Fig. 1: Estimation of positional error of morphogen gradients in the vertebrate neural tube.

A Schematic of opposing noisy SHH and BMP gradients with resulting gene expression domains. The patterning domain is confined between the floor plate (FP) and the roof plate (RP). B Gradients from different embryos (top, different colors) differ in amplitude and decay length. As a result, the threshold concentration C_θ is reached at different readout positions x_θ,i. The overall readout position, μ_x, is given by the mean of the readout positions, x_θ,i. The positional error, σ_x, is the standard deviation of the readout positions x_θ,i (Eq. (2)). At the threshold concentration C_θ, the readout (middle) of the gradient typically assumes its half-maximal value. If the precision of both gradient and readout is measured by the same error metric, there is a 1:1 correspondence between them. Noise in individual gradients can further widen the transition zone between two adjacent domains. With the FitEPM (bottom, Eq. (14)), individual exponential gradients are averaged (red solid line) and then lin-fitted with another exponential (red dotted line), to then apply linear error propagation to estimate the positional error. C In the linear error propagation method, the local tangent (green) to an averaged morphogen concentration gradient (blue) is used to translate the vertical gradient variability, σ_C, into a horizontal positional error, σ_x (black). D Comparison of the previously reported positional error of GBS-GFP (green squares)⁹ with the results obtained with 10⁴ randomly drawn gradients that match the reported gradient summary statistics. The positional error is calculated either directly, Eq. (2) (DEEM, black solid line), or with the linear error propagation method, Eq. (3), using the derivative of either the mean gradient (Eq. (13), NumEPM, blue dashed line) or of an exponential function fitted to the mean gradient (Eq. (14), FitEPM, green solid line). See Methods for details. E When applied to sets of exponential gradients with mean and standard deviation of C₀ and λ identical to those reported for GBS-GFP (green) and pSMAD (red), the FitEPM overestimates the positional error (Eq. (2), black) by orders of magnitude further away from the source. Source data are provided as a Source Data file.