Mechanism behind the peak of initial activity a_R^low at wave speeds exceeding the run speed in the large-wavelength regime

Determine the mechanism that causes the initial receptor activity at the start of rightward runs originating from low-attractant regions (a_R^low) to exhibit a peak as a function of the attractant wave speed v_w in the large-wavelength traveling-wave environment [L](x,t) = [L]_0 + A sin((2π/λ)(v_w t − x)), and explain why this peak occurs at v_w greater than the cell run speed v.

Background

The paper studies single-cell E. coli chemotaxis in a traveling-wave attractant environment modeled as L = [L]_0 + A sin((2π/λ)(v_w t − x)), focusing on how the steady-state chemotactic drift depends on the wave speed v_w. The authors differentiate responses between moderate and large wavelengths λ.

In the large-wavelength regime, the spatial attractant gradient is weak over a run, so run durations are largely determined by the cell’s initial receptor activity at run onset. Measuring the initial activity for rightward runs that begin in low-attractant regions (a_R^low), the authors observe that a_R^low exhibits a peak as a function of v_w, and that the peak is located at v_w > v (where v is the cell’s run speed). Although this observation is empirically documented, the paper does not provide a mechanistic explanation for the emergence of the peak or for its location relative to v, explicitly identifying this as an unresolved point.