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. $\frac{\partial P(y,t)}{\partial t}=-\frac{\partial}{\partial y}{a_1(y)P}+\frac{1}{2}\frac{\partial^2}{\partial y^2}{a_2(y)P}$, where $ a_\nu(y)=\int_{-\infty}^{\infty}r^\nu W(y;r)dr$.

  For degradation, one molecule dies per reaction -> $ r=-\frac{1}{V}$. $ W(x;r)=\gamma_2n\delta(-\frac{1}{V})$.

  • $ a_1(x)=\int_{-\infty}^{\infty}(-\frac{1}{V})\left(\gamma_2n\delta(r-\frac{1}{V})\right)dr=-\gamma_2x$ -> $-\frac{\partial}{\partial x}{a_1(x)P}=\frac{\partial}{\partial x}[\gamma_2xp(x)]=\gamma_2(p+\frac{n}{V}\frac{\partial}{\partial x}p)$
  • $ a_2(x)=\int_{-\infty}^{\infty}(-\frac{1}{V})^2\left(\gamma_2n\delta(r-\frac{1}{V})\right)dr=\gamma_2\frac{x}{V}$ -> $\frac{1}{2}\frac{\partial^2}{\partial x^2}{a_2(x)P}=\gamma_2(\frac{x}{2V}\frac{\partial^2}{\partial x^2}P+\frac{1}{V}\frac{\partial}{\partial x}P)$…???

• 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.