Error-suppression mechanism of PCR by blocker strands

Abstract

The polymerase chain reaction (PCR) is a central technique in biotechnology. Its ability to amplify a specific target region of a DNA sequence has led to prominent applications, including virus tests, DNA sequencing, genotyping, and genome cloning. These applications rely on the specificity of the primer hybridization and therefore require effective suppression of hybridization errors. A simple and effective method to achieve that is to add blocker strands, also called clamps, to the PCR mixture. These strands bind to the unwanted target sequence, thereby blocking the primer mishybridization. Because of its simplicity, this method is applicable to a broad nucleic-acid-based biotechnology. However, the precise mechanism by which blocker strands suppress PCR errors remains to be understood, limiting the applicability of this technique. Here, we combine experiments and theoretical modeling to reveal this mechanism. We find that the blocker strands both energetically destabilize the mishybridized complex and sculpt a kinetic barrier to suppress mishybridization. This combination of energetic and kinetic biasing extends the viable range of annealing temperatures, which reduces design constraint of the primer sequence and extends the applicability of PCR.

Significance

Applications of the PCR are growing by day, and there is a demand to suppress PCR errors to enhance its fidelity. However, the conventional approach that targets the stability of the correct priming has fundamental limitations. Alternative approaches based on a blocker method are often used to suppress mispriming. However, applications of these approaches are limited due to the unclear error suppression mechanism. Our experiments and theory successfully demonstrate that the addition of blocker strands suppresses errors by combining kinetic and energetic biasing, which enables circumventing the limitations of the conventional method. Our quantitative modeling also allows for optimizing the blocker sequence and protocol. Since the method targets hybridization, it would be readily applied to a wide range of biotechnologies.

Introduction

The polymerase chain reaction (PCR) is used in a broad, ever-expanding range of biotechnological applications (1). Fidelity of PCR is determined by the specificity of the primer hybridization. In applications, mishybridization leads to unwanted consequences, such as false positives in virus tests and sequencing errors. Given the importance and widespread nature of these applications, methods for suppressing hybridization errors are crucial. A simple and effective method to suppress mishybridization is to add blocker strands, also called clamps, to the reaction mixture (2,3,4,5). The blockers bind to the target mutants and suppress mishybridization to the mutant sequence. The hybridization of nucleic acids is essential in many biotechnologies such as genome editing (6), DNA nanotechnologies (7), and RNA interference (8). The blocker method may be applicable to these systems.

However, despite its effectiveness, the detailed mechanism by which blockers suppress errors remains unclear. This limits the rational design of the blocker sequence (4). One hypothesis is that the blocker strands energetically destabilize the mishybridized state. An alternative hypothesis is that the blocker strands effectively create a kinetic barrier to kinetically suppress the mishybridization. Discerning between these hypotheses would improve our understanding of how PCR works and suggest design principles for optimized protocols. In this article, we combine experiments and theory to show that PCR blockers suppress errors in PCR by a combination of energetic and kinetic discriminations. A nontrivial consequence of our theory is that, in the presence of blockers, the range of viable annealing temperatures for the PCR reactions is considerably broadened.

To illustrate the factors that determine the hybridization error, we consider the example of a PCR mixture containing a right template $\mathrm{R}$ and a contaminated wrong template $\mathrm{W}$ with a mutation in the primer-binding region (Fig. 1 a). During a PCR cycle, the temperature is lowered from a high denaturing temperature to the annealing temperature $T_{\mathrm{a}}$. Then, a primer strand $\mathrm{P}$ hybridizes to either $\mathrm{R}$ or $\mathrm{W}$. Since the primer hybridization is reversible, $\mathrm{P}$ repeatedly hybridizes to or dissociates from the template. Eventually, a polymerase binds to the hybridized complex $\mathrm{P}:\mathrm{R}$ or $\mathrm{P}:\mathrm{W}$ and elongates $\mathrm{P}$ to produce a complementary copy $\overline{\mathrm{R}}$ or $\overline{\mathrm{W}}$, respectively. Important quantities characterizing the PCR are the growth rates ${\alpha }_{\mathrm{R}}$ and ${\alpha }_{\mathrm{W}}$ and error rate η, defined by

$${\alpha }_{\mathrm{R}}=\frac{r}{\left[\mathrm{R}\right]},{\alpha }_{\mathrm{W}}=\frac{w}{\left[\mathrm{W}\right]},\eta =\frac{{\alpha }_{\mathrm{W}}}{{\alpha }_{\mathrm{R}}+{\alpha }_{\mathrm{W}}}\text{.}$$ (Equation 1)

Figure 1.

Energetic versus kinetic biasing in PCR. (a) Standard PCR scheme. P, R, and W are the primer, right template, and wrong template with a mismatch in the primer-binding region, respectively. (b) Energy landscape corresponding to the primer hybridization in PCR. Here, $\overline{\mathrm{R}}$ and $\overline{\mathrm{W}}$ denote the right and wrong products, respectively. (c) Energy landscape for energetic (left) versus kinetic (right) biasing. At an early time, $\left[\mathrm{P}:\mathrm{R}\right]$ and $\left[\mathrm{P}:\mathrm{W}\right]$ are similar because of the similar barrier height for the hybridization. On the bottom, the colored bars correspond to the fractions of $\left[\mathrm{P}:\mathrm{R}\right]$ (orange), $\left[\mathrm{P}:\mathrm{W}\right]$ (blue), and the free templates (gray). At thermodynamic equilibrium, the amount ratio of the wrong hybridization to the right one is $e^{-\Delta G/k_{\mathrm{B}}{T}_{\mathrm{a}}}$, where $\Delta G$ is the energetic bias and $T_{\mathrm{a}}$ is the annealing temperature. In the figure, $\Delta G^{‡}$ is the kinetic bias, and τ is the relaxation time of the binding dynamics. To see this figure in color, go online.

Here, $[·]$ denotes a concentration and r and w are the increases in concentrations of $\overline{\mathrm{R}}$ and $\overline{\mathrm{W}}$ in a cycle, respectively. Hence, ${\alpha }_{\mathrm{R}}$ and ${\alpha }_{\mathrm{W}}$ represent the fractions of copied strands per template in a cycle. Ideally, one wants to maximize the growth rate ${\alpha }_{\mathrm{R}}$, also called the PCR efficiency, and at the same time minimize the error rate η.

The accuracy of conventional PCR relies on primer hybridization to $\mathrm{R}$ being energetically more stable than hybridization to $\mathrm{W}$, as quantified by the free-energy difference $\Delta G$ between $\mathrm{P}:\mathrm{R}$ and $\mathrm{P}:\mathrm{W}$ (Fig. 1 b). This energetic bias can be increased to reduce η by carefully designing the primer sequence and increasing the annealing temperature $T_{\mathrm{a}}$ (1).

However, this approach has an inherent limitation in that fidelity and efficiency are subject to a trade-off. To see that, we consider the hybridization kinetics (Fig. 1 b). The DNA binding rate is usually diffusion limited and thus does not significantly depend on the sequence (9,10). Hence, we assume that $\mathrm{P}$ hybridizes to $\mathrm{R}$ and $\mathrm{W}$ at the same rate. On the other hand, the dissociation rates depend on $\Delta G$. Repeated hybridization and dissociation of $\mathrm{P}$ eventually bring the system to thermodynamic equilibrium, where the amount ratio of the wrong hybridization to the right one is $e^{-\Delta G/k_{\mathrm{B}}{T}_{\mathrm{a}}}$. Here, $k_{\mathrm{B}}$ is the Boltzmann constant. Thus, the error rate in this equilibrium limit is equal to

$${\eta }_{\text{eq}}=\frac{1}{1+\left(k_{\text{pol}}^{\mathrm{R}}/k_{\text{pol}}^{\mathrm{W}}\right)e^{\Delta G/k_{\mathrm{B}}{T}_{\mathrm{a}}}}$$ (Equation 2)

(supporting material section S4). $k_{\text{pol}}^{\mathrm{R}\left(\mathrm{W}\right)}$ is the elongation rate on R and W by the polymerase. In this case, one can show that the short time error rate is always larger than ${\eta }_{\text{eq}}$ (11) (see Fig. 1 c). In fact, one problem with this approach is that the enzymatic reaction is usually quite efficient and starts elongation before the binding equilibrates. This means that the error rate is usually not as small as one would expect from Eq. 2. Slowing down of the reaction by, for example, reducing the polymerase concentration would lower the error rate by allowing sufficient time for equilibration but at the cost of efficiency. This result thus demonstrates the aforementioned trade-off between fidelity and efficiency.

An alternative strategy is to sculpt a kinetic bias by building asymmetric barriers characterized by a difference $\Delta G^{‡}$ so that $\mathrm{P}$ preferably binds to $\mathrm{R}$ (Fig. 1 c, right). Theory predicts that such kinetic bias can reduce η without sacrificing efficiency (11).

Brief methods

To quantify the error rate, we perform PCR with only a single side of the primer set (Fig. S1). Hence, the product concentration increases linearly rather than exponentially, as in standard PCR (Fig. S2). We mix a primer strand $\mathrm{P}$, two variants of 72-nt template DNA strands $\mathrm{R}$ and $\mathrm{W}$, indicated concentrations of thermostable DNA polymerase, and necessary chemicals for the reactions. We also add blocker strands depending on the experiments. The $\mathrm{R}$ and $\mathrm{W}$ templates are mixed at the same concentrations ($\left[\mathrm{R}\right]=\left[\mathrm{W}\right]=2.5\text{nM}$), much smaller than the primer concentration ($\left[\mathrm{P}\right]=100\text{nM}$). The $\mathrm{P}$ strand binds to $\mathrm{R}$ without mismatches and to $\mathrm{W}$ with a single-base mismatch. In these conditions, the hybridization error is expected to be large. After hybridization, polymerases copy the template and produce $\overline{\mathrm{R}}$ or $\overline{\mathrm{W}}$. We repeat 10 or 40 thermal cycles to reduce statistical errors, measure r and w, and calculate the error rate and the efficiency by means of Eq. 1.

The blocker strands ${\mathrm{B}}_{\mathrm{R}}$ and ${\mathrm{B}}_{\mathrm{W}}$ are 16-nt chimeric strands of DNA and locked nucleic acid (LNA) bases. They hybridize to the primer-binding region of $\mathrm{R}$ and $\mathrm{W}$. The ${\mathrm{B}}_{\mathrm{R}\left(\mathrm{W}\right)}$ strand hybridizes to $\mathrm{R}\left(\mathrm{W}\right)$ without mismatches and to $\mathrm{W}\left(\mathrm{R}\right)$ with a single mismatch. Two bases at the 3′ end of the blocker strands are floating to prevent them from acting as primers. Blocker hybridization to the template is faster and more stable than primer binding to the template because of their high concentration ($\left[{\mathrm{B}}_{\mathrm{R}\left(\mathrm{W}\right)}\right]=20\left[\mathrm{P}\right]=2000\text{nM}$) and the four LNA bases placed in the vicinity of the mismatch position, which significantly increases hybridization specificity (12,13).

Materials and methods

Linear PCR experiment

DNA strands were synthesized by Eurofins Genomics (Louisville, KY, USA) and Integrated DNA Technologies (Coralville, IA, USA) (see supporting material section S1). DNA/LNA chimeric strands were synthesized by Aji Bio-Pharma (San Diego, CA, USA). The reaction mixture for polymerization contained Hot-start Taq DNA polymerase (New England Biolabs, Ipswich, MA, USA), Taq standard reaction buffer, and R, W, P, and blocker strands. We performed initial heating for 30 s at 95°C and then 10 or 40 cycles of 15 s at 95°C, 30 s at 60°C, and 5 s at 68°C using a PCR cycler. Immediately after the cycles, the mixture was cooled down on the ice to stop the enzyme reaction and used for the quantification. The number of cycles was 40 when the polymerase concentration was 0.17, 0.25, or 0.5 units/mL and 10 otherwise.

Quantification of ${\mathrm{P}}_{\mathrm{R}}$ and ${\mathrm{P}}_{\mathrm{W}}$

Additional quantitative PCR was performed on a real-time PCR cycler after the linear PCR experiment for quantifying r and w (see supporting material sections S1 and S2). The reaction mixture contains Luna Universal qPCR Master Mix (New England Biolabs), 200 nM each of the primers, and the diluted sample. The dilution rate is 1/250 in the final concentration. The thermal cycle consists of initial heating for 60 s at 95°C, 40 cycles of 15 s at 95°C, 30 s at 66°C, and 5 s at 72°C.

Melt curve analysis

We measured the melting curves for the hybridization of $\mathrm{P}$, ${\mathrm{B}}_{\mathrm{R}}$, and ${\mathrm{B}}_{\mathrm{W}}$ to $\mathrm{R}$ and $\mathrm{W}$ from 95°C to 20°C and calculated their $K_{\mathrm{d}}$ (supporting material section S3). We mixed 100 nM each of DNA and double-strand-specific fluorescent molecule EvaGreen (Biotium, San Francisco, CA, USA). The fluorescent profile was analyzed based on the exponential background method (14) to obtain the melting curve.

Results

PCR in the absence of blocker strands

We first characterized the performance of conventional PCR by measuring the efficiency ${\alpha }_{\mathrm{R}}$ and error rate η as a function of the annealing temperature $T_{\mathrm{a}}$. In the absence of the blocker strands, ${\alpha }_{\mathrm{R}}$ and ${\alpha }_{\mathrm{W}}$ were large at low $T_{\mathrm{a}}$ (Fig. 2, a and b). The efficiency ${\alpha }_{\mathrm{R}}$ decreased at $T_{\mathrm{a}}$ exceeding the melting temperature of $\mathrm{P}:\mathrm{R}\text{,}$ $T_{\mathrm{m}}^{\mathrm{P}:\mathrm{R}}=62.8^{\circ }\mathrm{C}$. $T_{\mathrm{m}}$ is defined as the temperature where half of the DNA strands form the complex. On the other hand, ${\alpha }_{\mathrm{W}}$ decreased at $T_{\mathrm{a}}>T_{\mathrm{m}}^{\mathrm{P}:\mathrm{W}}=59.3^{\circ }\mathrm{C}$. Accordingly, the error rate η was large at low $T_{\mathrm{a}}$ and decreased when $T_{\mathrm{a}}>T_{\mathrm{m}}^{\mathrm{P}:\mathrm{W}}$ (Fig. 2 c). Our results confirm that, in conventional PCR, $T_{\mathrm{a}}$ needs to be finely tuned in the range $T_{\mathrm{m}}^{\mathrm{P}:\mathrm{W}}<T_{\mathrm{a}}<T_{\mathrm{m}}^{\mathrm{P}:\mathrm{R}}$ for simultaneously achieving high ${\alpha }_{\mathrm{R}}$ and low η.

Figure 2.

Growth rates and error rates of PCR as a function of the annealing temperature. The growth rates of producing $\overline{\mathrm{R}}$ (a) and $\overline{\mathrm{W}}$ (b) as the function of the annealing temperature $T_{\mathrm{a}}$. (c) Error rate η calculated by Eq. 1. The inset is the magnification of $+{\mathrm{B}}_{\mathrm{W}}$. The error bars indicate the standard deviations. Polymerase concentration is 25 units/mL, as in the standard PCR protocol. To see this figure in color, go online.

Error suppression by blocker strands

We next studied the effect of the blocker strands on the efficiency and the error rate. Intuitively, we expect the blockers to affect the PCR dynamics in the following way. The blocker ${\mathrm{B}}_{\mathrm{W}}$ preferably hybridizes to $\mathrm{W}$. As the temperature is lowered to $T_{\mathrm{a}}$ during the thermal cycle, ${\mathrm{B}}_{\mathrm{W}}$ should quickly occupy most $\mathrm{W}$ while binding to a small fraction of $\mathrm{R}$ only. Hence, the hybridization of $\mathrm{P}$ should be significantly biased toward $\mathrm{R}$, thus suppressing the error without sacrificing the speed. On the other hand, the addition of ${\mathrm{B}}_{\mathrm{R}}$ prevents $\mathrm{P}$ from hybridizing to $\mathrm{R}$. Therefore, we expect an increased error rate in this case.

Indeed, the addition of ${\mathrm{B}}_{\mathrm{W}}$ drastically suppressed the errors at all the annealing temperatures we tested (Fig. 2 c) without significantly impairing efficiency, at least for large $T_{\mathrm{a}}$ (Fig. 2 a). At $T_{\mathrm{a}}\simeq T_{\mathrm{m}}^{{\mathrm{B}}_{\mathrm{W}}:\mathrm{R}}=56.3^{\circ }\mathrm{C}$, ${\alpha }_{\mathrm{R}}$ was reduced due to the hybridization of ${\mathrm{B}}_{\mathrm{W}}$ to $\mathrm{R}$. We found that the errors are still effectively suppressed at a $T_{\mathrm{a}}$ much lower than $T_{\mathrm{m}}^{\mathrm{P}:\mathrm{W}}$, meaning that fine-tuning of $T_{\mathrm{a}}$ is not needed in the presence of blockers.

In contrast, ${\mathrm{B}}_{\mathrm{R}}$ drastically reduced ${\alpha }_{\mathrm{R}}$ and increased η. At $T_{\mathrm{a}}<{60}^{\circ }\mathrm{C}$, η was larger than 50%, meaning that ${\mathrm{B}}_{\mathrm{R}}$ inverted the preference of P hybridization. This setup could be used to amplify rare sequences that would be otherwise difficult to sample.

We used chimeric DNA strands containing LNA bases for the blocker strands, which enhance the specificity of hybridization. The blocker strands with only DNA bases had a limited effect when used with the same concentration and strand length as the chimeric strands (Figs. S8 and S9).

Mathematical model

We quantified the PCR kinetics and in particular the role of blockers using a mathematical model (see supporting material section S4 for details). The model includes reversible hybridization rate of $\mathrm{P}$ to the template strands. In contrast, we assume that the blockers are always at chemical equilibrium, as their high concentrations make their hybridization dynamics very fast. For simplicity, polymerization is modeled as a single rate without explicitly including polymerase binding and dissociation.

Introducing blockers creates an effective kinetic bias and at the same time enhances the effective energetic bias between right and wrong targets (Fig. 3 a). These effects are quantified by

$$\begin{array}{l}\Delta G^{‡}\simeq k_{\mathrm{B}}{T}_{\mathrm{a}}\ln \left(1+\frac{\left[{\mathrm{B}}_{\mathrm{W}}\right]}{K_{\mathrm{d}}^{{\mathrm{B}}_{\mathrm{W}}:\mathrm{W}}}\right)\text{,}\\ \Delta G-\Delta G_0\simeq k_{\mathrm{B}}{T}_{\mathrm{a}}\ln \left(1+\frac{1}{1+\left[\mathrm{P}\right]/K_{\mathrm{d}}^{\mathrm{P}:\mathrm{W}}}\frac{\left[{\mathrm{B}}_{\mathrm{W}}\right]}{K_{\mathrm{d}}^{{\mathrm{B}}_{\mathrm{W}}:\mathrm{W}}}\right)\text{.}\end{array}$$ (Equation 3)

Figure 3.

Effective energetic and kinetic bias in the presence of blockers. (a) The free-energy landscape in the presence of ${\mathrm{B}}_{\mathrm{W}}$. (b) Dependence of energetic ($\Delta G,\Delta G_0$) and kinetic ($\Delta G^{‡}$) bias on $T_{\mathrm{a}}$ calculated by the model (supporting material section S4). To see this figure in color, go online.

Here, $\Delta G_0$ is the energetic bias in the absence of the blocker, and $K_{\mathrm{d}}$ is the dissociation constant of the specified hybridization.

We measured the hybridization melting curves and estimated the temperature dependence of $K_{\mathrm{d}}$ (Fig. S4) to evaluate the dependence of $\Delta G$ and $\Delta G^{‡}$ on $T_{\mathrm{a}}$ (Fig. 3 b). We found that $\Delta G\gg \Delta G^{‡}$ at high $T_{\mathrm{a}}$ and $\Delta G^{‡}\gg \Delta G$ at low $T_{\mathrm{a}}$. This result shows that the error suppression by blocker in a broad $T_{\mathrm{a}}$ range stems from the combined effects of the energetic and kinetic biasing pronounced at high and low $T_{\mathrm{a}}$, respectively.

Kinetics of error suppression

To analyze the detailed kinetics of error suppression by the blocker strands, we varied the polymerase concentration by more than two orders of magnitude while fixing $T_{\mathrm{a}}$ to 60°C, which is an appropriate temperature for our primer sequence (Fig. 4). Since elongation by polymerase quenches the hybridization dynamics, a change in the polymerase concentration tunes the timescale available for the hybridization dynamics to relax.

Figure 4.

Improved performance of PCR reaction in the presence of blockers is consistent with model predictions. Dependence of the efficiencies (a and b) and error rate (c) on the polymerase concentration. (d) The error rate is plotted against the growth rate ${\alpha }_{\mathrm{R}}$ (efficiency). Symbols denote the experimental data, and dashed lines correspond to the model fitting. Stars indicate the limiting values at high (open) or low (closed) polymerase concentrations, which correspond to ${\eta }_{\text{eq}}$ given by Eq. 2 and ${\eta }_{\text{fast}}$ given by Eq. 4, respectively. The polymerase concentration of the standard PCR protocol is 25 units/mL. $T_{\mathrm{a}}={60}^{\circ }\mathrm{C}$. We excluded two points with negative averages due to the statistical errors from (b) and (c) ($+{\mathrm{B}}_{\mathrm{W}}$ with 0.17 and 0.25 units/mL polymerase). The error bars indicate the standard deviations. See Fig. S6 for the plots in the linear scale. To see this figure in color, go online.

In the absence of blocker strands, η decreased as the polymerase concentration decreased (Fig. 4 c). In the low polymerase concentration limit, the limiting error rate ${\eta }_{\text{eq}}$, see Eq. 2, was lower in the presence of blockers ${\mathrm{B}}_{\mathrm{W}}$. Since the error is determined by the energetic bias in this limit, this result confirms our prediction that the blocker addition increases $\Delta G$.

The blocker addition lowered η at high polymerase concentration as well (Fig. 4 c), consistent with our prediction that the blockers also create an effective kinetic barrier for the wrong primer strands. The theory approximates the limiting error rate as

$${\eta }_{\text{fast}}\simeq \frac{1}{1+{\left(k_{\text{on}}\left[\mathrm{P}\right]t_{\mathrm{a}}\right)}^{-1}\exp \left(\Delta G^{‡}/k_{\mathrm{B}}{T}_{\mathrm{a}}\right)}$$ (Equation 4)

(supporting material section S4), depending on the barrier difference $\Delta G^{‡}$. Here, $k_{\text{on}}$ is the binding rate of the primer, and $t_{\mathrm{a}}=30\mathrm{s}$ is the duration of the annealing step. These results support that the blocker strands suppress errors both energetically and kinetically, as illustrated in Fig. 3 a.

The model successfully reproduced the experimental results (dashed lines in Fig. 4). The values of the three fitting parameters were comparable with estimates based on previous work (see supporting material section S5). Even without blocker strands, a slight reduction of polymerase concentration is effective at suppressing errors without affecting much the efficiency (black arrow in Fig. 4 d). However, this strategy requires fine-tuning of the polymerase concentration to maintain the efficiency. On the other hand, the addition of blocker strands is more effective at reducing errors and without reducing the efficiency (blue arrow in Fig. 4 d).

Kinetic regime

Our theoretical model suggests the presence of a novel error correction regime. This fact is related with the relative magnitude of $\Delta G$ and $\Delta G^{‡}$, which changes at a temperature $T_a\approx {58}^{\circ }\mathrm{C}$, where the $\Delta G$ and $\Delta G^{‡}$ curves intersect each other in Fig. 3 b. According to a theory (11), the relative magnitude of these two quantities determines the behavior of the primer-binding dynamics.

To illustrate this theoretical prediction, we ran a simulation of our model but without the elongation by polymerase. On the one hand, in the “energetic regime” characterized by $\Delta G>\Delta G^{‡}$ (60°C), the fraction $\gamma \equiv \left[\mathrm{P},:,\mathrm{W}\right]/\left(\left[\mathrm{P},:,\mathrm{R}\right],+,\left[\mathrm{P},:,\mathrm{W}\right]\right)$ decreases with the annealing time $t_{\mathrm{a}}$ and converges to an equilibrium value ${\gamma }_{\text{eq}}=1/\left(1+e^{\Delta G/k_{\mathrm{B}}{T}_{\mathrm{a}}}\right)$ (Fig. 5 a). On the other hand, in the “kinetic regime” characterized by $\Delta G<\Delta G^{‡}$ (57.5°C), the kinetic barrier blocks the primer binding to the wrong template (Fig. 1 c), and $\gamma <{\gamma }_{\text{eq}}$ is observed. At the limit of vanishing $t_{\mathrm{a}}$, γ converges to ${\gamma }_{\text{short}}=1/\left(1+e^{\Delta G^{‡}/k_{\mathrm{B}}{T}_{\mathrm{a}}}\right)$ (supporting material section S4).

Figure 5.

Dynamics of binding and error calculated by simulation. (a) Binding dynamics in the absence of polymerase. γ is the fraction of the complex of the primer and wrong template. ${\gamma }_{\text{eq}}$ is its equilibrium value. See the main text. ${\gamma }_{\text{eq}}=0.038$ for 60°C and 0.056 for 57.5°C. (b) Error rate dynamics in the presence of polymerase normalized by ${\eta }_{\text{eq}}$ given by Eq. 2. The polymerase concentration is 0.1 (dashed) or 25 units/mL (dotted). ${\eta }_{\text{eq}}=0.0032$ for 60°C and 0.0045 for 57.5°C. (c) The growth rates of the right product. For the simulation, the dissociation constants are obtained experimentally. $k_{\text{on}}$, $k_{\text{pol}}^{\mathrm{R}}$, and $k_{\text{pol}}^{\mathrm{W}}$ are obtained by the global fitting of the experimental results at each temperature. See supporting material section S5 for details. To see this figure in color, go online.

These binding dynamics imply that, in the presence of the polymerase, η may increase with $t_{\mathrm{a}}$ and be smaller than ${\eta }_{\text{eq}}$ in the kinetic regime. That is, we may reduce errors and cycle time simultaneously and also achieve significant error suppression. This prediction was confirmed in the case of dilute polymerase (0.1 units/mL) and relatively short $t_{\mathrm{a}}$ (Fig. 5 b). However, at $t_{\mathrm{a}}$ larger than $\sim 100\mathrm{s}$, η increased with $t_{\mathrm{a}}$ in both regimes. This increase is pronounced with a typical polymerase concentration for PCR (25 units/mL). The increase in η at large $t_{\mathrm{a}}$ is caused by the saturation of the growth; ${\alpha }_{\mathrm{W}}$ increases with $t_{\mathrm{a}}$, while ${\alpha }_{\mathrm{R}}$ is already saturated (Fig. 5 c). Nevertheless, we still obtained $\eta <{\eta }_{\text{eq}}$ at small $t_{\mathrm{a}}$. At the limit of vanishing $t_{\mathrm{a}}$, η converges to

$${\eta }_{\text{short}}=\frac{1}{1+\left(k_{\text{pol}}^{\mathrm{R}}/k_{\text{pol}}^{\mathrm{W}}\right)e^{\Delta G^{‡}/k_{\mathrm{B}}{T}_{\mathrm{a}}}}\text{,}$$ (Equation 5)

which has the same mathematical form as ${\eta }_{\text{eq}}$ given by Eq. 2 but with $\Delta G^{‡}$ appearing instead of $\Delta G$ (supporting material section S4).

It remains unclear whether the regime $\eta <{\eta }_{\text{eq}}$ is experimentally accessible. The expected timescale $t_{\mathrm{a}}\sim 1\mathrm{s}$ is significantly shorter than the typical timescale ($\sim 10\mathrm{s}$) necessary for cooling from $\sim {95}^{\circ }\mathrm{C}$ at the denaturing step to $\sim {60}^{\circ }\mathrm{C}$ at the annealing step. Our model assumes an instantaneous temperature jump for simplicity and may no longer be applicable for such a short $t_{\mathrm{a}}$.

The short $t_{\mathrm{a}}$ results in a significant reduction of the product amount (Fig. 5 c). However, the approach implementing the kinetic regime would be beneficial in applications such as genome editing, where error suppression is critical.

Multiple wrong sequences

In real-world applications of PCR, samples may contain multiple types of unwanted sequences. We study by numerical simulations of our model whether blockers could suppress replication errors in this case. For simplicity, we focus on a scenario in which the sample contains N types of wrong sequences, and we add N blocker sequences, each of which perfectly hybridizes to the corresponding wrong sequence. We fix the total concentration of the wrong sequences and the blocker sequences so that the concentration of each wrong sequence and blocker sequence is proportional to $1/N$. We note that the concentration of ${\mathrm{B}}_{\mathrm{W}}:\mathrm{W}$ is roughly proportional to $\left[\mathrm{W}\right]\left[{\mathrm{B}}_{\mathrm{W}}\right]$. Since $\left[\mathrm{W}\right]$ and $\left[{\mathrm{B}}_{\mathrm{W}}\right]$ decrease with N, blocking may become less effective with N. The error rate η is defined similarly to Eq. 1 but where ${\alpha }_{\mathrm{W}}$ is the total amount of wrong products (see supporting material section S4). We find that although the error increases with N, the blocker strands suppress η even in the large N limit (Fig. 6). Moreover, our model predicts that the addition of blockers should not significantly affect ${\alpha }_{\mathrm{R}}$ unless they strongly hybridize to $\mathrm{R}$.

Figure 6.

Error rate in the presence of multiple error sequences and blockers predicted by numerical simulation. The blue dotted line and black dashed line correspond to the result for $N\to \infty$ and without blocker strands, respectively. To see this figure in color, go online.

Discussion

Kinetic modeling of PCR reaction has contributed to quantitatively characterize the reaction performance (15,16,17) and other aspects such as amplification heterogeneity (18). However, modeling has been scarcely used to develop new guiding principles. The physics of information processing can provide such principles thanks to its progress in characterizing general biochemical reactions (11,19,20,21,22,23,24,25,26,27,28,29,30,31).

Yang et al. (4) empirically derived the design strategy of the blocker sequence, providing criteria for the optimal blocker length, number of LNA nucleotides, and relations among $T_{\mathrm{m}}$ of blocker, primer, and right and wrong templates. Such empirical criteria are helpful in the practice of the blocker method. However, their applicability is limited to particular conditions. The theoretical modeling proposed here provides a quantitative ground for these criteria and also extends the range of applicable conditions. For example, Eq. 3 reveals how the concentrations and $K_{\mathrm{d}}$ of the blocker affect the blocking efficacy as a function of temperature. Moreover, our model can be used to predict the efficacy of a designed blocker sequence.

DNA blocker strands were less effective in error-rate control than the LNA-DNA chimeric blocker strands at the same concentration (Figs. S8 and S9). The free-energy profile obtained from the model and experimentally measured thermodynamic parameters (Table S4) implies that the energetic biasing $\Delta G$ only slightly changes with the DNA blocker (Fig. S10). Thus, the blocking is not effective at high $T_{\mathrm{a}}$. On the other hand, we expect a substantial blocking at low $T_{\mathrm{a}}$ because of the significant kinetic biasing $\Delta G^{‡}$ at low $T_{\mathrm{a}}$. In addition, the DNA blocker could be effective even at higher temperatures by increasing $\Delta G$ and $\Delta G^{‡}$, which is possible by increasing the blocker concentration or decreasing $K_{\mathrm{d}}^{{\mathrm{B}}_{\mathrm{W}}:\mathrm{W}}$ as indicated by Eq. 3. One can decrease $K_{\mathrm{d}}^{{\mathrm{B}}_{\mathrm{W}}:\mathrm{W}}$ by using longer, and therefore more stably binding, blocker sequences.

We demonstrated that adding blocker strands discriminates the right and wrong sequences by combining energetic and kinetic biasing. The kinetic biasing is effective in decreasing the error rate without affecting the efficiency. An alternative setup we studied is the use of blocker strands targeting the right template. In this case, we could increase the error rate up to a value larger than 80%. This inverted error control cannot be achieved without kinetic biasing and may be helpful for sampling rare sequences. Previous studies suggested that some synthetic and biological molecular machines use kinetic biasing to control their reaction directionality (32,33). Our work suggests that kinetic biasing can also be used to discriminate errors in biotechnological applications.

Importantly, we found that error suppression is still effective at $T_{\mathrm{a}}$ much lower than $T_{\mathrm{m}}$ of the primer binding. This implies that we can suppress the hybridization errors in systems with limited temperature controllability, such as the reverse transcription, in which the optimal reaction temperature is not high, and the hybridization inside biological cells. This might also lead to a reduced cost of applications such as virus tests by using a low-cost cycler since the method we described does not require accurate temperature control.

Author contributions

H.A., S.P., and S.T. designed the research, developed the theory, and wrote the paper. H.A. and S.O. performed the experiments.

Editor: Jie Yan.

Supporting citations

References (34,35,36,37,38,39) are cited in the supporting material.