Scaling of stochasticity in gene cascades

Stochastic fluctuations in protein levels are inevitable due to the probabilistic nature of gene expression. These fluctuations are attributed to two factors: intrinsic noise (noise associated with random transcription and translation events) and extrinsic noise (noise associated with fluctuations in the reaction rates due to correspondent fluctuations in cellular enzyme levels). We present results on how stochastic noise in proteins scales in a cascade of genes both in the presence and absence of extrinsic noise. In particular, we derive analytical formulas relating the noise in the proteins to the length, per stage magnification of the cascade and the amount of extrinsic noise present in the cell. We show that when there is no extrinsic noise the gene cascade acts like a noise attenuator where downstream proteins exhibit reduced noise. Moreover, for two different cascades with the same average final protein level, the cascade with the larger number of stages will have lower stochastic noise in the final protein. We also show that adding extrinsic noise can change the qualitative behavior of the cascade from a noise attenuator to a noise magnifier where downstream protein have increased noise. Furthermore, there exists a critical level of extrinsic noise above which the cascade with lower number of stages will have lesser noise in the final protein.