Polyubiquitin ligand-induced phase transitions are optimized by spacing between ubiquitin units.

Biomolecular condensates form via multivalent interactions among key macromolecules and are regulated through ligand binding and/or posttranslational modifications. One such modification is ubiquitination, the covalent addition of ubiquitin (Ub) or polyubiquitin chains to target macromolecules. Specific interactions between polyubiquitin chains and partner proteins, including hHR23B, NEMO, and UBQLN2, regulate condensate assembly or disassembly. Here, we used a library of designed polyubiquitin hubs and UBQLN2 as model systems for determining the driving forces of ligand-mediated phase transitions. Perturbations to either the UBQLN2-binding surface of Ub or the spacing between Ub units reduce the ability of hubs to modulate UBQLN2 phase behavior. By developing an analytical model based on polyphasic linkage principles that accurately described the effects of different hubs on UBQLN2 phase separation, we determined that introduction of Ub to UBQLN2 condensates incurs a significant inclusion energetic penalty. This penalty antagonizes the ability of polyUb hubs to scaffold multiple UBQLN2 molecules and cooperatively amplify phase separation. The extent to which polyubiquitin hubs promote UBQLN2 phase separation is encoded in the spacings between Ub units. This spacing is modulated by chains of different linkages and designed chains of different architectures, thus illustrating how the ubiquitin code regulates functionality via the emergent properties of the condensate. The spacing in naturally occurring linear polyubiquitin chains is already optimized to promote phase separation with UBQLN2. We expect our findings to extend to other condensates, emphasizing the importance of ligand properties, including concentration, valency, affinity, and spacing between binding sites in studies and designs of condensates.

How Hierarchical Interactions Make Membraneless Organelles Tick Like Clockwork.

Biomolecular condensates appear throughout the cell, serving many different biochemical functions. We argue that condensate functionality is optimized when the interactions driving condensation vary widely in affinity. Strong interactions provide structural specificity needed to encode functional properties but carry the risk of kinetic arrest, while weak interactions allow the system to remain dynamic but do not restrict the conformational ensemble enough to sustain specific functional features. To support our opinion, we describe illustrative examples of the interplay of strong and weak interactions that are found in the nucleolus, SPOP/DAXX condensates, polySUMO/polySIM condensates, chromatin, and stress granules. The common feature of these systems is a hierarchical assembly motif in which weak, transient interactions condense structurally defined functional units.

Reduction of oligomer size modulates the competition between cluster formation and phase separation of the tumor suppressor SPOP.

Phase separation compartmentalizes many cellular pathways. Given that the same interactions that drive phase separation mediate the formation of soluble complexes below the saturation concentration, the contribution of condensates versus complexes to function is sometimes unclear. Here, we characterized several new cancer-associated mutations of the tumor suppressor speckle-type POZ protein (SPOP), a substrate recognition subunit of the Cullin3-RING ubiquitin ligase. This pointed to a strategy for generating separation-of-function mutations. SPOP self-associates into linear oligomers and interacts with multivalent substrates, and this mediates the formation of condensates. These condensates bear the hallmarks of enzymatic ubiquitination activity. We characterized the effect of mutations in the dimerization domains of SPOP on its linear oligomerization, binding to its substrate DAXX, and phase separation with DAXX. We showed that the mutations reduce SPOP oligomerization and shift the size distribution of SPOP oligomers to smaller sizes. The mutations therefore reduce the binding affinity to DAXX but unexpectedly enhance the poly-ubiquitination activity of SPOP toward DAXX. Enhanced activity may be explained by enhanced phase separation of DAXX with the SPOP mutants. Our results provide a comparative assessment of the functional role of complexes versus condensates and support a model in which phase separation is an important factor in SPOP function. Our findings also suggest that tuning of linear SPOP self-association could be used by the cell to modulate activity and provide insights into the mechanisms underlying hypermorphic SPOP mutations. The characteristics of cancer-associated SPOP mutations suggest a route for designing separation-of-function mutations in other phase-separating systems.

Crossing boundaries of light microscopy resolution discerns novel assemblies in the nucleolus.

The nucleolus is the largest membraneless organelle and nuclear body in mammalian cells. It is primarily involved in the biogenesis of ribosomes, essential macromolecular machines responsible for synthesizing all proteins required by the cell. The assembly of ribosomes is evolutionarily conserved and accounts for the most energy-consuming cellular process needed for cell growth, proliferation, and homeostasis. Despite the significance of this process, the substructural mechanistic principles of the nucleolar function in preribosome biogenesis have only recently begun to emerge. Here, we provide a new perspective using advanced super-resolution microscopy and single-molecule MINFLUX nanoscopy on the mechanistic principles governing ribosomal RNA-seeded nucleolar formation and the resulting tripartite suborganization of the nucleolus driven, in part, by liquid-liquid phase separation. With recent advances in the cryogenic electron microscopy (cryoEM) structural analysis of ribosome biogenesis intermediates, we highlight the current understanding of the step-wise assembly of preribosomal subunits in the nucleolus. Finally, we address how novel anticancer drug candidates target early steps in ribosome biogenesis to exploit these essential dependencies for growth arrest and tumor control.

Beneficial and detrimental effects of non-specific binding during DNA hybridization.

DNA strands have to sample numerous states to find the alignment that maximizes Watson-Crick-Franklin base pairing. This process depends strongly on sequence, which affects the stability of the native duplex as well as the prevalence of non-native inter- and intramolecular helices. We present a theory that describes DNA hybridization as a three-stage process: diffusion, registry search, and zipping. We find that non-specific binding affects each of these stages in different ways. Mis-registered intermolecular binding in the registry search stage helps DNA strands sample different alignments and accelerates the hybridization rate. Non-native intramolecular structure affects all three stages by rendering portions of the molecule inert to intermolecular association, limiting mis-registered alignments to be sampled, and impeding the zipping process. Once in-register base pairs are formed, the stability of the native structure is important to hold the molecules together long enough for non-native contacts to break.

Distinct growth regimes of α-synuclein amyloid elongation.

Addition of amyloid seeds to aggregation-prone monomers allows for amyloid fiber growth (elongation) omitting slow nucleation. We here combine Thioflavin T fluorescence (probing formation of amyloids) and solution-state NMR spectroscopy (probing disappearance of monomers) to assess elongation kinetics of the amyloidogenic protein, α-synuclein, for which aggregation is linked to Parkinson's disease. We found that both spectroscopic detection methods give similar kinetic results, which can be fitted by applying double exponential decay functions. When the origin of the two-phase behavior was analyzed by mathematical modeling, parallel paths as well as stop-and-go behavior were excluded as possible explanations. Instead, supported by previous theory, the experimental elongation data reveal distinct kinetic regimes that depend on instantaneous monomer concentration. At low monomer concentrations (toward end of experiments), amyloid growth is limited by conformational changes resulting in ß-strand alignments. At the higher monomer concentrations (initial time points of experiments), growth occurs rapidly by incorporating monomers that have not successfully completed the conformational search. The presence of a fast disordered elongation regime at high monomer concentrations agrees with coarse-grained simulations and theory but has not been detected experimentally before. Our results may be related to the wide range of amyloid folds observed.

Circuit Topology Approach for the Comparative Analysis of Intrinsically Disordered Proteins.

Intrinsically disordered proteins (IDPs) lack a stable native conformation, making it challenging to characterize their structure and dynamics. Key topological motifs with fundamental biological relevance are often hidden in the conformational noise, eluding detection. Here, we develop a circuit topology toolbox to extract conformational patterns, critical contacts, and timescales from simulated dynamics of intrinsically disordered proteins. We follow the dynamics of IDPs by providing a smart low-dimensionality representation of their three-dimensional (3D) configuration in the topology space. Such an approach allows us to quantify topological similarity in dynamic systems, therefore providing a pipeline for structural comparison of IDPs.

Amyloid assembly is dominated by misregistered kinetic traps on an unbiased energy landscape.

Atomistic description of protein fibril formation has been elusive due to the complexity and long time scales of the conformational search. Here, we develop a multiscale approach combining numerous atomistic simulations in explicit solvent to construct Markov State Models (MSMs) of fibril growth. The search for the in-register fully bound fibril state is modeled as a random walk on a rugged two-dimensional energy landscape defined by ß-sheet alignment and hydrogen-bonding states, whereas transitions involving states without hydrogen bonds are derived from kinetic clustering. The reversible association/dissociation of an incoming peptide and overall growth kinetics are then computed from MSM simulations. This approach is applied to derive a parameter-free, comprehensive description of fibril elongation of Aß16-22 and how it is modulated by phenylalanine-to-cyclohexylalanine (CHA) mutations. The trajectories show an aggregation mechanism in which the peptide spends most of its time trapped in misregistered ß-sheet states connected by weakly bound states twith short lifetimes. Our results recapitulate the experimental observation that mutants CHA19 and CHA1920 accelerate fibril elongation but have a relatively minor effect on the critical concentration for fibril growth. Importantly, the kinetic consequences of mutations arise from cumulative effects of perturbing the network of productive and nonproductive pathways of fibril growth. This is consistent with the expectation that nonfunctional states will not have evolved efficient folding pathways and, therefore, will require a random search of configuration space. This study highlights the importance of describing the complete energy landscape when studying the elongation mechanism and kinetics of protein fibrils.

Conformational entropy limits the transition from nucleation to elongation in amyloid aggregation.

The formation of ß-sheet-rich amyloid fibrils in Alzheimer's disease and other neurodegenerative disorders is limited by a slow nucleation event. To understand the initial formation of ß-sheets from disordered peptides, we used all-atom simulations to parameterize a lattice model that treats each amino acid as a binary variable with ß- and non-ß-sheet states. We show that translational and conformational entropy give the nascent ß-sheet an anisotropic surface tension that can be used to describe the nucleus with 2D classical nucleation theory. Since translational entropy depends on concentration, the aspect ratio of the critical ß-sheet changes with protein concentration. Our model explains the transition from the nucleation phase to elongation as the point where the ß-sheet core becomes large enough to overcome the conformational entropy cost to straighten the terminal molecule. At this point the ß-strands in the nucleus spontaneously elongate, which results in a larger binding surface to capture new molecules. These results suggest that nucleation is relatively insensitive to sequence differences in coaggregation experiments because the nucleus only involves a small portion of the peptide.

Low-dissipation self-assembly protocols of active sticky particles.

We use neuroevolutionary learning to identify time-dependent protocols for low-dissipation self-assembly in a model of generic active particles with interactions. When the time allotted for assembly is sufficiently long, low-dissipation protocols use only interparticle attractions, producing an amount of entropy that scales as the number of particles. When time is too short to allow assembly to proceed via diffusive motion, low-dissipation assembly protocols instead require particle self-propulsion, producing an amount of entropy that scales with the number of particles and the swim length required to cause assembly. Self-propulsion therefore provides an expensive but necessary mechanism for inducing assembly when time is of the essence.

Thermodynamics of Huntingtin Aggregation.

Amyloid aggregates are found in many neurodegenerative diseases, including Huntington's, Alzheimer's, and prion diseases. The precise role of the aggregates in disease progression has been difficult to elucidate because of the diversity of aggregated states they can adopt. Here, we study the formation of fibrils and oligomers by exon 1 of huntingtin protein. We show that the oligomer states are consistent with polymer micelles that are limited in size by the stretching entropy of the polyglutamine region. The model shows how the sequences flanking the amyloid core modulate aggregation behavior. The N17 region promotes aggregation through weakly attractive interactions, whereas the C38 tail opposes aggregation via steric repulsion. We also show that the energetics of cross-ß stacking by polyglutamine would produce fibrils with many alignment defects, but minor perturbations from the flanking sequences are sufficient to reduce the defects to the level observed in experiment. We conclude with a discussion of the implications of this model for other amyloid-forming molecules.

Protein Network Structure Enables Switching between Liquid and Gel States.

Biomolecular condensates are emerging as an important organizational principle within living cells. These condensed states are formed by phase separation, yet little is known about how material properties are encoded within the constituent molecules and how the specificity for being in different phases is established. Here we use analytic theory to explain the phase behavior of the cancer-related protein SPOP and its substrate DAXX. Binary mixtures of these molecules have a phase diagram that contains dilute liquid, dense liquid, and gel states. We show that these discrete phases appear due to a competition between SPOP-DAXX and DAXX-DAXX interactions. The stronger SPOP-DAXX interactions dominate at sub-stoichiometric DAXX concentrations leading to the formation of cross-linked gels. The theory shows that the driving force for gel formation is not the binding energy, but rather the entropy of distributing DAXX molecules on the binding sites. At high DAXX concentrations the SPOP-DAXX interactions saturate, which leads to the dissolution of the gel and the appearance of a liquid phase driven by weaker DAXX-DAXX interactions. This competition between interactions allows multiple dense phases to form in a narrow region of parameter space. We propose that the molecular architecture of phase-separating proteins governs the internal structure of dense phases, their material properties and their functions. Analytical theory can reveal these properties on the long length and time scales relevant to biomolecular condensates.

Effects of non-pairwise repulsion on nanoparticle assembly.

Electrostatic interactions provide a convenient way to modulate interactions between nanoparticles, colloids, and biomolecules because they can be adjusted by the solution pH or salt concentration. While the presence of salt provides an easy method to control the net interparticle interaction, the nonlinearities arising from electrostatic screening make it difficult to quantify the strength of the interaction. In particular, when charged particles assemble into clusters or aggregates, nonlinear effects render the interactions strongly non-pairwise. Here, we report Brownian dynamics simulations to investigate the effect that the non-pairwise nature of electrostatic interactions has on nanoparticle assembly. We compare these simulations to a system in which the electrostatics are modeled by a strictly pairwise Yukawa potential. We find that both systems show a narrow range in parameter space where the particles form well-ordered crystals. Bordering this range are regions where the net interactions are too weak to stabilize aggregated structures or strong enough that the system becomes kinetically trapped in a gel. The non-pairwise potential differs from the pairwise system in the appearance of an amorphous state for strongly charged particles. This state appears because the many-body electrostatic interactions limit the maximum density achievable in an assembly.

Ion Specificity and Nonmonotonic Protein Solubility from Salt Entropy.

The addition of salt to protein solutions can either increase or decrease the protein solubility, and the magnitude of this effect depends on the salt used. We show that these effects can be captured using a theory that includes attractive and repulsive electrostatic interactions, nonelectrostatic protein-ion interactions, and ion-solvent interactions via an effective solvated ion radius. We find that the ion radius has significant effects on the translational entropy of the salt, which leads to salt specificity in the protein solubility. At low salt, the dominant effect comes from the entropic cost of confining ions within the aggregate, whereas at high concentrations, the salt drives a depletion attraction that favors aggregation. Our theory explains the reversal in the Hofmeister series observed in lysozyme cloud point measurements and semi-quantitatively describes the solubility of lysozyme and chymosin crystals. We present a comparison of the contributions to the free energy and give guidelines for when salting in or salting out should be expected.

Theory of amyloid fibril nucleation from folded proteins.

We present a theoretical model for the nucleation of amyloid fibrils. In our model we use helix-coil theory to describe the equilibrium between a soluble native state and an aggregation-prone unfolded state. We then extend the theory to include oligomers with ß-sheet cores and calculate the free energy of these states using estimates for the energies of H-bonds, steric zipper interactions, and the conformational entropy cost of forming secondary structure. We find that states with fewer than ~10 ß-strands are unstable relative to the dissociated state and three ß-strands is the highest free energy state. We then use a modified version of Classical Nucleation Theory to compute the nucleation rate of fibrils from a supersaturated solution of monomers, dimers, and trimers. The nucleation rate has a non-monotonic dependence on denaturant concentration reflecting the competing effects of destabilizing the fibril and increasing the concentration of unfolded monomers. We estimate heterogeneous nucleation rates and discuss the application of our model to secondary nucleation.

Minimal physical requirements for crystal growth self-poisoning.

Self-poisoning is a kinetic trap that can impair or prevent crystal growth in a wide variety of physical settings. Here we use dynamic mean-field theory and computer simulation to argue that poisoning is ubiquitous because its emergence requires only the notion that a molecule can bind in two (or more) ways to a crystal; that those ways are not energetically equivalent; and that the associated binding events occur with sufficiently unequal probability. If these conditions are met then the steady-state growth rate is in general a non-monotonic function of the thermodynamic driving force for crystal growth, which is the characteristic of poisoning. Our results also indicate that relatively small changes of system parameters could be used to induce recovery from poisoning.

Stable, metastable, and kinetically trapped amyloid aggregate phases.

Self-assembly of proteins into amyloid fibrils plays a key role in a multitude of human disorders that range from Alzheimer's disease to type II diabetes. Compact oligomeric species, observed early during amyloid formation, are reported as the molecular entities responsible for the toxic effects of amyloid self-assembly. However, the relation between early-stage oligomeric aggregates and late-stage rigid fibrils, which are the hallmark structure of amyloid plaques, has remained unclear. We show that these different structures occupy well-defined regions in a peculiar phase diagram. Lysozyme amyloid oligomers and their curvilinear fibrils only form after they cross a salt and protein concentration-dependent threshold. We also determine a boundary for the onset of amyloid oligomer precipitation. The oligomeric aggregates are structurally distinct from rigid fibrils and are metastable against nucleation and growth of rigid fibrils. These experimentally determined boundaries match well with colloidal model predictions that account for salt-modulated charge repulsion. The model also incorporates the metastable and kinetic character of oligomer phases. Similarities and differences of amyloid oligomer assembly to metastable liquid-liquid phase separation of proteins and to surfactant aggregation are discussed.

Self-assembly at a nonequilibrium critical point.

We use analytic theory and computer simulation to study patterns formed during the growth of two-component assemblies in two and three dimensions. We show that these patterns undergo a nonequilibrium phase transition, at a particular growth rate, between mixed and demixed arrangements of component types. This finding suggests that principles of nonequilibrium statistical mechanics can be used to predict the outcome of multicomponent self-assembly, and suggests an experimental route to the self-assembly of multicomponent structures of a qualitatively defined nature.

Salting out the polar polymorph: analysis by alchemical solvent transformation.

We computationally examine how adding NaCl to an aqueous solution with α- and γ-glycine nuclei alters the structure and interfacial energy of the nuclei. The polar γ-glycine nucleus in pure aqueous solution develops a melted layer of amorphous glycine around the nucleus. When NaCl is added, a double layer is formed that stabilizes the polar glycine polymorph and eliminates the surface melted layer. In contrast, the non-polar α-glycine nucleus is largely unaffected by the addition of NaCl. To quantify the stabilizing effect of NaCl on γ-glycine nuclei, we alchemically transform the aqueous glycine solution into a brine solution of glycine. The alchemical transformation is performed both with and without a nucleus in solution and for nuclei of α-glycine and γ-glycine polymorphs. The calculations show that adding 80 mg/ml NaCl reduces the interfacial free energy of a γ-glycine nucleus by 7.7 mJ/m(2) and increases the interfacial free energy of an α-glycine nucleus by 3.1 mJ/m(2). Both results are consistent with experimental reports on nucleation rates which suggest: J(α, brine) < J(γ, brine) < J(α, water). For γ-glycine nuclei, Debye-Hückel theory qualitatively, but not quantitatively, captures the effect of salt addition. Only the alchemical solvent transformation approach can predict the results for both polar and non-polar polymorphs. The results suggest a general "salting out" strategy for obtaining polar polymorphs and also a general approach to computationally estimate the effects of solvent additives on interfacial free energies for nucleation.

Physical limits of cells and proteomes.

What are the physical limits to cell behavior? Often, the physical limitations can be dominated by the proteome, the cell's complement of proteins. We combine known protein sizes, stabilities, and rates of folding and diffusion, with the known protein-length distributions P(N) of proteomes (Escherichia coli, yeast, and worm), to formulate distributions and scaling relationships in order to address questions of cell physics. Why do mesophilic cells die around 50 °C? How can the maximal growth-rate temperature (around 37 °C) occur so close to the cell-death temperature? The model shows that the cell's death temperature coincides with a denaturation catastrophe of its proteome. The reason cells can function so well just a few degrees below their death temperature is because proteome denaturation is so cooperative. Why are cells so dense-packed with protein molecules (about 20% by volume)? Cells are packed at a density that maximizes biochemical reaction rates. At lower densities, proteins collide too rarely. At higher densities, proteins diffuse too slowly through the crowded cell. What limits cell sizes and growth rates? Cell growth is limited by rates of protein synthesis, by the folding rates of its slowest proteins, and--for large cells--by the rates of its protein diffusion. Useful insights into cell physics may be obtainable from scaling laws that encapsulate information from protein knowledge bases.