Jonathan A. Sherratt

Models of Epidermal Wound Healing

The spreading of cells across the surface of an epidermal wound enables epidermal migration to be studied independently of the wound contraction that occurs in deeper wounds. In particular, the stimulus for the increase in epidermal mitosis during would healing is uncertain. Our modelling suggests that biochemical regulation of mitosis is fundamental to the process, and that a single chemical with a simple regulatory effect can account for the healing of circular epidermal wounds. The model results compare well with experimental data.

Fibroblast migration and collagen deposition during dermal wound healing : mathematical modelling and clinical implications

The extent to which collagen alignment occurs during dermal wound healing determines the severity of scar tissue formation. We have modelled this using a multiscale approach, in which extracellular materials, for example collagen and fibrin, are modelled as continua, while fibroblasts are considered as discrete units. Within this model framework, we have explored the effects that different parameters have on the alignment process, and we have used the model to investigate how manipulation of transforming growth factor-β levels can reduce scar tissue formation. We briefly review this body of work, then extend the modelling framework to investigate the role played by leucocyte signalling in wound repair. To this end, fibroblast migration and collagen deposition within both the wound region and healthy peripheral tissue are considered. Trajectories of individual fibroblasts are determined as they migrate towards the wound region under the combined influence of collagen/fibrin alignment and gradients in a paracrine chemoattractant produced by leucocytes. The effects of a number of different physiological and cellular parameters upon the collagen alignment and repair integrity are assessed. These parameters include fibroblast concentration, cellular speed, fibroblast sensitivity to chemoattractant concentration and chemoattractant diffusion coefficient. Our results show that chemoattractant gradients lead to increased collagen alignment at the interface between the wound and the healthy tissue. Results show that there is a trade-off between wound integrity and the degree of scarring. The former is found to be optimized under conditions of a large chemoattractant diffusion coefficient, while the latter can be minimized when repair takes place in the presence of a competitive inhibitor to chemoattractants.

Periodic travelling waves in cyclic populations: field studies and reaction-diffusion models

Periodic travelling waves have been reported in a number of recent spatio-temporal field studies of populations undergoing multi-year cycles. Mathematical modelling has a major role to play in understanding these results and informing future empirical studies. We review the relevant field data and summarize the statistical methods used to detect periodic waves. We then discuss the mathematical theory of periodic travelling waves in oscillatory reaction–diffusion equations. We describe the notion of a wave family, and various ecologically relevant scenarios in which periodic travelling waves occur. We also discuss wave stability, including recent computational developments. Although we focus on oscillatory reaction–diffusion equations, a brief discussion of other types of model in which periodic travelling waves have been demonstrated is also included. We end by proposing 10 research challenges in this area, five mathematical and five empirical.

Mathematical analysis of a basic model for epidermal wound healing

The stimuli for the increase in epidermal mitosis during wound healing are not fully known. We construct a mathematical model which suggests that biochemical regulation of mitosis is fundamental to the process, and that a single chemical with a simple regulatory effect can account for the healing of circular epidermal wounds. The numerical results of the model compare well with experimental data. We investigate the model analytically by making biologically relevant approximations. We then obtain travelling wave solutions which provide information about the accuracy of these approximations and clarify the roles of the various model parameters.

Cellular pattern formation during Dictyostelium aggregation

The development of multicellularity in the life cycle of Dictyostelium discoideum provides a paradigm model system for biological pattern formation. Previously, mathematical models have shown how a collective pattern of cell communication by waves of the messenger molecule cyclic adenosine 3′5′-monophosphate (cAMP) arises from excitable local cAMP kinetics and cAMP diffusion. Here we derive a model of the actual cell aggregation process by considering the chemotactic cell response to cAMP and its interplay with the cAMP dynamics. Cell density, which previously has been treated as a spatially homogeneous parameter, is a crucial variable of the aggregation model. We find that the coupled dynamics of cell chemotaxis and cAMP reaction-diffusion lead to the break-up of the initially uniform cell layer and to the formation of the striking cell stream morphology which characterizes the aggregation process in situ. By a combination of stability analysis and two-dimensional simulations of the model equations, we show cell streaming to be the consequence of the growth of a small-amplitude pattern in cell density forced by the large-amplitude cAMP waves, thus representing a novel scenario of spatial patterning in a cell chemotaxis system. The instability mechanism is further analysed by means of an analytic caricature of the model, and the condition for chemotaxis-driven instability is found to be very similar to the one obtained for the standard (non-oscillatory) Keller-Segel system. The growing cell stream pattern feeds back into the cAMP dynamics, which can explain in some detail experimental observations on the time evolution of the cAMP wave pattern, and suggests the characterization of the Dictyostelium aggregation field as a self-organized excitable medium.

Diffusion driven instability in an inhomogeneous domain

Diffusion driven instability in reaction-diffusion systems has been proposed as a mechanism for pattern formation in numerous embryological and ecological contexts. However, the possible effects of environmental inhomogeneities has received relatively little attention. We consider a general two species reaction-diffusion model in one space dimension, with one diffusion coefficient a step function of the spatial coordinate. We derive the dispersion relation and the solution of the linearized system. We apply our results to Turing-type models for both embryogenesis and predator-prey interactions. In the former case we derive conditions for pattern to be isolated in one part of the domain, and in the latter we introduce the concept of “environmental instability”. Our results suggest that environmental inhomogeneity could be an important regulator of biological pattern formation.

Oncogenes, Anti-Oncogenes and the Immune Response to Cancer: A Mathematical Model

We develop a mathematical model for the initial growth of a tumour after a mutation in which either an oncogene is expressed or an anti-oncogene (i.e. tumour suppressor gene) is lost. Our model incorporates mitotic control by several biochemicals, with quite different regulatory characteristics, and we consider mutations affecting the cellular response to these control mechanisms. Our mathematical representation of these mutations reflects the current understanding of the roles of oncogenes and anti-oncogenes in controlling cell proliferation. Numerical solutions of our model, for biologically relevant parameter values, show that the different types of mutations have quite different effects. Mutations affecting the cell response to chemical regulators, or resulting in autonomy from such regulators, cause an advancing wave of tumour cells and a receding wave of normal cells. By contrast, mutations affecting the production of a mitotic regulator cause a slow localized increase in the numbers of both normal and mutant cells. We extend our model to investigate the possible effects of an immune response to cancer by including a first order removal of mutant cells. When this removal rate exceeds a critical value, the immune system can suppress tumour growth; we derive an expression for this critical value as a function of the parameters characterizing the mutation. Our results suggest that the effectiveness of the immune response after an oncogenic mutation depends crucially on the way in which the mutation affects the biochemical control of cell division.

Control of epidermal stem cell clusters by Notch-mediated lateral induction

Stem cells in the basal layer of human interfollicular epidermis form clusters that can be reconstituted in vitro. In order to supply the interfollicular epidermis with differentiated cells, the size of these clusters must be controlled. Evidence suggests that control is regulated via differentiation of stem cells on the periphery of the clusters. Moreover, there is growing evidence that this regulation is mediated by the Notch signalling pathway. In this paper, we develop theoretical arguments, in conjunction with computer simulations of a model of the basal layer, to show that regulation of differentiation is the most likely mechanism for cluster control. In addition, we show that stem cells must adhere more strongly to each other than they do to differentiated cells. Developing our model further we show that lateral-induction, mediated by the Notch signalling pathway, is a natural mechanism for cluster control. It can not only indicate to cells the size of the cluster they are in and their position within it, but it can also control the cluster size. This can only be achieved by postulating a secondary, cluster wide, differentiation signal, and cells with high Delta expression being deaf to this signal.

Theoretical models of wound healing: past successes and future challenges

The complex biology of wound healing is an area in which theoretical modelling has already made a significant impact. In this review article, the authors describe the key features of wound healing biology, divided into four components: epidermal wound healing, remodelling of the dermal extracellular matrix, wound contraction, and angiogenesis. Within each of these categories, previous modelling work is described, and the authors identify what they regard as the main challenges for future theoretical work.

Modelling the movement of interacting cell populations

Mathematical modelling of cell movement has traditionally focussed on a single population of cells, often moving in response to various chemical and environmental cues. In this paper, we consider models for movement in two or more interacting cell populations. We begin by discussing intuitive ideas underlying the extension of models for a single-cell population to two interacting populations. We then consider more formal model development using transition probability methods, and we discuss how the same equations can be obtained as the limiting form of a velocity-jump process. We illustrate the models we have developed via two examples. The first of these is a generic model for competing cell populations, and the second concerns aggregation in cell populations moving in response to chemical gradients.