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Position determination in biological systems is often achieved throughprotein concentration gradients. Measuring the local concentration of such aprotein with a spatially-varying distribution allows the measurement ofposition within the system. In order for these systems to work effectively,position determination must be robust to noise. Here, we calculate fundamentallimits to the precision of position determination by concentration gradientsdue to unavoidable biochemical noise perturbing the gradients. We focus ongradient proteins with first order reaction kinetics. Systems of this type havebeen experimentally characterised in both developmental and cell biologysettings. For a single gradient we show that, through time-averaging, greatprecision can potentially be achieved even with very low protein copy numbers.As a second example, we investigate the ability of a system with oppositelydirected gradients to find its centre. With this mechanism, positionalprecision close to the centre improves more slowly with increasing averagingtime, and so longer averaging times or higher copy numbers are required forhigh precision. For both single and double gradients, we demonstrate theexistence of optimal length scales for the gradients, where precision ismaximized, as well as analyzing how precision depends on the size of theconcentration measuring apparatus. Our results provide fundamental constraintson the positional precision supplied by concentration gradients in variouscontexts, including both in developmental biology and also within a singlecell.

Fundamental Limits to Position Determination by Concentration Gradients