Quantifying stochasticity in gene-expression with extrinsic parameter fluctuations

Homogeneous cell populations can exhibit considerable cell-to-cell variation in the level of a given protein. Both intrinsic and extrinsic sources of noise have been implicated in driving this variability in protein level. More specifically, intrinsic noise is the protein variability that is different across genes and arises from the inherent stochastic nature of biochemical processes involved in gene-expression. In contrast, extrinsic noise is the protein variability that is common across genes and comes from random fluctuations in the levels of shared cellular enzymes. These fluctuations in enzyme levels induce fluctuations in different gene-expression parameters such as the transcription and translation rate. Here, we derive exact analytical formulas quantifying the extent of variability in protein levels in the presence of both intrinsic noise and extrinsic parameter fluctuations. Consistent with previous results, we find that extrinsic fluctuations in the transcription rate enhance extrinsic noise in gene-expression but do not affect the intrinsic noise. Interestingly, analysis reveals that extrinsic fluctuations in the translation rate dramatically increase both intrinsic and extrinsic noise in gene-expression. Implications of our results in the context of quantifying intrinsic and extrinsic noise from two-color reporter systems are discussed. In summary, formulas developed here show how extrinsic parameter fluctuations propagate through the gene-expression process to create heterogeneity in protein level and have important implications for discriminating between alternative sources of stochasticity in gene-expression.