Toward universal models for collective interactions in biomolecular condensates Open Access

Our understanding of living matter has been revolutionized by the notion about higher-order protein structures^1–3 (Fig. 1). These non-stoichiometric assemblies regulate key cellular processes and can sample a wide range of dynamics and material states.⁴ Changes, in particular, in their biophysical properties may shift the physiological to an aberrant state, leading to dysregulation, as implicated in numerous human diseases.^5,6 The conversion of liquid-like to solid-like assemblies, for example to amyloid aggregates, is a hallmark in neurological disorders, such as Alzheimer's or Parkinson's disease or amyotrophic lateral sclerosis (ALS).^7,8 The contribution of dense, liquid-like clusters, often termed as biomolecular condensates, to physiological processes is increasingly recognized.^9,10 Nevertheless, our understanding of the molecular nature and interactions of condensates is incomplete. Growing experimental evidence also highlights that these clusters can also be formed below the critical concentration of liquid–liquid phase separation.^11–13 

In this regard, it is important to emphasize that most of the concepts related to phase separation and demixing (or liquid–liquid phase separation) are developed, in the vast majority of cases, for binary systems. In contrast, in cellular environments, the system is composed of a large number of different components, each of which is present in small quantities. In the formation of droplets of liquid “A” within the context of liquid “B,” the fluctuation in the number of molecular components plays a fundamental role. Since these components are present in small numbers of molecules, they may follow a non-Gaussian distribution and, in certain extreme cases, may be limited to one or even no molecules. This leads to variations in the properties of liquid A from droplet to droplet, which tends to “blur” the phase separation boundaries in the phase diagram.

There are fundamental open questions regarding the organization of condensates: (1) Are interactions driving condensate formation different from those that assemble stoichiometric protein complexes? (2) What is the molecular basis of their “collective” nature? (3) How the selectivity of condensate interactions is controlled? (4) What is the basis of their cellular context-dependent regulation, assembly/disassembly? (5) Which are the consequences of the condensate interactions in terms of macroscopic behavior and physical properties? (6) What is the effect of the large concentration fluctuation from droplet to droplet?

Numerous studies have attempted to delineate the interactions distinguished in formation of condensates and identify the key motifs, which drive this process.¹⁴ A wide variety of interactions including electrostatic¹⁵ and van der Waals interactions¹⁶ as well as pi–pi contacts^17,18 were found to be critical for droplet formation. Our motivation in this article is different. We focus on the global, physical features of the condensate interaction network, and aim to provide general models, which can account for these characteristics. We aim at understanding (1) the dynamic nature of the interactions as droplets fuse/fission; (2) the dependence of droplet formation/disassembly on the external factors; and (3) the possible change in biophysical properties of the droplets. Here, we aim to define the boundary conditions for interactions, which enable these emergent properties, and deduce the phenomenological consequences of the corresponding interaction network.

Our starting point is the recent analysis of experimental data of various proteins, which were observed to form dense, sizable droplets below the critical concentration for phase separation.^11–13 The droplet size distributions could be described by a scale-invariant lognormal function, which could also estimate the critical concentration for protein phase separation.¹⁹ The existence of scale invariance for the droplet size distribution suggests a universal behavior,²⁰ independent of the sequences and structures of the proteins undergoing phase separation. This is in line with the earlier notion on the generic nature of protein condensates.²¹ With respect to the interactions, scale invariance stems from self-similarity at different length scales,²² leading to universality.

The article will be organized as follows. First, we describe protein interactions at different scales (Sec. II). Second, we discuss the organization of collective, dynamic interactions (Sec. III). Third, we analyze the sensitivity of the interaction network to environmental conditions (Sec. IV). Fourth, we detail the phenomenological consequences such as the increased correlation length and scale invariance (Sec. V). The mathematical formalisms for collective interactions (Appendix A) and for scale invariance (Appendix B) can be found in the BioRXiv version of the article (https://www.biorxiv.org/content/10.1101/2024.10.17.618883v1). Finally, we conclude the article with the possible applications of our model, in particular, for therapeutic approaches and drug discovery.

Protein interactions appear to be distinct at small and large scales of molecular organization. Stoichiometric protein complexes are held together by well-defined contacts between specific residues, while biomolecular condensates are assembled through multiple valances, with a wide diversity of chemical nature and affinities. Which interactions can operate at both scales?

We initiate by asking how the external conditions (laboratory or cell) influence the activity of stoichiometric protein complexes. Fine-tuning of biological activities in accord with diverse signals often involves spatiotemporal variations at the interface formed by specific partners.^23,24 Cell cycle regulation by protein phosphatases, for example, is associated with dynamic, transient interactions by the regulatory loop, which can simultaneously bind different inhibitors.²⁵ In a similar vein, transcriptional co-activators may simultaneously engage with different activator domains through heterogeneous interactions by the same target site.^26,27 Intriguingly, transcription activity reduces upon increasing the stability of binding motifs, which reduces the heterogeneity of the bound complex.³ Although it seems counterintuitive, advanced structure determination and biophysical techniques demonstrate that specific association can be achieved through multiple conformations.²⁸ For example, through multiple orientations of a short secondary structure, which is accommodating into a shallow binding cleft.²⁹ Although each orientation generates a distinct contact pattern, the bound ensemble involves a specific set of residues, which interact in a combinatorial manner. High density of residues with similar physio-chemical properties, for example consecutive negatively charged amino acids, compete for the same target, such as a single positive residue in the interacting partner, resulting in ambiguous, sub-optimal bound states.³⁰ Indeed, interaction fuzziness is analogous to the rugged energy landscape of protein folding, which is detailed elsewhere.³¹ 

Fuzzy interactions are enabled by evolutionary conserved, simple biochemical features, while the amino acid types are only weakly constrained.^27,32 An evolutionary selection for fuzzy binding was proposed in complex processes, which simultaneously affect multiple pathways.³³ “Multivalency” appears to be a consistent and pertinent feature of proteins,³⁴ which form condensates, although the underlying binding motifs (i.e., “valence”) may not be recognized.¹⁴ Simple features such as patterning of residues with distinct physico-chemical characters can be sufficient to observe liquid-like droplets.³⁵ Although high resolution structural characterization of condensates is still lacking, conformational heterogeneity of the assembly is supported by various lines of experimental evidence (e.g., solution and single molecule methods).^36,37 Conformational heterogeneity can be generated by combinatorial binding mechanisms via multiple valances³⁸ as in the case of prion-like domains³⁹ or via other fuzzy binding mechanisms (e.g., multiple orientations using the same site) by more structured proteins.^40,41 In both cases, high binding entropy can be attained in line with the dynamics and often liquid-like character of the condensates.²¹ 

Taken together, variable, fuzzy interactions can generate self-similarity at different length scales of protein organization and may serve as a universal feature.

We stress that the exponents reported for standard percolation are α = 1.5 and φ = 2.⁴⁴ The deviation between the critical exponents suggests that proteins forming clusters below the critical concentration for phase separation may belong to a different universality class than standard percolation. Furthermore, the results are consistent with critical phenomena and continuous phase transitions, consistently with earlier experimental results.⁴⁵ 

Here, we analyze whether fuzzy interactions enable a continuous transition from the dynamic clusters formed at sub-critical concentration to macroscopic condensates (Fig. 2). The mathematical formalism for fuzzy interactions is detailed in Appendix A.⁴⁶ In the main text, we only discuss the key features of the model, which lead to the formation of condensates. Most importantly, a fuzzy interaction motif (Fig. 1) can interact with various other motifs (x₁, x₂, …, x_N) simultaneously with different membership functions m(x₁), m(x₂), …, m(x_N) (Fig. 1) within the framework of the fuzzy set theory.⁴⁷ An aromatic ring, for example, can interact both in parallel and perpendicular orientations with other pi groups.⁴⁸ This leads to a set of simultaneous, partial contacts, which mutually influence the pairwise binding affinities (Appendix A). The likelihood of contacting all these potential interaction motifs depends on the length and dynamical properties of the segments in between the motifs.⁴⁹ In case the linker is fully dynamic, all potential interactions can be sampled, while a drop in dynamics also restrains the available interaction space. This is a key point, which is capable of describing a gradual change in binding affinity as the cluster grows.⁴⁹ Furthermore, considering the dissociation probability of the motifs, and the diffusion of the particles (monomers, oligomers, or large clusters/polymers) out of the interaction zone, the affinity can be tuned on-the-fly following the changes in occupancy (also considering partial sites). Simulations based this framework (Appendix A) correctly recapitulate that the fraction of oligomers and large polymers converge to a limit, instead of showing an infinite growth⁴⁹ (Fig. 2). The simulations also describe the assembly/disassembly of the clusters as well as their growth in size as the system approaches the phase boundary. Linker dynamics, through its impact on the number of available interaction sites, influence the rate but not the extent of clustering (i.e., fraction of clusters)⁴⁹ (Fig. 2). Here, we also note that some membraneless organelles in the cell exhibit scale-free interaction networks, such as the Cajal body, nucleolus, nuclear pore complex, paraspeckle, and spliceosome.

In this section, we conclude that a model based on fuzzy interactions (Appendix A) can describe continuous and dynamic cluster formation, converging to a hypothetical phase separated system.

Condensates are often formed/dissolved upon cellular cues or changes in solution conditions.^16,50,51 Their cellular context sensitivity also induces changes in the biophysical properties, for example in the material state of condensates.⁵² This is of particular importance in case the biophysical properties perturb the physiological state.⁵³ Can the model based on fuzzy interactions recapitulate the context-dependent behavior of condensates? We follow two approaches: first based on the energy landscape framework adapted for protein interactions and second based on computer simulations using the fuzzy model.

In analogy with the energy landscape framework for protein folding, the energy landscape of protein interactions can also be considered as rugged, instead of sampling only the distinguished “native” state.³¹ Thus, proteins may sample alternative bound states while bound with their specific partners, which may be considered as non-native states. Each bound conformation is energetically sub-optimal, yet distinct with different proteins,⁵⁴ consistent with the notion of fuzzy interactions. Given this model, the bound state is represented by an ensemble of conformations and contact patterns, the populations of which depend on external factors. While variations in conditions shift the ensemble and may favor given bound forms, the bound state never confines to a unique form but remains to be heterogeneous.³¹ Therefore, the interaction between specific partners will respond to the conditions through changing the populations in the bound state ensemble.⁵⁵ Such change allows fine-tuning of biological activity, for example regarding substrate selectivity.⁵⁴ 

Now, we consider how the network of fuzzy interactions responds to external conditions. In the second part of the simulations based on the fuzzy interaction framework⁴⁹ (Appendix A), which represents a hypothetically phase separated system, monomers are still exchanging with the environment. The fraction of clusters remains stable, while their internal organization may undergo substantial changes through variations in occupancies of the interaction sites, considering also partial contacts (Appendix A). The heterogeneous interaction network occasionally starts forming regular interaction patterns, such as ladders of contacts formed by consecutive residues.⁴⁹ Depending on the linker dynamics, the network may transform from disordered to ordered interactions (Fig. 2). Small perturbations of the model result in distinct outcomes of the simulations, illustrating the sensitivity of the system. Intriguingly, dynamic systems without fuzzy interactions do not exhibit such extensive rearrangements of the interaction network. We also note that context-dependent controls are often achieved through fuzzy inference systems.⁵⁶ 

In this section, we conclude that the network of fuzzy interactions is sensitive to the perturbations by external factors, thus the model may serve as a basis of collective, context-dependent behaviors.

When a system develops a large correlation length—defined as the spatial range over which density fluctuations are strongly correlated—it behaves similarly to a system near criticality, at least on spatial scales smaller than the correlation length.^20,57 In the case of our droplet system, this correlation length could approach the size of individual cells as the critical density of the protein is approached from below. From statistical mechanics, it is well known that systems near criticality exhibit extreme sensitivity to external perturbations.⁵⁸ This means that even small changes can elicit prompt and widespread responses across both large spatial and temporal scales. In such a regime, fluctuations are not isolated; instead, they propagate rapidly through the system, allowing it to respond in a highly coordinated manner to external forces.

In biomolecular condensates, the situation is further complicated by the heterogeneity of the amino acids and their intra- and intermolecular microenvironments.⁵⁹ This suggests that if the system is indeed close to criticality—or equivalently, if it has developed a large correlation length—this molecular diversity might enable different parts of the system to respond distinctly and efficiently to various external stimuli. This may lead to distinct responses to different external signals, depending on the local conditions or the specific interactions between different protein components.

Within the framework of percolation theory, it is important to note that in standard percolation models, clusters form in an irreversible manner. Once a cluster forms, it remains intact, with no mechanism for the system to return to a prior state or for clusters to disintegrate. This lack of dynamical equilibrium distinguishes standard percolation from other processes where continuous formation and dissolution of structures occur. The scaling analysis suggests that the droplet formation observed in our system does not correspond to the behavior of standard percolation.¹⁹ Instead, the dynamics we are studying seem to involve a more complex, reversible process. A more accurate and realistic model of this system would need to account for the ability of clusters or droplets to not only form but also to disintegrate and re-form dynamically.

This indicates that the system likely exists in a state of dynamical equilibrium, where clusters of proteins or molecules can fluctuate between different states in response to varying conditions, such as concentration, temperature, or other external factors. Thus, unlike in traditional percolation where the formation of large, connected clusters marks a one-way transition, the system we are dealing with appears to have more fluidity, with clusters constantly forming and dissolving. This suggests that models incorporating reversible aggregation and disaggregation would provide a more accurate description of the droplet formation process in our system.

In this article, we argued that similar interaction mechanisms take place at various scales of protein assembly from oligomers to large clusters. We demonstrated that heterogeneous, variable interactions of specific partners may generate a collective network in a manner that is consistent with continuous phase transitions. This in line with the scale-invariant lognormal distributions of the droplet sizes, which hints on second-order phase transition. The recently found validity of scale invariance places proteins close to their phase boundary, thus maximizing the sensitivity to environmental conditions. This is in accord with the model we presented here, which can describe the conversion between different material states.

The importance of condensates in human physiology is increasingly recognized.⁹ Likewise, increasing number of diseases are associated with perturbations of the biophysical properties of the droplet state.^6,60 In particular, protein aggregation, a hallmark of various neurological disorders, were related to the material state conversion of liquid-like condensates.⁶¹ The premise is that understanding the driving forces and the nature of collective interactions may open novel approaches for target selection or drug development for condensation diseases. The results indicating the proximity of proteins close to their phase separation boundary also implies that the fluctuations in the system can be efficiently modulated, also by small-molecule drugs. Promising attempts toward this direction have been published.⁶² Another possible application is to exploit context sensitivity for modulating cellular pathways,⁶³ which eventually can be used as a readout in screening studies. While there is a long, rugged road ahead (such as the interaction energy landscape of the condensates), quantitative models may contribute to all these efforts.

This work was supported by AIRC IG 26229 (M.F.) and PRIN 2022EMZJL4 (M.F.). K.K.H.M. acknowledges support via a Ph.D. scholarship from the Scuola Superiore Meridionale, Napoli, Italy.

The authors have no conflicts to disclose.

No ethics statement was required for this work.

Edoardo Milanetti: Formal analysis (equal); Validation (equal); Writing – review & editing (equal). Karan K. H. Manjunatha: Data curation (lead). GianCarlo Ruocco: Conceptualization (lead); Writing – review & editing (equal). Amos Maritan: Writing – original draft (equal); Writing – review & editing (equal). Monika Fuxreiter: Conceptualization (lead); Funding acquisition (lead); Supervision (lead); Writing – original draft (lead); Writing – review & editing (lead).

All data presented in this manuscript are publicly accessible.