The Combinatorics of Feedback


If you don't see a dynamic simulation like in the picture above, please install Java.

This applet simulates the regulation of uptake and metabolism of small molecules in cells. The applet allows you to explore the different ways of combining negative and positive feedback to regulate the level and flux of the molecule.

There are four basic motifs involving two entangled feedback loops which can be chosen by clicking on the names at the top of the applet: socialist, consumer, fashion and collector. The radio buttons to the right of these four can be used to set the type of interaction between the small molecule (s) and the regulator (R). A normal arrow indicates activation, a barred arrow indicates repression. Two more buttons there cut the transport (T), or metabolism (E) loops.

The dynamic picture shows the influx of the small molecule (brown circles) through the transport "gate" (red) and their consumption by metabolic enzymes (blue pacman). The regulator molecules (green) sense the levels of the small molecule and regulate the transport and metabolism. The feedback is shown using a mechanical analogy where the levers open or close the transport gate and the pacman when weighed down by the regulators in the bag connected to the levers.

The text fields around the picture set various parameters:
Sigma: the extracellular level of the small molecule,
Gamma: the metabolic rate per enzyme per small molecule,
Kt, ht: the half-maximum and Hill coefficient for the regulation of the transport,
Ke, he: the half-maximum and Hill coefficient for the regulation of the metabolism.

Kt, ht, Ke, he set the "strength" of each feedback loop. If R activates T, then increasing Kt or decreasing ht weakens the loop. If R represses T, then increasing Kt or decreasing ht strengthens the loop.

The "Pause" button temporarily halts the simulation. "Reinitialize" sets the levels of s, R and T to zero. The scrollbar can be used to set the overall speed of the animation.


Cells contain many response systems designed to regulate the flux and concentration of small molecules, ranging from nutrients to posions. Often these systems consist of two feedback loops, one controlling the uptake of the small molecule and the other controlling its metabolism. The loops are entangled because both the transport proteins (T) and the metabolic enzymes (E) are controlled by a single regulator (R) that senses the concentration of the small molecule (s).

The two feedback loops can implement either negative or positive feedback. Therefore there are four possible motifs with distinct feedback logic:
Socialist (- -): negative T and E feedback,
Consumer (+ -): positive T and negative E feedback,
Fashion (- +): negative T and positive E feedback,
Collector (+ +): positive T and E feedback.
Each can be implemented either with s activating R, or s repressing R.

The names reflect the behaviour of the motifs. The socialist is good for keeping the s level relatively insensitive to changes in the extracellular level. Thus, the iron homeostasis systems in various organisms contain this motif. The consumer motif maximizes both the influx and the consumption. It is therefore ideal for nutrient molecules, and indeed this motif is found in the regulation of lactose, galactose, maltose, arabinose and other sugars in bacteria.

The other two motifs show stranger behaviour, that is not commonly observed in cells, but is seen in human behaviour. In the fashion motif, when s is small the system tries to increase uptake but when it is available in plenty it shuts down the intake. This is reminiscent of fashionable goods which people desire more for their scarcity than for any intrinsic value. The collector motif is bistable, having one state where it minimizes s and one where it accumulates as much as it can. Making the analogy to obesity, if a person's weight (s) increases, it affects their internal state (R) such that they are less likely to exercise (E) and more likely to eat more (T). The collector motif might also be relevant in biological organisms in the pre-hibernation state.

This classification also provides a framework for studying more complicated response systems with multiple feedback loops. The iron homeostasis system in E. coli, for instance, combines the socialist and fashion motifs.

Combinatorics of feedback in cellular uptake and metabolism of small molecules
Submitted to Proc. Natl. Acad. Sci. (USA)
Preprint available upon request, email