I am posting here some preliminary ideas on the design of a biochemical reaction network that produces concentration pulse trains with a powerlaw distribution of pulse widths. I don't aim to find the most elegant solution, a proof of principle would be ok.

The idea is to utilize the intrinsic randomness of single-molecule reactions and to amplify it into macroscopic population fluctuations using nonlinear effects. The concept goes as follows:

There is a huge reservoir of some substance X0. It can be converted into another molecular species X by the help of an (activated) growth factor G*. Another enzyme, called the decay factor D, reconverts X back into X0. The growth reaction is assumed to be autocatalytic:

X0 + X + G* --> 2X + G*

For the decay reaction we assume that the enzyme D is operating within the saturation regime, i.e. at constant conversion rate:

X + D --> X0 + D.

At the beginning of each pulse, the autocatalytic growth reaction produces an exponential "concentration explosion", as long as the activated growth factor G* is present. If we assume, in the extreme case, that we have only a single G*-molecule around which looses its activation status spontaneously (Poisson process)

G* --> G,

we have the exponential growth of X terminated after an exponentially distributed time interval DeltaT_grow. This creates a power-law-distributed maximum concentration of X. After this event, the decay reaction reduces the concentration X at constant rate. The time period DeltaT_decay until the concentration is back to "ground level" is, therefore, also power-law-distributed. This is the main mechanism to create long-time correlations.

(The decay reaction is already proceeding during the exponential growth phase. But this should be no problem as soon as the growth rate greatly exceeds the decay rate)

Since we want to create a more or less binary concentration pulse, we couple the heavily fluctuating X-concentration to another chemical species E:

X + E <---> XE

Assume that the total number #E_tot of E-molecules is small. Then, as long as the number of available X-molecules is larger than a certain threshold, basically all E will be bound in XE-complexes, i.e. the number of XEs saturates at the limit #E_tot. Vice versa, if the X are below threshold, the number of XEs falls to zero. So the XE-signal has the desired binary pulse character. The "on-time" of XE(t) should be power-law-distributed.

There remains (at least) one problem with this scheme: How is the next pulse initiated ? To do this, we only need to re-activate the growth factor. However, it is important that this does not happen before the last pulse is over, i.e. before X is decayed back to ground level. We therefore couple the activation to the binary switch molecule E/XE:

G + E --> G* + E

E is not available during the period when X is large, because all E is bound into XE-complexes.

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