Gene regulation and noise reduction by coupling of stochastic processes

Here we characterize the low-noise regime of a stochastic model for a negative self-regulating binary gene. The model has two stochastic variables, the protein number and the state of the gene. Each state of the gene behaves as a protein source governed by a Poisson process. The coupling between the two gene states depends on protein number. This fact has a very important implication: There exist protein production regimes characterized by sub-Poissonian noise because of negative covariance between the two stochastic variables of the model. Hence the protein numbers obey a probability distribution that has a peak that is sharper than those of the two coupled Poisson processes that are combined to produce it. Biochemically, the noise reduction in protein number occurs when the switching of the genetic state is more rapid than protein synthesis or degradation. We consider the chemical reaction rates necessary for Poisson and sub-Poisson processes in prokaryotes and eucaryotes. Our results suggest that the coupling of multiple stochastic processes in a negative covariance regime might be a widespread mechanism for noise reduction.

Figure 1

Binary negative self-regulating gene. The white (gray) circle represents the gene at the on (off) state, also indicated by $\alpha$ ($\beta$). The arrows indicate the chemical reactions of protein synthesis, protein degradation, and off-on and on-off gene switching with rates as indicated in the kinetic scheme (3). Protein molecules are represented by the rounded rectangles, their number by $n$, and their destruction by $⌀$. The bar-terminated line denotes repression.

Figure 2

Fano factor versus the mean protein number. Here we fixed $a=500$ with different colors standing for fixed values of $b$ as indicated by the key. Each curve corresponds to a fixed value of $b$ and variation of $z_0$. For fixed $a$ and $b$, $\langle n\rangle$ depends only on $z_0$.

Figure 3

Probability distribution of the protein number. The probability distribution ${\varphi }_n$ is shown for increasing levels of coupling. Curves A–E show probability distributions corresponding to increasing values of $f$ and $h$. As the coupling gets stronger the variance of the distribution decreases. The parameter values $(a,b,z_0)$ for each curve are A, $(1.0,2.0,0.99)$; B, $(1.0,15.0,0.95)$; C, $(14.0,70.0,0.5)$; D, $(50.0,50.0,0.5)$; and E,$(5\times {10}^3,1.0,{10}^{-4})$. Equation (7) shows that $f$ is linearly dependent on $a$, while $h$ is inversely proportional to $z_0$.