Dynamics of chemo-receptor activity with time-periodic attractant field

TL;DR

The paper investigates how E. coli chemoreceptor activity responds to a time-periodic attractant field by coupling cooperative receptor switching (MWC-like clusters) with a methylation/demethylation adaptation mechanism under a sinusoidal stimulus $[L](t)=[L]_0+[L]_1\sin(\omega t)$. Using Monte Carlo simulations and analytical results, it shows that the activity amplitude grows with frequency, plateaus, and then declines at large $\omega$, while the phase lag increases toward $\,3\pi/2$; plotting activity versus attractant over a period yields a hysteresis-like loop whose area exhibits two peaks and a mid-frequency minimum. The work identifies three key time-scales—activity switching, methylation, and input variation—that govern the response and derives large-$\omega$ analytical expressions that agree with simulations, revealing a breakdown of quasi-equilibrium at very high frequencies. These findings provide a quantitative framework for understanding how oscillatory chemical environments shape chemotactic signaling, with experimentally testable predictions (e.g., via FRET) and implications for the robustness of signaling under fluctuating stimuli, including extensions to non-sinusoidal or traveling-wave attractant fields.

Abstract

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

Figures (6)

• Figure 1: Schematic representation of (de)methylation reactions and activity switching of chemoreceptor cluster.
• Figure 2: Fraction of active receptors over three consecutive oscillation cycles measured in the steady state. Panel (a) shows the applied periodic stimulus with amplitude $[L]_1 = 10 \mu M$. Panels (b)--(d) display the corresponding activity response for frequencies $\omega = 0.5\,\mathrm{s}^{-1}$, $\omega = 0.02\,\mathrm{s}^{-1}$, and $\omega = 0.001\,\mathrm{s}^{-1}$, respectively. Both the amplitude and the phase lag depends on the frequency $\omega$.
• Figure 3: (a) Activity amplitude (green line) as a function of applied frequency $\omega$. The amplitude increases with $\omega$, reaches a plateau and then for large $\omega$ decreases again. In the same plot we show amplitude of $\langle [1 + \exp(F(t))]^{-1} \rangle$ (black line) which matches with activity for small and intermediate $\omega$ but deviates for large $\omega$. The maximum error-bar in these data is less than $2.5 \times 10^{-4}$. (b) Green curve shows the phase lag $\Delta \Phi$ between activity and applied stimulus as a function of $\omega$. $\Delta \Phi$ increases monotonically for small and intermediate $\omega$ and saturates at $3 \pi /2$ for large $\omega$. The black curve shows phase lag between $\langle [1 + \exp(F(t))]^{-1} \rangle$ and applied signal, which follows $\Delta \Phi$ for small and intermediate $\omega$ but saturates at $\pi$ when $\omega$ is large. The maximum error-bar in these data is less than $6 \times 10^{-3}$. All simulation parameters are as in Table \ref{['table:parameters']}.
• Figure 4: Average activity as a function of attractant concentration over one complete time period, for different $\omega$ values. Activity traces out a loop. The area, shape and orientation of the loop depends sensitively on frequency. The statistical errors in these data points are less than the symbol size. All simulation parameters are as listed in Table \ref{['table:parameters']}.
• Figure 5: (a) Area of the loop shows one peak at small $\omega$, another peak at large $\omega$ and decreases sharply to reach minimum for intermediate $\omega$. Main plot shows the variation in log-log scale and the inset shows it in linear scale for $y$-axis and log scale for $x$-axis. The loop area vanishes for a specific $\omega$ at which the phase lag $\Delta \Phi$ becomes equal to $\pi$. (b) The orientation $\theta$ of the loop shows a minimum with $\omega$. The error-bar is within the symbol size. All simulation parameters are listed in Table \ref{['table:parameters']}.