The protein is produced in bursts.

Supplemental Mathematics

The paper is 1.

• Derive Eq. (1).

  Note that here the master equation is approximated as a Fokker-Planck equation; see Stochastic Processes - Fokker-Planck equations. \(\displaystyle\frac{\partial P(y,t)}{\partial t}=-\frac{\partial}{\partial y}\{a_1(y)P\}+\frac{1}{2}\frac{\partial^2}{\partial y^2}\{a_2(y)P\}\), where \(\displaystyle a_\nu(y)=\int_{-\infty}^{\infty}r^\nu W(y;r)dr\).

  For degradation, one molecule dies per reaction -> \(\displaystyle r=-\frac{1}{V}\). \(\displaystyle W(x;r)=\gamma_2n\delta(-\frac{1}{V})\). \(\displaystyle a_1(x)=\int_{-\infty}^{\infty}(-\frac{1}{V})(\gamma_2n\delta(-\frac{1}{V}))dr=-\gamma_2x\) -> \(\displaystyle\frac{\partial}{\partial x}[\gamma_2xp(x)]\) term. I suspect the authors ignored \(a_2\) term from the smoothness of \(p(x)\).

• Derive Eq. (4) & (8) see note

Brief Summary

Model: protein is produced in bursts, where each time the produced number is exponentially distributed and uncorrelated with other burst. The degradation is first-order, where the dilution and degradation are considered together. Under resonable approximation, the steady-state protein distribution is a gamma distribution.

To compare these results with numerical simulations, the authors simulated with the cell cycle with binomial partitioning segregation, but I don’t get all the details from the text!

Then consider autoregulation - the burst rate depends on the current level of abundance. Then negative feedback -> distribution squeezed; potitive feedback -> gives rise to bistable distribution.

If a repressor \(R\) regulates the production of \(x\), the joint distribution

• In the limit were fluctuations in \(R\) are fast compared with the rate of transcription of \(x\): \(p(R,x)=p(R)p(x)\) (extrinsic noise is negligible because it’s averaged out);
• …slow…: \(p(R,x)=p(R)p(x\vert R)\) (every time \(R\) changes, \(x\) will relax to the corresponding new steady-state before \(R\) changes again).

The analytical form shows that the effect of extrinsic noise in this case is assymetric, increasing \(p(x)\) only for small value of \(x\).

References

1. Friedman, N., Cai, L., & Xie, X. S. (2006). Linking Stochastic Dynamics to Population Distribution: An Analytical Framework of Gene Expression. Physical review letters, 97(16), 168302.