Phase differences in reaction-diffusion-advection systems and applications to morphogenesis

The authors study the effect of advection on reaction-diffusion patterns. It is shown that the addition of advection to a two-variable reaction–diffusion system with periodic boundary conditions results in the appearance of a phase difference between the patterns of the two variables which depends on the difference between the advection coefficients. The spatial patterns move like a travelling wave with a fixed velocity which depends on the sum of the advection coefficients. By a suitable choice of advection coefficients, the solution can be made stationary in time. In the presence of advection a continuous change in the diffusion coefficients can result in two Turing-type bifurcations as the diffusion ratio is varied, and such a bifurcation can occur even when the inhibitor species does not diffuse. It is also shown that the initial mode of bifurcation for a given domain size depends on both the advection and diffusion coefficients. These phenomena are demonstrated in the numerical solution of a particular reaction–diffusion system, and finally a possible application of the results to pattern formation in Drosophila larvae is discussed

Pattern formation of reaction-diffusion system having self-determined flow in the amoeboid organism of Physarum plasmodium

The amoeboid organism, the plasmodium of Physarum polycephalum, behaves on the basis of spatio-temporal pattern formation by local contraction-oscillators. This biological system can be regarded as a reaction-diffusion system which has spatial interaction by active flow of protoplasmic sol in the cell. Paying attention to the physiological evidence that the flow is determined by contraction pattern in the plasmodium, a reaction-diffusion system having self-determined flow arises. Such a coupling of reaction-diffusion-advection is a characteristic of the biological system, and is expected to relate with control mechanism of amoeboid behaviours. Hence, we have studied effects of the self-determined flow on pattern formation of simple reaction-diffusion systems. By weakly nonlinear analysis near a trivial solution, the envelope dynamics follows the complex Ginzburg-Landau type equation just after bifurcation occurs at finite wave number. The flow term affects the nonlinear term of the equation through the critical wave number squared. Contrary to this, wave number isn't explicitly effective with lack of flow or constant flow. Thus, spatial size of pattern is especially important for regulating pattern formation in the plasmodium. On the other hand, the flow term is negligible in the vicinity of bifurcation at infinitely small wave number, and therefore the pattern formation by simple reaction-diffusion will also hold. A physiological role of pattern formation as above is discussed.Comment: REVTeX, one column, 7 pages, no figur

Cell-scale degradation of peritumoural extracellular matrix fibre network and its role within tissue-scale cancer invasion

Local cancer invasion of tissue is a complex, multiscale process which plays an essential role in tumour progression. Occurring over many different temporal and spatial scales, the first stage of invasion is the secretion of matrix degrading enzymes (MDEs) by the cancer cells that consequently degrade the surrounding extracellular matrix (ECM). This process is vital for creating space in which the cancer cells can progress and it is driven by the activities of specific matrix metalloproteinases (MMPs). In this paper, we consider the key role of two MMPs by developing further the novel two-part multiscale model introduced in [33] to better relate at micro-scale the two micro-scale activities that were considered there, namely, the micro-dynamics concerning the continuous rearrangement of the naturally oriented ECM fibres within the bulk of the tumour and MDEs proteolytic micro-dynamics that take place in an appropriate cell-scale neighbourhood of the tumour boundary. Focussing primarily on the activities of the membrane-tethered MT1-MMP and the soluble MMP-2 with the fibrous ECM phase, in this work we investigate the MT1-MMP/MMP-2 cascade and its overall effect on tumour progression. To that end, we will propose a new multiscale modelling framework by considering the degradation of the ECM fibres not only to take place at macro-scale in the bulk of the tumour but also explicitly in the micro-scale neighbourhood of the tumour interface as a consequence of the interactions with molecular fluxes of MDEs that exercise their spatial dynamics at the invasive edge of the tumour

Biological inferences from a mathematical model for malignant invasion.

Invasive cells show changes in adhesion, motility and the protease-antiprotease balance. In this paper the authors derive a model based on a continuum approach that describes the behaviour of the invasive cells. The invasive cells are studied in the context of their interaction with normal cells, noninvasive tumour cells, ECM proteins and the proteases. The authors briefly describe the methods of mathematical analysis used and then go on to highlight the biological inferences drawn from the mathematical analysis. Based on the results from the modelling the authors suggest that the movement of cells under the simultaneous effects of a haptotactic gradient and a concomitantly created chemotactic gradient is oscillatory both with respect to the speed of invasion and the wave profile of the invasive cells. They further demonstrate that the average speed of invasion can be computed as a measure of the phenotypic properties of the cell and the matrix. They use the model to suggest an intuitive explanation for the occurrence of noninvasion with high protease expression on the basis of chemotactic gradients that prevent invasion. The authors have studied the effect of the diffusivity of the protease on an invading cell and shown that increase in diffusivity initially results in enhanced invasion, but extreme increases in protease diffusivity can result in noninvasion

A Two Parameter Family of Travelling Waves With a Singular Barrier Arising From the Modelling of Extracellular Matrix Mediated Cellular Invasion

Invasive cells variously show changes in adhesion, protease production and motility. In this paper the authors develop and analyse a model for malignant invasion, brought about by a combination of proteolysis and haptotaxis. A common feature of these two mechanisms is that they can be produced by contact with the extracellular matrix through the mediation of a class of surface receptors called integrins. An unusual feature of the model is the absence of cell diffusion. By seeking travelling wave solutions the model is reduced to a system of ordinary differential equations which can be studied using phase plane analysis. The authors demonstrate the presence of a singular barrier in the phase plane and a "hole" in this singular barrier which admits a phase trajectory. The model admits a family of travelling waves which depend on two parameters, i.e. the tissue concentration of connective tissue and the rate of decay of the initial spatial profile of the invading cells. The slowest member of this family corresponds to the phase trajectory which goes through the "hole" in the singular barrier. Using a power series method the authors derive an expression relating the minimum wavespeed to the tissue concentration of the extracellular matrix which is arbitrary. The model is applicable in a wide variety of biological settings which combine haptotaxis with proteolysis. By considering various functional forms the authors show that the key mathematical features of the particular model studied in the early parts of the paper are exhibited by a wider class of models which characterise the behaviour of invading cells. c #1999 Elsevier Science B.V. All rights reserved