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

Abstract

Spatial boundaries formed during animal development originate from the pre-patterning of tissues by signaling molecules, called morphogens. The accuracy of boundary location is limited by the 2uctuations of morphogen concentration that thresholds the expressio level of target gene. Producing more morphogen molecules, which gives rise to smaller relative 2uctuations, would better serve to shape more precise target boundaries; however, it incurs more thermodynamic cost. In the classical diffusion-depletion model of morphogen pro1le formation, the morphogen molecules synthesized from a local source display an exponentially decaying concentration pro1le with a characteristic length γ. Our theory suggests that in order to attain a precise pro1le with the minimal cost, γ should be roughly half the distance to the target boundary position from the source. Remarkably, we 1nd that the pro1les of morphogens that pattern the Drosophila embryo and wing imaginal disk are formed with nearly optimal γ. Our 1nding underscores the thermodynamic cost as a key physical constraint in the morphogen pro1le formation in Drosophila development.