### 24 Protein expressions, a stochastic process

/*-------------Divider-------------*/
Proteins were produced by cells through transcriptions and translation. However, even under the same regulatory conditions, the expression level of a protein is not always a constant. It is therefore more proper to consider it as a stochastic process. It is expected that the randomness of protein expression level comes from the randomness of transcriptions and translations, but how could we describe it quantitatively? To put it in another way, how shall we expect the randomness of protein expression levels would change if we alter the speed of transcriptions and translation? The article we suggested today was the first one to systemically adjust the parameters associated with transcriptions and translations to observe their effect on the randomness of protein expression level. We will talk about the reasoning of the authers than move on to his experiment designs.

/*Theoretical Reasoning*/

Firstly, the author used the concept of Langevin dynamics (please refer to: #14
How enzymes propel themselves - IV). Assume r the copy number of mRNA, p the copy number of proteins,
γ the degradation rate, k the transcription or translation rate, and η the random force. We could write the following equations:

in which
η follows
qi is the strength of the stochastic forces. If we take the average of the above differential equations, we will get the steady state solution of our system:
Nonetheless, this is not the ultimate result we want since we want to know the behavior of r & p, rather than their average. Therefore we expand r = <r> + δr and we substitute it back to original differential equations:
This is a stochastic differential equation of δr and it is not easy to solve it directly, so we will perform Fourier transform on it:
If we further multiply both sides of the equation with their complex conjugate and perform inverse Fourier transform then we will get: (the original article has a small mistake by dropping the square):
If we assume that the copy number of mRNA satisfies Poisson distribution (please refer to #22 Fundamentals-Poisson distribution), then we will get

So we could know that
By the same way we will know that (we left the derivations for our readers)

Up to now we could define 'noise strength' ν as the following:

For a variable following Poisson distribution, ν=1. However, if we define
The physical meaning of φ is the ratio of mRNA half life to protein half life, and it is usually much smaller than 1. b represents how many protein each mRNA could produce on average, also called 'burst size.' So we predict that
/*End of Theoretical Reasoning*/

Since we have predicted that
ν = 1+b，<p> = k_R*b/γ_P, how could we verify it? The authors transferred a segment of gene, whose promoter could be induced by IPTG and produces green fluorescent proteins (GFP), into Bacillus subtilis. They further mutate these bacteria to get clones with various translational efficiency and transcriptional efficiency.

If we increase the number of GFP by increasing the concentration of IPTG, it will increase k_R but not b, and we should expect that <p> will increase but not ν. However, if we compare clones with different translational efficiency, the increased amount of GFP is caused by the change of b, so both <p> and ν will increase and they should be proportional to each other. That means cell could adjust the relative speed of transcription and translation to change the strength of noise in protein expressions. Cells utilizing 'high transcription speed with low translation speed' strategy and cells utilizing 'low transcription speed with high translation speed' strategy could have the same mean protein expression level, but their fluctuations are very different. Some key enzymes like ATP cyclase controls the concentration of cAMP, one of the key signaling molecules in cells which had better be well controlled. Therefore in E. coli, gene of ATP cyclase has a very low translation rate. However, if there is a protein we hope to have variable expression levels to maintain the diversity of our population, this gene would probably have a very high translation rate. This research also told us that, if we are going to build an artificial gene circuits, the relative transcription and translation rate would have large effect on signal noises.
Figure source: Fig. 3 of suggested reading. The figure shows the result of computer simulations. If we change the burst number, even the mean protein expression level could be the same, their noise strengths could be very different.

/*-------------Divider-------------*/