  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.
