🧬 biology

Navigation driven by bidirectional information transmission between sensing and actuation

This paper introduces "Behavioral Equations of State" (BESTs), a theoretical framework demonstrating that navigation performance in both spatial- and temporal-sensing biological systems is universally determined by the strength and timescales of bidirectional information transmission between sensing and actuation, a prediction experimentally validated through stochastic simulations of *E. coli* chemotaxis.

Original authors: Avishek Das, Pieter Rein ten Wolde

Published 2026-07-30

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CC BY 4.0

Original authors: Avishek Das, Pieter Rein ten Wolde

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

Imagine you are trying to navigate a crowded, foggy dance floor where the music suddenly changes tempo. You can't see the DJ, but you can feel the beat through the floor. To dance well, you need two things: first, you must listen carefully to the music (sensing), and second, you must move your feet to match that rhythm (actuation). But here's the tricky part: your dancing changes where you stand, which changes how you hear the music next. This creates a loop where listening and moving constantly talk to each other. This isn't just about dancing; it's how your immune system fights germs, how your brain reacts to stress, and how tiny living cells find food in a vast ocean. Scientists have long known that this "feedback loop" is crucial, but they've struggled to measure exactly how much the information flowing back and forth actually controls how well the system works. Is it just about having strong muscles or sharp ears, or is there a deeper rule connecting the flow of information to the success of the dance?

This paper dives into that mystery by looking at how cells navigate, specifically focusing on two types of "dancers": big cells that compare the environment on their left and right sides (spatial sensing), and tiny, fast bacteria that compare the environment now versus a split second ago (temporal sensing). The researchers, Avishek Das and Pieter Rein ten Wolde, wanted to see if they could find a universal rule—a "Behavioral Equation of State" or BEST—that links the amount of information flowing between sensing and moving directly to how well the cell navigates. They didn't just guess; they built mathematical models to solve this puzzle and then tested their ideas with computer simulations of Escherichia coli bacteria.

Here is what they found: In the world of shallow, gentle gradients (like a very mild slope of food), the success of navigation isn't determined by the specific details of the cell's parts, like how strong its sensors are or how fast its motors spin. Instead, performance is entirely controlled by two specific "information currents." The first is the feedforward flow: how well the cell translates the outside world into an internal signal. The second is the feedback flow: how well the cell's movement changes the future signals it receives. The authors discovered that if you know the strength of these two information flows, you can predict exactly how well the cell will navigate, regardless of the cell's specific hardware. It's as if the "currency" of navigation is information itself, not the physical gears of the machine.

However, the story has a twist. The paper suggests that having too much feedback information can actually be bad. Imagine a dancer who is so sensitive to the music that they start tripping over their own feet because they are reacting to every tiny, random noise in the room rather than the actual beat. The simulations showed that if the feedback loop is too strong, the cell gets overwhelmed by noise, and navigation performance drops. There is a "sweet spot" for feedback information where performance is maximized.

The researchers also made sure to clarify what doesn't work. They found that looking at just a single moment in time (a "single-step" view) isn't enough to understand navigation. You have to look at the whole history of the signal, the entire trajectory of the dance, to see the full picture. Furthermore, while their theory works perfectly for gentle slopes, they found that in steeper, more chaotic environments, the speed at which the cell processes information becomes an extra, critical factor.

To prove their theory wasn't just a mathematical fantasy, they ran detailed computer simulations of E. coli, a bacterium famous for its ability to swim toward food. Even though the real bacterial system is incredibly complex and non-linear (meaning it doesn't follow simple, straight-line rules), the simulation results collapsed perfectly onto the authors' predicted curve. Without tweaking any numbers or forcing the data to fit, the "Behavioral Equation of State" accurately predicted how well the bacteria would swim based solely on the measured information flows. This suggests that bidirectional information transmission is a fundamental organizing principle for navigation, acting as a universal rulebook that nature follows, even in the noisy, chaotic world of microscopic life.

Technical Summary: Navigation Driven by Bidirectional Information Transmission

Problem Statement
Biological navigation, such as cellular chemotaxis, relies on a feedback loop where a sensory system maps environmental inputs to an internal output, which drives an actuation response that subsequently alters future sensory inputs. While the field has extensively characterized information transmission from input to output in the absence of feedback, a quantitative understanding of how bidirectional information flow (sensorimotor feedback) controls navigation in the presence of stochastic noise is lacking. Specifically, it is unclear how the accuracy of this closed-loop information transmission constrains or enables navigation performance, and whether system-specific microscopic parameters (e.g., sensory gain, noise magnitudes) can be abstracted into universal information-theoretic variables.

Methodology
The authors employ a combination of analytical theory and stochastic simulations to investigate two generic models representing the two major classes of biological navigators:

Spatial-Sensing Model: A linear stochastic model describing cells (e.g., eukaryotes) that estimate spatial gradients by comparing concentrations across their length. The model includes sensory and actuation gains, relaxation rates, and Gaussian white noise.
Temporal-Sensing Model: A weakly nonlinear stochastic model describing cells (e.g., E. coli) that infer gradients from temporal changes in concentration while swimming. This model incorporates bilinear coupling between sensory output and velocity.

To quantify information flow, the authors utilize transfer entropy rates, specifically distinguishing between:

Feedforward Transfer Entropy ( $\dot{T}_{FF}$ ): Information transmitted from sensory input to sensory output.
Feedback Transfer Entropy ( $\dot{T}_{FB}$ ): Information transmitted from sensory output, through actuation, back to the future sensory input.

Crucially, the theory accounts for information encoded in full signal trajectories (multi-step transfer entropy) rather than single-step states. For the nonlinear temporal-sensing model and the realistic E. coli simulations, the authors employ nonlinear perturbation theory combined with stochastic filtering theory and the TE-PWS algorithm to compute transfer entropies exactly.

Key Contributions and Results

Derivation of Behavioral Equations of State (BESTs):
The primary theoretical result is the derivation of "Behavioral Equations of State" (BESTs). These are equalities that map navigation performance directly to bidirectional information flows, independent of the elementary system parameters (gain, noise, etc.).
- Spatial Sensing: In the linear-response regime of shallow gradients, navigation performance ( $P$ ) is fully determined by $\dot{T}_{FF}$ and $\dot{T}_{FB}$ :
  $\frac{P}{P_0} = \frac{\dot{T}_{FF}}{2} \left[ \sqrt{(2\dot{T}_{FB} + 1)^2 - 1} + \frac{1}{\sqrt{(2\dot{T}_{FB} + 1)^2 - 1}} \right]$
  Performance increases monotonically with feedforward information but exhibits a non-monotonic dependence on feedback information, peaking at an optimal $\dot{T}_{FB}^* = (\sqrt{2}-1)/2$ nats. Excessive feedback information degrades performance by amplifying sensory noise.
- Temporal Sensing: For weak nonlinearity and shallow gradients, the BEST takes the form:
  $\frac{P}{P_0} = 4 \frac{\rho}{1+\rho} \sqrt{\dot{T}_{FF}} \sqrt{\dot{T}_{FB}}$
  where $\rho = F/H$ is the ratio of sensory processing speed to actuation speed. Here, performance scales with the square root of both information flows.
Role of System Parameters:
The analysis reveals that elementary parameters (sensory gain, actuation strength, noise magnitudes) do not independently control navigation. Instead, they influence performance only through their effects on the magnitude and speed of bidirectional information transmission. Information acts as a "universal currency" for stochastic navigation.
Necessity of Multi-Step History:
The authors demonstrate that single-step transfer entropies are insufficient to predict navigation performance. The full history of signal trajectories must be accounted for, as the difference between single-step and full-history transfer entropy can be large or even divergent.
Validation in E. coli Chemotaxis:
The theory was tested against stochastic simulations of a realistic, nonlinear model of E. coli chemotaxis (involving run-and-tumble dynamics). Using the TE-PWS algorithm to compute transfer entropies, the authors found that the observed navigation performance obeys the BEST prediction without any fitting or scaling parameters. Furthermore, simulations confirmed the existence of an optimal feedback information level (~4 nats) beyond which performance declines, consistent with the trade-off between signal amplification and noise amplification.

Significance
The paper establishes bidirectional information transmission between sensing and actuation as an organizing principle for stochastic navigation. By deriving system-independent relations (BESTs), the work suggests that navigation performance can be predicted and understood through macroscopic information flows rather than microscopic parameter tuning. This provides a framework for analyzing diverse biological feedback systems where input sensing and actuation are stochastic, potentially extending beyond navigation to other biological functions like immune response and homeostasis. The findings offer an experimentally testable prediction: data from navigators with varying sensing and actuation parameters should collapse onto a single curve defined by the relationship between performance and bidirectional information.

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