On the stability of some biological schemes with cellular interactions

AbstractA model of multicellular growth in biological organisms is studied from the point of view of its ability to produce stable configurations, and the kinds of cell lineages which may occur inside such configurations are characterized. The kind of model considered is called a scheme, and the set of all stable configurations of a scheme is viewed as a formal language. In a stable configuration, individual cells may change state, die out, or divide into several daughter cells, but they must do so in such a way as to leave the total configuration unchanged.It is shown that for each scheme with cellular interactions there is a bound on the distance between a parent cell and its immediate daughter cells, but that a cell may have descendants, after many generations, which are arbitrarily far away from it. It is also shown that for each scheme there is a bound on the number of consecutive cells which may die out as a block, and be replaced by division of other cells, in one generation; a biologically reasonable property. This leads to the result that, viewed as languages, sets of stable configurations of schemes are regular sets