A computational method for the coupled solution of reaction–diffusion equations on evolving domains and manifolds: Application to a model of cell migration and chemotaxis

MacDonald, Grant; Mackenzie, J. A.; Nolan, M.; Insall, Robert H.

doi:10.1016/j.jcp.2015.12.038

Cited by 50 publications

(90 citation statements)

References 54 publications

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“…We will indicate the solutions at discrete time t n with a n and b n . We use a predictorcorrector finite difference method to approximate the time-derivatives (see for example [31]). To calculate the solution at each time point, we follow the steps outlined below.…”

Section: Temporal Discretisationmentioning

confidence: 99%

“…We present new three-dimensional results on regular and irregular geometries, exhibiting the wave pinning process on complex geometries. A key part of our study involves the numerical simulation of the BSWP model in three-dimensional geometries using a recently developed bulk-surface finite element method (BS-FEM) [10,12,31,32,33,34]. This numerical framework allows to compute the solutions of the BSWP model on complex convex and non-convex geometries.…”

Section: Introductionmentioning

confidence: 99%

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A coupled bulk-surface model for cell polarisation

Cusseddu

Edelstein‐Keshet

Mackenzie

et al. 2019

Journal of Theoretical Biology

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117

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Several cellular activities, such as directed cell migration, are coordinated by an intricate network of biochemical reactions which lead to a polarised state of the cell, in which cellular symmetry is broken, causing the cell to have a well defined front and back. Recent work on balancing biological complexity with mathematical tractability resulted in the proposal and formulation of a famous minimal model for cell polarisation, known as the wave pinning model. In this study, we present a three-dimensional generalisation of this mathematical framework through the maturing theory of coupled bulk-surface semilinear partial differential equations in which protein compartmentalisation becomes natural. We show how a local perturbation over the surface can trigger propagating reactions, eventually stopped in a stable profile by the interplay with the bulk component. We describe the behavior of the model through asymptotic and local perturbation analysis, in which the role of the geometry is investigated. The bulk-surface finite element method is used to generate numerical simulations over simple and complex geometries, which confirm our analysis, showing pattern formation due to propagation and pinning dynamics. The generality of our mathematical and computational framework allows to study more complex biochemical reactions and biomechanical properties associated with cell polarisation in multi-dimensions.

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Section: Temporal Discretisationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A coupled bulk-surface model for cell polarisation

Cusseddu

Edelstein‐Keshet

Mackenzie

et al. 2019

Journal of Theoretical Biology

Self Cite

117

View full text Add to dashboard Cite

show abstract

“…Weller et al [3] and McRae et al [24] solve a Monge-Ampére type equation on the surface of the sphere to generate optimally transported meshes that become equidistributed with respect to a suitable monitor function. MacDonald et al [23] devise a moving mesh method for the numerical simulation of coupled bulk-surface reaction-diffusion equations on an evolving two-dimensional domain. They use a one-dimensional moving mesh equation in arclength to concentrate mesh points along the evolving domain boundary.…”

Section: Introductionmentioning

confidence: 99%

A surface moving mesh method based on equidistribution and alignment

Kolasinski¹,

Huang²

2020

Journal of Computational Physics

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A surface moving mesh method is presented for general surfaces with or without explicit parameterization. The method can be viewed as a nontrivial extension of the moving mesh partial differential equation method that has been developed for bulk meshes and demonstrated to work well for various applications. The main challenges in the development of surface mesh movement come from the fact that the Jacobian matrix of the affine mapping between the reference element and any simplicial surface element is not square. The development starts with revealing the relation between the area of a surface element in the Euclidean or Riemannian metric and the Jacobian matrix of the corresponding affine mapping, formulating the equidistribution and alignment conditions for surface meshes, and establishing a meshing energy function based on the conditions. The moving mesh equation is then defined as the gradient system of the energy function, with the nodal mesh velocities being projected onto the underlying surface. The analytical expression for the mesh velocities is obtained in a compact, matrix form, which makes the implementation of the new method on a computer relatively easy and robust. Moreover, it is analytically shown that any mesh trajectory generated by the method remains nonsingular if it is so initially. It is emphasized that the method is developed directly on surface meshes, making no use of any information on surface parameterization. It utilizes surface normal vectors to ensure that the mesh vertices remain on the surface while moving, and also assumes that the initial surface mesh is given. The new method can apply to general surfaces with or without explicit parameterization since the surface normal vectors can be computed even when the surface only has a numerical representation. A selection of twoand three-dimensional examples are presented.

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“…Well-known application examples are the transport of surfactants on moving interfaces with application to multiphase flows [1][2][3][4][5][6] and the chemotaxis processes in which single colonies of bacteria cells merge together to form one larger colony on solid surfaces and biological films. [7][8][9][10] Surface convection and diffusion problems are also fundamental subproblems involved in the surface flow models with applications to geoscience. [11][12][13][14][15][16] While there are a lot of discussions on numerical methods in the literature for convection-diffusion-reaction PDEs in one, two, and three dimensions, it is challenging to develop efficient numerical methods for such PDEs on surfaces (or manifolds).…”

Section: Introductionmentioning

confidence: 99%

A gradient recovery–based adaptive finite element method for convection‐diffusion‐reaction equations on surfaces

Xiao

Feng

2019

Numerical Meth Engineering

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In this paper, we present an adaptive mesh refinement method for solving convection-diffusion-reaction equations on surfaces, which is a fundamental subproblem in many models for simulating the transport of substances on biological films and solid surfaces. The method considered is a combination of well-known techniques: the surface finite element method, streamline diffusion stabilization, and the gradient recovery-based Zienkiewicz-Zhu error estimator. The streamline diffusion method overcomes the instability issue of the finite element method for the dominance of the convection.The gradient recovery-based adaptive mesh refinement strategy enables the method to provide high-resolution numerical solutions by relatively fewer degrees of freedom. Moreover, the implementation detail of a surface mesh refinement technique is presented. Various numerical examples, including the convection-dominated diffusion problems with large variations of solutions, nearly singular solutions, discontinuous sources, and internal layers on surfaces, are presented to demonstrate the efficacy and accuracy of the proposed method. KEYWORDSadaptive strategy, recovery-based error estimator, streamline diffusion method, surface convection-diffusion-reaction equations, surface finite element method MSC CLASSIFICATION 65N30, 65N12, 65N50, 58J32, 76R05

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A computational method for the coupled solution of reaction–diffusion equations on evolving domains and manifolds: Application to a model of cell migration and chemotaxis

Cited by 50 publications

References 54 publications

A coupled bulk-surface model for cell polarisation

A coupled bulk-surface model for cell polarisation

A surface moving mesh method based on equidistribution and alignment

A gradient recovery–based adaptive finite element method for convection‐diffusion‐reaction equations on surfaces

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