Determination of real-time PCR amplification efficiency

Chapter 3 - Quantification strategies in real-time PCR
by Michael W. Pfaffl
in:  A-Z of quantitative PCR  (Editor: S.A. Bustin) 
International University Line (IUL), La Jolla, CA, USA
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Individual samples generate different and individual fluorescence histories in kinetic RT-PCR. The shapes of amplification curves differ in the steepness of any fluorescence increase and in the absolute fluorescence levels at plateau depending on background fluorescence levels. The PCR efficiency has a major impact on the fluorescence history and the accuracy of the calculated expression result and is critically influenced by PCR reaction components. Efficiency evaluation is an essential marker in gene quantification procedure. Constant amplification efficiency in all compared samples is one important criterion for reliable comparison between samples. This becomes crucially important when analyzing the relationship between an unknown sequence versus a standard sequence, which is performed in all relative quantification models. In experimental designs employing standardization with housekeeping genes, the demand for invariable amplification efficiency between target and standard is often ignored, despite the fact that corrections have been suggested. A correction for efficiency, as performed in efficiency corrected mathematically models, is strongly recommended and results in a more reliable estimation of the ‘real expression ratio’ compared to NO efficiency correction. Small efficiency differences between target and reference gene generate false expression ratio, and the researcher over- or under-estimates the ‘real’ initial mRNA amount.
The assessment of the exact amplification efficiencies of target and reference genes must be carried out before any calculation of the normalized gene expression is done. LightCycler Relative Expression Software, Q-Gene, REST and REST-XL software applications allow the evaluation of amplification efficiency plots. Different tissues exhibit different PCR efficiencies, caused by RT inhibitors, PCR inhibitors and by variations in the total RNA fraction pattern extracted.

Experimental comparison of relative RT-qPCR quantification approaches for gene expression studies in poplar.
Regier N, Frey B.
BMC Mol Biol. 2010 11: 57, 8 pages
Swiss Federal Research Institute WSL, Zürcherstrasse 111, 8903 Birmensdorf, Switzerland.

BACKGROUND: RT-qPCR is a powerful tool for analysing gene expression. It depends on measuring the increase in fluorescence emitted by a DNA-specific dye during the PCR reaction. For relative quantification, where the expression of a target gene is measured in relation to one or multiple reference genes, various mathematical approaches are published. The results of relative quantification can be considerably influenced by the chosen method.
RESULTS: We quantified gene expression of superoxide dismutase (SOD) and ascorbate peroxidase (APX) in the roots of two black poplar clones, 58-861 and Poli, which were subjected to drought stress. After proving the chosen reference genes actin (ACT), elongation factor 1 (EF1) and ubiquitin (UBQ) to be constantly expressed in the different watering regimes, we applied different approaches for relative quantification to the same raw fluorescence data. The results obtained using the comparative Cq method, LinRegPCR, qBase software and the Pfaffl model showed a good correlation, whereas calculation according to the Liu and Saint method produced highly variable results. However, it has been shown that the most reliable approach for calculation of the amplification efficiency is using the mean increase in fluorescence during PCR in each individual reaction. Accordingly, we could improve the quality of our results by applying the mean amplification efficiencies for each amplicon to the Liu and Saint method.
CONCLUSIONS: As we could show that gene expression results can vary depending on the approach used for quantification, we recommend to carefully evaluate different quantification approaches before using them in studies analysing gene expression.

Several methods are described in the literature to calculate real-time PCR efficiency:

Determination of PCR efficiencies in competitive RT-PCR
Various external effects on PCR amplification effiency

New added publications:

Efficiency of the Polymerase Chain Reaction.
Booth CS, Pienaar E, Termaat JR, Whitney SE, Louw TM, Viljoen HJ.
Department of Chemical and Biomolecular Engineering, University of Nebraska-Lincoln Lincoln, NE 68588-0643.
Chem Eng Sci. 2010 65(17): 4996-5006.

The polymerase chain reaction (PCR) has found wide application in biochemistry and molecular biology such as gene expression studies, mutation detection, forensic analysis and pathogen detection. Increasingly quantitative real time PCR is used to assess copy numbers from overall yield. In this study the yield is analyzed as a function of several processes: (1) thermal damage of the template and polymerase occurs during the denaturing step, (2) competition exists between primers and templates to either anneal or form dsDNA, (3) polymerase binding to annealed products (primer/ssDNA) to form ternary complexes and (4) extension of ternary complexes. Explicit expressions are provided for the efficiency of each process, therefore reaction conditions can be directly linked to the overall yield. Examples are provided where different processes play the yield-limiting role. The analysis will give researchers a unique understanding of the factors that control the reaction and will aid in the interpretation of experimental results.

pcrEfficiency: a Web tool for PCR amplification efficiency prediction.
Mallona I, Weiss J, Marcos EC.
Genetics, Institute of Plant Biotechnology (IBV), Technical University of Cartagena (UPCT), Campus Muralla del Mar, 30202 Cartagena, Spain
BMC Bioinformatics. 2011 12: 404

Relative calculation of differential gene expression in quantitative PCR reactions requires comparison between amplification experiments that include reference genes and genes under study. Ignoring the differences between their efficiencies may lead to miscalculation of gene expression even with the same starting amount of template. Although there are several tools performing PCR primer design, there is no tool available that predicts PCR efficiency for a given amplicon and primer pair.
We have used a statistical approach based on 90 primer pair combinations amplifying templates from bacteria, yeast, plants and humans, ranging in size between 74 and 907 bp to identify the parameters that affect PCR efficiency. We developed a generalized additive model fitting the data and constructed an open source Web interface that allows the obtention of oligonucleotides optimized for PCR with predicted amplification efficiencies starting from a given sequence.
pcrEfficiency provides an easy-to-use web interface allowing the prediction of PCR efficiencies prior to web lab experiments thus easing quantitative real-time PCR set-up. A web-based service as well the source code are provided freely at http://srvgen.upct.es/efficiency.html under the GPL v2 license.

Experimental Validation of a Fundamental Model for PCR Efficiency.
Louw TM, Booth CS, Pienaar E, Termaat JR, Whitney SE, Viljoen HJ.
Department of Chemical and Biomolecular Engineering, University of Nebraska-Lincoln, Lincoln, NE 68588-0643.
Chem Eng Sci. 2011 Apr 15;66(8): 1783-1789.

Recently a theoretical analysis of PCR efficiency has been published by Booth et al., (2010). The PCR yield is the product of three efficiencies: (i) the annealing efficiency is the fraction of templates that form binary complexes with primers during annealing, (ii)the polymerase binding efficiency is the fraction of binary complexes that bind to polymerase to form ternary complexes and (iii)the elongation efficiency is the fraction of ternary complexes that extend fully. Yield is controlled by the smallest of the three efficiencies and control could shift from one type of efficiency to another over the course of a PCR experiment. Experiments have been designed that are specifically controlled by each one of the efficiencies and the results are consistent with the mathematical model. The experimental data has also been used to quantify six key parameters of the theoretical model. An important application of the fully characterized model is to calculate initial template concentration from real-time PCR data. Given the PCR protocol, the midpoint cycle number (where the template concentration is half that of the final concentration) can be theoretically determined and graphed for a variety of initial DNA concentrations. Real-time results can be used to calculate the midpoint cycle number and consequently the initial DNA concentration, using this graph. The application becomes particularly simple if a conservative PCR protocol is followed where only the annealing efficiency is controlling.

Enhanced analysis of real-time PCR data by using a variable efficiency model: FPK-PCR.
Lievens A, Van Aelst S, Van den Bulcke M, Goetghebeur E.
Platform for Molecular Biology and Biotechnology, Scientific Institute of Public Health, J. Wytsmanstreet 14, B-1050 Brussels, Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281, S9 B-9000 Gent, Belgium and European Commission, Joint Research Center, Institute for Health and Consumer Protection, Molecular Biology and Genomics Unit, via E. Fermi 2749, 21027 Ispra (VA), Italy.
Nucleic Acids Res. 2011 Nov 18.

Current methodology in real-time Polymerase chain reaction (PCR) analysis performs well provided PCR efficiency remains constant over reactions. Yet, small changes in efficiency can lead to large quantification errors. Particularly in biological samples, the possible presence of inhibitors forms a challenge. We present a new approach to single reaction efficiency calculation, called Full Process Kinetics-PCR (FPK-PCR). It combines a kinetically more realistic model with flexible adaptation to the full range of data. By reconstructing the entire chain of cycle efficiencies, rather than restricting the focus on a 'window of application', one extracts additional information and loses a level of arbitrariness. The maximal efficiency estimates returned by the model are comparable in accuracy and precision to both the golden standard of serial dilution and other single reaction efficiency methods. The cycle-to-cycle changes in efficiency, as described by the FPK-PCR procedure, stay considerably closer to the data than those from other S-shaped models. The assessment of individual cycle efficiencies returns more information than other single efficiency methods. It allows in-depth interpretation of real-time PCR data and reconstruction of the fluorescence data, providing quality control. Finally, by implementing a global efficiency model, reproducibility is improved as the selection of a window of application is avoided.

Validation of kinetics similarity in qPCR.
Bar T, Kubista M, Tichopad A.
Labonnet Ltd., 2 Hamelacha St., Ramat-Hasharon, 47445, Israel, TATAA Biocenter, Odinsgatan 28, 411 03 Göteborg, Sweden, Biotechnology Institute, Academy of Science of the Czech Republic, Vídeňská 1083, 142 20 Prague 4, Czech Republic and Charles University, Medical Faculty Hospital in Pilsen, Dr. E. Beneše 13, 305 99 Pilsen - Bory, the Czech Republic.
Nucleic Acids Res. 2011 Oct 19.

Quantitative real-time PCR (qPCR) is the method of choice for specific and sensitive quantification of nucleic acids. However, data validation is still a major issue, partially due to the complex effect of PCR inhibition on the results. If undetected PCR inhibition may severely impair the accuracy and sensitivity of results. PCR inhibition is addressed by prevention, detection and correction of PCR results. Recently, a new family of computational methods for the detection of PCR inhibition called kinetics outlier detection (KOD) emerged. KOD methods are based on comparison of one or a few kinetic parameters describing a test reaction to those describing a set of reference reactions. Modern KOD can detect PCR inhibition reflected by shift of the amplification curve by merely half a cycle with specificity and sensitivity >90%. Based solely on data analysis, these tools complement measures to improve and control pre-analytics. KOD methods do not require labor and materials, do not affect the reaction accuracy and sensitivity and they can be automated for fast and reliable quantification. This review describes the background of KOD methods, their principles, assumptions, strengths and limitations. Finally, the review provides recommendations how to use KOD and how to evaluate its performance.

Statistical methods for efficiency adjusted real-time PCR quantification.
Yuan JS, Wang D, Stewart CN Jr.
UTIA Genomics Hub, The University of Tennessee, Knoxville, TN 37996, USA.
Biotechnol J. 2008 3(1): 112-23.

The statistical treatment for hypothesis testing using real-time PCR data is a challenge for quantification of gene expression. One has to consider two key factors in precise statistical analysis of real-time PCR data: a well-defined statistical model and the integration of amplification efficiency (AE) into the model. Previous publications in real-time PCR data analysis often fall short in integrating the AE into the model. Novel, user-friendly, and universal AE-integrated statistical methods were developed for real-time PCR data analysis with four goals. First, we addressed the definition of AE, introduced the concept of efficiency-adjusted Delta Delta Ct, and developed a general mathematical method for its calculation. Second, we developed several linear combination approaches for the estimation of efficiency adjusted Delta Delta Ct and statistical significance for hypothesis testing based on different mathematical formulae and experimental designs. Statistical methods were also adopted to estimate the AE and its equivalence among the samples. A weighted Delta Delta Ct method was introduced to analyze the data with multiple internal controls. Third, we implemented the linear models with SAS programs and analyzed a set of data for each model. In order to allow other researchers to use and compare different approaches, SAS programs are included in the Supporting Information. Fourth, the results from analysis of different statistical models were compared and discussed. Our results underline the differences between the efficiency adjusted Delta Delta Ct methods and previously published methods, thereby better identifying and controlling the source of errors introduced by real-time PCR data analysis.


A mechanistic model of PCR for accurate quantification of quantitative PCR data.
Boggy GJ, Woolf PJ.
Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan, United States of America.
PLoS One. 2010 5(8): e12355.

BACKGROUND: Quantitative PCR (qPCR) is a workhorse laboratory technique for measuring the concentration of a target DNA sequence with high accuracy over a wide dynamic range. The gold standard method for estimating DNA concentrations via qPCR is quantification cycle () standard curve quantification, which requires the time- and labor-intensive construction of a standard curve. In theory, the shape of a qPCR data curve can be used to directly quantify DNA concentration by fitting a model to data; however, current empirical model-based quantification methods are not as reliable as standard curve quantification.
PRINCIPAL FINDINGS: We have developed a two-parameter mass action kinetic model of PCR (MAK2) that can be fitted to qPCR data in order to quantify target concentration from a single qPCR assay. To compare the accuracy of MAK2-fitting to other qPCR quantification methods, we have applied quantification methods to qPCR dilution series data generated in three independent laboratories using different target sequences. Quantification accuracy was assessed by analyzing the reliability of concentration predictions for targets at known concentrations. Our results indicate that quantification by MAK2-fitting is as reliable as standard curve quantification for a variety of DNA targets and a wide range of concentrations.
SIGNIFICANCE: We anticipate that MAK2 quantification will have a profound effect on the way qPCR experiments are designed and analyzed. In particular, MAK2 enables accurate quantification of portable qPCR assays with limited sample throughput, where construction of a standard curve is impractical.


Shape based kinetic outlier detection in real-time PCR.
Sisti D, Guescini M, Rocchi MB, Tibollo P, D'Atri M, Stocchi V.
BMC Bioinformatics. 2010 12;11: 186

BACKGROUND:
Real-time PCR has recently become the technique of choice for absolute and relative nucleic acid quantification. The gold standard quantification method in real-time PCR assumes that the compared samples have similar PCR efficiency. However, many factors present in biological samples affect PCR kinetic, confounding quantification analysis. In this work we propose a new strategy to detect outlier samples, called SOD.
RESULTS:
Richards function was fitted on fluorescence readings to parameterize the amplification curves. There was not a significant correlation between calculated amplification parameters (plateau, slope and y-coordinate of the inflection point) and the Log of input DNA demonstrating that this approach can be used to achieve a "fingerprint" for each amplification curve. To identify the outlier runs, the calculated parameters of each unknown sample were compared to those of the standard samples. When a significant underestimation of starting DNA molecules was found, due to the presence of biological inhibitors such as tannic acid, IgG or quercitin, SOD efficiently marked these amplification profiles as outliers. SOD was subsequently compared with KOD, the current approach based on PCR efficiency estimation. The data obtained showed that SOD was more sensitive than KOD, whereas SOD and KOD were equally specific.
CONCLUSION:
Our results demonstrated, for the first time, that outlier detection can be based on amplification shape instead of PCR efficiency. SOD represents an improvement in real-time PCR analysis because it decreases the variance of data thus increasing the reliability of quantification.

WEB INTERFACE - Cy0 is a new method in Real-time PCR analysis that does not require the assumption of equal efficiency between unknowns and standard curve (Michele Guescini, Davide Sisti, & Renato Panebianco, 2010)

Quality control for quantitative PCR based on amplification compatibility test.
Tichopad A, Bar T, Pecen L, Kitchen RR, Kubista M, Pfaffl MW.
Methods. 2010 50(4): 308-312

Quantitative qPCR is a routinely used method for the accurate quantification of nucleic acids. Yet it may generate erroneous results if the amplification process is obscured by inhibition or generation of aberrant side-products such as primer dimers. Several methods have been established to control for pre-processing performance that rely on the introduction of a co-amplified reference sequence, however there is currently no method to allow for reliable control of the amplification process without directly modifying the sample mix. Herein we present a statistical approach based on multivariate analysis of the amplification response data generated in real-time. The amplification trajectory in its most resolved and dynamic phase is fitted with a suitable model. Two parameters of this model, related to amplification efficiency, are then used for calculation of the Z-score statistics. Each studied sample is compared to a predefined reference set of reactions, typically calibration reactions. A probabilistic decision for each individual Z-score is then used to identify the majority of inhibited reactions in our experiments. We compare this approach to univariate methods using only the sample specific amplification efficiency as reporter of the compatibility. We demonstrate improved identification performance using the multivariate approach compared to the univariate approach. Finally we stress that the performance of the amplification compatibility test as a quality control procedure depends on the quality of the reference set.

Efficiency clustering for low-density microarrays and its application to qPCR
Eric F Lock, Ryan Ziemiecke, J. S. Marron and Dirk P Dittmer
BMC Bioinformatics 2010, 11

Background
Pathway-targeted or low-density arrays are used more and more frequently in biomedical research, particularly those arrays that are based on quantitative real-time PCR. Typical QPCR arrays contain 96-1024 primer pairs or probes, and they bring with it the promise of being able to reliably measure differences in target levels without the need to establish absolute standard curves for each and every target. To achieve reliable quantification all primer pairs or array probes must perform with the same efficiency.
Results
Our results indicate that QPCR primer-pairs differ significantly both in reliability and efficiency. They can only be used in an array format if the raw data (so called CT values for real-time QPCR) are transformed to take these differences into account. We developed a novel method to obtain efficiency-adjusted CT values. We introduce transformed confidence intervals as a novel measure to identify unreliable primers. We introduce a robust clustering algorithm to combine efficiencies of groups of probes, and our results indicate that using n < 10 cluster-based mean efficiencies is comparable to using individually determined efficiency adjustments for each primer pair (N = 96-1024).
Conclusions
Careful estimation of primer efficiency is necessary to avoid significant measurement inaccuracies. Transformed confidence intervals are a novel method to assess and interprete the reliability of an efficiency estimate in a high throughput format. Efficiency clustering as developed here serves as a compromise between the imprecision in assuming uniform efficiency, and the computational complexity and danger of over-fitting when using individually determined efficiencies.

A new real-time PCR method to overcome significant quantitative inaccuracy due to slight amplification inhibition.
BMC Bioinformatics. 2008 30;9: 326
Guescini M, Sisti D, Rocchi MB, Stocchi L, Stocchi V.

BACKGROUND:
Real-time PCR analysis is a sensitive DNA quantification technique that has recently gained considerable attention in biotechnology, microbiology and molecular diagnostics. Although, the cycle-threshold (Ct) method is the present "gold standard", it is far from being a standard assay. Uniform reaction efficiency among samples is the most important assumption of this method. Nevertheless, some authors have reported that it may not be correct and a slight PCR efficiency decrease of about 4% could result in an error of up to 400% using the Ct method. This reaction efficiency decrease may be caused by inhibiting agents used during nucleic acid extraction or copurified from the biological sample. We propose a new method (Cy0) that does not require the assumption of equal reaction efficiency between unknowns and standard curve.
RESULTS:
The Cy0 method is based on the fit of Richards' equation to real-time PCR data by nonlinear regression in order to obtain the best fit estimators of reaction parameters. Subsequently, these parameters were used to calculate the Cy0 value that minimizes the dependence of its value on PCR kinetic. The Ct, second derivative (Cp), sigmoidal curve fitting method (SCF) and Cy0 methods were compared using two criteria: precision and accuracy. Our results demonstrated that, in optimal amplification conditions, these four methods are equally precise and accurate. However, when PCR efficiency was slightly decreased, diluting amplification mix quantity or adding a biological inhibitor such as IgG, the SCF, Ct and Cp methods were markedly impaired while the Cy0 method gave significantly more accurate and precise results.
CONCLUSION:
Our results demonstrate that Cy0 represents a significant improvement over the standard methods for obtaining a reliable and precise nucleic acid quantification even in sub-optimal amplification conditions overcoming the underestimation caused by the presence of some PCR inhibitors.

Amplification efficiency: linking baseline and bias in the analysis of quantitative PCR data.
Ruijter JM, Ramakers C, Hoogaars WM, Karlen Y, Bakker O, van den Hoff MJ, Moorman AF.
Nucleic Acids Res. 2009 37(6): e45

Despite the central role of quantitative PCR (qPCR) in the quantification of mRNA transcripts, most analyses of qPCR data are still delegated to the software that comes with the qPCR apparatus. This is especially true for the handling of the fluorescence baseline. This article shows that baseline estimation errors are directly reflected in the observed PCR efficiency values and are thus propagated exponentially in the estimated starting concentrations as well as 'fold-difference' results. Because of the unknown origin and kinetics of the baseline fluorescence, the fluorescence values monitored in the initial cycles of the PCR reaction cannot be used to estimate a useful baseline value. An algorithm that estimates the baseline by reconstructing the log-linear phase downward from the early plateau phase of the PCR reaction was developed and shown to lead to very reproducible PCR efficiency values. PCR efficiency values were determined per sample by fitting a regression line to a subset of data points in the log-linear phase. The variability, as well as the bias, in qPCR results was significantly reduced when the mean of these PCR efficiencies per amplicon was used in the calculation of an estimate of the starting concentration per sample.

Assessing the performance capabilities of LRE-based assays for absolute quantitative real-time PCR.
Rutledge RG, Stewart D.
PLoS One. 2010 5(3):e9731.

BACKGROUND:
Linear regression of efficiency or LRE introduced a new paradigm for conducting absolute quantification, which does not require standard curves, can generate absolute accuracies of +/-25% and has single molecule sensitivity. Derived from adapting the classic Boltzmann sigmoidal function to PCR, target quantity is calculated directly from the fluorescence readings within the central region of an amplification profile, generating 4-8 determinations from each amplification reaction.
FINDINGS:
Based on generating a linear representation of PCR amplification, the highly visual nature of LRE analysis is illustrated by varying reaction volume and amplification efficiency, which also demonstrates how LRE can be used to model PCR. Examining the dynamic range of LRE further demonstrates that quantitative accuracy can be maintained down to a single target molecule, and that target quantification below ten molecules conforms to that predicted by Poisson distribution. Essential to the universality of optical calibration, the fluorescence intensity generated by SYBR Green I (FU/bp) is shown to be independent of GC content and amplicon size, further verifying that absolute scale can be established using a single quantitative standard. Two high-performance lambda amplicons are also introduced that in addition to producing highly precise optical calibrations, can be used as benchmarks for performance testing. The utility of limiting dilution assay for conducting platform-independent absolute quantification is also discussed, along with the utility of defining assay performance in terms of absolute accuracy.
CONCLUSIONS:
Founded on the ability to exploit lambda gDNA as a universal quantitative standard, LRE provides the ability to conduct absolute quantification using few resources beyond those needed for sample preparation and amplification. Combined with the quantitative and quality control capabilities of LRE, this kinetic-based approach has the potential to fundamentally transform how real-time qPCR is conducted.

Bias in the Cq value observed with hydrolysis probe based quantitative PCR can be corrected with the estimated PCR efficiency value.
Tuomi JM, Voorbraak F, Jones DL, Ruijter JM.
Methods. 2010 50(4): 313-22

For real-time monitoring of PCR amplification of DNA, quantitative PCR (qPCR) assays use various fluorescent reporters. DNA binding molecules and hybridization reporters (primers and probes) only fluoresce when bound to DNA and result in the non-cumulative increase in observed fluorescence. Hydrolysis reporters (TaqMan probes and QZyme primers) become fluorescent during DNA elongation and the released fluorophore remains fluorescent during further cycles; this results in a cumulative increase in observed fluorescence. Although the quantification threshold is reached at a lower number of cycles when fluorescence accumulates, in qPCR analysis no distinction is made between the two types of data sets. Mathematical modeling shows that ignoring the cumulative nature of the data leaves the estimated PCR efficiency practically unaffected but will lead to at least one cycle underestimation of the quantification cycle (C(q) value), corresponding to a 2-fold overestimation of target quantity. The effect on the target reference ratio depends on the PCR efficiency of the target and reference amplicons. The leftward shift of the C(q) value is dependent on the PCR efficiency and with sufficiently large C(q) values, this shift is constant. This allows the C(q) to be corrected and unbiased target quantities to be obtained.



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