Physical Review X

Characterising equilibrium conformational ensembles with deep generative models requires assessing not only whether a model reproduces the target distribution, but also the mechanism of how it arrives here. Here we examine two distinct routes to generative conformational sampling-- stochastic relaxation and deterministic transport--through a study of denoising diffusion probabilistic models (DDPM) and rectified-flow (RF) models across molecular systems of increasing complexity. Using systems of increasing complexity, including a multimodal two-dimensional potential, the folded mini-protein Trp-cage, and a high-dimensional dihedral subspace of the intrinsically disordered protein -synuclein, we show that the key distinction between these paradigms lies not only in endpoint fidelity but in how distributional error is resolved during sampling. Diffusion models converge via pronounced late-stage stochastic relaxation and exhibits robust recovery of configurational breadth across neural architectures. Rectified flow approaches the target more gradually through deterministic transport and therefore depends much more strongly on architectural expressivity, particularly in heterogeneous high-dimensional landscapes. Analyses of entropy and moment evolution further show that diffusion more reliably restores both ensemble location and fluctuation structure, whereas RF requires Transformer-level feature mixing to represent the transport geometry accurately. These results establish convergence mechanism as a key design principle for generative sampling.

Understanding why specific metabolic states become stable in cancer has remained a fundamental challenge, as current pathway-centric frameworks lack a unifying physical principle governing global metabolic organization. We introduce the Metabolic Spin-Glass (MSG) model, which recasts cellular metabolism as a frustrated many-body system governed by a Hamiltonian that integrates reaction free energies, cofactor-mediated thermodynamic couplings, and patient-specific transcriptomic fields. The Hamiltonian is formulated as a binary optimization problem and solved using hybrid quantum annealing. Embedding gastric cancer transcriptomes (n=497) reveals that malignant phenotypes correspond to thermodynamically distinct ground states rather than isolated pathway perturbations. The Warburg effect emerges intrinsically as a thermodynamic phase transition, and stem-like tumors occupy the deepest attractor basin reflecting high energetic stability. A thermodynamic order parameter stratifies patients into prognostically distinct subtypes independently of transcriptomic classification, suggesting clinically applicable non-redundant biomarkers. This work establishes a spin-glass energy landscape framework for physically principled, patient-specific cancer metabolic stratification.

Quantifying task difficulty remains an open theoretical problem in neuroscience and artificial intelligence. While difficulty is often treated as a scalar property of stimuli or optimization landscapes, neural computation unfolds as a transient reconfiguration of high-dimensional dynamical systems. Here we propose a dynamical manifold theory of difficulty based on heterogeneous, modular FitzHugh-Nagumo networks subjected to structured task demand. Task difficulty is modeled as a conflict-driven control parameter that perturbs competing neural submodules. We define four dynamical metrics: (i) transition action (energetic cost), (ii) peak dispersion entropy, (iii) coherence recovery deficit, and (iv) mean-field trajectory curvature. Across systematic sweeps of task demand, we demonstrate that difficulty does not collapse to a single axis but instead emerges as a multidimensional manifold. Energetic cost and dispersion entropy form a dominant axis, while geometric curvature and integration recovery exhibit partial independence and nontrivial correlations. These results suggest that cognitive difficulty corresponds to structured reorganization in neural state space rather than mere increases in activation amplitude. The proposed framework provides a biophysically interpretable foundation for linking neural dynamics, cognitive effort, and difficulty estimation in artificial systems.

Whether species-rich communities erode gradually or collapse abruptly under environmental change is a central question in ecology [1]. Classical pairwise theory predicts that coexistence is always lost gradually, through smooth declines to extinction [2], yet real ecological interactions are often strongly state-dependent - shaped by nonlinearities that fixed pairwise coefficients cannot capture [3]. Here we show that higher-order (nonlinear) interactions make abrupt, irreversible loss of coexistence a typical route to community collapse: across diverse random communities, the equilibrium supporting coexistence disappears suddenly at a fold bifurcation. Using polynomial homotopy continuation [4] to track equilibria as environmental conditions change, we find that folds progressively dominate the boundary of the coexistence domain as nonlinearity strengthens, replacing the gradual extinctions of pairwise theory. Furthermore, the sign structure of higher-order interactions controls both the onset of tipping-points and whether biodiversity buffers or amplifies collapse. Because higher-order and nonlinear interactions are intimately linked, tipping points also arise generically in pairwise models with strong nonlinearity. Applying our continuation framework to a canonical model of plant-pollinator collapse [5], we formally resolve its bifurcation structure as fold-mediated, and we show that fold bifurcations are typical across published multispecies models spanning mutualistic, competitive, and consumer-resource interactions. These results challenge the expectation that monitoring abundances suffices to anticipate collapse, and unify structural-stability theory, which delineates the safe operating space for coexistence, with critical transition theory, which characterizes the nature of its boundaries.

Collective foraging in animal groups often relies on behavioral diversity, with individuals adopting different, sometimes complementary roles during shared tasks. However, most theoretical models predict that individuals responding to similar information cues in a shared environment should converge on a single optimal behavioral strategy. Using a spatially explicit multi-agent deep reinforcement learning model embedded in a three-species food chain, we show that stable behavioral diversity can emerge spontaneously among initially naive agents. Rather than converging on a single optimum, agents differentiate along a low-dimensional manifold of sensorimotor control, reflecting tradeoffs in speed regulation, spatial exploration, and deterministic turning rules. While multiple strategies yield comparable individual energetic returns, they are not interchangeable; group performance depends on how specific strategies combine to produce spatial resource partitioning and distributed directional influence. Replacing co-learned individuals with similarly competent agents trained in other groups disrupts these interaction structures and strongly reduces total energy acquisition. These results demonstrate that coordinated collective behavior and diverse, compatible strategies can arise endogenously from shared learning histories, but that this form of collective performance is path dependent and may be fragile to changes in group composition.

1Although resources are typically distributed continuously in space, species distributions often organize into discrete clusters. In his seminal paper [36], Turing demonstrated that such clusters can spontaneously arise in population densities, even when populations evolve in environments with continuously varying conditions. This phenomenon is known as Turing instability. In this work, we focus on two models grounded in population dynamics: a one-dimensional model based on the nonlocal Fisher-KPP equation, and a two-dimensional model involving an environmental gradient. We show that phenotypic clusters (sometimes referred to as "species") emerge in these models. We prove that they do not emerge because of Turing instability, but because of stochasticity, and that they disappear when stochasticity is reduced. First, for both models, we start our simulations with initial populations uniformly distributed in the state space. We show that phenotypic clusters quickly emerge and that the distances between them depend on the population size, that is, on the degree of stochasticity. Next, we start from already clearly defined phenotypic clusters. We identify three regimes in the connection between population size, the initial distances between clusters, and the distances between clusters at equilibrium. Last, on the two-dimensional model, we relax the hypothesis of complete clonality by varying the effective recombination rate, explore its effect on phenotypic clustering, and show that phenotypic clustering decays drastically with slight recombination.

We investigate the target search process by proteins locating specific target sites along DNA - a phenomenon fundamental to biological functions such as gene regulation, transcription, replication, recombination, and gene-editing technologies. This process proceeds through a repetitive sequence of stochastic motions: consisting of one-dimensional (1D) sliding along the DNA contour interspersed with detachment and three-dimensional (3D) excursions in the bulk, and then reattachment to a random location on DNA. Recognizing this sequence of random events as analogous to the resetting processes widely studied in statistical physics, we employ a first-passage-renewal framework and derive general expressions for both the mean and fluctuations of the total search time. Our results are completely generic and do not depend on the detailed microscopic dynamics of either the 1D or 3D phases. Quite interestingly, we find that intermittent detachment can not only accelerate the mean search but can also regulate fluctuations around it. Our analysis reveals a universal fluctuation inequality that links the variability and mean of the sliding time to the mean excursion time, thereby identifying the fundamental conditions under which target search process becomes efficient. Notably, we find that broad distributions of sliding times emerge as a universal characteristic for optimal search efficiency--a feature emanating from the slow dynamics along the DNA. Using the facilitated diffusion mechanism as a representative example, we validate the generality of our results. These findings provide a unified theoretical framework connecting stochastic search, resetting dynamics, and biological efficiency, while also highlighting the crucial role of DNA structure such as its contour length in modulating search performance.

Quantum effects in biology are unavoidable at the molecular scale; the unresolved question is whether they can remain functionally relevant across the timescale gap between femtosecond molecular dynamics and microsecond-to-millisecond biological function. Here we formalize this mismatch as an equilibrium-to-functionality gap and use tubulin as a stringent open-system test case. We combine secular Lindblad, Redfield, and hierarchical equations of motion (HEOM) treatments to quantify decoherence, non-perturbative relaxation, and the physical amplification required for functional relevance. Equilibrium dephasing yields a conservative [Formula] fs at 310 K, with a generic protein-bath baseline of {approx} 13 fs. A completed 30 ps HEOM trajectory for the full 1JFF tryptophan network shows distributed non-Markovian relaxation, with terminal purity Pur = 0.210 and stretched-exponential exponent {beta}KWW {approx} 0.44, confirming that Redfield is useful as a short-time perturbative comparator but not quantitatively interchangeable with HEOM in this intermediate-coupling regime. We introduce a coherence-utility criterion [U] = [K]{tau}coh/{tau}func, separating required amplification from empirically bounded gain. A thermodynamic uncertainty relation closure shows that neural-scale cascade amplification would require Pmin [~] 10-7 W, about five orders of magnitude above the local microtubule GTP budget. Frohlich pumping is found to be linewidth-gated rather than generically micron-scale; ordered-water cavity QED and geometric subradiance remain experimentally testable but severely constrained candidates. The result is not a model of consciousness, but a reproducible physical benchmark framework for evaluating biological quantum-coherence claims under explicit open-system, energetic, and experimental constraints. Six falsifiable experimental programmes are prioritized, and the full computational framework is released with a validation ledger, cryptographic audit trail, and living supplementary material. O_FIG O_LINKSMALLFIG WIDTH=200 HEIGHT=107 SRC="FIGDIR/small/724047v1_ufig1.gif" ALT="Figure 1"> View larger version (20K): org.highwire.dtl.DTLVardef@19e4f42org.highwire.dtl.DTLVardef@65a719org.highwire.dtl.DTLVardef@1bd63beorg.highwire.dtl.DTLVardef@df77d8_HPS_FORMAT_FIGEXP M_FIG O_FLOATNOGraphical abstract.C_FLOATNO Equilibrium tubulin coherence lies in the femtosecond regime, while functional neural timescales lie in the millisecond regime. Frohlich pumping, QED-cavity protection, and geometric subradiance remain experimentally discriminable non-equilibrium candidates requiring independently bounded amplification. C_FIG FundingThis research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. Versioned computational scope of this releaseThis manuscript reports the theoretical framework, calibrated equilibrium baseline, Redfield/HEOM validation ledger, stratified Bayesian evidence synthesis, classical comparators, and falsifiable experimental design. The release-specific reproduction audit, including the current validation-check total and the SHA-256 fingerprints of the binary production artefacts (.npz, .pkl), is documented in LIVING_SI.md and outputs_data/raw_json/structur al/validation_report.json. A completed 30 ps HEOM production trajectory has been validated on constrained hardware; the master dataset contains the full 8-site population trajectory. A summary of those results is provided in [§]2.2.5. All claims made below are restricted to the numerical and theoretical evidence reported in this manuscript and its associated repository artefacts. The public repository ships the calibrated phenomenological baseline for accessibility; the HEOM production artefacts serve as the non-perturbative validation benchmark. All source figure outputs associated with this release are maintained in the public repository under outputs_data/figures_final/.

Planarian fragments can regenerate with normal gross anatomy after a transient bioelectric perturbation yet display altered outcomes upon re-cutting, implying that regeneration can store a persistent hidden state. Here we formulate an open-path version of Tangential Action Spaces (TAS) for this setting. Regeneration after a given cut is represented as a prescribed coarse anatomical trajectory together with multiple physiological lifts in a higher-dimensional state space. A metric on physiological state space defines a baseline lift, an effective excess-cost functional, and a baseline-relative endpoint displacement that serves as written hidden regenerative state. Re-cutting converts this open-path construction into a challenge readout. Locally, the theory yields a cut-dependent memory co-metric that identifies latent directions that are easy, difficult, or inaccessible to rewrite. We show that this geometry is consistent with published observations of cryptic phenotypes, stable re-challenge ratios, and near-absorbing double-headed outcomes. A reduced rank-one latent-threshold realization fitted to published 8-OH immediate and re-challenge counts identifies a challenge-sensitive cryptic interval below the immediate double-headed threshold and predicts out-of-sample re-challenge penetrances near 15% for nigericin- and monensin-treated immediate single-headed survivors using only their immediate phenotype penetrances. As a mechanistic bridge, a local electrodiffusive in-silico example instantiates a local version of the physiological-state effort metric G. This metric defines the baseline lift and excess rewriting cost, in relative biophysical units, and yields explicit example local write geometry. An illustrative semimechanistic readout based on integrated wound-edge gap-junction contrast and Na/K-ATPase load reproduces the treated-family ordering and similar transfer predictions when the untreated baseline is softly anchored near zero. These quantitative layers are intended as proof-of-concept calibratability and mechanistic-grounding checks rather than full validation of the complete open-path model. The framework therefore turns cryptic regenerative memory into a geometric, costed, and experimentally testable object, yielding predictions about temporal-profile dependence, compensatory cancellation, sign-reversing controls, cut dependence, anisotropic rewriting, and multi-round accumulation of hidden regenerative state.

Molecular dynamics (MD) simulations provide detailed insights into biomolecular motion but are often limited by the prohibitive cost of sampling long-timescale behavior. Here, we present a Transformer-based framework that reconstructs temporally continuous dynamical trajectories from only a small fraction of the initial data, directly targeting time-ordered evolution rather than independent ensemble snapshots. Using three systems spanning distinct dynamical regimes (intrinsically disordered -Synuclein, Cytochrome P450 ligand-binding motion, and a synthetic three-well potential), we show that the model learns both local fluctuations and long-range temporal structure. At inference time, the model generates full trajectories autoregressively from an initial prefix as prompt, capturing metastable transitions, basin-to-basin movements, and system-specific dynamical signatures. Free-energy surfaces computed from generated trajectories closely match ground-truth landscapes and, in several cases, we observe enhanced sampling in generated trajectories relative to the trained trajectories--while preserving kinetically meaningful transition patterns. These results demon-strate that Transformer architectures can serve as efficient, system-agnostic tools for time-continuous molecular trajectory prediction, offering a data-driven complement to long MD simulations and enabling accelerated exploration of conformational space.

Network-based epidemic models have been instrumental in understanding how contact structure shapes infectious disease dynamics, yet widely used frameworks such as Erd[o]s-Renyi, configuration-model, and stochastic block networks do not explicitly capture the combination of fully accessible (saturated) within-group interactions and constrained between-group connectivity characteristic of many real-world settings. Here, we introduce the Multi-Clique (MC) network model, a generative framework in which individuals are organised into fully connected cliques representing stable contact groups (e.g., households, classrooms, or workplaces), with a limited number of external connections governing inter-group transmission. Using stochastic susceptible-infectious-recovered (SIR) simulations on degree-matched networks, we compare epidemic dynamics on MC networks with those on classical random graph models. Despite having an identical mean degree, MC networks exhibit systematically distinct behaviour, including slower epidemic growth, reduced peak prevalence, increased fade-out probability, and delayed time to peak. These effects arise from rapid within but constrained between clique transmission, creating structural bottlenecks that standard models do not capture. The MC framework provides an interpretable, data-driven representation of recurrent contact structure, with parameters that map directly to observable quantities such as household and classroom sizes. By isolating the role of intergroup connectivity, the model offers a basis for evaluating targeted intervention strategies that reduce between-group mixing while preserving within-group interactions. Our results highlight the importance of explicitly representing the real-life clique-based network structure in epidemic models and suggest that classical degree-matched networks may systematically overestimate epidemic speed and intensity in structured populations.

Cellular monolayers often exhibit orientational order, with nematic alignment of cell shape and cytoskeletal structures governing tissue-scale collective dynamics. Despite extensive studies, a unified analysis framework for characterizing active nematics in living systems remains partial, and key discrepancies with theory persist. Here, we present a systematic and comparative analysis of nematic order and tissue flow dynamics across twelve distinct cell types. We quantify the impact of analysis parameters and provide data-driven guidelines to improve reproducibility and cross-study comparability. Across all nematic systems, we uncover remarkably consistent static properties, supporting the universality of nematic behavior in living tissues. By combining orientation-field analysis with velocity-field measurements and numerical simulations, we show that all examined systems display contractile active nematic signatures, with characteristic flow structures around topological defects. However, direct tracking of individual defects reveals subdiffusive dynamics, in stark contrast with the superdiffusive, self-propelled motion predicted by the hydrodynamic theory of active nematics. Our results establish a standardized framework for nematic analysis in biological systems and highlight fundamental limitations of current active nematic models in describing defect dynamics in living tissues.

Understanding the dynamical mechanisms underlying epidemic wave formation remains a central problem in mathematical epidemiology. Population-level epidemic waves are commonly interpreted as emergent consequences of nonlinear transmission feedback between susceptible and infectious individuals. However, epidemic time series from different regions often display markedly different waveform regimes, ranging from sharply peaked epidemics with rapid post-peak decline to more prolonged plateau-like dynamics. Here we propose the SEVA (Seasonal/Environmental Viral Activity) framework as a parsimonious alternative dynamical interpretation of epidemic wave formation. In this formulation, epidemic waveforms arise from depletion of a finite vulnerable population under a temporally structured viral activity field. The activity function is represented by a monotonic logistic hazard describing the temporal evolution of viral activity. With activation timing and steepness held constant across regions, daily incidence emerges as the product of activity intensity and the remaining vulnerable population. The framework is applied to first-wave COVID-19 hospitalization and mortality data from selected European countries and U.S. states during spring 2020. With fixed activation parameters and region-specific activity intensity, the model provides a simple dynamical explanation for diverse epidemic waveform regimes--including sharply peaked waves and plateau-like dynamics--without modification of the underlying dynamical structure. When epidemic trajectories are expressed in normalized form, curves from regions with very different mortality burdens display closely similar temporal structures. Within the SEVA formulation, this behaviour arises naturally from the interaction between a common temporal activation profile and regionally varying activity intensity. In this perspective, sharply peaked epidemics and plateau-like trajectories represent different dynamical regimes of the same activity-driven depletion process.

While fundamental for understanding community assembly, dominant beta diversity metrics suffer from severe sample-size dependence and lack tractable generative predictions. Here, we generalize Whittakers multiplicative index into a novel abundance-weighted framework. By proving this index strictly equals the expectation of the Good-Turing sample coverage deficit, we unify species abundance and spatial occupancy into a single, design-unbiased parameter. Furthermore, deriving exact neutral expectations provides process-based baselines to dissect beta diversitys internal structure via additive decomposition into species (SCBD) and local (LCBD) contributions. Application to two large-scale forest plots empirically validates its strict sampling invariance and reveals non-neutral signatures. Crucially, our SCBD diagnostic reveals a striking ecological phase transition: rare-to-intermediate species are more spatially scattered than neutral drift predicts until reaching an aggregation onset threshold, beyond which dominant species exhibit supra-neutral aggregation, reflecting competition-colonization trade-offs. Concurrently, LCBD diagnostics successfully isolate the spatial footprints of environmental filtering from pervasive neutral noise. By avoiding the conflation of statistical estimation bias from genuine biological scaling and anchoring metrics in tractable models, this framework provides a rigorous toolkit that transforms our empirical understanding of spatial coexistence.

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.

Cells sense and respond to the mechanical properties of their environment, yet the minimal physical principles sufficient to reproduce mechanotransduction and durotaxis remain debated. This work introduces FraCeMM, a physics-first mechanochemical simulation framework coupling stochastic ligand-integrin-talin binding to a deformable soft-body cell model on an elastic substrate. Without imposed polarity, directional cues, or migration rules, the model reproduces hallmark mechanobiological behaviors including stiffness-dependent spreading, traction reinforcement, focal adhesion asymmetry, and directed durotaxis. A finite pool of adhesion molecules, mechanically coupled through elastic linkages, drives emergent force asymmetry and polarization via self-consistent feedback between stochastic binding, molecular availability, and substrate stiffness. Despite minimal assumptions and a coarse-grained molecular representation, resulting traction forces, adhesion loads, and migration speeds fall within experimentally reported ranges. These results support the view that local force balance, limited adhesion resources, and mechanically binding are sufficient to generate adaptive mechanosensing and directed migration, establishing a transparent and extensible foundation for computational mechanobiology.

This Perspective formally proposes Particle Biology as a unifying theoretical framework to address the critical bottleneck in current life science research. Current life science research has reached a critical bottleneck. While the field has advanced to the study of 3D genomic spatial configurations and chromosomal organization, it remains largely descriptive and confined to the macromolecular level. This approach lacks a first-principles understanding of the underlying physical forces that drive biological processes. This Perspective formally proposes Particle Biology as a unifying theoretical framework. We establish an axiomatic system positing that life phenomena are fundamentally emergent spatiotemporal patterns of electromagnetic forces among atoms, electrons, and nuclei operating far from thermodynamic equilibrium. By defining biological states through the Biological Hamiltonian and mapping biochemical pathways to multidimensional Potential Energy Surfaces (PES), we bridge the gap between descriptive biology and predictive physics. We categorize core research technologies into three modalities--seeing, computing, and controlling particles--facilitated by advancements in Cryo-EM, AlphaFold 3, and Boron Neutron Capture Therapy (BNCT). Ultimately, the trajectory of molecular biology has evolved from cells to DNA and onto the 3D spatial genome, yet it cannot go deeper within current paradigms. The next logical evolution is to move beyond the macromolecular bottleneck to focus on the electromagnetic interactions between atoms and ions--the true Particle Biology level--to redefine disease and intervention.

Epidemic growth rates, reproduction numbers and counts of new infections are universally used to guide public health intervention decisions. It is widely and reasonably believed that larger values of these indicators evidence the need for more urgent or stringent control. Here we show that this intuition can fail dramatically. We construct pairs of epidemics with indistinguishable growth rates, reproduction numbers and infection curves but fundamentally divergent responses to identical interventions, with one epidemic subsiding while the other grows exponentially. Conversely, we identify pairs in which one epidemic exhibits larger indicators and causes three times as many infections, yet both become suppressed with equal effectiveness under the same intervention. These paradoxical outcomes arise from structural uncertainties in transmission, which are invisible to standard outbreak indicators but become decisive under feedback control. Because structural uncertainty is unavoidable when representing real outbreaks, epidemic controllability and intervention performance cannot be reliably inferred without explicitly modelling this feedback between transmission and intervention.

Neural population activity typically evolves on low-dimensional manifolds and can be described as trajectories in attractor-like state spaces, including metastable switching among quasi-stable assembly states. Here we develop a unified definition of clustered neural networks with local excitatory-inhibitory balance in which enhanced within-cluster effective coupling can be realized by connection probability (structural clustering), synaptic efficacy (weight clustering), or any mixture of both. We introduce a single mixing parameter{kappa} [isin] [0, 1] that redistributes a defined clustering contrast between connection probabilities and synaptic efficacies while preserving the mean input of a balanced random network. Using mean-field theory and network simulations, we show that metastable dynamics are supported across the full{kappa} continuum. Shifting contrast between structural and weight clustering changes higher-order input structure, reshaping multistable regimes, neuronal correlations, and the balance between single- and multi-cluster episodes. Because real nervous systems jointly organize topology and synaptic strength, our approach provides a biologically realistic assembly definition and a basis for future models combining structural and functional plasticity. In practical terms,{kappa} offers a translation axis for neuromorphic and other constrained substrates, clarifying trade-offs between routing resources and synaptic weight resolution when implementing attractor-based computational primitives such as winner-take-all decisions and working-memory states for artificial agents.

Evolutionary therapies regulate heterogeneous populations by altering selective pressures through treatment sequences in cancer and infections. This letter develops an invariant-set framework for treatment-induced containment based on positive triangular invariant sets. For periodically switched systems, sufficient conditions are derived for the existence of such invariant regions. Robustness with respect to mutation is established by showing that the invariant simplex persists under small perturbations of the subsystem matrices. In the two-phenotype case, the analysis yields an explicit mutation threshold that separates regimes in which therapy cycling maintains containment from regimes in which mutation can enable evolutionary escape. Simulations illustrate the geometry of the invariant sets and the role of mutation and dwell time in containment robustness.