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

On the response of elongating ribosomes to forces opposing translocation

The elongation phase of protein synthesis is a cyclic, steady-state process. It follows that its directionality is determined by the thermodynamics of the accompanying chemical reactions, which strongly favor elongation. Its irreversibility is guaranteed by its coupling to those reactions, rather being a consequence of any of the conformational changes that occur as it unfolds. It also follows that, in general, the rate of elongation is not proportional to the forward rate constants of any of its steps, including its final, mechano-chemical step, translocation. Instead, the reciprocal of the rate of elongation should be linearly related to the reciprocal of those rate constants. When the results of experiments done a decade ago to measure the effect that forces opposing translocation have on the rate of elongation are reinterpreted in light of these findings, it becomes clear that translocation was rate limiting under conditions in which those experiments were done, and that it is likely to be a Brownian ratchet process, as was concluded earlier.

Stochastic thermodynamics and modes of operation of a ribosome: A network theoretic perspective

The ribosome is one of the largest and most complex macromolecular machines in living cells. It polymerizes a protein in a step-by-step manner as directed by the corresponding nucleotide sequence on the template messenger RNA (mRNA) and this process is referred to as "translation" of the genetic message encoded in the sequence of mRNA transcript. In each successful chemomechanical cycle during the (protein) elongation stage, the ribosome elongates the protein by a single subunit, called amino acid, and steps forward on the template mRNA by three nucleotides called a codon. Therefore, a ribosome is also regarded as a molecular motor for which the mRNA serves as the track, its step size is that of a codon and two molecules of GTP and one molecule of ATP hydrolyzed in that cycle serve as its fuel. What adds further complexity is the existence of competing pathways leading to distinct cycles, branched pathways in each cycle, and futile consumption of fuel that leads neither to elongation of the nascent protein nor forward stepping of the ribosome on its track. We investigate a model formulated in terms of the network of discrete chemomechanical states of a ribosome during the elongation stage of translation. The model is analyzed using a combination of stochastic thermodynamic and kinetic analysis based on a graph-theoretic approach. We derive the exact solution of the corresponding master equations. We represent the steady state in terms of the cycles of the underlying network and discuss the energy transduction processes. We identify the various possible modes of operation of a ribosome in terms of its average velocity and mean rate of GTP hydrolysis. We also compute entropy production as functions of the rates of the interstate transitions and the thermodynamic cost for accuracy of the translation process.

Structural conditions on complex networks for the Michaelis-Menten input-output response

The Michaelis-Menten (MM) fundamental formula describes how the rate of enzyme catalysis depends on substrate concentration. The familiar hyperbolic relationship was derived by timescale separation for a network of three reactions. The same formula has subsequently been found to describe steady-state input-output responses in many biological contexts, including single-molecule enzyme kinetics, gene regulation, transcription, translation, and force generation. Previous attempts to explain its ubiquity have been limited to networks with regular structure or simplifying parametric assumptions. Here, we exploit the graph-based linear framework for timescale separation to derive general structural conditions under which the MM formula arises. The conditions require a partition of the graph into two parts, akin to a "coarse graining" into the original MM graph, and constraints on where and how the input variable occurs. Other features of the graph, including the numerical values of parameters, can remain arbitrary, thereby explaining the formula's ubiquity. For systems at thermodynamic equilibrium, we derive a necessary and sufficient condition. For systems away from thermodynamic equilibrium, especially those with irreversible reactions, distinct structural conditions arise and a general characterization remains open. Nevertheless, our results accommodate, in much greater generality, all examples known to us in the literature.