times, the corresponding ratio of these rates scales with ${\mathrm{\Gamma }}^{(n_2+1)/2}$ (see Appendix 3, Equation (39) and Equation (40)). Thus, the aggregate growth inside compartment I is faster than in compartment II if there is enrichment of monomers in the condensed phase (${\displaystyle \mathrm{\Gamma }>1}$). Moreover, the relative magnitudes of growth rate at early times solely depends on the reaction order of primary nucleation, $n_1$, while at late times, relative growth is determined by the reaction order of secondary nucleation, $n_2$.

This difference in growth rates between the phases can be qualitatively explained by the rapid preference of monomers to recover phase equilibrium (Figure 3(a)). The enhanced monomer concentration in compartment I causes aggregates to nucleate first inside compartment I. As a consequence, elongation of aggregates is more pronounced inside compartment I leading to a stronger consumption of monomers. This difference in monomer consumption between the compartments couples to the flux (Equation 3), which forces more monomers to diffuse into compartment I to maintain partitioning equilibrium, even as aggregates grow. This positive feedback mechanism in compartment I is accompanied by negative feedback for compartment II, which continuously loses monomers leading to a slowing down of the aggregation kinetics outside. Thus, the coupling between the aggregation kinetics and phase separation, mediated by diffusion of monomers (Equation 3), is key to determine aggregate enrichment/depletion in each phase.

Figure 3

Download asset Open asset

To understand this feedback mechanism, we study the time evolution of the aggregate concentration inside each phase, ${\displaystyle c_{\mathrm{a}}^{\mathrm{I}}(t)}$ and ${\displaystyle c_{\mathrm{a}}^{\mathrm{I}\mathrm{I}}(t)}$ (Figure 3(b)). The first aggregates are initiated by primary nucleation and solely determined by the monomer concentration. Because monomer concentrations in the compartments are slaved due to the rapid flux that maintains partitioning equilibrium, the time evolution of the aggregate concentrations in the early regime of the aggregation kinetics are slaved as well, following ${\displaystyle c_{\mathrm{a}}^{\mathrm{I}}(t)/c_{\mathrm{a}}^{\mathrm{I}\mathrm{I}}(t)\propto {\mathrm{\Gamma }}^{n_1}}$. When aggregates start consuming monomers via elongation, the flux of monomers from compartment II to I causes a saturation of the aggregate concentration outside the compartment II, while the concentration of aggregates in compartment I increases significantly. This rapid increase of growth is facilitated by the continuous influx of monomers (positive feedback). As monomers get depleted in the entire system the growth of aggregates also saturates in compartment I. Most importantly, the resulting asymptotic concentrations at large time scales, ${\displaystyle c_{\mathrm{a}}^{\mathrm{I}}(\mathrm{\infty })}$ and ${\displaystyle c_{\mathrm{a}}^{\mathrm{I}\mathrm{I}}(\mathrm{\infty })}$, can differ by several orders of magnitude, even for modest values of $\mathrm{\Gamma }$ corresponding to weak relative interactions.

To elucidate the impact of the reaction orders on the aggregation kinetics, we first consider the enrichment of aggregates relative to the case of homogeneous aggregation, that is for $\mathrm{\Gamma }=1$. For large values of monomer partitioning, the asymptotic concentrations in compartments I and II at large times relative to the homogeneous aggregate concentration ${\displaystyle c_{\mathrm{a}}^{\mathrm{t}\mathrm{o}\mathrm{t}}{|}_{{\varphi }^{\mathrm{I}}={\varphi }^{\mathrm{I}\mathrm{I}}}}$ at large times read

where $w$ is a dimensionless numerical prefactor (Appendix 3, Equation 52). We see that for a large monomer partitioning factor $\mathrm{\Gamma }$, the partitioning of aggregates inside compartment I gets more pronounced, while aggregates in compartment II are more depleted relative to the homogeneous case (Figure 3(c)). Most importantly, the value of the terminal values of aggregate concentrations for given monomer partitioning factor are controlled by the reaction orders for primary and secondary nucleation, $n_1$ and $n_2$. The role of $n_1$ and $n_2$ results directly from the interplay between aggregate growth and nucleation and their dependence on the monomer concentration.

Having understood the role of the monomer partitioning factor $\mathrm{\Gamma }$ in aggregation kinetics, we now turn to how the asymptotic concentrations of aggregates in each compartment depend on the volume of the compartments. The dependence on compartment volume is given the partition degree $\xi (\overline{\varphi })$. From Equation 2, we see that the partition degree $\xi (\overline{\varphi })\in [1,{\mathrm{\Gamma }}^{-1}]$, where the value of one is relevant for small compartments (Figure 2(b)). Following Equations (4) and (5), we see that for a small volume of compartment I, enrichment and depletion exhibit an inverse scaling, i.e. ${\displaystyle c_{\mathrm{a}}^{\mathrm{I}}(\mathrm{\infty })\propto {\left(c_{\mathrm{a}}^{\mathrm{I}\mathrm{I}}(\mathrm{\infty })\right)}^{-1}\propto {\mathrm{\Gamma }}^{\frac{n_2+1}{2}}}$, which is solely dependent on the reaction order for secondary nucleation. Contrariwise, when the volume of compartment I is large, enrichment of aggregates inside I vanishes, while depletion inside compartment II then solely depends on the reaction order for primary nucleation, ${\displaystyle c_{\mathrm{a}}^{\mathrm{I}\mathrm{I}}(\mathrm{\infty })\propto {\mathrm{\Gamma }}^{-n_1}}$.

This switch between aggregate partitioning governed by secondary nucleation, to a partitioning solely determined by primary nucleation, arises from primary nucleation events occurring first inside compartment I due to a higher monomer concentration (${\displaystyle \mathrm{\Gamma }>1}$). Once the first aggregates have formed via primary nucleation inside compartment I, small and large compartments behave fundamentally differently. If compartment I is small, only a few aggregates can form via primary nucleation due to the small compartment size. As aggregates begin to grow earlier in compartment I, the unbalance of monomers causes a flux from II to I. As a consequence of this continuous flux, the secondary nucleation events quickly overwhelm primary nucleation events inside compartment I, while secondary nucleation is suppressed in compartment II. However, if compartment I is large, the aggregation kinetics is similar to that for a homogeneous system because the monomer mass concentration is very close to the total monomer mass in the system and there is only a negligible amount of monomers entering from compartment II. Additionally, in the smaller compartment II where aggregates grow via primary nucleation, the coupling flux continuously removes monomers suppressing primary nucleation. Since compartment I is large, it shows little or no enrichment of aggregates relative to the homogeneous case while inside the small compartment II, aggregates are depleted determined by the lack of primary nucleation events relative to the homogeneous case.

To quantify the switch in aggregate partitioning as a function of compartment volume, we define the asymptotic aggregate partitioning ratio

As the compartment volume enters the partitioning factor $\epsilon (\overline{\varphi })$ solely via the partition degree $\xi (\overline{\varphi })$, the sign of $n_2-n_1+1$ determines whether larger or smaller compartments lead to a larger partitioning (Figure 3(e)). Indeed, we find that the slope of the partitioning factor scales as $\epsilon {(\overline{\varphi })}^{\prime }\propto (n_1-n_2-1)$. Thus, for ${\displaystyle n_1>n_2+1}$, increasing the compartment volume by increasing the amount of $A$-material $\overline{\varphi }$ causes a larger relative partitioning. Conversely, for ${\displaystyle n_1<n_2+1}$, larger partitioning can be found for smaller compartment sizes. Consistently, if the nucleation coefficients obey $n_2=n_1-1$, compartment volume has no impact on the partitioning factor $\epsilon$.

This qualitative switch in the mechanism for aggregate partitioning raises the question which systems favor large or small compartment volumes in order to maximize aggregate partitioning $\epsilon (\overline{\varphi })$. Figure 4 depicts the regimes in terms of the reaction orders characterizing primary and secondary nucleation, $n_1$ and $n_2$, for which the maximal aggregate partitioning corresponds to smaller and larger compartment volumes. This prediction can be related to specific aggregating systems for which the values of the reaction orders $n_1$ and $n_2$ have been experimentally determined (References see caption of Figure 4). Using these values for the reaction orders, our model predicts that largest partitioning is obtained for large compartments in systems of aggregating tau and yeast prion Ure2p. These two examples belong primarily to the class of systems where the mechanism responsible for the formation of new aggregates in the late stage is fragmentation which has a zero secondary reaction order, $n_2=0$ (i.e. nucleation is monomer independent). For non-fragmenting systems with ${\displaystyle n_2>0}$, our model predicts different scenarios for aggregating systems: largest aggregate partitioning for large compartment volumes occurs in the case branching systems, such as actin in the presence of the complex Arp2/3, as well as systems proliferating through monomer dependent secondary nucleation with ${\displaystyle n_2<n_1-1}$, such as the Islet Amyloid Polypeptide (IAPP). In contrast, largest aggregate partitioning is reached for small compartments in the case of the 40- and 42-residue forms of Amyloid-β peptide (Aβ40 and Aβ42).

Figure 4

Download asset Open asset

Our results have demonstrated that aggregates can be effectively partitioned inside liquid-like compartments, raising the question: can compartments also control the total amount of aggregates or their average size? To test this possibility, we compute the difference between the total amount of aggregates formed in the presence of liquid compartments, ${\displaystyle c_{\mathrm{a}}^{\mathrm{t}\mathrm{o}\mathrm{t}}=\left(c_{\mathrm{a}}^{\mathrm{I}}(\mathrm{\infty })V^{\mathrm{I}}+c_{\mathrm{a}}^{\mathrm{I}\mathrm{I}}(\mathrm{\infty })V^{\mathrm{I}\mathrm{I}}\right)/V}$, compared to the number of aggregates formed in the homogeneous system without compartments, ${\displaystyle c_{\mathrm{a}}^{\mathrm{t}\mathrm{o}\mathrm{t}}{|}_{{\varphi }^{\mathrm{I}}={\varphi }^{\mathrm{I}\mathrm{I}}}}$. The homogeneous case can be studied by considering equal compositions of both compartments, that is ${\displaystyle {\varphi }^{\mathrm{I}}={\varphi }^{\mathrm{I}\mathrm{I}}}$. This difference between the homogeneous case and the case with compartments be quantified by introducing the relative asymptotic aggregate concentration, ${\displaystyle \mathcal{𝒞}(\overline{\varphi },\mathrm{\Gamma })=(c_{\mathrm{a}}^{\mathrm{t}\mathrm{o}\mathrm{t}}-c_{\mathrm{a}}^{\mathrm{t}\mathrm{o}\mathrm{t}}{|}_{{\varphi }^{\mathrm{I}}={\varphi }^{\mathrm{I}\mathrm{I}}})/c_{\mathrm{a}}^{\mathrm{t}\mathrm{o}\mathrm{t}}{|}_{{\varphi }^{\mathrm{I}}={\varphi }^{\mathrm{I}\mathrm{I}}}}$, which is positive for an increased pool of aggregates, and negative for a lowered pool of aggregates relative to the homogeneous state. We find that compartments can affect the total number of aggregates relative to the homogeneous system depending on the relative values of the reaction orders for secondary nucleation and aggregate growth, the value of monomer partitioning $\mathrm{\Gamma }$ and the amount of compartment material $\overline{\varphi }$ that in turn regulates compartment size $V^{\text{I}}$. In particular, for reaction orders ${\displaystyle n_2<1}$, the presence of the liquid droplet always reduces the total amount of aggregates formed relative to the homogeneous system for all values of $\overline{\varphi }$ and $\mathrm{\Gamma }$ and thereby always leading to larger aggregates (Figure 5(a)). However, for ${\displaystyle n_2>1}$, we find a different behaviour. For low partitioning factors $\mathrm{\Gamma }$, the presence of liquid compartments decreases the total number of aggregates, corresponding to a larger average aggregate size, while for larger values of $\mathrm{\Gamma }$, more and thereby shorter aggregates form compared to the homogeneous system (Figure 5(b)). This behavior is also affected by compartment volume; the corresponding boundary in the $\overline{\varphi }$-$\mathrm{\Gamma }$ diagram separates these two regimes corresponding to more but smaller or less but larger aggregates (Figure 5(c)). The role of the reaction order for secondary nucleation $n_2$ on total aggregate concentration and average size can be explained as follows. In a homogeneous system proliferating through secondary nucleation pathways, the average aggregate size in the saturating regime of the aggregation kinetics at long times scales as $\sqrt{k_{+}/k_2}{[M_{\mathrm{m}}^{\mathrm{tot}}]}^{(1-n_2)/2}$ (Michaels et al., 2015). Larger values of $\mathrm{\Gamma }$ lead to an increase of monomers in the compartment, favouring both secondary nucleation and aggregate growth by elongation inside the compartment. If ${\displaystyle n_2>1}$, the rate of secondary nucleation is increased by $\mathrm{\Gamma }$ more than elongation, which results in more numerous aggregates and hence shorter aggregates due to the limiting and fixed amount of total monomer mass in the system. The opposite trend is observed when $n_2\le 1$. In summary, a strong partitioning of aggregates inside compartments caused by a strong monomer partitioning (large $\mathrm{\Gamma }$) is accompanied by an increase of the total number of aggregates in the system in the presence of secondary nucleation, while in the absence of secondary nucleation, the total amount of aggregates decreases.

Figure 5

Download asset Open asset

In our model, we have considered the case that monomers and aggregates do not affect phase separation and phase separation is driven by the competition between the entropic tendency to mix and interactions favoring demixing. Future work could be devoted to extending our model by a coupling between aggregates and the liquid compartment or by entropically driven phase separation, relevant for the assembly of coacervates (Overbeek and Voorn, 1957) or mixtures with depletion interactions. Moreover, our model is restricted to time scales when aggregates hardly diffuse and monomer diffusion is not affected by rheological properties of the aggregates. The observed strong aggregate partitioning may diminish if aggregates significantly diffuse, or if they slow down diffusion of monomers. Furthermore, we have focussed on the case where both phases have the same reaction rates of aggregation. This assumption may be inaccurate for protein-rich phases (Wei et al., 2017) but can be scrutinized using our model (see Appendix 3). We find that due to the power law dependence of the monomer partitioning $\mathrm{\Gamma }$ (Equation 6) differences in aggregation rates must be very large to significantly affect the partitioning of aggregates. A lowered partitioning of aggregates could be caused by the coarsening dynamics of many droplets (Ostwald, 1897; Lifshitz and Slyozov, 1961; Bray, 1994). While coarsening via coalescence would not affect our results at all because aggregates remain confined inside the droplets, dissolving droplets undergoing Ostwald ripening would diminish the degree of aggregate partitioning. However, because the aggregation kinetics varies with compartment size, aggregates in droplets of different size may compete about monomers. This non-equilibrium competition could cause accumulation of more aggregates either in smaller or larger compartments. Overall, for systems where partitioning of monomers is fast relative to the aggregation kinetics, the mechanism underlying the strong partitioning of aggregates proposed in this study could be relevant for several phenomena in living cells. It could have impact on strategies of drug design or serve as a principle to speed up irreversible chemical reactions and can be tested experimentally.

Our quantitative predictions of strong aggregate enrichment inside a liquid-like compartment (Figure 3 (b–e) and Figure 4) are experimentally testable using recently developed bulk and microfluidic assays. For example, synthetic liquid biocompartments of tuneable size and composition can be used to locally affect reaction rates and partition proteins (Faltova et al., 2018) and thereby represent attractive platforms to investigate the partitioning and aggregation of different amyloidogenic peptides and proteins, including Amyloid-β. These synthetic compartments are highly flexible and allow to validate the effect of several parameters predicted in this work. For instance, the monomer partitioning factor $\mathrm{\Gamma }$ could be varied in vitro by changing the degree of phase separation (Equation (1) and Figure 2) or by conjugating the proteins with specific sequences capable of tuning recruitment into the liquid compartments (Faltova et al., 2018). Moreover, the compartment volumes can be adjusted by the initial supersaturation via changes in temperatures, which affect the kinetic rate constants only weakly (Cohen et al., 2018). Measuring the concentration or size of aggregates inside and outside of the compartment by epi-fluorescence spectroscopy as a function of time and parameters such as the partitioning factor and compartment volume will allow for tracking aggregate enrichment as a function of compartment volume and test both the scaling predictions and the crossover of the scaling exponent from $n_2+1$ at small volume to $n_1$ at large volumes. It would be particularly interesting to test this prediction for different amyloid-forming protein systems or for varying the reaction orders ${\displaystyle n_1}$ and ${\displaystyle n_2}$ by adjusting the total amount of monomers (Meisl et al., 2014); see Figure 4.

Our model may already provide a framework to explain the phenomena of aggregate partitioning inside living cells. An example of such phenomena could be the partitioning of pericentriolar material into centrosomes (Zwicker et al., 2014) and the spatial organization of aggregates inside stress granules (Molliex et al., 2015; Mateju et al., 2017). The propensity of aggregates to solidify the compartment as reported in Mateju et al. (2017) could be accounted for in our model through a gel-sol transition (Stockmayer, 1943; Harmon et al., 2017). Including the solidification induced by aggregates could lead to additional volume changes of the compartment which in turn may affect the aggregation kinetics. Furthermore, the enrichment of toxic aggregates inside liquid compartments may trigger new directions for drug design against aberrant protein aggregation. Our results suggest to design drugs not only with respect to their ability to interfere with the aggregation kinetics (Arosio et al., 2014) but also with respect to their partitioning properties into the liquid compartments. This strategy is reminiscent of quantifying the potency of low-molecular weighted anesthetics via the Meyer-Overton correlation based on solubility of the anaesthetics in oil (Meyer, 1937; Franks and Lieb, 1978).

The reported feedback mechanism of aggregate growth mediated by liquid compartments may represent a general principle to spatially confine and speed up other irreversible chemical processes or to control aggregate amount and average size. Examples may include precipitation of proteins or polymerization kinetics of actin and microtubules (see also Figure 4). Indeed, a speed up of the chemical reactions could be expected due to the increased concentration of educts inside the liquid compartments. Thus, liquid compartments are ideal biomolecular microreactors that enrich the amount of products by dynamically exchanging reactants with their surroundings.

We consider phase separation of an incompressible, ternary mixture composed of monomers, component $A$ and $B$ (we neglect the interactions between the aggregates and phase separation for simplicity) described by the following Flory-Huggins free energy density (Flory, 1942; Huggins, 1942)

where the logarithmic terms correspond to the mixing entropy. The interactions between $A$ and $B$ are described by the parameter $\mathrm{\Lambda }$ while the interactions with the monomers are characterized by ${\mathrm{\Lambda }}_{mi}$, $i=A,B$. These interaction parameters have the unit $[1/\text{volume}]$. Here we define a dimensionless interaction parameters by writing $\mathrm{\Lambda }=\chi /\nu$ and ${\mathrm{\Lambda }}_{mi}={\chi }_{mi}/\nu$ with $\nu ={\nu }_A$. Moreover, for simplicity, we consider equal molecular volumes of $A$ and $B$, $\nu ={\nu }_B$. Thus we arrive at

With ${\varphi }_B=1-{\varphi }_A-{\nu }_{\mathrm{m}}{c}_{\mathrm{m}}$ and monomers being dilute, ${\nu }_{\mathrm{m}}{c}_{\mathrm{m}}\ll 1$, we can expand $f$ in ${\nu }_{\mathrm{m}}{c}_{\mathrm{m}}$ up to the first order:

where we defined ${\varphi }_A\equiv \varphi$ and neglected all terms of order $𝒪\left[{\left({\nu }_{\mathrm{m}}{c}_{\mathrm{m}}\right)}^2\right]$. The corresponding chemical potentials read

At equilibrium the chemical potentials of each component inside (I) and outside (II) the compartment are balanced leading to relationships between the concentration values inside and outside. Specifically, if phase separation equilibrium is reached the following conditions are fulfilled

The relations above allow to calculate the equilibrium concentration in each phase, for component $A$, ${\displaystyle {\varphi }^{\mathrm{I}}}$ and ${\displaystyle {\varphi }^{\mathrm{I}\mathrm{I}}}$, and the monomers, ${\displaystyle c_{\mathrm{m}}^{\mathrm{I}}}$ and ${\displaystyle c_{\mathrm{m}}^{\mathrm{I}\mathrm{I}}}$. An analytic result of the equilibrium concentrations is very difficult to obtain. However, we can focus on the leading contributions for the balance of the chemical potentials inside and outside taking advantage that monomers are dilute and thereby obtain an approximation for the equilibrium values inside and outside.

Considering that the dimensionless interaction parameters $\chi$ are all of $𝒪(1)$, the impact of the dilute monomers on the phase equilibrium between $A$ and $B$ is negligible and we can approximate

The resulting chemical potential Equation (12) simply corresponds to the chemical potential of a binary, incompressible Flory-Huggins mixture. From the equilibrium condition Equation (11a), we can calculate the binodal line described by the condition

which solely depends on the interaction parameters between $A$ and $B$, $\chi$. For fixed interaction parameters $\chi$, the binodal gives the equilibrium volume fractions inside and outside the compartment, ${\displaystyle {\varphi }^{\mathrm{I}}}$ and ${\displaystyle {\varphi }^{\mathrm{I}\mathrm{I}}}$. For finite sized compartments, these equilibrium volume fractions are slightly increased due to the Laplace pressure. This increase is described by the Gibbs-Thomson relationships, ${\varphi }_{\text{eq}}^{\alpha }={\varphi }^{\alpha }(1+{\mathrm{\ell }}_{\gamma }^{\alpha }/R)$, where ${\mathrm{\ell }}_{\gamma }^{\alpha }$ denotes the capillary length for each phase. For strong phase separation of compartment I, ${\displaystyle {\varphi }^{\mathrm{I}}\gg {\varphi }^{\mathrm{I}\mathrm{I}}}$, the capillary lengths obey ${\displaystyle {\ell }_{\gamma }^{\mathrm{I}}\ll {\ell }_{\gamma }^{\mathrm{I}\mathrm{I}}}$ (Weber et al., 2018). Moreover, the capillary length in the minority phase II can be estimated by ${\displaystyle {\ell }_{\gamma }^{\mathrm{I}\mathrm{I}}\simeq 2\gamma \nu /({\varphi }^{\mathrm{I}}{k}_{\text{B}}T)}$, which is typically closer to the molecular size than the size of micron-sized compartments of radius $R$. Thus, for compartments significantly exceeding the size of the molecules we can neglect Laplace and surface tension effects on the equilibrium volume fraction and approximate ${\varphi }_{\text{eq}}^{\alpha }\simeq {\varphi }^{\alpha }$ in the following.

By means of equilibrium condition Equation (11b), we can calculate the impact of the phase separated compartment on the monomer distribution leading to the monomer partitioning factor

where the relative interaction strength reads $\mathrm{\Delta }\chi =\left({\chi }_{B,\text{m}}-{\chi }_{A,\text{m}}\right)$. Thus, there is partitioning of monomers into the condensed phase (${\displaystyle \mathrm{\Gamma }>1}$) if monomers favor the presence of $A$ relative to $B$, that is ${\displaystyle {\chi }_{A,\mathrm{m}}<{\chi }_{B,\mathrm{m}}}$.

This flux can be calculated for a maintained concentration difference ${\displaystyle {\varphi }^{\mathrm{I}}-{\varphi }^{\mathrm{I}\mathrm{I}}}$ using the chemical potential for the monomers ${\mu }_{\mathrm{m}}$ (Equation 10b). To this end, let us consider that the compartment is spherical of radius $R$. Perturbing the concentrations in both phases may lead to an unbalance of the chemical potential and thus a flux between the phases. The net radial flux of monomers ${\displaystyle J=J_{\text{r}}^{\mathrm{I}}-J_{\text{r}}^{\mathrm{I}\mathrm{I}}}$ [unit: mass per time] arises from a flux contribution of each phase through the interface, where the radial flux in each phase reads

and ${\mathit{𝒆}}_r$ is the radial unit vector pointing normal to the spherical interface. The local fluxes at the interface, ${{\mathit{𝒋}}^{\alpha }|}_R$, can be calculated in the limit where the aggregation kinetics is slow compared to diffusion in each phase. In this limit we can use a quasi static approximation, $\nabla \cdot \left({\xi }^{\alpha }\nabla {\mu }_{\text{m}}^{\alpha }(r)\right)\simeq 0$. For a constant mobility coefficient in each phase we can solve a Laplace equation for the chemical potential, ${\nabla }^2{\mu }_{\text{m}}^{\alpha }(r)\simeq 0$, with the following boundary conditions for the chemical potentials: ${\mu }_{\text{m}}^{\alpha }$ deeply inside each phase and ${{\mu }_{\text{m}}^{\alpha }|}_R$ at the interface $r=R$. For simplicity, we discuss the case of a small compartment I with a homogeneous chemical potential inside, thus ${\displaystyle {\mu }_{\mathrm{m}}^{\mathrm{I}}(r)={\mu }_{\mathrm{m}}^{\mathrm{I}}}$ is constant and equal to the chemical potential right inside the interface ${\displaystyle {\mu }_{\mathrm{m}}^{\mathrm{I}}{|}_R}$. For compartment II, we then find by solving the Laplace equation ${\displaystyle {\mu }_{\mathrm{m}}^{\mathrm{I}\mathrm{I}}(r)={\mu }_{\mathrm{m}}^{\mathrm{I}\mathrm{I}}+\left({\mu }_{\mathrm{m}}^{\mathrm{I}\mathrm{I}}{|}_R-{\mu }_{\mathrm{m}}^{\mathrm{I}\mathrm{I}}\right)R/r}$ where ${\displaystyle {\mu }_{\text{m}}^{\text{II}}}$ is the chemical potential far from the interface. The flux at the interface can be calculated by ${\displaystyle {\mathit{j}}^{\mathrm{I}\mathrm{I}}{|}_R=-{\mathit{e}}_r{m}_{\mathrm{m}}{\xi }^{\mathrm{I}\mathrm{I}}{\mathrm{\partial }}_r{\mu }_{\mathrm{m}}^{\mathrm{I}\mathrm{I}}(r){|}_{r=R}={\mathit{e}}_r{m}_{\mathrm{m}}{\xi }^{\mathrm{I}\mathrm{I}}\left({\mu }_{\mathrm{m}}^{\mathrm{I}\mathrm{I}}{|}_R-{\mu }_{\mathrm{m}}^{\mathrm{I}\mathrm{I}}\right)/R}$. The absence of aggregation at the interface allows that diffusion can equilibrate the chemical potentials, that is ${\displaystyle {\mu }_{\mathrm{m}}^{\mathrm{I}\mathrm{I}}{|}_R={\mu }_{\mathrm{m}}^{\mathrm{I}}{|}_R}$ which is also equal to the chemical potential deeply inside the phase ${\displaystyle {\mu }_{\mathrm{m}}^{\mathrm{I}}}$ because phase ${\displaystyle \text{I}}$ is considered to be homogeneous. The net flux of monomers, ${\displaystyle J=J_{\text{r}}^{\mathrm{I}}-J_{\text{r}}^{\mathrm{I}\mathrm{I}}}$ with ${\displaystyle J_{\text{r}}^{\text{I}}=0}$, is

Using the chemical potential of the monomers, Equation (10b), we can approximate the chemical potential difference as

where we expanded ${\displaystyle M_{\mathrm{m}}^{\mathrm{I}}=M_{\mathrm{m},\mathrm{e}\mathrm{q}}^{\mathrm{I}}+\delta M_{\mathrm{m}}^{\mathrm{I}}}$ and ${\displaystyle M_{\mathrm{m}}^{\mathrm{I}\mathrm{I}}=M_{\mathrm{m},\mathrm{e}\mathrm{q}}^{\mathrm{I}\mathrm{I}}+\delta M_{\mathrm{m}}^{\mathrm{I}\mathrm{I}}}$ up to linear order around the equilibrium concentrations ${\displaystyle M_{\mathrm{m},\mathrm{e}\mathrm{q}}^{\mathrm{I}}}$ and ${\displaystyle M_{\mathrm{m},\mathrm{e}\mathrm{q}}^{\mathrm{I}\mathrm{I}}=M_{\mathrm{m},\mathrm{e}\mathrm{q}}^{\mathrm{I}}/\mathrm{\Gamma }}$, respectively. Thus net change of monomers due to the exchange of material through the interface reads

where the diffusion constant in phase II is ${\displaystyle D_{\mathrm{m}}=k_{\mathrm{B}}T{m}_{\mathrm{m}}{\xi }^{\mathrm{I}\mathrm{I}}/M_{\mathrm{m},\mathrm{e}\mathrm{q}}^{\mathrm{I}}}$ and the rate to relax back to monomer partitioning at equilibrium is $k(R)=4\pi R{D}_{\mathrm{m}}$. To ease notation we omitted the “$\delta$” to indicate linear deviations from equilibrium.

In this appendix, we discuss the detail associated with the derivation of analytical solutions to the aggregation kinetics in the presence of liquid compartments, Equation(3), main text.

Initial layer dynamics

Due to the separation of timescales between monomer equilibration between the two compartments and protein aggregation, the system described by Equation (3) (main text) will develop initially through a rapid phase of equilibration, before any aggregation occurs in either compartment. During this initial layer phase, which is the temporal equivalent of a boundary layer (Astarita and Marucci, 1974; Danckwerts, 1951), the initial values of the monomer concentration in each compartment, ${\displaystyle M_{\mathrm{m}}^{\mathrm{I}}(0)}$ and ${\displaystyle M_{\mathrm{m}}^{\mathrm{I}\mathrm{I}}(0)}$, relax quickly to equilibrium such that the condition ${\displaystyle M_{\mathrm{m}}^{\mathrm{I}}(t)=\mathrm{\Gamma }{M}_{\mathrm{m}}^{\mathrm{I}\mathrm{I}}(t)}$ is satisfied before aggregation is initiated. This early equilibration kinetics is described by setting the aggregation terms in Equation (3) (main text) to zero, yielding the following equations:

(21a) ${\displaystyle {\displaystyle \frac{\text{d}{M}_{\mathrm{a}}^{\mathrm{I}}(t)}{\mathrm{d}\mathrm{t}}{\displaystyle =-k^{\mathrm{I}}[M_{\mathrm{m}}^{\mathrm{I}}(t)-\mathrm{\Gamma }{M}_{\mathrm{m}}^{\mathrm{I}\mathrm{I}}(t)],}}}$

(21b) ${\displaystyle {\displaystyle \frac{\mathrm{d}{\mathrm{M}}_{\mathrm{a}}^{\mathrm{I}\mathrm{I}}(\mathrm{t})}{\mathrm{d}\mathrm{t}}{\displaystyle =k^{\mathrm{I}\mathrm{I}}[M_{\mathrm{m}}^{\mathrm{I}}(t)-\mathrm{\Gamma }{M}_{\mathrm{m}}^{\mathrm{I}\mathrm{I}}(t)].}}}$

The solution to Equation (21) is:

(22a) ${\displaystyle {\displaystyle M_{\mathrm{m}}^{\mathrm{I}}{\displaystyle =\mathrm{\Gamma }B+\frac{k^{\mathrm{I}}[M_{\mathrm{m}}^{\mathrm{I}}(0)-\mathrm{\Gamma }{M}_{\mathrm{m}}^{\mathrm{I}\mathrm{I}}(0)]}{A}{e}^{-At},}}}$

(22b) ${\displaystyle {\displaystyle M_{\mathrm{m}}^{\mathrm{I}\mathrm{I}}{\displaystyle =B+\frac{k^{\mathrm{I}\mathrm{I}}[M_{\mathrm{m}}^{\mathrm{I}}(0)-\mathrm{\Gamma }{M}_{\mathrm{m}}^{\mathrm{I}\mathrm{I}}(0)]}{A}{e}^{-At},}}}$

where ${\displaystyle A=k^{\mathrm{I}}+\mathrm{\Gamma }{k}^{\mathrm{I}\mathrm{I}}}$, and ${\displaystyle B=[k^{\mathrm{I}\mathrm{I}}{M}_{\mathrm{m}}^{\mathrm{I}}(0)+k^{\mathrm{I}}{M}_{\mathrm{m}}^{\mathrm{I}\mathrm{I}}(0)]/A}$. Note that the kinetics described by Equation (22) ‘pushes’ the system towards the slow manifold, which is described by ${\displaystyle M_{\mathrm{m}}^{\mathrm{I}}(t)=\mathrm{\Gamma }{M}_{\mathrm{m}}^{\mathrm{I}\mathrm{I}}(t)}$. Hence, when ${\displaystyle M_{\mathrm{m}}^{\mathrm{I}}(0)=\mathrm{\Gamma }{M}_{\mathrm{m}}^{\mathrm{I}\mathrm{I}}(0)}$, there is no initial phase of ‘correction’ of the initial conditions. At the end of this initial boundary layer phase, the monomer concentrations in the two compartments are given by:

(23a) ${\displaystyle {\displaystyle M_{\mathrm{m}}^{\mathrm{I}}{\displaystyle =\mathrm{\Gamma }B,}}}$

(23b) ${\displaystyle {\displaystyle M_{\mathrm{m}}^{\mathrm{I}\mathrm{I}}{\displaystyle =B.}}}$

Since we are not very much interested in this initial phase of redistribution of the initial conditions, in the following we shall assume for simplicity that the initial monomer concentrations in compartments I and II satisfy the relationship ${\displaystyle M_{\mathrm{m}}^{\mathrm{I}}(0)=\mathrm{\Gamma }{M}_{\mathrm{m}}^{\mathrm{I}\mathrm{I}}(0)}$. This assumption does not affect the generality of our results. In fact, if this condition was not satisfied initially, then, according to Equation (22), rapid equilibration between the two compartments would correct these initial conditions, eventually leading to a ’corrected’ set of initial conditions that lie in the slow manifold.

Appendix 3—figure 1

Download asset Open asset

After an initial, rapid phase of monomer redistribution through the two compartments, the system enters a slower phase of dynamics, where, at leading order, the system stays on the slow manifold ${\displaystyle M_{\mathrm{m}}^{\mathrm{I}}=\mathrm{\Gamma }{M}_{\mathrm{m}}^{\mathrm{I}\mathrm{I}}}$ at all times. For simplicity, let us assume that the initial concentrations of monomers in the two compartments obey the relationship ${\displaystyle M_{\mathrm{m}}^{\mathrm{I}}(0)=\mathrm{\Gamma }{M}_{\mathrm{m}}^{\mathrm{I}\mathrm{I}}(0)}$ (otherwise there will be a fast equilibration of the initial conditions such that this relationship is satisfied). To describe the aggregation process in the slow manifold, we write ${\displaystyle M_{\mathrm{m}}^{\mathrm{I}}-\mathrm{\Gamma }{M}_{\mathrm{m}}^{\mathrm{I}\mathrm{I}}\simeq 0}$ for all times in Equation (3) (main text) and find:

It is useful to introduce the total monomer concentration in the system as:

Note that this concentration may vary in time as aggregates are nucleated and grow, however, the total mass concentration

is conserved at all times. Using the condition ${\displaystyle M_{\mathrm{m}}^{\mathrm{I}}=\mathrm{\Gamma }{M}_{\mathrm{m}}^{\mathrm{I}\mathrm{I}}}$, we can write the following relationships linking the concentrations of monomers in compartments I and II to the total concentration of monomers in the system:

where we abbreviated the partitioning degree

Thus, using Equation (27), we can re-write Equation (24) as: