Institute of Biotechnology, University of Lausanne, 1015 Lausanne, Switzerland

Max-Planck-Institute für Quantenoptik, 85748 Garching, Germany

Department of Mathematics, University of Fribourg, CH-1700 Fribourg, Switzerland

Abstract

Background

PCR has the potential to detect and precisely quantify specific DNA sequences, but it is not yet often used as a fully quantitative method. A number of data collection and processing strategies have been described for the implementation of quantitative PCR. However, they can be experimentally cumbersome, their relative performances have not been evaluated systematically, and they often remain poorly validated statistically and/or experimentally. In this study, we evaluated the performance of known methods, and compared them with newly developed data processing strategies in terms of resolution, precision and robustness.

Results

Our results indicate that simple methods that do not rely on the estimation of the efficiency of the PCR amplification may provide reproducible and sensitive data, but that they do not quantify DNA with precision. Other evaluated methods based on sigmoidal or exponential curve fitting were generally of both poor resolution and precision. A statistical analysis of the parameters that influence efficiency indicated that it depends mostly on the selected amplicon and to a lesser extent on the particular biological sample analyzed. Thus, we devised various strategies based on individual or averaged efficiency values, which were used to assess the regulated expression of several genes in response to a growth factor.

Conclusion

Overall, qPCR data analysis methods differ significantly in their performance, and this analysis identifies methods that provide DNA quantification estimates of high precision, robustness and reliability. These methods allow reliable estimations of relative expression ratio of two-fold or higher, and our analysis provides an estimation of the number of biological samples that have to be analyzed to achieve a given precision.

Background

Quantitative PCR is used widely to detect and quantify specific DNA sequences in scientific fields that range from fundamental biology to biotechnology and forensic sciences. For instance, microarray and other genomic approaches require fast and reliable validation of small differences in DNA amounts in biological samples with high throughput methods such as quantitative PCR. However, there is currently a gap between the analysis of the mathematical and statistical basis of quantitative PCR and its actual implementation by experimental laboratory users

Quantitative PCR amplifications performed in the presence of a DNA-binding fluorescent dye are typically represented in the form of a plot as shown in Figure

Representations of real-time PCR amplification curves

**Representations of real-time PCR amplification curves**. The three phases of the amplification reaction are shown either on a linear scale (panel A) or on a semi-log scale (panel B). Panel A represents a typical amplification curve, while panel B depicts amplification curves generated from serial dilutions of the same sample, either undiluted or diluted 10- or 1000-fold (indicated as 1, 0.1 or 0.001, respectively). During the lag phase (phase I), the fluorescence resulting from DNA amplification is undetectable above noise fluorescence in part A, while in part B, some data points take negative values and are not represented. This phase is used to evaluate the baseline "noise" of the PCR amplification. Exponential amplification of the DNA is detected in phase II (cycles 16 to 23, panel A). This phase of the amplification corresponds to the linear portion of the curve in panel B (closed circles). A threshold value is usually set by the user to cross the log linear portion of the curve, defining the threshold cycle value (

In a perfectly efficient PCR reaction, the amount or copy number of DNA molecules would double at each cycle but, due to a number of factors, this is rarely the case in experimental conditions. Therefore the PCR efficiency can range between 2, corresponding to the doubling of the DNA concentration at each cycle, to a value of 1, if no amplification occurs (Eq. 1 in methods). Furthermore, the efficiency of DNA amplification is not constant throughout the entire PCR reaction. The efficiency value cannot be measured during phase I, but it may be suboptimal during the first cycles because of the low concentration of the DNA template and/or sampling errors linked to the stochastic process by which the amplification enzymes may replicate only part of the available DNA molecules _{0 }values that are large enough so that sampling errors become negligible

Another assumption of qPCR is that the quantity of PCR product in the exponential phase is proportional to the initial amount of target DNA. This is exploited by choosing arbitrarily a fluorescence threshold with the condition that it lies within the exponential phase of the reaction. When fluorescence crosses this value, the cycle is termed the "Threshold cycle" (

One of the first and simple methods to process qPCR data remains a set of calculations based solely on _{t }method _{t }method may not accurately estimate relative DNA amounts from one condition or one sequence to the other. Consequently, other methods of data processing have been developed to estimate the efficiency of individual PCR amplifications

Methods to estimate amplification efficiency can be grouped in two approaches, both of which rely on the log-linearization of the amplification plot. The most commonly used method requires generating serial dilutions of a given sample and performing multiple PCR reactions on each dilution

Measurement of the efficiency of a PCR reaction

**Measurement of the efficiency of a PCR reaction**. A: Estimation of the efficiency using the Serial dilution (SerDil) method. Five dilutions of a cDNA sample were amplified using the fibronectin (FN) amplicon. Each dilution was analyzed with five replicates PCR reactions and each data point represents one

The other method currently used to measure efficiency is based on Eq. 3, which associates an efficiency value with each PCR reaction

None of the current qPCR data treatment methods is in fact fully assumption-free, and their statistical reliability are often poorly characterized. In this study, we evaluated whether known mathematical treatment methods may estimate the amount of DNA in biological samples with precision and reliability. This led to the development of new mathematical data treatment methods, which were also evaluated. Finally, experimental measurements were subjected to a statistical analysis, in order to determine the size of the data set required to achieve significant conclusions. Overall, our results indicate that current qPCR data analysis methods are often unreliable and/or unprecise. This analysis identifies novels strategies that provide DNA quantification estimates of high precision, robustness and reliability.

Results

Quantitative PCR usually relies on the comparison of distinct samples, for instance the comparison of a biological sample with a standard curve of known initial concentration, when absolute quantification is required

To evaluate the reproducibility of ^{5}, which is within the 4 to 8 logs dynamic range reported in other studies

**Additional tables**. Additional Table 1: Primer sequences and qPCR dataset description. Additional Table 2: Mean efficiency of each primer set. Additional Table 3: Induction ratio of extracellular matrix gene by TGF-β as assessed from 10 replicate assays. Additional Table 4: Induction ratio of extracellular matrix gene by TGF-β as assessed from 3 replicate assays

Click here for file

**Complete set of data and macro**. Excel file containing all raw qPCR data and the macro used into the present article.

Click here for file

**Additional Figures**. Additional Figure

Click here for file

Estimation of the efficiency of a PCR reaction

We compared estimates of the efficiency obtained from two distinct methods: the generally used serial dilution (Figure

The statistical equivalence of the LinReg and serial dilution methods was assessed using an analysis of variance (ANOVA, Table

Comparison between serial dilution and LinReg for the measurement of efficiency

Amplicon

Test of equality*

Test of variance**

Cav

p = 0.24

p < 0.001

CTGF

p < 0.001

p < 0.001

Eln

p = 1

p < 0.001

FN

p = 0.65

p < 0.001

L27

p = 0.8

p < 0.001

Perl

p = 1

p < 0.001

PAI-1

p = 0.93

p < 0.001

* Single one-way ANOVA was applied to the complete set of data to assess whether both the Serial dilution and the LinReg methods give similar averaged efficiencies (_{0}: efficiencies are equal, _{1}: efficiencies are different): a p-value below 0.05 indicates that measured efficiencies are significantly different. **F-test assessing whether the variance (error) induced by each method is the same [_{0}: variances are equal, _{1}:

Overall, we conclude that the two methods display comparable accuracy in measuring efficiency values of a set of reactions. Statistically, this implies that these methods provide acceptable estimator of the efficiency. However, LinReg appears to be more robust, as lower variances were obtained. Furthermore, LinReg can be mathematically justified when the PCR amplification is in the exponential phase (see Additional File

**Mathematical Justification of LinReg**. Justification of the LinReg method to estimate PCR efficiency, when PCR is considered as a branching process.

Click here for file

Experimental parameters influencing efficiency determination

Next, we wished to determine which of the experimental variables may affect the precision of the estimation of efficiency. This was evaluated on the complete set of quantitative PCR reactions. Figure

Distribution of efficiency values in the complete data set

**Distribution of efficiency values in the complete data set**. Efficiency of 704 PCR reactions encompassing different cDNA samples, primers and dilutions are shown as determined using the LinReg method, as done in Figure 2B.

First, we determined if single PCR parameters (amplicons, cDNA samples,

Parameters influencing the efficiency of qPCR reactions.

Parameter

Df

F-value

p-value

Amplicon

6

219.331

<0.001

Sample

10

7.227

<0.001

1

0.646

0.422

Dilution

4

2.111

0.078

Amplicon:Sample

15

4.525

<0.001

Amplicon:Ct

6

1.957

0.07

Amplicon:Dilution

20

2.576

<0.001

Sample:Ct

10

1.226

0.271

Sample:Dilution

37

1.512

0.029

4

1.872

0.114

Residuals

700

A multiple ANOVA test was performed on efficiencies obtained from 700 reactions. Dependence with amplicon sequence, sample, Ct and dilutions were tested as well as all combinations of co-dependence. Df indicates the degree of freedom. The larger the F-value, or the smaller the p value, the more significant is the effect of the corresponding parameter.

Overall, these results indicate that efficiencies are highly variable among PCR reactions and that the main factor that defines the efficiency of a reaction is the amplicon. This is consistent with the empirical knowledge that primer sequences must be carefully designed in quantitative PCR to avoid non-productive hybridization events that decrease efficiency, such as primer-dimers or non-specific hybridizations. Efficiency might also depend upon the dilution for a minority of the cDNA samples, indicating that dilute samples should be preferred to obtain reliable efficiency values.

DNA quantification models

The models we evaluated in this study can fall into two different groups: being derived from either linear or from non-linear fitting methods. Comparison of qPCR data using models based on non-linear fitting methods (Eq. 6 and Eq. 8) is done simply by calculating the ratio of the initial amount of target DNA of each amplicon (Eq. 7 and Eq. 9) as in the first part of Eq. 2. The standard deviation of the ratio on a pool of replicate is calculated using Eq. 10. Note that in this case, errors resulting from the non-linear fitting itself are not considered in the analysis.

Linear fitting methods also allow the estimation of the initial level of fluorescence induced by the target DNA. For instance, Eq. 3, upon which the LinReg method relies to determine efficiency, can also be used to determine _{0 }as the intercept to the origin of a linear regression of the log of fluorescence. This figure can then be used to calculate relative DNA levels (Eq. 2). This calculation method was termed _{0}.

However, even small errors on the determination of the efficiency will lead to a great dispersion of _{0 }values due to the exponential nature of PCR (Eq. 2). Therefore, we considered alternative calculation strategies, whereby the efficiency is averaged over several reactions rather than using individual values, which should provide more robust and statistically more coherent estimations. We therefore evaluated the use of efficiency values calculated in three different manners.

As the amplicon sequence is the main contributor to the efficiency, we used the efficiency averaged over all cDNA samples, dilutions and replicates of a given amplicon, as a more accurate estimator of the real efficiency than individual values. The error on the efficiency is no longer considered in the calculations of relative DNA concentrations, thus assuming that the estimator is sufficiently precise so that errors become negligible. This model is termed below (^{Ct}.

Alternatively, the small influence of the sample upon the efficiency was taken into account by averaging the efficiencies obtained for each dilutions and replicates of a given cDNA sample and a given amplicon. Thus, for a given cDNA sample and amplicon, one efficiency value is obtained from 24 PCR reactions. This value is used in further calculations, assuming again the average value to be a sufficiently good estimator of the efficiency so that the relative error may not be taken into account. This model was named (^{Ct}.

Finally, we tested a model in which the efficiency is estimated individually for each set of replicated reactions. This was addressed by averaging the efficiency of each replicates of a given amplicon, cDNA sample and dilution. This model is referred below as ^{ct}. These three models are summarized in Table

Models for the use of single reaction efficiencies

Grouping of individual efficiencies for average determination

Model

Amplicon (7)

cDNA (4)

Dilutions (5)

Replicates (5)

Reactions (700)

(^{Ct }*

individual

pooled

pooled

pooled

100

(^{Ct}

individual

individual

pooled

pooled

25

^{
ct
}

individual

individual

individual

pooled

5

*Each line indicates how single efficiency values are grouped in each of the calculation models. For instance, the (^{Ct }model line indicates that one individual amplicon is chosen out of the 7 available ones, then efficiencies estimated from the different cDNA samples, dilutions and replicates are all pooled, leading to the determination of one efficiency value from 100 reactions. In the (^{Ct }model, individual efficiency values are calculated for each cDNA sample and each amplicon averaging efficiency values from the 5 replicate reactions performed on the 5 dilutions (25 values overall).

Evaluation of the quantitative PCR calculation models

The dilutions of a given sample form a coherent set of data, with known concentration relationships between each dilution. Each calculation model was therefore used on each dilution series, using the undiluted sample for normalization. All data can be presented as measured relative concentrations, the undiluted dilution taking the relative concentration value of 1, the 10-fold dilution taking the value of 0.1, the 50-fold dilution a value of 0.02, and so on. The measured relative concentrations for all dilutions, samples and primers and the associated errors were calculated using each model from the complete dataset of 704 reactions. For the models giving a direct insight to the initial _{0 }values, _{0 }were averaged for each amplicon and cDNA sample, and they were plotted in comparison with the expected concentrations relative to the undiluted samples (Figure

Comparison of the different calculation models when applied to samples of known relative concentrations

**Comparison of the different calculation models when applied to samples of known relative concentrations**. Each cDNA samples serial dilutions were processed with the indicated models, measured concentration were expressed as relative to the undiluted sample. Then results of all amplicons were averaged for a given model.

We defined the resolution as the ability of a model to discriminate between two dilutions. Relative concentrations were compared pair-wise between adjacent dilutions. Typically, it can be seen in Figure ^{Ct }and (^{Ct }models performed well under this criterion, allowing easy discrimination of the 10-fold and 50-fold dilutions in this example.

The resolution was statistically evaluated with a coupled ANOVA-LSD t-test, which is a two step analysis of variance (ANOVA) coupled to a t-test run under the ^{Ct }and LR_{0 }p-value indicate that these models could discriminate the 10-fold from the 50-fold dilution, but not further dilutions. The (^{Ct }and (^{Ct }models were able to discriminate the 10-fold from the 50-fold dilution and were at the limit of significance when comparing the 50-fold with the 100 fold dilutions (significant, p < 0.1). Finally, none of the models were able to discriminate the 100 fold from the 1000 fold dilution. These comparisons indicated that (^{Ct }and (^{Ct }models performed equally well in this assay, followed by Δ^{Ct }and LR_{0}, while the sigmoid or exponential models were of low resolution. These results also illustrate that more dilute samples are generally more difficult to discriminate, as expected from the finding that variance increases with higher

Resolution of each calculation model

Relative concentrations

Δ

E^{Ct}

(Savrg E)^{Ct}

(Pavrg E)^{Ct}

LR N_{0}

Sigmoid

Exponential

t-value

p-value

t-value

p-value

t-value

p-value

t-value

p-value

t-value

p-value

t-value

p-value

t-value

p-value

1 – 0.1

122.7

0.000

18.6

0.000

141.7

0.000

141.3

0.000

18.9

0.000

29.1

0.000

18.2

0.000

0.1 – 0.02

7.3

0.000

1.9

0.030

12.6

0.000

12.6

0.000

3.5

0.000

-4.4

N/A

0.8

0.225

0.02 – 0.01

0.7

0.232

0.4

0.360

1.6

0.052

1.6

0.052

0.9

0.190

6.1

0.000

-0.4

N/A

0.01 – 0.001

0.5

0.325

0.3

0.375

1.3

0.105

1.2

0.107

0.5

0.306

0.1

0.478

2.7

0.004

A ANOVA coupled t-test was performed to determine which dilutions are statistically different from one another. The comparison of adjacent dilutions are shown. A p-value higher than 0.05 indicate that the model is unable to discriminate between the two adjacent dilutions. N/A stands for not available and indicate that the expected higher concentration of the comparison was in fact calculated to be lower from the qPCR results.

The precision of a model is defined by its ability to provide expected relative concentrations of the known dilutions. Again Figure ^{Ct }and (^{Ct }models provide precise relative concentration values over all dilutions, with the measured relative concentrations matching the expected ones. Estimations obtained by the Δ

**ΔCt systematic bias**. When not fulfilled, the Δ

Click here for file

We statistically evaluated the precision of each model by plotting the expected relative concentration against the measured relative concentration averaged from all primers and samples (Additional File ^{Ct }and (^{Ct }models outperformed all other models, being more precise than the and sigmoid models, the exponential, Δ_{0 }displaying lowest precision.

Precision of each calculation model

Model

Slope

SD

p-value

r^{2}

Sigmoid

0.085

1.225

4.4E-12

0.00004

LR N_{0}

1.830

0.621

4.7E-26

0.07441

Exponential

0.707

1.047

0.004

0.00421

E^{Ct}

1.103

0.127

1.2E-13

0.40965

ΔCt

0.689

0.013

1.4E-149

0.96041

(Pavrg E)^{Ct}

0.994

0.026

0.021

0.93213

(Savrg E)^{Ct}

0.996

0.027

0.170

0.92419

Expected and measured relative concentrations were plotted and a linear regression was performed (Additional File ^{2}) values indicate the width of the spreading of individual data around the linear regression line. Higher r^{2 }indicate more robust models.

Finally, the robustness is related to the variability of the results obtained from a given model, and it indicates whether trustable results may be obtained from a small collection of data. For instance, a model could be very precise (eg providing a slope of 1) with a large data set, but the distribution of the points around the regression line could be very dispersed. Such a model would not be robust as a small data set would not allow precise measurements. Thus, the robustness of a model was estimated from the standard deviation of the slope and the related correlation coefficient of the linear regression (^{2}), with higher ^{2 }values indicating more robust models. Three models showed high robustness, the Δ^{Ct }and (^{Ct}, followed by ^{Ct }(Table ^{Ct }and the (^{Ct }methods. However, only the slope of the (^{Ct }did not statistically differ from 1.

Model evaluation on a biological assay of gene expression regulation

Usually, experimenters are interested in the difference between two conditions (with versus without a drug, sane versus metastatic tissue, etc...) ^{Ct}, (^{Ct }and Δ

Ten replicate PCR reactions were performed for each condition (induced or non-induced) and normalized expression values obtained with (^{Ct }(^{Ct}, (^{Ct }or Δ^{Ct }(p < 0.05) but not with (^{Ct }(0.38) or Δ

Comparison of the (^{Ct}, (^{Ct }and ΔCt methods to quantify gene expression regulation

**Comparison of the ( SavrgE) ^{Ct}, (PavrgE)^{Ct }and ΔCt methods to quantify gene expression regulation**. cDNAs were prepared from RNA extracted from fibroblastic cells induced or not by TGF-β treatment, as described in the Materials and methods. Expression as determined from the mRNA levels of the plasminogen activator inhibitor 1 (PAI-1), fibronectin (FN) and connective tissue growth factor (CTGF) genes were normalized to those of the ribosomal L27 protein, used as an invariant internal reference. Normalized gene expression was calculated using the (

To assess whether the relative performance of the three models depends critically on the number of replicate assays, the analysis was repeated, but taking into account only the first three values obtained from the set of 10 replicates. Similar results were obtained (Figure ^{Ct }model. Thus, small differences in gene expression are also more reliably estimated from this model with a low number of replicates commensurate with usual experimental procedures.

Dataset size required to achieve statistical significance

In the above example, independent biological samples were mixed so as to decrease the variability associated with cell culture and mRNA isolation. Therefore, this study provides the statistical significance that may be expected just from the intra-assay variability in the qPCR process. However, statistical significance will also depend on the inter-assay, or biological variability. To assess the statistical significance associated to particular conclusions on gene expression regulation, replicates of induction experiments are usually generated and, in most experimental studies, the number of biological replicates is low, being typically obtained from 3–6 independent biological samples.

Thus, we wished to determine how many biological replicates may be necessary to obtain statistically reliable results, depending upon the variability of the assay (Eq. 15). Using the data from the 10 replicates to estimate the intra-assay variability, we found that the standard deviation is proportional to the induction ratio value (Additional File

Table

Number of measurement replicates needed to reach statistical significance

Range (%)

CV value (%)

10

20

30

40

50

15

9

3

1

1

1

30

35

9

4

3

2

40

62

16

7

4

3

50

97

25

11

7

4

The number of replicates needed to reach statistical significance depends on the reproducibility of the measures (CV) and the range of confidence one wants to achieve (the real value being within the indicated percentile from the estimator qPCR value). The confidence level α was set to 0.05 (_{0.975 }= 1.960).

Discussion

The simplicity of producing quantitative PCR data has overshadowed the difficulty of making a proper analysis of those data. Although the principle of qPCR is theoretically well described, analysis of the experimental data can become very difficult if one is not aware of the different assumptions that the different models are based on, and of their resulting limitations. Furthermore, a systematic evaluation of the relative performance of the models used for the treatment of experimental measurements and a description of their statistics are currently lacking. Thus, no single method has gained a general acceptance in the community of experimentalists.

In this study, we reviewed the mathematical basis and assumptions of previously described calculation methods and evaluated their ability to provide quantitative results from a practical dataset size. Initially, we first evaluated previously reported methods and concluded that an estimation of the PCR amplification efficiency is prerequisite to obtaining precise quantification, in agreement with other studies ^{Ct}), we generated new methods or variations (exponential fit, (^{Ct }and (^{Ct}, _{0}), and we compared all approaches with datasets generated from several independent genes and biological conditions.

Estimation of the PCR efficiency

The classical serial dilution and the newer LinReg methods used for measuring efficiency were both found to provide good estimator values of the efficiency, as based on an ANOVA analysis. However, the efficiency values were not uniformly equivalent when comparing both methods, as they were significantly different for some of the assay genes. The larger variability of efficiency values obtained with the classical serial dilution method with all test genes led us to conclude that it was not an estimator as accurate as LinReg. In addition, while both methods are very sensitive to changes of the concentration of potential inhibitors present in the sample upon serial dilutions, the serial dilution method does not allow the assessment of such effect while LinReg does

Factors affecting the efficiency value

The large variation associated with efficiency estimations, even from duplicate analysis of the same sample, led us to analyse the determinants of this variability. This analysis showed that efficiency is strongly dependent on the primer sequence. These results are in accordance with the common knowledge that careful design of the PCR primers is required to obtain useable PCR amplifications data and high efficiency values

Assay of independent biological samples was also found to significantly affect the efficiency, but to a lesser extent. Others have observed that sample to sample variations may predominate, which may reflect differences in sample preparation methods and/or distinct biological systems _{0 }concentration are not the main determinants of the PCR efficiency. Therefore, the distinct efficiencies obtained from independent samples should also reflect other properties of the sample that are not affected by dilution.

For instance, the presence of damaged or nicked cDNA in the sample has been shown to affect PCR efficiency

Evaluation of various qPCR calculation models

Models were evaluated under 3 criteria: resolution, precision and robustness. Resolution is a measure of the ability of a model to discriminate two successive dilutions. Precision is the correlation between measured and expected concentrations. Finally the robustness is a measure of the dispersion of the measured values around the expected concentrations.

Overall, two calculation models stand out: the (^{Ct }and the (^{Ct }models, as these are among the top scoring methods on the three evaluation criteria. However, (^{Ct }shows a small but statistically significant bias when comparing the obtained and expected values, suggesting that it slightly underestimates the more dilute DNA concentrations. In contrast, results calculated from (^{Ct }cannot be statistically distinguished from the expected data and are thus of higher precision. In addition, (^{Ct }displayed a higher resolution than (^{Ct }when assessed on biological samples. These models are followed by ^{Ct}, which is of lower but consistent resolution, robustness and precision.

The Δ^{Ct }or (^{Ct }models. It must be emphasized here that if PCR is performed with carefully designed and optimized primers that yield high efficiency

Finally the sigmoid, exponential and LR _{0 }models analysed here are least suitable for quantitative PCR analysis as they have a low resolution and/or precision, and because they display very low robustness. Improved versions of the original sigmoid model _{0 }value from its own set of parameters. Thus sigmoid fit methods such as the logistic model used above are purely descriptive, and biological conclusions drawn from the fitting parameters may be unreliable.

Perhaps surprisingly, the exponential fitting method also scored with very low performance, despite the expected exponential nature of DNA amplification by PCR. This may result in part from poorly characterized borders of the exponential phase, leading to the fitting of experimental points that are already in phase III. Alternatively it may result from the possible non-exponential nature of PCR that would result from both linear and exponential amplification, as discussed above. The exponential and sigmoid methods are based on descriptive models. They often produce outlier _{0 }values, suggesting that they might not be accurate mathematical models of the PCR process _{0}. An additional explanation for the inadequate performance of the sigmoid, exponential, and LR _{0 }models is that they do not explicitly determine the efficiency value, and therefore cannot make use of average efficiencies obtained from several independent measurements. These observations thus support the conclusion that the determination of a precise efficiency value is paramount to the success of qPCR, and it provides a rational explanation for this phenomenon.

Conclusion

Overall, three models stand out and may be used preferably depending on the experimental conditions and objectives: the Δ^{Ct }and (^{Ct }models. Δ^{Ct }and (^{Ct }rely on an averaged efficiency value, either performed from all data resulting from one amplicon but irrespective of the biological sample or condition, or performed over each sample and amplicon, respectively. Thus, (^{Ct }may be favoured when the same gene or sequence is to be amplified repeatedly from various biological treatments or specimens, or when following changes in the physiological or differentiation status of a cell population over time, to obtain comparative or relative estimates. In contrast, when absolute quantification of DNA and highest precision is needed, and/or when multiple sequences must be amplified from few biological samples or conditions, (^{Ct }will be the method of choice, and the statistical analysis provided in this study will allow the estimation of the dataset size required to achieve a given accuracy.

Methods

Cell culture and cDNA preparation

Primary mouse fibroblast and NIH-3T3 mouse fibroblasts were cultured in DMEM supplemented with 10% serum. Cells were exposed to 100 pM TGF-β or to the ethanol carrier for 4 hours before RNA extraction. Total RNA was extracted from confluent 75 cm^{2 }culture dish (approx 2 million cells) using Trizol reagent (Invitrogen) according to the manufacturer's protocol and resuspended in 20 μl RNAse-free water. Reverse transcription was performed with the GeneAmp Gold RNA PCR Core kit (PE Applied Biosystem) using 5 μl (approx 2.5 μg) of RNA in a 25 μl final volume using oligo-dT as a primer. The resulting cDNA solution was diluted 10-fold in deionized water and the solution thus obtained was considered as the undiluted sample (1-fold dilution) for the qPCR measurements. This final dilution step was found to be necessary to prevent inhibitory effects on the PCR efficiency that likely result from contaminant carry-over (data not shown).

Experimental set of qPCR data

To statistically qualify the quantitative PCR process and to evaluate the different models, we generated an experimental data set using 7 different amplicons. Expression of the genes listed in Additional File

At least 4–8 distinct biological cDNA samples were used in conjunction with each of the 7 different target genes (amplicon). Each of these samples was serially diluted to obtain 10-fold, 50-fold, 100-fold and for some samples 1000-fold dilutions from the undiluted (1-fold) sample. Each of these dilutions was measured in 5 replicate PCR reactions (4 replicates for the 1000× dilutions), using each of the seven amplicons. This produced a data set of 704 reactions. The complete raw data set is given in the Additional File

Quantitative PCR assays

SYBR green I technology was used for all quantitative PCR reactions, which were assembled using the Eurogentec kit RT-SN10-05 (Seraing, Belgium). Reactions were processed with 5.9 μl of cDNA samples in 25 μl final volume. One tip/well was used to distribute samples on the PCR plate in order to increase reproducibility of the data. Primers were all used at a final concentration of 100 nM and the specificity of the amplification product was verified for each reaction by examination of the corresponding dissociation curve. All PCR reactions were performed on an ABI Prism 7700 Sequence detector (PE Applied Biosystem, Foster City, CA, USA). For all reactions, cycling conditions were 95°C for 15 min (denaturation) and then 40 cycles of 95°C 15 sec – 62°C 1 min. Data acquisitions were performed with the SDS 1.9.1 software (PE Applied Biosystem, Foster City, CA, USA). Baseline limits were set as suggested by the manufacturer (i.e. at least two cycles before the rise of the earliest amplification). Threshold was set to lie in the middle of the exponential phase of the amplification plot, so that efficiency values truly reflect the reaction dynamic at the

Equations

The full mathematical development of the following equations can be found in the Additional File

**Equation development**. Detailed development of all equations 1–14 of the Methods section.

Click here for file

The exponential behaviour of DNA increase in the exponential phase is described as follows:

_{c }= _{0 }· ^{c}

where _{c }is the amount of PCR DNA product at cycle _{0 }the initial amount of target dsDNA and

When comparing distinct samples, the relative DNA concentrations can be calculated as:

where _{AB }represents the initial concentration ratio of sample A over B. Amplification efficiencies can be measured by taking the log of both side of Eq. 1, which gives a linear function of log _{c }=

_{c }= log _{0 }+

where the ordinate to the origin gives a direct estimate of _{0}, and the slope an estimate of the amplification efficiency. But in fact it must be noted that qPCR measures fluorescence that is proportional to the amount of DNA. Therefore Eq. 3 really measures _{0}, with _{0 }= _{0}. But this is not so important when measuring relative level of DNA since the ratio of initial fluorescence is equal to the initial ratio of target DNA.

When _{t}, Eq. 3 can be rearranged as:

This expresses _{0}), with a slope

Alternatively, sigmoid fitting of amplification can be performed using Eq. 6

where _{0 }are fitting parameters and

Exponential fitting can also be performed but it requires to first trim the data, removing values that are in phase III. Then the remaining data can be fitted using:

_{c }= exp[_{0})]

where _{0 }are fitting parameters and

_{0 }= exp[-_{0}]

The propagation of error was determined using a Taylor expansion to the first order. For the first part of the normalized ratio (Eq. 2), this led to:

where Δ_{AB }is the standard deviation of the ration of amplicon A over amplicon B, and Δ_{0 }and Δ_{0 }the standard deviation of the initial amount of target DNA of amplicon A and B.

For the second part of the normalized ratio (Eq. 2), the propagation of errors is described by:

Standard deviation on the efficiencies calculated with the Serial dilution were evaluated from the standard deviation of the slope of the regression using a Taylor expansion to the first order for error propagation:

where Δ

Standard deviation on the efficiencies measured with LinReg were obtained by averaging all efficiencies obtained from the same data set used for the Serial dilution.

When comparing the expression of a gene in different experimental conditions, the useful figure is the normalized induction ratio:

where _{1–2 }is the ratio of the expression of the gene of interest (induction) between condition 1 and condition 2 and _{AB(i) }is the normalized expression of the gene interest (Eq. 2) in condition

See Additional File

Finally, the number of replicate needed to reach statistical significance can be calculated as follows:

where n is the number of independent replicates,

**Statistical significance and required sample size**. Presentation of all of the equations leading to the development of eq.15 of the Methods section.

Click here for file

Authors' contributions

YK carried out the qPCR experiment, performed all of the statistical tests and wrote the article. AMN carried out some qPCR experiment. SP programmed the macro for the non-linear fitting of the amplification plots. CM provided the mathematical validation of the LinReg method. NM provided oversight of the work and helped finalize the article.

Acknowledgements

We thank prof S. Morgenthaler for help with the multi-ANOVA procedure and for helpful comments on the statistical test used herein. The financial support of the Swiss national Science Foundation and from the Etat de Vaud is gratefully acknowledged.