Pattern Formation And Wound Healing

One of the main immediate challenges in the biomedical sciences is the synthesis of the vast amount of data now available at the molecular and cellular levels for development, regulation and repair. This, in turn, requires an understanding of the interaction and coordination of a myriad of complex inter-related processes occurring on very different spatial and temporal scales. Mathematics provides the obvious language in which to develop and interpret these interactions, and a number of mathematical models have already been proposed to account for certain observed biological and medical phenomena. Here, we consider two areas of modelling, namely spatial patterning, and wound healing, both sharing the common underlying processes of cells creating and responding to signalling cues

Classifying general nonlinear force laws in cell-based models via the continuum limit

though discrete cell-based frameworks are now commonly used to simulate a whole range of biological phenomena, it is typically not obvious how the numerous different types of model are related to one another, nor which one is most appropriate in a given context. Here we demonstrate how individual cell movement on the discrete scale modeled using nonlinear force laws can be described by nonlinear diffusion coefficients on the continuum scale. A general relationship between nonlinear force laws and their respective diffusion coefficients is derived in one spatial dimension and, subsequently, a range of particular examples is considered. For each case excellent agreement is observed between numerical solutions of the discrete and corresponding continuum models. Three case studies are considered in which we demonstrate how the derived nonlinear diffusion coefficients can be used to (a) relate different discrete models of cell behavior; (b) derive discrete, intercell force laws from previously posed diffusion coefficients, and (c) describe aggregative behavior in discrete simulations

Refining self-propelled particle models for collective behaviour

Swarming, schooling, flocking and herding are all names given to the wide variety of collective behaviours exhibited by groups of animals, bacteria and even individual cells. More generally, the term swarming describes the behaviour of an aggregate of agents (not necessarily biological) of similar size and shape which exhibit some emergent property such as directed migration or group cohesion. In this paper we review various individual-based models of collective behaviour and discuss their merits and drawbacks. We further analyse some one-dimensional models in the context of locust swarming. In specific models, in both one and two dimensions, we demonstrate how varying the parameters relating to how much attention individuals pay to their neighbours can dramatically change the behaviour of the group. We also introduce leader individuals to these models with the ability to guide the swarm to a greater or lesser degree as we vary the parameters of the model. We consider evolutionary scenarios for models with leaders in which individuals are allowed to evolve the degree of influence neighbouring individuals have on their subsequent motion

Mesoscopic and continuum modelling of angiogenesis

Angiogenesis is the formation of new blood vessels from pre-existing ones in response to chemical signals secreted by, for example, a wound or a tumour. In this paper, we propose a mesoscopic lattice-based model of angiogenesis, in which processes that include proliferation and cell movement are considered as stochastic events. By studying the dependence of the model on the lattice spacing and the number of cells involved, we are able to derive the deterministic continuum limit of our equations and compare it to similar existing models of angiogenesis. We further identify conditions under which the use of continuum models is justified, and others for which stochastic or discrete effects dominate. We also compare different stochastic models for the movement of endothelial tip cells which have the same macroscopic, deterministic behaviour, but lead to markedly different behaviour in terms of production of new vessel cells.Comment: 48 pages, 13 figure

Modeling chemotaxis reveals the role of reversed phosphotransfer and a bi-functional kinase-phosphatase

Understanding how multiple signals are integrated in living cells to produce a balanced response is a major challenge in biology. Two-component signal transduction pathways, such as bacterial chemotaxis, comprise histidine protein kinases (HPKs) and response regulators (RRs). These are used to sense and respond to changes in the environment. Rhodobacter sphaeroides has a complex chemosensory network with two signaling clusters, each containing a HPK, CheA. Here we demonstrate, using a mathematical model, how the outputs of the two signaling clusters may be integrated. We use our mathematical model supported by experimental data to predict that: (1) the main RR controlling flagellar rotation, CheY6, aided by its specific phosphatase, the bifunctional kinase CheA3, acts as a phosphate sink for the other RRs; and (2) a phosphorelay pathway involving CheB2 connects the cytoplasmic cluster kinase CheA3 with the polar localised kinase CheA2, and allows CheA3-P to phosphorylate non-cognate chemotaxis RRs. These two mechanisms enable the bifunctional kinase/phosphatase activity of CheA3 to integrate and tune the sensory output of each signaling cluster to produce a balanced response. The signal integration mechanisms identified here may be widely used by other bacteria, since like R. sphaeroides, over 50% of chemotactic bacteria have multiple cheA homologues and need to integrate signals from different sources