Kevin J. Painter

A user’s guide to PDE models for chemotaxis

Mathematical modelling of chemotaxis (the movement of biological cells or organisms in response to chemical gradients) has developed into a large and diverse discipline, whose aspects include its mechanistic basis, the modelling of specific systems and the mathematical behaviour of the underlying equations. The Keller-Segel model of chemotaxis (Keller and Segel in J Theor Biol 26:399–415, 1970; 30:225–234, 1971) has provided a cornerstone for much of this work, its success being a consequence of its intuitive simplicity, analytical tractability and capacity to replicate key behaviour of chemotactic populations. One such property, the ability to display “auto-aggregation”, has led to its prominence as a mechanism for self-organisation of biological systems. This phenomenon has been shown to lead to finite-time blow-up under certain formulations of the model, and a large body of work has been devoted to determining when blow-up occurs or whether globally existing solutions exist. In this paper, we explore in detail a number of variations of the original Keller–Segel model. We review their formulation from a biological perspective, contrast their patterning properties, summarise key results on their analytical properties and classify their solution form. We conclude with a brief discussion and expand on some of the outstanding issues revealed as a result of this work.

Spatial pattern formation in chemical and biological systems

One of the central issues in developmental biology is the formation of spatial pattern in the embryo. A number of theories have been proposed to account for this phenomenon. The most widely studied is reaction diffusion theory, which proposes that a chemical pre-pattern is first set up due to a system of reacting and diffusing chemicals, and cells respond to this pre-pattern by differentiating accordingly. Such patterns, known as Turing structures, were first identified in chemical systems only recently. This article reviews the application of reaction diffusion theory to chemical systems and then considers a number of biological applications.

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.

Modelling cell migration strategies in the extracellular matrix

The extracellular matrix (ECM) is a highly organised structure with the capacity to direct cell migration through their tendency to follow matrix fibres, a process known as contact guidance. Amoeboid cell populations migrate in the ECM by making frequent shape changes and have minimal impact on its structure. Mesenchymal cells actively remodel the matrix to generate the space in which they can move. In this paper, these different types of movement are studied through simulation of a continuous transport model. It is shown that the process of contact guidance in a structured ECM can spatially organise cell populations. Furthermore, when combined with ECM remodelling, it can give rise to cellular pattern formation in the form of “cell-chains” or networks without additional environmental cues such as chemoattractants. These results are applied to a simple model for tumour invasion where it is shown that the interactions between invading cells and the ECM structure surrounding the tumour can have a profound impact on the pattern and rate of cell infiltration, including the formation of characteristic “fingering” patterns. The results are further discussed in the context of a variety of relevant processes during embryonic and adult stages.

Cryptic patterning of avian skin confers a developmental facility for loss of neck feathering.

Vertebrate skin is characterized by its patterned array of appendages, whether feathers, hairs, or scales. In avian skin the distribution of feathers occurs on two distinct spatial levels. Grouping of feathers within discrete tracts, with bare skin lying between the tracts, is termed the macropattern, while the smaller scale periodic spacing between individual feathers is referred to as the micropattern. The degree of integration between the patterning mechanisms that operate on these two scales during development and the mechanisms underlying the remarkable evolvability of skin macropatterns are unknown. A striking example of macropattern variation is the convergent loss of neck feathering in multiple species, a trait associated with heat tolerance in both wild and domestic birds. In chicken, a mutation called Naked neck is characterized by a reduction of body feathering and completely bare neck. Here we perform genetic fine mapping of the causative region and identify a large insertion associated with the Naked neck trait. A strong candidate gene in the critical interval, BMP12/GDF7, displays markedly elevated expression in Naked neck embryonic skin due to a cis-regulatory effect of the causative mutation. BMP family members inhibit embryonic feather formation by acting in a reaction-diffusion mechanism, and we find that selective production of retinoic acid by neck skin potentiates BMP signaling, making neck skin more sensitive than body skin to suppression of feather development. This selective production of retinoic acid by neck skin constitutes a cryptic pattern as its effects on feathering are not revealed until gross BMP levels are altered. This developmental modularity of neck and body skin allows simple quantitative changes in BMP levels to produce a sparsely feathered or bare neck while maintaining robust feather patterning on the body.

Continuous Models for Cell Migration in Tissues and Applications to Cell Sorting via Differential Chemotaxis

Chemotaxis, the guided migration of cells in response to chemical gradients, is vital to a wide variety of biological processes, including patterning of the slime mold Dictyostelium, embryonic morphogenesis, wound healing, and tumor invasion. Continuous models of chemotaxis have been developed to describe many such systems, yet few have considered the movements within a heterogeneous tissue composed of multiple subpopulations. In this paper, a partial differential equation (PDE) model is developed to describe a tissue formed from two distinct chemotactic populations. For a “crowded” (negligible extracellular space) tissue, it is demonstrated that the model reduces to a simpler one-species system while for an “uncrowded” tissue, it captures both movement of the entire tissue (via cells attaching to/migrating within an extracellular substrate) and the within-tissue rearrangements of the separate cellular subpopulations. The model is applied to explore the sorting of a heterogeneous tissue, where it is shown that differential-chemotaxis not only generates classical sorting patterns previously seen via differential-adhesion, but also demonstrates new classes of behavior. These new phenomena include temporal dynamics consisting of a traveling wave composed of spatially sorted subpopulations reminiscent of Dictyostelium slugs.

Mathematical modelling of glioma growth: the use of Diffusion Tensor Imaging (DTI) data to predict the anisotropic pathways of cancer invasion.

The nonuniform growth of certain forms of cancer can present significant complications for their treatment, a particularly acute problem in gliomas. A number of experimental results have suggested that invasion is facilitated by the directed movement of cells along the aligned neural fibre tracts that form a large component of the white matter. Diffusion tensor imaging (DTI) provides a window for visualising this anisotropy and gaining insight on the potential invasive pathways. In this paper we develop a mesoscopic model for glioma invasion based on the individual migration pathways of invading cells along the fibre tracts. Via scaling we obtain a macroscopic model that allows us to explore the overall growth of a tumour. To connect DTI data to parameters in the macroscopic model we assume that directional guidance along fibre tracts is described by a bimodal von Mises-Fisher distribution (a normal distribution on a unit sphere) and parametrised according to the directionality and degree of anisotropy in the diffusion tensors. We demonstrate the results in a simple model for glioma growth, exploiting both synthetic and genuine DTI datasets to reveal the potentially crucial role of anisotropic structure on invasion.

CONVERGENCE OF A CANCER INVASION MODEL TO A LOGISTIC CHEMOTAXIS MODEL

A characteristic feature of tumor invasion is the destruction of the healthy tissue surrounding it. Open space is generated, which invasive tumor cells can move into. One such mechanism is the urokinase plasminogen system (uPS), which is found in many processes of tissue reorganization. Lolas, Chaplain and collaborators have developed a series of mathematical models for the uPS and tumor invasion. These models are based upon degradation of the extracellular material through plasmid plus chemotaxis and haptotaxis. In this paper we consider the uPS invasion models in one-space dimension and we identify a condition under which this cancer invasion model converges to a chemotaxis model with logistic growth. This condition assumes that the density of the extracellular material is not too large. Our result shows that the complicated spatio-temporal patterns, which were observed by Lolas and Chaplain et al. are organized by the chaotic attractor of the logistic chemotaxis system. Our methods are based on energy estimates, where, for convergence, we needed to find lower estimates in Lγ for 0 < γ < 1. This is a new method for these types of PDE.

Travelling waves in hyperbolic chemotaxis equations

Mathematical models of bacterial populations are often written as systems of partial differential equations for the densities of bacteria and concentrations of extracellular (signal) chemicals. This approach has been employed since the seminal work of Keller and Segel in the 1970s (Keller and Segel, J. Theor. Biol. 30:235–248, 1971). The system has been shown to permit travelling wave solutions which correspond to travelling band formation in bacterial colonies, yet only under specific criteria, such as a singularity in the chemotactic sensitivity function as the signal approaches zero. Such a singularity generates infinite macroscopic velocities which are biologically unrealistic. In this paper, we formulate a model that takes into consideration relevant details of the intracellular processes while avoiding the singularity in the chemotactic sensitivity. We prove the global existence of solutions and then show the existence of travelling wave solutions both numerically and analytically.

Adding Adhesion to a Chemical Signaling Model for Somite Formation

Somites are condensations of mesodermal cells that form along the two sides of the neural tube during early vertebrate development. They are one of the first instances of a periodic pattern, and give rise to repeated structures such as the vertebrae. A number of theories for the mechanisms underpinning somite formation have been proposed. For example, in the “clock and wavefront” model (Cooke and Zeeman in J. Theor. Biol. 58:455–476, 1976), a cellular oscillator coupled to a determination wave progressing along the anterior-posterior axis serves to group cells into a presumptive somite. More recently, a chemical signaling model has been developed and analyzed by Maini and coworkers (Collier et al. in J. Theor. Biol. 207:305–316, 2000; Schnell et al. in C. R. Biol. 325:179–189, 2002; McInerney et al. in Math. Med. Biol. 21:85–113, 2004), with equations for two chemical regulators with entrained dynamics. One of the chemicals is identified as a somitic factor, which is assumed to translate into a pattern of cellular aggregations via its effect on cell–cell adhesion. Here, the authors propose an extension to this model that includes an explicit equation for an adhesive cell population. They represent cell adhesion via an integral over the sensing region of the cell, based on a model developed previously for adhesion driven cell sorting (Armstrong et al. in J. Theor. Biol. 243:98–113, 2006). The expanded model is able to reproduce the observed pattern of cellular aggregates, but only under certain parameter restrictions. This provides a fuller understanding of the conditions required for the chemical model to be applicable. Moreover, a further extension of the model to include separate subpopulations of cells is able to reproduce the observed differentiation of the somite into separate anterior and posterior halves.