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A crucial step in the early development of multicellular organisms involvesthe establishment of spatial patterns of gene expression which later directproliferating cells to take on different cell fates. These patterns enable thecells to infer their global position within a tissue or an organism by readingout local gene expression levels. The patterning system is thus said to encodepositional information, a concept that was formalized recently in the frameworkof information theory. Here we introduce a toy model of patterning in onespatial dimension, which can be seen as an extension of Wolpert's paradigmatic"French Flag" model, to patterning by several interacting, spatially coupledgenes subject to intrinsic and extrinsic noise. Our model, a variant of anIsing spin system, allows us to systematically explore expression patterns thatoptimally encode positional information. We find that optimal patterningsystems use positional cues, as in the French Flag model, together withgene-gene interactions to generate combinatorial codes for position which wecall "Counter" patterns. Counter patterns can also be stabilized against noiseand variations in system size or morphogen dosage by longer-range spatialinteractions of the type invoked in the Turing model. The simple setup proposedhere qualitatively captures many of the experimentally observed properties ofbiological patterning systems and allows them to be studied in a single,theoretically consistent framework.

Beyond the French Flag Model: Exploiting Spatial and Gene Regulatory Interactions for Positional Information