Expectation Invariants for Probabilistic Program Loops as Fixed Points

Many molecules that control genetic regulatory circuits act at extremely low intracellular concentrations. Resultant fluctuations (noise) in reaction rates cause large random variation in rates of development, morphology and the instantaneous concentration of each molecular species in each cell. To achieve regulatory reliability in spite of this noise, cells use redundancy in genes as well as redundancy and extensive feedback in regulatory pathways. However, some regulatory mechanisms exploit this noise to randomize outcomes where variability is advantageous.