How Does Cellular Contact Affect Differentiation Mediated Pattern Formation?

Abstract

In this paper, we present a two-population continuous integro-differential model of cell differentiation, using a non-local term to describe the influence of the local environment on differentiation. We investigate three different versions of the model, with differentiation being cell autonomous, regulated via a community effect, or weakly dependent on the local cellular environment. We consider the spatial patterns that such different modes of differentiation produce, and investigate the formation of both stripes and spots by the model. We show that pattern formation only occurs when differentiation is regulated by a strong community effect. In this case, permanent spatial patterns only occur under a precise relationship between the parameters characterising cell dynamics, although transient patterns can persist for biologically relevant timescales when this condition is relaxed. In all cases, the long-lived patterns consist only of stripes, not spots.

Iridophores as a source of robustness in zebrafish stripes and variability in Danio patterns

“…Iridophores were considered largely unnecessary, and a series of interactions in the form of short-range activation and long-range inhibition 17 , 18 was deduced 19 . Past mathematical models, whether discrete 20 – 24 or continuum 4 , 19 , 20 , 25 – 27 , have also explored the interactions of melanophores and dense xanthophores.

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A quantitative modelling approach to zebrafish pigment pattern formation

“…We do not directly model L-iridophores, since these appear after the adult pattern is formed and are more likely involved in pattern maintenance ( Frohnhöfer et al, 2013 ). Unlike previous models of stripe formation ( Nakamasu et al, 2009 ; Bullara and De Decker, 2015 ; Volkening and Sandstede, 2015 ; Painter et al, 2015 ; Bloomfield et al, 2011 ; Volkening and Sandstede, 2018 ), we include xanthoblasts as an independent cell-type in our model. This is because the larval xanthoblasts appear principally by dedifferentiation of the embryonic xanthophores, and most metamorphic xanthophores arise from the larval xanthoblasts ( Mahalwar et al, 2014 ; McMenamin et al, 2014 ; Budi et al, 2011 ; Singh et al, 2014 ; Dooley et al, 2013 ), whilst xanthoblasts that do not re-differentiate into xanthophores persist in the stripe regions where they play a role in consolidating melanocytes into stripes.…”

“…However, a potential limitation is that parameters do not always have a clear biological interpretation which, can sometimes make it difficult to link parameters to measurable data. In the context of zebrafish stripe formation, these models have not yet incorporated S-iridophores ( Watanabe and Kondo, 2015 ; Kondo, 2017 ; Painter et al, 2015 ; Bloomfield et al, 2011 ; Binder and Simpson, 2013 ; Volkening and Sandstede, 2015 ; Kondo, 2017 ; Nakamasu et al, 2009 ; Moreira and Deutsch, 2005 ; Bullara and De Decker, 2015 ; Yamaguchi et al, 2007 ; Asai et al, 1999 ). They suggest that the role for iridophores is restricted to simply orienting stripes ( Volkening and Sandstede, 2015 ; Nakamasu et al, 2009 ; Binder and Simpson, 2013 ).…”

Aggregation patterns from nonlocal interactions: Discrete stochastic and continuum modeling

“…It follows that the scaled density ρ(x,t) can be interpreted as a probability density. The conservation equation (1) serves as a model for various phenomena in biology and physics such as swarming and flocking, aggregation and collective behavior of cell cultures, chemotactic or nanotubular collective behavior of cells, mesh arrangement of filaments and crystallization [1][2][3][4][5][6][7][8][9][10][11][12][13][14].…”

“…(1) to model cell-adhesion effects describing cell aggregation patterns (with generalizations to a nonlinear convolution operator). Such equations have also been used to model somitogenesis [12], cellular pattern formation [13], and cell renewal in mosaic tissues [14].…”