I need a biological process that can be described as a stochastic process in statistical physics. I am familiar with some processes such as birth-death or gene expression, but now I need a process where a specific reset process occurs. In this reset process, there is a counter that counts and increments an observable as the system evolves over time. The reset occurs randomly after a certain amount of time has passed, during which the counter does not count anything.Is it possible? Thanks for your attention.
I think bacterial chemotaxis may qualify, depending on how the phenomenon gets aliased into the signal and time-counter variables.
The bacterial flagellar rotor can turn in two directions. Because the flagellum is chiral, clockwise and counter-clockwise rotor directions produce two different types of movement. One type is running in which the motion is roughly linear, the other is tumbling in which the bacterium basically rotates.
It's easy to show that random switching between these modes leads, over time, to a random position in space. So, if one (say) puts a single bacterium at the center of a 2D playpen, its coordinates after an interval are random.
The ratio of runs/tumbles is set by a kind of olfactory system, so that when the bacterium is moving towards an attractant there are more runs to close the distance, and less tumbles to scramble the bearing. But even when traveling straight to an attractant, tumbles still occasionally occur. This is summarized below in a figure and its caption from 1 (with minor edits, and my boldface):
Attractants bind to the receptor-transducer proteins ( Tar, tsr, Trg, Tap) in the membrane. The receptors also mediate the response to certain repellent. This activates a CheW-dependent change in the rate of autophosphorylation of CheA and the proteins that are phosphorylated by CheA (CheB and CheY). Attractants reduce the rate of autophosphorylation, and repellents increase the rate of autophosphorylation. The phosphoryl group is transferred from CheA-P to CheY and CheB. CheY-P interacts with the switch proteins (FilM, FliN, FliG) to cause clockwise rotation of the motor and tumbling. Thus repellents increase tumbling because they increase the level of CheY-P, and attractants reduce tumbling because they reduce the level of CheY-P. CheB-P is a methylesterase whose activity results in demethylation of the receptor-transducer proteins. As CheB-P goes up, methylation goes down. Thus repellents reduce the level of methylation because they increase the level of CheB-P and attractants increase methylation because they reduce the level of CheB-P. CheB-P autodephosphorylates. CheR is a methyltransferase that methylates the receptor-transducer proteins. The more highly methylated receptor-transducer protein does not transmit the chemoattracant signal to CheA. Hence adapation to a chemoattracant is due to increased methylation of the receptor-transducer proteins. Adapation to a repellent is due to undermethylation of the receptor-transducer protein. The relative activities of CheA-P determine the level of CheY-P, hence the frequency of tumbling .
When a run is initiated, the protein CheY is in a dephosphorylated state. The phosphorylation of this protein is a second stochastic process driving the chemotaxis. When it is eventually dephosphorylated again, also a stochastic process, this comprises a reset of that timing system.
As a result, the time in current run or tumble is a stochastic variable that near-instantaneously resets. It could be appropriate to act as the "counter" time-index variable that could be paired with the bacterium coordinates random variable.