The polarisation of cells and tissues is fundamental for tissue morphogenesis
during biological development and regeneration. A deeper understanding of
biological polarity pattern formation can be gained from the consideration of
pattern reorganisation in response to an opposing instructive cue, which we
here consider by example of experimentally inducible body axis inversions in
planarian flatworms. Our dynamically diluted alignment model represents three
processes: entrainment of cell polarity by a global signal, local cell-cell
coupling aligning polarity among neighbours and cell turnover inserting
initially unpolarised cells. We show that a persistent global orienting signal
determines the final mean polarity orientation in this stochastic model.
Combining numerical and analytical approaches, we find that neighbour coupling
retards polarity pattern reorganisation, whereas cell turnover accelerates it.
We derive a formula for an effective neighbour coupling strength integrating
both effects and find that the time of polarity reorganisation depends linearly
on this effective parameter and no abrupt transitions are observed. This allows
to determine neighbour coupling strengths from experimental observations. Our
model is related to a dynamic 8-Potts model with annealed site-dilution and
makes testable predictions regarding the polarisation of dynamic systems, such
as the planarian epithelium.Comment: Preprint as prior to first submission to Journal of the Royal Society
Interface. 25 pages, 6 figures, plus supplement (18 pages, contains 1 table
and 7 figures). A supplementary movie is available from
https://dx.doi.org/10.6084/m9.figshare.c388781
