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Workshop Programme

for period 31 October - 1 November 2005

Pattern Formation in Growing Domains

31 October - 1 November 2005

Timetable

Monday 31 October
11:00-11:30 Coffee
11:30-12:30 Painter, K (Heriot-Watt)
  Modelling the development of pigmentation patterns on angelfish and zebrafish Sem 1
 

The process by which the early embryo is transformed from an undifferentiated, and effectively homogeneous, population of cells into a fully formed and complex organism has long fascinated both experimentalists and theoreticians alike.

Skin pigmentation markings are a particularly elegant and striking example of embryonic pattern formation. In certain fish, these patterns can be observed from early early embryonic through to full adult stages, thus providing a model for studying the coupled effect of patterning and embryonic growth. In this talk I will discuss the application of mathematical models to explain the development and growth of pigmentation patterns in anglefish and zebrafish.

 
12:30-13:30 Lunch
14:00-15:00 Gaffney, E (Birmingham)
  Biological pattern formation: the effects of time delays Sem 1
 

The incorporation of time delays can greatly affect the behaviour of partial differential equations and dynamical systems. In addition, there is evidence that time delays in gene expression due to transcription and translation play an important role in the dynamics of cellular systems. In this talk we investigate the effects of incorporating gene expression time delays into a one dimensional putative reaction diffusion pattern formation mechanism on both stationary domains and domains with spatially uniform exponential growth. Oscillatory behaviour is rare though we find that the time taken to initiate and stabilise patterns increases dramatically as the time delay is increased. In addition, we observe that on rapidly growing domains the time delay can induce a failure of the Turing instability which cannot be predicted by a naive linear analysis of the underlying equations about the homogeneous steady state. The dramatic lag in the induction of patterning, or even its complete absence on occasions, highlights the importance of considering explicit gene expression time delays in models for cellular reaction diffusion patterning.

 
15:30-16:00 Tea
16:00-17:00 Rademacher, J (Berlin)
  Pulse motion on a uniformly growing interval in the semi-strong limit of the Schnakenberg model Sem 1
17:30-18:30 Wine and Beer reception
Tuesday 1 November
09:30-10:30 Maini, P (Oxford)
  Modelling biological pattern formation: the effect of domain growth Sem 1
 

The Turing reaction-diffusion model serves as a paradigm model for self-organised pattern formation in biology. We will present the background to this modelling approach and illustrate some of its successes and failures. Then we will consider its behaviour under domain growth, with particular emphasis on robustness, and the effects of noise. Application to areas such as pigmentation patterning in butterflies, ligament development in bivalues, and limb development in mouse will be discussed.

 
11:00-11:30 Coffee
11:30-12:30 Comanici, A (Houston)
  Patterns on growing square domains via mode interactions Sem 1
12:30-13:30 Lunch
14:00-15:00 Matthews, P (Nottingham)
  Interactions between pattern formation and domain growth Sem 1
 

We develop a general framework for investigating pattern formation in reaction-diffusion systems for which the tissue on which the spatial pattern resides is growing at a rate which is regulated by the diffusible chemicals that establish the pattern. There is a complex interplay between the effects of the chemicals on the domain size and the influence of the domain size on the formation of patterns. The nature of this interaction is revealed by a weakly nonlinear analysis, which yields a pair of nonlinear equations for the amplitude of the spatial pattern and the domain size. The domain is found to grow (or shrink) at a rate that depends quadratically on the pattern amplitude.

 
15:30-16:00 Tea

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