PRX Life

Quantifying how cooperative, many-body relationships drive allostery in protein networks remains a major challenge. To address this, we develop the Laplacian minor hierarchy, a mathematical framework that characterizes the geometric invariants of a protein network. Lower-order minors yield standard metrics including the partition function and effective distances, whereas higher-order minors define novel topological measures: cooperation indices, each bounded between zero and one, that characterize pathway correlations at increasing levels of complexity, the third-order minor determines whether allosteric pathways are correlated or uncorrelated, and the fourth-order minor quantifies how distinct pathways communicate through intermediary residues. We apply this framework to analyze the evolutionary adaptation of the PSD95pdz3 domain from Class I to Class II ligand specificity via mutations G330T and H372A. The cooperation index demonstrates a distinct evolutionary hierarchy: the G330T mutation establishes distributed pathway couplings that the H372A mutation subsequently exploits, whereas H372A alone produces minimal global changes. Furthermore, the fourth-order analysis identifies His317 as a critical intermediary node bridging the class-switching (330-372) and class-bridging (330-400) allosteric pathways. These results demonstrate that allosteric dependencies emerge only when mutations accumulate in specific combinations, with a hierarchical organization of pathways structured around position 330 and intermediary nodes His317 and Phe400. Rather than predicting allosteric mechanisms, this framework provides a mechanistic explanation for why and how allostery emerges during protein evolution.

During vertebrate development, the flat surface of the gut epithelium undergoes a dramatic transformation into densely packed arrays of finger-like projections called intestinal villi. Recent studies show that the villus formation relies on a tissue dewetting process, in which mesenchymal tissues buckle the overlying epithelial layer into periodic folds. However, the mechanisms driving subsequent elongation of these folds into finger-like villi remain largely unexplored. Here, we propose a simple mechanism for villus elongation that couples tissue dewetting to cell differentiation, which emerged as a repeated outcome of multiple independent simulations of an evolutionary-developmental Cellular Potts Model. In this mechanism, a liquid-like mesenchymal tissue continuously differentiates into a solid-like mesenchymal tissue at the interface between them. This differentiation drives the liquid-like tissue to continuously retract from the solid-like tissue in the opposite direction of the interface through dewetting, ultimately creating a finger-like projection. A merit of our proposed mechanism is that it only requires two tissues with different viscosities, high surface tension, and cell differentiation. We develop a simplified phase-field model to determine exactly how villus morphology depends on these three requirements. Since these requirements are satisfied not only in intestinal villi but also in many other developing tissues, we propose that the same mechanism could also drive the elongation of other tissues.

Biomolecular condensates are dynamic assemblies whose functions depend on continuous exchange of molecular components with the surrounding environment. While scaffold molecules drive phase separation and condensate architecture, many functional components are clients that are recruited through interactions with the scaffold-rich environment. Despite their prevalence, how client-scaffold interactions shape client exchange dynamics remains poorly understood. Here, we develop a reaction-diffusion model for client exchange in scaffold-driven condensates, in which clients switch between a scaffold-bound state and an unbound state. Bound clients exchange through scaffold-mediated transport, whereas unbound clients diffuse through the pore space of the condensate. Using the fluorescence recovery of fully photobleached condensates as a measure of client exchange, we compare transport through these two pathways with bound-unbound conversion and identify three limiting regimes. In the slow-conversion regime, bound and unbound clients recover through distinct scaffold- and pore-mediated pathways. In the intermediate-conversion regime, recovery of bound clients becomes limited by client unbinding. In the fast-conversion regime, local equilibrium between bound and unbound clients produces an effective single-state recovery. We further propose a unifying description that connects these regimes and quantitatively captures the apparent recovery timescales extracted from numerical simulations across condensate sizes. Our results provide a framework for interpreting component-specific exchange dynamics, and highlight client size, client-scaffold binding, and condensate porosity as key regulators of client turnover in multicomponent condensates.

Phages must locate and bind to bacterial surface receptors to initiate infection. Their tail fiber configuration critically influences this process. We develop a stochastic model describing surface search as a renewal process, incorporating attachment, detachment, and target-finding steps. Using both numerical simulations and analytical calculations, we quantify how tail fiber number, attachment-detachment rates, and geometric constraints impact the mean and the distribution of time to successful adsorption. Notably, the search efficiency shows a nonmonotonic dependence on tail fibers number, governed by a trade-off between binding stability and diffusion-mediated mobility. This optimum shifts depending on the effective bacterial density, target radius, and fiber reach. Short fiber reach imposes severe geometric constraints, reducing mobility at high tail fiber counts and leading to performance degradation. Our findings suggest that phage adsorption strategies are shaped by a balance between anchoring and exploration, with evolutionary implications for tail fiber design and infection efficiency.

We introduce a 3D mechanically adaptive viscoelastic cell-network model that links single-cell interactions to emergent tissue rheology. Unlike existing continuum or cell-based models, viscoelasticity is embedded within discrete, mechanically adaptive intercellular connections, allowing tissue-scale rheology and phenomena such as swirling and jamming to arise from single-cell behaviors and connection remodeling. The framework is motivated by recent advances in three-dimensional imaging and structural analysis that resolve single-cell behaviors within aggregates. It is validated against two gold-standard bulk assays performed on spherical aggregates: micropipette aspiration and Hertzian plate compression. Under aspiration, the model demonstrates a transition from elastic deformation to viscous creep governed by localized packing and emergent jamming at the aspirated neck, accompanied by increased mechanically adaptive remodeling. Under compression, core rheology determines deformation mode: liquid-like aggregates exhibit enhanced swirling, consistent with experimental observations, whereas solid-like aggregates exhibit affine, Poisson-like deformation. These results bridge cell-scale dynamics and quantifiable tissue rheology including elastic modulus and vicosity, providing a framework to interpret emerging 3D measurements of multicellular mechanics.

Biomolecular condensates are thought to play a pivotal role in cellular organization by regulating biochemical reactants in space and time. Sustained molecular fluxes across condensate boundaries, together with the participation of phase-separating molecules in active chemical reactions such as ATP hydrolysis, call for a nonequilibrium description. Here, we propose a self-consistent framework in which diffusion-drift dynamics and chemical reactions are coupled through a conditional free energy, defined as the excess contribution to the chemical potential. Self-consistency is achieved by deriving this quantity from the same free-energy functional that governs molecular interactions and phase separation. We apply the framework to a minimal client-scaffold system and investigate how active chemical processes and phase separation interact at steady state. In doing so, our approach recovers the fundamental rules previously identified for the emergence of nonequilibrium steady-state fluxes. The model shows that active reactions involving the scaffold molecules can regulate the phase behavior of the condensate. Moreover, nonequilibrium steady-state fluxes are maximal near the boundary between the phase-separated and homogeneous regimes, suggesting that condensates sustaining molecular transport may operate close to their stability threshold. In the same region, client fluxes are also enhanced, revealing an indirect coupling between scaffold activity and client transport. These results provide a baseline for developing more detailed theories of chemically active condensates.

The rapid emergence of spatial transcriptomics offers unprecedented opportunities to study cell-cell communication (CCC) by capturing gene expression alongside spatial context. However, existing CCC inference methods often rely on static, heuristic models that overlook the inherently spatiotemporal dynamics and mechanistic complexity of intercellular signaling, limiting both accuracy and biological interpretability. Here, we present SpaGRD, a first-principles-based method that explicitly models ligand-receptor interactions through partial differential equations derived from Fick law of diffusion and the mass action law. Leveraging graph signal processing techniques, SpaGRD solves these equations on spatial graphs, providing a principled and generalizable approach to CCC inference. Through extensive simulations, SpaGRD demonstrates superior accuracy and robustness compared to existing methods. Applications to multiple datasets across diverse tissues and platforms reveal dynamic CCC patterns with spatially resolved signaling heterogeneity, providing biologically meaningful insights into cellular coordination and developmental processes. By bridging physical modeling with spatial transcriptomics, SpaGRD provides an accurate, interpretable, and mechanistically grounded framework for advancing quantitative studies of spatiotemporal cell-cell communication.

Contact mediated cell-cell communication where direct physical contact between adjacent ligand cells and receptor cells trigger signal output is important during growth, development and regeneration of organisms. While the molecular machinery underlying contact mediated cell signaling is well explored, how the local spatial context of cells affect cell-cell contact mediated gene expression is not clear. Here, we present a vertex-based computational model to study spatial and temporal behavior of contact mediated signal output (which we refer to as output) in growing cell collectives. We consider cell-cell contact length dependent output synthesis and output degradation in receptor cells together with cell division to understand how dynamics at the scale of single cells lead to heterogeneous signal output. By tracking single receptor cells over time in growing cell collectives in silico, we show that cell growth and division lead to continuous and dynamic rearrangement of cell-cell contact between receptor and ligand cells which in turn affect the output levels. Our model predicts that the orientation of cell division plays a key role in the heterogeneity of signal output. We elucidate the link between cell mechanical properties that control cell shape, growth, and division, with signal output in receptor cells during contact mediated signaling processes.

Cell shape critically influences function, yet how complex and reproducible morphologies emerge from stochastic cellular dynamics remains unclear. Here, we investigate dendritic morphogenesis of two classes of Drosophila mechanosensory neurons with contrasting architectures, combining in vivo live imaging, quantitative analysis, cytoskeletal perturbations, and computational modeling. We show that despite sharing similar local stochastic branching rules, the two classes exhibit divergent growth dynamics that cannot be explained by standard, diffusive growth models. This discrepancy arises because Class I neurons display subdiffusive branch dynamics over long timescales, unlike Class IV. Based on these findings, we develop a minimal model with only four parameters that separates short-and long-term branch behaviors, and successfully recapitulates growth dynamics and final morphologies in both classes. Cytoskeletal perturbations reveal a functional separation between actin, which drives short-term exploratory branch fluctuations and arbor expansion, and microtubules, which tune long-term branch diffusivity and determine class-specific morphology. Together, these results establish a parsimonious, generalizable framework linking local cytoskeletal regulation to global neuronal architecture and reveal how stochastic dynamics encode reproducible cell shapes.

The primary morphological transition in Hydra regeneration, from an initially quasi-spherical tissue fragment to an elongated body, the hallmark of a mature Hydra, is preceded by a prolonged period of modest shape changes. Here, we ask whether this early stage already contains signatures of morphogenetic organization consistent with canalization toward the main morphological transition. We analyzed shape fluctuations during this period in tissue fragments with different initial and physiological conditions. Using principal component analysis, we quantified the effective dimension of the dynamical morphological fluctuation modes. We find that this effective dimension decreases progressively during the preparatory stage, well before the onset of significant elongation, indicating a progressive restriction of the accessible fluctuation manifold. This decrease is not explained by a single global measure of shape and persists when early and late states are compared at approximately matched shapes. We further show that calcium activity is associated with both the visible morphological changes and this hidden dynamical state. Tissues retaining positional cues from the parent Hydra exhibit lower effective dimensions, whereas tissues lacking such cues or subjected to mechanochemical perturbation maintain higher effective dimensions. These results identify an early, hidden dynamical phase of canalization in Hydra regeneration.

We perform a large-scale computational characterization of the map of protein primary to secondary structure using an AVR3a class protein effector domain from the plant pathogen P. palmivora as a case study. We formulate a modified site-scanning approach for exploring the neutral component of secondary structure phenotypes based on predictions from the machine-learning algorithm Porter 5 and apply it to the AVR3a phenotype. We predict a set of sensitive sites within the effector domain that are generally located at or near the boundaries of structured regions, with restrictions on the possible amino acid residues at these sites dictated by the secondary structure type that they participate in within the WT. We characterize a set of mutated phenotypes derived through the exploration of the neutral component of the WT effector domain, selecting them so that they span a range including both very rarely and very commonly seen secondary structures, and that they include both secondary structures nearly identical to the WT and ones far removed from it. We find that all these diverse phenotypes have an estimated robustness of the same order as that of the WT, and that the robustness scales logarithmically phenotype frequency, as seen in other genotype-to-phenotype maps. Furthermore, we observe that the dependence of the estimated phenotype frequency on the Kolmogorov complexity indicates simplicity bias in the protein secondary structure map.

Animal behavior unfolds across many timescales, from fast movement patterns to slow changes in internal states such as hunger, arousal, and circadian phase. These slow variables are rarely measured directly and must instead be inferred from their effects on the faster movements that can be observed. Here we propose treating timescale itself as an explicit coordinate of the state representation, constructing a time-frequency state space where fast movements and slow modulations appear simultaneously. We find that slow modes emerge as linear arms radiating from a stationary-weighted hub in the leading non-trivial eigenvectors of the transfer operator, with one arm per metastable basin across three systems of increasing complexity. In a synthetic system, the framework recovers a hidden bistable driver across nearly three decades of dwell time, while a fixed-timescale analysis of the same trajectory finds no separable slow modes. In nematode locomotion, it reproduces the canonical run-pirouette organization. In freely moving fruit flies, where fast leg kinematics are orders of magnitude faster than the behavioral states they compose, the multi-timescale operator identifies four metastable behavioral basins directly from the postural time series, without first decomposing into a sequence of stereotyped actions. We further find that these basins exhibit a broad, heavy-tailed distribution of residence times. Treating timescale as a state coordinate thus exposes a predictable geometric form for the slow organization of behavior, providing a general route for extracting collective modes from partially observed biological time series without first organizing the dynamics into discrete events.

For more than a billion years, microorganisms have evolved complex strategies for navigating aquatic habitats, despite the fundamental limitations and constraints imposed by their physical environment. A common theme across these strategies is the use of active slender appendages (cilia, flagella, archaella) to generate self-propulsion. Diverse selection pressures and evolutionary trajectories have driven the emergence of drastically different morphologies of biological microswimmers, each tailored for distinct functions ranging from motility to taxis to prey capture to feeding. Despite the biological and ecological significance of these intricate microscale processes, realistic computational modelling of these organisms and their behaviours is still in its infancy. Here, we present a comprehensive open-source simulation platform for motile microswimmers, that faithfully captures the universal hydrodynamic principles shared by such systems. We illustrate the predictive power and versatility of this approach to resolve and explore morphology-function relationships across different microswimmer species and provide new insights into the diversification of locomotion strategies in early eukaryotes.

Traditional protein mutation detection and pathogenicity prediction pipelines rely on static single-conformation structural modeling, inherently ignoring conformational flexibility, dynamic ensemble evolution, and the underlying manifold geometry of protein dynamics. This induces systematic detection failures in flexible regions, allosteric sites, and metastable functional domains, yet lacks a rigorous mathematical characterization of such failure mechanisms. In this work, we establish a theorem-driven geometric-algebraic framework for dynamic protein mutation modeling. Starting from a dynamic conformational Riemannian manifold, we construct the latent representation space via representation-induced completion of operator-valued observations, rather than pre-assumed embedding structures. Within this setting, algebraic constraints are not imposed axiomatically but relaxed into learnable approximate Lie algebra regularization, enabling statistical verification of structural consistency. By integrating Levi-Civita connection, geodesic deviation, and heat kernel asymptotics, we introduce a Lipschitz-stable topological spectral defect (TSD,{delta} spec) index that quantifies the intrinsic inconsistency between static representations and dynamic geometric invariants, linking it to curvature-induced instability and Lie algebra deformation. Under a functorial compatibility principle, we design a dual-branch architecture for pathogenicity prediction and defect awareness, realized via local Lie algebra encoding and low-rank spectral approximation. On multi-source datasets (108 curated PDB structures, 1060 validated residues from ClinVar, DMS, MaveDB, and gnomAD), we establish three fundamental theorems and validate key findings: TSD effectively distinguishes pathogenic/functional variants (PTM: {micro} = 0.386, OMIM: {micro} = 0.443, Clin-Var: {micro} = 0.302) from neutral ones (gnomAD: {micro} = -0.660) with high significance (P = 6.67 x 10 -18) and strong classification performance (AUC=0.82-0.86), while correlating strongly with protein stability ({Delta}{Delta}G, Spearman=0.9794, P = 5.38 x 10 -28). TSD further reveals PTM sites as topological hubs and neutral variants as evolutionary topological redundancy, enabling a paradigm shift from sequence alignment to geometric dynamics and providing a physics-based biomarker for variants of uncertain significance (VUS). These results upgrade protein mutation modeling from empirical static prediction to provable dynamic mechanism analysis. The source code of this work is publicly available at https://github.com/Harmenlv/LieFold-AI/tree/main.

Protein folding is the process by which a polypeptide chain organizes into its three-dimensional structure through a balance of stabilizing and destabilizing interactions encoded by the sequence. A central question in protein biophysics is how thermodynamic factors guide a polypeptide toward its native folded state despite the rugged energy landscape and the competing influence of nonnative interactions. In many biomolecular processes, cooperativity provides a mechanism by which multiple weak interactions act collectively to generate a robust response. In the context of protein folding, such cooperative effects may arise when the formation of one native contact enhances the stability or likelihood of nearby native contacts, thereby promoting collective organization toward the folded state. At the same time, folding is opposed by the much larger number of non-native interactions, whose heterogeneity can introduce frustration and destabilize folding even when the average native bias favors the folded phase. The interplay of these competing effects in determining foldability remains unclear in statistical-mechanical models. Here, we address this problem using a one-dimensional spin-glass model of protein folding with explicit shared-residue cooperative interactions encoded through wedge-based motifs. We show that modest cooperative bias can stabilize folding even where the noncooperative system remains unfolded, whereas non-native energetic fluctuation suppresses folding and shifts the transition to higher cooperative strengths. We further find that partial cooperative coverage is sufficient to lower the folding threshold. Therefore, the model provides a mean-field framework for incorporating cooperative interaction strength into the native one-dimensional model of protein folding and for describing how local cooperativity reshapes the folding transition.

This study introduces a statistical mechanical framework for allosteric communication in proteins based on the spanning-tree ensemble of residue contact networks. By representing protein structures as weighted graphs, we identify each spanning tree as a topological microstate. The canonical partition function is evaluated exactly via the determinant of the reduced weighted Kirchhoff (Laplacian) matrix, allowing for the derivation of global thermodynamic functions (including Helmholtz free energy, internal energy, entropy, and heat capacity) without approximation. Allosteric channels between specific residue pairs are defined as sub-ensembles containing unique simple paths. Using the Burton-Pemantle theorem and the Moore-Penrose pseudoinverse of the graph Laplacian, we compute exact path probabilities and channel-specific thermodynamics. This methodology enables a decomposition of channel heat capacity into energetic and topological components and quantifies residue-level allosteric importance through fractional contributions to the channel partition function. The framework was applied to the G12D mutation in KRAS, comparing wild-type (PDB: 6GOD) and mutant (PDB: 6GOF) proteins. Results show that while the mutation minimally affects mean internal energy and entropy, it reduces global heat capacity by 27.3%. This indicates a topological stiffening where the mutant occupies a significantly narrower landscape of spanning-tree configurations. At the channel level, the mutation maintains distributional stability across six functional routes but triggers a substantial internal redistribution of allosteric importance. Specific residues, such as Q61 and F156, shift occupancy by up to 35.5%. These findings suggest that the G12D mutation does not destroy communication pathways but reorganizes internal information traffic to favor a catalytically impaired state. This approach provides a rigorous, parameter-free metric for understanding how point mutations perturb distal protein signaling.

Vesicle internalization proceeds through a series of multivesicular topologies essential for endocytic transport and cellular compartmentalization. The energetic landscapes of related transitions, including vesicle budding and pearling, are known to be governed by the coupling of spontaneous curvature, leaflet area asymmetry, and reduced volume. However, the physical principles driving the structural transformation of hemifused intermediates remain unresolved. Using a continuum elastic model, we identify a morphological phase transition in hemifused invaginating vesicles, from an initial lens-like geometry to an elongated "kettle" geometry. This transition is discontinuous as long as the invaginating vesicles reduced volume is below a critical threshold, but continuous otherwise. The kettle-like morphology is metastable across a broad range of leaflet area asymmetries, potentially enabling a hysteretic externalization pathway. Increasing either the spontaneous curvature of the shared outer leaflet or the size of the invaginating vesicle, alone or in tandem with the host vesicle, turns the kettle morphology into the global free energy minimum. Notably, simply scaling up the size of both vesicles does not eliminate the free energy barrier. This quantitative characterization provides a structural reference for identifying internalization intermediates witnessed in experimental imaging, and maps the morphological evolution of the internalization pathway across its physical parameter space.

How can simple organisms lacking nervous systems encode and transmit environmental signals to generate complex, adaptive behaviours? Using the unicellular organism Physarum polycephalum as a model, we identify a unifying mechanochemical mechanism that links intracellular calcium oscillations to large-scale behavioural coordination. We first demonstrate experimentally that local perturbation of the actomyosin cortex is sufficient to induce symmetry breaking and directed migration, even in the absence of nutrient cues. Building on evidence linking calcium concentration to actin depolymerization and contractile relaxation, we develop a mechanochemical tubule model in which self-sustained calcium oscillations are coupled to pressure-driven mechanics. We show that environmental cues, encoded through the local modulation of these oscillations, give rise to directed transport and the redistribution of biomass. By extending this framework to a two-dimensional phase-field model, we demonstrate that this mechanism is sufficient to generate a diverse set of slime mould behaviours, including chemotaxis, network formation, and balancing exploration-exploitation trade-offs. In doing so, we provide a single mechanistic framework linking intracellular dynamics to organism-scale behaviour across spatial and temporal scales. Our work shows that these sophisticated behaviours can emerge from the modulation of self-sustained oscillations coupled by diffusion, providing a physically grounded mechanism for information processing in non-neural organisms and offering insight into the evolutionary origins of coordinated behaviour.

Transcription factors organize into liquid-like condensates to facilitate gene expression, yet the physical mechanisms governing their formation and properties remain poorly understood. We study the size statistics of transcriptional condensates in human HAP1 cells using widefield and super-resolution microscopy tagging the epigenetic reader BRD4. We find that hubs that appear monolithic in widefield resolve into clusters of smaller droplets that resist coarsening. We link this size control to Active Model B+, a non-equilibrium field theory that captures a regime of reverse Ostwald ripening out of thermal equilibrium. In this regime, chemically driven currents cause larger droplets to transfer mass back to smaller ones, stabilizing a state of microphase segregation. The observed exponential size distribution of BRD4 foci quantitatively matches our numerical simulations, suggesting a universal physical picture for the non-equilibrium self-limitation of cellular condensates.

Mechanical properties of epithelial tissues play essential roles in morphogenesis and physiological function. In this study, we analytically derived the in-plane bulk modulus, shear modulus, and Poissons ratio of a three-dimensional cell vertex model of epithelial monolayers. We showed that the model can robustly reproduce a near-zero in-plane Poissons ratio, a mechanical feature reported in cultured epithelial tissues. Numerical simulations further confirmed that the theoretically predicted Poissons ratio accurately describes the response of the model under finite, biologically relevant strains. In addition, the model exhibits not only morphological bistability between squamous-like and columnar-like states, but also mechanical bistability characterized by distinct elastic responses. Together, these results provide a minimal three-dimensional framework that links cell-scale mechanical interactions and epithelial morphology to tissue-scale elastic properties.