Bulletin of Mathematical Biology

We propose a minimal mathematical framework to describe the go-or-grow dynamics of tumor cells comprising two phenotypically distinct populations. One population is migratory and undergoes constant diffusion, while the other proliferates in an oxygen-dependent manner. The local oxygen concentration governs transitions between these phenotypes. We then ask whether the time scale of cyclic hypoxia, where oxygen is spatially homogeneous but fluctuates between normoxic and hypoxic levels over time, alters the dynamics of the reduced generalist model in the fast phenotype-switching regime compared with the full go-or-grow model. We observe that the reduced generalist phenotype model fails to capture the go-or-grow dynamics under rapid oxygen fluctuations. Our results predict that the tumor growth rate increases in the generalist-phenotype model under rapid oxygen fluctuations compared with tumors of the proposed go-or-grow model. We then explore whether oxygen consumption, which leads to spatial heterogeneity in oxygen concentration, plays a role in distinguishing the dynamics of the reduced model from the proposed go-or-grow model. We demonstrate that the tumor in the proposed go-or-grow model exhibits a higher growth rate than the tumor in the reduced generalist-phenotype model under a slow oxygen consumption rate.

Fractional differential equations are increasingly used to represent temporal memory in tumour immune and pharmacokinetic models, but changing the derivative order without a consistent time scale can make comparisons difficult to interpret. We study a dimensionless, commensurate Caputo system for tumour, immune, normal cell, and drug states. We establish positivity, boundedness, the tumour invasion threshold, and local stability conditions. We then use a fractional Adams Bashforth Moulton predictor corrector method with an explicit mesh convergence study. Two conclusions correct potentially misleading interpretations of the model. First, the equilibrium locations are independent of the fractional order: decreasing the order changes transient time scales but does not by itself alter the model's asymptotic states. Second, after a unit bolus the drug equation has the exact solution C({tau})=E(-K{tau}). Its maximum is therefore the initial concentration, not a delayed peak. For 0<<1, the algebraic tail makes the infinite-horizon area under the concentration curve diverge. In the illustrative coupled simulations, the apparent late tumour reduction at small is consequently driven by persistent modelled drug exposure and must not be read as evidence of superior therapy. All parameters are dimensionless and illustrative; no patient data are fitted. Source code and generated tables are included to make every numerical statement reproducible.

Moment-closure methods provide a widely used approach for approximating low-order statistics of stochastic biochemical reaction networks, but the resulting closed equations may exhibit physically meaningless dynamics. Here, we systematically investigate the stability of second-order moment-closure approximations for a bursty gene expression model with nonlinear protein degradation. We consider the Gaussian, log-normal, and negative-binomial closures. By analyzing the resulting dynamical systems, we show that the Gaussian closure fails the proposed stability criterion for all parameter values, whereas both the log-normal and negative-binomial closures possess a unique physically accessible fixed point and are stable if and only if kB(1 + 2B) [&ge;] 1, where B is the mean burst size and k is the dimensionless protein synthesis rate. Moreover, this condition is automatically satisfied when the steady-state mean protein copy number is at least one, suggesting that it is mild in biologically relevant regimes. Numerical comparisons over a broad parameter range further show that the negative-binomial closure is the most accurate, while the Gaussian closure is the least accurate. We further develop higher-order negative-binomial moment-closure schemes and demonstrate that increasing the closure order substantially improves predictive accuracy. These results highlight the importance of considering dynamical stability, in addition to numerical accuracy, when assessing moment-closure approximations.

Biological populations have been evolving in multi-dimensional trait spaces. Ecological interactions can induce the evolutionary diversification of a biological population into distinct multiple populations, which is called evolutionary branching in adaptive dynamics theory. If possible mutations are restricted to certain directions due to developmental, physiological, or physical constraints, the resulting evolutionary dynamics may be restricted to subspaces with fewer dimensions than the original trait spaces. In general, such constraints are also evolving slowly, due to rare and slight mutational deviations from the constraint subspaces. In this sense, constraints are not strict. However, the method for describing non-strict constraints that is globally consistent has yet to be developed. This study proposes a general and flexible method for constructing a non-strict constraint subspace from the corresponding strict constraint subspace. This method enables efficient analysis of adaptive evolution including evolutionary branching under non-strict constraints in the framework of adaptive dynamics theory. Applying this method to a resource competition model shows that the shape of the carrying capacity distribution can be treated as a non-strict constraint curve, and that the likelihood of evolutionary branching is affected not only by the shape of the constraint curve but also by its slow evolution.

Buruli ulcer (BU) is a neglected tropical skin disease caused by infection with Mycobacterium ulcerans. The mode of its transmission into the skin has not been fully clarified, but it is believed to involve mechanical transfer, perhaps by insect bites. M. ulcerans secretes a toxin called mycolactone which causes necrosis of the skin and formation of large skin lesions. Understanding the development of these skin lesions over time is crucial to understand the pathogenesis of BU. We formulate a mathematical model to describe the space-time dynamics of M. ulcerans bacteria, mycolactone and skin cell densities as well as fibrin deposition. We investigate the relative contributions to ulcer formation of direct cytotoxicity of mycolactone and ischaemia following the loss of coagulation control. Our results are consistent with existing biological evidence that higher mycolactone concentrations accelerate cell death. Furthermore, the model indicates that the rate of wound expansion is primarily controlled by the diffusion and net birth rate of bacteria, which sets the overall scaling for the speed of wound expansion, along with two dimensionless constants that define the shape of the trajectory of wound edge position against time, namely the ratio of the mycolactone diffusion coefficient to the bacterial diffusion coefficient, and the ratio of the mycolactone degradation rate to the bacterial net birth rate. When mycolactone diffuses significantly faster than bacteria spread or when mycolactone degrades more slowly than bacteria reproduce, the initial development of the wound is faster. In contrast, the long-term wound expansion speed appears to be relatively unchanged across a wide range of the dimensionless parameters.

Pseudoprogression in response to immune checkpoint inhibitor (ICI) therapy can be a complex challenge during treatment response evaluation. Erroneous classification of progression may result in stopping effective treatment for patients, while delayed confirmation of progression can prolong an ineffective treatment course. We develop a mechanistic mathematical model of tumor-immune dynamics under ICI treatment in which observed tumor volume includes contributions from both tumor burden and immune infiltration. We show analytically how pseudoprogression can arise from increased immune infiltration and proliferation as well as continued tumor growth during the ``ramp-up" phase. The model also incorporates circulating tumor DNA (ctDNA) dynamics, which we demonstrate are able to closely fit existing longitudinal tumor and ctDNA data and extract realistic parameter ranges to generate representative in silico data. Using model-generated cohorts, we train a random forest (RF) classifier using longitudinal tumor and ctDNA measurements to distinguish pseudoprogression and true progression, and demonstrate that it improves response classification relative to the Response Evaluation Criteria in Solid Tumors (RECIST 1.1) and provides classification earlier than the immunotherapy-specific iRECIST, even in the presence of tumor measurement noise. This work predicts that paired longitudinal tumor-ctDNA measurements may enable earlier discrimination of true progression from pseudoprogression, and motivates collection of imaging and ctDNA data synchronously during ICI therapy, especially in the setting of suspected disease progression.

Non-exponential residence times give rise to history-dependent or memory effects in epidemic dynamics, yet their mathematical formulations remain fragmented. Here we develop a unified framework for a history-dependent SEIR model with general latent- and infectious-period distributions. We first establish the equivalence between the stage-age and renewal formulations and show that both admit a unified transition-flow representation. We then introduce the concept of a transition-flow multiplier, which provides a direct link between residence-time distributions and transition flows. When the multiplier is a weighted sum of shifted power functions, the model admits an exact representation in terms of Riemann-Liouville or tempered Riemann-Liouville fractional operators, providing a first-principles foundation for fractional epidemic models. Finally, using Euler-Lotka equations and stochastic-order arguments, we characterize how the latent- and infectious-period distributions shape epidemic growth and decay. A longer latent period slows both early growth and late decay, while, at fixed means, greater variability in the latent period accelerates early growth but slows late decay, whereas greater variability in the infectious period slows both.

Host immunity acquired through prior infections is a key factor shaping the spread of emerging variants. However, accounting for complete infection histories in mathematical models is challenging because the number of possible histories grows exponentially with the number of variants. Here, we present a flexible framework for modeling the co-circulation of multiple variants emerging at arbitrary times while incorporating cross-immunity and variant-specific characteristics. We introduce three tractable approximations: truncation, which retains only infection histories up to a given number of strains; one variant at a time, which is exact when variants circulation does not overlap; and two variants at a time, which is exact when at most two strains co-circulate. We systematically evaluate these approximations across a range of cross-immunity levels and inter-variant emergence delays. Truncation approximations perform best under short emergence delays and strong cross-immunity, whereas the other approaches outperform them at longer delays. When applied to SARS-CoV-2 variant dynamics, the second-order truncation and the two variants at a time approximation accurately capture incidence patterns, whereas the other approaches perform well only for specific subsets of variants. These results identify the parameter regimes in which simplifying assumptions remain valid, provide guidance for model selection, and highlight the key role of infection history in shaping epidemic trajectories.

Devireddy et al. [1] previously showed that an enthalpy-based, macroscale model simulated tissue freezing histories are in close agreement with a coupled model incorporating cellular water transport and intracellular ice formation (IIF). That comparison, however, was performed for a limited set of tissue geometries, cooling conditions, and biophysical parameters. Here, we revisit that conclusion by reimplementing the published model from its governing equations and systematically expanding the parameter space to determine when microscale processes alter the predicted macroscale freezing response. We incorporate both surface catalyzed nucleation (SCN) and volume catalyzed nucleation (VCN) of intracellular ice with cellular osmotic dehydration and examine a broad range of tissue dimensions, cooling rates, convective boundary conditions, membrane permeabilities, cell sizes, activation energies, intracellular water fractions, and nucleation rates. Linearization of the cellular water transport equation about osmotic equilibrium yields an osmotic relaxation timethat can be compared with the local residence time of tissue undergoing phase change. Their ratio defines a dimensionless group, that organizes the transition between regimes in which the uncoupled enthalpy formulation and the coupled micro and macroscale formulation produce similar or substantially different results. Across the one-dimensional parameter space examined here, the two models remain within a prescribed thermal-error bound for the dimensionless group below approximately 0.1, whereas the discrepancy increases systematically above this threshold. Model divergence also depends on intracellular ice nucleation, i.e., when nucleation is sufficiently slow, cellular water transport can become rate limiting for latent heat release. Conversely, sufficiently rapid intracellular ice formation reduces the macroscopic discrepancy by providing an additional pathway for latent-heat release. Multidimensional simulations further show that geometry alone can move a nominally conventional freezing protocol into the coupled regime. In particular, multidirectional cooling can reduce the local residence time sufficiently to produce substantial differences in predicted frozen volume even when the corresponding thermal histories appear similar. Finally, we formulate an inverse approach for estimating IIF parameters from tissue scale thermal histories. Synthetic simulations indicate that parameter recovery is feasible within a finite range of cooling conditions, providing a testable strategy for future experimental measurements. Together, these results provide a mechanistic criterion for deciding when microscale biophysical processes must be retained in macroscale tissue-freezing models.

Regular environmental cycles such as day and night or the lunar cycle are ubiquitous and drive many biological processes. Many organisms have evolved endogenous time-keeping mechanisms to anticipate these cycles to accurately time their behavior or synchronize physiological processes. The evolution of such synchronization to environmental cycles and their variability is not very well understood. As an example, we use the marine insect Clunio (Diptera, Chironomidae), which synchronize adult emergence and reproduction with the lunar cycle, and show different emergence patterns across populations. We use a model constructed with replicator-mutator dynamics to provide a novel outlook at the evolution of these emergence patterns. In particular, we investigate the effect of two traits of individual phenotypic variability of emergence, the spread and shape, on population-wide emergence patterns. We find that the semilunar and lunar emergence patterns are alternatively stable and occur when the phenotypic distribution has a narrow and is roughly flat. Arrhythmic emergence occurs when the phenotypic distribution is wide and nearly flat. The semi-lunar pattern can only be observed when the parameter of rhythmicity is changed. With our model, we introduce the concepts of frequency-dependent selection and individual phenotypic variability into the theory of chronobiology, which gives us insights into the evolution of rhythmicity in these organisms.

Mechanistic models of infectious disease transmission provide a framework for understanding and predicting disease dynamics, but statistical inference is often computationally prohibitive. This challenge is especially pronounced for models with high-dimensional structures and many unknown parameters, where trajectory matching may require thousands of evaluations of an expensive likelihood function. We developed a surrogate-assisted optimization framework that uses Bayesian optimization to reduce the number of evaluations of the underlying mechanistic model required for parameter inference. We evaluated the approach using simulated fitting problems based on age-structured susceptible-exposed-infected-recovered models with 4, 8, and 16 age classes and a 210-compartment age-structured rotavirus transmission model. Applying surrogate-assisted optimization substantially reduced the number of likelihood evaluations required to identify good parameter estimates, with larger improvements as model dimensionality increased. In the rotavirus model, the surrogate Bayesian optimization approach reached the benchmark after approximately 0.7 hours, compared with approximately 9.4 hours using traditional Nelder-Mead optimization. These results demonstrate that surrogate-assisted optimization can substantially reduce the computational burden of trajectory matching for complex mechanistic transmission models, particularly for higher-dimensional inference problems.

Oncogene amplification on circular extrachromosomal DNA (ecDNA) has been linked to poor prognosis and higher treatment resistance in multiple types of human cancer. ecDNA are mobile genetic elements lacking centromeres that are partitioned unevenly into daughter cells at mitosis. While random segregation of ecDNA contributes to gene-copy-number heterogeneity among tumour cells, how ecDNA contribute to phenotypic heterogeneity, via mRNA and proteins, and how this affects targeted therapy outcomes is still not understood. In fact, cancer cell populations with ecDNA show remarkable heterogeneity in mRNA concentrations, which is not explained by gene-copy-number heterogeneity alone. In this paper, we use a stochastic model of gene transcription to mathematically model mRNA production and decay in cancer cells with ecDNA, extending a method that has been used to study phenotypic heterogeneity in eurokaryotic and prokaryotic cells. Using our model, we explore the space of possible mRNA distributions from a given ecDNA distribution, under different model parameter regimes. Our model is able to produce realistic mRNA distributions and we use it to estimate transcriptional parameters from single-cell measurements of ecDNA and mRNA counts in cell-line data. We discover a remarkable similarity in estimated parameters across cell lines and amplification types, indicating similar processes underlying transcription in ecDNA and chromosomal amplifications. We also model genotoxic therapy and use our model to predict the mRNA composition of persisting cells, showing the usefulness of such a modelling framework to understanding the origins of treatment resistance in ecDNA-containing cancers.

A major transition in the evolution of eukaryotic cells was phagocytosis: the ability of one cell to engulf another. In some cases, the engulfed cell persists, forming an endosymbiosis which is often mutually beneficial. However, it is unclear whether the interaction between the ancestors was also cooperative. There are clear benefits to developing the ability to eat a competitor, but can it ever be beneficial to eat a cooperator? We use a dynamical systems framework to identify the ecological conditions under which phagocytosis can evolve from a system of cooperating microbes.Beginning with a system of two cooperating species, we use invasion analysis to identify the parameter regimes under which a phagocytosing mutant can invade and the subsequent population dynamics. We identify a critical threshold that determines whether an invading predator will coexist with the cooperative community or replace members. Repeatedly invading with different predators, we find that the system saturates when there are three coexisting species, though this community does not have a single optimal composition. However, if we incorporate additional costs, we find there is a single evolutionarily stable community of two cooperators and a pure predator. Ultimately, we show that it is possible for phagocytosis to evolve from a cooperative community and, moreover, coexist with the original cooperating species.

Species at the highest trophic levels are generally the least resilient to perturbations, yet the factors that determine their persistence remain poorly understood. We ask how the intrinsic self-regulation of a basal host governs the persistence of the trophic levels above it. To address this question, we use a minimal discrete-time model of a host-parasitoid-hyperparasitoid system motivated by the biology of trigonalid wasps, obligate hyperparasitoids that complete development only when their caterpillar host is also attacked by a primary parasitoid. We compare two forms of host self-regulation in the absence of natural enemies: scramble (Ricker) fecundity, appropriate when the parasitoid is the host's principal enemy and which recovers the classical model of Beddington and Hammond as a special case; and contest (Beverton-Holt) fecundity, appropriate when the host is held in check by many additional enemies. Using numerical continuation, Lyapunov exponents, and direct computation of the model's trajectories, we find that the form of host regulation is decisive for persistence at the top trophic level. Under scramble host dynamics, coexistence is extremely fragile with respect to both parameter values and initial conditions. Under contest host dynamics, coexistence is robust across broad parameter ranges and loses stability only gradually, through a supercritical Neimark--Sacker bifurcation and a subsequent torus-doubling route to chaos. Therefore, in at least some trophic networks, the resilience of the highest levels can be determined at the base of the network.

Allergic conjunctivitis is an inflammation of the conjunctiva caused by allergen; it is common disorder affecting up to 40% of the population. In this work, we study seasonal allergic conjunctivitis (SAC), also called "hay fever eyes", which is caused by exposure to airborne pollens. We develop a mathematical model quantifying the ocular immune system response to the allergens. First, we present a simplified qualitative description of the immunopathogenesis of SAC. Then, we express each chosen immunopathological mechanism mathematically to construct a system of thirty-one ordinary differential equations. We compare summary statistics of the predicted observable immune signals to experimental measurements and find our model captures key qualitative features of SAC progression. We then compare our predicted time series of histamine concentration to symptom scores and find a strong correlation suggesting the model predicts relevant clinically trends. Next, we calibrate the model through multi-step process. We find the most influential parameters are the production and depletion rates of IL-4, and the production rates of IL-5 and IL-8. These cytokines are targeted in treatments for asthma, atopic dermatitis, and severe eosinophilic associated disorder and suggest potential therapeutic targets for SAC. Our calibrated model mimics most of the summary statistics of the experimentally observable immune signals with discrepancies for IL-5 and IL-13 indicating that additional immunopathological mechanisms could be important.

Spatial expansion is a common feature of many ecological processes, including biological invasions, range shifts, population recovery, and the spread of pathogens and human activities. Although such expansion often arises from diffusive processes, there remain few general statistical methods to quantify these dynamics. Here, we present a diffusion-enhanced spatio-temporal model (DESTM) that approximates diffusive process using Gaussian Markov random fields (GMRF). The model works by incorporating the statistical interaction between spatial diffusion and temporal lags into a joint precision structure of the latent field. To demonstrate, we simulate data from a diffusion movement process using a continuous-time Markov chain (CTMC), and then approximate this mechanistic process using the DESTM. We show that separable spatio-temporal models can lead to large bias in estimates of the local density. In contrast, the DESTM did not appear to bias inferences, and reverts to the separable model when diffusion is not apparent. Finally, we apply the model to two large-scale examples of spatial expansion: (1) the historical spread of the Japanese distant-water longline fleet across the Pacific, representing anthropogenic spatial expansion in the ocean, and (2) the recovery and geographic expansion of Bald Eagle populations across North America. Estimates from both case studies, supported by model comparisons and ecological plausibility, suggest that the DESTM provides a better description of the underlying expansion dynamics, although improvements in forecast performance were more apparent for the Japanese longline fleet. We highlight that the model provides a fast and efficient framework for modeling spatial expansion dynamics, with the potential to represent increasingly complex ecological dynamics through future extensions. We propose that future research explore this framework as a quantitative tool to explain how spatial expansion occurs.

Diffuse midline gliomas are a group of tumours for which no effective therapies exist. These tumours have substantial intratumoural heterogeneity and harbour beneficial subclonal interactions. Therapeutic failure due to resistance is a major concern in these tumours. We demonstrate how distinct subpopulations derived from the same tumour can interact with one another to alleviate the impacts of therapeutic intervention. This is achieved through the establishment of a mathematical extension to the Lotka-Volterra competition model, where interactions perturbing the Allee effect were demonstrated to exist. These interactions had substantial downstream consequences as they were determined to be perturbed by therapeutic interventions and strikingly, when simulating a two-drug paradigm, the interactions were demonstrated to substantially later the outcomes of treatment scheduling.

Cancer progression often requires the accumulation of multiple driver mutations, yet the same set of drivers may be acquired in different orders. How these alternative mutation-order pathways jointly shape tumor clonal structure remains unclear. We develop a multitype branching-process model in which malignant transformation requires two driver mutations and distinguish malignant cells both by their mutation order and by the independent transformation event that founded their clone. Under a successive exponential approximation, we establish point-process limits for the pathway-specific clone-size processes and derive a closed-form expression for the limiting expected Simpson's index of the combined malignant population. When the two mutation orders produce malignant cells with the same net growth rate, the limiting index separates into effective pathway weights, determined by mutation rates and birth-death dynamics at preceding stages, and within-pathway concentration terms, determined by intermediate-to-malignant growth-rate ratios. This decomposition shows that a driver's effect on heterogeneity depends critically on when it is acquired. A strong driver acquired early expands the intermediate lineage and increases the supply of independent malignant founders, whereas the same driver acquired at the final transition strengthens the growth and age advantage of early-founded malignant clones. Under additive fitness effects, these counteracting mechanisms can produce a non-monotone relationship between selective advantage and clonal concentration. We further show that threshold-like non-additive fitness effects can generate highly concentrated malignant populations, while order-dependent terminal fitness leads the faster-growing pathway to dominate asymptotically. Together, these results reveal how mutation order, mutational accessibility, selection, and epistasis jointly determine lineage-level intratumor heterogeneity.

Bacterial mutation rates are traditionally inferred from phenotypic data using fluctuation assays. Mutation rate computation from these assays relies on a mathematical model describing the emergence of mutants during population growth. However, this standard model relies on restrictive assumptions that are often transgressed in real biological setups. Key assumptions include equal growth rate of the wild-type and the mutant population, no cell death during the growth phase, and no post-growth sampling. Although several refined mathematical models have been proposed over the last decades to circumvent some of these assumptions, none can fully account for the complex ecological scenarios frequently observed in the lab or in the wild, such as the combination of cell death and an unknown fitness effect of the mutation, or non-constant death rates. In this work, we propose to replace these standard mathematical models by a stochastic simulation framework, which can circumvent all these assumptions. We provide a fast and accurate implementation of this stochastic model, and combine it with an Approximate Bayesian Computation scheme to perform simulation-based, likelihood-free parameter inference. Our approach can simultaneously infer mutation rate and fitness effect of the mutation while accounting for death and partial plating, and could be extended to arbitrary complex conditions that can be efficiently simulated.

In this study, we develop a mathematical model of the hypothalamic pituitary adrenal (HPA) axis and systematically investigate the model complexity required to reproduce observed hormone dynamics. The model is fitted to ACTH and cortisol measurements from five healthy volunteers reported in the literature. We show that simultaneous reproduction of both hormone profiles requires a delay differential equation model with at least five state variables among the model architectures considered. Following fitting the volunteer data, we investigate the response of the model to sustained stress using bifurcation analysis. The coupled system exhibits bifurcations over a range of stress intensities. This analysis reveals bifurcations whose critical stress thresholds depend strongly on the modulatory state of the system. These results provide a mathematical framework for investigating transitions between physiological and dysregulated HPA axis dynamics, which may be related to states of health and disease.