Enhanced potency of aggregation inhibitors mediated by liquid condensates

Liquid condensates are membraneless organelles that form via phase separation in living cells. These condensates provide unique heterogeneous environments that have much potential in regulating a range of bio-chemical processes from gene expression to ﬁlamentous protein aggregation—a process linked to Alzheimer’s and Parkinson’s diseases. Here we theoretically study the physical interplay between protein aggregation, its inhibition, and liquid-liquid phase separation. Our key ﬁnding is that the action of protein aggregation inhibitors can be strongly enhanced by liquid condensates. The physical mechanism of this enhancement relies on the partitioning and colocalization of inhibitors with their targets inside the liquid condensate. Our theory uncovers how the physicochemical properties of condensates can be used to modulate inhibitor potency, and we provide experimentally testable conditions under which drug potency is maximal. Our ﬁndings suggest design principles for protein aggregation inhibitors with respect to their phase-separation properties. DOI


I. INTRODUCTION
The proteostasis network in living cells uses chaperones as key players for protein quality control [1,2]. The main role of chaperones is to stabilize the native fold of proteins, but they are also involved in degradation processes such as lysosome-mediated degradation of dysfunctional proteins [3,4]. A decline of the proteostasis network, due to ageing, for example [2], causes the accumulation of misfolded proteins and their subsequent aggregation into filamentous aggregates and amyloid fibrils-a process linked to over 50 devastating disorders, including Alzheimer's and Parkinson's diseases [5][6][7][8][9]. It is therefore important to obtain insights into the mechanisms underlying filamentous protein aggregation.
A new class of organelles that has recently received significant attention are protein condensates. They form via phase separation from the cyto-or nucleoplasm and share most physical properties with liquid-like, condensed droplets [32][33][34][35]. Liquid condensates provide physicochemical environments distinct from the surrounding cytoplasm and nucleoplasm, allowing a range of biochemical reactions associated with biological function or dysfunction to be spatially regulated. Recent experimental [35][36][37] and theoretical [38] studies show that condensates can affect amyloid aggregation. For example, stress granules concentrate monomers causing aggregation to occur only inside the condensates [39,40]. Condensates can accumulate chaperones capable of either suppressing aberrant phase transition of condensates [41] or driving amyloid formation by sequestering misfolded proteins [42][43][44]. Chaperones are also involved in the control of condensate stability [45] and the surveillance of stress granule quality [46,47].
The relevance of liquid condensates for the spatial organization of chaperones and filamentous aggregates suggests that spatial compartmentalization may play an important role in regulating and suppressing protein aggregation. Similarly, the action of synthetic aggregation inhibitors may be affected by the presence of liquid condensates. Abbreviating FIG. 1. How liquid-like condensates impact protein aggregation inhibition. (a) We determine the physical conditions when the presence of a phase-separated condensate (phase I, right) yields a higher degree of inhibition of aggregation (e.g., fewer aggregates at the end of the reaction) compared with a homogeneous system (left). (b) Definition of partitioning of monomers and drug components and associated nonequilibrium diffusive fluxes across the condensate interface. We neglect diffusive exchange of aggregates through the interface. (c) The reaction network of filamentous protein aggregation and three established mechanisms of inhibition [10,[28][29][30].
both chaperones and synthetic inhibitors as "drugs," we raise the following question: How do liquid condensates affect drug-mediated inhibition of protein aggregation [ Fig. 1(a)]? Here, we address this question by presenting a physical model of protein aggregation inhibition coupled to liquidliquid phase separation. We find that liquid condensates can significantly amplify drug-mediated inhibition of protein aggregation. The mechanism for this amplification builds on the colocalization inside the condensate of the inhibiting drug and its target [ Fig. 1(a)]. The key control parameters are the volume of the condensate and the partitioning coefficients of drug and monomers. Our results reveal a simple physical mechanism that may allow liquid condensates to control protein aggregation in cells and are likely to inspire new drug discovery strategies against protein aggregation disorders that could improve drug efficacy.

II. AGGREGATION KINETICS WITH INHIBITORS AND LIQUID CONDENSATES
We consider a system of volume V containing a phaseseparated liquid-like condensate of volume V I (phase I) coexisting with the surrounding, dilute phase (phase II). Phase I is rich in some protein A, while phase II is dilute in A but is rich in another protein, lipids, or water, shortly denoted as the B component [ Fig. 1(a)]. In addition to the phase-separating components A and B the system contains a concentration M tot m of monomeric protein and a concentration c d of an inhibitor of aggregation. We assume that all aggregating species and the drug are sufficiently dilute with respect to A and B such that their impact on AB-phase separation can be neglected (see Supplemental Material [48] Sec. 1.3.1). The coexisting phases I and II thus create distinct biochemical environments for aggregation and can give rise to diffusive fluxes across the interface that partition aggregating monomers and drug components, which we denote with i (i.e., i denotes monomers or inhibitors). By equating chemical potentials of component i between the two phases, we can calculate the equilibrium partitioning coefficient of species i as p i = c I i | eq /c II i | eq . Here, c I i and c II i are the concentrations of component i in phases I and II, respectively. p i is determined by the degree of phase separation and the relative strength of interaction between species i and A or B (see Supplemental Material [48] Sec. 1.3.1 and Ref. [38]). Favoring interactions between i and A, for instance, leads to p i > 1, i.e., enrichment of i in the condensate. Deviations from partitioning equilibrium lead to interphase fluxes of species i that reestablish equilibrium [ Fig. 1(b)]. For small deviations from equilibrium, these fluxes depend linearly on the concentrations in phases I and II (see Supplemental Material [48] Sec. S1.3.2 and Ref. [38]) where p i is the equilibrium partitioning coefficient and k diff,i = 4π RD i is the relaxation rate toward equilibrium, with R being the radius of the condensate and D being the diffusion coefficient of species i. Note that these fluxes vanish at partitioning equilibrium, i.e., when c I i /c II i = p i . In the presence of a liquid condensate, aggregation occurs in both phase I and phase II involving a number of different microscopic events [49][50][51][52][53][54][55][56], which are illustrated in Fig. 1(c). Aggregation is initiated by primary nucleation, i.e., the formation of the smallest aggregates directly from the monomers. Aggregates then grow by elongation. In addition, aggregation is often accelerated by secondary nucleation mechanisms [57], including fragmentation, lateral branching, or surfacecatalyzed secondary nucleation [ Fig. 1(c)]. These secondary nucleation processes are in general characterized by a dependence on the current population of aggregates and introduce therefore an autocatalytic cycle in the system [ Fig. 1 We focus here on suppression of aggregation by the inhibitor through three established mechanisms [10,[28][29][30], including inhibitor binding to (i) free monomers, (ii) aggregate growth ends, and (iii) aggregate surface sites, as depicted in Fig. 1(c) (see Supplemental Material [48] Fig. S1 for examples of inhibitors). For simplicity, we neglect drug binding to other species, including intermediate oligomers, but note that our framework can in principle be generalized to account for these inhibition mechanisms.
To couple the aggregation kinetics with liquid-liquid phase separation, we consider an approximate case where the partitioning of monomers and inhibitors is close to equilibrium at all times. This approximation is valid if these species diffuse fast compared with the characteristic timescales of aggregation and inhibitor binding (see Supplemental Material [48] Sec. S1.4). Typical values for the aggregation rates and condensate sizes suggest that our approximation is consistent with experimental conditions (see Supplemental Material [48] Table S1). For simplicity, we restrict ourselves to monomers and drug molecules diffusing through the condensate interface 043173-2 and neglect the diffusive exchange of aggregates. Aggregate diffusion is slow due to their large molecular weight, which limits their diffusion via slow reptation in an environment of entangled filaments. Under these assumptions, deviations from the equilibrium partitioning, caused by slow aggregation, are quasistatically balanced by rapid diffusive fluxes of monomers and drug molecules across the condensate interface [Eq. (1a)]. These fluxes ensure that the partitioning of the respective species stays approximately close to equilibrium at all times. Moreover, as suggested by recent inhibition experiments in vitro [10], inhibitor binding to its target is often much faster than aggregation, such that this binding kinetics can be considered to be in pre-equilibrium (see Supplemental Material [48] Sec. S2 and Table S1). In this case we can capture the impact of the drug on aggregation by means of effective rate parameters (see Supplemental Material [48] Secs. S1.2 and S1.3).
where k 1 , k 2 , and k + denote the aggregation rate constants for primary nucleation, secondary nucleation, and growth in the absence of the inhibitor and K i are the equilibrium binding constants of the drug to monomers (i = m), fibril ends (i = e), and fibril surface sites (i = s). By combining the above ingredients, a coupled set of kinetic equations describing inhibited aggregation in both phases α = I, II can be written as (see Supplemental Material [48] Sec. S2) where c (α) a and M (α) a denote the number and mass concentrations of aggregates, respectively, in phase α, and M (α) m is the monomer concentration in phase α, and c (α) d is the drug concentration in phase α. Equation (1e) describes the formation of new aggregates by primary and secondary nucleation, with n 1 and n 2 denoting the respective reaction orders. Depending on the value of n 2 , the secondary nucleation rate describes different processes [56]: fragmentation (n 2 = 0), branching (n 2 = 1), and surface-catalyzed secondary nucleation (n 2 > 1) (see Supplemental Material [48] Fig. S1 for typical reaction orders). These reaction terms can be combined to describe the competition of multiple nucleation processes. Equation (1f) captures the buildup of aggregate mass by monomer pickup, where k + is the rate constant for elongation and the factor 2 accounts for two growing ends per aggregate. Equation (1g) accounts for monomer depletion by growth and the interphase flux maintaining the monomer equilibrium partitioning. Equation (1h) describes the partitioning dynamics of the drug.
The validity of Eqs. (1a)-(1h) relies on a few implicit assumptions. For example, phase coexistence and the aggregation kinetics are usually coupled since both are determined by the chemical potentials of the components [58]. However, for irreversible aggregation processes it can shown that both actually decouple [59]. Moreover, focusing on irreversible aggregation in Eqs. (1a)-(1h), we neglected the effect of monomer dissociation (inverse process of elongation) and filament coagulation (inverse process of fragmentation). These processes may be easily included in the kinetic equations [60][61][62]. However, for typical amyloid-forming systems their effect on the aggregate concentration is negligible except in the very late stages of aggregation, i.e., past the plateau phase when the aggregate size distribution slowly relaxes to thermodynamic equilibrium. Finally, we consider system and condensate volumes, V and V I , to be sufficiently large (>1 pL) such that stochastic effects on primary nucleation are negligible [63].
Overall, our model has three aggregation rate parameters, reaction orders for primary and secondary nucleation, binding equilibrium constants for each inhibition mechanism, and three phase-separation parameters: compartment volume V I , and the partitioning coefficients for drug and monomers, p d and p m . In the following, we will choose the aggregation parameters such that they are consistent with in vitro experiments [53] and vary the phase-separation parameters in ranges such that they are consistent with protein condensates found in vitro and in vivo (see Table I for a summary of parameters in the model and associated values).

A. Condensates enhance inhibition of secondary nucleation
To study the impact of a liquid condensate on aggregation inhibition, we numerically studied Eqs. (1a)-(1h) and determined asymptotic analytical solutions for the number and mass concentrations of aggregates inside and outside the liquid condensate (Fig. 2(a); see Supplemental Material [48] Sec. S2.5). As an illustrative example, we focus on the inhibition of secondary nucleation by drugs that bind to the surface of existing fibrils. In Figs. 2(a) and 2(b) we consider the uninhibited, homogeneous system (no condensates, no drug) as a reference and compare the number concentration of aggregates formed for three systems: (i) no condensates with drug, (ii) condensates, no drug, and (iii) condensates with drug; systems (i) and (iii) have the same total drug concentration. Strikingly, while in all three systems the terminal (t → ∞) number concentration of aggregates is reduced relative to the reference, we see that the system with condensates and drugs is most effective in inhibiting aggregation. Moreover, adding the drug to a heterogeneous system results in a significantly more pronounced reduction compared with adding the same amount of drug to the homogeneous system [ Fig. 2(b)]. Therefore the presence of liquid condensates can enhance the 043173-3  Fig. 2. Note that in the limit of fast inhibitor binding and rapid establishment of phase equilibrium the exact values of the binding or unbinding and diffusion rates do not affect the plots in Fig. 2, as long as these rates are fast compared with aggregation.

Symbol
Meaning Value (1) a homogeneous system without an inhibitor, (2) a homogeneous system with an inhibitor of secondary nucleation, (3) a heterogeneous system with a liquid condensate in the absence of the drug, and (4) same as system (3), but in the presence of the drug. A list of parameters used to generate the plots can be found in Table 1. (b) Number concentration of aggregates at the end of the aggregation reaction for the four systems [systems (1)-(4)] considered in (a). (c) Aggregation time course inside (phase I, top part of plot) and outside (phase II, bottom part of plot) the liquid condensate without and with the drug. We see that adding the drug causes a reduction of aggregate concentration inside the condensate (phase I) but leads to an increase in aggregate concentration outside the condensate (phase II). (d) Schematic representation of the positive feedback mechanism causing enhanced inhibition inside the liquid condensate. For p m > 1, aggregation occurs primarily in phase I due to the diffusive flux of monomers maintaining equilibrium. As drug molecules in phase I bind to their target (p d > 1), their concentration decreases. The resulting concentration imbalance across the condensate interface causes a diffusive flux of drug molecules from phase II to phase I to restore phase equilibrium. As a result, inhibition is enhanced inside the condensate (phase I), but the reaction proceeds in an uncontrolled fashion outside (phase II). inhibitory action of the drug compared with the homogeneous system.

B. A positive feedback mechanism underlies enhanced inhibition in the presence of condensates
To understand the mechanism underlying this enhancement effect, we consider the impact of the drug on the aggregate concentration inside (phase I) and outside (phase II) the condensate [Fig. 2(c)]. We find that, while the drug inhibits aggregation in phase I, aggregation in phase II yields more aggregates in the presence of the drug compared with not having the drug. This counterintuitive effect is a direct consequence of the rapid tendency of monomers and drug components to reestablish partitioning equilibrium [ Fig. 2(d)]. Consider the case where monomers and drugs partition preferentially into the condensate (p m > 1, p d > 1). Because p m > 1, aggregation in phase I is accelerated compared with the continuous phase II [38]. As aggregation occurs in phase I, further monomers diffuse from phase II to phase I to maintain the monomer partitioning equilibrium, causing aggregation to occur primarily in phase I [38]. Partitioning of inhibitors into the condensate then leads to an increase in binding events of drug molecules to the aggregates inside the condensate. Therefore further drug molecules diffuse from the outside into the condensate to maintain the chemical potential of inhibitors constant across the interface of the condensate. This positive feedback in phase I couples to negative feedback in phase II, where drug molecules are continuously depleted in favor of the condensate. As a result, aggregation in phase I is suppressed, while aggregation in phase II potentially remains "uncontrolled" due to positive feedback. In other words, inhibition is enhanced in phase I, while it is lowered in phase II.

C. Physical conditions for enhanced inhibition of secondary nucleation by condensates
To understand which of these competing effects dominate, we need to quantify how the phase-separation parameters of volume and partitioning coefficients affect the total amount of aggregates in the system. To this end, we introduce an enhancement function E as The enhancement E compares the final aggregate concentration formed in a homogeneous system in the presence of the drug, c a (∞)| homo , with the average aggregate concentration formed with the same amount of drug in the phase-separated system (see Supplemental Material [48] Sec. S3): For E > 1, the condensate enhances inhibition in the entire system compared with a homogeneous solution. For large condensates (V I V ), we find that inhibition is enhanced for p d 1. For p d 1, we find E < 1, i.e., the phase-separated system yields more aggregates than the homogeneous one. Enhanced inhibition therefore occurs if both the inhibitor and the target are concentrated inside the condensate (colocalization). For small condensates (V I V ), the contribution of c I a (∞) to the average concentration c a (∞) is reduced, making the system more susceptible to uncontrolled aggregation in phase II. Thus, in the limit of small condensates, the drug should preferably partition outside the condensate (p d 1). Overall, our results suggest that that there are two competing effects at play: Effective enhancement of inhibition by the condensate requires colocalization of the drug with the aggregates inside the condensate while avoiding uncontrolled aggregation outside. Consistently, the drug should partition in phase I for large condensates, while for smaller droplets the drug should suppress aggregation outside the condensate.

D. Optimal enhancement
This qualitative change of mechanism with condensate volume raises the question of which system yields the most effective reduction of aggregate concentration. To this end, we compare the terminal aggregate concentrations for different values of drug partitioning p d and condensate volumes V I /V and for the different systems [ Fig. 3(b)]: (i) no condensates with drug, (ii) condensates, no drug, and (iii) condensates with drug. For large condensates the optimal system depends on p d . When p d 1, the optimal choice is to add the drug to a homogeneous solution, while if the drug is enriched in the condensate (p d 1), the most effective suppression of aggregation occurs in the phase-separated system in the presence of the drug. For intermediate condensate volumes, adding the drug to the heterogeneous system is the best strategy for all values of p d . By contrast, for smaller condensates, adding the drug to the phase-separated system causes an increase in the overall aggregate concentration due to uncontrolled aggregation in phase II. In this case, the optimal solution is the phase-separated system without the drug. The resulting phase diagram, shown in Fig. 3(c), thus provides a practical strategy for choosing the most effective system to suppress aggregation depending on the phase-separation parameters.

E. Condensates increase potency of aggregation inhibitors
Our theory predicts under which conditions condensates promote the activity of protein aggregation inhibitors relative to a homogeneous system. This implies that an equivalent degree of inhibition compared with the homogeneous system can be realized using a smaller amount of drug in the presence of condensates. In other words, liquid condensates can enhance the potency of a drug. To illustrate this effect, we define for both the homogeneous and the phase-separated system a drug-response curve R(c d ) given as the relative difference between asymptotic fibril concentrations with and without drug, Fig. 4(a)]. On the basis of the response function, we introduce the drug potency P = 1/EC 50 as the inverse of the drug concentration necessary to suppress 50% of the aggregates, denoted as EC 50 . The smaller EC 50 is, the more potent is the drug. In Fig. 4(b), we show the relative potency, defined as the ratio between the EC 50 values in the absence and in the presence of condensates, respectively (see Supplemental Material [48] Sec. S4). We see that the relative potency increases with decreasing drop 043173-5 volume in the limit of large condensates. In this limit, we find The relative potency P rel increases from unity for large condensates to a value equal to the drug partitioning p d when the condensate volume is decreased. Hence reducing the volume of the condensate V I /V and/or increasing the drug enrichment factor p d lead to a strong increase in the relative potency. This increase corresponds to a strong reduction of the amount of drug necessary to inhibit aggregation. For small condensates, the optimal system becomes the phase-separated system without drug [ Fig. 3(c)]; correspondingly, the potency decreases due to uncontrolled aggregation in phase II.

IV. DISCUSSION AND CONCLUSIONS
Heterogeneous environments have much potential in regulating biochemical reactions [32][33][34][35]. Our present study shows that liquid-like condensates can strongly affect drugmediated inhibition of filamentous protein aggregation by providing such a heterogeneous environment composed of two coexisting phases. A key finding is that the potency of inhibitors can be increased due to the presence of a liquid condensate. This increase in potency implies that a significantly smaller drug dose is required to equally inhibit aggregation compared with the corresponding homogeneous system. We find that this potency increase depends on the colocalization of the inhibitor with its target within the liquid condensate leading to a positive feedback on the inhibition of aggregates. In a Flory-Huggins model of phase separation, we can calculate the degree of drug or monomer partitioning in the dilute limit (see Supplemental Material [48] Eq. (S12)). The partitioning of species i depends on parameters such as the degree of phase separation, molecular volumes, and the relative interaction parameter between species i and the phase-separating components A and B, derived from multiple microscopic processes and specific interactions. These parameters may be tuned experimentally to affect the tendencies of inhibitors and aggregates to colocalize within the droplet phase.
To experimentally scrutinize our key findings, we propose considering protein-rich condensates which recruit aggregation-prone monomers as well as aggregation inhibitors (chaperones or synthetic drugs that are both dilute compared to the condensate components, and determining the response as a function of inhibitor concentration with and without condensates [ Fig. 4(a)].
In living cells, the colocalization of inhibitors and aggregates may not only increase the therapeutic outcome but also reduce the toxic effects on the organism by preventing drug molecules from interacting with other sensitive cellular domains. Thus our finding of an enhanced drug potency due to the presence of condensates suggests revisiting nominally efficacious drugs that have been disregarded in a homogeneous setting due to their high toxicity and shifting the focus to how drugs act specifically in the context of a spatially organized, intracellular environment.