Numerical approximation of a 3D mechanochemical interface model for skin patterning

Vilaca, Luis Miguel De Oliveira; Milinkovitch, Michel C.; Ruiz–Baier, Ricardo

doi:10.1016/j.jcp.2019.01.023

Cited by 10 publications

(6 citation statements)

References 60 publications

(102 reference statements)

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“…Another direction is to consider the influence of the interface shape and mechanical structure on patterning. For instance, a fundamental question is whether a local deformation of the interface could have an influence on patterning, noting that the self-organisation is associated with a local mechanical compression and deformation of the epithelium [25, 28], as well as motivating earlier and more recent theoretical work [11, 13].…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Turing pattern formation in reaction-cross-diffusion systems with a bilayer geometry

Diez

Krause

Maini

et al. 2023

Preprint

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Conditions for self-organisation via Turing's mechanism in biological systems represented by reaction-diffusion or reaction-cross-diffusion models have been extensively studied. Nonetheless, the impact of tissue stratification in such systems is under-explored, despite its ubiquity in the context of a thin epithelium overlying connective tissue, for instance the epidermis and underlying dermal mesenchyme of embryonic skin. In particular, each layer can be subject to extensively different biochemical reactions and transport processes, with chemotaxis - a special case of cross-diffusion - often present in the mesenchyme, contrasting the solely molecular transport typically found in the epidermal layer. We study Turing patterning conditions for a class of reaction-cross-diffusion systems in bilayered regions, with a thin upper layer and coupled by a linear transport law. In particular, the role of differential transport through the interface is explored together with the presence of asymmetry between the homogeneous equilibria of the two layers. A linear stability analysis is carried out around a spatially homogeneous equilibrium state in the asymptotic limit of weak and strong coupling strengths, where quantitative approximations of the bifurcation curve can be computed. Our theoretical findings, for an arbitrary number of reacting species, reveal quantitative Turing conditions, highlighting when the coupling mechanism between the layered regions can either trigger patterning or stabilize a homogeneous equilibrium regardless of the independent patterning state of each layer. We support our theoretical results through direct numerical simulations, and provide an open source code to explore such systems further.

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Section: Conclusion and Discussionmentioning

confidence: 99%

Turing pattern formation in reaction-cross-diffusion systems with a bilayer geometry

Diez

Krause

Maini

et al. 2023

Preprint

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“…The linear stability analysis for the nonlinearly coupled system from Section 4 could be carried out extending our results from Section 3 following the ideas from [6]. As the main application is on skin patterning for feather development, we could also incorporate growth models specifically targeted for shells or thin plates as in [14], and concentrate on bi-layered structures extending the formulations in [13].…”

Section: Discussionmentioning

confidence: 99%

Coupling chemotaxis and growth poromechanics for the modelling of feather primordia patterning

A.¹,

Vilaca²,

Milinkovitch³

et al. 2022

Preprint

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We propose a new mathematical model for the interaction of skin cell populations with fibroblast growth factor and bone morphogenetic protein, occurring within deformable porous media. The equations for feather primordia pattering are based on the work by K.J. Painter et al. [J. Theoret. Biol., 437 (2018) 225-238]. We perform a linear stability analysis to identify relevant parameters in the coupling mechanisms, focusing in the regime of infinitesimal strains. We also extend the model to the case of nonlinear poroelasticity and include solid growth by means of Lee decompositions of the deformation gradient. We present a few illustrative computational examples in 2D and 3D, and briefly discuss the design of tailored efficient solvers.

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“…The linear stability analysis for the nonlinearly coupled system from Section 4 could be carried out extending our results from Section 3 following the ideas from [38]. As the main application is on skin patterning for feather development, we could also incorporate growth models specifically targeted for shells or thin plates as in [44], and concentrate on bi-layered structures extending the formulations in [45]. Viscous effects might be required in the rheological assumptions leading to the construction of stress, where relevant works in that context include [46,47].…”

Section: Discussionmentioning

confidence: 99%

Coupling Chemotaxis and Growth Poromechanics for the Modelling of Feather Primordia Patterning

Vilaca

Milinkovitch

et al. 2022

Mathematics

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In this paper we propose a new mathematical model for describing the complex interplay between skin cell populations with fibroblast growth factor and bone morphogenetic protein, occurring within deformable porous media describing feather primordia patterning. Tissue growth, in turn, modifies the transport of morphogens (described by reaction-diffusion equations) through diverse mechanisms such as advection from the solid velocity generated by mechanical stress, and mass supply. By performing an asymptotic linear stability analysis on the coupled poromechanical-chemotaxis system (assuming rheological properties of the skin cell aggregates that reside in the regime of infinitesimal strains and where the porous structure is fully saturated with interstitial fluid and encoding the coupling mechanisms through active stress) we obtain the conditions on the parameters—especially those encoding coupling mechanisms—under which the system will give rise to spatially heterogeneous solutions. We also extend the mechanical model to the case of incompressible poro-hyperelasticity and include the mechanisms of anisotropic solid growth and feedback by means of standard Lee decompositions of the tensor gradient of deformation. Because the model in question involves the coupling of several nonlinear PDEs, we cannot straightforwardly obtain closed-form solutions. We therefore design a suitable numerical method that employs backward Euler time discretisation, linearisation of the semidiscrete problem through Newton–Raphson’s method, a seven-field finite element formulation for the spatial discretisation, and we also advocate the construction and efficient implementation of tailored robust solvers. We present a few illustrative computational examples in 2D and 3D, briefly discussing different spatio-temporal patterns of growth factors as well as the associated solid response scenario depending on the specific poromechanical regime. Our findings confirm the theoretically predicted behaviour of spatio-temporal patterns, and the produced results reveal a qualitative agreement with respect to the expected experimental behaviour. We stress that the present study provides insight on several biomechanical properties of primordia patterning.

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Numerical approximation of a 3D mechanochemical interface model for skin patterning

Cited by 10 publications

References 60 publications

Turing pattern formation in reaction-cross-diffusion systems with a bilayer geometry

Turing pattern formation in reaction-cross-diffusion systems with a bilayer geometry

Coupling chemotaxis and growth poromechanics for the modelling of feather primordia patterning

Coupling Chemotaxis and Growth Poromechanics for the Modelling of Feather Primordia Patterning

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