Analytical treatment of gradient detection when gradients are confined within a single sensory interval

Derive an analytical expression for the signal-to-noise ratio and the resulting sensory horizon for bacterial chemotactic detection in the steady 1/r concentration field around a spherical phytoplankton cell in the regime where the sharp increase in gradient amplitude is confined within a single sensory interval T (i.e., when the bacterium’s displacement UT exceeds its distance to the cell surface). The analysis should explicitly account for dynamic noise arising from bacterial motion and finite integration time, avoiding reliance on a phenomenological low-pass filter.

Background

The paper models gradient sensing noise by extending linear-gradient results to non-linear 1/r fields and introduces a dynamic correction for motion. However, for very small phytoplankton, the gradient increase may occur within a single sensory window, creating a tightly confined, high-frequency signal that proved analytically intractable.

To proceed, the authors imposed a phenomenological low-pass filter in the signal-to-noise ratio to model loss of high-frequency information. An analytical, first-principles derivation for this regime would remove this modeling assumption and clarify limits of chemotactic detection for the smallest cells.