Bifurcation Structure and Control Interpretation in pH-Responsive Tumor– Immune Dynamics: A Reaction–Diffusion Framework

Bifurcation Structure and Control Interpretation in pH-Responsive Tumor– Immune Dynamics: A Reaction–Diffusion Framework | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Bifurcation Structure and Control Interpretation in pH-Responsive Tumor– Immune Dynamics: A Reaction–Diffusion Framework Enrique Rosario Aloma, Luis Rodriguez, Maymunah Ray, Benjamin Hoffmeister This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8746034/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 4 You are reading this latest preprint version Abstract Living systems often regulate their microenvironment through feedback mechanisms that couple biochemical activity, transport processes, and energetic constraints. In tumors, extracellular acidity represents a critical environmental variable that suppresses immune activity and shapes therapeutic response. Here, we analyze the dynamical and spatial structure of pH-responsive tumor–immune interactions using a biologically motivated reaction–diffusion framework with nonlinear feedback. We model tumor growth, immune effector dynamics, and extracellular acidity as a coupled control system in which alkalinization acts as an environmental modulation mechanism subject to transport and metabolic constraints. Using bifurcation analysis and numerical phase-space exploration, we identify distinct dynamical regimes corresponding to immune suppression, partial control, and tumor clearance. These regimes are organized by well-defined bifurcation boundaries that depend on kinetic suppression strength, diffusive transport scales, and alkalinization efficiency. Our analysis reveals that tumor control is not determined by biological potency alone, but instead emerges from the interplay between nonlinear feedback, spatial diffusion, and energetic limitations. In particular, we show how diffusion-limited transport and metabolic cost impose hard constraints on the effectiveness and scalability of pH-modulating strategies, leading to sharp transitions between controllable and uncontrollable regimes. Sensitivity analyses demonstrate that the qualitative phase structure is robust across broad parameter ranges. This work provides a physical interpretation of tumor–immune regulation as a constrained reaction–diffusion control problem and offers a unifying dynamical framework for understanding environmentally mediated control in biological systems. The results have implications for the design of pH-modulating immunotherapies and more broadly for biological systems in which organisms actively modify their environment under transport and energetic constraints. biological control systems nonlinear dynamics bifurcation analysis feedback regulation reaction-diffusion systems synthetic biology circuits tumor-immune interactions Full Text Additional Declarations No competing interests reported. Supplementary Files sensitivitycode.zip sensitivitydata.zip Cite Share Download PDF Status: Under Review Version 1 posted Reviewers invited by journal 11 Feb, 2026 Editor assigned by journal 05 Feb, 2026 Submission checks completed at journal 05 Feb, 2026 First submitted to journal 30 Jan, 2026 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8746034","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":589890649,"identity":"4dd9c4f5-4f32-4636-9925-bb6d982b3ff4","order_by":0,"name":"Enrique Rosario 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In tumors, extracellular acidity represents a critical environmental variable that suppresses immune activity and shapes therapeutic response. Here, we analyze the dynamical and spatial structure of pH-responsive tumor\u0026ndash;immune interactions using a biologically motivated reaction\u0026ndash;diffusion framework with nonlinear feedback.\u003c/p\u003e \u003cp\u003eWe model tumor growth, immune effector dynamics, and extracellular acidity as a coupled control system in which alkalinization acts as an environmental modulation mechanism subject to transport and metabolic constraints. Using bifurcation analysis and numerical phase-space exploration, we identify distinct dynamical regimes corresponding to immune suppression, partial control, and tumor clearance. These regimes are organized by well-defined bifurcation boundaries that depend on kinetic suppression strength, diffusive transport scales, and alkalinization efficiency.\u003c/p\u003e \u003cp\u003eOur analysis reveals that tumor control is not determined by biological potency alone, but instead emerges from the interplay between nonlinear feedback, spatial diffusion, and energetic limitations. In particular, we show how diffusion-limited transport and metabolic cost impose hard constraints on the effectiveness and scalability of pH-modulating strategies, leading to sharp transitions between controllable and uncontrollable regimes. Sensitivity analyses demonstrate that the qualitative phase structure is robust across broad parameter ranges.\u003c/p\u003e \u003cp\u003eThis work provides a physical interpretation of tumor\u0026ndash;immune regulation as a constrained reaction\u0026ndash;diffusion control problem and offers a unifying dynamical framework for understanding environmentally mediated control in biological systems. The results have implications for the design of pH-modulating immunotherapies and more broadly for biological systems in which organisms actively modify their environment under transport and energetic constraints.\u003c/p\u003e","manuscriptTitle":"Bifurcation Structure and Control Interpretation in pH-Responsive Tumor– Immune Dynamics: A Reaction–Diffusion Framework","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-02-17 05:09:14","doi":"10.21203/rs.3.rs-8746034/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"reviewersInvited","content":"","date":"2026-02-11T22:39:47+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-02-05T23:31:33+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-02-05T12:39:09+00:00","index":"","fulltext":""},{"type":"submitted","content":"Journal of Biological Physics","date":"2026-01-31T03:16:11+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"journal-of-biological-physics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jobp","sideBox":"Learn more about [Journal of Biological Physics](http://link.springer.com/journal/10867)","snPcode":"10867","submissionUrl":"https://submission.nature.com/new-submission/10867/3","title":"Journal of Biological Physics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"5aee6781-ae9e-4095-bf5d-9a22c6858ac8","owner":[],"postedDate":"February 17th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2026-02-17T05:09:14+00:00","versionOfRecord":[],"versionCreatedAt":"2026-02-17 05:09:14","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8746034","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8746034","identity":"rs-8746034","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}