# A standard additions method reduces inhibitor-induced bias in quantitative real-time PCR

## Abstract

A method of calibration for real-time quantitative polymerase chain reaction (qPCR) experiments based on the method of standard additions combined with non-linear curve fitting is described. The method is tested by comparing the results of a traditionally calibrated qPCR experiment with the standard additions experiment in the presence of 2 mM EDTA, a known inhibitor chosen to provide an unambiguous test of the principle by inducing an approximately twofold bias in apparent copy number calculated using traditional calibration. The standard additions method is shown to substantially reduce inhibitor-induced bias in quantitative real-time qPCR.

### Keywords

Nucleic acids Quantitative PCR Calibration Standard additions## Introduction

An accurate value for the amount of material present in a given sample depends on the assumption that the response of the method to the material in the sample is the same as that for the material used for calibration. The sample matrix—the bulk material of which the sample is composed—may cause substantial differences in response between sample and reference standard; this is the “matrix effect”. In qPCR, matrix effects are particularly likely to arise from inhibitors present in the sample but not in the (usually ‘cleaner’) calibrant [1, 2, 3, 4, 5]. Such effects vary significantly, and often greatly, from one sample to the next, and may also vary from one qPCR reaction to another [6]. A variety of approaches, mostly concentrating on fitting the observed amplification curve for each reaction, has accordingly been applied to reduce the adverse effects of sample-specific inhibition [7, 8, 9, 10, 11, 12, 13].

Here, we demonstrate a different approach to those so far described for qPCR. A recent review [14] has provided a summary of the main techniques presently available for reducing matrix effects, together with methods of uncertainty estimation. One of the more general tools applied in chemical measurement, particularly for elemental analysis, is the method of standard additions. Here, we describe the theory and practice of standard additions as applied to quantification using real-time qPCR, and demonstrate the principle by comparing the results of a traditionally calibrated qPCR experiment with a standard additions experiment in the presence of 2 mM EDTA, a known inhibitor chosen here to provide an unambiguous test of the principle.

## Standard additions methods

*y*=

*b*

_{0}+

*b*

_{1}

*x*, the extrapolated concentration \( \hat{x} \) in the original test solution is given by

*b*

_{0}and

*b*

_{1}.

The utility of the procedure arises from the fact that many of the common matrix effects in analytical chemistry have the effect of changing the slope of the calibration line. In the standard additions procedure, the estimated slope is specific to the test sample in question, and the extrapolated value is consequently corrected for any differences in slope due to matrix effects in the test material.

*x*

_{0}and added amount

*δ*

_{x}, with corresponding mean responses

*y*

_{0}and

*y*

_{δ}, and again assuming linear response,

## Application to real-time qPCR

*C*

_{t}required to reach a given fluorescence threshold and to plot the resulting

*C*

_{t}values against log(

*N*), where

*N*stands for target concentration. The actual dependence of

*C*

_{t}on added target depends on the amplification efficiency

*e*expressed as the fractional increase in amount of target with each cycle, and the fluorescence threshold

*T*

_{f}as follows. With the test material target concentration at

*N*

_{0}(expressed as copy number for convenience

^{1}), the copy number

*N*

_{t}at cycle

*C*

_{t}is given by

*T*

_{f}, which corresponds to a DNA concentration

*N*

_{T}. Substituting into Eq. 4 and rearranging gives the usual fitted qPCR calibration model:

Since the exact relationship between *N*_{T} and the fluorescence threshold *T*_{f} is rarely known, and further because the first two amplification cycles can be expected to have different efficiencies, the term log(*N*_{t})/log(1 + *e*) in Eq. 5 is generally replaced by an arbitrary intercept.

*e*between calibration standards and test materials will affect the relationship between

*N*

_{0}and

*C*

_{t}, and hence affect the resulting estimates of

*N*

_{0}. If

*e*is the efficiency in the test sample and

*e*′ that for calibration standards, the estimated target concentration \( \hat{N} \) given an observed

*C*

_{t}value

*C*

_{t}(obs) will be

*C*

_{t}approach, it has been estimated from reported variations in efficiency of sample and reference amplicon (0.8–1.0) that a true tenfold difference in amount of input DNA between the sample and the control may result in an observed fold difference of between 0.7 and 210, while more modest differences in efficiency between the sample and reference (from 0.78 to 0.82) can induce fourfold errors in estimation [7]. It is also clear that the relationship between estimated and actual concentrations for the test sample is non-linear in efficiency. In principle, a standard additions method could provide the sample-specific values of

*e*and

*C*

_{t}required to eliminate these effects.

### Classical standard additions in qPCR

*C*

_{t,δ}indicates the

*C*

_{t}value with an amount

*δN*of added target.

In principle, plotting \((1 + e)^{{ - C_{t} }} \) against *δN* provides a linear relationship, so if a reliable estimate of the efficiency *e* were available, the standard additions formula of Eq. 1 could be applied to obtain *N*_{0} (though weighted regression becomes important because of the rapidly increasing dispersion in the linearised values). A single-point standard additions approach would also be applicable. However, *e* is generally not known for the test material.

Since neither *N*_{T} nor *e* are known and there is no closed form solution for the best fit parameters, the only option is to apply a numerical approach such as non-linear least squares fitting to estimate *e*, *N*_{T} and *N*_{0} simultaneously from at least three independent and replicated observations of *δN* and *C*_{t}. This is the most general approach, and requires no experiments other than a series of additions of target DNA to the test sample following a preliminary observation to guide the amount added at each step; that is, the classical standard additions experiment. This is the approach tested in this paper.

### Practical implementation and proof of principle

A standard additions experiment traditionally involves observations on a test material or extract with progressively increasing amounts of calibration material added. In performing the addition, however, it is important to keep the concentrations of any interfering or inhibiting species constant across the experiment, that is, to arrange for a constant volume for all test suspensions. This is shown schematically in Fig. 1, in which the calibration material in a buffer or background is added to a fixed volume of extracted sample DNA and additional buffer added to keep the total volume constant. For larger ranges of added calibrant, it is more convenient to prepare a series of dilutions of the calibrant in buffer and add a constant volume; in the demonstration below, this was the approach used.

To test the principle, we analysed a DNA extract from a modified canola species and the same extract with sufficient EDTA added to provide substantial inhibition. If the standard additions method is effective, we expect to observe a substantial difference in estimated copy number between the inhibited and non-inhibited materials when using traditional external calibration, and a much reduced or negligible difference in estimated copy number when using standard additions.

## Experimental

### PCR materials

Genomic canola DNA was extracted from ground genetically modified canola seed (AOCS 0304-B Certified Reference Material, 99.83% RoundUp Ready canola Event RT73, AOCS, Urbana, IL) as described previously [15] except that the incubation step with Nuclei Lysis Solution (Promega Corp, WI, USA) was undertaken overnight at 65 °C. Copy number concentration was assigned by UV determination and use of a factor of 1.225 pg per haploid genome [16]. Primers designed to amplify a 76-base pair (bp) fragment of the canola endogenous *fatA* gene and the Taqman® probe (Taqman is a registered trademark of Roche Molecular Systems, Inc., Pleasanton, CA, USA) for detection have been described previously [15], and were obtained from Geneworks (Adelaide, Australia) and Operon (Cologne, Germany), respectively.

### PCR set-up

To evaluate the effectiveness of standard addition in the presence of PCR inhibitors, a set of six serially diluted RT73 DNA standards was prepared. Four experiments were undertaken using these standards together with a no template control (NTC) which contained 5 μL of water instead of DNA. For the first experiment, each PCR (20 μL final volume) comprised 500 nM of both the forward and reverse primers, 250 nM Taqman probe, 1× Taqman Universal Mastermix and either 62, 16, 4, 1, 0.2, 0.1 or 0 ng RT73 DNA standard. This experiment effectively formed an external calibration set for traditional qPCR. The second experiment was identical to the first experiment except that each PCR contained 2 mM EDTA. The third and fourth experiments used the standard addition method and were identical to the first and second experiments, respectively, except that 1 ng RT73 DNA was added to each reaction to represent the Sample DNA. Experiments one and two provided data using traditional external calibration whilst experiments three and four provided a directly comparable estimate using standard additions. All standards and NTCs were analysed in quadruplicate. The complete series of four experiments was repeated three times in separate runs. Real-time PCR was performed on a Stratagene MX3000P (Integrated Sciences, Sydney, Australia) for 50 cycles using previously described thermocycling conditions [15]. The serial dilutions, master mix preparations and reaction set-up were completed using a CAS-1200N robotic liquid handling system (Corbett Research, Sydney, Australia).

### Computations

All statistical analysis, including non-linear fitting, was performed using the statistical software program R version 2.12.2 [17] running under Microsoft Windows 7.

Curve fitting used standard non-linear least squares estimation. The optimisation used a quasi-Newton method, though essentially identical results could be obtained using Nelder–Mead optimisation. Standard errors were computed from the diagonal elements of the estimated covariance matrix obtained by inversion of the Hessian and multiplication by \( 2s_{\text{res}}^{{2}} \) where \( s_{\text{res}}^{{2}} \) is the residual variance.

## Results

*x*-axis is plotted on a log scale and (b) the response,

*C*

_{t}, decreases with log(concentration) as in the usual qPCR calibration. The similarity between the two figures is that both plot response against

*added*target. Comparing Fig. 2 with a familiar external qPCR calibration, while the usual calibration would show a linear decrease in

*C*

_{t}response with increasing log([DNA]), Fig. 2 shows clear non-linearity in the

*C*

_{t}response: the ‘plateau’ in

*C*

_{t}response at low concentrations occurs at the level corresponding to the amount of DNA that was present in the test material.

*p*= 3 × 10

^{−4}for a two-tailed

*t*test for difference between the two sets of three mean values). This is because the inhibitor is present in the test material but not in the calibration standards (as would normally be the case with an unsuspected inhibitor). By contrast, the results from the standard additions method, shown to the right of the figure, show no significant effect of inhibitor (

*p*> 0.05); the estimated copy number is close to 700 whether inhibitor is present or not. It seems safe to conclude that the standard additions method has been effective in removing the substantial effect of inhibition.

## Discussion

### Comparison with alternative strategies

It is useful to compare the standard additions method with existing strategies which mitigate the effects of inhibition or, more generally, the effects of sample-specific amplification efficiency. A number of strategies are available for the purpose; all with at least some drawbacks. The majority of recent approaches have concentrated on obtaining good sample-specific estimates of efficiency. Approaches to this have included successive dilution [8], estimation from alternative assays included in the PCR [9] and a variety of methods for estimation of sample-specific efficiency from the real-time PCR fluorescence curve [see, for example, 7, 10, 11, 12, 13].

Successive dilution as implemented by, for example, Swillens et al. [8] involves diluting by known increments and fitting a straight line to the *C*_{t} versus log(dilution) plot. This strategy is expected to be effective when the initial copy number is high and the efficiency is unrelated to concentration; otherwise, either the results become very variable as dilution increases or the dependence of efficiency on dilution results in non-linear behaviour and imperfect estimation of efficiency [18]. One alternative dilution strategy, progressive dilution to estimate a ‘limiting value’ for the original concentration by arbitrary fitting and extrapolation, has been suggested in other sectors but rarely applied for nucleic acid quantification; although less affected by assumptions about the relationship of efficiency and dilution, it too is not appropriate for lower copy numbers.

Meijerink et al. [9] suggested use of an efficiency compensation control; essentially the inclusion of two reference reactions included in multiplex assays in the test sample and calculation of a sample-specific efficiency correction based on the difference in *C*_{t} for the reference reactions. This is effective when the two reference assays and the amplification for the target amplicon are all equally affected by sample-specific effects; it is not appropriate when assays differ appreciably in amplification efficiency or inhibition effect [6].

Rutledge and Stewart recently reviewed strategies based on efficiency estimation from the fluorescence curve [18]. Before considering these in detail, it is useful to review the main parts of the typical PCR amplification curve. The fluorescence curve consists of an initial region essentially indistinguishable from the baseline fluorescence, usually extending to well over half the final cycle time, in which the majority of amplification actually occurs. This is followed by a detectable rapid rise which often appears nearly exponential, leading into a nearly linear portion around an inflection point and a subsequent asymptotic approach to a maximum or ‘plateau’. The most important point to make about this curve is that the majority of the amplification occurs in the near-baseline region which is—because it contains essentially no useful signal—largely uninformative about the efficiency; only the location of the later phases provides direct evidence of the amplification efficiency in the first part of the curve. Essentially, by the time the fluorescence signal is detectable, it is already likely to be adversely influenced by the effects that eventually cause the amplification to stop. Despite this, several studies [e.g. 7, 8, 9, 10, 11] have shown that efficiency correction based on modelling the visibly increasing and near-exponential part of the fluorescence curve is effective at reducing the adverse effects of sample-specific amplification efficiency, perhaps in part because behaviour late in the amplification correlates with effects in the earlier phases. Rutledge and Stewart, however, demonstrated that the efficiency deduced from the visibly ‘exponential’ part of the curve generally underestimated the average amplification efficiency [18], making the approach unsuitable for accurate quantification. Modelling the complete curve based on an assumed dependence of amplification efficiency on cycle time as suggested separately by Rutledge and Stewart [12] is intuitively more appealing and apparently at least as successful, though it does rely on a realistic model of amplification behaviour. A similar approach used an alternative model of amplification efficiency based on fluorescence intensity and a different dependence for the efficiency [13].

Finally, Batsch et al. [13] suggested an improved indicator of cycle time, termed \( {C_{{{y_0}}}} \), based on extrapolation through the linear region of the amplification curve as a correction for minor changes in amplification efficiency; this corrects for differences in *C*_{t} that would otherwise occur due to differences in gradient in the later part of the curve.

The standard additions approach demonstrated in the present paper does not use a model for the whole amplification curve, at least in the present implementation. Nor does it assume independence of inhibition and dilution; the dilution is kept constant. Instead, the correction for inhibition arises because the calibration curve is subject to exactly the same inhibition effects as the target amplicon in the test sample. This relaxes many assumptions about amplification behaviour. An additional advantage is that, because the technique relies on addition of target, it is suitable at comparatively low copy numbers; the calibration curve is always observed at higher copy numbers and observations on the fortified solutions are not less precise than the observation on the test sample itself. The experiment conducted here demonstrates that this is sufficient to remove an approximately twofold inhibitor-induced bias (see Fig. 4). The approach therefore offers an alternative calibration strategy that does not rely heavily on a model for the most important parts of the amplification.

### Limitations of the standard additions approach

The theory used here relies on Eq. 8, which uses the same fundamental assumption as the usual real-time qPCR calibration methods and indeed most other compensation methods: that the amplification efficiency is essentially constant during the exponential phase of amplification. In considering the possibility of inhibition, it is important to be aware that not all inhibition mechanisms necessarily induce a constant change in efficiency during amplification. Failures in this assumption would lead to significant lack of fit and correspondingly poor results. Checking for good model fit and in particular for a linear dependence of calculated total DNA concentration (available from the fitted curve parameters) on added DNA is therefore important.

The standard additions approach also has some practical disadvantages. For both external calibration and standard addition procedures a reliable reference value for the calibrant and construction of a calibration series are required—involving additional work—and in the present proof-of-principle implementation it assumes that any cycle time estimation errors are at least consistent across the calibration range. In addition, standard addition requires replicates of the dilution series of calibrant to be run per sample, increasing the reagent requirements and number of observations required for each test material. This makes it better suited to the determination of reference values than to routine clinical practice. The present demonstration also uses a numerical curve fitting method instead of a simple linear calibration.

Further improvements in accuracy or ease of use are possible; for example, if an external calibration were also run, we intuitively expect the apparent copy number in the standard additions sequence to be linear in added target, offering the possibility of a linear standard additions experiment based on plotting the apparent copy number obtained from calibration against added target, as in Eq. 1. There is also room for improvement in the estimation of cycle time *C*_{t}. Nonetheless, it seems safe to conclude that the standard additions approach offers a useful means of substantially reducing inhibitor-induced bias in realistic test samples.

## Conclusion

A method of calibration for real-time PCR experiments based on the method of standard additions and non-linear curve fitting has been suggested. The method successfully reduced inhibitor-induced bias in a quantitative real-time PCR example.

## Footnotes

- 1.
‘Copy number’ used here is ‘copy number per experiment’. This gives a convenient formulation for a standard additions experiment in which a known volume of test material extract is used, the volume is held constant and known amounts (in copy number) of DNA are added as recommended here. Given a known volume of test material extract, copy number concentration in the test material extract can be calculated from the volume used. An essentially identical formulation can be used if DNA per experiment is expressed throughout in mass units, for example as nanograms of DNA.

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