This page has been doing the rounds on some of the computer tech blogs and social media and I wanted to share it here.
It’s a pretty deep technical dive into how E. coli chemotaxis works.
The Baffling Intelligence of a Single Cell: The story of E. coli chemotaxis
Hi Mark
This is, indeed, a fascinating paper, even though it’s often difficult to follow (for me, anyway). But the basic molecular process that allows E. coli to successfully navigate to an attractant via a biased random walk is pretty clear and the molecular/mechanical details of the structure and mechanism of the motor that drives the flagella that propel the bacterium are amazing to see. What we have here is a control system that is implemented via molecules rather than neurons. It would be an interesting exercise to map out this molecular control process in terms of the main functional components of a control system: sensor, comparator, reference, error, output, feedback function.
One thing I didn’t understand is how the adaptation process works that increases the dynamic range of the attractant sensor. It seems to me that this adaptation process might be functionally equivalent to a second level control system that keeps the sensor reaction range appropriate to the range of the attractant concentration through which E. coli is moving. If you understand how this adaptation mechanism works could you please explain it and, even better, describe the code that would allow a simulation of E. coli which can move to a nutrients that are surrounded by attractant gradients that vary in density by several orders of magnitude.
Very interesting stuff. It’s amazing how much is known about the details of the E. coli navigation process.
The increase of dynamic range of the attractant sensor is a mechanism known as Fold Change Detection. The sensitivity of the receptors is modulated by an internal chemical feedback loop, essentially functioning as a memory. An excellent article by Miri Adler and Uri Alon, “Fold Change Detection in Biological Systems” presents the main ideas and gives examples of biological networks employing the principle.
I find it useful to employ mathematical notation, so please forgive me. In chemotaxis modelling, u(t) represents sensed attractant (in these models as a concentration), x an internal variable (memory?, again as a concetration), and y the output, which represents frequency of tumble.
Consider the network motif designated Type 1 Incoherent Feed Forward Loop (there is a taxonomy of such network motifs, see book by Uri Alon and his research group). In this motif, u attractant positively activates the production of both x and increases y, while the intermediary x inhibits or reduces y. This is modelled with the coupled differential equations
x’(t)=-x(t)+u(t);
Sy’(t)=-y(t)+u(t)/x(t).
Here the prime denotes derivative with respect to time and S is a scale constant (which absorbs the details of various rate to keep the equations dimensionless). The equations indicate that x tracks u in. When x is equal to u, y is constant, i.e. tumbles occur at regular intervals. However, a change in u will elicit a rapid change in y, and x will slowly “catch” up. As it does, the frequency of tumbles slows down until x reaches the value of u and y returns to its base rate of tumbling.
This mechanism yields a variable clock, which is tuned by the sensor which itself then habituates. This mechanism, if tuned properly, keeps the non-linear sensor in its linear region which then enables finer local “discrimination” of the surround.
Powers’ re-organization concept can now utilize this mechanism by arranging the sensors and the “velocities” to do what Powers describes: if things don’t get better, change course!
I developed a similar mechanism to model reorganization in PCT because I was dissatisfied with the level of detail provided by Powers. I found the software demo convincing but not compelling; the digital aspect and the counter embedded in the code of the simulated e. coli seemed to me a sort of cheat.
There are other simple network motifs (see Adler and Alon paper) that I have played around with.
I might write another post related to these findings because I seem to have strayed afar from the topic of dynamic range.
I will attempt to give a synopsis of the referenced paper if there is interest. In the meantime, I found it very fruitful to watch lectures from Alon lab, in particular the Systems Biology Course 2018. Fascinating stuff! And more compatible with Powers’ ideas that might at first seem.
Leo(?) Geontoro and Eduardo Sontag are both researchers who have published control-theoretic papers in this area for those that are mathematically inclined.
This is brilliant, and tantalising to see the physiological / chemical substrates described. So the concentration (u) would modify the reorganisation ‘strength’ and x would be a chemical signifying intrinsic error?
For the 1IFFL (type 1 Incoherent Feed Forward Loop) it is a little difficult to formulate in PCT terms.
On the one hand, the attractant u might be thought of as the perceptual signal; one path continues upward toward the reference node x which is then fed back down to the output y. Both the attractant u and the memory x act on y, so the comparison or error is in some sense calculated by or implicit in y. The error, r-p, is analogous to u/x term in the above equation.
On the other hand, the perception directly affects the reference, which is perhaps more in the spirit of Powers’ notion of modes; here the x term is more directly analogous to a (content addressable?) memory which is then fed as reference to a lower perceptual control unit. The reference signal is both created or recorded along the upward path' and this memory is automatically recalled and fed as reference signal along the downward path’.
However, this model is truly feed forward: it does not capture the entire control loop. It might be more accurate to think of these equations as modelling a variation of the ‘inside’ portion of the standard PCT control loop diagrams.
There is another prominent network motif more similar in spirit to Powers which also exhibits fold change detection. It is the NonLinear Integral Feedback Loop, NLIL, and has very similar equations to the previous post.
x’(t)=x(t)(y(t)-1)
Sy’(t)=-y(t)+u(t)/x(t)
In contrast to the 1IFFL here the attractant u acts on y directly by excitation and does not affect x directly. The output y, or frequency of tumble is inhibited by x as before. Here the output frequency is normalized so that the base rate of tumble is once per unit of time. When u increases, y also increases. But the increase in y causes an increase in x, which is also self-activating. When y deviates from its baseline 1, there is an `error’ which is integrated by x and fed back to y as inhibition.
In this model, it might be appropriate to think of the reference signal as the attractor of y, it is constant. But then y also acts as the perception, and the error term, r-p, is analogous to the (y-1) term above. So in some sense y is the output, but also functions as the lower level comparator and perceptual signal. lt also provides the perceptual signal for a `higher-order’ control unit whose output is x and reference is fixed at 1.
The following is an attempt to partially address the original question regarding dynamic range of sensors:
Sensors have characteristic sigmoidal shape: input causes no change for low intensities and begin to rise once some threshold is met. Input intensity then induces sensor activity in some proportion until sensor activity saturates. Both systems above effectively adapt their dynamics so that the center (or beginning) of this linear region of sensitivity of the sensor ``shifts" according to the typical or habituated or familiar surround.
Fold change detection yields exact adaptation and Weber’s Law. It gets its name from the fact that response is invariant to scale. E. Coli in particular can steer' toward a high concentration which is localized, but will become blind’ after some time if the concentration suddenly goes up everywhere. After habituation, the sensors are now sensitive to local variations of concentration and again `seek’ out the (relatively) higher concentration locales.
Your irises utilize similar mechanisms and your vision habituates regardless of how bright it is outside or dark inside.