Dynamics of chemoreceptor activity with time-periodic attractant field

When exposed to a time-periodic chemical signal, an E.coli cell responds by modulating its receptor activity in a similar time-periodic manner. But there is a phase lag between the applied signal and activity response. We study the variation of activity amplitude and phase lag as a function of applied frequency $\omega$ , using numerical simulations. The amplitude increases with $\omega$ , reaches a plateau and then decreases again for large $\omega$ . The phase lag increases monotonically with $\omega$ and finally saturates to $3\pi /2$ when $\omega$ is large. The activity is no more a single-valued function of the attractant signal, and plotting activity vs attractant concentration over one complete time period generates a loop. We monitor the loop area as a function of $\omega$ and find two peaks for small and large $\omega$ and a sharp minimum at intermediate $\omega$ values. We explain these results from an interplay between the timescales associated with adaptation, activity switching and applied signal variation. In particular, for very large $\omega$ the quasi-equilibrium approximation for activity dynamics breaks down, which has not been explored in earlier studies. We perform analytical calculation in this limit and find good agreement with our simulation results.