Long-range seed dispersal enables almost stationary patterns in a model for dryland vegetation.

Spatiotemporal patterns of vegetation are a ubiquitous feature of semi-arid ecosystems. On sloped terrain, vegetation patterns occur as stripes perpendicular to the contours. Field studies report contrasting long-term dynamics between different observation sites; some observe slow uphill migration of vegetation bands while some report stationary patterns. In this paper, we show that long-range seed dispersal provides a mechanism that enables the occurrence of both migrating and stationary patterns. We utilise a nonlocal PDE model in which seed dispersal is accounted for by a convolution term. The model represents vegetation patterns as periodic travelling waves and numerical continuation shows that both migrating and almost stationary patterns are stable if seed dispersal distances are sufficiently large. We use a perturbation theory approach to obtain analytical confirmation of the existence of almost stationary patterned solutions and provide a biological interpretation of the phenomenon.

Spatial self-organisation enables species coexistence in a model for savanna ecosystems.

The savanna biome is characterised by a continuous vegetation cover, comprised of herbaceous and woody plants. The coexistence of species in arid savannas, where water availability is the main limiting resource for plant growth, provides an apparent contradiction to the classical principle of competitive exclusion. Previous theoretical work using nonspatial models has focussed on the development of an understanding of coexistence mechanisms through the consideration of resource niche separation and ecosystem disturbances. In this paper, we propose that a spatial self-organisation principle, caused by a positive feedback between local vegetation growth and water redistribution, is sufficient for species coexistence in savanna ecosystems. We propose a spatiotemporal ecohydrological model of partial differential equations, based on the Klausmeier reaction-advection-diffusion model for vegetation patterns, to investigate the effects of spatial interactions on species coexistence on sloped terrain. Our results suggest that species coexistence is a possible model outcome, if a balance is kept between the species' average fitness (a measure of a species' competitive abilities in a spatially uniform setting) and their colonisation abilities. Spatial heterogeneities in resource availability are utilised by the superior coloniser (grasses), before it is outcompeted by the species of higher average fitness (trees). A stability analysis of the spatially nonuniform coexistence solutions further suggests that grasses act as ecosystem engineers and facilitate the formation of a continuous tree cover for precipitation levels unable to support a uniform tree density in the absence of a grass species.

A Nonlocal Model for Contact Attraction and Repulsion in Heterogeneous Cell Populations.

Instructing others to move is fundamental for many populations, whether animal or cellular. In many instances, these commands are transmitted by contact, such that an instruction is relayed directly (e.g. by touch) from signaller to receiver: for cells, this can occur via receptor-ligand mediated interactions at their membranes, potentially at a distance if a cell extends long filopodia. Given that commands ranging from attractive to repelling can be transmitted over variable distances and between cells of the same (homotypic) or different (heterotypic) type, these mechanisms can clearly have a significant impact on the organisation of a tissue. In this paper, we extend a system of nonlocal partial differential equations (integrodifferential equations) to provide a general modelling framework to explore these processes, performing linear stability and numerical analyses to reveal its capacity to trigger the self-organisation of tissues. We demonstrate the potential of the framework via two illustrative applications: the contact-mediated dispersal of neural crest populations and the self-organisation of pigmentation patterns in zebrafish.

How does cellular contact affect differentiation mediated pattern formation?

In this paper, we present a two-population continuous integro-differential model of cell differentiation, using a non-local term to describe the influence of the local environment on differentiation. We investigate three different versions of the model, with differentiation being cell autonomous, regulated via a community effect, or weakly dependent on the local cellular environment. We consider the spatial patterns that such different modes of differentiation produce, and investigate the formation of both stripes and spots by the model. We show that pattern formation only occurs when differentiation is regulated by a strong community effect. In this case, permanent spatial patterns only occur under a precise relationship between the parameters characterising cell dynamics, although transient patterns can persist for biologically relevant timescales when this condition is relaxed. In all cases, the long-lived patterns consist only of stripes, not spots.

Alterations in proteolytic activity at low pH and its association with invasion: a theoretical model.

The extracellular pH (pHe) of solid tumours is often lower than in normal tissues, with median pH values of about 7.0 in tumours and 7.5 in normal tissue. Despite this more acidic tumour microenvironment, non-invasive measurements of intracellular pH (pHi) have shown that the pHi of solid tumours is neutral or slightly alkaline compared to normal tissue (pHi 7.0-7.4). This gives rise to a reversed cellular pH gradient between tumours and normal tissue, which has been implicated in many aspects of tumour progression. One such area is tumour invasion: the incubation of tumour cells at low pH has been shown to induce more aggressive invasive behaviour in vitro. In this paper the authors use mathematical models to investigate whether altered proteolytic activity at low pH is responsible for the stimulation of a more metastatic phenotype. The authors examined the effect of culture pH on the secretion and activity of two different classes of proteinases: the metalloproteinases (MMPs), and the cysteine proteinases (such as cathepsin B). The modelling suggests that changes in MMP activity at low pH do not have significant effects on invasive behaviour. However, the model predicts that the levels of active-cathepsin B are significantly altered by acidic pH. This result suggests a critical role for the cysteine proteinases in tumour progression.

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.

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.

Dictyostelium discoideum: cellular self-organization in an excitable biological medium.

The dynamics which govern the establishment of pattern and form in multicellular organisms remain a key problem of developmental biology. We study this question in the case of morphogenesis during aggregation of the slime mould Dictyostelium discoideum. Here detailed experimental information allows the formulation of a mechanistic model in which the central element is the coupling of the previously much-studied intracellular cyclic AMP signalling with the chemotactic cell response in cyclic AMP gradients. Numerical simulations of the model show quantitatively how signal relay, chemotactic movement and adaptation orchestrate the collective modes of cell signalling and migration in the aggregating cell layer. The interaction of chemotaxis with the cyclic AMP excitation waves causes the initially homogeneous cell layer to become unstable towards the formation of a branching cell stream pattern with close cell-cell contacts as observed in situ. The evolving cell morphology in turn leads to a pattern of non-homogeneous excitability of the medium and thus feeds back into the cAMP dynamics. This feedback can explain the decrease in signalling period and propagation speed with time, as well as observations on the structure of the spiral wave core in this self-organized excitable medium.

Generation of periodic waves by landscape features in cyclic predator-prey systems.

The vast majority of models for spatial dynamics of natural populations assume a homogeneous physical environment. However, in practice, dispersing organisms may encounter landscape features that significantly inhibit their movement. We use mathematical modelling to investigate the effect of such landscape features on cyclic predator-prey populations. We show that when appropriate boundary conditions are applied at the edge of the obstacle, a pattern of periodic travelling waves develops, moving out and away from the obstacle. Depending on the assumptions of the model, these waves can take the form of roughly circular 'target patterns' or spirals. This is, to our knowledge, a new mechanism for periodic-wave generation in ecological systems and our results suggest that it may apply quite generally not only to cyclic predator-prey interactions, but also to populations that oscillate for other reasons. In particular, we suggest that it may provide an explanation for the observed pattern of travelling waves in the densities of field voles (Microtus agrestis) in Kielder Forest (Scotland-England border) and of red grouse (Lagopus lagopus scoticus) on Kerloch Moor (northeast Scotland), which in both cases move orthogonally to any large-scale obstacles to movement. Moreover, given that such obstacles to movement are the rule rather than the exception in real-world environments, our results suggest that complex spatio-temporal patterns such as periodic travelling waves are likely to be much more common in the natural world than has previously been assumed.

A mathematical model for collagen fibre formation during foetal and adult dermal wound healing.

Adult dermal wounds, in contrast to foetal wounds, heal with the formation of scar tissue. A crucial factor in determining the nature of the healed tissue is the ratio of collagen 1 to collagen 3, which regulates the diameter of collagen fibres. We develop a mathematical model which focuses on the stimulus for collagen synthesis due to the secretion of the different isoforms of the regulatory chemical transforming growth factor beta. Numerical simulations of the model lead to a value of this ratio consistent with that of healthy tissue for the foetus but corresponding to scarring in adult wound healing. We investigate the effect of topical application of TGF beta isoforms during healing and determine the key parameters which control the difference between adult and foetal repair.

How far can a juxtacrine signal travel?

Juxtacrine signalling is the process of cell communication in which ligand and receptors are both anchored in the cell membrane. We develop three mathematical models for this process, involving different mathematical representations of the dynamics of membrane-bound ligand and free and bound receptors, within an epithelial sheet. We consider the dynamics of this system following a localized disturbance, such as would be provided by a source of ligand or by the generation of a free edge via wounding. We study the ability of the juxtacrine mechanism to transmit a signal away from this disturbance, and show analytically that the spatial half-life of the signal can in fact be arbitrarily large. This result is quite general, since we use a generic reaction kinetic scheme; the key assumption is that ligand and receptor production are both upregulated by binding. Moreover, the result applies to all three of our model formulations. We conclude by discussing applications of the result to the particular case of the transforming growth factor alpha binding to epidermal growth factor receptor in epidermal wound healing.

Modelling tumour acidity and invasion.

The intracellular pH (pHi) of mammalian cells is tightly regulated by the concerted action of a number of different pumps in the plasma membrane. Despite the acidic extracellular environment (pHe 6.8-7.0) of some tumours, the pHi of solid tumours is neutral or slightly alkaline compared to normal tissues (pHi 7.0-7.4). This gives rise to a reversed pH gradient across the cell membrane between tumours and normal tissue, which has been implicated in many aspects of tumour progression. One such area is tumour invasion: the incubation of tumour cells at low pH has been shown to induce more aggressive invasive behaviour in vitro. In this paper we use mathematical models to investigate whether altered proteolytic activity at low pH is responsible for the stimulation of a more metastatic phenotype. We examined the effect of culture pH on the secretion and activity of two different classes of proteinases: the metalloproteinases (MMPs), and the cysteine proteinases (such as cathepsin B). The modelling predicts that, in addition to metalloproteinase activity, the pH-induced peripheral redistribution of cathepsin B could be a major factor involved in the acquisition of a more metastatic phenotype in malignant cells at low pHe.

Epidermal wound healing: the clinical implications of a simple mathematical model.

The role of biochemical regulation in the healing of epidermal wounds remains the subject of much biological debate. We have previously developed a mathematical model which focuses on the role of mitogenic autoregulation in reepithelialization (23-25). Here, we discuss some predictions of our model and their clinical relevance. We investigate both the effects of adding mitotic regulators to healing wounds and the dependence of healing time on wound shape. The latter study suggests a possible mechanism for the control of changes in wound shape during healing. The predictions we make are amenable to experimental verification, and suggest new ideas for experimental research.

Predictive mathematical modeling in metastasis.

Mathematical modeling is emerging as a powerful predictive tool in many areas of biology and medicine, with applications to cancer metastasis increasingly widespread and effective. This type of modeling involves quantitatively accurate representations of specific cellular activities, and is quite different from more traditional applications of mathematics to cancer, such as the simple fitting of experimental data to Gompertzian growth curves. It is made possible by the twin revolutions in molecular biology and nonlinear mathematics over the last two decades, and involves using experimental data at the cell and molecular level to construct mathematical models, which can then be used to predict the macroscopic implications of this data.Mathematical models have been in use for biological prediction since the early part of the century, initially in ecology and embryology. These early models were phenomenological, that is, they acted as a convenient way to express and explore theories, but did not represent particular postulated mechanisms. Establishment of mathematical biology as a recognized scientific field was achieved by a number of major successes for these early models. Most notable amongst these is the work of Hodgkin and Huxley on electrical signaling in nerve axons, which underlies much of neurophysiology, and for which they were awarded the Nobel Prize for Physiology and Medicine in 1963. More recently, the ability to isolate biological mechanisms at the molecular level has led to a new type of mathematical model, which represents specific low-level mechanisms, either known or hypothesised.

Mathematical modelling of juxtacrine cell signalling.

Juxtacrine signalling is emerging as an important means of cellular communication, in which signalling molecules anchored in the cell membrane bind to and activate receptors on the surface of immediately neighbouring cells. We develop a mathematical model to describe this process, consisting of a coupled system of ordinary differential equations, with one identical set of equations for each cell. We use a generic representation of ligand-receptor binding, and assume that binding exerts a positive feedback on the secretion of new receptors and ligand. By linearising the model equations about a homogeneous equilibrium, we categorize the range and extent of signal patterns as a function of parameters. We show in particular that the signal decay rate depends crucially on the form of the feedback functions, and can be made arbitrarily small by appropriate choice of feedback, for any set of kinetic parameters. As a specific example, we consider the application of our model to juxtacrine signalling by TGF alpha in response to epidermal wounding. We demonstrate that all the predictions of our linear analysis are confirmed in numerical simulations of the non-linear system, and discuss the implications for the healing response.

Keratinocyte growth factor signalling: a mathematical model of dermal-epidermal interaction in epidermal wound healing.

A wealth of growth factors are known to regulate the various cell functions involved in the repair process. An understanding of their therapeutic value is essential to achieve improved wound healing. Keratinocyte growth factor (KGF) seems to have a unique role as a mediator of mesenchymal-epithelial interactions: it originates from mesenchymal cells, yet acts exclusively on epithelial cells. In this paper, we study KGF's role in epidermal wound healing, since its production is substantially up-regulated after injury. We begin by modelling the dermal-epidermal signalling mechanism of KGF to investigate how this extra production affects the signal range. We then incorporate the effect of KGF on cell proliferation, and using travelling wave analysis we obtain an approximation for the rate of healing. Our modelling shows that the large up-regulation of KGF post-wounding extends the KGF signal range but is above optimal for the rate of wound closure. We predict that other functions of KGF may be more important than its role as a mitogen for the healing process.

Spatially varying equilibria of mechanical models: application to dermal wound contraction.

Mechanochemical models based on the Oster-Murray continuum framework have been applied to a variety of biological settings to obtain an understanding of the morphogenesis of living tissues. Wound-healing in mammalian skin is an important example, because a complex sequence of biochemical and biomechanical responses are orchestrated to close a wound by a combination of new tissue formation and wound contraction. Mechanical interactions between dermal fibroblastic cells and the collagen-rich extracellular matrix are crucial in the development of a contracted wound state. We and others have previously proposed mechanochemical models for wound repair to gain a greater understanding of both normal and abnormal healing. In the present work, the existence of spatially varying equilibria of these models is investigated by using a small-stain approximation and phase-plane techniques, with numerical simulations to confirm the analytical predictions. These results are sources of novel insight into the roles of key biological parameters in determining the mechanical properties of a contracted wound. These methods may also be relevant to other morphogenetic scenarios for which similar mechanochemical models have been proposed.

Mathematical modeling of corneal epithelial wound healing.

We propose a reaction-diffusion model of the mechanisms involved in the healing of corneal surface wounds. The model focuses on the stimulus for increased mitotic and migratory activity, specifically the role of epidermal growth factor. Analysis of the model equations elucidates the interaction and roles of the model parameters in determining the speed of healing and the shape of the traveling wave solutions which correspond to the migration of cells into the wound during the initial phase of healing. We determine an analytic approximation for the speed of traveling wave solutions of the model in terms of the parameters and verify the results numerically. By comparing the predicted speed with experimentally measured healing rates, we conclude that serum-derived factors can alone account for the overall features of the healing process, but that the supply of growth factors by the tear film in the absence of serum-derived factors is not sufficient to give the observed healing rate. Numerical solutions of the model equations also confirm the importance of both migration and mitosis for effective would healing. By modifying the model we obtain an analytic prediction for the healing rate of corneal surface wounds when epidermal growth factor is applied topically to the wound.

Mathematical modelling of anisotropy in fibrous connective tissue.

We present two modelling frameworks for studying dynamic anistropy in connective tissue, motivated by the problem of fibre alignment in wound healing. The first model is a system of partial differential equations operating on a macroscopic scale. We show that a model consisting of a single extracellular matrix material aligned by fibroblasts via flux and stress exhibits behaviour that is incompatible with experimental observations. We extend the model to two matrix types and show that the results of this extended model are robust and consistent with experiment. The second model represents cells as discrete objects in a continuum of ECM. We show that this model predicts patterns of alignment on macroscopic length scales that are lost in a continuum model of the cell population.

Healing by numbers: mathematics meets medicine.

The molecular biology revolution is providing vast quantities of information about the cell biology and biochemistry underlying clinical medicine. How can the pieces of this jigsaw puzzle be put together? This is currently a major biological challenge, and help is at hand from an unlikely source - mathematics.