Explain the inversion of orientation dependence of MLE variance at weak fields

Determine the origin of the observed inversion in orientation dependence of the maximum-likelihood estimator’s circular variance for electric-field direction sensing in the elliptical-cell electrophoretic sensor-redistribution model at small κ, in which the estimator’s variance is maximal when the applied field is aligned with the cell’s major axis (ψ = 0) despite the Fisher-information-based bound predicting minimal variance in that orientation. Establish conditions under which the estimator’s variance can exceed and trend opposite to the Fisher-information-based lower bound in this model.

Background

The paper develops a stochastic sensing model in which transmembrane sensors redistribute on an elliptical cell via electromigration and diffusion. Using Fisher information, the authors find that—in principle—information about field direction is maximal when the field is parallel to the cell’s major axis.

However, simulations of the maximum-likelihood estimator reveal that, at weak fields, the circular variance of the estimator is actually maximal when the field is aligned with the major axis (ψ = 0), implying better accuracy when the long axis is perpendicular to the field—opposite to the Fisher-information-based prediction. The authors explicitly note that they lack a clear explanation for this discrepancy, which persists even when considering a more physically plausible vector-sum estimator.