Breast Unit, Western general Hospital, Edinburgh, UK

Novartis Pharmaceuticals, Biostatistics, CH – 4002 Basel, Switzerland

Breast Unit, Edinburgh University, Edinburgh, UK

Abstract

Background

Currently real time PCR is the most precise method by which to measure gene expression. The method generates a large amount of raw numerical data and processing may notably influence final results. The data processing is based either on standard curves or on PCR efficiency assessment. At the moment, the PCR efficiency approach is preferred in relative PCR whilst the standard curve is often used for absolute PCR. However, there are no barriers to employ standard curves for relative PCR. This article provides an implementation of the standard curve method and discusses its advantages and limitations in relative real time PCR.

Results

We designed a procedure for data processing in relative real time PCR. The procedure completely avoids PCR efficiency assessment, minimizes operator involvement and provides a statistical assessment of intra-assay variation.

The procedure includes the following steps. (I) Noise is filtered from raw fluorescence readings by smoothing, baseline subtraction and amplitude normalization. (II) The optimal threshold is selected automatically from regression parameters of the standard curve. (III) Crossing points (CPs) are derived directly from coordinates of points where the threshold line crosses fluorescence plots obtained after the noise filtering. (IV) The means and their variances are calculated for CPs in PCR replicas. (V) The final results are derived from the CPs' means. The CPs' variances are traced to results by the law of error propagation.

A detailed description and analysis of this data processing is provided. The limitations associated with the use of parametric statistical methods and amplitude normalization are specifically analyzed and found fit to the routine laboratory practice. Different options are discussed for aggregation of data obtained from multiple reference genes.

Conclusion

A standard curve based procedure for PCR data processing has been compiled and validated. It illustrates that standard curve design remains a reliable and simple alternative to the PCR-efficiency based calculations in relative real time PCR.

Background

Data processing can seriously affect interpretation of real time PCR results. In the absence of commonly accepted reference procedures the choice of data processing is currently at the researcher's discretion. Many different options for data processing are available in software supplied with different cyclers and in different publications

The standard curve method simplifies calculations and avoids practical and theoretical problems currently associated with PCR efficiency assessment. Widely used in many laboratory techniques this approach is simple and reliable. Moreover, at the price of a standard curve on each PCR plate it also provides the routine validation for methodology. To benefit from the advantages of the standard curve approach and to evaluate its practical limitations we designed a data processing procedure implementing this approach and validated it for relative real time PCR.

Results

Description of the data processing procedure

Source data

Raw fluorescence readings were exported from Opticon Monitor software and processed in MS Excel using a VBA script (the mathematical formulae, script and samples of source data are attached to the electronic version of publication, see Additional files

Pdf file with formulae.

Click here for file

ZIP file containing VBA macros (PCR1.xls), test data for the above macros (Target1.csv, Target2.csv, Target3.csv, Target4.csv, Target5.csv, Reference1.csv, Reference2.csv) and instruction to the above macros (Instructions.pdf). Unzip file into a separate folder on your PC and follow the instructions.

Click here for file

Noise filtering

The random cycle-to-cycle noise was reduced by smoothing with a 3 point moving average (two-point average in the first and the last data points). Background subtraction was performed using minimal value through the run. If significant scattering in plateau positions was observed it was removed by amplitude normalization (normalizing by maximal value in the cell over the whole PCR run). The noise filtering is illustrated in the Figure

Noise filtering

**Noise filtering**. Axes: vertical – fluorescence, horizontal – cycle number, A Source data, B Smoothing, C Baseline subtraction, D Amplitude normalization

Crossing points calculation

The crossing points (CPs) were calculated directly as the coordinates of points in which the threshold line actually crossed the broken lines representing fluorescence plots obtained after the noise filtering (Figure

Direct calculation of crossing points

Direct calculation of crossing points.

Standard curve calculation

A standard curve was derived from the serial dilutions by a customary way. Relative concentrations were expressed in arbitrary units. Logarithms (base 10) of concentrations were plotted against crossing points. Least square fit was used as the standard curve.

Threshold selection

The optimal threshold was chosen automatically. The VBA script examined different threshold positions calculating coefficient of determination (r^{2}) for each resulting standard curve. The maximum coefficient of determination pointed to the optimal threshold (typically the maximum r^{2 }was larger than 99%).

Calculating means and variances of means for crossing points in PCR replicas

The optimal threshold was used to calculate CPs for unknown samples. Means and variances of means were then calculated for CPs in PCR replicas.

Derivation of non-normalized values from crossing points

The non-normalized values were calculated from the CPs' means by the standard curve equation followed by exponent (base 10). The variances were traced by the law of error propagation.

Summarizing data from several reference genes to a single normalizing factor

Two options are available in the VBA script to summarize data from multiple reference genes:

- Arithmetic mean (deprecated),

- Geometric mean (recommended).

Calculation of normalized results for target genes

The final results representing relative expression of target genes were calculated by dividing the non-normalized values by the above normalization factor. The normalized results' variances were derived by the law of error propagation.

When confidence intervals or coefficients of variation were needed they have been calculated from the corresponding variances (see

Procedure testing and validation

We tested this procedure on the measurement of expression of 6 genes in 42 breast cancer biopsies (Figure

Expression of Cyclin B1 mRNA in breast cancer biopsies

**Expression of Cyclin B1 mRNA in breast cancer biopsies**. The observed decrease of Cyclin B1 expression after treatment was expected in most but not all cases. Bars show actual 95% confidence intervals estimated by the described statistical procedure in a set of real clinical specimens (

Primers' sequences

**Short name**

**Full name**

**GenBank number**

**Primers**

SCGB2A2

Mammaglobin 1 (Secretoglobin, family 2A, member 2)

NM_002411

TCC AAG ACA ATC AAT CCA CAA G

AAA ATA AAT CAC AAA GAC TGC TG

SCGB2A1

Mammaglobin 2 (Secretoglobin, family 2A, member 1)

NM_002407

AAG ACC ATC AAT TCC GAC ATA

CAC CAA ATG CTG TCG TAC ACT

CCNB1

Cyclin B1

NM_031966

CAT GGT GCA CTT TCC TCC TT

CAG GTG CTG CAT AAC TGG AA

CKS2

CDC28 protein kinase regulatory subunit 2

NM_001827

TTC ATG AGC CAG AAC CAC AT

CTC GTG CAC AGG TAT GGA TG

PTN

Pleiotrophin (heparin binding growth factor 8, neurite growth-promoting factor 1)

NM_002825

GTG CAA GCA AAC CAT GAA GA

GCT CGC TTC AGA CTT CCA GT

LPIN2

Lipin 2

NM_014646

TTG TTG CTG CAG ATT GAT CC

CCA AAT GGC AAT GGA TTT TC

ACTB

Actin, beta

NM_001101

GGA GCA ATG ATC TTG ATC TT

CCT TCC TGG GCA TGG AGT CCT

GAPD

glyceraldehyde-3-phosphate dehydrogenase

NM_002046

TGC ACC ACC AAC TGC TTA GC

GGC ATG GAC TGT GGT CAT GAG

Primers for

To validate the assumption of a Normal distribution for the initial data (

Distribution of crossing points in PCR replicas

**Distribution of crossing points in PCR replicas**. Axes: vertical – relative frequency (%), horizontal – crossing points. Histogram represents a typical crossing points' distribution in 96× replica (Plate 1 from Table 2). The Kolmogorov-Smirnov test has not revealed significant deviations from the Normal distribution. The red line shows a Normal fit.

Crossing points' distributions observed in PCR replicas

**Plate**

**Number of replicates**

**Mean CP**

**SD**

**CV**

**Skewness**

**Kurtosis**

**Kolmogorov-Smirnov test**

1

96

21.48

0.06

0.3%

0.1

-0.1

Normal

2

94

18.09

0.07

0.4%

1.5

5.7

Sharper than normal

3

96

20.09

0.04

0.2%

0.1

-0.3

Normal

4

96

18.13

0.10

0.5%

0.5

1.0

Normal

Transformation of the Normal distribution through PCR data processing was analyzed by a computer simulation. It showed that the shape of resulting distributions significantly depends on the initial data dispersion. At low variation in crossing points (SD < 0.2 or CV < 1%) the distributions remain close to Normal through all steps of data processing (Figure

Transformation of normal distribution through data processing

**Transformation of normal distribution through data processing**. Axes: vertical – relative frequency (%), horizontal – results. Red lines show Normal fits. **A: **At CPs' CV 0.5% the deviations from normality were not detectable using the Kolmogorov-Smirnov test. **B: **At CPs' CV 1% the deviations from normality were not detectable in non-normalized values though moderate deviations were detectable in final results. **C: **At CPs' CV 2% deviations from normality were detectable in both non-normalized values and in final results.

Addressing the use of amplitude normalization we studied several factors potentially affecting PCR plateau level. On the gels run immediately after PCR the weak bands initially visible without staining because of SYBR Green originated from PCR mixes were remarkably increased after additional staining with SYBR Green (Figure

Effect of staining with SYBR Green 1 on PCR gel

**Effect of staining with SYBR Green 1 on PCR gel**. A: Before staining. B: After staining. Before electrophoresis SYBR Green1 was added to marker but not to samples.

Effect of different factors on plateau position

**Effect of different factors on plateau position**. A: More enzyme in blue than in red samples B: More primers in blue than in red samples C: Domed and plain caps

Discussion

PCR data processing is a complex procedure that includes a number of steps complementing each other. Many different options have been suggested by different authors at each step of PCR data processing. In the discussion below we go through our procedure on a step-to-step basis shortly discussing the available options and explaining our choices. In general, we preferred the simplest functioning solutions. In statistical treatment we looked for valid practical estimations rather than for mathematically exact solutions. Because of lack of relevant theoretical data we paid especial attention to the amplitude normalisation and to statistical processing of intra-assay PCR replicas. To validate these sections of our procedure we had to address some basic theoretical issues.

PCR data processing may need to be optimized for specific PCR machines and chemistry. The discussed processing was optimized for data obtained on an Opticon Monitor 2 machine (MJ Research) using the QuantiTect SYBR Green PCR kit (Qiagen).

Smoothing

Smoothing is necessary if noticeable non-specific scattering from cycle to cycle is observed on the raw fluorescence plots. Apart from moving averages there are other more sophisticated mathematical approaches to filter this kind of noise

Background subtraction

Background subtraction is a common step in PCR data processing. Often it requires operator's involvement to choose between several available options (

Amplitude normalization

Amplitude normalization unifies plateau positions in different samples. Although amplitude normalization was available in some versions of Light-Cycler software and has been used by some researchers

Amplitude normalization is based on the suggestion that in ideal PCR, output is determined by the initially available PCR resources. In this case PCRs prepared from the same master mix will run out of the same limiting resource in different samples. The resource can run out sooner (abundant template) or later (rare template) but finally the same amount of PCR products will be produced in all samples. This assumption is valid for ideal PCR but in practice it may not always hold (for example, non-specific PCR products may also consume PCR resources). The factors potentially leading PCR to the plateau include utilization of primers or nucleotides, thermal inactivation of DNA polymerase, competition between primers and PCR products for annealing, enzyme inactivation by PCR products and accumulation of inhibitors

In this work we used QuantiTect SYBR Green PCR kit (Qiagen). With this kit neither increase of primers nor addition of enzyme notably affected the plateau positions (Figure

What may cause the plateau scattering in fluorescence plots? In certain cases, it may be optical factors. Freshwater

Optical factors affect the plateau scattering

**Optical factors affect the plateau scattering**. SYBR Green real time PCR in frosted plates (green) and white plates (blue). Frosted plates cause increased plateau scattering because of inconsistent reflection and refraction (Reproduced from [18], with ABgene^{® }permission).

So far, lack of understanding of the PCR plateau nature makes the amplitude normalization an optional step. When used, amplitude normalization should be empirically validated in each individual plate. Linearity of the standard curve may act as an empirical test for amplitude normalization,

Effect of amplitude normalization on standard curve

Effect of amplitude normalization on standard curve.

Finally, a "PCR-specific" explanation of plateau scattering can not explain the scattering observed in PCR replicas (Figure

Effect of amplitude normalization on plateau scattering in 96× replica

**Effect of amplitude normalization on plateau scattering in 96× replica**. Axes: vertical – Fluorescence, horizontal – Cycle. Data for plate 3 from Table 2.

Threshold selection

As long as the standard curve provides both basis and empirical validation for PCR results the threshold may be put at any level where it produces a satisfactory standard curve. At the same time, the linearity of standard curve is theoretically explained at exponential phase of PCR only. Therefore, the common practice is to put the threshold as low as possible to cross the fluorescence plots in the exponential phase. For this reason we usually restrict the search of the optimal threshold position to the lower half of the fluorescence plot.

Crossing point calculation

Currently the most established methods of crossing point calculations are the fit point method and the second derivative maximum method

Our calculation method also produces good results. In addition, it is simple and does not alter the initial mathematical definition of crossing points.

Statistical treatment of PCR replicas

The next step in the data processing is derivation of results from crossing points. Two separate issues need to be addressed during this step: (i) best-fit values and (ii) errors in replicates. Calculation of best-fit values is simple with standard curve methodology (see formulae in

Description and interpretation of intra-assay PCR variation

PCR uncertainty is usually characterized by coefficient of variation. This reflects the fact that the errors propagated to non-normalized values and to final results are higher at higher best-fit values. This is not always the case with the crossing points. However, coefficients of variation still may be used for rough comparison of CPs' dispersions because the CPs' absolute values vary in quite a limited range (typically between 20 and 30 cycles).

Importantly, that during PCR interpretation the statistical significance of differences between samples should not be based on intra-assay variation. Intra-PCR replicates account only for errors originated from PCR. At the same time the uncertainty in final results is usually more affected by pre-PCR steps

Starting point for statistical assessment

Two different approaches may be utilized for initial statistical handling of intra-assay PCR replicates. Either CP values are first averaged and then transformed to non-normalized values or

Crossing point distribution in PCR replicas

To choose appropriate statistical methods to deal with crossing points, we started from the assessment of crossing points' distributions in PCR replicates. Distributions of crossing points were studied in four PCR plates each of those represented a 96× replicate. The distributions were close to the Normal (Table

Error propagation

The CPs' variances were traced to final results by the law of error propagation. This assumed the normality of distributions not only in crossing points but also at the later steps of data processing. Strictly speaking, this assumption is not completely true: the data processing deforms normal distribution. Three functions are used to calculate results from crossing points: linear function (linear standard curve), exponent (calculation of non-normalized values) and ratio (normalizing by reference genes). Among them only linear function keeps normality of distribution. Exponent and ratio distort it. At the same time, the degree of the introduced distortion depends on particular numeric parameters. Analyzing the deformation of normal distribution at the parameters typical for real time PCR we found that at low initial dispersions the resulting distributions remain close to normal (Figure

Magnitude of propagated error at different steps of data processing

**SD in crossing points**

**CV in crossing points**

**CV in non-normalized values**

**CV in normalized results**

0.1

0.5%

7%

10%

0.2

1.0%

14%

20%

0.3

1.5%

22%

31%

0.4

2.0%

28%

40%

0.6

3.0%

45%

66%

In all instances mean values are 20 in crossing points, 10 in non-normalized values and 1 in final results. See Figures 5 and 13 for more details.

Additionally the analysis confirmed the remarkable increase of relative variation at each step of data processing.

Standard curves

In line with the common practice, we interpreted the standard curve as an ordinary linear function ignoring its statistical nature and uncertainty because the uncertainty was usually quite small (typical coefficient of determination above 99%). With sufficient number and range of standard dilutions and proper laboratory practice it is always should be possible to produce the standard curve of sufficient quality.

Specific design of standard curves may differ for different genes depending on the variability of their expression. For relatively stabile genes (

Even though the standard curves could be quite reproducible

Summarizing data from several reference genes

Several reference genes are required for accurate relative quantification

Arithmetic mean is the most "intuitive" way. However, it has a major disadvantage: it depends on arbitrary choice of the absolute values for reference genes. For example, the normalizing factor will differ, if a reference gene is described either as a fraction of 1 (absolute values from 0 to 1) or in percents (values 0% to 100%). Importantly, this can change the

Obviously, the different ways of summarizing data from reference genes will produce different results. At the same time, at truly stable expression of reference genes the general tendencies in results should be similar. Currently we calculate the single normalizing factor by geometric mean, because it better fits to the relative nature of measurements as well as to the logarithmic scale of gene expression changes

Unfortunately common practice tends to ignore the uncertainty of normalizing factor. Our procedure estimates this uncertainty using the law of error propagation (see formulae in

Methods based on PCR efficiency and individual shapes of fluorescent plots

Standard curve approach was chosen for our procedure because currently PCR efficiency assessment may complicate data processing. The main complication is that actual efficiency of replication is not constant through the PCR run being high at exponential phase and gradually declining toward the plateau phase. However, most current methods of PCR efficiency assessment report "overall" efficiency as a single value. Additionally, PCR efficiency may be calculated in different ways that can "overestimate" or "underestimate" the "true" PCR efficiency

At present the most popular method of PCR efficiency assessment is based on the slope of standard curve. This method does not account for PCR efficiencies in individual target samples. In contrast, recent publications on PCR efficiency assessment were concentrated on the analysis of individual shapes of fluorescence plots

Limitations of our data processing

This section summarizes conditions that must be adhered to in order to obtain valid results with our data processing:

• all PCRs must achieve doubtless plateau and no non-specific PCR products should be observed to use amplitude normalization;

• standard curves with coefficient of determination above 99% are required to ignore uncertainty of regression and to validate the use of amplitude normalization;

• low dispersion in PCR replicates (crossing points' CV < 1% or SD < 0.2) is required to use the conventional statistical methods.

These limitations are linked: amplitude normalization provides the low dispersion in replicas needed for statistical treatment.

Conclusion

In this article we described a procedure for relative real time PCR data processing. The procedure is based on the standard curve approach, does not require PCR efficiency assessment, can be performed in fully automatic mode and provides statistical assessment of intra-assay PCR variation. The procedure has been carefully analyzed and tested. The standard curve approach was found a reliable and simple alternative to the PCR-efficiency based calculations in relative real time PCR.

Methods

Tissue samples, RNA extraction, reverse transcription

Breast cancer biopsies were taken from 21 patients before and after treatment with an aromatase inhibitor. Samples were obtained in the Edinburgh Breast Unit (Western General Hospital, Edinburgh) with patients' informed consent and ethical committee approval. Biopsies were snap frozen and stored in liquid nitrogen until RNA extraction. Before RNA extraction the frozen tissue was defrosted and stabilized in RNA-later-ICE reagent (Ambion). Total RNA was extracted with RNeasy-mini columns (Qiagen). Amount and purity of RNA were evaluated by spectrophotometer. RNA integrity was confirmed by agarose gel electrophoresis.

cDNA was synthesised with SuperScript III reverse transcriptase (Invitrogen) in accordance with the manufacturer's recommendations. Briefly:

1) oligo(dT)_{20 }primers and dNTPs were added to total RNA,

2) the mix was heated to 65°C for 5 min and then chilled on ice,

3) first-Strand buffer, DDT, RNase inhibitor (RNaseOUT, Invitrogen) and Reverse transcriptase were added to specimens,

4) reverse transcription was carried out for 60 minutes at 50°C.

PCR

Calibrator preparation, cDNA dilution and PCR plate set up were performed as illustrated in Figure

PCR set up

PCR set up.

1. Aliquots of cDNA samples running on the same plate were pooled and the pool was used as calibrator.

2. cDNAs were diluted with water prior PCR.

3. The set of samples consisting of the diluted cDNAs and the dilutions of the calibrator were used for several PCR plates: one plate for each gene.

4. For each sample the whole PCR mix including primers and cDNA was prepared before dispensing into the plate.

5. Samples were loaded to 96× PCR plates by 15 μl per cell in triplicates or quadruplicates.

Primer's sequences are given in Table

Several additional PCRs were run with different amount of primers (0.1 μM, 0.3 μM, 0.9 μM), different amount of enzyme (0.8U, 1.5U and 3.1U of HotStarTaq, Qiagen were added to 15 μl PCRs made with QuantiTect SYBR Green PCR mix, Qiagen) and different caps (domed and plain caps, MJ Research).

PCR product electrophoresis

Electrophoreses were run immediately after PCRs. 10 μl of PCR products were mixed with 2 μl of loading buffer. 6 μl of the mix per well was loaded into 10% PAAG (TBE Ready Gel, Biorad). Electrophoresis was run at 100 V for ~1 hr using MiniProtean-II cell (Biorad).

Prior electrophoresis 1 μl of 1:100 Sybr-Green-1 (Molecular Probes) was added into molecular weight marker (PCR Low Ladder Set, Sigma) but not into the PCR samples. After electrophoresis the gels were stained for 10 min in fresh prepared 1:10000 SybrGreen-1 (Molecular Probes). Photos were taken before and after staining using the GelDocMega4 gel documentation system (Uvitec).

96× PCR replicas

To study distributions of crossing points in PCR replicas four PCR plates have been run with a 96× replica on each. The distributions were evaluated using histograms, skewness and kurtosis measures, and the Kolmogorov-Smirnov test for Normality (see Table

Normal distribution transformation through the data processing

The transformation of Normal distribution through data processing was studied by computer simulation (Figure

Computer simulation of PCR data processing

**Computer simulation of PCR data processing**. Computer simulation of PCR data processing at 1% CV in crossing points (see Methods for details).

Basing on the above empirical observations (Table

Parameters used in calculations were close to actual parameters typically observed in our PCRs (MeanCP = 20, Slope = -0.3, Intercept = 7). The resulted true values for non-normalized and normalized results were 10 and 1 correspondingly.

To study error propagation at different initial dispersions we performed simulations using the Normal distributions with different variances (CV 0.5%, 1%, 1.5%, 2%, 3%, 4% and 5%; the means were always 20). Detailed illustration for CV 1% is presented in Figure

Excel VBA macros

The calculations where performed using MS Excel VBA script included to the electronic version of publication (see Additional file

List of abbreviations

CP (CPs) – crossing point (crossing points)

SD – standard deviation

CV – coefficient of variation

r^{2 }– coefficient of determination in linear regression

Authors' contributions

AL carried out the main body of the project including PCR, statistics and programming.

WM conceived of the study and participated in its design and co-ordination.

AK verified statistical methods and mathematical calculations.

All co-authors contributed to the manuscript preparation.

Acknowledgements

The study was supported by an educational grant from Novartis. Preliminary results were presented at 1^{St }International qPCR Symposium (3–6 March, 2004, Freising-Weihenstephan, Germany,