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

Abstract

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.

Appendices

Appendix A: Master Equations and Steady-State Probabilities

The master equations governing the time evolution of the probabilities can be written as:

$$\begin{aligned} \dfrac{\mathrm{d}P_{1}(t)}{\mathrm{d}t}= & {} -\left( \omega _{a}+\omega ^{\prime }_{a}+\omega ^{\prime \prime }_{a} +\omega ^{\prime \prime \prime }_{a}\right) P_{1}(t)\nonumber \\&+\, \omega _{r1} P_{2}(t) + \omega ^{\prime }_{r1}P^{\prime }_{2}(t) + \omega ^{\prime \prime }_{r1}P^{\prime \prime }_{2}(t) \nonumber \\&+\, \omega ^{\prime \prime \prime }_{r1}P^{\prime \prime \prime }_{2}(t) + \omega _{r2}P_{3}(t) + \omega ^{\prime }_{r2}P^{\prime }_{3}(t) \nonumber \\&+\, \omega ^{\prime \prime }_{r2}P^{\prime \prime }_{3}(t) + \omega ^{\prime \prime \prime }_{r2}P^{\prime \prime \prime }_{3}(t) \nonumber \\&+ \omega _{h2}P_{5}(t) + \Omega _{h2}P^{*}_{5}(t) \nonumber \\ \dfrac{\mathrm{d}P_{2}(t)}{\mathrm{d}t}= & {} \omega _{a}P_{1}(t) - \left( \omega _{r1} + \omega _{h1}\right) P_{2}(t) \nonumber \\ \dfrac{\mathrm{d}P_{3}(t)}{\mathrm{d}t}= & {} \omega _{h1}P_{2}(t) - \left( \omega _{p} + \omega _{r2}\right) P_{3}(t) \nonumber \\ \dfrac{\mathrm{d}P_{4}(t)}{\mathrm{d}t}= & {} \omega _{p}P_{3}(t) + \omega ^{\prime }_{p}P^{\prime }_{3}(t) + \omega _{br}P_{5}(t) - \omega _{bf}P_{4}(t) \nonumber \\ \dfrac{\mathrm{d}P_{5}(t)}{\mathrm{d}t}= & {} \omega _{bf}P_{4}(t) - \left( \omega _{h2} + \omega _{br}\right) P_{5}(t) \nonumber \\ \dfrac{\mathrm{d}P^{\prime }_{2}(t)}{\mathrm{d}t}= & {} \omega ^{\prime }_{a}P_{1}(t) - \left( \omega ^{\prime }_{r1} + \omega ^{\prime }_{h1}\right) P^{\prime }_{2}(t) \nonumber \\ \dfrac{\mathrm{d}P^{\prime }_{3}(t)}{\mathrm{d}t}= & {} \omega ^{\prime }_{h1}P^{\prime }_{2}(t) - \left( \omega ^{\prime }_{r2} + \omega ^{\prime }_{p}\right) P^{\prime }_{3}(t) \nonumber \\ \dfrac{\mathrm{d}P^{\prime \prime }_{2}(t)}{\mathrm{d}t}= & {} \omega ^{\prime \prime }_{a}P_{1}(t) - \left( \omega ^{\prime \prime }_{r1} + \omega ^{\prime \prime }_{h1}\right) P^{\prime \prime }_{2}(t) \nonumber \\ \dfrac{\mathrm{d}P^{\prime \prime }_{3}(t)}{\mathrm{d}t}= & {} \omega ^{\prime \prime }_{h1}P^{\prime \prime }_{2}(t) - \left( \Omega _{p} + \omega ^{\prime \prime }_{r2}\right) P^{\prime \prime }_{3}(t) \nonumber \\ \dfrac{\mathrm{d}P^{\prime \prime \prime }_{2}(t)}{\mathrm{d}t}= & {} \omega ^{\prime \prime \prime }_{a}P_{1}(t) - \left( \omega ^{\prime \prime \prime }_{h1} + \omega ^{\prime \prime \prime }_{r1}\right) P^{\prime \prime \prime }_{2}(t) \nonumber \\ \dfrac{\mathrm{d}P^{\prime \prime \prime }_{3}(t)}{\mathrm{d}t}= & {} \omega ^{\prime \prime \prime }_{h1}P^{\prime \prime \prime }_{2}(t) - \left( \Omega ^{\prime }_{p} + \omega ^{\prime \prime \prime }_{r2}\right) P^{\prime \prime \prime }_{3}(t) \nonumber \\ \dfrac{\mathrm{d}P^{*}_{4}(t)}{\mathrm{d}t}= & {} \Omega _{p}P^{\prime \prime }_{3}(t) + \Omega _{br} P^{*}_{5}(t) + \Omega ^{\prime }_{p}P^{\prime \prime \prime }_{3}(t) - \Omega _{bf}P^{*}_{4}(t) \nonumber \\ \dfrac{\mathrm{d}P^{*}_{5}(t)}{\mathrm{d}t}= & {} \Omega _{bf}P^{*}_{4}(t) - \left( \Omega _{br} + \Omega _{h2}\right) P^{*}_{5}(t) \end{aligned}$$

(26)

with the normalization condition

$$\begin{aligned}&P_{1}(t)+P_{2}(t)+P_{3}(t)+P_{4}(t)+P_{5}(t) +P'_{2}(t)+P'_{3}(t)+P''_{2}(t)+P''_{3}(t)\nonumber \\&\quad +P'''_{2}(t)+P'''_{3}(t)+P^{*}_{4}(t)+P^{*}_{5}(t)=1. \end{aligned}$$

(27)

The steady-state solutions $P_{\mu }$ of equations (26) are given by

$$\begin{aligned} P_{1}=\dfrac{1}{X1} \end{aligned}$$

(28)

where

$$\begin{aligned} X1= & {} A \bigg [\dfrac{1}{\omega _{h1}}\bigg (1+\dfrac{\omega _{r2}}{\omega _p}\bigg )+\dfrac{1}{\omega _p}+\dfrac{1}{\omega _{bf}} \bigg (1+\dfrac{\omega _{br}}{\omega _{h2}}\bigg ) +\dfrac{1}{\omega _{h2}}\bigg ] \\&+\, B \bigg [\dfrac{1}{\omega _{h1}'}\bigg (1 +\dfrac{\omega _{r2}'}{\omega _p'}\bigg )+\dfrac{1}{\omega _p'} +\dfrac{1}{\omega _{bf}}\bigg (1+\dfrac{\omega _{br}}{\omega _{h2}}\bigg ) +\dfrac{1}{\omega _{h2}}\bigg ] \\&+\, C \bigg [\dfrac{1}{\omega _{h1}''} \bigg (1+\dfrac{\omega _{r2}''}{\Omega _p}\bigg ) +\dfrac{1}{\Omega _p}+\dfrac{1}{\Omega _{bf}}\bigg (1+\dfrac{\Omega _{br}}{\Omega _{h2}}\bigg )+\dfrac{1}{\Omega _{h2}}\bigg ] \\&+\, D \bigg [\dfrac{1}{\omega _{h1}'''} \bigg (1+\dfrac{\omega _{r2}'''}{\Omega _p'}\bigg ) +\dfrac{1}{\Omega _p'}+\dfrac{1}{\Omega _{bf}}\bigg (1+\dfrac{\Omega _{br}}{\Omega _{h2}}\bigg )+\dfrac{1}{\Omega _{h2}}\bigg ] \end{aligned}$$

$$\begin{aligned} P_{2}= & {} A \bigg [\dfrac{1}{\omega _{h1}} \bigg (1+\dfrac{\omega _{r2}}{\omega _p}\bigg ) \bigg ]P_{1} \end{aligned}$$

(29)

$$\begin{aligned} P_{3}= & {} A \bigg [\dfrac{1}{\omega _{p}}\bigg ] P_{1} \end{aligned}$$

(30)

$$\begin{aligned} P_{4}= & {} (A+B) \bigg [\dfrac{1}{\omega _{bf}} \bigg (1+\dfrac{\omega _{br}}{\omega _{h2}}\bigg )\bigg ]P_{1} \end{aligned}$$

(31)

$$\begin{aligned} P_{5}= & {} (A+B) \bigg [\dfrac{1}{\omega _{h2}}\bigg ] P_{1} \end{aligned}$$

(32)

$$\begin{aligned} P_{2}'= & {} B \bigg [\dfrac{1}{\omega _{h1}'} \bigg (1+\dfrac{\omega _{r2}'}{\omega _p'}\bigg )\bigg ]P_{1}\end{aligned}$$

(33)

$$\begin{aligned} P_{3}'= & {} B \bigg [\dfrac{1}{\omega _{p}'}\bigg ] P_{1}\end{aligned}$$

(34)

$$\begin{aligned} P_{2}''= & {} C \bigg [\dfrac{1}{\omega _{h1}''} \bigg (1+\dfrac{\omega _{r2}''}{\Omega _p}\bigg )\bigg ] P_{1}\end{aligned}$$

(35)

$$\begin{aligned} P_{3}''= & {} C \bigg [\dfrac{1}{\Omega _{p}}\bigg ] P_{1} \end{aligned}$$

(36)

$$\begin{aligned} P_{2}'''= & {} D \bigg [\dfrac{1}{\omega _{h1}'''} \bigg (1+\dfrac{\omega _{r2}'''}{\Omega _p'}\bigg )\bigg ] P_{1}\end{aligned}$$

(37)

$$\begin{aligned} P_{3}'''= & {} D \bigg [\dfrac{1}{\Omega _{p}'}\bigg ] P_{1}\end{aligned}$$

(38)

$$\begin{aligned} P^{*}_{4}= & {} (C+D) \bigg [\dfrac{1}{\Omega _{bf}} \bigg (1+\dfrac{\Omega _{br}}{\Omega _{h2}}\bigg )\bigg ] P_{1}\end{aligned}$$

(39)

$$\begin{aligned} P^{*}_{5}= & {} (C+D) \bigg [\dfrac{1}{\Omega _{h2}}\bigg ] P_{1} \end{aligned}$$

(40)

Appendix B: Intuitive Derivation of the Expression for Average Rate of Translation

Following Cleland’s approach Cleland (1975) for replacing complex network of biochemical pathways by a simpler equivalent network and deriving the effective rates of the transitions of that network, we derive the expression for the average velocity of the ribosome in our model. To illustrate the method, we consider a simpler reaction

$$\begin{aligned} X \mathop {\rightleftharpoons }^{k_{1}}_{k_{-1}} Y \mathop {\rightarrow }^{k_{2}} Z \end{aligned}$$

The effective rate constant, $k_{1}'$, for $X \xrightarrow {k_{1}'} Y,$ is given by

$$\begin{aligned} k_{1}'=\dfrac{k_{1}k_{2}}{k_{-1}+k_{2}} \end{aligned}$$

The same treatment can be applied to our model.

Let us first assume that only correctly charged cognate tRNAs are present (i.e., assuming that mis-charged cognate, near-cognate and non-cognate tRNAs are absent in the surrounding). For the five consecutive steps of the cycle, we denote the transit times by $T_{1}$, $T_{2}$, $T_{3}$, $T_{4}$ and $T_{5}$, respectively. It is straightforward to see that

$$\begin{aligned} T_{1}= & {} \dfrac{1}{\omega _{a}}\bigg (1+\dfrac{\omega _{r1}}{\omega _{h1}}\bigg )\bigg (1+\dfrac{\omega _{r2}}{\omega _p}\bigg ) = \frac{1}{A} \end{aligned}$$

(41)

$$\begin{aligned} T_{2}= & {} \dfrac{1}{\omega _{h1}} \bigg (1+\dfrac{\omega _{r2}}{\omega _p}\bigg )\end{aligned}$$

(42)

$$\begin{aligned} T_{3}= & {} \dfrac{1}{\omega _{p}}\end{aligned}$$

(43)

$$\begin{aligned} T_{4}= & {} \dfrac{1}{\omega _{bf}} \bigg (1+\dfrac{\omega _{br}}{\omega _{h2}}\bigg )\end{aligned}$$

(44)

$$\begin{aligned} T_{5}= & {} \dfrac{1}{\omega _{h2}} \end{aligned}$$

(45)

Therefore, for the above cycle, i.e., when only correctly charged cognate tRNA molecules are present in the surrounding, the average velocity of the ribosome would be

$$\begin{aligned} \dfrac{1}{V_{c1}}= & {} T_{c1}= T_{1}+T_{2}+T_{3}+T_{4}+T_{5} \end{aligned}$$

(46)

Similarly, for the other three cycles we can specify the transit times in a similar manner. For mis-charged cognate tRNA, the corresponding symbols are $T'_{1},T'_{2},T'_{3},T'_{4},T'_{5}$, respectively, while for near-cognate tRNA, ths symbols are $T''_{1},T''_{2},T''_{3},T''_{4},T''_{5}$, respectively. For non-cognate tRNA, we have $T'''_{1},T'''_{2},T'''_{3},T'''_{4},T'''_{5}$, respectively.

Next, let us consider the general case when all the four different substrates are present in the surrounding; the kinetics of the system is described by full model shown in Fig. 3. The transit times are analogous to resistances in electrical circuits, which means that for a series of reaction, the transit times are additive and for parallel reaction pathways, the reciprocals of the transit times are additive. Hence, the average velocity for the complete model is

$$\begin{aligned} \dfrac{1}{V_\mathrm{tot}}= & {} T_\mathrm{tot}=\dfrac{A}{A+B+C+D} \Bigg (T_{1,\mathrm{eff}}+T_{2}+T_{3}+T_{4}+T_{5} \Bigg )\end{aligned}$$

(47)

$$\begin{aligned}&+\dfrac{B}{A+B+C+D}\Bigg (T_{1,\mathrm{eff}}+T'_{2}+T'_{3}+T'_{4}+T'_{5} \Bigg )\end{aligned}$$

(48)

$$\begin{aligned}&+ \dfrac{C}{A+B+C+D}\Bigg (T_{1,\mathrm{eff}}+T''_{2}+T''_{3}+T''_{4}+T''_{5} \Bigg )\end{aligned}$$

(49)

$$\begin{aligned}&+ \dfrac{D}{A+B+C+D}\Bigg (T_{1,\mathrm{eff}}+T'''_{2}+T'''_{3}+T'''_{4}+T'''_{5} \Bigg ) \end{aligned}$$

(50)

where

$$\begin{aligned} \dfrac{1}{T_{1,\mathrm{eff}}}= \dfrac{1}{T_{1}}+\dfrac{1}{T_{1}'}+\dfrac{1}{T_{1}''}+\dfrac{1}{T_{1}'''} \equiv A+B+C+D \end{aligned}$$

(51)

Appendix C: Master Equations and Derivation of Dwell Time Distribution

The master equations governing the time evolution of the probabilities are identical to those given in Appendix A, except for the following:

$$\begin{aligned} \dfrac{\mathrm{d}P_{1}(t)}{\mathrm{d}t}= & {} -\left( \omega _{a}+\omega ^{\prime }_{a} +\omega ^{\prime \prime }_{a}+\omega ^{\prime \prime \prime }_{a}\right) P_{1}(t)\nonumber \\&+\, \omega _{r1} P_{2}(t) + \omega ^{\prime }_{r1}P^{\prime }_{2}(t) + \omega ^{\prime \prime }_{r1}P^{\prime \prime }_{2}(t) \nonumber \\&+\, \omega ^{\prime \prime \prime }_{r1}P^{\prime \prime \prime }_{2}(t) + \omega _{r2}P_{3}(t) + \omega ^{\prime }_{r2}P^{\prime }_{3}(t) \nonumber \\&+\, \omega ^{\prime \prime }_{r2}P^{\prime \prime }_{3}(t) + \omega ^{\prime \prime \prime }_{r2}P^{\prime \prime \prime }_{3}(t) \nonumber \\ \dfrac{\mathrm{d}\widetilde{P_{1}}(t)}{\mathrm{d}t}= & {} \omega _{h2}P_{5}(t) + \Omega _{h2}P^{*}_{5}(t) \end{aligned}$$

(52)

and the normalization condition becomes

$$\begin{aligned}&P_{1}(t)+P_{2}(t)+P_{3}(t)+P_{4}(t)+P_{5}(t) +P'_{2}(t)+P'_{3}(t)+P''_{2}(t)+P''_{3}(t)\nonumber \\&\quad +P'''_{2}(t)+P'''_{3}(t)+P^{*}_{4}(t) +P^{*}_{5}(t)+\widetilde{P_{1}}(t)=1. \end{aligned}$$

(53)

Here, we take the initial conditions to be $P_{1}(0)=1$, and $P_{2}(0)=P_{3}(0)=P_{4}(0)=P_{5}(0)=P'_{2}(0) =P'_{3}(0)=P''_{2}(0)=P''_{3}(0)=P'''_{2}(0) =P'''_{3}(0)=P^{*}_{4}(0)=P^{*}_{5}(0)= \widetilde{P_{1}}(0)=0$.