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, nam ed "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 addit ion, there are two kinds of single cones, sensitive to blue and to UV, resp ectively. We study characteristics of cell-differentiation rules that reali ze stable formation of cone mosaic. Assumptions are: undifferentiated cells are arranged in a regular square lattice, and they are one of the three ty pes (B, U, and D cells). A D cell has two parts (G and R-parts) and takes o ne of the four directions. The cells change their cell type and orientation following a continuous-time Markovian chain. The state transtion occurs fa ster if it increases the stabilities of the focal cell, in which the stabil ity is the sum of affinities with neighboring cells. After the transient pe riod, the system may reach a stable pattern (pre-pattern). The pattern beco mes 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 sta tes that can generate the same cone mosaic patterns as in zerbrafish retina . Successful transition rules give (1) zero or small affinity with the pair s 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 affi nity 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 latt er 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 i s shortened considerably if the pre-pattern formation occurs only in a narr ow band of morphogenetic eel layer that sweeps from one end of the lattice to the other. (C) 1999 Academic Press.