Diffusion driven instability in an inhomogeneous domain

Benson, Debbie L.; Sherratt, Jonathan A.; Maini, Philip K.

doi:10.1007/bf02460888

Cited by 83 publications

(59 citation statements)

References 17 publications

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“…Thus, for aggregation to form, we require this state to be unstable to spatially inhomogeneous perturbations (Benson et al, 1993), i.e. the maximum real part of the eigenvalues to be greater than zero.…”

Section: Of 36mentioning

confidence: 99%

Lyapunov–Schmidt and Centre Manifold Reduction Methods for Nonlocal PDEs Modelling Animal Aggregations

Buono

Eftimie

2016

Springer Proceedings in Mathematics &Amp; Statistics

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[Date] Cells adhere to each other and to the extracellular matrix (ECM) through protein molecules on the surface of the cells. The breaking and forming of adhesive bonds, a process critical in cancer invasion and metastasis, can be influenced by the mutation of cancer cells. In this paper, we develop a nonlocal mathematical model describing cancer cell invasion and movement as a result of integrin-controlled cell-cell adhesion and cell-matrix adhesion, for two cancer cell populations with different levels of mutation. The partial differential equations for cell dynamics are coupled with ordinary differential equations describing the extracellular matrix (ECM) degradation and the production and decay of integrins. We use this model to investigate the role of cancer mutation on the possibility of cancer clonal competition with alternating dominance, or even competitive exclusion (phenomena observed experimentally). We discuss different possible cell aggregation patterns, as well as travelling wave patterns. In regard to the travelling waves, we investigate the effect of cancer mutation rate on the speed of cancer invasion.

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Section: Of 36mentioning

confidence: 99%

Lyapunov–Schmidt and Centre Manifold Reduction Methods for Nonlocal PDEs Modelling Animal Aggregations

Buono

Eftimie

2016

Springer Proceedings in Mathematics &Amp; Statistics

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“…Such a system has been studied in one dimension by one of us [11], and it has been shown that the dispersion relation does not admit an analytic solution, even in the simple case of a step-like variation of the diffusion constant and zero-flux external conditions. In general, the Laplacian should be replaced by ∇ · (D(x, y)∇u).…”

Section: The Modelmentioning

confidence: 99%

“…Former studies in one dimension of a Schakenberg system with a step-like variation of the diffusion constants [11] show that there are three types of patterns, depending on the value of the ratio D o /D i . Roughly speaking, when this ratio is large, isolated patterns (type A) are obtained in each region, and there is a discontinuity at the border.…”

Section: Numerical Calculationsmentioning

confidence: 99%

Size-dependent symmetry breaking in models for morphogenesis

Barrio

Maini

Aragón

et al. 2002

Physica D: Nonlinear Phenomena

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A general property of dynamical systems is the appearance of spatial and temporal patterns due to a change of stability of a homogeneous steady state. Such spontaneous symmetry breaking is observed very frequently in all kinds of real systems, including the development of shape in living organisms. Many nonlinear dynamical systems present a wide variety of patterns with different shapes and symmetries. This fact restricts the applicability of these models to morphogenesis, since one often finds a surprisingly small variation in the shapes of living organisms. For instance, all individuals in the Phylum Echinodermata share a persistent radial fivefold symmetry. In this paper, we investigate in detail the symmetry-breaking properties of a Turing reaction-diffusion system confined in a small disk in two dimensions. It is shown that the symmetry of the resulting pattern depends only on the size of the disk, regardless of the boundary conditions and of the differences in the parameters that differentiate the interior of the domain from the outer space. This study suggests that additional regulatory mechanisms to control the size of the system are of crucial importance in morphogenesis.

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“…However, as experimental evidence suggests, some embryological systems may exhibit environmental inhomogeneities. The form of Turing patterns in such cases was analysed by Benson et al (1993). They considered a two-species reaction-diffusion system where the dispersal rate of one species varied in a simple step-wise manner, and discussed its application to the development of cartilage pattern in embryonic chick limb.…”

Section: Continuum Models For Pattern Formationmentioning

confidence: 99%

Mathematical Modelling of Juxtacrine Patterning

Wearing

2000

Bulletin of Mathematical Biology

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Spatial pattern formation is one of the key issues in developmental biology. Some patterns arising in early development have a very small spatial scale and a natural explanation is that they arise by direct cell-cell signalling in epithelia. This necessitates the use of a spatially discrete model, in contrast to the continuum-based approach of the widely studied Turing and mechanochemical models. In this work, we consider the pattern-forming potential of a model for juxtacrine communication, in which signalling molecules anchored in the cell membrane bind to and activate receptors on the surface of immediately neighbouring cells. The key assumption is that ligand and receptor production are both up-regulated by binding. By linear analysis, we show that conditions for pattern formation are dependent on the feedback functions of the model. We investigate the form of the pattern: specifically, we look at how the range of unstable wavenumbers varies with the parameter regime and find an estimate for the wavenumber associated with the fastest growing mode. A previous juxtacrine model for Delta-Notch signalling studied by Collier et al. (1996, J. Theor. Biol. 183, 429-446) only gives rise to patterning with a length scale of one or two cells, consistent with the fine-grained patterns seen in a number of developmental processes. However, there is evidence of longer range patterns in early development of the fruit fly Drosophila. The analysis we carry out predicts that patterns longer than one or two cell lengths are possible with our positive feedback mechanism, and numerical simulations confirm this. Our work shows that

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Diffusion driven instability in an inhomogeneous domain

Cited by 83 publications

References 17 publications

Lyapunov–Schmidt and Centre Manifold Reduction Methods for Nonlocal PDEs Modelling Animal Aggregations

Lyapunov–Schmidt and Centre Manifold Reduction Methods for Nonlocal PDEs Modelling Animal Aggregations

Size-dependent symmetry breaking in models for morphogenesis

Mathematical Modelling of Juxtacrine Patterning

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