A generalized Michaelis-Menten equation in protein synthesis: effects of mis-charged cognate tRNA and mis-reading of codon

The sequence of amino acid monomers in the primary structure of a protein is decided by the corresponding sequence of codons (triplets of nucleic acid monomers) on the template messenger RNA (mRNA). The polymerization of a protein, by incorporation of the successive amino acid monomers, is carried out by a molecular machine called ribosome. We develop a stochastic kinetic model that captures the possibilities of mis-reading of mRNA codon and prior mis-charging of a tRNA. By a combination of analytical and numerical methods we obtain the distribution of the times taken for incorporation of the successive amino acids in the growing protein in this mathematical model. The corresponding exact analytical expression for the average rate of elongation of a nascent protein is a `biologically motivated' generalization of the {\it Michaelis-Menten formula} for the average rate of enzymatic reactions. This generalized Michaelis-Menten-like formula (and the exact analytical expressions for a few other quantities) that we report here display the interplay of four different branched pathways corresponding to selection of four different types of tRNA.