The Chemotactic Index for Spatial Gradient Sensing
We consider the problem of quantifying the chemotactic efficiency of single cells as measured by the chemotactic index $Ψ$. Previous work in a model framework for direct sensing of spatial gradients indicated that $Ψ$ depends on a single dimensionless group $s$, which plays the role of the square of the signal to noise ratio in the problem. We revisit this problem theoretically and demonstrate that the cumulants in the model can be calculated exactly. We derive explicit results for the multivariate cumulants up to third order in terms of the diffusive current density and Gaunt coefficients. We discuss the machinery required to translate Burg-Purcell style limits on concentration gradient uncertainty into results for the chemotactic index. We compute the leading corrections to $Ψ$ in an Edgeworth expansion, and identify a dimensionless group $λ$ in the problem which is a ratio of concentrations that captures the effects of the non-Gaussianity. By careful consideration of experimental results on slime mold chemotaxis, we demonstrate that the explanatory success of the original Gaussian approximation for the chemotactic index stems in part from the fact the concentration gradients were shallow, $|λ| \ll s$.