Cell-differentiation rules that generate regular mosaic patterns: Modelling motivated by cone mosaic formation in fish retina

We study characteristics of cell-differentiation rules that realize stable formation of regularly arranged checker-board patterns, exemplified by cone 'mosaic' zebrafish retina, or the regular arrangement of cone photoreceptor cells. We consider the situation in which cells are arranged on a square lattice and are initially undifferentiated. Later each cell becomes one of the two differentiated states, affected by the state of the neighboring cells. The cells that undergo differentiation form a 'morphogenetic cell row' which sweeps from one end to the other end of the lattice through time. This models an outward sweep of the margin of expanding mosaic region of the retina which occurs as undifferentiated photoreceptor cells become differentiated in concentric circles, joining the mosaic. We introduce an index to measure the ability of cell-differentiation rules to generate regular checker-board patterns from irregular initial patterns, and attempt to characterize the successful rules. We first show the importance of six 'preservation conditions' which guarantee perfectly regular photoreceptor arrangement for all the rows after a regular row. Then we select an additional six 'optimizing conditions' for responses to configurations that are consistently shown by the rules of high average scores. We also examine the effect of interaction between responses to different configurations. Finally we examine the concept of morphogenetic row precedence, i.e. that the successful rules generating a high score tend to treat the consistency with neighbors in the newly differentiated cells (those in the morphogenetic cell row) as more important than the consistency with previously differentiated neighbors.