Why do you need to use the real PCR efficiency in the analysis of your qPCR data?

Shortly back to the qPCR basics that we described in our first blog. The exponential phase of the PCR reaction is determined by the kinetic equation of PCR: $N_c = N_0 \cdot E^{c}$. The common way to analyse the observed fluorescence data is that, after baseline correction, the user, or the PCR instrument software, sets a quantification threshold at a chosen fluorescence value and determines where this threshold intersects each of the amplification curves. The line from this intersection down to the X-axis intersects this cycle axis at the so-called quantification cycle or Cq value (still often called the threshold cycle, or Ct). The Cq value is thus the fractional number of cycles that the reaction needed to reach the fluorescence threshold. Unfortunately, in most clinical and many experimental research reports, this Cq value is reported as the outcome of a PCR despite the fact that this Cq value is dependent on the PCR efficiency, as we showed in the first blog (Fig. 1).

To include the PCR efficiency in this analysis per reaction, the fluorescence at the start of the reaction ($F_0$) can be calculated with the threshold ($F_q$), Cq, the efficiency value ($E$) and the inverse of the kinetic PCR equation ($F_0 = F_q / E^{Cq}$). This $F_0$ is in fact the primary outcome of the PCR reaction and, because of the linear relation between number of targets and their associated fluorescence, $F_0$ is a direct measure of $N_0$, the number of targets at the start of the reaction.

However, for most of you, $F_0$ and $N_0$ will be new concepts because you are used to see the outcome of a PCR analysis reported as an expression ratio between target-of-interest and reference genes, calculated as $2^{-dCq}$, or maybe even as a fold difference, or treatment effect, between an experimental group and a control group, calculated as $2^{-ddCq}$. The latter is the comparative Ct method, or Livak method, better known as $2^{-\Delta\Delta Ct}$ (Livak and Schmittgen, 2001). How do we go from the above defined $F_0$ to these dCq equations?

In the history of qPCR data analysis, we can distinguish different simplification steps that were taken in the analysis; here we will discuss these 'solutions' with respect to their assumptions, problems and caveats. The first problem is the variation between different clinical or biological samples due to differences in sample size and yield of pre-PCR processing of the samples. Assuming that sampling and pre-processing affect every target in a sample in the same way, qPCR results are preferably reported as an expression ratio (ER) between target-of-interest (toi) and reference (ref) genes [Eq. 1]. With the kinetic equation for $F_0$ we can write this equation for ER as [Eq. 2]. In the dCq approach, the equation for ER is simplified to [Eq. 3].

Note that this simplification assumes two things: first, that the quantification threshold ($F_q$) is the same for both targets and, second, that both assays amplify with 100% efficiency: both $E_{toi}$ and $E_{ref}$ are replaced by 2. The validity of the reported ER depends on the validity of both assumptions. In this blog, we show equations with only 1 reference but in good PCR practice (according to the MIQE guidelines) you need a stable set of at least two validated reference genes.

The second simplification that entered the field of qPCR analysis involved the practice to mainly report the fold-difference between the gene expression ratios in a control group and a group that underwent an experimental condition. This was triggered by the use of qPCR in experimental research studies about the difference in gene expression in various experimental groups.

Starting with the above expression ratio (ER), such a fold-difference (FD) between groups can be written as [Eq. 4].

When we would have started this derivation of FD with the original $F_0$ equation, you would have seen that there is no longer the prerequisite that $F_{q,toi}$ and $F_{q,ref}$ are set at the same fluorescence level. This cancelling out of the threshold setting from the equations can be considered a bonus of reporting a fold difference.

Soon after the publication of the $2^{-ddCq}$ approach, a paper by Pfaffl (2001) recognised that in qPCR practice the assumption that all targets were amplified with 100% efficiency did not hold. This paper, therefore, recommended an efficiency-corrected approach based on the difference of Cq values between target-of-interest and reference genes in control and experimental groups [Eq. 5].

Note that in this equation, the difference in Cq values is not between target and reference, as in the $2^{-ddCq}$ equation, but between treated and control groups per assay. Although the 'Pfaffl' equation thus does not ignore differences in amplification efficiency, it uses mean Cq values for each target in each group. This averaging of Cq values leads to loss of information on failed reactions and information on biological variation within the experimental groups.

The above description of the main simplification steps in qPCR analysis shows that fundamentally the $2^{-dCq}$ and $2^{-ddCq}$ approaches are based on the basic equation for PCR kinetics and its inverse equation to calculate the $F_0$ per reaction: $F_0 = F_q / E^{Cq}$. However, for convenience and simplification, these approaches assume the amplification efficiency, $E$, to be 2, or 100%, in all reactions for all targets in every experimental condition. To further muddle the issue, in many papers the '2 to the power minus' part of these equations is omitted as a further simplification and just ddCq values are published as if they are a fold-difference. Such papers leave it to the reader to figure out whether the Y-axis annotation saying 'fold difference' is indeed showing fold difference values or just the ddCq value. In case of the latter, the reader has to assume a PCR efficiency of 2, decide whether the axis shows ddCq or −ddCq, and do the math. It is of the utmost importance to realize that even when the efficiency values are mentioned in the paper, as required by the original MIQE guidelines, these values cannot be applied because the reported ddCq values result from two different targets. To do so, the individual Cq values are required.

To illustrate the effect of ignoring the PCR efficiency in qPCR data analysis, we performed a simulation with two targets and two experimental conditions. In this simulation, the PCR efficiencies of the target-of-interest (toi) and the reference target (ref) both varied from 1.7 to 2.0 (or from 70% to 100%). In the control condition, the number of starting copies of the toi and ref are set to 50 and 250 copies, respectively (Fig. 2A, left). In the treated condition, due to a sampling mistake introduced in the simulation, the sample input is 5-times larger, and the PCR of the reference target, by definition not affected by the treatment, thus starts with 1250 copies. Therefore, because of the treatment effect of 25 times, the toi starts at 6250 copies (Fig. 2A, right). To also illustrate the effect of setting $F_q$ at different levels, we simulated thresholds at three fluorescence levels corresponding to $10^{9}$, $10^{10}$ and $10^{11}$ copies of the amplicons (Fig. 2A).

Using those quantification thresholds, we determined all Cq values and calculated the dCq values between toi and ref for every combination of efficiency values in the control and treated condition (Fig. 2B). In these graphs we see that the dCq values depend on the threshold level and that even when the PCR efficiencies are the same, the dCq values are different rather than constant. Each dCq value was used in the $2^{-dCq}$ equation to calculate the toi/ref expression ratio in the control and treated condition (Fig. 2C). Remember that in this simulation for the control condition this ratio was set at 0.2 (= 50/250) and in the treated condition this ratio is 5 (= 6250/1250). Note that only when both PCR efficiencies are 2, the calculated ratios are indeed 0.2 and 5, respectively. For all other efficiency combinations, the expression ratio calculated with $2^{-dCq}$ is not correct and results in a 20,000-fold range of ER values in both conditions. Note that the reported expression ratio can be in the wrong direction, suggesting over-expression rather than the true under-expression of the target in the control condition, or vice versa in the treated condition.

The dCq values (Fig. 2B) were then used to calculate the ddCq value for every combination of efficiency values (Fig. 2D, left) and these ddCq values were used in the $2^{-ddCq}$ equation to calculate the fold difference between treated and control conditions. Remember that the treatment effect set in the simulation was 25 times (Fig. 2A). This expected fold difference value is only found when both the target-of-interest and the reference are amplified with 100% or 2.0 efficiency. For all other efficiency values, the result is wrong: the lower the actual efficiency, the larger this error is. Because the error in the expression ratio in treated and control conditions is correlated (visible as the similarity of the graphs in the left and right panel of Fig. 2C), the error in fold difference is less than the error in expression ratio (Fig. 2C) but still ranges between 40% too low to over 4 times too high. Because the actual error is dependent on both the actual efficiency values and gene expression levels, the direction of the error in fold difference cannot be inferred from the actual efficiency values.

Taken together, the simulation and graphs in Figure 2 show that assuming that the PCR efficiency is 2, or 100%, for all assays while that is not the case, will always give a wrong result for the toi/ref expression ratio as well as for the fold difference, or treatment effect, between control and treated conditions. Note that similar wrong results would occur when the simulation was carried out with a set of reference genes, each with their own amplification efficiency. Because these errors compound exponentially, what begins as a seemingly small methodological assumption becomes an analytical distortion. The simulation shows that in some cases this can even lead to a conjectured treatment effect that is in the reverse direction of the true effect when no efficiency correction is applied. Although it is often stated that these errors are acceptable when PCR efficiency differences are small, there is no reason at all to accept these errors. With the same data and similar effort, the analysis can be carried out with the true PCR efficiencies and reveal the correct result, as will be shown below.

Using the inverse of the kinetics equation of PCR, the fluorescence associated with the number of target copies at the start of the reaction, $F_0$, can be calculated easily from the Cq value per reaction and the PCR efficiency of the assay (see a future blog on how to determine the PCR efficiency). Figure 3, based on true qPCR data, shows how this analysis is performed. The experiment consists of two experimental groups with four biological replicate samples in each group. There are no technical PCR replicates. The amplification curves of these reactions are shown (Fig. 3, graphs). Starting with the $F_0$ values for each target per reaction, the toi/ref ratio per sample, the mean ratio per group and the treated/control fold difference between groups (Fig. 3, table) can be calculated. Note that the toi/ref ratios per group can be compared with a Mann-Whitney test to determine whether the expression ratio in the treated group has been significantly affected by the experiment.

Although this test already answers the biological or clinical question, it is common practice to report a fold difference or treatment effect. This fold difference (FD) can be calculated by dividing the geometric mean toi/ref ratios in the treated and control group (Fig. 3, table). Error propagation can then be used to calculate an SD of this fold difference from the SD and the mean expression ratio ($\overline{ER}$) of each group [Eq. 6].

Note that in this example the SD of the fold difference is rather large due to one very high toi/ref ratio in the treated group (Fig. 3, yellow cell in the table). The ability to identify this deviating sample shows an additional advantage of starting the analysis with the $F_0$ per sample, compared to the ddCq or Pfaffl's efficiency-corrected approach. In the latter approaches, the mean Cq value per target and group is used which leads to loss of information on individual samples and loss of variation within the groups. With the illustrated $F_0$ approach, this loss of information does not happen and a quality control and outlier detection at every level of the experiment can be carried out.

We hope that this blog convincingly showed you that when you do NOT use the real PCR efficiency values in your qPCR analysis, the results will suffer from an error of unknown magnitude and may even be in the wrong direction. As a consequence, the results that you are reporting are meaningless. Efficiency-correction is the only way to avoid these errors and to reach correct and reliable results. So to report valid and meaningful results from your experiment in the future, you have to use the Cq and efficiency values that you obtain from your qPCR analysis and apply these in the efficiency-corrected analysis approach that we described here. Because this efficiency-corrected analysis is initially performed per sample, the added bonus of this approach is that you will be able to detect deviating samples before losing this information by calculating mean expression levels per group.