A Mathematical Model for Prokaryotic Protein Synthesis

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Abstract
A kinetic model for the synthesis of proteins in prokaryotes is presented and analysed. This model is based on a Markov model for the state of the DNA strand encoding the protein. The states that the DNA strand can occupy are: ready, repressed, or having a mRNA chain of length i in the process of being completed. The case i = 0 corresponds to the RNA polymerase attached, but no nucleotides attached to the chain. The Markov model consists of differential equations for the rates of change of the probabilities. The rate of production of the mRNA molecules is equal to the probability that the chain is assembled to the penultimate nucleotide, times the rate at which that nucleotide is attached. Similarly, the mRNA molecules can also be in different states, including: ready and having an amino acid chain of length j attached. The rate of protein synthesis is the rate at which the chain is completed. A Michaelis-Menten type of analysis is done, assuming that the rate of protein degradation determines the ’slow’ time, and that all the other kinetic rates are ‘fast’. In the self-regulated case, this results in a single ordinary differential equation for the protein concentration.

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