Pattern formation in dense populations studied by inference of nonlinear diffusion‐reaction mechanisms
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Summary: Reaction‐diffusion systems have been proposed as a model for pattern formation and morphogenesis. The Fickian diffusion typically employed in these constructions model the Brownian motion of particles. The biological and chemical elements that form the basis of this process, like cells and proteins, occupy finite mass and volume and interact during migration. We propose a Reaction‐Diffusion system with Maxwell‐Stefan formulation to construct the diffusive flux. This formulation relies on inter‐species force balance and provides a more realistic model for interacting elements. We also present a variational system inference‐based technique to extract these models from spatiotemporal data for these processes. We show that the inferred models can capture the characteristics of local Turing instability that instigates the pattern formation process. Moreover, the equilibrium solutions of the inferred models form similar patterns to the observed data.
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Cell morphology and fluorescent reporters have been used to measure or infer cell states in migrating cells.
“Cell morphology42 and fluorescent reporters44 have been used to measure or infer cell states in migrating cells.”
From Inference of weak-form partial differential equations describing migration and proliferation mechanisms in wound healing experiments on cancer cells (2025) · cites this paper as [42]
The Introduction discusses cellular migration, and the Abstract describes inferring diffusion–reaction models from spatiotemporal data. Neither passage addresses using cell morphology or fluorescent reporters to measure or infer states in migrating cells. Searches for those measurement methods found no relevant passage in our copy. (LLM, conf=0.72)
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References 39 · Cited by 2
Research works printed but not linked: 24
- Garikipati, K. Perspectives on the mathematics of biological patterning and morphogenesis. Journal of the Mechanics and Physics of Solids 99, 192-210 (2017).
- Kondo, S. & Miura, T. Reaction-diffusion model as a framework for understanding biological pattern formation. Science 329, 1616-1620 (2010). URL https://www.science.org/doi/abs/ 10.1126/science.1179047. https://www.science.org/doi/pdf/10.1126/science.1179047. · DOI · Paper page
- Turing, A. M. The chemical basis of morphogenesis. Bulletin of mathematical biology 52, 153-197 (1990).
- Madzvamuse, A., Ndakwo, H. S. & Barreira, R. Cross-diffusion-driven instability for reaction- diffusion systems: analysis and simulations. Journal of mathematical biology 70, 709-743 (2015).
- Krishna, R. & Wesselingh, J. The maxwell-stefan approach to mass transfer. Chemical Engineering Science 52, 861-911 (1997). URL https://www.sciencedirect.com/science/ article/pii/S0009250996004587.
- Darken, L. S. Diffusion, mobility and their interrelation through free energy in binary metallic systems. Trans. Aime 175, 184-201 (1948).
- D'Agostino, C., Mantle, M., Gladden, L. & Moggridge, G. Prediction of binary diffusion coefficients in non-ideal mixtures from nmr data: Hexane-nitrobenzene near its consolute point. Chemical Engineering Science 66, 3898-3906 (2011). URL https://www.sciencedirect.com/ science/article/pii/S0009250911003150.
- Duncan, J. B. & Toor, H. An experimental study of three component gas diffusion. AIChE Journal 8, 38-41 (1962).
- Bothe, D. On the Maxwell-Stefan Approach to Multicomponent Diffusion, 81-93 (Springer Basel, Basel, 2011). URL https://doi.org/10.1007/978-3-0348-0075-4_5. · DOI
- Mazumder, A., Dobyns, B. M., Howard, M. P. & Beckingham, B. S. Theoretical and experimental considerations for investigating multicomponent diffusion in hydrated, dense polymer membranes. Membranes 12 (2022). URL https://www.mdpi.com/2077-0375/12/10/942.
- Giacomazzi, E., Picchia, F. & Arcidiacono, N. A review of chemical diffusion: Criticism and limits of simplified methods for diffusion coefficient calculation. Combustion Theory and Modelling 12, 135-158 (2008).
- Brunton, S. L., Proctor, J. L. & Kutz, J. N. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences 113, 3932-3937 (2016). URL https://www.pnas.org/doi/abs/10.1073/pnas.1517384113. https://www.pnas.org/doi/pdf/10.1073/pnas.1517384113. · DOI · Paper page
- Wang, Z., Huan, X. & Garikipati, K. Variational system identification of the partial differential equations governing microstructure evolution in materials: Inference over sparse and spatially unrelated data. Computer Methods in Applied Mechanics and Engineering 377, 113706 (2021).
- Nikolov, D. P. et al. Ogden material calibration via magnetic resonance cartography, parameter sensitivity and variational system identification. Philosophical Transactions of the Royal Society A 380, 20210324 (2022).
- Wang, Z., Carrasco-Teja, M., Zhang, X., Teichert, G. H. & Garikipati, K. System Inference Via Field Inversion for the Spatio-Temporal Progression of Infectious Diseases: Studies of COVID-19 in Michigan and Mexico. Archives of Computational Methods in Engineering 28, 4283-4295 (2021). URL https://link.springer.com/10.1007/s11831-021-09643-1. · DOI · Paper page
- Wang, Z., Huan, X. & Garikipati, K. Variational system identification of the partial differential equations governing the physics of pattern-formation: Inference under varying fidelity and noise. Computer Methods in Applied Mechanics and Engineering 356, 44-74 (2019). URL https://linkinghub.elsevier.com/retrieve/pii/S0045782519304037.
- Bothe, D. & Étienne Druet, P. On the structure of continuum thermodynamical diffusion fluxes-a novel closure scheme and its relation to the maxwell-stefan and the fick-onsager approach. International Journal of Engineering Science 184, 103818 (2023). URL https: //www.sciencedirect.com/science/article/pii/S0020722523000095.
- Logg, A. & Wells, G. N. Dolfin: Automated finite element computing. ACM Trans. Math. Softw. 37 (2010). URL https://doi.org/10.1145/1731022.1731030. · DOI · Paper page
- Logg, A., Mardal, K.-A. & Wells, G. Automated solution of differential equations by the finite element method: The FEniCS book, vol. 84 (Springer Science & Business Media, 2012).
- Pedregosa, F. et al. Scikit-learn: Machine learning in Python. Journal of Machine Learning Research 12, 2825-2830 (2011).
- Krishna, R. & Wesselingh, J. The maxwell-stefan approach to mass transfer. Chemical Engineering Science 52, 861-911 (1997). URL https://www.sciencedirect.com/science/ article/pii/S0009250996004587.
- Brauns, F., Halatek, J. & Frey, E. Phase-space geometry of mass-conserving reaction-diffusion dynamics. Physical Review X 10, 041036 (2020).
- Czirok, A. Endothelial cell motility, coordination and pattern formation during vasculogenesis. Wiley Interdisciplinary Reviews: Systems Biology and Medicine 5, 587-602 (2013).
- Khain, E. & Sander, L. M. Dynamics and pattern formation in invasive tumor growth. Phys. Rev. Lett. 96, 188103 (2006). URL https://link.aps.org/doi/10.1103/PhysRevLett.96.188103. · DOI
- Authors
- Srivastava, S., & Garikipati, K.
- Affiliations
- University of Michigan (US)
- Corresponding author
- Garikipati, Krishna
- Venue
- International Journal for Numerical Methods in Engineering
- Publisher
- Wiley
- Published
- 2024-03-06
- Volume
- 125
- Issue
- 12
- ISSN
- 0029-5981 (ISSN-L)
- 1097-0207
- OpenAlex
- W4392760958
- Cited by
- 2
- References
- 39
- Citations by year
- 2025: 2
- FWCI
- 0.33
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Licence
- Locations
- International Journal for Numerical Methods in Engineering (publishedVersion)
- arXiv (Cornell University) (submittedVersion, cc-by-nc-sa)
- Deep Blue (University of Michigan) (submittedVersion, public-domain)
- arXiv (Cornell University)
- Funders
- W. M. Keck Foundation
- Topics
- Physical Sciences › Computer Science › Computer Networks and Communications › Nonlinear Dynamics and Pattern Formation
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Model Reduction and Neural Networks
- Physical Sciences › Mathematics › Modeling and Simulation › Mathematical Biology Tumor Growth
- Keywords
- pattern formation, reaction-diffusion systems, nonlinear diffusion, morphogenesis, Turing instability, variational inference, equilibrium solutions
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