Pattern Scaling Achieved by Oppositely Directed Morphogen Gradients

Morphogens are proteins, often produced in a localised region, whose concentrations spatially demarcate regions of differing gene expression in developing embryos. The boundaries of expression must be set accurately and in proportion to the size $L$ of the developing field; this cannot be accomplished by a single one-dimensional gradient. Here, we show how a pair of morphogens produced at opposite ends of a developing field can solve the pattern-scaling problem. In the most promising scenario, two morphogens interact according to $A+B\rightarrow\emptyset$ and the switch occurs according to the absolute concentration of the first gradient. In this case scaling occurs in a window of developing-field sizes centred at a few times the morphogen decay length.