Living systems must maintain robust biochemical function despite fluctuations that span a wide range of timescales. Biomolecular condensates formed by liquid-liquid phase separation (LLPS) have been shown to buffer concentration fluctuations, but the principles governing their dynamic regulation remain unclear. We address this by probing the response of LLPS to oscillatory perturbations that mimic fluctuations across different timescales, establishing the first systematic frequency-domain analysis of concentration buffering by condensates. We find that condensates act as frequency-selective filters: The perturbed dilute phase behaves as a high-pass filter, while the dense phase attenuates both low- and high-frequency perturbations. We establish quantitative links between LLPS parameters, including interaction strength, droplet size, and interphase exchange rates, and the timescale range over which condensates effectively buffer concentration fluctuations. These findings establish fundamental dynamical limits of concentration buffering by LLPS, with implications for how cells may use LLPS to adapt to fluctuating environments and for the design of synthetic condensates with programmable control properties.

Many biological functions and dysfunctions rely on two fundamental processes, molecular assembly and the formation of condensed phases such as biomolecular condensates. Condensed phases generally form via phase separation, while molecular assemblies are clusters of molecules of various sizes, shapes, and functionality. We developed a theory that relies on thermodynamic principles to understand the interplay between molecular assembly and phase separation. We propose two prototypical classes of protein interactions and characterize their different equilibrium states and relaxation dynamics. We obtain results consistent with recent in vitro experimental observations of reconstituted proteins, including anomalous size distribution of assemblies, the gelation of condensed phases, and the change in condensate volume during ageing. Our theory provides the framework to unravel the mechanisms underlying physiological assemblies essential for cellular function, and aberrant assemblies which are associated with several neurodegenerative disorders.

Electrostatically driven liquid-liquid phase separation underlies complex coacervation in solutions of oppositely charged macromolecules and plays a central role in the phase behavior of charged polymers such as nucleic acids and intrinsically disordered proteins. The Voorn-Overbeek model provides a minimal mean-field description of this phenomenon by combining polymer mixing entropy with electrostatic interactions captured at the Debye-Hückel level. Despite its long-standing importance, the Voorn-Overbeek theory does not admit closed-form analytical solutions for phase coexistence, and its phase behavior has, therefore, been studied primarily using numerical approaches or near-critical expansions. Here, we derive a self-consistent analytical solution for the binodal concentrations of the simplest Voorn-Overbeek model, describing two oppositely charged polymer species in a neutral solvent under local electroneutrality. By reformulating the coexistence conditions as a fixed-point problem, we obtain explicit analytical expressions for the phase boundaries that remain accurate across the entire phase-separated regime. These results establish an analytically tractable framework for complex coacervation and offer a foundation for future extensions incorporating additional electrostatic and compositional effects.

In the bacterial Csr/Rsm system, non-coding RNAs activate mRNA translation by removing homodimeric Csr/Rsm proteins from ribosome-binding sites of mRNAs. In Pseudomonas protegens, each RsmZ ncRNA sequesters up to five RsmE dimers sequentially and specifically within a narrow affinity range, functioning as a 'protein sponge'. Although the RsmE binding cascade is cooperative, binding of the highest affinity stem-loop RNA in RsmZ (SL2) reduces RNA binding affinity at the second site 10- to 30-fold. This unusual negative cooperativity may facilitate RsmE release from tightly bound mRNA for handover to the non-coding RNA, yet the underlying mechanisms remain unclear. Using Isothermal Titration Calorimetry, NMR spectroscopy and Molecular Dynamics, we reveal an allosteric mechanism resembling a Newton's cradle, coupling the binding at one site to conformational and dynamic changes at the second site, explaining the reduced affinity of the second binding event, and handover of RsmE dimer from mRNA to the ncRNA RsmZ.

The aggregation of the 42-residue form of the amyloid-β peptide (Aβ42) is important in Alzheimer's disease (AD). Preclinical and clinical findings support that glucagon-like peptide-1 receptor agonists (GLP-1RAs) can protect against neuroinflammation and neurodegeneration with potential therapeutic relevance for AD, but studies of their direct effects on Aβ42 are limited. Herein, we investigated five FDA-approved GLP-1RAs, and show semaglutide, tirzepatide, and liraglutide inhibit Aβ42 aggregation. Semaglutide and tirzepatide delayed Aβ42 aggregation by targeting the primary nucleation microscopic step, with submicromolar IC50 values for primary nucleation (KIP). Liraglutide was highly effective at suppressing primary nucleation with a very low KIP value, and it demonstrated an additional modest inhibition of secondary nucleation. Consistent with a dominant effect on primary nucleation, Aβ42 formed β-sheet-rich fibrils in the presence of these GLP-1RAs. Aβ42 fibrils formed with semaglutide or tirzepatide had morphological properties and templating efficiencies that were similar to unmodified fibrils, while liraglutide significantly reduced fibril maturity, increased fibril tortuosity and length, and attenuated the ability of fibrils to passively self-replicate whether they were formed in the presence of liraglutide or exposed to this GLP-1RA after their formation. These results provide molecular-level insight into how specific GLP-1RAs can selectively target the fundamental steps governing toxic Aβ42 aggregation. Further studies are warranted to determine if current or next-generation anti-amyloid GLP-1RAs can delay or prevent AD through multifaceted protective mechanisms, including the direct inhibition of Aβ42 aggregation.

Formation of new amyloid fibrils and oligomers from monomeric protein on the surfaces of existing fibrils is an important driver of many disorders such as Alzheimer's and Parkinson's diseases. The structural basis of this secondary nucleation process, however, is poorly understood. Here, we ask whether secondary nucleation sites are found predominantly at rare growth defects: irregularities in the fibril core structure incorporated during their original assembly. We first demonstrate using the specific inhibitor of secondary nucleation, Brichos, that secondary nucleation sites on Alzheimer's disease-associated fibrils composed of Aβ40 and Aβ42 peptides are rare compared to the number of protein molecules they contain. We then grow Aβ40 fibrils under conditions designed to eliminate most growth defects while leaving the regular fibril morphology unchanged, and confirm the latter using cryo-electron microscopy. We measure both the ability of these annealed fibrils to promote secondary nucleation and the stoichiometry of their secondary nucleation sites, finding that both are greatly reduced as predicted. Re-analysis of published data for other proteins suggests that fibril growth defects may also drive secondary nucleation generally across most amyloids. These findings could unlock structure-based drug design of therapeutics that aim to halt amyloid disorders by inhibiting secondary nucleation sites.

The interface of biomolecular condensates plays a critical role in regulating many biochemical processes, such as protein aggregation and enzymatic activity. Targeted modulation of these interfaces offers a promising route for engineering condensates and correcting aberrant behaviors. However, the physical principles underlying the preferential localization of molecules at condensate interfaces are poorly understood. In this study, we develop an analytical theory for the enrichment of client molecules at the interface of phase-separated systems. We derive minimal conditions for interfacial localization based on pairwise interaction parameters. Our results reveal a general mechanism for passive interface-driven enrichment, which could guide the design of macromolecules that localize at the interface of biomolecular condensates.

Alzheimer’s disease (AD) is marked by the abnormal aggregation of amyloid-beta peptides within the central nervous system. The formation of amyloid fibrils from amyloid-beta peptides is a hallmark of AD Here, we demonstrate that the aggregation of amyloid-beta 42 spreads both spatially and temporally. By measuring the spatial propagation of amyloid-beta in macroscopic capillaries and performing Monte Carlo simulations, we show that this spreading occurs through a diffusion mechanism involving oligomers in solution. These species, catalytically produced through spontaneous secondary nucleation, significantly accelerate the propagation velocity of the reaction wavefront. Our findings suggest that, in addition to their potential role in toxicity, these oligomers in solution are key drivers of the spatial spreading of aggregation and can therefore be considered key targets for therapeutic intervention. Amyloid fibril formation from amyloid-beta peptides is a key feature of Alzheimer’s disease, yet the mechanisms of its spatial and temporal spread remain unclear. Here, the authors reveal that amyloid-beta 42 aggregation propagates via diffusion of oligomers in solution, highlighting these species as key drivers of the spatial spreading of aggregation.

Enzymatic reactions in biomolecular condensates are often assumed to be regulated through local enrichment of reactants. However, condensates also reshape molecular transport and reaction kinetics, making it unclear how phase separation controls catalysis in living cells. Here, we develop a quantitative theory of biomolecular catalysis in phase-separated systems and find that liquid condensates can act as tunable catalytic switches, transitioning between regimes of enhanced and suppressed enzymatic activity, exhibiting optimal responses at biologically relevant condensate sizes. We show that condensate-mediated catalysis cannot be understood from reactant enrichment alone, but instead emerges from the coupled interplay of molecular partitioning, diffusive transport, and phase-dependent reaction kinetics. The strongest regulatory effects occur under rapid interphase exchange, where the spatially heterogeneous catalytic network admits a system-level Michaelis–Menten description governed by system-averaged concentrations and reaction kinetics. Our framework predicts that micron-sized condensates can either enhance or suppress enzymatic activity by up to two orders of magnitude, and that optimal catalytic regulation can emerge at condensate sizes comparable to many biomolecular condensates. These results provide experimentally testable predictions for condensate-mediated catalysis and establish quantitative principles for understanding and engineering enzyme-catalysed reactions in biomolecular condensates.

The onset and development of Alzheimer’s disease is linked to the accumulation of pathological aggregates formed from the normally monomeric amyloid-β peptide within the central nervous system. These Aβ aggregates are increasingly successfully targeted with clinical therapies at later stages of the disease, but the fundamental molecular steps in early stage disease that trigger the initial nucleation event leading to the conversion of monomeric Aβ peptide into pathological aggregates remain unknown. Here, we show that the Aβ peptide can form biomolecular condensates on lipid bilayers both in molecular assays and in living cells. Our results reveal that these Aβ condensates can significantly accelerate the primary nucleation step in the amyloid conversion cascade that leads to the formation of amyloid aggregates. We show that Aβ condensates contain phospholipids, are intrinsically heterogeneous, and are prone to undergo a liquid-to-solid transition leading to the formation of amyloid fibrils. These findings uncover the liquid–liquid phase separation behavior of the Aβ peptide and reveal a molecular step very early in the amyloid-β aggregation process.

The aggregation of proteins into amyloid fibrils is a hallmark of several neurodegenerative disorders, including Parkinson's disease. A growing body of experimental evidence highlights the significant role lipid membranes play in modulating this aggregation process, particularly for proteins such as α-synuclein. Despite this, there has been a lack of quantitative theoretical frameworks capable of describing the kinetics of lipid-induced protein aggregation. In this work, we develop an analytical theoretical model that explicitly incorporates lipid-mediated interactions into the aggregation kinetics. By formulating rate equations in terms of lipid surface coverage and applying a fixed-point analysis, we derive self-consistent solutions for the full timecourse of aggregation. Our model captures both one-step and two-step nucleation mechanisms and enables the prediction of key kinetic observables, including half-times and maximal growth rates. These results provide a quantitative foundation for interpreting experimental data and offer new mechanistic insights into how lipids influence the self-assembly of amyloidogenic proteins.

Parkinson's disease (PD) is a neurological disorder characterized by neuronal loss and the deposition of α-synuclein-lipid coaggregates in the brain of patients as well as disruptions in lipid metabolism. Mutations in the gene GBA, which encodes the lysosomal glycoprotein Glucocerebrosidase, are together the most important genetic risk factor for PD and have been associated with lysosomal dysfunction, accumulation of pathological α-synuclein as well as major changes in both the levels and properties of lipids. Ambroxol, a small molecule chaperone capable of binding and stabilizing Glucocerebrosidase, was found to revert changes in lipid levels and increase in α-synuclein levels due to GBA mutations potentially via restoring lysosomal function. Here, we show that Ambroxol also has a direct effect on α-synuclein-lipid coaggregation by inhibiting the primary nucleation step in the aggregation process. We find that Ambroxol not only displaces α-synuclein from negatively charged membranes but also prevents the formation of early α-synuclein-lipid coaggregates during primary nucleation. These results suggest that Ambroxol may have beneficial effects on other synucleinopathies, such as multiple system atrophy and dementia with Lewy Bodies, that are also characterised by the aggregation of α-synuclein into amyloid fibrils.

Analyzing kinetic experiments on protein aggregation using integrated rate laws has led to numerous advances in our understanding of the fundamental chemical mechanisms behind amyloidogenic disorders such as Alzheimer's and Parkinson's diseases. However, the description of biologically relevant processes may require rate equations that are too complex to solve using existing methods, hindering mechanistic insights into these processes. An example of significance is coaggregation in environments containing multiple amyloid-beta (Aβ) peptide alloforms, which may play a crucial role in the biochemistry of Alzheimer's disease but whose mechanism is still poorly understood. Here, we use the mathematics of Lie symmetry to derive a general integrated rate law valid for most plausible linear self-assembly reactions. We use it in conjunction with experimental data to determine the mechanism of coaggregation of the most physiologically abundant Aβ alloforms: Aβ42, Aβ40, Aβ38 and Aβ37 peptides. We find that Aβ42 fibril surfaces catalyze the formation of co-oligomers, which accelerate new Aβ40, Aβ38 and Aβ37 fibril formation whilst inhibiting secondary nucleation of new Aβ42 fibrils. The simplicity, accuracy and broad applicability of our general integrated rate law will enable kinetic analysis of more complex filamentous self-assembly reactions, both with and without coaggregation.

Biomolecular condensates are increasingly recognized as pivotal regulators of cellular physiology, yet their pathological aging underlies numerous diseases. However, the mechanisms governing this condensate aging remain largely elusive. Liquid-liquid phase separation of minimal biomolecular building blocks, such as small peptides or nucleotides, offers ideal model systems to dissect the mechanisms underlying condensate formation and aging. Here, we report that condensates formed by Adenosine 5’-triphosphate (ATP) undergo liquid-to-solid phase transitions (LSPT) in macromolecularly crowded environments, evolving from dynamic liquid droplets into urchin-like fibrillar aggregates. In the initial stages of aging, small aggregates actively engulf surrounding droplets via direct contact and rapid wetting-driven merging. As aging progresses, internal flows emerge within condensates, with velocities oriented toward the nearest aggregate core. These flows arise endogenously from ATP fibrillization through the Marangoni effect and result in a long-range chemotaxis ripening process that facilitates transport of ATP molecules from peripheral droplets to a central aggregate. The Marangoni effect also drives the long-range motion of liquid droplets on hydrophobic surfaces towards the aggregates, representing a novel form of dialytaxis. These findings provide crucial and previously unrecognized dynamic behaviors and insights into the physical principles underlying condensate aging. ### Competing Interest Statement Ho Cheung Shum is a scientific advisor of EN Technology Limited, MicroDiagnostics Limited, PharmaEase Tech Limited, Upgrade Biopolymers Limited and Multera Limited, in which he owns some equity, and is a founding director and co-director of the research center, Advanced Biomedical Instrumentation Centre Limited. The works in this paper are, however, not directly related to the works of these entities, as far as we know. The authors declare no other competing interests. Research Grants Council, Nos. 17306221, 17317322, and C7165-20GF Swiss National Science Foundation, SNF 219703

Low-molecular-weight oligomers formed from amyloidogenic peptides and proteins have been identified as key cytotoxins across a range of neurodegenerative disorders, including Alzheimer's disease and Parkinson's disease. Developing therapeutic strategies that target oligomers is therefore emerging as a promising approach for combating protein misfolding diseases. As such, there is a great need to understand the fundamental properties, dynamics, and mechanisms associated with oligomer formation. In this review, we discuss how chemical kinetics provides a powerful tool for studying these systems. We review the chemical kinetics approach to determining the underlying molecular pathways of protein aggregation and discuss its applications to oligomer formation and dynamics. We discuss how this approach can reveal detailed mechanisms of primary and secondary oligomer formation, including the role of interfaces in these processes. We further use this framework to describe the processes of oligomer conversion and dissociation, and highlight the distinction between on-pathway and off-pathway oligomers. Furthermore, we showcase on the basis of experimental data the diversity of pathways leading to oligomer formation in various in vitro and in silico systems. Finally, using the lens of the chemical kinetics framework, we look at the current oligomer inhibitor strategies both in vitro and in vivo.

Information theory has long been integrated into the study of biological ageing, for example, in examining the roles of genetic and epigenetic fidelity in cellular and organismal longevity. Here, we introduce a theoretical model that interprets ageing in multicellular systems through the lens of Fisher information. Previous theories have suggested that the ageing of multicellular organisms is an inevitable consequence of the inherent tension between individual cell reproduction and the homeostasis of the multicellular system. Utilising concepts from information theory and statistical mechanics, we show that Fisher information parametrises the dynamics of this tension through non-monotonic behaviour, which depends on an optimal balance of competition and cooperation between cells. Moreover, Fisher information suggests that the ability to infer true biological age from a sample evolves through complex dynamics over an organism’s lifespan.

The aggregation of α -synuclein into amyloid fibrils is a hallmark of Parkinson’s disease. This process has been shown to directly involve interactions between proteins and lipid surfaces when the latter are present. Despite this importance, the molecular mechanisms of lipid-induced amyloid aggregation have remained largely elusive. Here, we present a global kinetic model to describe lipid-induced amyloid aggregation of α -synuclein. Using this framework, we find that α -synuclein fibrils form via a two-step primary nucleation mechanism and that lipid molecules are directly involved in both the nucleation and fibril elongation steps, giving rise to lipid–protein coaggregates. To illustrate the applicability of this kinetic approach to drug discovery, we identify the mechanism of action of squalamine, a known inhibitor of lipid-induced α -synuclein aggregation, revealing that this small molecule reduces the rate of lipid-dependent primary nucleation. Our work will likely guide the rational design of α -synuclein aggregation inhibitors.

As the largest organelle, the nucleus endures significant mechanical stresses over the cellular lifespan, and mechanostability – the ability to resist deformation – is critical for genome integrity and function. Here, we reveal that nuclear mechanostability is an emergent property arising from the coalescence of satellite DNA repeats into nuclear condensates known as chromocenters. Targeted chromocenter disruption in Drosophila testes subjected to natural and artificial mechanical stress compromises nuclear mechanostability, leading to deformed nuclei, DNA damage, and chromosome breaks. Conversely, enhancing chromocenter formation through genetic means improves mechanostability. Molecular dynamics simulations suggest that chromocenters enable physically linked chromosomes to respond collectively, rather than individually, to mechanical challenge, and dissipate external forces over a larger nuclear surface. We propose that the satellite DNA-dependent mechanostability framework described here likely extends to other cells and tissues facing mechanical stress, and offers an explanation for the evolutionary success of these non-coding repeats across eukaryotes. ### Competing Interest Statement The authors have declared no competing interest. Swiss National Science Foundation, https://ror.org/00yjd3n13, 310030_189131, 320030_228043, 310030_207588, 310030_205199 European Research Council, 101018257, 101002094 Dutch Research Council, OCENW.KLEIN.200

Phase transitions are important to understand cell dynamics, and the maturation of liquid droplets is relevant to neurodegenerative disorders. We combined NMR and Raman spectroscopies with microscopy to follow, over a period of days to months, droplet maturation of the protein fused in sarcoma (FUS). Our study reveals that the surface of the droplets plays a critical role in this process, while RNA binding prevents it. The maturation kinetics are faster in an agarose-stabilized biphasic sample compared with a monophasic condensed sample, owing to the larger surface-to-volume ratio. In addition, Raman spectroscopy reports structural differences upon maturation between the inside and the surface of droplets, which is comprised of β-sheet content, as revealed by solid-state NMR. In agreement with these observations, a solid crust-like shell is observed at the surface using microaspiration. Ultimately, matured droplets were converted into fibrils involving the prion-like domain as well as the first RGG motif.

Self-replication of amyloid fibrils via secondary nucleation is an intriguing physicochemical phenomenon in which existing fibrils catalyze the formation of their own copies. The molecular events behind this fibril surface-mediated process remain largely inaccessible to current structural and imaging techniques. Using statistical mechanics, computer modeling, and chemical kinetics, we show that the catalytic structure of the fibril surface can be inferred from the aggregation behavior in the presence and absence of a fibril-binding inhibitor. We apply our approach to the case of Alzheimer’s A β 42 amyloid fibrils formed in the presence of proSP-C Brichos inhibitors. We find that self-replication of A β 42 fibrils occurs on small catalytic sites on the fibril surface, which are far apart from each other, and each of which can be covered by a single Brichos inhibitor.