Mathematical Modelling of Juxtacrine Patterning
AUTHORS:
Helen J. Wearing (1), Markus R. Owen (2) & Jonathan A. Sherratt (1)
1: Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, UK.
2: Department of Mathematics, University of Utah,
Salt Lake City, Utah 84112, USA.
ABSTRACT:
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
wavenmumbers varies with the parameter regime and find an estimate for
the wavenumber assosciated with the fastest growing mode. A previous
juxtacrine model for Delta-Notch signalling studied by Collier et
al. (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 many 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 juxtacrine signalling
provides a novel and robust mechanism for the generation of spatial
patterns.
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