A comparison of reaction diffusion and mechanochemical models for limb development

Several theoretical models have been proposed to attempt to elucidate the underlying mechanisms involved in the spatial patterning of skeletal elements in the limb. Here, I briefly compare two such models - reaction diffusion (RD) and mechanochemical (MC) - and highlight their properties and predictions

Reaction–Diffusion Finite Element Model of Lateral Line Primordium Migration to Explore Cell Leadership

Collective cell migration plays a fundamental role in many biological phenomena such as immune response, embryogenesis and tumorigenesis. In the present work, we propose a reaction–diffusion finite element model of the lateral line primordium migration in zebrafish. The population is modelled as a continuum with embedded discrete motile cells, which are assumed to be viscoelastic and able to undergo large deformations. The Wnt/ß-catenin–FGF and cxcr4b–cxcr7b signalling pathways inside the cohort regulating the migration are described through coupled reaction–diffusion equations. The coupling between mechanics and the molecular scenario occurs in two ways. Firstly, the intensity of the protrusion–contraction movement of the cells depends on the cxcr4b concentration. Secondly, the intra-synchronization between the active deformations and the adhesion forces inside each cell is triggered by the cxcr4b–cxcr7b polarity. This influences the inter-synchronization between the cells and results in two main modes of migration: uncoordinated and coordinated. The main objectives of the work were (i) to validate our assumptions with respect to the experimental observations and (ii) to decipher the mechanical conditions leading to efficient migration of the primordium. To achieve the second goal, we will specifically focus on the role of the leader cells and their position inside the population

Preface

n/

Reaction-diffusion models for biological pattern formation

We consider the use of reaction-diffusion equations to model biological pattern formation and describe the derivation of the reaction-terms for several illustrative examples. After a brief discussion of the Turing instability in such systems we extend the model formulation to incorporate domain growth. Comparisons are drawn between solution behaviour on growing domains and recent results on self-replicating patterns on domains of fixed size

Enzyme kinetics for a two-step enzymic reaction with comparable initial enzyme-substrate ratios

We extend the validity of the quasi-steady state assumption for a model double intermediate enzyme-substrate reaction to include the case where the ratio of initial enzyme to substrate concentration is not necessarily small. Simple analytical solutions are obtained when the reaction rates and the initial substrate concentration satisfy a certain condition. These analytical solutions compare favourably with numerical solutions of the full system of differential equations describing the reaction. Experimental methods are suggested which might permit the application of the quasi-steady state assumption to reactions where it may not have been obviously applicable before

Cluster formation for multi-strain infections with cross-immunity

Many infectious diseases exist in several pathogenic variants, or strains, which interact via cross-immunity. It is observed that strains tend to self-organise into groups, or clusters. The aim of this paper is to investigate cluster formation. Computations demonstrate that clustering is independent of the model used, and is an intrinsic feature of the strain system itself. We observe that an ordered strain system, if it is sufficiently complex, admits several cluster structures of different types. Appearance of a particular cluster structure depends on levels of cross-immunity and, in some cases, on initial conditions. Clusters, once formed, are stable, and behave remarkably regularly (in contrast to the generally chaotic behaviour of the strains themselves). In general, clustering is a type of self-organisation having many features in common with pattern formation

Modelling aspects of solid cancer growth

The modelling of cancer provides an enormous mathematical challenge because of its inherent multiscale nature. For example, in vascular tumours, nutrient is transported by the vascular system, which operates on a tissue level. However, it affects processes occurring on a molecular level. Molecular and intra-cellular events in turn effect the vascular network and therefore the nutrient dynamics. Our modelling approach is to model, using partial differential equations, processes on the tissue level and couple these to the intercellular events (modelled by ordinary differential equations) via cells modelled as automaton units. Thusfar, within this framework we have modelled structural adaptation at the vessel level and the effects of growth factor production in response to hypoxia. We have also investigated the effects of acid production, mutation and phenotypic evolution driven by tissue environment. These results will be presented