The CME model for protein synthesis in a single cell

The stochastic behavior of biochemical reactions has been observed in single living cells as well as in in vitro experiments. Sunney Xie and colleagues have observed a stochastic, randomly timed bursting behavior in the production of proteins, i.e., translation of an mRNA [26, 217]. The observed number of the protein molecules produced in each burst, np, follows a geometric distribution, 

Based on the CME framework it is possible to demonstrate that these observations are exactly consistent with a very simple kinetic model for translation. In this model we let R be a ribosome that carries out the protein synthesis. We further assume that a ribosome and an mRNA can form a complex that continues to synthesize copies of the protein until the complex dissociates. The kinetic scheme is 

We define pseudo-first order rate constants k = A] [mRNA], k3 = [AA] . In 

The solution to Equation (11.37) can be obtained in two parts. From the two-state transition R = mRNA R it follows that the duration between two bursts of protein synthesis is exponentially distributed with mean time 1 / ,. The duration of each burst is also exponentially distributed with mean time 1 / k2. Thus the X in Equation (11.35) is related to the kinetic constant A = k.  

For each burst, we can obtain the probability distribution for the number of proteins produced from Equation (11.37) with k°x = 0  

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