Mathematical model of real-time PCR kinetics

Analytical Biochemistry, 2008

Quantitative real-time PCR remains a cornerstone technique in gene expression analysis and sequence characterization. Despite the importance of the approach to experimental biology the confident assignment of reaction efficiency to the early cycles of real-time PCR reactions remains problematic. Considerable noise may be generated where few cycles in the amplification are available to estimate peak efficiency. An alternate approach that uses data from beyond the log-linear amplification phase is explored with the aim of reducing noise and adding confidence to efficiency estimates. PCR reaction efficiency is regressed to estimate the per-cycle profile of an asymptotically departed peak efficiency, even when this is not closely approximated in the measurable cycles. The process can be repeated over replicates to develop a robust estimate of peak reaction efficiency. This leads to an estimate of the maximum reaction efficiency that may be considered primer-design specific. Using a series of biological scenarios we demonstrate that this approach can provide an accurate estimate of initial template concentration.

Biotechnology and Bioengineering, 2005

Real-time polymerase chain reaction (PCR) is one of the most sensitive and accurate methods for quantifying transcript levels especially for those expressed at low abundance. The selective amplification of target DNA over multiple cycles allows its initial concentration to be determined. The amplification rate is a complex interplay of the operating conditions, initial reactant concentrations, and reaction rate constants. Experimentally, the compounded effect of all factors is quantified in terms of an effective efficiency, which is estimated by curve fitting to the amplification data. We present a comprehensive model of PCR to study the effect of various reactant concentrations on the amplification efficiency. The model is used to calculate the kinetic progression of the target DNA concentration with cycle number under conditions when different species are stoichiometrically or kinetically limiting. The reaction efficiency remains constant for the initial cycles. As the primer concentration becomes limiting, the efficiency is marked by a gradual decrease. This is in contrast to a steep decline under nucleotide limiting conditions. Under some conditions, commonly used experimentally, increasing primer concentration has the adverse effect of reducing the final amplified template concentration. This phenomenon seen at times experimentally is explained by the simulation results under rate limiting enzyme concentrations. Primer dimer formation is shown to significantly affect the reaction rates, effective efficiency, and the estimated initial concentrations. This model, by describing the interplay of the many operating variables, will be a useful tool in designing PCR conditions and evaluating its results.

PLoS ONE, 2012

While many decisions rely on real time quantitative PCR (qPCR) analysis few attempts have hitherto been made to quantify bounds of precision accounting for the various sources of variation involved in the measurement process. Besides influences of more obvious factors such as camera noise and pipetting variation, changing efficiencies within and between reactions affect PCR results to a degree which is not fully recognized. Here, we develop a statistical framework that models measurement error and other sources of variation as they contribute to fluorescence observations during the amplification process and to derived parameter estimates. Evaluation of reproducibility is then based on simulations capable of generating realistic variation patterns. To this end, we start from a relatively simple statistical model for the evolution of efficiency in a single PCR reaction and introduce additional error components, one at a time, to arrive at stochastic data generation capable of simulating the variation patterns witnessed in repeated reactions (technical repeats). Most of the variation in C q values was adequately captured by the statistical model in terms of foreseen components. To recreate the dispersion of the repeats' plateau levels while keeping the other aspects of the PCR curves within realistic bounds, additional sources of reagent consumption (side reactions) enter into the model. Once an adequate data generating model is available, simulations can serve to evaluate various aspects of PCR under the assumptions of the model and beyond.

Nucleic Acids Research, 2011

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.

Nucleic Acids Research, 1998

The present studies demonstrate a theoretical and practical framework for the accurate quantitation of gene expression in RNA extracted from microscopic tissue samples. The approaches are developed around competitive RT-PCR techniques. Assay performance has been examined and validated at both the RT and PCR steps. Our analysis of RT transcription efficiency for a number of native and competitor combinations shows that this property can differ, even for very similar templates. However, this difference is consistent and, once identified and measured, can be removed as an obstacle to accuracy. Using mathematical modeling, we have examined the simulated co-amplification of native and competitor templates in PCR. Useful insights have emerged from such modeling which indicate that differences in initial amplification efficiency and the rate of decay of amplification efficiency during the reaction can rapidly lead to inaccuracy, even while the slope and linearity of log plots of the competitor input and reaction product ratios are close to ideal. Finally, we show here that competitive RT-PCR reactions do not have to remain in the log-linear phase of PCR in order to accomplish accurate and precise quantification. Using appropriate competitors sharing primer binding sites and high internal sequence similarity, identical amplification efficiencies are preserved throughout the reaction. Reaction products, including heteroduplexes formed between native and competitor templates as reactions progress to plateau, can be identified and quantified accurately using the new technique of denaturing HPLC (dHPLC). This analytical technique allows the accuracy of competitive RT-PCR to be preserved beyond the linear phase. The technique has high sensitivity and precision and target abundances as low as 100 copies could be reliably estimated.

Analytical Biochemistry, 2009

Mathematical Biosciences and Engineering, 2010

A mathemati cal model for PCR (Polymerase Chain Reaction) is developed using the law of mass action. Differential equations are written from the chemical equations, preserving th e detail of the complementary DNA single strand being extended one bas e pair at a time. Th e equations for th e annealing stage are solved analytically. The method of multiple scales is used to approximate solutions for the extension stage. A map is then developed from the solutions to simulate PCR. The advantage of this model is the ability to use the map to optimize the process. Our result s suggest that dynamically optimizing the extension and annealing stages ma y significantly reduc e the total time for a PCR run,

Chemical Engineering Science, 2011

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.

Applied Mathematics and Computation, 2006

Earlier work by Saha et al. 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.

umt.edu

Abstract. PCR (Polymerase Chain Reaction), a method which replicates a selected sequence of DNA, has revolutionized the study of genomic material, but mathematical study of the process has been limited to simple deterministic models or descriptions relying on stochastic ...