Bulletin of Mathematical Biology

, Volume 79, Issue 5, pp 1005–1027 | Cite as

A Generalized Michaelis–Menten Equation in Protein Synthesis: Effects of Mis-Charged Cognate tRNA and Mis-Reading of Codon

  • Annwesha Dutta
  • Debashish ChowdhuryEmail author
Original Article


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 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.


Michaelis–Menten equation Master equation Translation Ribosome Dwell time 



DC thanks Joachim Frank, Ruben Gonzalez Jr. and Michael Ibba for useful correspondence. We also thank Joachim Frank, Adil Moghal and Alex Mogilner for valuable comments on a draft of this manuscript. We thank the anonymous referees for useful suggestions and for drawing our attention to a very recent paper. This work is supported by “Prof. S. Sampath Chair” professorship (DC) and a J.C. Bose National Fellowship (DC). DC also acknowledges hospitality of the Biological Physics Group of the Max-Planck Institute for the Physics of Complex Systems at Dresden, under the Visitors Program, during the initial stages of this work.

Funding was provided by Science and Engineering Research Board.


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Copyright information

© Society for Mathematical Biology 2017

Authors and Affiliations

  1. 1.Department of PhysicsIndian Institute of TechnologyKanpurIndia

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