Gene Expression Transcription and Translation Stochastic Description

4 Gene Expression (Transcription and Translation) Stochastic Description 

In the reactions above M and P respectively denote messenger RNA and protein, while hm and np are the corresponding molecular counts Sm and Sp are the substrates out of which RNA and protein molecules are made of, Ym and Yp correspond to the products into which RNA and proteins are degraded, Um and kp are the RNA and protein effective synthesis rates (accounting for the abundance of substrate moiecuies as well as of polymerases or ribosomes), and finally ym and yp are the RNA and protein degradation rates. 

If we assume that the RNA dynamics (production and degradation) are much faster than the protein dynamics, then we can make aquasi-stationary approximation as in Chap. 5. After doing it, we obtain two separated master equations, one for the RNA and the other for the protein dynamics  

Despite its elegance and simplicity, the above analyzed approximation is incorrect because it relays on a wrong assumption. Although the protein decaying process is indeed much slower than RNA synthesis and decay, protein synthesis is as fast, if not faster, than the RNA-related processes. In what follows we shall construct a better approximation, but in order to do so it is useful to analyze Eqs. (8.36) and (8.37) in a more detailed way. In the model in (8.31)-(8.34) the RNA production and degradation processes are completely independent of the protein count, thus we expect the master equation for P(nm t) to remains the same in the improved model as in Eq. (8.36). Regarding Eq. (8.37), in order to interpret it, it is more convenient to write it as 

Let us focus on the factor kj ikp/ym), appearing in the first row. This factor, which accounts for the rate of protein synthesis, can be split as the rate of RNA production, ka, times the probability that a single RNA is translated once during its life time, kp/YM- To understand this last assertion, notice that translation and degradation are carried out by two different molecules that compete for the RNA  

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