Journal of The Royal Society Interface

Kinetic parameters in engineered biomolecular circuits are typically known only within intervals and are not precisely tunable; controller synthesis is therefore naturally posed over sets of gains rather than at a single nominal design. We formulate robust controller tuning under parametric plant uncertainty as a universally quantified set-inversion problem: given a plant-parameter box, compute the controller gains that stabilize every plant in the box. Robust stability is decided by Hurwitz sign conditions, which are bounded over the joint gain-parameter domain using validated range enclosures, yielding machine-checkable certificates of local exponential stability of the deterministic equilibrium. Numerical experiments show that the certified region depends strongly on the range enclosure. For an antithetic integral feedback (AIF) controller acting on a two-species gene-expression cascade, a tensor-product Bernstein enclosure certifies 83.2% of the design plane at bisection tolerance epsilon=0.05 with 324 terminal boxes, whereas four baseline enclosures certify between 50.9% and 58.9% on the same paving. For a fifth-order loop, the certified fraction increases from 34.4% to 73.8%, and the certified and excluded terminal boxes confine the robustly stabilizing area to a window of about 2.1 percentage points. Finally, for a refining variant of the algorithm in which the plant box is refined alongside the design box, the undecided area satisfies |U(epsilon)|=O(epsilon) under a non-degeneracy assumption on the stability boundary.

Understanding how animals respond to their physical environment requires environmental observations at the scale at which behavioural decisions are made. For soaring birds, the coarse resolution of weather products has long hindered the analysis of their behavioural response to fine-scale atmospheric dynamics, forcing uplift sources to be inferred largely from behaviour itself. Here, we combined high-resolution movement data from 24 golden eagles with the kilometre-scale COSMO weather model. We first classified thermal, orographic, and gravity-wave uplifts using independent atmospheric predictors and then quantified the birds' use of each uplift type and their fine-scale behavioural responses. Eagles relied predominantly on thermals, but opportunistically adjusted their use of uplift sources seasonally. The birds' flight behaviour could not reliably indicate which uplift type was primarily used, and thus suggests that both atmospheric processes and behavioural responses are better described as continua than discrete categories. Finally, we compared vertical wind velocities derived from eagles soaring behaviour with those modelled by the COSMO weather model, showing that most of the thermals exploited by eagles remain unresolved at kilometre-scale model resolution. Our results demonstrate how high-resolution weather models provide new insights into bird movement decisions, while also highlighting the potential of soaring birds as biologically embedded atmospheric sensors that could help closing the resolution gap in atmospheric models.

Forecasting and causal inference in complex, non-linear dynamical systems are fundamentally challenging due to noisy and high-dimensional time-series data. While Empirical Dynamic Modeling addresses this via state-space reconstruction in Euclidean space, it suffers from the "curse of dimensionality," distorting distance measurements as dimensions increase. This study proposes Forest-Weighted S-map (FORWS) and Forest-Weighted Causal Inference (FORWC), utilizing adaptive "forest weights" derived from random forest ensembles instead of Euclidean metrics. Using simulated and empirical ecological datasets, I demonstrate that these methods exhibit comparable or improved forecasting skill relative to conventional tools, while offering substantial resilience against dynamic process noise and mitigating the curse of dimensionality. Notably, FORWC enables multimodal causal inference, successfully extracting directed interactions between high-dimensional acoustic vectors and scalar temperature data monitored in a honeybee hive. By bridging chaotic physics with machine learning, this framework accurately extracts intrinsic manifold topology, providing a robust paradigm for uncovering causal networks in complex real-world environments.

Tissue shape and function are defined by the mechanical interactions of cellular and extracellular components. Lymph nodes cyclically remodel in response to immune challenges whilst preserving essential stromal structures. However, the relative contributions of the fibroblastic reticular stromal cell network and the ensheathed extracellular matrix, remain undefined. We quantified the contribution of ECM to the viscoelastic properties of lymph nodes to parameterise an in silico model exploring the FRC network's adaptation to pressure-driven tissue expansion. The balance between tissue pressure, FRC contractility and ECM stiffness permit robust remodelling and growth, while maintaining physiological geometries and balancing force distribution. Local perturbation of ECM stiffness or FRC contractility disrupts force distribution globally and impacts FRC proliferation and tissue expansion. Spatially dispersed perturbations exert higher impact on tissue architecture than equivalent localised perturbations, with effects propagating across the network. The integration of cellular and extracellular mechanics thereby enables robust lymph node remodelling.

Catalysis and allostery are complementary principles of biological function: catalysis accelerates bio-chemical reactions, whereas allosteric regulation dynamically controls molecular interactions. The emergence and persistence of early autocatalytic ribozyme systems likely depended not only on template-directed self-replication but also on ATP production, utilization, and recycling through prebiotically plausible energy-conversion processes. We previously proposed that resistance to molecular parasitism in autocatalytic RNA networks can emerge through hyperparasitic regulation, whereby hyperparasitic ribozymes compete with parasitic ribozymes for binding to the host replicase while additionally binding parasitic ribozymes, thereby redirecting competitive interactions away from host exploitation. Here, we develop a minimal mathematical model of an autocatalytic RNA-world system in which replication, ATP/ADP-based energy conversion, and ATP-dependent allosteric regulation become dynamically coupled. Building on a general reaction framework describing ribozyme replication, catalytic interactions, and ATP/ADP cycling, we identify the minimal regulatory architecture required for stable RNA replication in the presence of parasitic mutants. Our simulations reveal a sequential evolutionary transition in which control precedes optimization. ATP-dependent allosteric regulation first evolves to tame molecular parasitism through hyperparasitic binding, whereby ATP-loaded parasitic ribozymes compete with ATP-free parasites for binding to the host replicase while also binding directly to parasitic ribozymes. This stabilizes RNA replication but simultaneously imposes an energetic cost by sequestering ATP and thereby reducing ATP-ADP turnover. The resulting regulatory burden creates selective pressure for the evolution of ATP synthase/ATPase ribozymes that accelerate ATP-ADP cycling and restore the metabolic flux required for sustained RNA replication. We further identify two plausible evolutionary routes to parasite control: parasitic ribozymes either become intrinsically allostery-prone or are converted into allostery-prone forms by an evolved allosterase ribozyme. Once ATP turnover is sufficiently rapid, both mechanisms confer long-term resistance to recurrent parasitic invasion. These results suggest that stable RNA-based evolution required the progressive integration of information replication, molecular regulation, and increasingly efficient metabolic energy conversion. More generally, the model identifies ATP-dependent allosteric regulation as a plausible evolutionary bridge linking autocatalytic RNA replication to the emergence of regulated proto-metabolism, transforming a parasite-limited replicating system into a self-regulating proto-biological organization capable of sustained evolutionary dynamics. HighlightsO_LIA minimal autocatalytic RNA network achieves stable self-replication despite recurrent parasitic invasion C_LIO_LIATP-dependent allostery tames molecular parasitism through hyperparasitic binding C_LIO_LIControl of molecular parasitism precedes optimization of metabolic energy conversion C_LIO_LIATP sequestration creates an energetic burden that drives increased ATP-ADP turnover C_LIO_LIStable parasite control evolves through either intrinsic allostery or allosterase-mediated regulation C_LIO_LIStable RNA autocatalysis emerges through the sequential integration of replication, molecular regulation, and energy conversion C_LI

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.

Background: Immigrant patients with limited dominant-language proficiency may face intersecting challenges when navigating cancer care and long-term oral anticancer treatment. Although studies have reported lower medication adherence among migrant and ethnic minority populations, less is known about how migration-related, linguistic, experiential and contextual factors shape treatment engagement from patients own perspectives. This study explored how immigrant patients experience illness, navigate treatment and engage with oral anticancer medication within the broader context of cancer care. Methods: Thirteen immigrant patients with limited dominant-language proficiency receiving oral anticancer medication for haematological malignancies were recruited from the haematology outpatient clinic of a Belgian university hospital. Semi-structured interviews were conducted in participants native languages using an adapted version of the McGill Illness Narrative Interview, with professional interpreters or intercultural mediators. Interviews were analysed using inductive reflexive thematic analysis within an interpretivist framework. Results: Analysis of patients illness narratives generated five experiential dimensions: 1) bodily, biographical and identity rupture; 2) temporal disruption and uncertainty; 3) linguistic vulnerability shaping the illness experience; 4) meaning-making and explanatory frameworks; and 5) resources sustaining treatment engagement. Linguistic vulnerability shaped access to biomedical knowledge, participation in healthcare encounters and patient autonomy, while patients mobilised personal, relational, existential, linguistic and institutional resources to sustain treatment continuity. Treatment engagement emerged as a dynamic and relational process embedded within broader migration-related, linguistic and healthcare contexts. Rather than representing fixed determinants or sequential stages, the five dimensions formed an evolving configuration whose relative salience varied throughout the illness trajectory. Conclusion: This study proposes a multidimensional interpretive model of engagement with oral anticancer medication among immigrant patients with limited dominant-language proficiency. Rather than conceptualising adherence as an isolated individual behaviour, the findings show how migration-related contexts shape the conditions under which treatment engagement becomes possible, difficult or fragile. By foregrounding immigrant patients lived experiences, the study identifies experiential, linguistic, relational and structural dimensions of cancer care that are difficult to capture through behavioural adherence measures alone and offers insights for more equitable, context-sensitive and patient-centred oncology care.

Motile microorganisms explore complex environments in search of nutrients, hosts and favourable ecological niches. Plant-pathogenic oomycetes, for instance, undergo such an exploratory phase through biflagellate zoospores that actively swim through water-filled soil pores before infecting host tissues. Linking individual zoospore swimming behaviour to emergent dispersal remains challenging. Here, we present an end-to-end, data-driven framework that transforms time-lapse microscopy image sequences into generative agent-based simulations of zoospore dispersal by inferring local behavioural rules directly from experimental trajectories. Using Phytophthora nicotianae as a model system, we isolated nearly 60,000 zoospore trajectories and quantified both local behavioural descriptors and emergent trajectory properties. Local behavioural measurements were first used to infer an empirical two-state model distinguishing SLOW and FAST swimming regimes while capturing temporal memory and the coupling between speed and turning. Implemented within an agent-based cellular automaton, this model reproduced the principal emergent properties of experimental dispersal. We then independently inferred the behavioural organisation of zoospore swimming using hidden Markov models. The most parsimonious two-state HMM recovered a closely related behavioural organisation, while revealing that the inferred states jointly reflected swimming speed, turning dynamics and directional persistence rather than speed alone. Finally, we challenged the inferred behavioural rules in an independent obstacle-filled environment. Combined with simple collision hypotheses, the model reproduced emergent dispersal without recalibrating the swimming rules and identified transient post-collision slowdown as a key response required to account for the experimental trajectories. Together, these results demonstrate that experimentally inferred local behavioural rules possess predictive power beyond the conditions used for their calibration. More broadly, this work establishes a predictive framework linking quantitative microscopy, behavioural-rule inference and generative modelling of microbial dispersal.

Public-health surveillance systems rely on downstream indicators to infer latent infection incidence, but delays and observation noise provide only an indirect and temporally distorted view of the underlying epidemic process. Reconstructing upstream epidemic trajectories from these observations is therefore an ill-posed inverse problem, in which different reconstruction assumptions may produce different trajectories that remain consistent with the observed data. Here, we develop a general spectral framework that quantifies the statistical distinguishability of candidate upstream trajectories under delayed and noisy observations. We show that epidemiological delay distributions impose a frequency-dependent temporal resolution limit on epidemic surveillance, fundamentally constraining the distinguishability of rapid upstream variation. This limitation propagates to epidemiological inference, making some quantities substantially more sensitive to reconstruction assumptions than others and rendering distinct event-impact profiles difficult to distinguish from downstream observations.

Interception of moving objects is a universal pattern of animal behavior with many industrial applications. Inspired by the behavior of the characid fish Brycon guatemalensis, which catches fruits falling onto the surface of a river, we study one-shot (i.e., open-loop) interceptions using both stochastic optimal control theory and reinforcement learning. In the first instance, an agent actuates a control (wait time before launch and heading angle) to meet its target, given a noisy estimate of the target initial state. In a minimal setting, where launch velocity and actuated control are constants, we show that the optimal wait time is attenuated by noise in both the target measurement and control actuation, as well as agent energy conservation. We then derive an upper bound for the heading angle variance which scales as the ratio of the non-vanishing optimal interception error to the initial planar separation. Complementing our control framework, our deep reinforcement learning approach shows that the neural network representing the fish first learns the value function for the interception, then learns how to aim, and finally learns the optimal wait time. Beyond the specific problem at hand, our results may also have applications to problems with little time for feedback or corrections such as perching, landing, and docking.

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.

The high metabolic cost of walking in individuals with a transtibial amputation is often attributed to reduced ankle propulsion. However, the cost of stabilizing walking may also increase after amputation, potentially explaining why restoring ankle propulsion does not always normalize metabolic cost. Because the contributions of propulsion and stabilization are difficult to dissociate experimentally, we used simulations to probe the effects of transtibial amputation and different prostheses on walking cost. We simulated gait using a conceptual planar torque-driven model. Leg joint torques were controlled by feedforward and linear, time-varying full state feedback. We computed feedforward torques associated with propulsion and feedback gains for stabilizing walking that minimized expected effort (sum of torques squared) in the presence of sensorimotor noise while imposing stability. We simulated walking for an intact model and three walkers with a prosthesis: a passive, a feedforward-controlled active, and a feedforward and local feedback-controlled active prosthesis. We solved a deterministic approximation of the resultant stochastic optimal control problems for different levels of sensorimotor noise. Feedforward and expected feedback torques were higher for the passive prosthesis walker than for the intact walker. An active prosthesis restored total biological effort (i.e., effort of all but the prosthesis joint) to the level of the intact walker, but this effort was generated by fewer biological joints. Biological feedback effort was higher in prosthesis walkers than in the intact walker. Adding local feedback to an active prosthesis reduced biological feedback effort, but only at low noise levels. Our results suggest that walking with a passive prosthesis increases effort related to both propulsion and stabilization. While active prostheses can support propulsion, local feedback control only marginally reduced stabilization-related effort, suggesting that controllers require sensory information from the rest of the body to fully restore the metabolic cost of walking.

Behavioral variability reflects differences in internal and external parameters and provides a substrate for natural selection. However, uncovering its structure is challenging because behavior evolves nonlinearly across multiple timescales. Our goal is to reconstruct phenotypic variation from finite behavioral observations without prescribing the scales at which individuals should be compared. To achieve this, we introduce a multiscale dissimilarity that compares dynamics across all statistically resolvable resolutions by constructing reduced transfer operators at progressively finer scales and weighting their differences by finite-sampling uncertainty. This dissimilarity defines a geometry of dynamical phenotypes, which we reconstruct using diffusion maps. We show that the resulting phenotypic coordinates locally follow the parameter directions to which dynamics are most sensitive. We validate our approach in a stochastic model and apply it to bacterial and larval zebrafish behavior. In bacteria, we identify run speed as a major component of inter-individual variability, associated with the chemotaxis protein CheB. In zebrafish, we find that prior prey experience reshapes behavioral dynamics and phenotypic variability, with paramecia experience concentrating fish in a prey-capture-associated region of phenotypic space. Together, these results provide a framework for identifying internal and environmental variables that structure dynamical phenotypes across scales.

Organisms rely on feedback control mechanisms to maintain key biological variables within functional ranges despite persistent perturbations. Understanding how effectively these mechanisms maintain homeostasis is fundamental, yet quantitative evaluation of feedback performance remains challenging in nonlinear biological systems. Our previously developed framework, Control Ratio (CoRa), addresses this challenge by isolating the contribution of a feedback interaction through controlled comparison with an otherwise identical system in which that interaction has been removed. However, because CoRa evaluates adaptation solely through steady-state responses, it cannot distinguish controllers that ultimately recover to the same state but follow markedly different transient trajectories, despite the potentially profound physiological consequences of those dynamics. Here, we introduce CoRaDyn, a framework for evaluating adaptation as a dynamic process rather than solely as a steady-state outcome. Building on the comparative strategy introduced in CoRa, CoRaDyn quantifies the cumulative effect of feedback throughout the post-perturbation response, generating a time-dependent characterization of feedback performance. This approach reveals how the contribution of feedback depends not only on system parameters but also on the time horizon over which adaptation is evaluated, allowing distinct physiological objectives -- such as rapid recovery or the generation of transient pulses -- to be systematically compared. We demonstrate CoRaDyn using a gene regulatory circuit implementing proportional-integral-derivative (PID) control. Whereas CoRa predicts identical perfect adaptation for all controllers containing integral feedback, CoRaDyn discriminates their transient performance, quantifies the dynamic contributions of proportional and derivative control, and reveals trade-offs that are invisible to steady-state analyses. By extending feedback evaluation from endpoints to the full adaptation process, CoRaDyn broadens the scope of biological questions that can be addressed using the CoRa framework and provides a general approach for comparing feedback architectures when transient dynamics are central to biological function.

From animal societies to self-organizing multi-agent systems, collectives adapt their group structure to tasks and environments. However, how they determine appropriate group sizes and the number of subgroups to form remains unclear. We formulate the Group Size and Number Regulation Problem (GSNRP), which asks how individuals regulate group sizes and numbers using only local information. In a first step, we establish a graph-theoretic model demonstrating that simple following behavior suffices to form group structures that match theoretical expectations, but is insufficient for active regulation of group size and number. In a second step, we operationalize individual group-size preferences in a decentralized fission-fusion mechanism based on perceived group size. Through multi-agent simulations, we validate that this mechanism achieves stable convergence across three signaling regimes, from position-only sensing to continuous group-size communication. Using tracking data from wild white-nosed coatis (mammals in the raccoon family), we calibrate individual group-size preferences and show that the controller recovers selected group-size, subgroup-count, and transition statistics. This in-sample case study demonstrates descriptive consistency with natural fission-fusion dynamics without establishing the underlying behavioral mechanism. These results suggest that natural and engineered collectives may share local principles of perception, preference, and response for regulating group structure.

Microbial populations generally evolve in fluctuating environments under time-varying conditions. These are often described by binary switching models, sometimes seen as coarse-grained feast-famine cycles, in which resource availability switches abruptly between abundant and scarce conditions. However, experimental studies suggest that feast-famine environments actually exhibit more complex temporal dynamics. Here, we study how two strains, one growing slightly slower than the other, compete for the same resources in fluctuating environments comprising a finite number of intermediate states, each having its own carrying capacity. Environmental switching between these states and their carrying capacities represents gradual changes in nutrient availability. This class of multi-state stochastic switching models can be interpreted as a coarse-grained description of feast-famine cycles and allows us to investigate strain competition under the gradual recovery and depletion of resources. By computational and analytical means, we characterise the population dynamics in these multi-state fluctuating environments. In particular, we study how the switching rates and distribution of carrying capacities affect the population-size statistics, fixation probability, and mean fixation time. By comparing these results with their counterparts in binary environments, we clarify how the frequency and amplitude of environmental fluctuations influence population dynamics in coarse-grained feast-famine cycles.

Can observations distinguish a bloom supplied from within a study volume from one supplied across its boundary? We develop a theoretical framework for that question at plankton bloom onset, conditional on a predeclared, observed or calibrated onset event and a declared set of environmental paths, biological responses, and model forms. The estimand follows source-event labels through forcing-dependent survival and genotype-specific growth. Its central certificate asks whether the local onset fraction is invariant over every source history that produces the same time-expanded observation record. For polyhedral history fibers, a Charnes-Cooper transformation computes both sharp dynamic-data endpoints as linear programs. When each source instead has a fixed normalized onset signature, the certificate reduces to a row-space test; uncertain signatures require a joint lifted program. For a finite compatible scenario ensemble, admissible fractions are the union across scenarios, and a point is justified only when every nonempty scenario gives the same singleton. A synthetic two-genotype witness gives the same observed total but local fractions of 2/3 and 1/3 under reversed forcing-response gains. The observer, mixture, and optimization ingredients are established; the contribution is their target-specific synthesis around source at onset. The framework is diagnostic rather than predictive. It specifies what a study must measure--local sources, boundary inflow, forcing, response, timing, and carrier signatures on one declared window--and returns an interval when missing components have justified bounds, including [0, 1] when they remain unconstrained.

Rhizomorphs are specialised, root-like fungal structures whose hierarchical organisation may offer a route to reinforcing mycelium-based materials, yet the environmental regulation of their network formation remains poorly understood. Here, we develop an image-based framework to characterise the longitudinal growth and organisation of Armillaria gallica rhizomorphs under varying nutrient availability and light exposure. Time-lapse imaging was combined with image segmentation, skeleton-based network analysis, optical measurements and Gompertz growth modelling. Nutrient availability produced a distinctly non-monotonic response. Moderate nutrient limitation (0.5 x standard concentration) favoured rapid and coherent exploratory growth, whereas intermediate enrichment (1.5 x) produced the greatest eventual network extent, reaching approximately 975 mm total strand length; network extent and radial expansion differed significantly across nutrient levels (padj = 0.0012). Further enrichment maintained substantial fungal coverage without additional network elaboration, consistent with a shift from long-range exploration towards more locally consolidating growth. By contrast, exclusion of ambient light produced no significant differences after multiple-testing correction. These results reveal a resource-dependent trade-off between exploration and network elaboration, demonstrate that fungal coverage and organised network formation are distinct outcomes, and provide a quantitative basis for controlling self-organised biological architectures for bio-derived material design.

Understanding the relationship between weather conditions and infectious disease transmission has important public health implications; nevertheless, the functional form of this relationship often remains poorly understood. Humidity has previously been suggested to have a U-shaped effect on the transmission of respiratory pathogens, with optimal transmission occurring at both low and high levels. Here, to assess the impact of this functional form and of its potential misspecification, we first simulate transmission rates based on the U-shaped relationship and then test whether an exponential function accurately estimates these rates across a range of climates. We further use a transmission model to simulate case count data for a range of hypothetical pathogens, comparing outbreak dynamics under the U-shaped forcing function to those under the best-fit exponential approximation. In our simulations, the model fit for the exponential function is climate-dependent, performing best in very dry and very humid environments. In locations where specific humidity values centre around the vertex of the U-shaped function, the exponential model underestimates the amplitude and frequency of incidence peaks. These findings demonstrate the potential consequences of assuming an incorrect functional form for weather-infectious disease relationships and highlight the need for more careful consideration of this functional form in modelling practice.

Trigger waves are positive feedback-mediated traveling waves that propagate without slowing down or diminishing in amplitude, enabling biochemical coordination over distances where diffusion alone would be insufficient. Apoptosis can spread as trigger waves of caspase activity, as demonstrated in Xenopus eggs and egg extracts. We measured apoptotic wave speeds in extracts at two cell cycle states and found that waves propagated more slowly in M-phase arrested cytostatic factor (CSF) extracts than in interphase extracts (21 versus 27 m min-1). Luther's equation, v=2{surd}D{kappa}, predicts wave speed from the diffusion coefficient D and the positive-feedback doubling rate {kappa}. However, D and {kappa} were comparable between the two extracts and could not explain the speed difference. This discrepancy highlighted a key limitation of the classical equation: it does not account for an activation threshold in trigger wave initiation, and in fact the threshold required to trigger apoptosis was found to differ between cell cycle states. We therefore extended Luther's equation to include such a threshold, obtaining v{propto}{surd}(AD/Uc ), where A is the reaction rate and Uc is the activation threshold. Perturbing apoptotic sensitivity confirmed the predicted threshold-speed relationship. These findings establish the activation threshold as a key determinant of trigger wave speed and provide a broadly applicable scaling law for predicting the spread of switch-like biochemical signals.