Biomolecular condensates are membraneless compartments that organize biochemical processes in cells. In contrast to well-understood mechanisms describing how condensates form and dissolve, the principles underlying condensate patterning—including their size, number, and spacing in the cell—remain largely unknown. We hypothesized that RNA, a key regulator of condensate formation and dissolution, influences condensate patterning. Using nucleolar fibrillar centers (FCs) as a model condensate, we found that inhibiting ribosomal RNA synthesis significantly alters the patterning of FCs. Physical theory and experimental observations support a model whereby active RNA synthesis generates a non-equilibrium state that arrests condensate coarsening and thus contributes to condensate patterning. Altering FC condensate patterning by expression of the FC component treacle ribosome biogenesis factor 1 (TCOF1) impairs ribosomal RNA processing, linking condensate patterning to biological function. These results reveal how non-equilibrium states driven by active chemical processes regulate condensate patterning, which is important for cellular biochemistry and function.

We numerically revisit the transfer-matrix formulation of directed polymers in random media and show that a common finite-dimensional framework organizes the canonical one-point fluctuation laws in ( 1 + 1 ) dimensions. For a fixed realization of the bulk disorder, full-space partition functions are obtained from the same time-ordered product W ( t ) through endpoint contractions or a Brownian-weighted initial vector, while the half-space construction modifies only the transfer rule at the absorbing boundary. These choices yield distributions consistent with the standard KPZ subclasses: Tracy-Widom GUE for point-to-point geometry, Tracy-Widom GOE for point-to-line geometry, Tracy-Widom GSE for half-space point-to-point geometry, and Baik-Rains for the stationary line-to-point construction. In all four cases, the free-energy fluctuations grow as t 1 / 3 , and the low-order cumulants approach the corresponding universal benchmarks. The matrix-product formulation also provides access to intrinsic spectral observables. For the leading eigenvalue λ 1 ( t ) , the fluctuations of ln λ 1 ( t ) exhibit an intermediate t 1 / 3 regime, while the standardized distribution remains distinct from the canonical benchmark laws over the studied time range.

We study correlation functions in generic non-equilibrium mixtures, including multi-temperature systems and non-reciprocal field theories. The corresponding linear theory is short-ranged, and nonlinearities are irrelevant in the renormalization-group sense. Nonetheless, we find that these nonlinearities generate long-ranged three-point correlations in the isotropic disordered phase. Our analytical predictions, which are based on a phenomenological theory, are confirmed by numerical simulations of Brownian colloids in contact with thermal baths at different temperatures. Dangerously irrelevant nonlinearities in non-equilibrium mixtures thus offer a new route to long-range correlations, supporting the hypothesis that such correlations are not the exception but the rule out of equilibrium.

A finite density of mutually avoiding directed polymers in a quenched random medium is a minimal model of glassy line matter. The dilute theory, solved by replica Bethe ansatz, predicts an interaction free energy proportional to ρ^2 and disorder cumulants with distinct power-law dependences on the density ρ, but direct numerical tests have been hindered by the combinatorially large many-polymer transfer matrix. We recast the problem as filling logarithmic eigenvalues of a single-polymer transfer-matrix product, obtaining the quenched free energy, its cumulants, and a disorder-induced linear spectral edge consistent with the replica prediction.

When competing species grow into new territory, the population is dominated by descendants of successful ancestors at the expansion front. Successful ancestry depends on both the reproductive advantage (fitness), as well as ability and opportunity to colonize new domains. We present a model that integrates both elements by coupling the classic description of one-dimensional competition (Fisher equation) to the minimal model of front shape (KPZ equation). Macroscopic manifestations of these equations are distinct growth morphologies controlled by expansion rates, competitive abilities, or spatial anisotropy. In some cases the ability to expand in space may overcome reproductive advantage in colonizing new territory. When new traits appear with accumulating mutations, we find that variations in fitness in range expansion may be described by the Tracy–Widom distribution.

Classical ecological models predict that large, diverse communities should be unstable, presenting a central challenge to explaining the stable biodiversity seen in nature. We revisit this long-standing problem by extending the generalized Lotka-Volterra model to include both spatial structure and environmental fluctuations across space and time. We find that neither space nor environmental noise alone can resolve the tension between diversity and stability, but that their combined effects permit arbitrarily many species to stably coexist despite strongly disordered competitive interactions. We analytically characterize the noise-induced transition to coexistence, showing that spatiotemporal noise drives an anomalous scaling of abundance fluctuations, known empirically as Taylor's law. At the community level, this manifests as an effective sublinear self-inhibition that renders the community stable and asymptotically neutral in the high-diversity limit. Spatiotemporal noise thus provides a novel resolution to the diversity-stability paradox and a generic mechanism by which complex communities can persist.

Electromagnetic fluctuations can transfer not only energy but also angular momentum, leading to forces, torques, heat currents, and friction in out-of-equilibrium setups. In enclosing configurations, we show that if at least one of two objects is rotationally symmetric, the torque is bounded by heat transfer, since both arise from photon transfers with angular momentum ħ n and energy . With only one object assumed to be rotationally symmetric, it may be possible to obtain a nonzero torque with reciprocal media, but nonreciprocal media are required to break the symmetry between n and -n and produce a nonzero torque if both objects are rotationally symmetric. We then specialize to concentric cylinders with a nonreciprocal dielectric response and use Rytov fluctuational electrodynamics to express heat transfer and torque in terms of an angular-momentum-resolved flux density, Φ_n(ω). We also analyze the conditions for stable levitation of the inner cylinder using the proximity force approximation, in the process obtaining a new analytic formula for the normal Casimir force between dilute plates at different temperatures. Finally, to find the extracted work in a contactless engine setup, we compute the fluctuation-induced friction for a slowly rotating inner cylinder, and we find a bound between torque, friction, and heat transfer. Due to this bound, the efficiency of the heat engine remains bounded by the Carnot limit.

We describe a contactless heat engine in which quantum and thermal electromagnetic fluctuations act as the working medium. The setup consists of two concentric cylinders held at different temperatures. The inner cylinder stably levitates within the outer one due to repulsive nonequilibrium Casimir forces. The chirality of the setup is broken by using nonreciprocal dielectric materials, akin to application of a magnetic field along the common cylinder axis. Using Rytov fluctuational electrodynamics, we show that heat transfer and torque can be expressed in terms of an angular-momentum-resolved heat flux density, Φn(ω): each exchanged photon carries energy ℏω and angular momentum ℏn. In reciprocal media, contributions from modes n and -n cancel and there is no net torque; nonreciprocity breaks this symmetry and powers rotation of the inner cylinder. Even in the absence of contact, electromagnetic fluctuations produce a frictional torque opposing rotation that we compute. This enables computation of characteristic steady state rotations, and estimation of the engine efficiency (which remains bounded by the Carnot limit). The cylindrical setup provides a natural realization of fluctuation-induced angular-momentum transfer and a possible route toward nanoscale contactless engines.

In complex ecological communities, species may self-organize into clusters or clumps where highly similar species can coexist. The emergence of such species clusters can be captured by the interplay between neutral and niche theories. Based on the generalized Lotka-Volterra model of competition, we propose a minimal model for ecological communities in which the steady states contain self-organized clusters. In this model, species compete only with their neighbors in niche space through a common interaction strength. Unlike many previous theories, this model does not rely on random heterogeneity in interactions. Even in this minimal model where only the common interaction strength is varied, we find an exponentially large set of states that exhibit a rich variety of cluster patterns with different sizes and combinations. There are sharp phase transitions into the formation of clusters. There are also multiple phase transitions between different sets of possible cluster patterns, many of which accumulate near a small number of critical points. We analyze this phase structure using both numerical and analytical methods. In addition, the special case with only nearest neighbor interactions is exactly solvable using the method of transfer matrices from statistical mechanics. We analyze the critical behavior of these systems.

We revisit the transfer-matrix approach to directed polymers in random media and show that a single ensemble of random transfer-matrix products provides a unified realization of the canonical one-point fluctuation laws in (1+1) dimensions. For a fixed disorder realization, the polymer partition function is obtained as a contraction of the same product matrix W(t), and different contractions reproduce the standard KPZ subclasses: Tracy-Widom GUE (point-to-point), GOE (point-to-line), GSE (half-space point-to-point), and Baik-Rains (stationary line-to-point). In each case, we observe t^1/3 free-energy fluctuation growth and convergence of standardized distributions with low-order cumulants close to the corresponding universal benchmarks. Viewing geometry-dependent subclasses as projections of a single matrix-product ensemble naturally suggests additional observables intrinsic to W(t). As an example, we examine the leading eigenvalue λ_1(t) whose logarithm exhibits t^1/3 scaling, while its standardized statistics remain distinct from the canonical Tracy-Widom laws within the accessible range. This transfer-matrix perspective thus organizes known KPZ one-point subclasses within a finite-dimensional matrix framework and highlights matrix-level fluctuation observables beyond geometry-selected universality classes.

The formation of condensates by the Linker for the Activation of T-cells (LAT) is a key signal gating and amplification step in the T-cell receptor signaling pathway. LAT condensation is challenging to study in-vivo and is therefore often investigated using reconstitution experiments. While these experiments recapitulate key aspects of LAT condensation, they also exhibit some puzzling features. Here, we describe the mechanisms underlying these observations using two complementary models. First, we employ a Smoluchowski aggregation model to show that the delay time before condensation is observed arises from a low effective binding probability between LAT monomers. Second, we propose a field-theoretic model that reproduces all condensate morphologies observed in experiments, showing that they can arise from common underlying dynamics modulated by variations in experimental conditions. This result unifies different experimental observations reported previously. While this article addresses open questions regarding the formation of LAT condensates, our results also provide a common framework for understanding condensation of other multivalent membrane proteins such as EGFR, FGFR2, and nephrin.

Competition during range expansions is of great interest from both practical and theoretical viewpoints. Experimentally, range expansions are often studied in homogeneous Petri dishes, which lack spatial anisotropy that might be present in realistic populations. Here, we analyze a model of anisotropic growth, based on coupled Kardar-Parisi-Zhang and Fisher-Kolmogorov-Petrovsky-Piskunov equations that describe surface growth and lateral competition. The anisotropy is encoded in how strongly genetic boundaries between strains are moved as a result of the expansion front morphology. We completely characterize spatial patterns and invasion velocities in this generalized model. In particular, we find that strong anisotropy results in a distinct morphology of spatial invasion with a kink in the displaced strain ahead of the boundary between the strains. This morphology of the outcompeted strain is similar to a shock wave and serves as a signature of anisotropic growth. We confirm these predictions with a commonly employed reaction-diffusion model of anisotropic growth.

We demonstrate how center-of-mass (COM) motion influences polymer segment fluctuations. Cancellation of internal forces, together with spatially uncorrelated external noise, generally yields COM diffusivity scaling as $1/s$ with segment length $s$, regardless of fractal dimension, viscoelasticity, or activity. This introduces distinct dynamic scaling corrections to two-point fluctuations and quenched-induced tangential correlations, validated by theory, simulations, and chromatin imaging data. In the latter, the extracted dynamic exponent reveals topological constraints, thereby resolving the discrepancy between chromatin's crumpled structure and its Rouse-like dynamics.

The formation of condensates by the Linker for the Activation of T-cells (LAT) is a key signal gating and amplification step in the T-cell receptor signaling pathway. LAT condensation is challenging to study in-vivo and is therefore often investigated using reconstitution experiments. While these experiments recapitulate key aspects of LAT condensation, they also exhibit some puzzling features. Here, we describe the mechanisms underlying these observations using two complementary models. First, we employ a Smoluchowski aggregation model to show that the delay time before condensation is observed arises from a low effective binding probability between LAT monomers. Second, we propose a field-theoretic model that reproduces all condensate morphologies observed in experiments, showing that they can arise from common underlying dynamics modulated by variations in experimental conditions. This result unifies different experimental observations reported previously. While this article addresses open questions regarding the formation of LAT condensates, our results also provide a common framework for understanding condensation of other multivalent membrane proteins such as EGFR, FGFR2, and nephrin.

We show that cancellation of internal forces together with spatially uncorrelated noise generically enforces a universal 1/s scaling of the center-of-mass (COM) diffusivity for polymer segments of contour length s, independent of model details. Within an active generalized Rouse-Langevin (A-GRLE) framework for fractal polymers in a viscoelastic bath with weak activity, we derive closed-form links between segment COM motion and the short-time behavior of two-point fluctuations. The same COM law predicts transient s^{−3} tangent-tangent correlations along the contour following a temperature quench. Reanalyzing recent two-locus chromatin tracking, we extract a dynamic COM exponent ξ=0.77±0.16, suggesting topological constraints. These results identify model-independent signatures of collective polymer dynamics and clarify that the observed Rouse-like scaling of chromatin loci reflects transient, unentangled dynamics with an inevitable crossover to an entangled regime at longer times.

We consider inhomogeneous polymers driven by energy-consuming active processes which encode temporal patterns of athermal kicks. We find that such temporal excitation programs, propagated by tension along the polymer, can effectively couple distinct polymer loci. Consequently, distant loci exhibit correlated motions that fold the polymer into specific conformations, as set by the local actions of the active processes and their distribution along the polymer. Interestingly, active kicks that are canceled out by a time-delayed echo can induce strong compaction of the active polymer.

A persistent public health challenge is finding immunization schemes that are effective in combating highly mutable pathogens such as HIV and influenza viruses. To address this, we analyze a simplified model of affinity maturation, the Darwinian evolutionary process B cells undergo during immunization. The vaccination protocol dictates selection forces that steer affinity maturation to generate antibodies. We focus on determining the optimal selection forces exerted by a generic time-dependent vaccination protocol to maximize production of broadly neutralizing antibodies (bnAbs) that can protect against a broad spectrum of pathogen strains. The model lends itself to a path integral representation and operator approximations within a mean-field limit, providing guiding principles for optimizing time-dependent vaccine-induced selection forces to enhance bnAb generation. We compare our analytical mean-field results with the outcomes of stochastic simulations and discuss their similarities and differences.

Kinetic traps are a notorious problem in equilibrium statistical mechanics, where temperature quenches ultimately fail to bring the system to low energy configurations. Using multifarious self-assembly as a model system, we introduce a mechanism to escape kinetic traps by utilizing nonreciprocal interactions between components. Introducing nonequilibrium effects offered by broken action-reaction symmetry in the system pushes the trajectory of the system out of arrested dynamics. The dynamics of the model is studied using tools from the physics of interfaces and defects. Our proposal can find applications in self-assembly, glassy systems, and systems with arrested dynamics to facilitate escape from local minima in rough energy landscapes.