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Martinson WD, Volkening A, Schmidtchen M, Venkataraman C, Carrillo JA. Linking discrete and continuous models of cell birth and migration. ROYAL SOCIETY OPEN SCIENCE 2024; 11:232002. [PMID: 39021774 PMCID: PMC11252671 DOI: 10.1098/rsos.232002] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/22/2023] [Accepted: 05/10/2024] [Indexed: 07/20/2024]
Abstract
Self-organization of individuals within large collectives occurs throughout biology. Mathematical models can help elucidate the individual-level mechanisms behind these dynamics, but analytical tractability often comes at the cost of biological intuition. Discrete models provide straightforward interpretations by tracking each individual yet can be computationally expensive. Alternatively, continuous models supply a large-scale perspective by representing the 'effective' dynamics of infinite agents, but their results are often difficult to translate into experimentally relevant insights. We address this challenge by quantitatively linking spatio-temporal dynamics of continuous models and individual-based data in settings with biologically realistic, time-varying cell numbers. Specifically, we introduce and fit scaling parameters in continuous models to account for discrepancies that can arise from low cell numbers and localized interactions. We illustrate our approach on an example motivated by zebrafish-skin pattern formation, in which we create a continuous framework describing the movement and proliferation of a single cell population by upscaling rules from a discrete model. Our resulting continuous models accurately depict ensemble average agent-based solutions when migration or proliferation act alone. Interestingly, the same parameters are not optimal when both processes act simultaneously, highlighting a rich difference in how combining migration and proliferation affects discrete and continuous dynamics.
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Affiliation(s)
| | | | - Markus Schmidtchen
- Institute of Scientific Computing, Technische Universität Dresden, Dresden, Germany
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Owen JP, Kelsh RN, Yates CA. A quantitative modelling approach to zebrafish pigment pattern formation. eLife 2020; 9:52998. [PMID: 32716296 PMCID: PMC7384860 DOI: 10.7554/elife.52998] [Citation(s) in RCA: 18] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/23/2019] [Accepted: 06/21/2020] [Indexed: 12/14/2022] Open
Abstract
Pattern formation is a key aspect of development. Adult zebrafish exhibit a striking striped pattern generated through the self-organisation of three different chromatophores. Numerous investigations have revealed a multitude of individual cell-cell interactions important for this self-organisation, but it has remained unclear whether these known biological rules were sufficient to explain pattern formation. To test this, we present an individual-based mathematical model incorporating all the important cell-types and known interactions. The model qualitatively and quantitatively reproduces wild type and mutant pigment pattern development. We use it to resolve a number of outstanding biological uncertainties, including the roles of domain growth and the initial iridophore stripe, and to generate hypotheses about the functions of leopard. We conclude that our rule-set is sufficient to recapitulate wild-type and mutant patterns. Our work now leads the way for further in silico exploration of the developmental and evolutionary implications of this pigment patterning system.
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Affiliation(s)
- Jennifer P Owen
- Department of Biology and Biochemistry and Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, United Kingdom
| | - Robert N Kelsh
- Department of Biology and Biochemistry and Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, United Kingdom
| | - Christian A Yates
- Department of Biology and Biochemistry and Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, United Kingdom
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Volkening A, Abbott MR, Chandra N, Dubois B, Lim F, Sexton D, Sandstede B. Modeling Stripe Formation on Growing Zebrafish Tailfins. Bull Math Biol 2020; 82:56. [PMID: 32356149 DOI: 10.1007/s11538-020-00731-0] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/17/2019] [Accepted: 04/03/2020] [Indexed: 12/26/2022]
Abstract
As zebrafish develop, black and gold stripes form across their skin due to the interactions of brightly colored pigment cells. These characteristic patterns emerge on the growing fish body, as well as on the anal and caudal fins. While wild-type stripes form parallel to a horizontal marker on the body, patterns on the tailfin gradually extend distally outward. Interestingly, several mutations lead to altered body patterns without affecting fin stripes. Through an exploratory modeling approach, our goal is to help better understand these differences between body and fin patterns. By adapting a prior agent-based model of cell interactions on the fish body, we present an in silico study of stripe development on tailfins. Our main result is a demonstration that two cell types can produce stripes on the caudal fin. We highlight several ways that bone rays, growth, and the body-fin interface may be involved in patterning, and we raise questions for future work related to pattern robustness.
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Affiliation(s)
- A Volkening
- NSF-Simons Center for Quantitative Biology, Northwestern University, Evanston, IL, USA.
| | - M R Abbott
- Mathematics, Statistics, and Computer Science, Macalester College, St. Paul, MN, USA
| | - N Chandra
- Division of Applied Mathematics, Brown University, Providence, RI, USA
| | - B Dubois
- Division of Applied Mathematics, Brown University, Providence, RI, USA
| | - F Lim
- Division of Applied Mathematics, Brown University, Providence, RI, USA
| | - D Sexton
- Department of Mathematics, University of Idaho, Moscow, ID, USA
| | - B Sandstede
- Division of Applied Mathematics, Brown University, Providence, RI, USA
- Data Science Initiative, Brown University, Providence, RI, USA
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Volkening A, Sandstede B. Iridophores as a source of robustness in zebrafish stripes and variability in Danio patterns. Nat Commun 2018; 9:3231. [PMID: 30104716 PMCID: PMC6089994 DOI: 10.1038/s41467-018-05629-z] [Citation(s) in RCA: 28] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/20/2017] [Accepted: 07/13/2018] [Indexed: 11/11/2022] Open
Abstract
Zebrafish (Danio rerio) feature black and yellow stripes, while related Danios display different patterns. All these patterns form due to the interactions of pigment cells, which self-organize on the fish skin. Until recently, research focused on two cell types (melanophores and xanthophores), but newer work has uncovered the leading role of a third type, iridophores: by carefully orchestrated transitions in form, iridophores instruct the other cells, but little is known about what drives their form changes. Here we address this question from a mathematical perspective: we develop a model (based on known interactions between the original two cell types) that allows us to assess potential iridophore behavior. We identify a set of mechanisms governing iridophore form that is consistent across a range of empirical data. Our model also suggests that the complex cues iridophores receive may act as a key source of redundancy, enabling both robust patterning and variability within Danio.
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Affiliation(s)
- Alexandria Volkening
- Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI, 02912, USA.
- Mathematical Biosciences Institute, Ohio State University, 1735 Neil Avenue, Columbus, OH, 43210, USA.
| | - Björn Sandstede
- Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI, 02912, USA
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Volkening A, Sandstede B. Modelling stripe formation in zebrafish: an agent-based approach. J R Soc Interface 2015; 12:20150812. [PMID: 26538560 PMCID: PMC4685853 DOI: 10.1098/rsif.2015.0812] [Citation(s) in RCA: 51] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/11/2015] [Accepted: 10/13/2015] [Indexed: 11/12/2022] Open
Abstract
Zebrafish have distinctive black stripes and yellow interstripes that form owing to the interaction of different pigment cells. We present a two-population agent-based model for the development and regeneration of these stripes and interstripes informed by recent experimental results. Our model describes stripe pattern formation, laser ablation and mutations. We find that fish growth shortens the necessary scale for long-range interactions and that iridophores, a third type of pigment cell, help align stripes and interstripes.
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Affiliation(s)
| | - Björn Sandstede
- Division of Applied Mathematics, Brown University, Providence, RI 02912, USA
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Hackett-Jones EJ, Landman KA, Fellner K. Aggregation patterns from nonlocal interactions: Discrete stochastic and continuum modeling. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:041912. [PMID: 22680503 DOI: 10.1103/physreve.85.041912] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/17/2011] [Indexed: 06/01/2023]
Abstract
Conservation equations governed by a nonlocal interaction potential generate aggregates from an initial uniform distribution of particles. We address the evolution and formation of these aggregating steady states when the interaction potential has both attractive and repulsive singularities. Currently, no existence theory for such potentials is available. We develop and compare two complementary solution methods, a continuous pseudoinverse method and a discrete stochastic lattice approach, and formally show a connection between the two. Interesting aggregation patterns involving multiple peaks for a simple doubly singular attractive-repulsive potential are determined. For a swarming Morse potential, characteristic slow-fast dynamics in the scaled inverse energy is observed in the evolution to steady state in both the continuous and discrete approaches. The discrete approach is found to be remarkably robust to modifications in movement rules, related to the potential function. The comparable evolution dynamics and steady states of the discrete model with the continuum model suggest that the discrete stochastic approach is a promising way of probing aggregation patterns arising from two- and three-dimensional nonlocal interaction conservation equations.
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Affiliation(s)
- Emily J Hackett-Jones
- Department of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia
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