PRX Life

A variety of connected systems, ranging from the cytoskeleton to human organizations, dynamically rearrange themselves in order to move through physical or abstract space. However, our understanding of how systems-level behaviors arise from local restructuring actions remains limited, necessitating comparison of real-world data to models that predict network structure and dynamics. To understand these systems, we study an accessible example, the branching slime mold Physarum polycephalum, by imaging the organism as it travels and extracting key fundamental quantities from its continuously remodeling tubular network. By using these quantities as input parameters to a traveling network model, we find that with no further fitting, the model quantitatively matches key emergent properties from P. polycephalum dynamics including path length, relocation time, and search efficiency at different spatial resolutions. These findings demonstrate how a traveling network model can capture P. polycephalum behaviors, highlighting the potential to use traveling networks more broadly for understanding and predicting connected dynamic systems by linking local measurements to emergent, system-wide behaviors.

Contact inhibition of proliferation (CIP) enables epithelial tissues to self-regulate growth and maintain tissue homeostasis. However, how cell-level mechanical contact, tissue-scale structural order, and proliferation kinetics interplay remains a fundamental open question in living matter physics. Here, we present a particle-based model of a confluent epithelial monolayer governed by overdamped dynamics, where individual cells interact via a two-dimensional hard core- soft shoulder potential. By comparing structural evolution during quasistatic densification with previously reported experimental division kinetics, we find that the dynamics of proliferation arrest mimics the onset of direct steric contacts between the hard cores of the shell. Identifying hard core contacts as the physical driver of CIP, we couple our mechanical model with a stochastic Monte Carlo division scheme in which the instantaneous division rate decreases to zero from an intrinsic value as the number of hard core contact increases to six from zero. We demonstrate that for high intrinsic division rates, the cellular densification outpaces mechanical relaxation. This kinetic mismatch drives premature hard-core contact formation, shifts the onset of jamming and contact inhibition to lower packing fractions, and induces increasingly disordered transient configurations before the tissue universally converges to a hexagonal close-packed limit. Our model's predicted division kinetics and structural order evolution are consistent with epithelial monolayer experiments, both reported and our own. This minimal physical framework links single-cell steric contact mechanics directly to tissue-scale growth regulation and structural evolution.

Allosteric communication in biomacromolecules is fundamentally governed by thermal fluctuation gradients, yet standard Gaussian Network Models (GNMs) treat atomic contacts as uniform, binary couplings without differentiating core constraints from solvent-exposed surface flexibility. Here, we present an analytical matrix framework that incorporates continuous distance-dependent weighting into the Kirchhoff matrix L. This formulation captures the steep steric constraints of hydrophobic core packing versus peripheral surface loops while strictly recovering the classic unweighted GNM as a high-temperature limit (T [->]{infty}). Using Schur complements of partitioned joint covariance matrices, we show that conditional fluctuation variances and higher-order entropy-transfer terms reduce analytically to exact ratios of submatrix determinants (covariance minors), eliminating the need for fitting parameters or molecular dynamics trajectories. Applied to KRAS (PDB: 6GOD), this framework constructs an integrated directional entropy-transfer asymmetry map. Order-1 minors (h(i) = Kii) establish a single-node fluctuation potential gradient, while order-2 minors (Rij) define pairwise channel bandwidths. Higher-order minors show multi-body spatial coupling: order-4 minors identify rigid core residues such as Phe156 as strategic interlobe relay hubs linking Lobe 1 and Lobe 2, and an order-3 triad cooperation index demonstrates that signal transmission from Switch II (Gln61) to Gly60 and Phe156 converges on a single, mechanically integrated allosteric sector. By deriving directional information flow directly from experimental atomic displacement parameters, this approach establishes a rigorous, computationally efficient framework for mapping allosteric networks across structural ensembles.

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.

In linear polysomes, excluded-volume interactions among ribosomes can induce dimensional reduction of mRNA. Yet linear architectures allow steric stress to relax at open ends-- limiting how strongly crowding can remodel the mRNA's structure and dynamics. Using coarse-grained molecular-dynamics simulations, we compare circular and linear polysomes over a range of ribosome densities. Circular closure selects a predominantly quasi-planar global conformational ensemble, as indicated by a shape dimensionality dshape {approx} 2 over a range of ribosome densities. Crucially, circular topology and ribosome crowding act cooperatively to suppress structural fluctuations. While closure alone or linear crowding reduces relative global size fluctuations ({Delta}Rg/Rg) only to {approx} 0.16, their combined effect drives this fluctuation down to {approx} 0.07. Within this stabilized architecture, increasing ribosome density drives a distinct in-plane reorganization: the ring becomes more isotropic, global size fluctuations are strongly suppressed, and the scaling exponent increases toward {nu} [~=] 0.74 - 0.77, consistent with two-dimensional self-avoiding walk-like value over the accessible finite-size window, 1000 [≤] N [≤] 4969. Closure shortens the radius-of-gyration decorrelation time of circular polysomes by 40-fold relative to matched linear systems, reflecting the topological elimination of free ends. Within this closureselected ensemble, ribosome crowding further reduces the decorrelation time by up to 20% at the highest density. A fluctuation-informed crossover model links the density dependence of the global scaling exponent to inter-ribosomal subchain statistics. These results distinguish the geometric role of circular closure from the density-dependent steric response that it enables, revealing a confined yet dynamically responsive conformational regime for circular polysomes.

Intrinsically disordered proteins and regions are found across all kingdoms of life, yet the computational characterisation of their conformational ensembles has remained almost entirely confined to the human proteome. Whether the physics-based force fields developed on eukaryotic sequences remain reliable for taxonomically distant organisms, and whether the sequence ensemble relationships they reveal reflect conserved physical laws or the peculiarities of a single evolutionary window, are questions fundamental to the field. Here we introduce BENDER, a dataset of 11,533 IDP sequences spanning 13 taxonomic groups, each simulated under CALVADOS 2 molecular dynamics and annotated with ensemble-level geometric and novel contact-network properties, together with per-sequence pi pi and cation pi contact frequencies linked to phase-separation propensity. We show that CALVADOS 2 ensembles agree strongly with an orthogonal structural reference across the full dataset, with both held-out taxa performing above the dataset median, and that direct comparison against a second independently parameterised force field reveals no systematic scaling-exponent bias. We find that cross-taxon training data improves out-of-distribution ensemble prediction in two independent architectures, and that ensemble contact-network global efficiency is accurately predictable from sequence alone on held-out viral sequences. Positive degree assortativity is conserved across all taxonomic groups, suggesting that hub topology in disordered protein contact networks is a conserved physical feature of sequence-encoded disorder rather than an evolutionary contingency.

The collective spatial phenomenon of complement protein opsonization on nanoparticle surfaces is a key component of the immune response to viruses, engineered nanoparticles, and diseased cells. Recent work showed this opsonization follows a sharp, percolation-like transition versus the spacing d between surface-bound attachment sites, leaving two open questions: 1) whether the transition exhibits hallmarks of true criticality, such as diverging susceptibility, and 2) whether it can be distinguished from an alternative first-order cooperative (Hill-type) process producing an equally sharp threshold without true criticality. Here, we resolve both questions using a hierarchical statistical-mechanics treatment spanning stochastic, mean-field, and spatial reaction-diffusion models. The variance of two order parameters, peak complement activity and activation lifetime, diverges near threshold and sharpens systematically with system size, the defining signature of a critical point rather than a smooth cooperative response. Extending the analysis across site spacing and intrinsic kinetic rate constants traces a two-dimensional locus of critical points with consistent critical exponents throughout, establishing a single, robust universality class. The mean-field dynamic exponent for activation lifetime agrees quantitatively with the exact value predicted for the general epidemic process. Finally, a reaction-diffusion model of the nanoparticle surface shows the critical locus is set by a diffusion-limited length scale, establishing complement percolation as a fundamentally transport-limited surface reaction. These results place complement activation within the percolation universality class and identify the physical parameters, diffusion, catalysis, and decay, that govern its critical threshold, with direct implications for rational design of complement-evading nanomaterials, immunology, and evolutionary biology.

Solitary animals face a tradeoff when detecting threats: faster detection means accepting more false alarms. We show that groups can manage this tradeoff better by treating an undisturbed neighbor as evidence against a threat, becoming both faster and more accurate than lone individuals. Modeling each animal as a noisy evidence-accumulator that flees when its belief crosses a threshold, we find that a neighbor's flight signals danger while its stillness signals safety. A naive responder reacts only to flights and inflates false alarms as the group grows; a Bayesian responder weighs both, approximated by a single social discounting rate that interpolates between these limits. This yields closed-form expressions for group performance, including cascade branching ratios that stay strongly subcritical in safety and turn supercritical under threat, so the rate at which an animal discounts a threat while its neighbors stay still can be inferred from behavior alone, and it sets a ceiling on how many neighbors an animal can attend before discounting alone can no longer hold its false-alarm rate. Wild sulphur molly shoals under bird attack are best described by discounting rates well above what individually Bayesian updating supplies over any neighborhood they could plausibly attend, and the same model, at the inferred value, predicts a false-alarm rate that stays constant as shoals grow.

Structural network representations of metastatic dissemination typically focus on static topology without resolving transport dynamics, relaxation timescales, or steady-state behaviour. Here, we formulate a discrete Markovian transport model on a directed higher-order network with transition rates derived from qualitative clinical affinity classes. By constructing a non-Hermitian row-stochastic transfer operator, we characterise the relaxation dynamics through its spectral decomposition. The system exhibits a fast-mixing regime characterised by a spectral gap of {gamma} {approx} 0.67, corresponding to a characteristic relaxation timescale of {tau} {approx} 1.49 discrete steps, with the influence of the primary tumour origin progressively attenuated during dissemination. Convergence towards a non-equilibrium steady state (NESS) is accompanied by a reduction in Shannon entropy, concentrating probability mass within specific topological sinks. This spectral relaxation delineates two distinct dynamical regimes: early transient dissemination (n < {tau}), dominated by local organ-specific transition probabilities (organotropism), and the asymptotic regime (n > {tau}), determined increasingly by the global transport architecture of the network. Comparison with independent clinical and autopsy observations across 21 primary tumours and 23 target organs indicates that the predicted stationary distribution is consistent with the observed hierarchy of metastatic organ involvement.

Cellular activity drives epithelial fluidization -- a widespread phenomenon observed during tissue development, remodeling, and repair both in vivo and in vitro. Yet the physical origins and spatial organization of active forces vary widely across biological systems and are often represented by a single generic mechanism in theoretical models. Here, using an active vertex model, we systematically compare four modes of epithelial activity spanning subcellular to tissue scales: apolar motility, polar motility, fluctuating contractility, and mechanochemical regulation. Although all four mechanisms drive the same global transition from a solid-like rectangular tissue to a fluid-like circular morphology, they reach this state through distinct pathways -- differing in the rates and topology of junctional rearrangements, cell elimination, and collective motion and leave distinguishable signatures in tissue architecture, cell dynamics, and mechanical relaxation. Among these observables, spatial velocity correlations directly capture the spatial organization of activity: their correlation length and functional form together resolve all four mechanisms. The robustness of these signatures across activity strengths suggests that spatial velocity correlations offer an experimentally accessible means of identifying the physical origin of epithelial activity from live-cell imaging alone.

Focal adhesions maintain force-bearing attachment between cells and the extracellular matrix but can also undergo dynamic remodeling. Their assembly and actomyosin tension are coupled through mechanochemical feedback. The processes underlying this feedback are not instantaneous and therefore involve a time delay. However, how this delayed feedback gives rise to stable adhesion maintenance or dynamic remodeling remains unclear. Here, paired time-lapse measurements of vinculin fluorescence and traction stress revealed distinct local adhesion-force dynamics, including low-fluctuation and recurrent fluctuation patterns. To examine how these patterns could arise, we formulated a minimal mechanochemical model coupling focal adhesion assembly and actomyosin force through delayed reciprocal feedback. The model exhibited stable and oscillatory modes depending on feedback strength, the balance of opposing feedback effects, and the effective feedback delay. Bistability and hysteretic switching also occurred in a subset of parameter space, and the oscillation period followed a power-law relation with the delay. These results suggest that stable adhesion maintenance and dynamic remodeling can emerge from a common mechanochemical feedback architecture.

Missense mutations and post-translational modifications (PTMs) are major molecular perturbations that reshape protein function but are traditionally studied independently. Current computational approaches largely rely on sequence conservation or static structural features, limiting our understanding of how perturbations alter intrinsic protein dynamics. We present DynGeo-Pheno, a unified geometric deep learning framework that integrates protein language model representations with anisotropic network model-derived dynamics to jointly capture evolutionary, structural, and biophysical information. DynGeo-Pheno predicts disease-associated phosphosites and pathogenic missense mutations with high accuracy on independent test datasets. Ablation analyses indicate that protein dynamics provide complementary information beyond sequence evolution and structural topology for pathogenicity prediction. Beyond predictive performance, DynGeo-Pheno reveals that disease-associated perturbations preferentially localize to functional structural regions, including ligand-binding pockets and PPI interfaces. Mechanistically, phosphosites and missense mutations appear to exhibit distinct yet convergent dynamic signatures. Phosphosites preferentially occur in flexible regulatory regions, whereas pathogenic mutations are enriched in ordered structural elements. Despite these differences, both perturbation types display enhanced long-range coupling, increased perturbation responsiveness, and elevated mechanical stability, indicating that pathogenic residues preferentially occupy mechanically constrained and allosteric regulatory sites. This study provides compelling evidence that intrinsic protein dynamics is an important complementary determinant of pathogenicity and establishes a unified framework for interpretable AI predictions and mechanistic understanding of how genetic and regulatory perturbations may shape protein function.

Long-range correlated motions couple distant regions of a protein, providing a physical basis for allosteric communication, cooperative conformational change, and the balance between structural stability and sensitivity to perturbations. Yet membrane proteins operate within a strongly anisotropic lipid bilayer, and how this environment reshapes such system-spanning coordination remains unclear. Using an implicit-membrane anisotropic network model, we perform a proteome-wide analysis of more than 3,000 human transmembrane proteins. We find that long-range correlations remain scale free under membrane constraints but become strongly direction dependent. Across protein sizes and topologies, their correlation lengths continue to scale with the corresponding molecular dimensions, while increasing membrane anisotropy extends in-plane correlations and shortens those along the membrane normal. Because spontaneous correlations and perturbation responses arise from the same underlying mechanics, we further resolve residue-level responses into in-plane and normal components. The resulting directional mechanical susceptibility provides new predictions of mutation-sensitive sites in GPCRs beyond those captured by conventional scalar flexibility measures. Together, these results show how environmental symmetry breaking can organize protein mechanics across scales, linking collective dynamics to the functional sensitivity of individual residues and connecting a general physical mechanism to experimentally measurable protein function.

Computational design has transformed many fields of engineering, where simulators can explore millions of candidate design configurations before experimental development and testing. Therapeutic design in biomedicine has resisted computational design approaches because disease progression and therapeutic response emerge from interactions among many cell types within human tissue, governed by biochemical parameters that are largely unknown and potentially unknowable. Here, we introduce the Cell Interaction Foundation Model (CIFM), a virtual tissue model that forward-simulates the transcriptional dynamics of cells in human tissue under arbitrary therapeutic conditions based upon a spatial transcriptomic seed. CIFM is a geometric graph neural network trained by self-supervised masked-transcriptome prediction on millions of cellular microenvironments spanning human tissue types and disease states; generative, auto-regressive, monte-carlo play-out, then, simulates transcriptional dynamics under combinatorial perturbations from a spatial transcriptomic seed. We validate CIFM by showing accuracy gains in gene expression prediction and imputation, disease classification, recapitulation of perturbation responses in prostate cancer models, and recovery of T cell-tumor signaling measured in cell-cell sequencing experiments. Beyond such conventional tasks, CIFM enables target identification and therapeutic design through generative tissue simulation play-outs. Analyzing over 106 single and combinatorial perturbations, CIFM designs immunotherapy strategies for cancer and autoimmune disease that exploit combinatorial manipulation of signaling pathways to induce or suppress immune activation. Broadly, CIFM shows how generative artificial intelligence methods can be applied to model emergent behavior in highly interacting biological systems, yielding new approaches to fundamental understanding of tissue behavior as well as large-scale therapeutic design.

The folding thermodynamics of proteins are dominated by two opposing forces, the loss in backbone entropy and the packing of hydrophobic groups. The same forces are major contributors to the extension thermodynamics of elastic proteins with the distinction that both processes act in concert, favoring the higher chain and solvent entropy of a relaxed conformation. The relative entropic contributions specify the recoil mechanism; human elastin recoil is primarily driven by hydrophobic forces, whereas fly resilin has a rubber-like mechanism driven by backbone entropy. Despite the importance of elastic proteins to tissue biomechanics, few have been identified, let alone characterized to the same extent as elastin and resilin. We develop a thermodynamic framework that maps proteins by sequence-derived estimates of extension-induced backbone and solvent entropy changes. Putative elastic proteins are proposed and classified by recoil mechanism based on estimated thermodynamic features. Proteins that map to elastic regions are overrepresented by the skin proteome. The set of predicted elastic domains is further extended by incorporating sequence context embedded in protein language models. Protein domains with distinct thermodynamic recoil mechanisms cluster on the latent space manifold. Some of these domains are anticipated to have roles within molecular machines, expanding the scope of elastic protein function beyond mechanical materials like elastin and resilin.

Many biological processes rely on mechanical forces, with protein molecules acting as key mediators. Understanding how proteins respond to mechanical stress is essential for conditions including cardiomyopathy and muscular dystrophy. Natural proteins such as dystrophin and utrophin are composed of heterogeneous folding domains with distinct mechanical properties; deciphering domain-level behavior provides insights into disease mechanisms and informs therapeutic strategies. Single-molecule force spectroscopy (SMFS) enables probing the mechanical properties of entire proteins, yet current approaches struggle to identify heterogeneous folding domains, particularly without prior knowledge. Here, we present the first automated framework to identify heterogeneous folding domains in SMFS data, applying both existing clustering methods and a novel physics-aware deep clustering architecture, LatentUnfold. LatentUnfold learns complementary latent representations from force magnitude and the force-extension physical relationship through dual autoencoders, jointly optimized for clustering assignments. We apply our framework to experimental SMFS data collected from a synthetic two-domain protein (ddFLN4-Titin I27) as well as natural protein constructs of dystrophin and utrophin, with Monte Carlo simulated datasets serving as controlled validation. For the synthetic protein, we recover mechanical properties consistent with previously reported values for each domain. For the natural proteins, we uncover two mechanically distinct domain populations - corresponding to the N-terminal domain and spectrin-like repeats - with differences in both unfolding force and contour length increase, and reveal different unfolding order between them for the first time. This work enables domain-level biological inference, overcoming prior limitations that relied on averaging and overlooked heterogeneity, thus advancing the understanding of mechanical behavior in protein unfolding.

Eukaryotic organisms rely on post-translational modifications to chromatin to maintain stable patterns of gene silencing. These modifications include trimethylation at histone H3 lysine 27 (H3K27me3), which is deposited by Polycomb Repressive Complex 2 (PRC2) and accumulates on the genome during embryogenesis. While this process underlies the proper specification of cell types, we lack the ability to quantitatively predict the de novo establishment of Polycomb states. The kinetics of H3K27 methylation is difficult to quantify in vivo, and further, the network of molecular interactions that influences Polycomb states is complex. Here, leveraging the Drosophila embryonic system, we measure H3K27me3 dynamics with ChIP-seq and extract the rate the modification spreads along chromatin in vivo. To provide a mechanistic explanation for this rate, we build a reaction-diffusion framework that models how PRC2 establishes states of gene silencing de novo. The reaction-diffusion system recapitulates experimental observations in wild-type and mutant embryos, and suggests that PRC2 can diffuse in 1D along chromatin at a rate enhanced by Polycomb Repressive Complex 1. Through this work, we define a minimal set of parameters that dictate in vivo Polycomb dynamics, and provide evidence that the early embryo creates a super-charged environment for epigenetic modification.

Mechanical properties of biological tissues, driven by passive and active forces, play a vital role in several processes ranging from development to cancer metastasis. However, the dynamical responses of cells in tissues, subject to mechanical deformations such as shear and the associated rheological properties, are not well characterized. Here, we use three-dimensional agent-based models for normal and cancer tissues to investigate their responses to simple shear as a function of cell stiffness and stochastic active forces. In the normal epithelium, with uniform strength of active force, the yield stress as a function of shear rate follows the Herschel-Bulkley form over a range of cell volume fraction. Strikingly, the shear rate dependence and the elasticity-dependent changes in the yield stress fall on master curves upon suitable scaling. To model cancer-like behavior, a certain fraction (Np) of cells was chosen to have enhanced activity and decreased stiffness. As Np increases, the extent of collective cell movement decreases, transitioning from affine (collective) to non-affine (individualistic) movement, a finding that is in accord with imaging experiments. Simulations of a model of a stiff solid tumor, with radius Rs embedded in normal tissue, show that as Rs increases, the yield stress increases. Interestingly, the cells migrate collectively as Rs increases. A Gaussian Mixture Model (GMM) and a mean field theory quantitatively account for the simulation as well as experimental results on cancerous, non-cancerous, and a mixture of these two types. The combined theoretical and experimental study establishes that heterogeneity in stiffness and activity determines non-affine movements in normal and cancer tissues.

Protein engineering is limited less by generating variants than by the cost of evaluating them, so designing under a tight budget demands sequence features that let a model learn fitness from very few examples. We introduce ALSEBO (Active Learning Sequence Exploration via Bayesian Optimization), which couples a generative latent sequence landscape to Bayesian optimization and featurizes candidates with direct-coupling-analysis (DCA) coevolutionary statistics. This representation carries a specific inductive bias: it places the dominant organizer of the fitness landscape along a single linear coordinate, producing a smooth, funnel-like objective that a low-data surrogate navigates efficiently. On a virtual avGFP fluorescence benchmark, ALSEBO reaches the optimum in [~]40 evaluations and outpaces protein-language-model embeddings and raw latent coordinates; controls with representation-neutral oracles confirm that the advantage is intrinsic, not an artifact of the benchmark. Molecular dynamics of the optimized variant recovers structural hallmarks of fluorescence, and ALSEBO transfers to divergent GFP orthologs and to a non-GFP enzyme, establishing a data-efficient route to protein design.

Radiation-induced DNA double-strand breaks (DSBs) drive cellular mortality, mutagenesis, and severe evolutionary bottlenecks. While classical phenomenological models, such as the Linear-Quadratic (LQ) framework, reliably predict macroscopic population survival, they obscure the intrinsic single-cell stochasticity that governs critical rare events like tumor recurrence or the emergence of radioresistant persisters. To bridge this divide, we develop a mathematically exact stochastic differential equation (SDE) framework that models continuous DSB induction and repair as a Feller square-root process. By deriving exact closed-form expressions for the foci moments, we establish a highly efficient Maximum Likelihood Estimation (MLE) pipeline that circumvents computationally exhaustive Monte Carlo simulations, allowing the direct extraction of deterministic repair velocities and intrinsic molecular noise from empirical single-cell $\gamma$-H2AX data. Integrating this kinetic model with a cumulative damage hazard via the Feynman-Kac formalism, our framework seamlessly recovers the classic macroscopic LQ survival topology from microscopic first principles. Furthermore, systematic sensitivity analysis uncovers a fundamental evolutionary duality: while initial physical damage operates additively, ultimate cellular fate is driven by a nonlinear survival response governed by the trade-off between the damage hazard rate and intrinsic molecular noise strength. Crucially, we demonstrate that this molecular noise inherently enhances population survival. Governed by Jensen's inequality, stochastic variance acts as a non-genetic bet-hedging mechanism that buffers the population by favoring cells with transiently low damage loads. Ultimately, this exact stochastic framework bridges microscopic biophysics and macroscopic demographics, offering deep mechanistic insights into the evolutionary roots of radioresistance.