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.

A new interpretation of the Keller-Segel model based on multiphase modelling

In this paper an alternative derivation and interpretation are presented of the classical Keller-Segel model of cell migration due to random motion and chemotaxis. A multiphase modelling approach is used to describe how a population of cells moves through a fluid containing a diffusible chemical to which the cells are attracted. The cells and fluid are viewed as distinct components of a two-phase mixture. The principles of mass and momentum balance are applied to each phase, and appropriate constitutive laws imposed to close the resulting equations. A key assumption here is that the stress in the cell phase is influenced by the concentration of the diffusible chemical. By restricting attention to one-dimensional cartesian geometry we show how the model reduces to a pair of nonlinear coupled partial differential equations for the cell density and the chemical concentration. These equations may be written in the form of the Patlak-Keller-Segel model, naturally including density-dependent nonlinearities in the cell motility coefficients. There is a direct relationship between the random motility and chemotaxis coefficients, both depending in an inter-related manner on the chemical concentration. We suggest that this may explain why many chemicals appear to stimulate both chemotactic and chemokinetic responses in cell populations. After specialising our model to describe slime mold we then show how the functional form of the chemical potential that drives cell locomotion influences the ability of the system to generate spatial patterns. The paper concludes with a summary of the key results and a discussion of avenues for future research.

Sepsis research in the next millennium: concentrate on the software rather than the hardware

Today the basic principles of septic conditions are understood. Nevertheless, sepsis research has reached a critical point. To integrate our knowledge towards a consistent theory of the disease process and to derive effective therapies, new perspectives for future research that fit the complexity of the problem have to be found. We conducted a review of the literature concerning systemic inflammatory response syndrome (SIRS) and sepsis with particular reference to liver pathophysiology. And compared our findings with characteristic features of complex systems. The complexity of sepsis is broadly recognized. A review of the different aspects of liver inflammation during SIRS and sepsis, i.e. endotoxin challenge, cytokine induced dysfunction, the mechanisms of leukocyte transmigration, and hormonal and neuroendocrine regulatory mechanisms is given. Key aspects of complex systems, including parallelism, locality, emergence, and cross-scale interactions are introduced. We conclude that sepsis research needs new perspectives that allow us to handle the complex interactions occurring during the disease process. We propose to focus research on the interactions between the constituents of the system rather than only describing isolated aspects of the disease process. We also conclude that the ideas and techniques of non-linear systems theory are suitable tools for the analysis of complex and dynamic diseases like SIRS and sepsis.

Extracellular matrix concentration exerts selection pressure on invasive cells

The mortality associated with cancer depends upon the ability of malignant cells to invade and metastasise into adjacent and distant regions. Such malignant spread is caused by the acquisition of an invasive phenotype which involves variable changes in cell-cell adhesion, proteolysis of adjoining extracellular matrix molecules, and the ability to move in a directed fashion in response to fixed and soluble gradients. Whilst the degree of variability in the pattern of metastasis is large, several cancers show regional predilections for invasion. Tumour cell heterogeneity and extracellular matrix composition have been shown to account for some regional variations. In this study, an invasion assay was used to assess the invasiveness of HT1080 tumour cells migrating through a collagen gel. Our experiments showed that, in the absence of externally imposed chemical gradients, HT1080 invasiveness was related in a biphasic manner to collagen concentration. Using a mathematical model, developed to study this phenomenon, we predicted that tumour cell proliferation may also be related in a biphasic manner to collagen concentration. This model prediction was confirmed using a combination of collagen gel invasion and proliferation assays. Investigation of the mathematical model suggests that interactions between haptotaxis and proliferation of the HT1080 cells may be responsible for the biphasic dependence of the penetration depth and proliferation on collagen gel concentration. In conclusion, we showed how mathematical methods can be combined with experimental work to provide new and valuable insight into important biological issues.