Chemotaxing E. coli do not count single molecules

Problem Statement
The fundamental principles determining the accuracy of biological information processing remain largely unknown. In the context of Escherichia coli chemotaxis, previous research established that gradient-climbing speed is bounded by the information cells acquire from their environment, and that E. coli operate near this bound. However, it was unclear what prevents them from acquiring more information. A prevailing hypothesis, rooted in the classic work of Berg and Purcell, posits that E. coli chemosensing is limited by the fundamental physics of stochastic ligand molecule arrivals at cell receptors. This view suggests bacteria approach the physical limit of sensing accuracy. Yet, no direct comparison has been made between bacterial chemosensing and these physical limits because quantifying a real cell's uncertainty about external signals has remained methodologically challenging. This study addresses whether E. coli are limited by the physics of molecule diffusion or by internal constraints within their signal processing machinery.

Methodology
The authors developed a theoretical framework to quantify "behaviorally-relevant" information rates, comparing the physical limit of sensing against the actual information encoded in the cell's signaling pathway.

1. Theoretical Formulation:

   • Behaviorally-Relevant Signal: Instead of the absolute concentration $c(t)$, the study focuses on the time derivative of the log-concentration, $s(t) = \frac{d}{dt}\log(c(t))$, which drives chemotactic decisions in shallow gradients.
   • Physical Limit ($\dot{I}^*_{s \to r}$): The authors derived the maximum information rate available from the stochastic arrival rate of ligand molecules, $r(t)$. Using causal Wiener filtering theory, they calculated the transfer entropy rate (equivalent to predictive information rate) from past particle arrivals to the future signal. This establishes the theoretical upper bound set by diffusion physics.
   • Biological Information Rate ($\dot{I}^*_{s \to a}$): They derived the information rate encoded in the dynamics of intracellular CheA kinase activity, $a(t)$. This required modeling the kinase response to ligand arrivals (using a linear response function with fast and slow adaptation times) and characterizing the internal noise in kinase activity.

2. Experimental Measurements:

   • Single-Cell Tracking: Cells were tracked in microfluidic devices across multiple background concentrations of aspartate (0.1, 1, and 10 $\mu$M) to measure swimming statistics (run duration, velocity variance) and extract signal correlation parameters.
   • Single-Cell FRET: Using Förster Resonance Energy Transfer (FRET) between CheY-mRFP1 and CheZ-mYFP, the authors measured kinase activity dynamics in single cells.
     • Response Functions: Kinase responses to step changes in aspartate concentration were measured to determine the gain ($G_r$) and adaptation times ($\tau_1, \tau_2$).
     • Noise Statistics: Fluctuations in kinase activity under constant background concentrations were analyzed to quantify the diffusivity ($D_n$) and correlation time ($\tau_n$) of internal noise.
   • Parameter Integration: The experimentally measured parameters were plugged into the derived analytical expressions to calculate both the physical limit ($\dot{I}^*_{s \to r}$) and the actual information rate ($\dot{I}^*_{s \to a}$) for E. coli.

Key Contributions and Results

• Quantitative Comparison: The study provides the first direct comparison between the physical limit of chemosensing and the actual performance of E. coli. The results show that E. coli encode approximately two orders of magnitude less information than the physical limit allows across all tested background concentrations.
• Internal Noise Dominance: The discrepancy is driven by internal noise in signal processing. The power spectral density of slow fluctuations in kinase activity significantly exceeds the noise level expected from filtered molecule arrival noise. Consequently, the accuracy of signal estimation from kinase activity is far lower than the accuracy achievable from molecule arrival rates.
• Functional Consequences: Simulations of run-tumble motion demonstrate that this loss of information has a direct functional impact. Ideal cells (hypothetical agents observing molecule arrivals directly) climb gradients significantly faster than E. coli-like cells. The ratio of drift speeds corresponds to the square root of the ratio of information rates, confirming that the internal limitations of the signaling pathway constrain the bacterium's chemotactic performance.
• Re-evaluation of Optimality: The findings challenge the long-held belief that E. coli chemotaxis approaches the physical limit. The authors argue that the previous assumption—that cells must infer gradient direction with high accuracy in every single run—is too stringent. Instead, gradient climbing is an accumulation of inferences over many runs, allowing for lower per-step accuracy.

Significance
The paper concludes that E. coli chemosensing is limited by internal constraints (specifically, noise in signal processing) rather than the fundamental physics of ligand diffusion. This revises the understanding of bacterial chemotaxis, suggesting that the system is not optimized to the absolute physical limit of information acquisition.

The authors posit that this gap between physical limits and biological performance likely arises from trade-offs imposed by other biological constraints, such as the need to operate over a wide range of background concentrations, the costs of energy and resources required for higher fidelity, or selection pressures for other tasks like localizing to concentration peaks. The study motivates a shift in perspective from asking how close biology is to physical limits to investigating the specific physical and biological constraints that shape the evolution of sensory systems. The authors suggest that E. coli serves as a valuable template for studying these constraints in higher organisms.