Model based analysis of real-time PCR data from DNA binding dye protocols
- Mariano J Alvarez^{1, 2, 4}Email author,
- Guillermo J Vila-Ortiz^{1},
- Mariano C Salibe^{1},
- Osvaldo L Podhajcer†^{2} and
- Fernando J Pitossi†^{3}
https://doi.org/10.1186/1471-2105-8-85
© Alvarez et al; licensee BioMed Central Ltd. 2007
Received: 05 January 2007
Accepted: 09 March 2007
Published: 09 March 2007
Abstract
Background
Reverse transcription followed by real-time PCR is widely used for quantification of specific mRNA, and with the use of double-stranded DNA binding dyes it is becoming a standard for microarray data validation. Despite the kinetic information generated by real-time PCR, most popular analysis methods assume constant amplification efficiency among samples, introducing strong biases when amplification efficiencies are not the same.
Results
We present here a new mathematical model based on the classic exponential description of the PCR, but modeling amplification efficiency as a sigmoidal function of the product yield. The model was validated with experimental results and used for the development of a new method for real-time PCR data analysis. This model based method for real-time PCR data analysis showed the best accuracy and precision compared with previous methods when used for quantification of in-silico generated and experimental real-time PCR results. Moreover, the method is suitable for the analyses of samples with similar or dissimilar amplification efficiency.
Conclusion
The presented method showed the best accuracy and precision. Moreover, it does not depend on calibration curves, making it ideal for fully automated high-throughput applications.
Background
The reverse transcription polymerase chain reaction (RT-PCR) is the most sensitive method for the detection of specific mRNAs [1]. However, due to the exponential nature of the PCR amplification process, small differences in amplification efficiency among samples may led to very different product yields, making RT-PCR unsuitable for quantitative purposes. The use of exogenously added standard sequences as internal competitors has overcome partially this issue [2–4], but the setting up of the system must be done for each target sequence taking into account multiple error sources [5], making competitive PCR unsuitable for high-throughput applications. The recent introduction of fluorescence techniques and instruments able to quantify the DNA content in each cycle had lead in the last years to the development of real-time PCR [6–8]. Because the high sensitivity of fluorescent product detection, real-time PCR does not rely on end-point analyses. Moreover, cycle-by-cycle data generated by real-time PCR provides information about the kinetics of the amplification process, overcoming the limitations of classical RT-PCR. Recently, the introduction of double-stranded DNA specific dyes [9] allowed the quantification of multiple targets without the need of specific fluorescent probes, making real-time PCR a popular method for microarray data validation [10].
Most currently used real-time PCR data analysis methods are based on determining the threshold cycle (CT), which is the cycle number at which a fixed amount of product is formed [11]. The comparative CT method, a broadly used semi-quantitative one, is based on the exponential description of the PCR process assuming constant amplification efficiency equal to 1 [12]. However, in our applications of SYBR-Green I real-time PCR, we have found amplification efficiency always lower that 1. Moreover, it has been shown that a difference as small as 4% in PCR efficiency could be translated to 400% error in comparative CT method based quantifications [13]. Thus, reliable quantitative real-time PCR depends on good estimations of PCR efficiency. Several methods have been proposed for amplification efficiency estimation. One of them uses a dilution curve to estimate the amplification efficiency of each target sequence [14], while the others estimate the amplification efficiency from single reaction data [13, 15–17]. Here we introduce a new mathematical model based on the classic exponential description of the PCR, in which amplification efficiency was modelled as a sigmoid function of the product yield. The model was validated with experimental data and it was used for the development of a new method for real-time PCR data analysis. This model based method for real-time PCR data analysis (MoBPA) estimates PCR amplification efficiency from single sample reaction data, eliminating the need of calibration curves, which is a drawback for high-throughput implementations [18]. Moreover, MoBPA showed the highest accuracy and precision compared with previous methods when used for quantification of samples amplified with similar or dissimilar efficiency.
Results and discussion
The model
According to its discrete nature, the PCR process can be expressed by the difference equation,
T_{n+1}= T_{ n }·(1 +E _{ n }); E_{ n }∈ (0,1) (1)
where Tm and b are parameters to be fitted by non-linear regression. There is no relationship between these parameters and kinetic parameters such as Fmax and k [18], since our model is defined as a function of the product yield instead of the PCR cycle number.
The intrinsic amplification efficiency Ei (i.e. the putative amplification efficiency for a product (or template) amount equal zero)[5], is useful for T_{0} estimation (see below), and can be obtained from Eq. (2) as,
Model based estimation of the initial template amount (T0)
In dsDNA binding dye protocols, real-time thermocyclers generate fluorescence intensity data. For most applications in which the dsDNA binding dye is in great excess, it can be assumed that the fluorescence intensity is proportional to the amount of double stranded DNA (product yield) [18]. However, it has been shown that this is not always the case [21]. Our model parameters are fitted with fluorescence intensity data, thus the product yield (T_{ n }) will be expressed in arbitrary fluorescent units, as the initial template amount (T_{0}). Since the fluorescent dye will be in great excess when compared to the initial template amount, we can assume that T_{0} is proportional to the initial template amount, and so the semiquantitative comparison between different samples using T_{0} is valid.
The simplest way to estimate T_{0} is assuming that amplification efficiency at cycle CT (E_{ CT }) is very similar to Ei. This assumption is usually correct because CT is defined at small amounts of PCR product. Then, solving Eq. (1) for n = CT and assuming constant amplification efficiency we obtain:
T_{ CT }= T_{0} (1 + E)^{ CT } (4)
from which,
T_{0} = T_{ CT }(1 + E)^{-CT} (5)
Ei can be estimated from Eq. (3), and the efficiency parameters b and Tm can be obtained fitting Eq. (2) to real-time PCR data by non-linear regression (see methods for details). Then, T_{0} is calculated assuming E = Ei. Since Eq. (5) is a power function of the amplification efficiency, small errors in the estimation of Ei can be translated to strong errors in T_{0} estimation. These errors can be minimized using CT values as small as possibly, and estimating the amplification parameters with pooled data from replicates to improve E_{ i }estimation accuracy. Replicates must be done by splitting a master-mix containing all components of the PCR to minimize the variability introduced be the operator.
Estimation of model parameters.
A | B | |||||||
---|---|---|---|---|---|---|---|---|
Mean ± | Correlation | Mean ± | ||||||
Dilution | T _{0} | SE | SEM | T_{0} × b | T_{0} × Tm | T _{0} | SE | SEM |
10 | 8.52 | 2.27 | 8.63 ± | 0.963 | -0.97 | 12.4 | 0.29 | 12.17 ± |
10 | 7 | 2 | 0.98 | 0.963 | -0.977 | 11.2 | 0.34 | 0.52 |
10 | 10.4 | 1.95 | 0.963 | -0.969 | 12.9 | 0.36 | ||
1 | 0.42 | 0.06 | 0.967 | -0.962 | 0.981 | 0.018 | ||
1 | 1.08 | 0.56 | 1 ± 0.32 | 0.964 | -0.948 | 0.804 | 0.024 | 1 ± 0.12 |
1 | 1.51 | 0.74 | 0.965 | -0.977 | 1.21 | 0.043 | ||
0.1 | 0.09 | 0.04 | 0.073 ± | 0.968 | -0.965 | 0.121 | 0.0032 | 0.12 ± |
0.1 | 0.065 | 0.015 | 0.0087 | 0.969 | -0.974 | 0.111 | 0.0026 | 0.0028 |
0.1 | 0.063 | 0.023 | 0.968 | -0.921 | 0.117 | 0.0028 |
Comparison of MoBPA with previous methods
Next, we evaluated the performance of different methods for quantification when amplification efficiency for different samples is not the same. For this, we analysed two different datasets: 1) in-silico generated data of real-time PCR runs starting at the same initial template amount but differing in their intrinsic amplification efficiency; and 2) real experimental data obtained by amplifying the same amount of mouse midbrain cDNA with β_{2}-microglobulin specific primers, but with different amounts of Taq DNA polymerase (Fig. 2D).
Simulation data was generated in-silico from eqs. (1) and (2), using parameters that resemble real PCR runs. Results from amplification of mouse midbrain cDNA with β-actin and β_{2}-microglobulin specific primers were used to estimate plausible values for the simulation parameters. The analysis of these simulation results must be taken with care, because we used the same model for generating and fitting the data, so some overfitting is expected. However, we also evaluated the performance of the different methods on real PCR data. To obtain different amplification efficiency in our PCR runs, we used two different amount of DNA polymerase. Efficiencies estimated with our approach from triplicate results were 0.855 and 0.915 for 0.1 and 0.25 units of Taq DNA polymerase, respectively. The use of more than 0.25 units of DNA polymerase did not led to additional increments in amplification efficiency, while PCR with less that 0.1 units of DNA polymerase showed no product. We also tried to partially reduce the PCR amplification efficiency by adding Mg^{2+} chelating agents like EDTA, by increasing the amount of dNTPs, by lowering the amount of Mg^{2+}, and by adding known DNA polymerase inhibitors like phenol. However, we could not find the appropriate conditions to achieve the partial inhibition of DNA polymerase in a reproducible way.
Both in-silico and experimental real-time PCR data were analysed with different methods. The simpler and widespread used methods are based on CT value determinations, assuming constant amplification efficiency during the exponential phase of the PCR and among different samples [11, 22]. Then, from Eq. (5) is easy to solve the initial template amount ratio between two samples as:
where T_{0} is the initial template amount, E is the amplification efficiency, and CT_{ i }the threshold cycle for each sample. The simplest form of threshold-based methods even assumes amplification efficiency equal 1. Thus, the ratio of initial template amounts between two samples will be 2^{ΔCT}[12]. Because of the exponential nature of this expression, amplification efficiencies below 1 will lead to unreliable quantifications. Amplification efficiency below 1 is frequent and indeed, most of our real-time PCR reactions showed amplification efficiencies between 0.7 and 0.95. To overcome this limitation, a dilution series based method for amplification efficiency estimation has been developed [14, 23]. A curve is constructed by amplification of serial dilutions of one reference sample and plotting the resulting CT values against the base 10 logarithm of the dilution factor. Assuming constant amplification efficiency between dilutions and over the number of thermocycles required to reach CT, the amplification efficiency can be obtained from Eq. (4) as E = 10^{-1/slope}-1. However, sample contamination with salt, phenol, chloroform, etc. may result in a lower-than-expected PCR efficiency [13, 24, 25]. Dilution of this sample will also dilute the contaminant decreasing its effect on the PCR reaction, thereby increasing the PCR efficiency with each dilution step. Indeed, we usually observe such dilution effect on PCR amplification efficiency, see for example data presented in Fig. 2A, in which estimated efficiency from the dilution series was 0.853, while estimations of the intrinsic amplification were 0.835, 0.852 and 0.856 for dilutions 100, 10 and 1, respectively. Amplification efficiency estimations from such dilution series will be inaccurate, introducing a bias in the quantification. Moreover, contaminants can quantitatively and qualitatively differ between samples, thus, the assumption of equal amplification efficiencies between samples can lead to strong errors in quantifications.
Four different methods for single reaction amplification efficiency estimation have been proposed. All of them use the kinetic data generated by real-time PCR cyclers, they do not assume equivalent amplification efficiency among samples and have the additional advantage that no dilutions curve is needed. Three of them assume constant amplification efficiency during the exponential phase of the PCR and estimate it from the few data points that fall into this phase [13, 15, 16]. The simplest one is based on the product yield measured at two thresholds along the exponential phase of the PCR, from which the amplification efficiency is estimated as E = (T_{1} /T_{2})^{1/(CT 1-CT 2)}- 1 [15]. Then, the initial template amount is calculated from Eq. (5). Based on the same model for the PCR reaction, Ramakers et. al. proposed the use of all the points of the background subtracted and logarithm transformed PCR data that fall into a "window-of-linearity", for amplification efficiency estimation by linear regression [13]. Recently, Tichopad et.al. introduced a similar approach based on a new statistical method for the identification of the exponential phase and estimation of amplification efficiency by non-linear regression [16]. Analysis of in-silico generated results with different amplification efficiencies showed better accuracy of these three approaches when compared with the threshold-based methods, since they introduced only 2.3 – 4.8 fold bias in quantifications (Fig. 3A). However, since T_{0} determinations depend on both efficiency and CT estimations, precision in quantification of replicates was very poor (mean CV: 58%, range: 19–134%). Indeed, quantification of samples that differ less than 10% in amplification efficiency were more accurate when analysed with the Pfaffl or efficiency 1 threshold-based methods [12, 14], most likely due to the poor precision of single reaction based efficiency estimations. These observations were further confirmed with experimental real-time PCR data (Fig. 3B).
Recently, Liu et.al. proposed a three parameters sigmoidal function for modelling the whole kinetic process of real-time PCR. Then, the initial template amount is estimated after fitting the sigmoidal model to background subtracted real-time PCR results [17]. Analysis of in-silico PCR results showed a good accuracy and moderate precision of this method for most PCR simulations, with a fold bias < 1.4 and a mean CV of 12% (range: 5–21%) (Fig. 3A). However, the model presented systematic deviations from the data that were particularly noted with amplification efficiencies above 0.9, both with experimental and in-silico generated data. These deviations had a strong effect on T_{0} estimation, leading to unreliable quantifications (Fig. 3B). Similar deviations were recently described in the plateau phase by Rutledge, who showed that elimination of these data points improve the accuracy of this method [18].
Finally, quantification from the same in-silico and experimental real-time PCR results with our model based approach showed the highest accuracy and precision. Analysis of in-silico data showed that 20% difference in amplification efficiency between samples only led to 0.31 fold bias, with a mean CV of only 2.6% (range: 0.8–4.2%) (Fig. 3A). Moreover, an increase of 7% in experimental real-time PCR amplification efficiency only led to 13% under-estimated quantification (mean CV: 10.5%) (Fig. 3B).
Analysis of real-time PCR results with similar amplification efficiency among samples. Data represent the quantification of dilutions 0.1 and 10 as the mean ± SEM for 12 experiments performed in triplicate.
Dilution | Analysis method | ||||||
---|---|---|---|---|---|---|---|
1 | 2 | 3 | 4 | 5 | 6 | 7 | |
0.1 | 0.084 ± 0.003 | 0.11 ± 0.0033 | 0.57 ± 0.30 | 0.37 ± 0.19 | 0.75 ± 0.45 | 0.71 ± 0.63 | 0.12 ± 0.022 |
1 | 1 ± 0.019 | 1 ± 0.017 | 1 ± 0.11 | 1 ± 0.20 | 1 ± 0.12 | 1 ± 0.10 | 1 ± 0.018 |
10 | 14 ± 0.680 | 10 ± 0.35 | 242 ± 206 | 83 ± 52 | 15 ± 5.5 | 48 ± 1.0 | 10 ± 1.1 |
Our model assumes that the signal is proportional to the amount of product, which is often the case for SYBR-Green I real-time PCR performed with saturating concentrations of dye. In such conditions centrally symmetric amplification curves are expected. However, in TaqMan applications, where the Taq DNA polymerase digests a probe labelled with a fluorescent reporter and quencher dye, the signal diverges from the product resulting in non-symmetric amplification curves (Supplementary Fig. 1A in Additional file 1) [8]. To test whether our approach is also suitable for TaqMan data analysis, we quantified β-actin and hypoxanthine phosphoribosyl transferase (HPRT) in 1/10 dilutions of the same cDNA sample using TaqMan probes. The error of quantifying a 10-fold concentrated sample was only slightly higher using TaqMan when compared to SYBR-Green I (Fig. 2C and Supplementary Fig. 1B in Additional file 1). These results suggest that MoBPA is robust enough to deal with non-symmetric amplification curves, possibly because it makes use of only the data points that fall into the exponential-lineal growth phases of the PCR.
Conclusion
Following the emergence of functional genomic methodologies, the development of high-throughput methods for microarray-derived data validation is becoming indispensably. Nevertheless, high-throughput quantification by real-time PCR is difficult to achieve, primarily due to deficiencies of the threshold-based methodologies, which require reliable estimation of amplification efficiencies [27]. In addition to the technical challenge of generating dilution curves for each target sequence, these methods assume similar amplification efficiencies between samples, and this assumption has been reported to be invalid [13, 24, 25]. Here we introduce a new method for the analysis of real-time PCR results termed: Model Based method for real-time PCR data Analysis (MoBPA). The new method produced the most accurate and precise quantifications when compared to previous methods regardless of whether samples were amplified with similar or different amplification efficiencies. Moreover, no calibration curve is needed for quantifications, since amplification efficiency is estimated directly from real-time PCR data. MoBPA provides many of the fundamental capabilities required for fully automated high-throughput quantification [18], including routine assessment of amplification efficiency within individual samples and no need of PCR-generated standard curves. In addition, quantitative data can be easily derived from arbitrary fluorescence units, simply applying a calibration factor that relates fluorescence to DNA mass [18]. In conclusion, MoBPA combines the well accepted exponential model of the PCR reaction with a sigmoid description of amplification efficiency to obtain a powerful method for real-time PCR data analysis that expand the applicability of real-time PCR to fully automated high-throughput applications.
Methods
RNA extraction and reverse transcription
Mice were killed by cervical dislocation and brains removed. Midbrains were dissected, snap frozen in liquid nitrogen and stored at -80°C until RNA extraction. Total RNA was isolated using TRIzol Reagent (Invitrogen, MD), genomic DNA contaminant was removed using DNAse I (Ambion, Inc., TX), and mRNA was purified by MicroPoly(A)Pure kit (Ambion, Inc., TX). First-strand complementary DNA was synthesized at 42°C by priming with oligo-dT_{12–18} (Invitrogen, MD) and using SuperScriptII reverse transcriptase according to the protocol provided by the manufacturer (Invitrogen, MD).
Polymerase chain reaction
PCR amplifications were obtained using an Icycler IQ Real-Time PCR Detection System (BioRad, CA). cDNA samples were assayed by triplicate. PCR reactions were performed in a final volume of 25 μl containing 1 μl of cDNA, 2.5 μl of the reaction buffer (200 mM Tris-HCl pH 8.4, 500 mM KCl), 3 mM MgCl_{2}, 0.3 mM of dNTPs mix, 0.2 nM of each primer, 0.3 × SYBR-Green I (Molecular Probes, OR), 100 μg/ml BSA, 0.25 μl ROX Reference Dye (Invitrogen, MD), 1% glycerol, and 1.25 U of Taq Platinum Polymerase (Invitrogen, MD). The primer sequences used were: β2-microglobulin, sense: TGA CCG GCT TGT ATG CTA TC and antisense: CAG TGT GAG CCA GGA TAT AG; β-actin, sense: CAA TGT GGC TGA GGA CTT TG and antisense: ACA GAA GCA ATG CTG TCA CC. PCR was performed as follows: one initial cycle of 94°C for 2.5 min, 40 cycles of 94°C for 30 sec, 58°C for 30 sec, and 72°C for 15 sec. TaqMan real-time PCRs were performed as the SYBR-Green I assays, but with no addition of SyBrGreen I and with the following primers and probes: β-actin sense: AGA AAA TCT GGC ACC ACA CC, antisense: CAG AGG CGT ACA GGG ATA GC, and probe: ACC GCG AGA AGA TGA CCC AGA TCA T; HPRT sense: AGA CTG AAG AGC TAT TGT AAT, antisense: CAG CAA GCT TGC GAC CTT GAC, and probe: TGC TTT CCT TGG TCA GGC AGT ATA.
Analysis of real-time PCR data
ROX base-line corrected real-time PCR results were analysed with different methods as described [12–17] using the R-System v2.2.0 [29] or the LinRegPCR software [13]. For the LinRegPCR software, we used the default fit option, which iteratively searches for lines consisting in 4 to 6 data points with the highest R^{2} value. Then we inspected the fit for each PCR curve and manually corrected the windows of linearity when needed.
The implementation of our method comprise the following steps: 1) Identification of ground, exponential, lineal growth and plateau phases, 2) background subtraction, 3) effective amplification efficiency estimation and fitting eq (2) to experimental data, 4) initial template amount calculation.
The ground phase was identified as described [16], using a p-value cut-off of 0.01, while the beginning and end of the lineal growth phase was identified as the second derivative maximum and minimum, respectively, of a four parameters sigmoid function (eq. 7) fitted to the data (Supplementary Figure 2A in Additional file 1).
Here, y is the PCR product yield (fluorescence units), x is the cycle number, mi and ma are the signal-offset and saturation value, respectively, c is the point of inflexion and h is the exponent parameter. Non-linear least square regressions were performed with a Gauss-Newton algorithm implemented in the nls function of the R-system software.
The background level was calculated as the last ground phase data point, which in tern was estimated by a linear regression over the last five data points of this phase [16].
The effective amplification efficiency for each PCR cycle (E_{ n }) was solved from Eq. (1) as E_{ n }= T_{n+1}/T_{ n }- 1 [5] and calculated using the background subtracted data. Then, the parameters b and Tm were estimated by fitting Eq (2) to the experimental effective amplification efficiency by non-linear regression. We only used data points that fall into the exponential and linear growth phase of the amplification curve, because efficiency calculated at the early ground phase and late plateau phase showed strong dispersion due to experimental background noise or showed no information, respectively (Supplementary Figure 2 in Additional file 1). Finally, the initial template amount T_{0} was estimated by fitting our discrete model, shown below in R-code:
T <- function(x,t0,b,Tm) {
output <- NULL
for (i in 1:length(x)) {
t00 <- t0
for (ii in 1:x [i]) t00 <- t00*(1+1/(1+exp((t00-Tm)/b)))
output <- c(output,t00)
}
return(output)
}
to background subtracted experimental data by the nls function of R-system, using the values for the parameters b and Tm estimated previously.
Our method was implemented in the R-System. The source code and Windows binary of MoBPA package are available for non-commercial research use (see Additional files 2 and 3).
Quantification error and Akaike's Information Criterion
For calculating the quantification error, we defined a reference sample (RS) and a target sample (TS), which are derived from the same original sample and differ by a known dilution factor or PCR amplification efficiency. We calculated the relative bias as RE = (TS - RS)/RS, and the quantification error as SQE = RE * 100 for RE ≥ 0, or SQE = -100 * RE /(1 + RE) for RE < 0.
For comparing alternative models we used a corrected form of the Akaike's Information Criterion (AIC), defined as:
where N is the number of data points, K is the number of parameters fitted by the regression plus one, and SS is the sum of the square of the vertical distances of the points from the curve [20].
In silico generation of PCR data
Simulation data was generated in silico from eqs. (1) and (2), using parameters that resemble real PCR runs. Results from amplification of mouse midbrain cDNA with β-actin and β_{2}-microglobulin specific primers were used to estimate plausible values for the simulation parameters. In such a way, the initial template amount in fluorescent arbitrary units (T_{0}) was set to 10^{-3}, and the product amount at the PCR plateau phase (Tmax) was set to 1200. The amplification parameters b and Tm were solve from eq. (2) and (3) as,
Then, for T_{ n }= Tmax, E_{ n }tends to zero, so we calculated approximated values for b and Tm using T_{ n }= Tmax = 1200 and E_{ n }= 0.001.
Zero mean, normally distributed random noise was added to the in-silico results. The background mean (Bg) and intra- and inter-replicates standard deviation (intraSD and interSD) were estimated from 48 triplicate samples to be 625, 17.94 and 19.06 respectively. The mean standard deviation for fluorescence intensity in the early ground phase of each PCR reaction (mSD), also estimated from experimental data, was 4.9. We found no correlation between the standard deviation of fluorescence intensity in the early ground phase of each PCR reaction and Bg values, thus, random noise generated from a normal distribution with mean 0 and standard deviation mSD was added to in-silico PCR data regardless of the background fluorescence level.
Notes
Declarations
Acknowledgements
We thank Dr. Andrea Califano for helpful comments on the manuscript. We specially thank Cynthia Serra and Daniela Celi for technical assistance. We are in debt with María Romina Girotti and Andrea Sabina Llera for facilitating their TaqMan PCR results.
Authors’ Affiliations
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