Comprehensive Algorithm for Quantitative Real-Time Polymerase Chain Reaction

SHENG ZHAO and RUSSELL D. FERNALD

JOURNAL OF COMPUTATIONAL BIOLOGY, Volume 12, Number 8, 2005

Quantitative real-time polymerase chain reactions (qRT-PCR) have become the method of choice for rapid, sensitive, quantitative comparison of RNA transcript abundance. Useful data from this method depend on fitting data to theoretical curves that allow computation of mRNA levels. Calculating accurate mRNA levels requires important parameters such as
reaction efficiency and the fractional cycle number at threshold (CT) to be used; however, many algorithms currently in use estimate these important parameters. Here we describe an objective method for quantifying qRT-PCR results using calculations based on the kinetics of individual PCR reactions without the need of the standard curve, independent of any assumptions or subjective judgments which allow direct calculation of efficiency and CT. We use a four-parameter logistic model to fit the raw fluorescence data as a function of PCR cycles to identify the exponential phase of the reaction. Next, we use a three-parameter simple exponent model to fit the exponential phase using an iterative nonlinear regression algorithm. Within the exponential portion of the curve, our technique automatically identifies candidate regression values using the P-value of regression and then uses a weighted average to compute a final efficiency for quantification. For CT determination, we chose the first positive second derivative maximum from the logistic model. This algorithm provides an objective and noise-resistant method for quantification of qRT-PCR results that is independent of the specific equipment used to perform PCR reactions.

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Evaluation of Real-Time PCR Amplification Efficiencies to Detect PCR Inhibitors

Elias J. Kontanis and Floyd A. Reed.

J Forensic Sci, July 2006, Vol. 51, No. 4

Real-time PCR analysis is a sensitive template DNA quantitation strategy that has recently gained considerable attention in the forensic community. However, the utility of real-time PCR methods extends beyond quantitation and allows for simultaneous evaluation of template DNA extraction quality. This study presents a computational method that allows analysts to identify problematic samples with statistical reliability by comparing the amplification efficiencies of unknown template DNA samples with clean standards. In this study, assays with varying concentrations of tannic acid are used to evaluate and adjust sample-specific amplification efficiency calculation methods in order to optimize their inhibitor detection capabilities. Kinetic outlier detection and prediction boundaries are calculated to identify amplification efficiency outliers. Sample-specific amplification efficiencies calculated over a four-cycle interval starting at the threshold cycle can be used to detect reliably the presence of 0.4 ng of tannic acid in a 25 mL PCR reaction. This approach provides analysts with a precise measure of inhibition severity when template samples are compromised. Early detection of problematic samples allows analysts the opportunity to consider inhibitor mitigation strategies prior to genotype or DNA sequence analysis, thereby facilitating sample processing in high-throughput forensic operations.

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Estimation of the reaction efficiency in polymerase chain reaction.

Nadia Lalam
Journal of Theoretical Biology
EURANDOM, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

Polymerase chain reaction (PCR) is largely used in molecular biology for increasing the copy number of a specific DNA fragment. The succession of 20 replication cycles makes it possible to multiply the quantity of the fragment of interest by a factor of 1 million. The PCR technique has revolutionized genomics research. Several quantification methodologies are available to determine the DNA replication efficiency of the reaction which is the probability of replication of a DNA molecule at a replication cycle. We elaborate a quantification procedure based on the exponential phase and the early saturation phase of PCR. The reaction efficiency is supposed to be constant in the exponential phase, and decreasing in the saturation phase. We propose to model the PCR amplification process by a branching process which starts as a Galton–Watson branching process followed by a size-dependent process. Using this stochastic modelling and the conditional least-squares estimation method, we infer the reaction efficiency from a single PCR trajectory.

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Molelling the PCR amplification process by a size-dependent branching process and estimation of the efficiency.

Lalam N, Jacob C. and Jagers P.
Adv. Appl. Prob. (2004) 36: 601-612

We propose a stochastic modelling of the PCR amplification process by a size dependent branching process starting as a supercritical Bienayme-Galton-Watson transient phase and then having a saturation near-critical size-dependent phase. This model allows us to estimate the probability of the replication of the DNA molecule at each cycle of a single PCR trajectory with a very good accuracy.

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Statistical Inference for Quantitative Polymerase Chain Reaction Using a Hidden Markov Model:
A Bayesian Approach

Nadia Lalam, Chalmers University of Technology, Sweden
Statistical Applications in Genetics and Molecular Biology: Vol. 6  : Iss. 1, Article 10.

Quantitative Polymerase Chain Reaction (Q-PCR) aims at determining the initial quantity of specific nucleic acids from the observation of the number of amplified DNA molecules. The most widely used technology to monitor the number of DNA molecules as they replicate is based on fluorescence chemistry. Considering this measurement technique, the observation of DNA amplification by PCR contains intrinsically two kinds of variability. On the one hand, the number of replicated DNA molecules is random, and on the other hand, the measurement of the fluorescence emitted by the DNA molecules is collected with some random error. Relying on a stochastic model of these two types of variability, we aim at providing estimators of the parameters arising in the proposed model, and, more specifically, of the initial amount of molecules. The theory of branching processes is classically used to model the evolution of the number of DNA molecules at each replication cycle. The model is a binary splitting Galton-Watson branching process. Its unknown parameters are the initial number of DNA molecules and the reaction efficiency of PCR, which is defined as the probability of replication of a DNA molecule. The number of DNA molecules is indirectly observed through noisy fluorescence measurements resulting in a so-called Hidden Markov Model. We aim at inference of the parameters of the underlying branching process, and the parameters of the noise from the fluorescence measurements in a Bayesian framework. Using simulations and experimental data, we investigate the performance of the Bayesian estimators obtained by Markov Chain Monte Carlo methods.

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Mathematical Model of Real-Time PCR Kinetics.

Jana L. Gevertz,1 Stanley M. Dunn,1 Charles M. Roth1,2
1Department of Biomedical Engineering, Rutgers University, Piscataway, New Jersey
2Department of Chemical & Biochemical Engineering, Rutgers University,
98 Brett Road, Piscataway, New Jersey 08854

Biotechnol Bioeng. 2005 Nov 5;92(3):346-55.

Abstract: Several real-time PCR (rtPCR) quantification techniques are currently used to determine the expression levels of individual genes from rtPCR data in the form of fluorescence intensities. In most of these quantification techniques, it is assumed that the efficiency of rtPCR is constant. Our analysis of rtPCR data shows, however, that even during the exponential phase of rtPCR, the efficiency of the reaction is not constant, but is instead a function of cycle number. In order to understand better the mechanisms belying this behavior, we have developed a mathematical model of the annealing and extension phases of the PCR process. Using the model, we can simulate the PCR process over a series of reaction cycles. The model thus allows us to predict the efficiency of rtPCR at any cycle number, given a set of initial conditions and parameter values, which can mostly be estimated from biophysical data. The model predicts a precipitous decrease in cycle efficiency when the product concentration reaches a sufficient level for template–template reannealing to compete with primer-template annealing; this behavior is consistent with available experimental data. The quantitative understanding of rtPCR provided by this model can allow us to develop more accurate methods to quantify gene expression levels from rtPCR data.

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Evaluation of absolute quantitation by nonlinear regression in probe-based real-time PCR.

Rasmus Goll, Trine Olsen, Guanglin Cui and Jon Florholmen
  Institute of Clinical Medicine, University of Tromso, Tromso, Norway
Department of gastroenterology, University hospital of Northern Norway, Tromso, Norway
BMC Bioinformatics 2006, 7:107 doi:10.1186/1471-2105-7-107

In real-time PCR data analysis, the cycle threshold (CT) method is currently the gold standard. This method is based on an assumption of equal PCR efficiency in all reactions, and precision may suffer if this condition is not met. Nonlinear regressionanalysis (NLR) or curve fitting has therefore been suggested as an alternative to the cycle threshold method for absolute quantitation. The advantages of NLR are that the individual sample efficiency is simulated by the model and that absolute quantitation is possible without a standard curve, releasing reaction wells for unknown samples. However, the calculation method has not been evaluated systematically and has not previously been applied to a TaqMan platform. Aim: To develop and evaluate an automated NLR algorithm capable of generating batch production regression analysis.Total RNA samples extracted from human gastric mucosa were reverse transcribed and analysed for TNFA, IL18 and ACTB by TaqMan real-time PCR. Fluorescence data were analysed by the regular CT method with a standard curve, and by NLR with a positive control for conversion of fluorescence intensity to copy number, and for this purpose an automated algorithm was written in SPSS syntax. Eleven separate regression models were tested, and the output data was subjected to Altman-Bland analysis. The Altman-Bland analysis showed that the best regression model yielded quantitative data with an intra-assay variation of 58% vs. 24% for the CT derived copy numbers, and with a mean inter-method deviation of x0.8. NLR can be automated for batch production analysis, but the CT method is more precise for absolute quantitation in the present setting. The observed inter-method deviation is an indication that assessment of the fluorescence conversion factor used in the regression method can be improved. However, the versatility depends on the level of precision required, and in some settings the increased cost effectiveness of NLR may justify the lower precision.

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Quantitative real-time RT-PCR based transcriptomics: Improvement of evaluation methods.

PhD. Thesis - Ales Tichopad
Lehrstuhl für Physiologie, Fakultät Wissenschaftszentrum Weihenstephan, Technische Universität München, Germany.

SUMMARY
Quantitative real-time polymerase chain reaction (qRT-PCR) is a new method for reliable quantification of low-abundance mRNA in biological samples. Since the strength of the fluorescence signal emitted by the report dye should be proportional to the produced DNA amount, the fluorescence monitoring enables visualisation of the full reaction trajectory. The reaction trajectory can be then extrapolated back to an input concentration.
RNA extraction can introduce unwanted contaminants into the sample, inhibiting the reverse transcription (RT) as well as the PCR reaction. These inhibitions cause then the reaction to precede sample-specific. In addition, the amplification efficiency varies not only between samples, but also along the recorded amplification trajectory of a single sample. Consequently, a correct determination of each probe’s PCR efficiency as well as a good standardization of the raw expression estimators is of great importance for a correct interpretation of results.
To find a solution to above problems a series of biological experiments with RNA extracted from various ovine and bovine tissues and from cultured leukocytes was carried out. Constant amount of RNA was then reverse–transcribed to cDNA. All PCR runs were performed on a LightCycler instrument and Fluorescence data was saved in the LightCycler software.
Based on this data, mathematical models together with statistical procedures were developed and validated. These can investigate the optimal quantification range and exactly determine its real-time PCR efficiency. Additionally, methods were developed to disclose heterogeneity between probes. All these procedures contribute to better quality of results obtained. Resulting from these standardisations, a decision algorithm for a proper analysis of the qRT-PCR data was designed.