Bulletin of Mathematical Biology最新文献

{"title":"Gene Regulatory Networks that Support Multi-fate Cellular Decisions.","authors":"B V Harshavardhan, Sarah Adigwe, Mohit Kumar Jolly, Tomáš Gedeon","doi":"10.1007/s11538-026-01791-4","DOIUrl":"10.1007/s11538-026-01791-4","url":null,"abstract":"<p><p>Cell fate decisions are driven by gene regulatory networks (GRNs). While the mutually inhibitory toggle switch effectively models binary fate decisions, fully connected inhibitory networks with more than two nodes fail to capture multi-fate decisions due to the low prevalence of \"single high states\", where only a single master regulator is highly expressed. The goal of this study is to find network structures that support all single high states. We find that the only network that attains the highest possible prevalence of all single high states within the set of monotone Boolean (MB) models is completely disconnected. Since biological networks typically require connectivity, we investigate network structures that support equipotency, where all single high states have equal prevalence within MB models. Finally, we characterize the networks that support multistability between all single high states, finding that it is possible only in networks in which each node either has self-activations or is inhibited by every other network node. Our findings provide a theoretical framework for understanding the network design principles that can support simultaneous differentiation into multiple distinct cell types.</p>","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 12","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149357141","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Eco-Evolutionary Feedback and Basin-Dependent Survival in a Ricker Model with a Strong Allee Effect.","authors":"Karima Mokni, Lahsen Ourbaa, Saber Elaydi, Amina Eladdadi, Mohamed Ch-Chaoui","doi":"10.1007/s11538-026-01713-4","DOIUrl":"https://doi.org/10.1007/s11538-026-01713-4","url":null,"abstract":"<p><p>Trait evolution can be both a population's rescue and its ruin. This raises a central eco-evolutionary question: under a strong Allee effect, can the evolution of a quantitative trait help a sparse population escape extinction, or can it instead destabilize an otherwise viable population and increase extinction risk? To address this, we study a discrete-time Darwinian extension of the Ricker model coupling population density to the evolution of a mean trait: a low-density reproductive limitation term generates the strong Allee effect, and the trait evolves along the fitness gradient. The model is a two-dimensional map with eco-evolutionary feedback: density shapes selection, while the evolving trait alters reproduction and competition. After determining the equilibrium structure and the threshold separating extinction from positive-density survival, we analyze how the upper positive equilibrium loses stability. Period-doubling (PD) and Neimark-Sacker (NS) bifurcations replace steady survival by alternating-generation and oscillatory dynamics. Codimension-two phenomena-the <math><mrow><mn>1</mn> <mspace></mspace> <mo>:</mo> <mspace></mspace> <mn>2</mn></mrow> </math> point where the PD and NS curves meet, together with the <math><mrow><mn>1</mn> <mspace></mspace> <mo>:</mo> <mspace></mspace> <mn>3</mn></mrow> </math> and <math><mrow><mn>1</mn> <mspace></mspace> <mo>:</mo> <mspace></mspace> <mn>4</mn></mrow> </math> strong resonances-organize transitions toward two-, three-, and four-generation cycles, with quasiperiodic and more complex fluctuations. These mechanisms arise from the non-monotone Ricker-type recruitment term and are absent from the monotone one-dimensional Beverton-Holt recruitment map. We show that evolutionary response and trait-mediated competition can either favor persistence or promote destabilization, depending on parameters. Basin computations reveal coexistence between positive-density attractors and extinction, so long-term survival depends on both parameters and initial conditions. Overall, the model clarifies how Darwinian trait evolution and low-density demographic limitation jointly shape extinction thresholds, oscillatory survival, and population fate.</p>","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 12","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149340933","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Phylogenetics in a Warm Place: Computational Aspects of the Tropical Grassmannian.","authors":"Samir Bhatt, John Sabol, Papri Dey, Matthew J Penn, David Duchene, Ruriko Yoshida","doi":"10.1007/s11538-026-01773-6","DOIUrl":"10.1007/s11538-026-01773-6","url":null,"abstract":"<p><p>Phylogenetic trees provide a fundamental representation of evolutionary relationships, yet the combinatorial explosion of possible tree topologies renders inference computationally challenging. Classical approaches to characterizing tree space, such as the Billera-Holmes-Vogtmann (BHV) space, offer elegant geometric structure but suffer from statistical and computational limitations. An alternative perspective arises from tropical geometry: the tropical Grassmannian ( <math> <mrow><mrow><mspace></mspace> <mtext>tropGr</mtext> <mspace></mspace></mrow> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mi>n</mi> <mo>)</mo></mrow> </math> ), which Speyer and Sturmfels showed coincides with the space of phylogenetic trees. In this paper, we review the structure of the tropical Grassmannian and present algorithmic methods for its computational study, including procedures for sampling from the tropical Grassmannian. Our aim is to provide a computational perspective on these ideas for readers interested in phylogenetic tree space, and to motivate new research at the interface of tropical geometry and phylogenetic inference.</p>","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 12","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13740000/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149340934","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Multistationarity in Semi-open Phosphorylation-Dephosphorylation Cycles.","authors":"Praneet Nandan, Beatriz Pascual-Escudero, Diego Rojas La Luz","doi":"10.1007/s11538-026-01771-8","DOIUrl":"https://doi.org/10.1007/s11538-026-01771-8","url":null,"abstract":"<p><p>Multistationarity underlies biochemical switching and cellular decision-making. We study how multistationarity in the sequential n-site phosphorylation-dephosphorylation cycle is affected when only some species are open, meaning allowed to exchange with the environment (so-called semi-open networks). Working under mass action kinetics, we obtain two complementary structural results for <math><mrow><mi>n</mi> <mo>≥</mo> <mn>2</mn></mrow> </math> . First, opening any nonempty subset of the substrate species preserves the network's capacity for nondegenerate multistationarity. Second, opening the enzyme species (both kinase and phosphatase), possibly together with any subset of substrates, always destroys multistationarity. The latter result is proved by a general reduction framework combining the detection of absolute concentration robustness (ACR) with projection onto the remaining species; when the projection produces a monostationary network, the full semi-open system is monostationary. We also illustrate the general method on multi-layer cascade variants and discuss biological implications.</p>","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 11","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149318311","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Sensitivity of Evolutionary Entropy and Reproductive Potential in Lefkovitch and Open-Group Leslie Matrices.","authors":"Henrique M Oliveira","doi":"10.1007/s11538-026-01774-5","DOIUrl":"https://doi.org/10.1007/s11538-026-01774-5","url":null,"abstract":"<p><p>We develop a sensitivity theory for the Malthusian parameter <math><mi>r</mi></math> , evolutionary entropy <math><mi>H</mi></math> , and reproductive potential <math><mi>Φ</mi></math> in Lefkovitch matrices, with open-group Leslie matrices as a special case. Starting from the dominant eigenvalue and the Markov chain associated with the projection matrix, we derive explicit formulas for the long-run stage distribution, generation time, evolutionary entropy, and reproductive potential. Evolutionary entropy and reproductive potential both admit exact transition-retention decompositions, with distinct interpretations: for <math><mi>H</mi></math> the separation concerns sources of demographic uncertainty, whereas for <math><mi>Φ</mi></math> it partitions the demographic links themselves. For small changes to positive coefficients, we obtain closed-form sensitivities for fertility, progression, and retention. The sensitivity of <math><mi>r</mi></math> follows from the standard formula for a simple dominant eigenvalue, while that of <math><mi>Φ</mi></math> follows from the exact identity <math><mrow><mi>r</mi> <mo>=</mo> <mi>H</mi> <mo>+</mo> <mi>Φ</mi> <mo>.</mo></mrow> </math> The formulas specialise to open-group Leslie matrices, in which retention occurs only in the terminal stage, and reduce to the classical Leslie theory when retention vanishes. We also distinguish a change to an existing demographic link from the creation of a fertility or retention link whose coefficient was zero. When a new link is created, <math><mi>r</mi></math> still has a finite one-sided derivative, whereas the leading changes in <math><mi>H</mi></math> and <math><mi>Φ</mi></math> are opposite terms of order <math><mrow><mi>ε</mi> <mo>log</mo> <mi>ε</mi></mrow> </math> . Empirical examples show that growth, evolutionary entropy, and reproductive potential can respond differently to the same demographic change. They also show that stage retention may account for most of <math><mi>H</mi></math> without determining whether <math><mi>H</mi></math> rises or falls when retention is increased.</p>","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 11","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149294669","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Approximating Peak Prevalence in Multistage SIR Epidemics.","authors":"Denis Tverskoi, Andrew Gothard, Grzegorz A Rempala","doi":"10.1007/s11538-026-01785-2","DOIUrl":"10.1007/s11538-026-01785-2","url":null,"abstract":"<p><p>Estimating peak prevalence is a central problem in epidemic modeling because it determines the period of greatest infectious burden and is closely linked to health-care demand. In multistage SIR models, however, peak prevalence is generally less tractable than in the classical model with exponentially distributed infectious periods. Motivated by the use of weighted infectious-stage aggregates as surrogates for prevalence, we investigate the relationship between the prevalence peak and the maximum of a weighted stage functional in a deterministic SIR-like epidemic model with k identical infectious stages, which we call an SI(k)R model. We show that this relationship depends critically on how the stage-progression rate is scaled as the number of infectious stages increases. Under naive scaling, in which the progression rate remains fixed, the weighted peak is asymptotically equivalent to the prevalence peak and the commonly used factor-two approximation fails. Under Erlang scaling, which preserves the mean infectious period, the multistage model converges to a delay formulation in which prevalence and the weighted stage functional become unweighted and triangularly weighted moving averages of incidence. This limiting representation provides a theoretical basis for the factor-two approximation and identifies the regimes in which it is accurate. It also explains why this approximation deteriorates as epidemic waves become more sharply peaked. We derive analytical error bounds and develop curvature-based and parameter-based corrections that substantially improve accuracy. Numerical studies confirm these improvements across a broad range of epidemiological parameters. Overall, the results show when weighted-stage peaks can be used reliably as proxies for peak prevalence and how the resulting estimates can be refined when the standard approximation loses accuracy.</p>","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 11","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13719725/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149294587","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"<ArticleTitle xmlns:ns0=\"http://www.w3.org/1998/Math/MathML\">Same <ns0:math><ns0:msub><ns0:mi>R</ns0:mi> <ns0:mn>0</ns0:mn></ns0:msub> </ns0:math> , Different Risk: Colonization-Age Timing and Heterogeneity Drive ICU MRSA Establishment and Burden.","authors":"Taye Samuel Faniran","doi":"10.1007/s11538-026-01770-9","DOIUrl":"10.1007/s11538-026-01770-9","url":null,"abstract":"<p><p>Intensive care unit (ICU) control of methicillin-resistant Staphylococcus aureus (MRSA) occurs over short horizons, whereas <math><msub><mi>R</mi> <mn>0</mn></msub> </math> summarizes average transmission at the importation-free disease-free state without encoding timing. We couple within-host bacterial and immune dynamics to an ICU model structured by colonization age, generating transmission and progression kernels that vary with colonization age. A two-stage framework uses a branching process to quantify finite-threshold crossing and a deterministic ward model to quantify conditional burden after the threshold is reached. Calibration fixes <math><mrow><mi>η</mi> <mo>=</mo> <mn>0.10</mn></mrow> </math> and distributes colonized admissions across pre-admission colonization ages. Of 1, 000, 000 parameter sets, 45, 250 satisfy cumulative <math> <mrow><msup><mi>R</mi> <mn>2</mn></msup> <mo>≥</mo> <mn>0.90</mn></mrow> </math> . The representative calculation is subcritical, with <math> <mrow><msub><mi>R</mi> <mn>0</mn></msub> <mo>=</mo> <mn>0.907957</mn></mrow> </math> ; across accepted sets, the median and 95% behavioral range are 0.917464 and <math><mrow><mo>[</mo> <mn>0.814593</mn> <mo>,</mo> <mspace></mspace> <mn>1.032975</mn> <mo>]</mo></mrow> </math> , with <math><mrow><mn>92.460</mn> <mo>%</mo></mrow> </math> below one. At <math> <mrow><msub><mi>Z</mi> <mtext>crit</mtext></msub> <mo>=</mo> <mn>5</mn></mrow> </math> , exact threshold-reaching probabilities are 0.114659 for age-dependent transmission and 0.117074 for matched constant transmission. Their accepted-set difference range crosses zero, so this contrast has no robust direction. By contrast, age-dependent transmission reaches 50% of reproductive potential earlier in <math><mrow><mn>98.378</mn> <mo>%</mo></mrow> </math> of accepted sets. In 200-day conditional simulations, delaying <math><msub><mi>τ</mi> <mtext>crit</mtext></msub> </math> from 5 to 15 days increases the age-dependent peak total MRSA burden from 5.629 to 7.298 patients and cumulative total burden from 683.8 to 1398.0 patient-days. At the 9-day baseline and equal <math><msub><mi>R</mi> <mn>0</mn></msub> </math> , age dependence gives a <math><mrow><mn>7.372</mn> <mo>%</mo></mrow> </math> higher peak but a <math><mrow><mn>9.304</mn> <mo>%</mo></mrow> </math> lower cumulative burden than constant transmission. Increasing <math><msub><mi>κ</mi> <mo>max</mo></msub> </math> from 0.6 to 1.4 times baseline reduces <math><msub><mi>R</mi> <mn>0</mn></msub> </math> by <math><mrow><mn>4.133</mn> <mo>%</mo></mrow> </math> . Thus, matching <math><msub><mi>R</mi> <mn>0</mn></msub> </math> does not preserve timing-sensitive outcomes, but their magnitude and direction depend on the endpoint; over the examined ranges, clearance timing has greater leverage than immune-strength intensification.</p>","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 11","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13719866/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149294970","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Modeling the Spread of Crimean Congo Hemorrhagic Fever with Tick Co-feeding.","authors":"Jessica L Kingsley, Amber A Young, Stella A Atim, Stephen Balinandi, Simbarashe Chitanga, Faraimunashe Chirove, Hannah E Correia, F Stephen Dobson, Christina J Edholm, Paul D Kasiaja, Praise-God Madueme, Kelly Joëlle Gatore Sinigirira, Michael L Washington, Holly Gaff, Suzanne Lenhart","doi":"10.1007/s11538-026-01777-2","DOIUrl":"10.1007/s11538-026-01777-2","url":null,"abstract":"<p><p>Crimean Congo hemorrhagic fever (CCHF) is a viral disease in animals and humans. Ticks transmit the virus that causes this disease to susceptible hosts by feeding on them, and ticks can acquire the virus by feeding on an infected host. The tick species most commonly associated with CCHF feed on different hosts at different stages in their life cycle: larvae and nymphs feed on small mammals, while adults feed on large mammals, like humans and cattle. An additional transmission pathway, co-feeding, occurs when infected and susceptible ticks feed in close proximity on the same host, allowing viral transmission between ticks even if the host remains uninfected. In this work, we developed the first system of ordinary differential equations capturing CCHF transmission that includes co-feeding dynamics, seasonal variation in tick populations, and all three feeding life stages of ticks with their subsequent hosts. We adapted the model to Uganda's Cattle Corridor, a region with frequent human-cattle interactions and reported CCHF activity. Simulations explored both baseline and immigration event scenarios, capturing realistic ecological and epidemiological dynamics. By varying the co-feeding transmission, we found notable effects on the hosts and ticks, including a shift in timing of seasonal infection peaks and changes to the number of human infections from ticks or cattle. The inclusion of additional tick life stages demonstrates how co-feeding strengthens viral circulation within ticks and increases host infectivity. These results highlight the importance of including ecological complexity in models of CCHF transmission and to illustrate the effects of co-feeding.</p>","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 11","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13719780/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149294575","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Tutorial on Symbolic Structural Identifiability Analysis of ODE Models in Julia.","authors":"Abdallah Alsammani","doi":"10.1007/s11538-026-01779-0","DOIUrl":"10.1007/s11538-026-01779-0","url":null,"abstract":"<p><p>Structural identifiability analysis determines whether the parameters of a mechanistic ordinary differential equation (ODE) model can be uniquely recovered from ideal observations and is a prerequisite for reliable parameter estimation. This tutorial presents a reproducible computational framework for symbolic structural identifiability analysis using StructuralIdentifiability.jl. We introduce the concepts of local and global identifiability, observability, parameter-to-output mappings, and identifiable parameter combinations within a unified computational workflow. The methodology is illustrated through six case studies from epidemiology, pharmacokinetics, and within-host viral dynamics, demonstrating globally identifiable models, locally but not globally identifiable models, structurally non-identifiable models, identifiable parameter combinations, and strategies for restoring identifiability through additional measurements, known initial conditions, fixed parameters, and model reparameterization. Every identifiability verdict reported here is reproduced by companion scripts run in a pinned and tested Julia environment and, where possible, confirmed by an explicit input-output or symmetry argument. The tutorial also shows how models built with ModelingToolkit.jl or Catalyst.jl, or imported from SBML, enter the analysis, how several experimental conditions can be pooled, and how structural identifiability fits into the subsequent steps of experimental design, parameter estimation, and practical identifiability analysis, providing a practical guide to structural identifiability analysis for mechanistic ODE models.</p>","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 11","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13696614/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149262852","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Identifiability of Large Phylogenetic Mixtures for Many Phylogenetic Model Structures.","authors":"Bryson Kagy, Seth Sullivant","doi":"10.1007/s11538-026-01782-5","DOIUrl":"10.1007/s11538-026-01782-5","url":null,"abstract":"<p><p>Identifiability of phylogenetic models is a necessary condition to ensure that the model parameters can be uniquely determined from data. Mixture models are phylogenetic models where the probability distributions in the model are convex combinations of distributions in simpler phylogenetic models. Mixture models are used to model heterogeneity in the substitution process in DNA sequences. While many basic phylogenetic models are known to be identifiable, mixture models in generality have only been shown to be identifiable in certain cases. We expand the main theorem of [Rhodes, Sullivant 2012] to prove identifiability of mixture models in equivariant phylogenetic models, specifically the Jukes-Cantor, Kimura 2-parameter model, Kimura 3-parameter model and the Strand Symmetric model.</p>","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 11","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13694390/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"149242965","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}