Stability analysis of reaction-diffusion models on evolving domains: the effects of cross-diffusion

This article presents stability analytical results of a two component reaction-diffusion system with linear cross-diffusion posed on continuously evolving domains. First the model system is mapped from a continuously evolving domain to a reference stationary frame resulting in a system of partial differential equations with time-dependent coefficients. Second, by employing appropriately asymptotic theory, we derive and prove cross-diffusion-driven instability conditions for the model system for the case of slow, isotropic domain growth. Our analytical results reveal that unlike the restrictive diffusion-driven instability conditions on stationary domains, in the presence of cross-diffusion coupled with domain evolution, it is no longer necessary to enforce cross nor pure kinetic conditions. The restriction to activator-inhibitor kinetics to induce pattern formation on a growing biological system is no longer a requirement. Reaction-cross-diffusion models with equal diffusion coefficients in the principal components as well as those of the short-range inhibition, long-range activation and activator-activator form can generate patterns only in the presence of cross-diffusion coupled with domain evolution. To confirm our theoretical findings, detailed parameter spaces are exhibited for the special cases of isotropic exponential, linear and logistic growth profiles. In support of our theoretical predictions, we present evolving or moving finite element solutions exhibiting patterns generated by a short-range inhibition, long-range activation reaction-diffusion model with linear cross-diffusion in the presence of domain evolution

Stability analysis and parameter classification of a reaction-diffusion model on an annulus

This work explores the influence of domain-size on the evolution of pattern formation modelled by an activator-depleted reactiondiffusion system on a flat-ring (annulus). A closed form expression is derived for the spectrum of the Laplace operator on the domain. Spectral method is used to depict the close form solution on the domain. The bifurcation analysis of activator-depleted reactiondiffusion system is conducted on the admissible parameter space under the influence of domain-size. The admissible parameter space is partitioned under a set of proposed conditions relating the reactiondiffusion constants with the domain-size. Finally, the full system is numerically simulated on a two dimensional annular region using the standard Galerkin finite element method to verify the influence of the analytically derived domain-dependent conditions

Domain-growth-induced patterning for reaction-diffusion systems with linear cross-diffusion

In this article we present, for the first time, domain-growth induced pat- tern formation for reaction-diffusion systems with linear cross-diffusion on evolving domains and surfaces. Our major contribution is that by selecting parameter values from spaces induced by domain and surface evolution, patterns emerge only when domain growth is present. Such patterns do not exist in the absence of domain and surface evolution. In order to compute these domain-induced parameter spaces, linear stability theory is employed to establish the necessary conditions for domain- growth induced cross-diffusion-driven instability for reaction-diffusion systems with linear cross-diffusion. Model reaction-kinetic parameter values are then identified from parameter spaces induced by domain-growth only; these exist outside the classical standard Turing space on stationary domains and surfaces. To exhibit these patterns we employ the finite element method for solving reaction-diffusion systems with cross-diffusion on continuously evolving domains and surfaces

Domain-dependent stability analysis of a reaction-diffusion model on compact circular geometries

In this work an activator-depleted reaction-diffusion system is investigated on polar coordinates with the aim of exploring the relationship and the corresponding influence of domain size on the types of possible diffusion-driven instabilities. Quantitative relationships are found in the form of necessary conditions on the area of a disk-shape domain with respect to the diffusion and reaction rates for certain types of diffusion-driven instabilities to occur. Robust analytical methods are applied to find explicit expressions for the eigenvalues and eigenfunctions of the diffusion operator on a disk-shape domain with homogenous Neumann boundary conditions in polar coordinates. Spectral methods are applied using chebyshev non-periodic grid for the radial variable and Fourier periodic grid on the angular variable to verify the nodal lines and eigensurfaces subject to the proposed analytical findings. The full classification of the parameter space in light of the bifurcation analysis is obtained and numerically verified by finding the solutions of the partitioning curves inducing such a classification. Spatio-temporal periodic behaviour is demonstrated in the numerical solutions of the system for a proposed choice of parameters and a rigorous proof of the existence of infinitely many such points in the parameter plane is presented under a restriction on the area of the domain, with a lower bound in terms of reaction-diffusion rates

Stability analysis and simulations of coupled bulk-surface reaction–diffusion systems

In this article, we formulate new models for coupled systems of bulk-surface reaction–diffusion equations on stationary volumes. The bulk reaction–diffusion equations are coupled to the surface reaction–diffusion equations through linear Robin-type boundary conditions. We then state and prove the necessary conditions for diffusion-driven instability for the coupled system. Owing to the nature of the coupling between bulk and surface dynamics, we are able to decouple the stability analysis of the bulk and surface dynamics. Under a suitable choice of model parameter values, the bulk reaction–diffusion system can induce patterning on the surface independent of whether the surface reaction–diffusion system produces or not, patterning. On the other hand, the surface reaction–diffusion system cannot generate patterns everywhere in the bulk in the absence of patterning from the bulk reaction–diffusion system. For this case, patterns can be induced only in regions close to the surface membrane. Various numerical experiments are presented to support our theoretical findings. Our most revealing numerical result is that, Robin-type boundary conditions seem to introduce a boundary layer coupling the bulk and surface dynamics

Mathematical modelling and numerical simulations of actin dynamics in the eukaryotic cell

The aim of this article is to study cell deformation and cell movement by considering both the mechanical and biochemical properties of the cortical network of actin filaments and its concentration. Actin is a polymer that can exist either in fil- amentous form (F-actin) or in monometric form (G-actin) (Chen et al. 2000) and the filamentous form is arranged in a paired helix of two protofilaments (Ananthakrish- nan et al. 2006). By assuming that cell deformations are a result of the cortical actin dynamics in the cell cytoskeleton, we consider a continuum mathematical model that couples the mechanics of the network of actin filaments with its bio-chemical dy- namics. Numerical treatment of the model is carried out using the moving grid finite element method (Madzvamuse et al. 2003). Furthermore, by assuming slow deforma- tions of the cell, we use linear stability theory to validate the numerical simulation results close to bifurcation points. Far from bifurcation points, we show that the math- ematical model is able to describe the complex cell deformations typically observed in experimental results. Our numerical results illustrate cell expansion, cell contrac- tion, cell translation and cell relocation as well as cell protrusions. In all these results, the contractile tonicity formed by the association of actin filaments to the myosin II motor proteins is identified as a key bifurcation parameter

A note on how to develop interdisciplinary collaborations between experimentalists and theoreticians

This special issue is inspired by and based on the six-month research programme held at the Isaac Newton Institute (INI) for Mathematical Sciences, Cambridge, UK between 13 July and 18 December 2015 entitled ‘Coupling geometric partial differential equations with physics for cell morphology, motility and pattern formation’. The research programme was the first of its kind to bring together at the INI world-leading theoreticians, experimentalists, biomedical practitioners and statisticians. This diverse and large group came together to share paired goals: understanding how current mathematical techniques, including mathematical modelling and numerical and statistical analysis, can be used to formulate and analyse topical problems in cell motility and pattern formation, and conversely, how diverse experimental results can be translated into predictive mathematical and computational models across several spatio-temporal scales. Recent advances in cell motility and pattern formation, including high-resolution imaging techniques in three dimensions, necessitate new mathematical and computational theories to help guide, suggest, refine and sharpen further experimental hypotheses. The research programme laid down premises for topical research that mandated coupling molecular, cellular, tissue and fluid dynamics in a multi-scale interdisciplinary environment thereby enabling the generation of new scientific knowledge across several disciplines. The six-month research programme included three workshops and an Open for Business event at the INI, a satellite meeting at the University of Sussex, and a unique hands-on experimental workshop in Germany on cell migration and advanced microscopy, hosted jointly by RWTH Aachen University and Forschungszentrum Jülich. Hence, with the goal of breaking barriers between these disciplines, the programme was tailored in a way that best harnessed expertise and knowledge between experimental and theoretical sciences