Models for cellular interactions in development without polarity of individual cells II. Problems of synchronization and regulation

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Abstract
The notion of polarity is often used in developmental biology for the explanation of various phenomena. Some authors have considered that polarity is absolutely essential for certain kinds of phenomena to appear. In this paper, we identify polarity of an individual cell with its ability to distinguish in somo sense between its ‘ left’ and its ‘ right ’ side. It has been shown, using a developmental model originally due to Lindenmayer, that oven if we restrict ourselves to models in which individual cells have no polarity, the models remain general enough to realize any effective procedure. In other words, they have a universal computing ability. However, it has been conjectured that, in a linear array of cells, the individual cells must have polarity in order to exhibit certain kinds of regulatory behaviour, i.e. the regaining of certain global properties of the array after a significant external disturbance (for example, the breaking of the array into two disconnected arrays). We demonstrate that one such conjecture is false, by solving the French flag problem for cells without polarity. To demonstrate that synchronization can also be achieved, we give a solution for cells without polarity to the firing squad synchronization problem. Thus, we provide further evidence for the thesis that polarity of individual cells is not a necessary mechanism for morphogenetic phenomena. Our solutions are also interesting because they do not make use of gradients or of positional information.