Cost-precision trade-off relation determines the optimal morphogen gradient for accurate biological pattern formation

Spatial boundaries growing into macroscopic structures through animal development originate from the pre-patterning of tissues by signaling molecules, called morphogens. To establish accurate boundaries, the morphogen concentration which thresholds the expression of target gene at the boundary should be precise enough, exhibiting large gradient and small fluctuations. Producing more morphogens would better serve to shape more precise target boundaries; however, it incurs more thermodynamic cost. In the classical diffusion-degradation model of morphogen profile formation, the morphogens synthesized from a local source display an exponentially decaying concentration profile with a characteristic length {lambda}. Our theory suggests that in order to attain a precise morphogen profile with the minimal cost, {lambda} should be roughly half the distance to the target boundary position from the source, so that the boundary is formed at the position where the morphogen concentration is [~]10 % of the value at the source. Remarkably, we find that the well characterized morphogens that pattern the fruit fly embryo and wing imaginal disk form profiles with nearly optimal {lambda}, which underscores the thermodynamic cost as a key physical constraint in the morphogen profile formation.