Error-Speed Correlations in Biopolymer Synthesis

Synthesis of biopolymers such as DNA, RNA, and proteins are biophysical processes aided by enzymes. The performance of these enzymes is usually characterized in terms of their average error rate and speed. However, because of thermal fluctuations in these single-molecule processes, both error and speed are inherently stochastic quantities. In this Letter, we study fluctuations of error and speed in biopolymer synthesis and show that they are in general correlated. This means that, under equal conditions, polymers that are synthesized faster due to a fluctuation tend to have either better or worse errors than the average. The error-correction mechanism implemented by the enzyme determines which of the two cases holds. For example, discrimination in the forward reaction rates tends to grant smaller errors to polymers with faster synthesis. The opposite occurs for discrimination in monomer rejection rates. Our results provide an experimentally feasible way to identify error-correction mechanisms by measuring the error-speed correlations.

Figure 1

(a) An enzyme reads an existing heteropolymer as a template and sequentially incorporates monomers to copy it. Each incorporated monomer can either be a right ($r$) or wrong ($w$) match with the template polymer. (b) Because of thermal fluctuations, the polymer length $L$ and error $\eta$ are random quantities at fixed completion time $T$. (c) When an enzyme produces a copy polymer with fixed length, the error $\eta$ and the time $T$ fluctuate. Scatterplots in (b) and (c) represent $N=5000$ realizations of the same polymerization process where incorporation occurs via two sequential irreversible reactions; see Supplemental Material [16]. Data skewness indicates correlations in the observables.

Figure 2

Reaction networks for polymer synthesis. (a) Hopfield model. The kinetic rates satisfy the relations $k_r=k\exp [\mathrm{\Delta }{E}_r/k_{B}T]$, $k_w=k\exp [\mathrm{\Delta }{E}_w/k_{B}T]$, $k_p^r=m\exp [\mathrm{\Delta }{E}_r/k_{B}T]$ and $k_p^w=m\exp [\mathrm{\Delta }{E}_w/k_{B}T]$ with $k\gg 1$, $m=1$, and $n\ll 1$, so that the model operates in the proofreading regime [25]. (b) Protein translation model from Ref. [24] with rates extracted from Ref. [10]. Same line thickness marks reaction rates of the same order of magnitude.

Figure 3

The Hopfield model and the protein translation model have opposite error-speed correlations. (a) Hopfield model. The gray shaded region defines the allowed values of $r_{\eta L}$ for a given error probability ${\eta }_0$, see Eq (11). Black crosses are estimates of ${\eta }_0$ and $r_{\eta L}$ for 60 random sets of reaction rates in the proofreading regime (see caption of Fig. 2, Supplemental Material [16], and Table S2). (b) Protein translation. To test whether Eq. (13) (gray line) is a good approximation of simulated values of the error-speed coefficient, we computed $r_{\eta L}$ corresponding to the kinetic rates in Ref. [24] for wild type E. coli, a hypercorrective and an error-prone mutation (blue crosses). We also evaluated $r_{\eta L}$ for randomly generated sets of the reaction rates in Fig. 2 (black squares). For all points in both panels, correlation coefficients are evaluated by means of Eqs. (7)–(8) upon computing moments of incorporation times with first passage time techniques [24]. See the Supplemental Material [16] for details of numerical calculations.

Figure 4

Proofreading suppresses the error-speed correlations. (a) Incorporation with a core network complemented by proofreading reactions. Each block in the figure represents an arbitrary subnetwork with an average flux in the direction of the arrows. (b) Comparison of Eq. (14) (solid line) with computation of error-speed correlations from measured kinetic rates, see Ref. [16]. We considered the ribosomes in three strains of E. coli: wild type, hypercorrective, and error prone [10, 24]. For each strain, we built the core network by removing the proofreading reactions and computed the relative change in $r_{\eta L}$ and ${\eta }_0$ between the original and core networks. We performed the same analysis for a model of T7 DNA polymerase (blue square, see Ref. [16] and [12]). The data qualitatively agree with Eq. (14).