Formation of cone mosaic of zebrafish retina

In the zebrafish retina, four types of cone photoreceptor cells (or cones) with different sensitive frequencies are arranged in a regular pattern, named "cone mosaic". A pair of small cones, one sensitive to red and the other sensitive to green, is in close contact and forms a "double cone". In addition, there are two kinds of single cones, sensitive to blue and to UV, respectively. We study characteristics of cell-differentiation rules that realize stable formation of cone mosaic. Assumptions are: undifferentiated cells are arranged in a regular square lattice, and they are one of the three types (B, U, and D cells). A D cell has two parts (G and R-parts) and takes one of the four directions. The cells change their cell type and orientation following a continuous-time Markovian chain. The state transtion occurs faster if it increases the stabilities of the focal cell, in which the stability is the sum of affinities with neighboring cells. After the transient period, the system may reach a stable pattern (pre-pattern). The pattern becomes fixed later when the cells are fully differentiated in which B cells, U cells, and D cells become blue-sensitive, UV-sensitive, and double cones, respectively. We search for the combinations of affinities between cell states that can generate the same cone mosaic patterns as in zerbrafish retina. Successful transition rules give (1) zero or small affinity with the pairs of cell states that are absent in the zebrafish cone mosaic (lambda(UR), lambda(BG)and the contact of two cells of the same type); (2) a large affinity between a part of D cells and a non-D cell (lambda(UG)and lambda(BR)); and (3) a positive affinity of an intermediate magnitude between two non-D cells (lambda(BU)) and between two parts of D cells (lambda(GR)). The latter should be of a magnitude of about 60-90% of the former. The time needed to form a regular pattern increases with the lattice size if all the cells start pre-pattern formation simultaneously. However, the convergence time is shortened considerably if the pre-pattern formation occurs only in a narrow band of morphogenetic cell layer that sweeps from one end of the lattice to the other.