Fundamental limits to position determination by concentration gradients

Abstract

Position determination in biological systems is often achieved through protein concentration gradients. Measuring the local concentration of such a protein with a spatially varying distribution allows the measurement of position within the system. For these systems to work effectively, position determination must be robust to noise. Here, we calculate fundamental limits to the precision of position determination by concentration gradients due to unavoidable biochemical noise perturbing the gradients. We focus on gradient proteins with first-order reaction kinetics. Systems of this type have been experimentally characterised in both developmental and cell biology settings. For a single gradient we show that, through time-averaging, great precision potentially can be achieved even with very low protein copy numbers. As a second example, we investigate the ability of a system with oppositely directed gradients to find its centre. With this mechanism, positional precision close to the centre improves more slowly with increasing averaging time, and so longer averaging times or higher copy numbers are required for high precision. For both single and double gradients, we demonstrate the existence of optimal length scales for the gradients for which precision is maximized, as well as analyze how precision depends on the size of the concentration-measuring apparatus. These results provide fundamental constraints on the positional precision supplied by concentration gradients in various contexts, including both in developmental biology and also within a single cell.

Figure 1. Single-Gradient Model in d = 2

(A) Variation of the estimated threshold position with averaging time, with x_T = 2 μm and λ = 2 μm. (B) Variation of the width as a function of averaging time. (C) Data collapse of the width at large τ for a range of parameter values. Full line shows the prediction of Equation 7 with k _2d = 0.40 and α = 2.5. (D) w(τ) as a function of decay length, with x_T = 2 μm. Results for three different averaging times are shown: ×, τ = 10 s; circle, τ = 15 s; and +, τ = 22.5 s. The full line shows the prediction from Equation 7. At large λ, the simulation results deviate from the prediction since the assumption that L ≫ λ is no longer valid. (E) Plot of the probability distribution for measuring the threshold at position x with an averaging time τ = 45 s. The full line shows a normal distribution. (F) Scaling of the crossover time, τ_×, according to Equation 13. In (A), (B), and (E), the standard parameter values given in the text were used. In (C) and (F), * indicates the standard parameter values. For the other datasets, one parameter value was changed as follows: open circle, D = 0.5 μm²s⁻¹; open square, J = 6.25 μm⁻¹s⁻¹; ×, Δx = 0.02 μm; closed circle, μ = 1 s⁻¹; +, μ = 0.11 s⁻¹; open diamond, x_T = 1 μm; and inverted triangle, x_T = 3 μm.

Figure 2. Two-Gradient Model in d = 2

(A) The mean threshold position fluctuates about L/2 due to the symmetry of the system. (B) Variation of the width w as a function of averaging time. (C) Data collapse of the width as a function of averaging time, at long times, for a range of parameter values. The full line shows Equation 19 with k~ _2d = 0.63 and Α~ = 2.5. * indicates the standard parameter values. For the other datasets, parameter values were changed as follows: open circle, D = 0.5 μm²s⁻¹; open square, J = 9 μm⁻¹s⁻¹; ×, Δx = 0.02 μm; closed circle, μ = 1 s⁻¹; +, μ = 0.25 s⁻¹; diamond, L = 7.5 μm; and inverted triangle, L = 15 μm and Δx = 0.02 μm. (D) Plot of width as a function of decay length for averaging times: ×, τ = 30 s; open circle, τ = 45 s; and +, τ = 60 s. The full line shows the prediction from Equation 19.