Stepwise Kinetic Equilibrium Models of Quantitative Polymerase Chain Reaction

ABSTRACT: BACKGROUND: Numerous models for use in interpreting quantitative PCR (qPCR) data are present in recent literature. The most commonly used models assume the amplification in qPCR is exponential and fit an exponential model with a constant rate of increase to a select part of the curve. Kinetic theory may be used to model the annealing phase and does not assume constant efficiency of amplification. Mechanistic models describing the annealing phase with kinetic theory offer the most potential for accurate interpretation of qPCR data. Even so, they have not been thoroughly investigated and are rarely used for interpretation of qPCR data. New results for kinetic modeling of qPCR are presented. RESULTS: Two models are presented in which the efficiency of amplification is based on equilibrium solutions for the annealing phase of the qPCR process. Model 1 assumes annealing of complementary targets strands and annealing of target and primers are both reversible reactions and reach a dynamic equilibrium. Model 2 assumes all annealing reactions are nonreversible and equilibrium is static. Both models include the effect of primer concentration during the annealing phase. Analytic formulae are given for the equilibrium values of all single and double stranded molecules at the end of the annealing step. The equilibrium values are then used in a stepwise method to describe the whole qPCR process. Rate constants of kinetic models are the same for solutions that are identical except for possibly having different initial target concentrations. Analysis of qPCR curves from such solutions are thus analyzed by simultaneous non-linear curve fitting with the same rate constant values applying to all curves and each curve having a unique value for initial target concentration. The models were fit to two data sets for which the true initial target concentrations are known. Both models give better fit to observed qPCR data than other kinetic models present in the literature. They also give better estimates of initial target concentration. Model 1 was found to be slightly more robust than model 2 giving better estimates of initial target concentration when estimation of parameters was done for qPCR curves with very different initial target concentration. Both models may be used to estimate the initial absolute concentration of target sequence when a standard curve is not available. CONCLUSIONS: It is argued that the kinetic approach to modeling and interpreting quantitative PCR data has the potential to give more precise estimates of the true initial target concentrations than other methods currently used for analysis of qPCR data. The two models presented here give a unified model of the qPCR process in that they explain the shape of the qPCR curve for a wide variety of initial target concentrations.     

Instant evaluation of the absolute initial number of cDNA copies from a single real-time PCR curve

Amplification of a cDNA product by quantitative PCR (qPCR) is monitored by a fluorescent signal proportional to the amount of produced amplicon. The qPCR amplification curve usually displays an exponential phase followed by a non-exponential phase, ending with a plateau. Contrary to prevalent interpretation, we demonstrate that under standard qPCR conditions, the plateau can be explained by depletion of the probe through Taq polymerase- catalysed hydrolysis. Knowing the probe concentration and the fluorescence measured at the plateau, a specific fluorescence can thus be calculated. As far as probe hydrolysis quantitatively reflects amplicon synthesis, this, in turn, makes it possible to convert measured fluorescence levels in the exponential phase into concentrations of produced amplicon. It follows that the absolute target cDNA concentration initially engaged in the qPCR can be directly estimated from the fluorescence data, with no need to refer to any calibration with known concentrations of target DNA.     

A kinetic-based sigmoidal model for the polymerase chain reaction and its application to high-capacity absolute quantitative real-time PCR

Background

Based upon defining a common reference point, current real-time quantitative PCR technologies compare relative differences in amplification profile position. As such, absolute quantification requires construction of target-specific standard curves that are highly resource intensive and prone to introducing quantitative errors. Sigmoidal modeling using nonlinear regression has previously demonstrated that absolute quantification can be accomplished without standard curves; however, quantitative errors caused by distortions within the plateau phase have impeded effective implementation of this alternative approach.

Results

Recognition that amplification rate is linearly correlated to amplicon quantity led to the derivation of two sigmoid functions that allow target quantification via linear regression analysis. In addition to circumventing quantitative errors produced by plateau distortions, this approach allows the amplification efficiency within individual amplification reactions to be determined. Absolute quantification is accomplished by first converting individual fluorescence readings into target quantity expressed in fluorescence units, followed by conversion into the number of target molecules via optical calibration. Founded upon expressing reaction fluorescence in relation to amplicon DNA mass, a seminal element of this study was to implement optical calibration using lambda gDNA as a universal quantitative standard. Not only does this eliminate the need to prepare target-specific quantitative standards, it relegates establishment of quantitative scale to a single, highly defined entity. The quantitative competency of this approach was assessed by exploiting "limiting dilution assay" for absolute quantification, which provided an independent gold standard from which to verify quantitative accuracy. This yielded substantive corroborating evidence that absolute accuracies of ± 25% can be routinely achieved. Comparison with the LinReg and Miner automated qPCR data processing packages further demonstrated the superior performance of this kinetic-based methodology.

Conclusion

Called "linear regression of efficiency" or LRE, this novel kinetic approach confers the ability to conduct high-capacity absolute quantification with unprecedented quality control capabilities. The computational simplicity and recursive nature of LRE quantification also makes it amenable to software implementation, as demonstrated by a prototypic Java program that automates data analysis. This in turn introduces the prospect of conducting absolute quantification with little additional effort beyond that required for the preparation of the amplification reactions.     

Effects of the turn-back primer on intermediate product generation in isothermal DNA amplification

In DNA amplification, the initial step of copying a target sequence from the template DNA--the so-called intermediate product generation step--is very important. In examining the turn-back primer (TP)-dependent isothermal DNA amplification (TIA) method, we determined the actual time point of intermediate product generation by extrapolating dsDNA amplification curves. Our results indicate that intermediate product creation is the rate-limiting step in TIA, and good TP design is advantageous for improving the intermediate production process.     

Robustness of a loop-mediated isothermal amplification reaction for diagnostic applications

We evaluated the robustness of loop-mediated isothermal amplification (LAMP) of DNA for bacterial diagnostic applications. Salmonella enterica serovar Typhi was used as the target organism and compared with a real-time quantitative PCR (qPCR) for testing assay performance and reproducibly, as well as the impact of pH and temperature stability. This isothermal amplification method appeared to be particularly robust across 2 pH units (7.3-9.3) and temperature values (57-67 °C). The detection limit was comparable to that observed using optimized home-brew qPCR assays. The specificity of the amplification reaction remained high even at temperatures markedly different from the optimal one. Exposing reagents to the ambient temperature during the preparation of the reaction mixture as well as prolonging times for preparing the amplification reaction did not yield false-positive results. LAMP remained sensitive and specific despite the addition of untreated biological fluids such as stool or urine that commonly inhibit PCR amplification. Whereas the detection of microorganisms from whole blood or a blood-culture medium typically requires extensive sample purification and removal of inhibitors, LAMP amplification remained more sensitive than conventional qPCR when omitting such preparatory steps. Our results demonstrate that LAMP is not only easy to use, but is also a very robust, innovative and powerful molecular diagnostic method for both industrialized and developing countries.     

Incorporation of measurement of DNA integrity into qPCR assays

Optimal accuracy of quantitative PCR (qPCR) requires correction for integrity of the target sequence. Here we combine the mathematics of the Poisson distribution and exponential amplification to show that the frequency of lesions per base (which prevent PCR amplification) can be derived from the slope of the regression line between cycle threshold (Ct) and amplicon length. We found that the amplifiable fraction (AF) of a target can be determined from this frequency and the target length. Experimental results from this method correlated with both the magnitude of a damaging agent and with other measures of DNA damage. Applying the method to a reference sequence, we determined the values for lesions/base in control samples, as well as in the AFs of the target sequence in qPCR samples collected from leukemic patients. The AFs used to calculate the final qPCR result were generally >0.5, but were <0.2 in a few samples, indicating significant degradation. We conclude that DNA damage is not always predictable; quantifying the DNA integrity of a sample and determining the AF of a specific qPCR target will improve the accuracy of qPCR and aid in the interpretation of negative results.     

Standardized determination of real-time PCR efficiency from a single reaction set-up

We propose a computing method for the estimation of real-time PCR amplification efficiency. It is based on a statistic delimitation of the beginning of exponentially behaving observations in real-time PCR kinetics. PCR ground fluorescence phase, non-exponential and plateau phase were excluded from the calculation process by separate mathematical algorithms. We validated the method on experimental data on multiple targets obtained on the LightCycler platform. The developed method yields results of higher accuracy than the currently used method of serial dilutions for amplification efficiency estimation. The single reaction set-up estimation is sensitive to differences in starting concentrations of the target sequence in samples. Furthermore, it resists the subjective influence of researchers, and the estimation can therefore be fully instrumentalized.     

Serious Overestimation in Quantitative PCR by Circular (Supercoiled) Plasmid Standard: Microalgal pcna as the Model Gene

Quantitative real-time PCR (qPCR) has become a gold standard for the quantification of nucleic acids and microorganism abundances, in which plasmid DNA carrying the target genes are most commonly used as the standard. A recent study showed that supercoiled circular confirmation of DNA appeared to suppress PCR amplification. However, to what extent to which different structural types of DNA (circular versus linear) used as the standard may affect the quantification accuracy has not been evaluated. In this study, we quantitatively compared qPCR accuracies based on circular plasmid (mostly in supercoiled form) and linear DNA standards (linearized plasmid DNA or PCR amplicons), using proliferating cell nuclear gene (pcna), the ubiquitous eukaryotic gene, in five marine microalgae as a model gene. We observed that PCR using circular plasmids as template gave 2.65-4.38 more of the threshold cycle number than did equimolar linear standards. While the documented genome sequence of the diatom Thalassiosira pseudonana shows a single copy of pcna, qPCR using the circular plasmid as standard yielded an estimate of 7.77 copies of pcna per genome whereas that using the linear standard gave 1.02 copies per genome. We conclude that circular plasmid DNA is unsuitable as a standard, and linear DNA should be used instead, in absolute qPCR. The serious overestimation by the circular plasmid standard is likely due to the undetected lower efficiency of its amplification in the early stage of PCR when the supercoiled plasmid is the dominant template.     

Validation and estimation of parameters for a general probabilistic model of the PCR process.

Earlier work rigorously derived a general probabilistic model for the PCR process that includes as a special case the Velikanov-Kapral model where all nucleotide reaction rates are the same. In this model, the probability of binding of deoxy-nucleoside triphosphate (dNTP) molecules with template strands is derived from the microscopic chemical kinetics. A recursive solution for the probability function of binding of dNTPs is developed for a single cycle and is used to calculate expected yield for a multicycle PCR. The model is able to reproduce important features of the PCR amplification process quantitatively.With a set of favorable reaction conditions, the amplification of the target sequence is fast enough to rapidly outnumber all side products. Furthermore, the final yield of the target sequence in a multicycle PCR run always approaches an asymptotic limit that is less than one. The amplification process itself is highly sensitive to initial concentrations and the reaction rates of addition to the template strand of each type of dNTP in the solution. This paper extends the earlier Saha model with a physics based model of the dependence of the reaction rates on temperature, and estimates parameters in this new model by nonlinear regression. The calibrated model is validated using RT-PCR data.     

Sequence specific detection of DNA using nicking endonuclease signal amplification (NESA)

We have developed a new method for identifying specific single- or double-stranded DNA sequences called nicking endonuclease signal amplification (NESA). A probe and target DNA anneal to create a restriction site that is recognized by a strand-specific endonuclease that cleaves the probe into two pieces leaving the target DNA intact. The target DNA can then act as a template for fresh probe and the process of hybridization, cleavage and dissociation repeats. Laser-induced fluorescence coupled with capillary electrophoresis was used to measure the probe cleavage products. The reaction is rapid; full cleavage of probe occurs within one minute under ideal conditions. The reaction is specific since it requires complete complementarity between the oligonucleotide and the template at the restriction site and sufficient complementarity overall to allow hybridization. We show that both Bacillus subtilis and B. anthracis genomic DNA can be detected and specifically differentiated from DNA of other Bacillus species. When combined with multiple displacement amplification, detection of a single copy target from less than 30 cfu is possible. This method should be applicable whenever there is a requirement to detect a specific DNA sequence. Other applications include SNP analysis and genotyping. The reaction is inherently simple to multiplex and is amenable to automation.     

Enumerating ammonia-oxidizing bacteria in environmental samples using competitive PCR

Primers targeting part of the ammonia-monooxygenase gene (amoA) have been used to detect and characterize ammonia-oxidizing bacteria (AOB) in different environments. In this study, a quantitative polymerase chain reaction (PCR) technique using a competitive template for the amoA primer pair is described and evaluated. The method is based on addition of an internal standard to the PCR, a competitive template, which is amplified together with the template in the environmental sample. By adding different amounts of competitive template to the sample and observing the relative intensity of environmental amplificate and competitive amplificate, the number of amoA gene copies can be determined. Different tests were made to evaluate the competitive PCR method (cPCR) with respect to equal amplification efficiency of the two templates, degeneracy of the priming site and the importance of flanking regions surrounding the competitive template. Calibration curves made by addition of known amounts of Nitrosomonas europaea to soil samples revealed a detection limit for this technique of less than 1000 cells g(-1) soil and a linear response over a wide range of cell additions. Cloning and sequencing of amoA amplificates have confirmed the specificity of the primers, as we have not detected any false positives among the more than 200 clones investigated. The vertical distribution of ammonia-oxidizers in the upper cm of a waterlogged rice paddy soil was compared to nitrate and oxygen concentration profiles determined with microsensors and to net process rates derived from these profiles.     

Comparison of microfluidic digital PCR and conventional quantitative PCR for measuring copy number variation

One of the benefits of Digital PCR (dPCR) is the potential for unparalleled precision enabling smaller fold change measurements. An example of an assessment that could benefit from such improved precision is the measurement of tumour-associated copy number variation (CNV) in the cell free DNA (cfDNA) fraction of patient blood plasma. To investigate the potential precision of dPCR and compare it with the established technique of quantitative PCR (qPCR), we used breast cancer cell lines to investigate HER2 gene amplification and modelled a range of different CNVs. We showed that, with equal experimental replication, dPCR could measure a smaller CNV than qPCR. As dPCR precision is directly dependent upon both the number of replicate measurements and the template concentration, we also developed a method to assist the design of dPCR experiments for measuring CNV. Using an existing model (based on Poisson and binomial distributions) to derive an expression for the variance inherent in dPCR, we produced a power calculation to define the experimental size required to reliably detect a given fold change at a given template concentration. This work will facilitate any future translation of dPCR to key diagnostic applications, such as cancer diagnostics and analysis of cfDNA.     

Use of Droplet Digital PCR for Estimation of Fish Abundance and Biomass in Environmental DNA Surveys

An environmental DNA (eDNA) analysis method has been recently developed to estimate the distribution of aquatic animals by quantifying the number of target DNA copies with quantitative real-time PCR (qPCR). A new quantitative PCR technology, droplet digital PCR (ddPCR), partitions PCR reactions into thousands of droplets and detects the amplification in each droplet, thereby allowing direct quantification of target DNA. We evaluated the quantification accuracy of qPCR and ddPCR to estimate species abundance and biomass by using eDNA in mesocosm experiments involving different numbers of common carp. We found that ddPCR quantified the concentration of carp eDNA along with carp abundance and biomass more accurately than qPCR, especially at low eDNA concentrations. In addition, errors in the analysis were smaller in ddPCR than in qPCR. Thus, ddPCR is better suited to measure eDNA concentration in water, and it provides more accurate results for the abundance and biomass of the target species than qPCR. We also found that the relationship between carp abundance and eDNA concentration was stronger than that between biomass and eDNA by using both ddPCR and qPCR; this suggests that abundance can be better estimated by the analysis of eDNA for species with fewer variations in body mass.     

Revisiting the sigmoidal curve fitting applied to quantitative real-time PCR data

Amplification of a cDNA product by quantitative polymerase chain reaction (qPCR) gives rise to fluorescence sigmoidal curves from which absolute or relative target gene content of the sample is inferred. Besides comparative C(t) methods that require the construction of a reference standard curve, other methods that focus on the analysis of the sole amplification curve have been proposed more recently. Among them, the so-called sigmoidal curve fitting (SCF) method rests on the fitting of an empirical sigmoidal model to the experimental amplification data points, leading to the prediction of the amplification efficiency and to the calculation of the initial copy number in the sample. The implicit assumption of this method is that the sigmoidal model may describe an amplification curve quantitatively even in the portion of the curve where the fluorescence signal is hidden in the noise band. The theoretical basis of the SCF method was revisited here for defining the class of experimental amplification curves for which the method might be relevant. Applying the SCF method to six well-characterized different PCR assays illustrated possible pitfalls leading to biased estimates of the amplification efficiency and, thus, of the target gene content of a sample.     

Improved HF183 Quantitative Real-Time PCR Assay for Characterization of Human Fecal Pollution in Ambient Surface Water Samples

Quantitative real-time PCR (qPCR) assays that target the human-associated HF183 bacterial cluster within members of the genus Bacteroides are among the most widely used methods for the characterization of human fecal pollution in ambient surface waters. In this study, we show that a current TaqMan HF183 qPCR assay (HF183/BFDrev) routinely forms nonspecific amplification products and introduce a modified TaqMan assay (HF183/BacR287) that alleviates this problem. The performance of each qPCR assay was compared in head-to-head experiments investigating limits of detection, analytical precision, predicted hybridization to 16S rRNA gene sequences from a reference database, and relative marker concentrations in fecal and sewage samples. The performance of the modified HF183/BacR287 assay is equal to or improves upon that of the original HF183/BFDrev assay. In addition, a qPCR chemistry designed to combat amplification inhibition and a multiplexed internal amplification control are included. In light of the expanding use of PCR-based methods that rely on the detection of extremely low concentrations of DNA template, such as qPCR and digital PCR, the new TaqMan HF183/BacR287 assay should provide more accurate estimations of human-derived fecal contaminants in ambient surface waters.     

Kinetic analysis of the entire RNA amplification process by Qbeta replicase

The kinetics of the RNA replication reaction by Qbeta replicase were investigated. Qbeta replicase is an RNA-dependent RNA polymerase responsible for replicating the RNA genome of coliphage Qbeta and plays a key role in the life cycle of the Qbeta phage. Although the RNA replication reaction using this enzyme has long been studied, a kinetic model that can describe the entire RNA amplification process has yet to be determined. In this study, we propose a kinetic model that is able to account for the entire RNA amplification process. The key to our proposed kinetic model is the consideration of nonproductive binding (i.e. binding of an enzyme to the RNA where the enzyme cannot initiate the reaction). By considering nonproductive binding and the notable enzyme inactivation we observed, the previous observations that remained unresolved could also be explained. Moreover, based on the kinetic model and the experimental results, we determined rate and equilibrium constants using template RNAs of various lengths. The proposed model and the obtained constants provide important information both for understanding the basis of Qbeta phage amplification and the applications using Qbeta replicase.     

Accuracy in copy number calling by qPCR and PRT: a matter of DNA

The possible implication of copy number variation (CNV) in the genetic susceptibility to human disease needs to be assessed using robust methods that can be applied at a population scale. In this report, we analyze the performance of the two major techniques, quantitative PCR (qPCR) and paralog ratio test (PRT), and investigate the influence of input DNA amount and template integrity on the reliability of both methods. Analysis of three genes (PRELID1, SYNPO and DEFB4) in a large sample set showed that both methods are prone to false copy number assignments if sufficient attention is not paid to DNA concentration and quality. Accurate normalization of samples is essential for reproducible qPCR because it avoids the effect of differential amplification efficiencies between target and control assays, whereas PRT is generally more sensitive to template degradation due to the fact that longer amplicons are usually needed to optimize sensitivity and specificity of paralog sequence PCR. The use of normalized, high quality genomic DNA yields comparable results with both methods.