Physical limits on size precision in growing organs and cells

Developing organs maintain robust size control in a noisy environment. Measurements of fluctuating asymmetry (FA) in adult Drosophila wings and adult human arms indicate a developmental precision of about 1% in final organ size. This observation invites the question of what sets the 1% level. To address this question theoretically, we investigate fundamental physical limits on setting the size of growing organs. An important feature of these systems is that the organ must decide when to stop growing based on a necessarily noisy estimate of its own size. We use a first-passage formalism to investigate the termination of growth and find that a Kalman filter that minimizes the estimate's dynamical tracking error also comes close to minimizing the organ's final size variance in most parameter regimes. For a simple model in which the concentration of a diluted chemical species is used to measure organ size, the calculated minimum variance works out to be an order of magnitude smaller than the experimental constraint. This suggests either that other downstream noise sources are important or that other considerations impose a noisier mechanism of size measurement. We can also apply our formalism to investigate variance in single cell sizes at division.