Mechanism for the stabilization of protein clusters above the solubility curve: the role of non-ideal chemical reactions

Statistics of carrier-cargo complexes

We explore the statistics of assembling soft-matter building blocks to investigate the uptake and encapsulation of cargo particles by carriers engulfing their load. While the such carrier-cargo complexes are important for many applications out of equilibrium, such as drug delivery and synthetic cell encapsulation, we uncover here the basic statistical physics in minimal hard-core-like models for particle uptake. Introducing an exactly solvable equilibrium model in one dimension, we demonstrate that the formation of carrier-cargo complexes can be largely tuned by both the cargo concentration and the carriers' interior size. These findings are intuitively explained by interpreting the internal free space (partition function) of the cargo inside a carrier as its engulfment strength, which can be mapped to an external control parameter (chemical potential) of an additional effective particle species. Assuming a hard carrier membrane, such a mapping can be exactly applied to account for multiple cargo uptake involving various carrier or cargo species and even attractive uptake mechanisms, while soft interactions require certain approximations. We further argue that the Boltzmann occupation law identified within our approach is broken when particle uptake is governed by nonequilibrium forces. Speculating on alternative occupation laws using effective parameters, we put forward a Bose-Einstein-like phase transition associated with polydisperse carrier properties.

Perspective: New directions in dynamical density functional theory

Classical dynamical density functional theory (DDFT) has become one of the central modeling approaches in nonequilibrium soft matter physics. Recent years have seen the emergence of novel and interesting fields of application for DDFT. In particular, there has been a remarkable growth in the amount of work related to chemistry. Moreover, DDFT has stimulated research on other theories such as phase field crystal models and power functional theory. In this perspective, we summarize the latest developments in the field of DDFT and discuss a variety of possible directions for future research.

Nucleation of the destruction complex on the centrosome accelerates degradation of β-catenin and regulates Wnt signal transmission

Liquid–liquid phase separation (LLPS) governs a variety of mesoscale cellular processes. However, less is known about how cells utilize LLPS to drive cellular function. Here, we examined the destruction complex (DC), an organelle which controls Wnt signaling and whose components phase separate. Through a combination of advanced microscopy, CRISPR, computational modeling, and optogenetics, we find that the DC is nucleated by the centrosome and that this nucleation drives efficient signal transduction. Our work not only uncovers a biological function for LLPS but also highlights nucleation as a general method for controlling the function of intracellular condensates. Finally, our findings suggest a thermodynamic coupling between Wnt signal transduction and the cell cycle which could lead to insights into Wnt-driven cancers.

Wnt signal transduction is controlled by the destruction complex (DC), a condensate comprising scaffold proteins and kinases that regulate β-catenin stability. Overexpressed DC scaffolds undergo liquid–liquid phase separation (LLPS), but DC mesoscale organization at endogenous expression levels and its role in β-catenin processing were previously unknown. Here, we find that DC LLPS is nucleated by the centrosome. Through a combination of CRISPR-engineered custom fluorescent tags, finite element simulations, and optogenetic tools that allow for manipulation of DC concentration and multivalency, we find that centrosomal nucleation drives processing of β-catenin by colocalizing DC components to a single reaction crucible. Enriching GSK3β partitioning on the centrosome controls β-catenin processing and prevents Wnt-driven embryonic stem cell differentiation to mesoderm. Our findings demonstrate the role of nucleators in controlling biomolecular condensates and suggest tight integration between Wnt signal transduction and the cell cycle.

Nucleation of glucose isomerase protein crystals in a nonclassical disguise: The role of crystalline precursors

The ability of proteins to self-assemble into complex, hierarchical structures has been the inspiration for the bottom-up design of a class of biomaterials with proteins as their building blocks. The earliest stages of formation often involve the passing of an activation barrier under the form of nucleus formation, a quaternary protein complex that templates incoming molecules to proper registry. For protein crystallization, the consensus has emerged that the fastest route toward a nucleus follows a winding path: first, densification, followed by symmetry formation. In this contribution, we show that this need not be the case for the protein glucose isomerase, which seems to follow the simplest path to a nucleus, making crystalline clusters from the earliest detectable beginnings.

Protein crystallization is an astounding feat of nature. Even though proteins are large, anisotropic molecules with complex, heterogeneous surfaces, they can spontaneously group into two- and three-dimensional arrays with high precision. And yet, the biggest hurdle in this assembly process, the formation of a nucleus, is still poorly understood. In recent years, the two-step nucleation model has emerged as the consensus on the subject, but it still awaits extensive experimental verification. Here, we set out to reconstruct the nucleation pathway of the candidate protein glucose isomerase (GI), for which there have been indications that it may follow a two-step nucleation pathway under certain conditions. We find that the precursor phase present during the early stages of the reaction process is nanoscopic crystallites that have lattice symmetry equivalent to the mature crystals found at the end of a crystallization experiment. Our observations underscore the need for experimental data at a lattice-resolving resolution on other proteins so that a general picture of protein crystal nucleation can be formed.

Nucleation of protein mesocrystals via oriented attachment

Self-assembly of proteins holds great promise for the bottom-up design and production of synthetic biomaterials. In conventional approaches, designer proteins are pre-programmed with specific recognition sites that drive the association process towards a desired organized state. Although proven effective, this approach poses restrictions on the complexity and material properties of the end-state. An alternative, hierarchical approach that has found wide adoption for inorganic systems, relies on the production of crystalline nanoparticles that become the building blocks of a next-level assembly process driven by oriented attachment (OA). As it stands, OA has not yet been observed for protein systems. Here we employ cryo-transmission electron microscopy (cryoEM) in the high nucleation rate limit of protein crystals and map the self-assembly route at molecular resolution. We observe the initial formation of facetted nanocrystals that merge lattices by means of OA alignment well before contact is made, satisfying non-trivial symmetry rules in the process. As these nanocrystalline assemblies grow larger we witness imperfect docking events leading to oriented aggregation into mesocrystalline assemblies. These observations highlight the underappreciated role of the interaction between crystalline nuclei, and the impact of OA on the crystallization process of proteins.

Liquid-Liquid Phase Separation in the Presence of Macromolecular Crowding and State-dependent Kinetics

Biomolecular condensates formed via liquid-liquid phase separation (LLPS) are increasingly being shown to play major roles in cellular self-organization dynamics in health and disease. It is well established that macromolecular crowding has a profound impact on protein interactions, particularly those that lead to LLPS. Although synthetic crowding agents are used during in vitro LLPS experiments, they are considerably different from the highly crowded nucleo-/cytoplasm and the effects of in vivo crowding remain poorly understood. In this work, we applied computational modeling to investigate the effects of macromolecular crowding on LLPS. To include biologically relevant LLPS dynamics, we extended the conventional Cahn-Hilliard model for phase separation by coupling it to experimentally derived macromolecular crowding dynamics and state-dependent reaction kinetics. Through extensive field-theoretic computer simulations, we show that the inclusion of macromolecular crowding results in late-stage coarsening and the stabilization of relatively smaller condensates. At a high crowding concentration, there is an accelerated growth and late-stage arrest of droplet formation, effectively resulting in anomalous labyrinthine morphologies akin to protein gelation observed in experiments. These results not only elucidate the crowder effects observed in experiments, but also highlight the importance of including state-dependent kinetics in LLPS models, and may help in designing further experiments to probe the intricate roles played by LLPS in self-organization dynamics of cells.

Effects of social distancing and isolation on epidemic spreading modeled via dynamical density functional theory

For preventing the spread of epidemics such as the coronavirus disease COVID-19, social distancing and the isolation of infected persons are crucial. However, existing reaction-diffusion equations for epidemic spreading are incapable of describing these effects. In this work, we present an extended model for disease spread based on combining a susceptible-infected-recovered model with a dynamical density functional theory where social distancing and isolation of infected persons are explicitly taken into account. We show that the model exhibits interesting transient phase separation associated with a reduction of the number of infections, and allows for new insights into the control of pandemics.

A mechanism for reversible mesoscopic aggregation in liquid solutions

Solutions of proteins and other molecules exhibit puzzling, mesoscopically sized inclusions of a solute-rich liquid, well outside the region of stability of the solute-rich phase. This mesoscopic size is in conflict with existing views on heterophase fluctuations. Here we systematically work out a microscopic mechanism by which a metastable solute-rich phase can readily nucleate in a liquid solution. A requisite component of the mechanism is that the solute form long-lived complexes with itself or other molecules. After nucleated in this non-classical fashion, individual droplets grow until becoming mechanically unstable because of a concomitant drop in the internal pressure, the drop caused by the metastability of the solute-rich phase. The ensemble of the droplets is steady-state. In a freshly prepared solution, the ensemble is predicted to evolve in a way similar to the conventional Ostwald ripening, during which larger droplets grow at the expense of smaller droplets.

Physical principles of intracellular organization via active and passive phase transitions

Exciting recent developments suggest that phase transitions represent an important and ubiquitous mechanism underlying intracellular organization. We describe key experimental findings in this area of study, as well as the application of classical theoretical approaches for quantitatively understanding these data. We also discuss the way in which equilibrium thermodynamic driving forces may interface with the fundamentally out-of-equilibrium nature of living cells. In particular, time and/or space-dependent concentration profiles may modulate the phase behavior of biomolecules in living cells. We suggest future directions for both theoretical and experimental work that will shed light on the way in which biological activity modulates the assembly, properties, and function of viscoelastic states of living matter.

Nonclassical nucleation pathways in protein crystallization

Classical nucleation theory (CNT), which was established about 90 years ago, has been very successful in many research fields, and continues to be the most commonly used theory in describing the nucleation process. For a fluid-to-solid phase transition, CNT states that the solute molecules in a supersaturated solution reversibly form small clusters. Once the cluster size reaches a critical value, it becomes thermodynamically stable and favored for further growth. One of the most important assumptions of CNT is that the nucleation process is described by one reaction coordinate and all order parameters proceed simultaneously. Recent studies in experiments, computer simulations and theory have revealed nonclassical features in the early stage of nucleation. In particular, the decoupling of order parameters involved during a fluid-to-solid transition leads to the so-called two-step nucleation mechanism, in which a metastable intermediate phase (MIP) exists between the initial supersaturated solution and the final crystals. Depending on the exact free energy landscapes, the MIPs can be a high density liquid phase, mesoscopic clusters, or a pre-ordered state. In this review, we focus on the studies of nonclassical pathways in protein crystallization and discuss the applications of the various scenarios of two-step nucleation theory. In particular, we focus on protein solutions in the presence of multivalent salts, which serve as a model protein system to study the nucleation pathways. We wish to point out the unique features of proteins as model systems for further studies.

Nucleation and growth of a core-shell composite nucleus by diffusion

The critical radius of a core-shell-type nucleus grown by diffusion in a phase-separated solution is studied. A kinetic critical radius rather than the thermodynamic critical radius of standard classical nucleation theory can be defined from the diffusional growth equations. It is shown that there exist two kinetic critical radii for the core-shell-type nucleus, for which both the inner-core radius and the outer-shell radius will be stationary. Therefore, these two critical radii correspond to a single critical point of the nucleation path with a single energy barrier even though the nucleation looks like a two-step process. The two radii are given by formulas similar to that of classical nucleation theory if the Ostwald-Freundlich boundary condition is imposed at the surface of the inner nucleus and that of the outer shell. The subsequent growth of a core-shell-type postcritical nucleus follows the classical picture of Ostwald's step rule. Our result is consistent with some of the experimental and numerical results which suggest the core-shell-type critical nucleus.