Embryonic 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 of the one-dimensional developing field; this cannot be accomplished by a single 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, the morphogens effectively interact according to the annihilation reaction $A+B\to\emptyset$ and the switch occurs according to the absolute concentration of $A$ or $B$. In this case embryonic markers across the entire developing field scale approximately with system size; this cannot be achieved with a pair of non-interacting gradients that combinatorially regulate downstream genes. This scaling occurs in a window of developing-field sizes centred at a few times the morphogen decay length.