A critical survey on the kinetic assays of DNA polymerase fidelity from a new theoretical perspective

Fig.2 shows a three-step kinetic model of the competitive incorporation of a single dRTP or dWTP to site $i+1$. The true fidelity is precisely given by the ratio of the final product $D_{i}R$ to $D_{i}W$ when the substrate DNA are totally consumed, i.e.,

which can be calculated by FP method.

A part of the kinetic equations for this model are given below ,

here $\alpha=$R,W. The dNTP binding rate is denoted as $k_{1,R\alpha}=k^{0}_{1,R\alpha}[d\alpha TP]$. The basic idea of FP method is not to directly solve the kinetic equations rigorously ($e.g.$ in the transient-state analysis) or approximately by imposing extra assumptions ($e.g.$ the steady-state assumption) . Instead, the two equations are integrated to give the products at time $t$,

The second term approaches to zero with $t$ increases to infinity. dNTP is usually in large excess to template DNA either $in\ vivo$ or $in\ vitro$, so [dNTP] remains approximately a constant during the reaction. Then the fidelity is simply given by

$f_{true}$ is exactly the initial discrimination defined by Eq.(3) with the two effective incorporation rates $\overline{k}_{RR}$ and $\overline{k}_{RW}$.

In practice, when the reaction time $t$ is large enough for sufficient product accumulation (i.e., the second term on the right side of Eq.(8) is far smaller than the first term), the measured $[D_{i}R](t)/[D_{i}W](t)$ becomes nearly time-invariant, and thus it is a good measure of $f_{true}$. In the direct competition assay conducted by Bertram et.alBertram et al. (2010), the incorporation reaction was terminated when about half of the substrate DNA were reacted. This termination criteria per se does not meet the above requirement. Other evidences should be considered. For instance, $[D_{i}R](t)/[D_{i}W](t)$ is proportional to $\textmd{[dRTP]}/\textmd{[dWTP]}$ if the reaction time $t$ is large, so one can decide whether $t$ is sufficient large by examining whether $[D_{i}R](t)\textmd{[dWTP]}/[D_{i}W](t)\textmd{[dRTP]}$ becomes nearly a constant when [dWTP] or [dRTP] is changed. Combined with these evidences, Bertram et.al were able to show that $[D_{i}R](t)/[D_{i}W](t)$ measured under their termination condition is really a good measure of the true fidelity.

The above FP treatment can be directly extended to multi-state incorporation schemes like Fig.3 to get the effective incorporation rate, as below,

Details of the calculation can be found in Supplementary Materials (SM) Sec.I B. Here $k_{1}$ is proportional to dNTP concentration $k_{1}=k^{0}_{1}\textmd{[dNTP]}$, so $k^{*}=k^{*0}\textmd{[dNTP]}$. The true fidelity is still given by Eq.(3) where $\overline{k}_{RR}$ and $\overline{k}_{RW}$ are effective rates defined here. Fig.3 describes the processive dNTP incorporation by DNAP without dissociation from the substrate DNA. If the dissociation is considered, the reaction scheme will be more complex, but the effective incorporation rates can still be given as above, as will be shown in RESULTS AND DISCUSSION Sec.2.2.

The steady-state assays measure the initial velocity of product generation under the condition that the substrate is in large excess to the enzyme. The normalized velocity per enzyme is in general given by the Michaelis-Menten equation

Here the superscript $pol$ indicates the polymerase activity, the subscript $s\cdot s$ indicates the steady state. Fitting the experimental data by this equation, one can get the specific constant $k_{cat}/K_{m}$ either for dRTP incorporation or dWTP incorporation and estimate the fidelity (the initial discrimination) by $f_{s\cdot s}=(k_{cat}/K_{m})_{R}\textmd{[dRTP]}\big{/}(k_{cat}/K_{m})_{W}\textmd{[dWTP]}$. What is the relation between $k_{cat}/K_{m}$ and the effective incorporation rate $\overline{k}$ in Eq.(3)?

To understand the exact meaning of $k_{cat}/K_{m}$, the complete multi-step incorporation reaction scheme Fig.(4) must be considered, which explicitly includes the DNAP binding step and the dissociation step. The last dissociation step is reasonably assumed irreversible, since the enzyme will much unlikely rebind to the same substrate molecule after dissociation because the substrate is in large excess to the enzyme. Under the steady-state condition, it can be easily shown

Here $k^{*}$ is defined in Eq.(1.2.). $K_{s\cdot s}=1+k_{off}/k_{on}$, $k_{on}=k^{0}_{on}\textmd{[DNA]}$. $K_{s\cdot s}$ is exactly the same for either dRTP incorporation or dWTP incorporation. So the fidelity is given as

This is understandable: the steps before dNTP binding should not contribute to the initial discrimination. However, it does not mean that those steps do not contribute to the total fidelity. Actually they can affect the proofreading efficiency, as will be demonstrated in RESULTS AND DISCUSSION Sec.2.

The transient-state assay often refers to two different methods, the pre-steady-state assay or the single-turnover assay. Since the theoretical foundations of these two methods are the same, we only discuss the latter below for simplicity.

In single-turnover assays, the enzyme is in large excess to the substrate, and so the dissociation of the enzyme from the product is neglected. The time course of the product accumulation or the substrate consumption is monitored. The data is then fitted by exponential functions (single-exponential or multi-exponential) to give one or more exponents (i.e. the characteristic rates). In DNAP fidelity assay, these rates are complex functions of all the involved rate constants and dNTP concentration, which in principle can be analytically derived for any given kinetic model. For instance, for the commonly-used simple model including only substrate binding and the subsequent irreversible chemical step, one can directly solve the kinetic equations to get two rate functions. It was proved by K. Johnson that the smaller one obeys approximately the Michaelis-Menten-like equationsJohnson (1992).

The subscript $t\cdot s$ indicates the transient state. Similar to the steady-state assays, $k_{pol}/K_{d}$ is regarded as the specific constant and thus DNAP fidelity is defined as the enzyme specificity $f_{t\cdot s}=(k_{pol}/K_{d})_{R}\textmd{[dRTP]}/(k_{pol}/K_{d})_{W}\textmd{[dWTP]}$. It was also shown that $k_{pol}/K_{d}$ equals to $k_{cat}/K_{m}$ for the two-step modelJohnson (1992), so $f_{t\cdot s}=f_{s\cdot s}$.

The equality $k_{pol}/K_{d}=k_{cat}/K_{m}$ actually holds for more general models like Fig.5. The rigorous proof is too lengthy to be presented here (details can be found in SM Sec.III B). Below we only give some intuitive explanations.

Since there are $N$ states in the reaction scheme Fig.5, the time evolution of the system can be described by $N$ exponentially-decay functions with $N$ characteristic rates. If the smallest rate $v^{pol}_{t\cdot s}$ is much smaller than the others, it can be easily proven that $v^{pol}_{t\cdot s}$ follows the same form as Eq.(14) in general. Thus $k_{pol}/K_{d}$ can be obtained by mathematically extrapolating [dNTP] to zero, $v^{pol}_{t\cdot s}\approx k_{pol}\textmd{[dNTP]}/K_{d}$. Intuitively, when [dNTP] approaches to zero, dNTP binding is the rate-limiting step, and all the steps after it will be so slow that the accumulation of each intermediate state is almost zero, i.e. they are approximately in steady state. This gives the velocity per substrate DNA, $v^{pol}_{t\cdot s}\approx k^{*}[E_{1}\cdot D_{i-1}]/[D_{0}]$. $k^{*}$ is defined by Eq.(1.2.), $[D_{0}]$ is the total concentration of DNA. On the other hand, all the steps before dNTP binding are relatively much faster and approximately in equilibrium, which leads to $[E_{1}\cdot D_{i-1}]\approx(k_{on}/k_{off})[D_{i-1}]$, i.e. $[E_{1}\cdot D_{i-1}]\approx[D_{0}]/(1+k_{off}/k_{on})$. Here $k_{on}=k^{0}_{on}[E]$, $[E]$ equals almost to the total DNAP concentration $[E_{0}]$ since DNAP is in large excess to DNA. So the normalized velocity per substrate is $v^{pol}_{t\cdot s}\approx k^{*}/K_{t\cdot s}$, which leads to

$K_{t\cdot s}=1+k_{off}/k_{on}$. This is exactly the same as Eq.(12). So the fidelity can be given by

As stated above, neither the steady-state assay nor the transient-state assay can give the effective incorporation rates. This is not a problem for fidelity assay of $exo^{-}$-DNAP, but is indeed a serious problem for $exo^{+}$-DNAP (as shown in later sections). Then, how can one estimate the effective rates by a general method ? A possible way is to directly dissect the reaction mechanism, i.e. measuring the rate constants of each step by transient-state experiments Wong et al. (1991); Tsai and Johnson (2006); Purohit et al. (2003); Joyce et al. (2008); Santoso et al. (2010); Hohlbein et al. (2013), and then one can calculate the effective rate according to Eq.(1.2.). This is a perfect approach but needs heavy work. Are there direct measurements of the effective rates? Here we suggest a possible single-molecule approach based on the FP analysis.

In a typical single-molecule experiment, the different states of the enzyme or the substrate can be distinguished by techniques such as smFRETSantoso et al. (2010); Hohlbein et al. (2013). So, if the state $E_{1}\cdot D_{i-1}$ and $E^{{\dagger}}_{1}\cdot D_{i}$ in Fig.4 can be properly identified, the following single-molecule experiment can be done to measure the effective incorporation rates.

1. Initiate the nucleotide incorporation reaction by adding $exo^{-}$-DNAP and dNTP to the substrate $D_{i-1}$ and begin to record the state-switching trajectory of a single enzyme-DNA complex. Here dNTP can be dRTP or dWTP, and the primer terminal can be matched(R) or mismatched(W). When a single DNAP is captured by the substrate DNA, it can catalyze the incorporation of one or more nucleotides, depending on the template sequence context and the dNTP used. Then one can select a particular time window from the recorded trajectory, starting from the first-arrival at $E_{1}\cdot D_{i-1}$ (denoted as starting point) and ending at the first-arrival at $E_{1}\cdot D_{i}$ (denoted as ending point).

2. In this time window, the system may make multiple visits to $E_{1}\cdot D_{i-1}$. Count the total time the system resides at $E_{1}\cdot D_{i-1}$. This so-called residence time may be clearly measured under low concentrations of dNTP.

3. Collect sufficient samples to get the averaged residence time $\Gamma_{1,i-1}$, which gives directly the required effective incorporation rate $k^{*}_{\alpha_{i-1}\alpha_{i}}=1/{\Gamma_{1}}_{\alpha_{i-1}}$. Here $k^{*}_{\alpha_{i-1}\alpha_{i}}=k^{*0}_{\alpha_{i-1}\alpha_{i}}[d\alpha_{i}TP]$, $\alpha=$A,T,G,C. The proof of this equality is given in SM Sec.IV A.

The advantage of this single-molecule analysis is its model-independence. Since $k^{*}_{i}=1/\Gamma_{1,i-1}$ holds in general, the measurement of $\Gamma_{1,i-1}$ does not depend on any hypothesis about the details of the reaction scheme (in fact, steps after dNTP binding are often unclear). So this method is hopefully an alternative of the conventional ensemble assays, particularly in cases where the latter may fail (see later sections).

Applying the FP method to the kinetic equations for the reaction scheme Fig.6, one can reduce this complex scheme to the simplified scheme Fig.1, with rigorously defined effective rates given below. The logic of the reduction is the same as that in RESULTS AND DISCUSSION Sec.1.2. Details can be found in SM Sec.I C.

The rate constants can be written more explicitly such as $\overline{k}_{RR}=k^{*}_{RR}$, if the states of the base pairs at site $i,i-1,etc.$ are explicitly indicated. All the rate constants in the same formula have the same state-subscript. $k^{*}$ is defined by Eq.(1.2.). $k_{p\rightarrow e}$ and $k_{e\rightarrow p}$ define the effective intermolecular transfer rates between Pol and Exo. So $\widetilde{k}_{pe}$ and $\widetilde{k}_{ep}$ represent the total transfer rates via both intramolecular and intermolecular ways. With these effective rates, the real initial discrimination and the real proofreading efficiency can be calculated by $f_{true,ini}=k^{*}_{RR}/k^{*}_{RW}$ and $f_{true,pro}=1+\overline{r}_{RW}/k^{*}_{WR}$ respectively.

$f_{true,pro}$ differs much from that given by K.Johnson et al. who may be the first to discuss the contribution of the two transfer pathways to the proofreading efficiency of T7 DNAP. Without a rigorous theoretical foundation, they gave intuitively $f_{pro}=1+(k_{pe}+\theta k^{p}_{off})_{W}/k_{el,W}$ Donlin et al. (1991). The effective elongation rate $k_{el,W}$ was interpreted as the steady-state incorporation velocity $v^{pol}_{s\cdot s,WR}$ (at certain [dNTP]), which is incorrect as will be explained in the section below. The ambiguous quantity $\theta$ was supposed between 0 and 1 (depending on the fate of the DNA after dissociation) and, unlike $\overline{r}$, can not be expressed explicitly in terms of $k^{p}_{on},k^{p}_{off},k^{e}_{on},k^{e}_{off}$,etc. So $f_{pro}$ is not equivalent to $f_{true,pro}$.

Because of the co-existence of the polymerase activity and the exonuclease activity, reaction schemes consisting of a single-nucleotide incorporation and a single-nucleotide excision are theoretically unacceptable and also impossible to implement in experiments. So the usual steady-state assay does not apply to $exo^{+}$-DNAP. It’s also improper to define the elongation probability by imposing steady-state assumptions to such unrealistic reaction models, as given by Eq.(6) in Ref.Fersht (1979). Nevertheless, the steady-state assay can still be employed to study the polymerase and exonuclease separately.

When mixed with the $exo^{+}$-DNAP, the substrate DNA can bind either to the polymerase domain or to the exonuclease domain. For some $exo^{-}$-DNAP, the exonuclease domain may exist and still be able to bind (but not excise) the substrate DNA, which is not discussed in preceding sections. For the steady-state assays of dNTP incorporation by such DNAPs, the reaction scheme becomes complicated (Fig.7. Again, the last enzyme dissociation steps are reasonably assumed irreversible under steady-state condition).

Under the steady-state condition, one can easily compute the specific constant

$K^{{}^{\prime}}_{s\cdot s}=1+k_{pe}/k_{ep}+k^{p}_{off}/k^{p}_{on}$, $k^{p}_{on}=k^{p0}_{on}\textmd{[DNA]}$. So the fidelity is given by

which is exactly $f_{true,ini}$. Combined with Eq.(12), we conclude that the form of Eq.(18) is universal: $k^{*}$ represents the effective incorporation rate of the subprocess beginning from dNTP binding (DNAP dissociation is not involved), and $K^{{}^{\prime}}_{s\cdot s}$ is a simple function of the equilibrium constants of all steps before dNTP binding, no matter how complex the reaction scheme is. Since $K^{{}^{\prime}}_{s\cdot s}$ depends only on the identity and the state of the primer terminal but not on the next incoming dNTP, the enzyme specificity is indeed equal to $f_{true,ini}$ in general.

There were also some studies trying the steady-state assay to define the effective elongation rate $\overline{k}_{WR}$ and the effective excision rate $\overline{r}_{RW}$. For instance, some works used the specific constantDonlin et al. (1991); Wong et al. (1991) or the maximal turn-over rate $k_{cat}$ Wingert et al. (2013) as $\overline{k}_{WR}$. As shown above, however, $\overline{k}$ is not equal to the specific constant or $k_{cat}$. So the steady-state assay fails in principle to measure $\overline{k}_{WR}$, unless $K^{{}^{\prime}}_{s\cdot s}\approx 1$. This condition is met by T7 DNAP ($k_{pe}\ll k_{ep}$ and $k^{p}_{off}\ll k^{p}_{on}$ were observed for the mismatched terminalDonlin et al. (1991)), and may even be generally met since replicative DNAPs are believed to be highly processive (i.e. $k^{p}_{off}\ll k^{p}_{on}$ ) and always tend to bind DNA preferentially at the polymerase domain (i.e. $k_{pe}/k_{ep}\ll 1$). So the the specific constant, but not $k_{cat}$, might be used in practice as $\overline{k}_{WR}$.

In the steady-state assay of the excision reaction (Fig.8), one may measure the initial velocity $v^{exo}_{s\cdot s}$ and interpret it as the effective excision rate $\overline{r}$ Wingert et al. (2013). Whereas $v^{exo}_{s\cdot s}$ is determined by all the rate constants in Fig.8, some rate constants like $k^{p{\dagger}}_{off}$ and $k^{{\dagger}}_{pe}$ are absent from $\overline{r}$. So in principle $v^{exo}_{s\cdot s}$ is not equal to $\overline{r}$. Under some conditions, $e.g.$ $k^{e}_{on}\ll k^{p}_{on}$ and the dissociation of the enzyme from the substrate is fast enough after the excision, the initial velocity $v^{exo}_{s\cdot s}$ may be approximately equal to $\overline{r}$ (details can be found in SM Sec.II C). But these conditions may not be met by real DNAP, $e.g.$, $k^{e}_{on}>k^{p}_{on}$ was observed for T7 polymeraseDonlin et al. (1991). Unless there are carefully designed control tests to provide compelling evidence, $v^{exo}_{s\cdot s}$ itself is not a good measure for $\overline{r}$.

One can also change the concentration of the substrate DNA to obtain the specific constant of the excision reaction in experimentsVashishtha and Kuchta (2015), as can be shown theoretically

which is just irrelevant to $\overline{r}_{RW}$. It can be shown further in any case

Details can be found in SM Sec.II C. In the experiment to estimate $f_{pro}$ for ap-polymerseWingert et al. (2013), the authors wrongly interpreted $\overline{k}_{WR}$ and $\overline{r}_{RW}$ as $k_{cat}$ and $v^{exo}_{s\cdot s}$ respectively , and gave that $excision/elongation=(v^{exo}_{s\cdot s})_{R_{i-1}W_{i}}/(k_{cat})_{W_{i}R_{i+1}}$. They thought this measure roughly reflects the true fidelity. Now it is clear that the two quantities are completely unrelated.

When DNA can bind to either the polymerase domain or the deficient exonuclease domain, the scheme for the transient-state assay of the dNTP incorporation is depicted in Fig.9. By the same logic presented in preceding sections, the specificity constant defined by transient-state assays can be written as

$K^{{}^{\prime}}_{t\cdot s}=1+k_{pe}/k_{ep}+k^{p}_{off}/k^{p}_{on}$. $k^{p}_{on}=k^{p0}_{on}[E_{0}]$ . This is exactly the same as Eq.(18). So, like the steady-state assay, the transient-state assay also applies in general to estimate $f_{true,ini}$. Additionally, the specificity constant, but not $k_{pol}$, can be used to estimate $\overline{k}$ when $K^{{}^{\prime}}_{t\cdot s}\sim 1$, as mentioned in the above section.

The transient-state assay of the exonuclease activity is often done under single-turnover conditions. The time course of product accumulation or substrate consumption is fitted by a single exponential or a double exponential to give one or two characteristic rates. In the single exponential case, the rate is simply taken as the effective excision rate $\overline{r}$. In the double exponential case, however, there is no criteria which one to select. This causes large uncertainty since the two rates often differ by one or more orders of magnitude. In the experiment of T7 polymeraseDonlin et al. (1991), two types of excision reactions were conducted, with or without preincubation of DNA and DNAP. A single characteristic rate was obtained in the former, while two rates were obtained in the latter where the smaller one almost equals to the rate in the former case. So this smaller one was selected as $\overline{r}$. In the experiment of human mitochondrial DNAPJohnson and Johnson (2001a); Johnson and Johnson (2001b), however, the larger one of the two fitted exponents was selected in some cases. For instance, two fitted exponents of the excision reaction of the substrate 25x1/45 (DNA contains a single mismatch in the primer terminal) are $1.1s^{-1}$ and $0.04s^{-1}$ and the former was selected as $\overline{r}$Johnson and Johnson (2001a). Different choices of the exponents can result in estimates of $\overline{r}$ differing by orders of magnitude. In the following, we show that the smallest of the fitted exponents may probably be equal to $\overline{r}$ under some conditions.

The minimal scheme for the transient-state assay of the excision reaction is depicted in Fig.10. By solving the corresponding kinetic equations, one can get three characteristic rates and the smallest one is given by

Here $\epsilon=k_{pe}q+k_{ep}k^{p}_{off}+k_{pe}k^{e}_{off}+qk^{p}_{off}$, $k_{on}=k^{0}_{on}[E]$. When the concentration of DNAP is large enough to ensure $k^{p}_{on}>k^{p}_{off},k^{e}_{on}>k^{e}_{off},k^{p}_{on}>k_{pe}$ and $\epsilon\approx 0$ (compared to other terms in the denominator), Eq.2.3. can be simplified as

If $\widetilde{k}_{ep}>\widetilde{k}_{pe}$, which is met if DNA binds preferentially to the polymerase domain, then we get $v^{exo}_{t\cdot s}\approx\overline{r}$ (of the same order of magnitude), $\overline{r}$ is defined in Eq.(2.1.). So, if the real excision reaction follows the minimal scheme, $v^{exo}_{t\cdot s}$ may be interpreted as $\overline{r}$. In the experiment of human mitochondrial DNAP Johnson and Johnson (2001a); Johnson and Johnson (2001b), the author adopted this interpretation, but used $k_{pol}$ as the effective elongation rate, and calculated the proofreading efficiency as $(v^{exo}_{t\cdot s})_{RW}/(k_{pol})_{WR}$. It is now clear that this quantity is not a proper measure of $f_{true,pro}\big{(}=(v^{exo}_{t\cdot s})_{RW}/k^{*}_{WR}\big{)}$. Here $k^{*}_{WR}$ may probably be replaced by $(k_{pol}\textmd{[dNTP]}/K_{d})_{WR}$, if $k^{p}_{on}>k^{p}_{off}$ and $k_{ep}>k_{pe}$ .

It is worth emphasizing that the interpretation of $v^{exo}_{t\cdot s}$ is severely model-dependent. The reaction scheme could be more complicated than the minimal model in Fig.10, $e.g.$ there may be multiple substeps in the intramolecular transfer process since the two domains are far apart (2-4 nmBȩbenek and Ziuzia-Graczyk (2018)), particularly when there are buried mismatches in the primer terminal. For any complex scheme, one can calculate the smallest characteristic rate $v^{exo}_{t\cdot s}$ and the real excision rate $\overline{r}$. These two functions always differ greatly (examples can be found in SM Sec.III D). So the single-turnover assay per se is not a universally reliable method to measure the effective excision rate. Is there a model-independent method to measure such excision rates? Below we suggest a possible single-molecule assay.

Similar to RESULTS AND DISCUSSION Sec.1.5, a single-molecule assay can be proposed to directly measure the effective rates $\overline{k}$ and $\overline{r}$, if the states in Fig.9 and Fig.10 can be well defined in the experiments. For instance, the states $E_{p}\cdot D$, $E_{e}\cdot D$ and $E+D$ can be clearly resolved by smFRET Markiewicz et al. (2012); Lamichhane et al. (2013).

To measure $\overline{k}$, the experiment is initiated by mixing DNAP and dNTP to the single molecule DNA. If $E_{p}\cdot D_{i}$, $E_{e}\cdot D_{i}$, $E+D_{i}$ and $E_{p}\cdot D_{i+1}$ in Fig.9 can be distinguished in the nucleotide incorporation process, then the residence time at $E_{p}\cdot D_{i}$ can be counted from a time window of the state-switching trajectory of the enzyme-DNA complex with the starting point $E_{p}\cdot D_{i}$ and the ending point $E_{p}\cdot D_{i+1}$. Collecting sufficient samples to obtain the averaged residence time $\Gamma_{p,i}$, one can get $k^{*}_{i+1}=1/\Gamma_{p,i}$ or $k^{*0}_{i+1}=1/(\Gamma_{p,i}[d\alpha_{i+1}TP])$ by the FP analysis