Department of Biology, University of Louisville, Louisville, Kentucky, 40292, USA

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.

Background

Quantitative Polymerase Chain Reaction, qPCR, has become a common tool of molecular biology to determine the absolute or relative concentrations of particular DNA sequences in samples. The method gives fluorescent values for each of a number of consecutive cycles beginning with cycle 1. Before cycle 1 the initial concentration of double-stranded DNA target sequence is T_{0} and its concentration is amplified in successive cycles to produce a high concentration of the target sequence. At each cycle, the efficiency of amplification (E) is the ratio of the amount of newly synthesized target at the end of the cycle to the amount present at the beginning, thus the amount of target at the end of the cycle is (1 + E) times the amount at the beginning. If every single-stranded target molecule re-associates with exactly one primer molecule and all these structures are extended by polymerase to completely synthesize the complementary target strand, the value of E is 1, which is its theoretical upper limit. The increase in DNA concentration is monitored by a detection system which generates fluorescence in proportion to the concentration of target DNA sequence. Several different types of detection systems are in common usage that generate fluorescence through different types of reactions. I present here two models of qPCR for use when the detection systems uses DNA binding dyes such as SYBR green or SYTO-13. During early cycles of the amplification, the concentration of target is too small to produce measurable fluorescence and is called the lag phase (Figure _{0}) is so small that even with repeated increases of (1 + E) fold in every cycle the target concentration is not high enough to be measurable. Thus, no information on E can be obtained from the amplification curve during lag phase. When target is present in quantities sufficient to be measured, the increase in fluorescence is approximately exponential over a number of successive cycles and the reaction is said to be in the exponential phase (Figure

A typical amplification curve resulting from a qPCR experiment

**A typical amplification curve resulting from a qPCR experiment**
.

Models of qPCR assume total fluorescence, F, is due to two sources: Baseline fluorescence, _{
b
}
_{
T
}
_{
b
}
_{
T
}
_{
T
}
_{
f
}
_{
f
} is a constant that converts the target concentration to fluorescence. Total fluorescence at the end of the **
n
**

where _{
n
} is target concentration at the end of the reassociation/extension step of cycle **
n
**. At the beginning of the

which has the solution:

Since

where _{
T
}
_{
0
} is the amount of fluorescence that could be produced by the target sequence before cycle 1. Values of _{
n
} are defined as _{
n
} may be estimated from the observed fluorescence values as:

Equation 4 is useful when _{
n
} can be reliably distinguished from _{
b
}
_{
n
} changes with **
n
** and use equation 2 to determine the target concentration or equation 3 to determine target fluorescence at each step of the process. Some models of qPCR model

Numerous approaches to modeling the qPCR process are present in recent literature and involve three different general approaches. First the C_{t} and Linear Regression approaches assume constant efficiency and estimate it by linear regression of ln(_{
n
}
_{
b
}
_{
T
}
_{
0
} as the intercept of a sigmoidal function

Current models

The models presented here may be regarded as an extension of the models presented by Smith **
A
**

Re-association step

At the end of the dissociation step of cycle **
A
**

**At end of:**

**
A
**

**
A
**

**
a
**

**
a
**

**
A
**

**
A
**

**
A
**

dissociation step

_{
n,0
}

_{
n,0
}

_{
n,0
}

_{
n,0
}

0

0

0

re-association step

_{
n,e
}

_{
n,e
}

_{
n,e
}

_{
n,e
}

_{
n,e
}

_{
n,e
}

_{
n,e
}

Model 1 reversible re-association

Reversible dissociation/re-association reactions and rate constants are shown in Figure **
A
**

**Finding equilibrium for Model 1.**

Click here for file

Panel A: Dissociation/re-association reactions and rate constants for Model 1. Panel B: Differential equations for model 1

**Panel A: Dissociation/re-association reactions and rate constants for Model 1. Panel B**: Differential equations for model 1.

Model 2 non-reversible re-association

Non-reversible re-association reactions and rate constants are shown in Figure _{
n,e
}, _{
n,e
} cannot be found as the simultaneous solution of equations in Figure _{
a
} > 0, _{
a12
} > 0, _{
n
} ≥0 and _{
n
} ≥0 then _{
n,e
} = 0 is the only possible equilibrium solution and with _{
n,e
} = 0, any value of _{
n
} will satisfy the equilibrium equations. This result is also expected on purely scientific grounds as annealing without dissociation will always eventually reduce the single strand template concentration to a concentration of zero. The equilibrium value, _{
n,e
}, is found by expressing the rate of change in _{
n
} as a function of _{
n
} and then integrating over _{
n
} as it goes from _{
n,0
} to zero. The solutions for _{
n,e
}
_{
n,e
} and _{
n,e
} are given as equations A2.4, A2.5 and A2.6, respectively, in Additional file _{
a
}, _{
a12
}
_{
n,0
} and _{
n,0
} (see Additional file

**Finding equilibrium for Model 2.**

Click here for file

Panel A: Dissociation/re-association reactions and rate constants for Model 2. Panel B: Differential equations for model 2

**Panel A: Dissociation/re-association reactions and rate constants for Model 2. Panel B**: Differential equations for model 2.

Extension step

The proportion of target/primer duplexes **
A
**

The whole process

The whole qPCR process is modeled by iteration of equations 5.1 and 5.2 with _{
n,e
} determined by equation A1.9 for model 1 or equation A2.5 for model 2 (see Additional files _{
0,0
} and _{
0,0
}, respectively. The Efficiency of amplification for single stranded template at cycle _{
n+1,0
} /_{
n,0
})-1 which may be obtained from equation 5.1 as

and fluorescence at cycle n is found using equation 1 as

Since equation 7 is defined only for n = 1,2,3… it cannot be used directly to obtain a value of _{
T
}
_{
0
}, however, fitting equation 7 to an actual qPCR curve provides an estimate of _{
0,0
} which is the initial concentration of each of **
A
**

Relating Model 2 to Boggy & Woolf (2010)

Model 2 annealing kinetics have been analyzed by Boggy & Woolf

**Relating Model 2 to the results of Boggy and Woolf (2010) [14].**

Click here for file

Restricting model fit to only part of the qPCR curve

Since models 1 & 2 both include the effect of declining primer concentration during PCR they predict a different shape of the qPCR curve between lag and stationary phase for solutions that differ in primer concentration. Consider two solutions, one with low _{
0,0
} and one with high _{
0,0
} and with identical primer concentrations. Allow the one with low _{
0,0
} to undergo enough cycles so its current target concentration is equal to the initial target concentration for the solution with higher _{
0,0
} . If primer concentration is not limiting in any way then the two solutions would have identical qPCR curves from that point on. If primer concentration is limiting in some way then the qPCR curve for the solution with lower _{
0,0
} will at that point rise more slowly than that for the solution with higher S_{0,0} due to lowered primer concentration. When fitting models 1 and 2 to data it is thus desirable to include as much of the exponential phase as possible to allow changing primer concentrations to affect the analysis. Including most of the exponential phase is desired, however, in late exponential phase effects not included in the model also begin to occur and may cause the model to have poor fit. Effects in late exponential phase not modeled include reduced amount of dNTPs, reduced amount of polymerase due to decay and possibly partially replicated templates. In order to restrict analyses to only part of the exponential phase, models were fit to qPCR curves for cycles in the lag and exponential phase for which (F_{n} - _{b}F)/(F_{max} - _{b}F) ≤ L, where _{b}F, and F_{max} are baseline fluorescence and maximum fluorescence achieved in stationary phase. The value, L, is the proportion of increase of F_{n} above _{b}F and is a cut off value used to restrict the analysis to the first parts of the qPCR curve up to a point in the exponential phase where the model is thought to be valid. When choosing L, a compromise between these opposing effects is necessary and is done here by choosing the highest value for L that allows good fit to qPCR curves. To accomplish this, the models were fit to qPCR curves using a range of L values from 0.1 to 0.98 and the goodness of fit and estimated initial concentration determined for each L value. Both model 1 and 2 generally gave poor fits for L ≥ 0.95. The value of 0.8 was chosen for presentation here because the MSresidual values for most qPCR curves increased for values of L higher than 0.8 and also because the accuracy of estimates of initial target concentration was better for L values near 0.8. The question of how much of the qPCR curve to use in analysis is a problem inherent to most methods used for interpretation of qPCR curves. Boggy and Woolf (2010) _{t} method uses a different cut-off for every curve which is invoked when the ‘window-of-linearity’ is chosen.

Methods for incorporating experimental design into the data analysis

Kinetic models of qPCR allow more sophisticated and powerful analyses than are possible with other models. If a group of solutions have exactly the same chemical components except for possibly differing in initial target concentration, then all the kinetic parameters should be the same for all the solutions. The rate constants may then be estimated by fitting several or many qPCR curves simultaneously to get better estimates of rate constants and initial target concentrations. In such fits the kinetic parameters are common to all the different qPCR curves in the analysis but each curve may have a unique value for initial target concentration. Also kinetic models such those presented here allow an analyst to adjust the model to accommodate different experimental designs. I present four different estimation methods (A, B, C, and D) which are used here to fit models to qPCR data which have different experimental designs.

Standard curve method (Method A)

This method may be used only when the true initial target concentrations are known for each curve in the analysis (a standard curve). All initial conditions of the qPCR, other than initial target concentration are assumed to be the same for all curves. All curves are analyzed simultaneously in a non-linear fit in which the estimated rate constants apply to all curves and the initial target concentration is held constant at its known value for each curve. The fitting process minimizes the cost function over all cycles in all curves. The advantage of this method is that it pools the information of all the curves to give a single best estimate for each model parameter. The predicted fluorescence values provided by the analysis may be compared to the observed fluorescence values to assess how well each qPCR curve is fit by the model. The value of this method is that it may be used for estimation of initial target concentration for samples with unknown initial target concentration provided all other conditions of the PCR are the same as those used in the standard curve. This is done by doing a non-linear fit to the qPCR curve of the unknown sample in which only the initial target concentration, _{
0,0
} , is estimated while using the estimated parameters from the standard curve analysis as constants. To assess how well the models presented here estimate _{
0,0
} , each qPCR curve in the known samples was treated as an unknown sample and the estimated _{
0,0
} value compared to the known value for _{
0,0
}.

Dilution curve method (Method B)

This method may be used when a standard curve is not available but a dilution series of the unknown sample is available. Here the absolute target concentration in the undiluted sample is unknown and denoted _{
0,0,max
}. The absolute target concentration of the i^{th} diluted sample is d_{i}
_{
0,0,max
} where d_{i} is the dilution ratio for the i^{th} sample. The qPCR curves from the diluted samples are analyzed simultaneously using non-linear curve fitting. Rate constants apply to all curves in the analysis and a single concentration parameter _{
0,0,max
} is estimated with the initial concentration for each sample set at d_{i}
_{
0,0,max
}. In general usage of this method the estimated _{
0,0,max
} would be the end result. To assess the accuracy of this method in this study the known absolute target concentrations for the data sets used here are ignored but their ratios are used to obtain values for d_{i}. Simultaneous fitting of all the curves was done as described above to estimate _{
0,0,max
} and all the rate constants. Next the estimated rate constants are treated as constants in a second non-linear fit of each qPCR curve separately in which an _{
0,0
} value is estimated for qPCR curve. These estimated _{
0,0
} values are compared to the known values to assess the accuracy of method B.

Simultaneous curves method (Method C)

This method may be used when two or more samples are to be analyzed that may have different initial concentrations and a standard curve is not available nor are any dilutions of either sample available. The samples are assumed to be subject to qPCR with identical conditions except for the fact they may have different initial target concentrations. In this method all samples are analyzed simultaneously with estimated rate constants applying to all samples, but each sample has a unique _{
0,0
} value to be estimated. To assess the accuracy of this method with the data sets analyzed here all information on initial target concentration is ignored and the estimated S_{0,0} values for each sample is then compared to the known values.

Separate curves method (Method D)

This method does not assume any similarity of rate constants among any of the samples and does not analyze samples simultaneously as methods A, B, and C do. Each sample is analyzed separately with all parameters estimated independently for each qPCR curve. Here the accuracy of method C is assessed by comparing the actual _{
0,0
} values to the estimated ones. This is the type of analysis that is conventionally done when analyzing qPCR curves.

Methods

Data sets

Both models 1 and 2 were fit to two different data sets for which the actual initial concentrations of target are known. The model of Boggy and Woolf (2010) ^{3} to 5 × 10^{8} molecules per reaction volume at 10 fold increments. The data of Rutledge (2004) ^{2} to 4.17 × 10^{7} molecules per reaction volume at 10 fold increments. Data set 1 has 2 replicate qPCR curves for each target concentration and data set 2 has 5 replicates for each target concentration.

Determining baseline fluorescence

Before fitting a model the baseline fluorescence was determined for each qPCR curve by fitting the sigmoidal function given in equation 8 below.

The first term on the right side of equation 8 is the baseline fluorescence, _{
b
}
_{
max
} is the maximum fluorescence, _{
1/2
} is the cycle number at which _{
n
} is midway between _{
b
}
_{
max
}
_{
s
} is a scale parameter. Parameters in equation 8 and bounds used in the non-linear estimation program are _{
b
}
_{
max
} > _{
b
}
_{1/2} > 1, and _{
s
} > 0. Equation 8 is fit to each entire qPCR curve by nonlinear estimation of the parameters using least squares as the criterion for fit with the PROC NLIN procedure of the SAS program version 9.3 _{
b
}
_{t} method and this same effect is likely occurring in fits of the models here. They also point out inherent problems with estimating baseline fluorescence from a fixed number of points in early cycles. Rutledge and Stewart (2008)

Estimation of model parameters

Methods described here are used to determine how well each model explains the shape of each qPCR curve and also to determine how accurately each model estimates initial target concentration for each of the two data sets. Each model, 0, 1 or 2, was fit to qPCR data by nonlinear curve fitting with PROC NLIN of SAS version 9.3 _{
0,0
} > 0, k > 0 for model 0, _{
0,0
} > 0 , _{
f
} >0, _{
s
} > _{
D12
} >0 for model 1, and _{
0,0
} >0 , _{
f
} >0 , and

**SAS commands for Models 1 and 2.**

Click here for file

Effect of varying L

To determine the effect of the choice of the value of L on the results of the analysis, the entire analysis was done separately for each of a range of values for L. Specifically, each of the three models (0, 1, 2) was fit using each of the estimation methods (A, B, C, D) to each of the two data sets (1, 2) for each of the L values of 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.95, 0.98. The MSresidual and Fold Error for each fit were averaged over replicates 1 and 2 for each data set and the resulting means plotted versus L for each dilution separately.

Effect of errors in estimation of baseline fluorescence

Analyses were done to assess the effect of errors in the estimation of baseline fluorescence, _{b}F, on the goodness of fit of the model and the estimated initial target concentration. The estimation procedure used to estimate _{b}F provides the standard error of the estimated value. Fits of the models and estimation of initial target concentration was done using _{b}F values 1 and 2 standard deviation units above and below the estimated value. Plots of the mean MSresidual and Fold Error in estimation of initial target concentration versus the _{b}F value used were used to assess the effect of errors in estimation of _{b}F.

Results

For each of the three models (0,1,2), each of four methods for parameter estimation were used for each data set. Here every model and estimation method was used to fit the same data and MSresidual is used to compare models and methods in their goodness of fit to the data. A value of MSresidual was computed for every qPCR curve for each model and for each estimation method. Plots of MSresidual versus log_{10}(T_{0}) are given in Figure ^{6} molecules in data set 1 and is representative of the fit found for all samples in both data sets. Note that in Figure

Goodness of fit to qPCR curves for data set 1 with L = 0.8

**Goodness of fit to qPCR curves for data set 1 with L = 0.8.** Plots are MS_{residual} vs known initial log_{10} target concentration, T_{0}, in nM/L. Panels A,B,C and D contain plots for the Standard curve, Dilution curve, Simultaneous curves, and Separate curves estimation methods, respectively. Plot symbols are: blue circle = model 0, red plus = model 1, green X = model 2.

Goodness of fit to qPCR curves for data set 2 with L = 0.8

**Goodness of fit to qPCR curves for data set 2 with L = 0.8.** Plots are MS_{residual} vs known initial log_{10} target concentration, T_{0}, in nM/L. Panels A,B,C and D contain plots for the Standard curve, Dilution curve, Simultaneous curves, and Separate curves estimation methods, respectively. Plot symbols are: blue circle = model 0, red plus = model 1, green X = model 2.

Fit of models to qPCR curves for each model and each estimation method for data set 1, rep = 1, dilution = 1E-2, and L = 0.8

**Fit of models to qPCR curves for each model and each estimation method for data set 1, rep = 1, dilution = 1E-2, and L = 0.8.** Plot symbols for predicted fluorescence are:blue circle = model 0, red plus = model 1, green X = model 2. The plot symbol for observed fluorescence is delta (**δ**).

Estimation of absolute target concentration

The estimates of initial target concentrations and the known initial target concentrations are denoted S_{0,0} and T_{0}, respectively. Plots of log_{10}(mean S_{0,0}) vs log_{10}(T_{0}) for each model and estimation method are given in Figure _{0,0} values that have a good linear relation to the true values, T_{0}, for estimation methods A and B with both data sets. For estimation methods C and D the linear relation between S_{0,0} and T_{0} holds well for all three models when applied to data set 1 ,however, for data set 2 model 1 and 2 are very linear, but model 0 gives non-linear results. The high linearity of these log-log plots is not a good indication of the accuracy of the estimation because the estimates can be linear but at the same time be biased. A measure of the accuracy of the estimation is the Fold Error, which is the ratio of the estimated value to the true value (S_{0}/T_{0}). Fold error is computed for each model and estimation method and given in Figure _{0} than the other two models. Estimation method C generally gave less accurate estimates than methods A and B and model 2 performed best. When using estimation method D, again model 2 gave the most accurate estimates over both data sets and model 0 was the least accurate for both data sets.

Relation between log_{10}(S_{0}_Mean) and log_{10}(T_{0}) for analysis of data set 1 with L = 0.8

**Relation between log**_{10}**(S**_{0}**_Mean) and log**_{10}**(T**_{0}**) for analysis of data set 1 with L = 0.8.** log10S_{0}_Mean denotes base 10 log of mean of S_{0} for the two replicates and the solid line denotes log_{10}(S_{0}_Mean) = log_{10}(T_{0}) which indicates perfect estimation. Panels A,B,C and D contain plots for the Standard curve, Dilution curve, Simultaneous curves, and Separate curves estimation methods, respectively. Plot symbols are: blue circle = model 0, red plus = model 1, green X = model 2.

Relation between log_{10}(S_{0}) and log_{10}(T_{0}) for analysis of data set 2 with L = 0.8

**Relation between log**_{10}**(S**_{0}**) and log**_{10}**(T**_{0}**) for analysis of data set 2 with L = 0.8.** log10S_{0}_Mean denotes base 10 log of mean of S_{0} for the 5 replicates and the solid line denotes log_{10}(S_{0}_Mean) = log_{10}(T_{0}) which indicates perfect estimation. Panels A,B,C and D contain plots for the Standard curve, Dilution curve, Simultaneous curves, and Separate curves estimation methods, respectively. Plot symbols are: blue circle = model 0, red plus = model 1, green X = model 2.

Relation between Fold Error and log_{10}(T_{0}) for analysis of data set 1 with L = 0.8

**Relation between Fold Error and log**_{10}**(T**_{0}**) for analysis of data set 1 with L = 0.8.** Fold Error is the ratio S0/T0 and a value of one indicates perfect estimation. Panels A,B,C and D contain plots for the Standard curve, Dilution curve, Simultaneous curves, and Separate curves estimation methods, respectively. Plot symbols are: blue circle = model 0, red plus = model 1, green X = model 2.

Relation between Fold Error and log_{10}(T_{0}) for analysis of data set 2 with L = 0.8

**Relation between Fold Error and log**_{10}**(T**_{0}**) for analysis of data set 2 with L = 0.8.** Fold Error is the ratio S0/T0 and a value of one indicates perfect estimation. Panels A,B,C and D contain plots for the Standard curve, Dilution curve, Simultaneous curves, and Separate curves estimation methods, respectively. Plot symbols are: blue circle = model 0, red plus = model 1, green X = model 2.

Effect of varying L

The plots of the Mean MS residual vs L and Mean log(Fold Error) vs L for the samples with dilution factor 0.1 in data set 2 are given in Figures

**Effect of varying L on MSresidual and Fold Error.**

Click here for file

Goodness of fit to qPCR curves for analysis of data set 2 with a range of L values

**Goodness of fit to qPCR curves for analysis of data set 2 with a range of L values.** Plots are mean MS_{residual} vs the value of L used in the analysis. Panels A,B,C and D contain plots for the Standard curve, Dilution curve, Simultaneous curves, and Separate curves estimation methods, respectively. Plot symbols are: blue circle = model 0, red plus = model 1, green X = model 2.

Relation between log_{10}(Fold Error) and L for analysis of data set 2 with a range of L values

**Relation between log**_{10}**(Fold Error) and L for analysis of data set 2 with a range of L values.** log10S_{0}_Mean denotes base 10 log of mean of S_{0} for the replicates. A value of 0 for log_{10}(Fold Error) indicates perfect estimation. Panels A,B,C and D contain plots for the Standard curve, Dilution curve, Simultaneous curves, and Separate curves estimation methods, respectively. Plot symbols are: blue circle = model 0, red plus = model 1, green X = model 2.

Effect of errors in estimation of baseline fluorescence

Plots of mean MSresidual and fold error in estimated initial target concentration versus baseline fluorescence for estimation methods A, B, and D are affected very little by variation in the value of _{b}F used (plots not reported here). When using estimation method C, the MSresidual values were not affected much by variations in the _{b}F but the initial target concentrations were over-estimated when using low values of _{b}F. Increases in Fold Error by a factor of up to 10 was found in some plots when a _{b}F value two standard error units below the estimated value was used.

Discussion

Fit to qPCR curves

The two models presented here have both been shown to give good fit to the lag and exponential phase of qPCR curves over a wide range of initial target concentrations while using a single set of rate constants. The lack of fit of model 0 to qPCR curves seen in Figures _{a}/k_{a12} where k_{a} and k_{a12} are the rate constants of model 2 (Figure

Estimation of T_{0}

The results here indicate the best methods for estimation of target concentration in an unknown sample is to use a standard curve that has a range of values for initial target concentration or to use a dilution curve composed of a series of dilutions of the original sample and analyze the qPCR curve with estimation methods A and B, respectively. Figures

The poor fit of the Boggy and Woolf (2010)

Effect of varying L

Figures

Effect of errors in estimation of baseline fluorescence

Estimation methods A, B and D are robust to errors in baseline fluorescence. Estimation method C is less robust to such errors.

General comments

The models presented here assume the annealing of single-stranded DNA to double-stranded molecules reaches equilibrium at each cycle. This assumption may not be true, however, the ability of the present models to fit qPCR curves as well as they do suggests it may be approximately true. Model 1 gave slightly better fits to qPCR curves and slightly better estimates of initial concentration than Model 2 when estimation methods A and B were used. This result suggests the reversible reactions assumed in model 1 (Figure

An advantage of the mechanistic approach to modeling qPCR curves is that models describe actual events of the process and thus may be expanded to include other effects that may improve accuracy. Kinetic models naturally allow simultaneous estimates of common parameters which will increase accuracy. This is extremely difficult or impossible with other methods used for modeling qPCR. Because the kinetic models account for variable efficiency by kinetic theory they may give better estimates of initial target concentration than other approaches. Another advantage of kinetic models is that parameters of the model may be estimated by dedicated experiments distinct from the qPCR curve experiment. For example, the parameter _{
f
} present in both models 1 and 2 converts double-stranded DNA concentration into amount of fluorescence for a particular experimental system. The value of _{
f
} could be determined in experiments separate from qPCR and then used as a constant in the analysis of a qPCR curve, thus increasing the accuracy of estimation of the kinetic parameters and target concentrations. Lastly, kinetic models uniquely allow estimation of absolute initial concentrations of target sequence without use of a standard curve of any type. The accuracy of estimation with kinetic models is enhanced greatly by the use of a standard curve, but it is not required. In fact the dilution curve method gave fits essentially as good as the standard curve method. The mechanistic models developed here offer a more complete description of the amplification occurring in qPCR, fit observed data very well, and allow more accurate estimation of initial target concentration than other methods.

Conclusions

Two stepwise kinetic equilibrium models of qPCR are presented and analytical solutions are given for equilibrium values during annealing of single stranded to double stranded molecules. The models are amenable to different types of non-linear fitting which include fitting several curves simultaneously when they have common parameter values. Both models are shown to give very good fit to qPCR data with a wide range of initial target concentrations with a single set of values for rate constants. The two models also give accurate estimates of initial absolute target concentration using several different methods for estimation. Using the models with data from a standard curve gives accurate estimates of initial absolute target concentration. In the absence of a standard curve, a dilution curve method also provided accurate estimates of the initial absolute target concentration. In the absence of either a standard or dilution curve the models provide estimates, though less accurate, of absolute initial target concentration. These models presently give the best unified theory for the interpretation of qPCR data in that they explain well the shape of the qPCR curve and how it is affected by variation in the initial target concentration.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Gary Cobbs did the entire project.

Acknowledgments

I thank Dr. James Alexander, Dr. Robert Page, Dr. David Schultz and 2 reviewers for critically reviewing earlier versions of this manuscript.