Going from microscopic to macroscopic on nonuniform growing domains
Throughout development, chemical cues are employed to guide the functional specification of underlying tissues while the spatiotemporal distributions of such chemicals can be influenced by the growth of the tissue itself. These chemicals, termed morphogens, are often modeled using partial differenti...
| Main Authors: | , , , |
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| Format: | Journal article |
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American Physical Society
2012
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| _version_ | 1797078146279800832 |
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| author | Yates, C Baker, R Erban, R Maini, P |
| author_facet | Yates, C Baker, R Erban, R Maini, P |
| author_sort | Yates, C |
| collection | OXFORD |
| description | Throughout development, chemical cues are employed to guide the functional specification of underlying tissues while the spatiotemporal distributions of such chemicals can be influenced by the growth of the tissue itself. These chemicals, termed morphogens, are often modeled using partial differential equations (PDEs). The connection between discrete stochastic and deterministic continuum models of particle migration on growing domains was elucidated by Baker, Yates, and Erban [ Bull. Math. Biol. 72 719 (2010)] in which the migration of individual particles was modeled as an on-lattice position-jump process. We build on this work by incorporating a more physically reasonable description of domain growth. Instead of allowing underlying lattice elements to instantaneously double in size and divide, we allow incremental element growth and splitting upon reaching a predefined threshold size. Such a description of domain growth necessitates a nonuniform partition of the domain. We first demonstrate that an individual-based stochastic model for particle diffusion on such a nonuniform domain partition is equivalent to a PDE model of the same phenomenon on a nongrowing domain, providing the transition rates (which we derive) are chosen correctly and we partition the domain in the correct manner. We extend this analysis to the case where the domain is allowed to change in size, altering the transition rates as necessary. Through application of the master equation formalism we derive a PDE for particle density on this growing domain and corroborate our findings with numerical simulations. |
| first_indexed | 2024-03-07T00:28:20Z |
| format | Journal article |
| id | oxford-uuid:7ee5d902-4fa6-4e83-bfba-ec34965d6121 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T00:28:20Z |
| publishDate | 2012 |
| publisher | American Physical Society |
| record_format | dspace |
| spelling | oxford-uuid:7ee5d902-4fa6-4e83-bfba-ec34965d61212022-03-26T21:13:10ZGoing from microscopic to macroscopic on nonuniform growing domainsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:7ee5d902-4fa6-4e83-bfba-ec34965d6121Mathematical Institute - ePrintsAmerican Physical Society2012Yates, CBaker, RErban, RMaini, PThroughout development, chemical cues are employed to guide the functional specification of underlying tissues while the spatiotemporal distributions of such chemicals can be influenced by the growth of the tissue itself. These chemicals, termed morphogens, are often modeled using partial differential equations (PDEs). The connection between discrete stochastic and deterministic continuum models of particle migration on growing domains was elucidated by Baker, Yates, and Erban [ Bull. Math. Biol. 72 719 (2010)] in which the migration of individual particles was modeled as an on-lattice position-jump process. We build on this work by incorporating a more physically reasonable description of domain growth. Instead of allowing underlying lattice elements to instantaneously double in size and divide, we allow incremental element growth and splitting upon reaching a predefined threshold size. Such a description of domain growth necessitates a nonuniform partition of the domain. We first demonstrate that an individual-based stochastic model for particle diffusion on such a nonuniform domain partition is equivalent to a PDE model of the same phenomenon on a nongrowing domain, providing the transition rates (which we derive) are chosen correctly and we partition the domain in the correct manner. We extend this analysis to the case where the domain is allowed to change in size, altering the transition rates as necessary. Through application of the master equation formalism we derive a PDE for particle density on this growing domain and corroborate our findings with numerical simulations. |
| spellingShingle | Yates, C Baker, R Erban, R Maini, P Going from microscopic to macroscopic on nonuniform growing domains |
| title | Going from microscopic to macroscopic on nonuniform growing domains |
| title_full | Going from microscopic to macroscopic on nonuniform growing domains |
| title_fullStr | Going from microscopic to macroscopic on nonuniform growing domains |
| title_full_unstemmed | Going from microscopic to macroscopic on nonuniform growing domains |
| title_short | Going from microscopic to macroscopic on nonuniform growing domains |
| title_sort | going from microscopic to macroscopic on nonuniform growing domains |
| work_keys_str_mv | AT yatesc goingfrommicroscopictomacroscopiconnonuniformgrowingdomains AT bakerr goingfrommicroscopictomacroscopiconnonuniformgrowingdomains AT erbanr goingfrommicroscopictomacroscopiconnonuniformgrowingdomains AT mainip goingfrommicroscopictomacroscopiconnonuniformgrowingdomains |