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  • 1
    Electronic Resource
    Electronic Resource
    Pattern formation in a generalized chemotactic model (1998)
    Springer
    Bulletin of mathematical biology 60 (1998), S. 1-26 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Many models have been proposed for spatial pattern formation in embryology and analyzed for the standard case of zero-flux boundary conditions. However, relatively little attention has been paid to the role of boundary conditions on the form of the final pattern. Here we investigate, numerically, the effect of nonstandard boundary conditions on a model pattern generator, which we choose to be of a cell-chemotactic type. We specifically focus on the role of boundary conditions and the effects of scale and aspect ratio, and study the spatiotemporal dynamics of pattern formation. We illustrate the properties of the model by application to the spatiotemporal sequence of skeletal development.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1006/bulm.1997.0010
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  • 2
    Electronic Resource
    Electronic Resource
    Enzyme kinetics at high enzyme concentration (2000)
    Springer
    Bulletin of mathematical biology 62 (2000), S. 483-499 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract We re-visit previous analyses of the classical Michaelis-Menten substrate-enzyme reaction and, with the aid of the reverse quasi-steady-state assumption, we challenge the approximation d[C]/dt ≈ 0 for the basic enzyme reaction at high enzyme concentration. For the first time, an approximate solution for the concentrations of the reactants uniformly valid in time is reported. Numerical simulations are presented to verify this solution. We show that an analytical approximation can be found for the reactants for each initial condition using the appropriate quasi-steady-state assumption. An advantage of the present formalism is that it provides a new procedure for fitting experimental data to determine reaction constants. Finally, a new necessary criterion is found that ensures the validity of the reverse quasi-steady-state assumption. This is verified numerically.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1006/bulm.1999.0163
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  • 3
    Electronic Resource
    Electronic Resource
    A chemotactic model for the advance and retreat of the primitive streak in avian development (2000)
    Springer
    Bulletin of mathematical biology 62 (2000), S. 501-525 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract The formation of the primitive streak in early avian development marks the onset of gastrulation, during which large scale cell movement leads to a trilaminar blastoderm comprising prospective endodermal, mesodermal and ectodermal tissue. During streak formation a specialized group of cells first moves anteriorly as a coherent column, beginning from the posterior end of the prospective anterior-posterior axis (a process called progression), and then reverses course and returns to the most posterior point on the axis (a process called regression). To date little is known concerning the mechanisms controlling either progression or regression. Here we develop a model in which chemotaxis directs the cell movement and which is capable of reproducing the principal features connected with progression and regression of the primitive streak. We show that this model exhibits a number of experimentally-observed features of normal and abnormal streak development, and we propose a number of experimental tests which may serve to illuminate the mechanisms. This paper represents the first attempt to model the global features of primitive streak formation, and provides an initial stage in the development of a more biologically-realistic discrete cell model that will allow for variation of properties between cells and control over movement of individual cells.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1006/bulm.1999.0166
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  • 4
    Electronic Resource
    Electronic Resource
    Bifurcating spatially heterogeneous solutions in a chemotaxis model for biological pattern generation (1991)
    Springer
    Bulletin of mathematical biology 53 (1991), S. 701-719 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract We consider a simple cell-chemotaxis model for spatial pattern formation on two-dimensional domains proposed by Oster and Murray (1989,J. exp. Zool. 251, 186–202). We determine finite-amplitude, steady-state, spatially heterogeneous solutions and study the effect of domain growth on the resulting patterns. We also investigate in-depth bifurcating solutions as the chemotactic parameter varies. This numerical study shows that this deceptively simple-chemotaxis model can produce a surprisingly rich spectrum of complex spatial patterns.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02461550
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  • 5
    Electronic Resource
    Electronic Resource
    A two-dimensional numerical study of spatial pattern formation in interacting Turing systems (1999)
    Springer
    Bulletin of mathematical biology 61 (1999), S. 483-505 
    Details
    ISSN: 1522-9602
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract For many years Turing systems have been proposed to account for spatial and spatiotemporal pattern formation in chemistry and biology. We extend the study of Turing systems to investigate the rôle of boundary conditions, domain shape, non-linearities, and coupling of such systems. We show that such modifications lead to a wide variety of patterns that bear a striking resemblance to pigmentation patterns in fish, particularly those involving stripes, spots and transitions between them. Using the Turing system as a metaphor for activator—inhibitor models we conclude that such a mechanism, with the aforementioned modifications, may play a rôle in fish patterning.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1006/bulm.1998.0093
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  • 6
    Electronic Resource
    Electronic Resource
    Enzyme kinetics for a two-step enzymic reaction with comparable initial enzyme-substrate ratios (1988)
    Springer
    Journal of mathematical biology 26 (1988), S. 689-703 
    Details
    ISSN: 1432-1416
    Keywords: Michaelis Menten approximation ; Quasi-steady state assumption ; Scaling ; Singular perturbations ; Fast and slow timescales
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract We extend the validity of the quasi-steady state assumption for a model double intermediate enzyme-substrate reaction to include the case where the ratio of initial enzyme to substrate concentration is not necessarily small. Simple analytical solutions are obtained when the reaction rates and the initial substrate concentration satisfy a certain condition. These analytical solutions compare favourably with numerical solutions of the full system of differential equations describing the reaction. Experimental methods are suggested which might permit the application of the quasi-steady state assumption to reactions where it may not have been obviously applicable before.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00276148
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  • 7
    Electronic Resource
    Electronic Resource
    Spatial and spatio-temporal patterns in a cell-haptotaxis model (1989)
    Springer
    Journal of mathematical biology 27 (1989), S. 507-522 
    Details
    ISSN: 1432-1416
    Keywords: Morphogenesis ; Spatio-temporal pattern ; Mechanochemical ; Haptotaxis ; Singular dispersion relation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract We investigate a cell-haptotaxis model for the generation of spatial and spatio-temporal patterns in one dimension. We analyse the steady state problem for specific boundary conditions and show the existence of spatially hetero-geneous steady states. A linear analysis shows that stability is lost through a Hopf bifurcation. We carry out a nonlinear multi-time scale perturbation procedure to study the evolution of the resulting spatio-temporal patterns. We also analyse the model in a parameter domain wherein it exhibits a singular dispersion relation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00288431
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  • 8
    Electronic Resource
    Electronic Resource
    Nonlinear pattern selection in a mechanical model for morphogenesis (1986)
    Springer
    Journal of mathematical biology 24 (1986), S. 525-541 
    Details
    ISSN: 1432-1416
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract We present a numerical study of the nonlinear mechanical model for morphogenesis proposed by Oster et al. (1983) with the aim of establishing the pattern forming capability of the model. We present a technique for mode selection based on linear analysis and show that, in many cases, it is a reliable predictor for nonlinear mode selection. In order to determine the set of model parameters that can generate a particular pattern we develop a technique based on nonlinear least square fitting to a dispersion relation. As an application we present a scenario for sequential pattern formation of dermal aggregations in chick embryos which leads to the hexagonal array of cell aggregations observed in feather germ formation in vivo.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00275682
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  • 9
    Electronic Resource
    Electronic Resource
    Travelling waves in a tissue interaction model for skin pattern formation (1994)
    Springer
    Journal of mathematical biology 33 (1994), S. 193-210 
    Details
    ISSN: 1432-1416
    Keywords: Travelling waves ; Tissue interaction ; Perturbation procedure ; Wavespeed
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Tissue interaction plays a major role in many morphogenetic processes, particularly those associated with skin organ primordia. We examine travelling wave solutions in a tissue interaction model for skin pattern formation which is firmly based on the known biology. From a phase space analysis we conjecture the existence of travelling waves with specific wave speeds. Subsequently, analytical approximations to the wave profiles are derived using perturbation methods. We then show numerically that such travelling wave solutions do exist and that they are in good agreement with our analytical results. Finally, the biological implications of our analysis are discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00160179
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  • 10
    Electronic Resource
    Electronic Resource
    Pattern formation in generalized Turing systems (1994)
    Springer
    Journal of mathematical biology 32 (1994), S. 345-393 
    Details
    ISSN: 1432-1416
    Keywords: Spatial pattern formations ; Bifurcation ; Turing systems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract Turing's model of pattern formation has been extensively studied analytically and numerically, and there is recent experimental evidence that it may apply in certain chemical systems. The model is based on the assumption that all reacting species obey the same type of boundary condition pointwise on the boundary. We call these scalar boundary conditions. Here we study mixed or nonscalar boundary conditions, under which different species satisfy different boundary conditions at any point on the boundary, and show that qualitatively new phenomena arise in this case. For example, we show that there may be multiple solutions at arbitrarily small lengths under mixed boundary conditions, whereas the solution is unique under homogeneous scalar boundary conditions. Moreover, even when the same solution exists under scalar and mixed boundary conditions, its stability may be different in the two cases. We also show that mixed boundary conditions can reduce the sensitivity of patterns to domain changes.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00160165
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