Protocols for Copying and Proofreading in Template-Assisted Polymerization

Abstract

We discuss how information encoded in a template polymer can be stochastically copied into a copy polymer. We consider four different stochastic copy protocols of increasing complexity, inspired by building blocks of the mRNA translation pathway. In the first protocol, monomer incorporation occurs in a single stochastic transition. We then move to a more elaborate protocol in which an intermediate step can be used for error correction. Finally, we discuss the operating regimes of two kinetic proofreading protocols: one in which proofreading acts from the final copying step, and one in which it acts from an intermediate step. We review known results for these models and, in some cases, extend them to analyze all possible combinations of energetic and kinetic discrimination. We show that, in each of these protocols, only a limited number of these combinations leads to an improvement of the overall copying accuracy.

Appendix: Exact Expressions

In this Appendix, we report the exact expression for the solutions of the proofreading models, that have been omitted or simplified in the Results section for ease of reading.

1.1 Template-Assisted Polymerization Without Intermediate States

The exact equation for the error is in the Results section. The copying speed is equal to

$$\begin{aligned} v=k_{10}^w+k_{10}^r-k_{01}^w\eta -k_{01}^r(1-\eta ). \end{aligned}$$

(17)

1.2 Double-Step Copying

Also in this case, the exact equation for the error is in the Results section. The expression for the speed is

$$\begin{aligned} v=\mathcal {N}[p_1^w k_{21}^w+p_1^r k_{21}^r-k_{12}^w\eta -k_{12}^r(1-\eta ) ] \end{aligned}$$

(18)

where we introduced the normalization factor for the occupancies $\mathcal {N}$, which in this case is equal to $\mathcal {N}=(1+p_1^w+p_1^r)^{-1}$.

1.3 Kinetic Proofreading

The error rate of the proofreading model satisfies the equation

$$\begin{aligned} \frac{\eta }{1-\eta }&=\frac{\left[ e^{\frac{\delta _{10}}{T}}+re^{\frac{\delta _{21}+\mu _{21}}{T}}\right] }{\left( 1+r e^{\frac{\mu _{21}}{T}}\right) } \\&\quad \times \, \frac{r e^{\frac{\mu _{10}+\mu _{21}}{T}}+r_p\left( 1+r e^{\frac{\mu _{21}}{T}}\right) -\eta e^{\frac{\Delta E_2^w}{T}} \left[ r +(1+r) r_p e^{\frac{\mu _{02}}{T}}\right] }{r e^{\frac{\mu _{10}+\mu _{21}+\delta _{21}+\delta _{10}}{T}}+r_p e^{\frac{\delta _{20}}{T}}\left( e^{\frac{\delta _{10}}{T}}+r e^{\frac{\mu _{21}+\delta _{21}}{T}}\right) -(1-\eta )e^{\frac{\Delta E_2^r}{T}}A} \nonumber \end{aligned}$$

(19)

where $r_p=\omega _{02}/\omega _{10}$ and $A=\left( r e^{\frac{\delta _{10}+\delta _{21}}{T}} +r_p e^{\frac{\delta _{10}+\delta _{02}+\mu _{02}}{T}} +r r_p e^{\frac{\delta _{21}+\delta _{02}+\mu _{02}}{T}} \right)$.

The copying speed is equal to

$$\begin{aligned} v=\mathcal {N}[p_1^w k_{21}^w+k_{20}^w + p_1^r k_{21}^r+k_{20}^r-(k_{12}^w+k_{02}^w)\eta -(k_{12}^r+k_{02}^w)(1-\eta )] \end{aligned}$$

(20)

where $\mathcal {N}$ is defined as previously.

1.4 Proofreading/Accommodation

The error rate satisfies

$$\begin{aligned} \frac{\eta }{1-\eta }&=e^{-\frac{\delta _{21}}{T}}\frac{\left[ e^{\frac{\delta _{10}}{T}}+\bar{r}e^{\frac{\bar{\delta }_{10}}{T}}+re^{\frac{\delta _{21}+\mu _{21}}{T}}\right] }{\left( 1+\bar{r}+r e^{\frac{\mu _{21}}{T}}\right) } \nonumber \\&\quad \times \, \frac{e^{\frac{\mu _{21}}{T}}\left( e^{\frac{\mu _{10}}{T}}+\bar{r}\right) -\eta e^{\frac{\Delta E_2^w}{T}}\left( 1+\bar{r}e^{\frac{\bar{\mu }_{10}}{T}}\right) }{e^{\frac{\mu _{21}}{T}}\left( e^{\frac{\mu _{10}+\delta _{10}}{T}}+\bar{r}e^{\frac{\bar{\delta }_{10}}{T}}\right) -(1-\eta ) e^{\frac{\Delta E_2^r}{T}} \left( e^{\frac{\delta _{10}}{T}}+\bar{r}e^{\frac{\bar{\mu }_{10}+\bar{\delta }_{10}}{T}}\right) } \end{aligned}$$

(21)

and the copying speed is

$$\begin{aligned} v=\mathcal {N}[p_1^w k_{21}^w+p_1^r k_{21}^r-k_{12}^w\eta -k_{12}^r(1-\eta )]. \end{aligned}$$

(22)