Abstract
Background: Realtime PCR analysis is a sensitive DNA quantification technique that has recently gained considerable attention in biotechnology, microbiology and molecular diagnostics. Although, the cyclethreshold (Ct) method is the present "gold standard", it is far from being a standard assay. Uniform reaction efficiency among samples is the most important assumption of this method. Nevertheless, some authors have reported that it may not be correct and a slight PCR efficiency decrease of about 4% could result in an error of up to 400% using the Ct method. This reaction efficiency decrease may be caused by inhibiting agents used during nucleic acid extraction or copurified from the biological sample. We propose a new method (Cy_{0}) that does not require the assumption of equal reaction efficiency between unknowns and standard curve.Conclusion: Our results demonstrate that Cy_{0} represents a significant improvement over the standard methods for obtaining a reliable and precise nucleic acid quantification even in suboptimal amplification conditions overcoming the underestimation caused by the presence of some PCR inhibitors.
Background
In the last few years, the realtime polymerase chain reaction (PCR) has rapidly become the most widely used technique in modern molecular biology [14].Methods
Experimental design
The absolute quantification method relies on the comparison of distinct samples, such as the comparison of a biological sample with a standard curve of known initial concentration [21].Quantitative RealTime PCR
The DNA standard consisted of a pGEMT (Promega) plasmid containing a 104 bp fragment of the mitochondrial gene NADH dehydrogenase 1 (MTND1) as insert. This DNA fragment was produced by the ND1/ND2 primer pair (forward ND1: 5 ' ACGCCATAAAACTCTTCACCAAAG3 ' and reverse ND2: 5 ' TAGTAGAAGAGCGATGGTGAGAGCTA3 ' ). This plasmid was purified using the Plasmid Midi Kit (Qiagen) according to the manufacturer ' s instructions. The final concentration of the standard plasmid was estimated spectophotometrically by averaging three replicate A_{260} absorbance determinations.Description of the SCF method
Fluorescence readings were used to fit the following 4parameter sigmoid function using nonlinear regression analysis:Eq. 1 
Eq. 2 
Description of the Cy_{0} method
The C_{y0} value is the intersection point between the abscissa axis and tangent of the inflection point of the Richards curve obtained by the nonlinear regression of raw data (Fig. 1).Figure 1
Example of modelling PCR amplification with a 5parameter Richards function Effectiveness of this model is illustrated by the predicted values generated by Eq. 3 (open circles) that agree with the observed fluorescence (dot and line). Curvefitting of experimentally derived fluorescence dataset to Eq. 3 generates values for the kinetic parameters from which the inflection point (solid black rhombus) and the slope of the curve can be derived. The quantitative entity Cy_{0 }(solid black dot), used in the proposed method, shows the cross point between the xaxis and the tangent crossing the inflection point of realtime PCR fluorescence curve.Eq. 3 
Eq. 4 
Eq. 5 
Eq. 6 
Statistical data analysis
Nonlinear regressions (for 4parameter sigmoid and 5parameter Richards functions) were performed determining unweighted least squares estimates of parameters using the LevenbergMarquardt method. Accuracy was calculated using the following equation: , where was the relative error, while and were the estimated and the true number of DNA molecules for each combination of input DNA (n_{Dna}) and amplification mix percentage (%_{mix}) used in the PCR. Precision was calculated as: , where was the coefficient of variation, and were the mean and the standard deviation for each combination of n_{Dna} and %_{mix}. In order to verity that the Richards curves, obtained by nonlinear regression of fluorescence data, were not significantly different from the sigmoidal curves, the values of d parameter were compared to the expected value d= 1, using t test for one sample. For each combination of n_{Dna}, %_{mix}, the t values were calculated as follow: , where and were the mean and the standard error of d values for each combination of n_{Dna} and %_{mix}, with p(t) < 0.05 for significance level. values were reported using 3d scatterplot graphic, a complete second order polinomial regression function was shown to estimate the trend of accuracy values. where also reported using 3d contour plots using thirdorder polynomials spline fitting. All elaborations and graphics were obtained using Excel (Microsoft), Statistica (Statsoft) and Sigmaplot 10 (Systat Software Inc.).Results
Experimental system 1: reduction of amplification mix percentage
With our experimental set up, the mean PCR reaction efficiency was 88% under optimal amplification conditions and slightly decreased in the presence of smaller amplification mix up to 84%. Moreover, for decreasing amplification mix amounts, the PCR reaction efficiencies showed higher dispersion levels than optimal conditions leading to increasing quantitative errors (Variation Interval, VI_{100%}= 1.9211.852 and VI_{60%}= 1.9031.776; Fig. 2). Subsequently, the fluorescence data obtained in these reactions were used to calculate the initial DNA amount using four different procedures: SCF, Ct, Cp and Cy_{0}.Precision and accuracy of the SCF method
Previous studies have shown that the SCF approach can lead to quantification without prior knowledge of amplification efficiency [18, 19, 26]; therefore, we evaluated the performance of this method on our data set. To assess the effect of unequal efficiencies on accuracy, the calculated input DNA, expressed as molecular number, was compared to the expected value obtaining the relative error (RE). The precision was further evaluated measuring the variation coefficient (CV%) of the estimated initial DNA in the presence of different PCR efficiencies and input DNA.Figure 2
Estimation of PCR efficiency using LinReg method Efficiency values were determined from 420 independent reactions using a combination of 3.14 x 10^{7} x 3.14 x 10^{1 }DNA molecules as starting template and amplification mix quantities ranging from 60% to 100%. The graph shows the distribution of PCR efficiencies in relation to the percentage of amplification mix used in the reaction. The solid black squares (?) represent the mean of each distribution.The Cy_{0} method
The SCF model assumes that the fluorescence signal is proportional to the amount of product, which is often the case for SYBRGreen I realtime PCR performed with saturing concentrations of dye. In such conditions, centrally symmetric amplification curves are expected. However, in our experience, we found several nonsymmetric amplification curves shown to have good amplification efficiency using standard curve analysis (Additional file 1 and 3). In order to find a suitable mathematical representation of the complete PCR kinetic curve we compared the standard error of estimate obtained by several equations that generate Sshaped curves (Tab. 1). As shown in Figure 1, these results demonstrated that realtime PCR readouts can be effectively modelled using the 5parameter Richards function (Eq. 3). The Richards equation is an extension of the sigmoidal growth curve; specifically, when d coefficient is equal to 1, the sigmoidal and Richards curves are the same. Hence, we analysed the variation of the d coefficient in the presence of different input DNA and PCR efficiencies. Figure 3 shows that the d value is close to 1 at amplification mix percentages ranging from 100% to 90% while at lower amplification mix contents, where PCR efficiency decreases, the d coefficient was significantly higher than 1 regardless of the starting DNA content (Fig. 3; Tab. 2). These data demonstrate that sigmoidal fitting represents a good approximation of realtime PCR kinetic only in the presence of optimal amplification conditions while the Richards curve is more suited when PCR is inhibited. Since the Richards growth equation includes sigmoidal amplification curves, when d = 1, this nonlinear fitting was used in our method.Figure 3
Distribution of Richards coefficients (d) estimated from PCR fluorescence curves using Eq. 3 in nonlinear fitting procedure. Richards coefficient values were determined from 420 independent PCR reactions. The data have been reported in Log_{10}scale, and represented as mean and standard deviation.Table 1
Comparison of five Sshaped models to fit the PCR curve: Sigmoid, Richards, Gompertz, Hill and Chapman. In this table, f is the fluorescence at cycle x; F_{max }represents the maximum fluorescence value; F_{b }is the background reaction fluorescence; b, c and d determine the shape of each curve. For each model the determination coefficient (R^{2}), the adjusted determination coefficient (Adj R^{2}) and the standard error of estimate have been calculated.Name  Equation  Estimated Parameters  R^{2}  Adj R^{2}  Standard Error of Estimate  
F_{max}  b  c  F_{b}  d  
Sigmoid  f = F_{b}+F_{max}/(1+exp((xc)/b))  45.11  1.49  22.37  0.03  1  1  0.1354  
Richards  f = F_{b}+(F_{max}/(1+exp((1/b)*(xc)))^d)  45.11  1.58  21.95  0.02  1.20  1  1  0.0926 
Gompertz  f = F_{b}+F_{max }*exp(exp((xc)/b))  45.19  2.15  21.45  0.29  0.9992  0.9992  0.6006  
Hill  f = F_{b}+F_{max }*x^b/(d^b+x^b)  45.18  14.95  0.08  22.34  1  1  0.1351  
Chapman  f = F_{b}+F_{max }*(1exp(b*x))^d  45.19  0.46  0.29  20615  0.9992  0.9992  0.6006 
Table 2
tstatistic values obtained for all variable combinations. When t < 0 the Richards coefficient is lower than 1, while for t > 0 the Richards coefficient is higher than 1.Amplification mix percentage  


Log_{10}input DNA  100%  90%  80%  70%  60% 
7.5  0.28348  1.15431  2.9303*  5.43493**  4.26067** 
6.5  3.0233*  0.5329  7.8552**  8.68609**  7.28178** 
5.5  2.2195*  2.70419*  4.7185**  8.61406**  4.60465** 
4.5  0.97856  1.32162  2.34*  16.5192**  17.5903** 
3.5  1.00647  1.038  2.3307*  13.2572**  4.65683** 
2.5  1.731  0.5995  5.8385**  6.90378**  6.13465** 
1.5  0.14417  1.25452  0.898  1.87978  3.69668** 
Figure 4
Plot of fluorescence observations versus cycle number obtained from the same starting DNA but in presence of decreasing amounts of amplification mix. This slight PCR inhibition produces curves which are less steep than controls and shifted towards the right. When analysed by the threshold method, these curves showed higher Ct values with a CV% of 1.45% (A). An example of Cy_{0 }procedure has been reported for the same data set (B). In this method, the amplification reactions are described by the tangent crossing the inflection point of fluorescence curves. As shown in this figure, the straightlines originating from PCRs, characterized by slightly different PCR efficiency and the same starting amounts, tend to cross into a common point near the xaxis leading to small variations in the Cy_{0 }values (CV% = 0.6%).Precision and accuracy of the Ct, Cp and Cy_{0} methods.
The performance of the Ct, Cp and Cy_{0} methods was compared in terms of precision and accuracy over a wide range of input DNA concentrations and under different reaction efficiencies obtained by decreasing the amount of amplification mix as reported in Liu and Saint [18, 27]. As shown in Figure 5A, the Ct method is highly rigorous at maximum reaction efficiency regardless of the starting DNA template. However, the absolute value of RE increased almost linearly with the decrease of efficiency regardless of the template concentrations resulting in an underestimation of the unknown of about 50% at the lowest amplification efficiencies. The Cp was more accurate than the Ct method in the presence of different amounts of amplification mix. Indeed, the relative error in the presence of 100% amplification mix tended towards zero as it did using the Ct method. However, when the efficiency declined, the RE increased initially in the same manner at low and high input DNA concentrations, while at 6070% of the amplification mix, this method markedly underestimated at low concentrations (mean RE_{60% mix;} = 0.58; Fig. 5C). Finally, the Cy_{0} method was more accurate than the Cp method (mean RE 0.12 versus 0.18, respectively; Fig. 5C, E), which in turn was better than the Ct method (mean RE = 0.31). Notably, at optimal amplification conditions (90100% of the amplification mix) the Cp and Cy_{0} methods were equivalent, but at decreasing efficiencies, the Cy_{0} accuracy was more stable than that of the Cp in the concentration range from 3.14x10^{7} to 3.14x10^{5 }molecules. At lower DNA concentrations, from 3.14x10^{4} to 3.14x10^{2 }molecules, the RE proportionally increased with the efficiency decline, but this underestimate was less marked than that of the Cp method at the same starting DNA (Fig. 5C, E). Regarding the precision of the three methods, the variation coefficients were determined for each combination of initial template amount and amplification mix percentage. The random error of quantification achieved by the Cp and Cy_{0} method was similar (mean CV% 21.8% and 22.5%, respectively), while the Ct procedure produced an overall CV% of about 39.7% (Tab. 3). When the CV was analysed in relation to PCR efficiency and input DNA, an area of low variation coefficients for the three methods was found between 3.14x10^{4} and 3.14x10^{7 }molecules as starting material (Fig. 5B, D, F). With DNA amounts ranging from 3.14x10^{3} to 3.14x10^{2 } molecules, the precision progressively decreased in each analysis procedure. These variations were not efficiencydependent, but were related to initial DNA quantity as shown by the shapes of level curves reported in figure 5B, D and F, which were perpendicular to the input template amounts.Figure 5
Comparison of the Ct, Cp and Cy_{0 }methods in terms of precision and accuracy. The accuracy of each method has been reported as Relative Error (RE = expected value ? estimated value) while the precision was evaluated measuring the variation coefficient (CV%). The 3D plots show the variation of relative error in relation to amplification mix percentage and log_{10 }input DNA for the Ct (A), Cp (C) and Cy_{0 }(E) methods. The areas in the level curve graphs represent the CV% values obtained for each amplification mix percentage and Log_{10 }input DNA combination using the Ct (B), Cp (D) and Cy_{0 }(F) methods.Table 3
Comparison of mean Relative Error and mean Variation Coefficient among the Ct, Cp, Cy_{0 }and SCF methods. The reported data were calculated on 420 PCRs except for ^{a}) in which the reaction number was 210.Ct  Cp  Cy_{0}  SCF  Log_{10}SCF  
Mean CV%  39.70%  21.80%  22.52%  594.74%^{a}  66.12%^{a} 
Mean RE  0.318  0.184  0.128  5.058^{a}  0.205^{a} 
Experimental system 2: Realtime PCR quantification in the presence of the inhibitor IgG
The realtime amplification plot of 4.05x10^{6} DNA molecules with increasing concentrations of IgG demonstrates the effects of PCR inhibition on amplification efficiency and accumulated fluorescence (Fig. 6A). As inhibitor concentrations increased, the amplification curves showed lower plateau fluorescence levels and a shift towards the right and the bottom of the inflection points, leading to amplification curves that were less steep and not as symmetric as those obtained in absence of the inhibitor agent (Fig. 6A). As shown in figure 6A the amplification curves inhibited by IgG showed a shape very similar to those resulting from the system of amplification mix reduction (system 1; Fig. 4A). Quantitative data analysis of these amplification plots showed that the estimated DNA quantities were systematically underestimated in the presence of IgG concentrations higher than 0.25 µg/ml and 1 µg/ml using Ct and Cp methods, respectively. However, the Cy_{0} method was able to adjust this bias minimizing the RE at high IgG concentrations (RE = 4.98%; CV = 4.33%; Fig. 6B). Furthermore, in presence of high IgG concentrations, the SCF approach, modified according to Rutledge 2004 [26], was inapplicable because it was impossible to minimize F_{0} value (Additional file 5).Discussion
None of the current quantitative PCR data treatment methods is in fact fully assumptionfree, and their statistical reliability are often poorly characterized. In this study, we evaluated whether known realtime elaboration methods could estimate the amount of DNA in biological samples with precision and accuracy when reaction efficiencies of the unknown are different from those of the standard curve.Figure 6
Realtime PCR amplification plots obtained from the same starting DNA in the presence of IgG acting as reaction inhibitor This inhibition system produces curves which are progressively less steep than noninhibited reactions with increasing IgG concentrations (A). When analysed by the Ct, Cp and Cy_{0 }methods these curves showed a RE% of 25.37%, 9.02% and 4.98% and a CV% of 25.62%, 10.66% and 4.33%%, respectively (B).Conclusions
Realtime PCR analysis is becoming increasingly important in biomedical research because of its accuracy, sensitivity and high efficiency. Although, realtime PCR analysis has gained considerable attention, it is far from being a standard assay. The standard methods are quite stable and straightforward but the accuracy of estimates is strongly impaired if efficiency is not equal in all reactions. Furthermore, the assumption of uniform efficiency has been reported to be invalid in many cases regarding medical diagnostics. In fact, the biological samples may contain inhibitors that could lead to different amplification efficiencies among samples.List of abbreviations used
Cp: crossing point; Ct: threshold cycle; CV: coefficient of variation; IgG: immunoglobulin G; RE: relative error; SCF: sigmoidal curve fitting.Authors' contributions
MG and DS carried out the design of the study, participated in data analysis, developed the Cy_{0} method and drafted the manuscript. MBLR participated in data collection and analysis and critically revised the manuscript. LS carried out the realtime PCR. VS participated in the design of the study and critically revised the manuscript. All authors read and approved the final manuscript.Acknowledgements
We thank Dr. Pasquale Tibollo for technical assistance and Dr. Giosué Annibalini for helpful comments on the manuscript.References
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Supplementary Material