### Experimental design

The absolute quantification method relies on the comparison of distinct samples, such as the comparison of a biological sample with a standard curve of known initial concentration [21]. We wondered how accuracy and precision change when a standard curve is compared with unknown samples characterized by different efficiencies. A natural way of studying the effect of efficiency differences among samples on quantification would be to compare the amounts of a quantified gene.

A slight amplification inhibition in the quantitative real-time PCR experiments was obtained by using two systems: decreasing the amplification mix used in the reaction and adding varying amounts of IgG, a known PCR inhibitor.

For the first system, we amplified the MT-ND1 gene by real-time PCR in reactions having the same initial amount of DNA but different amounts of SYBR Green I Master mix. A standard curve was performed over a wide range of input DNA (3.14 × 10^{7}–3.14 × 10^{1}) in the presence of optimal amplification conditions (100% amplification mix), while the unknowns were run in the presence of the same starting DNA amounts but with amplification mix quantities ranging from 60% to 100%. This produced different reaction kinetics, mimicking the amplification inhibition that often occurs in biological samples [17, 22].

Furthermore, quantitative real-time PCR quantifications were performed in the presence of an optimal amplification reaction mix added with serial dilutions of IgG (0.0625 – 2 μg/ml) thus acting as the inhibitory agent [23].

The reaction efficiency obtained was estimated by the LinReg method [24]. This approach identifies the exponential phase of the reaction by plotting the fluorescence on a log scale. A linear regression is then performed leading to the estimation of the efficiency of each PCR reaction.

### Quantitative Real-Time PCR

The DNA standard consisted of a pGEM-T (Promega) plasmid containing a 104 bp fragment of the mitochondrial gene NADH dehydrogenase 1 (MT-ND1) as insert. This DNA fragment was produced by the ND1/ND2 primer pair (forward ND1: 5'-ACGCCATAAAACTCTTCACCAAAG-3' and reverse ND2: 5'-TAGTAGAAGAGCGATGGTGAGAGCTA-3'). This plasmid was purified using the Plasmid Midi Kit (Qiagen) according to the manufacturer's instructions. The final concentration of the standard plasmid was estimated spectophotometrically by averaging three replicate A_{260} absorbance determinations.

Real time PCR amplifications were conducted using LightCycler^{®} 480 SYBR Green I Master (Roche) according to the manufacturer's instructions, with 500 nM primers and a variable amount of DNA standard in a 20 μl final reaction volume. Thermocycling was conducted using a LightCycler^{®} 480 (Roche) initiated by a 10 min incubation at 95°C, followed by 40 cycles (95°C for 5 s; 60°C for 5 s; 72°C for 20 s) with a single fluorescent reading taken at the end of each cycle. Each reaction combination, namely starting DNA and amplification mix percentage, was conducted in triplicate and repeated in four separate amplification runs. All the runs were completed with a melt curve analysis to confirm the specificity of amplification and lack of primer dimers. *Ct* (fit point method) and *Cp* (second derivative method) values were determined by the LightCycler^{®} 480 software version 1.2 and exported into an MS Excel data sheet (Microsoft) for analysis after background subtraction (available as Additional file 1). For *Ct* (fit point method) evaluation a fluorescence threshold manually set to 0.5 was used for all runs.

### Description of the *SCF* method

Fluorescence readings were used to fit the following 4-parameter sigmoid function using nonlinear regression analysis:

{F}_{x}=\frac{{F}_{\mathrm{max}}}{1+{e}^{(-\frac{1}{b}\left(x-c\right))}}+{F}_{b}

(1)

where *x* is the cycle number, *F*_{
x
}is the reaction fluorescence at cycle *x*, *F*_{
max
}is the maximal reaction fluorescence, *c* is the fractional cycle at which reaction fluorescence reaches half of *F*_{
max
}, *b* is related to the slope of the curve and *F*_{
b
}is the background reaction fluorescence. *F*_{
max
}quantifies the maximal fluorescence read by the instrument and does not necessarily indicate the amount of DNA molecules present at the end of the reaction. The fact that *F*_{
max
}does not necessarily represent the final amount of DNA might be due to un-saturating dye concentration or to fluorescence quenching by inhibitors. For each run a nonlinear regression analysis was performed and these four parameters were evaluated. A simple derivative of Eq. 1 allowed us to estimate *F*_{
0
}, when *x* = 0:

{F}_{0}=\frac{{F}_{\mathrm{max}}}{1+{e}^{(\frac{c}{b})}}

(2)

where *F*_{
0
}represents the initial target quantity expressed in fluorescence units. Conversion of *F*_{
0
}to the number of target molecules was obtained by a calibration curve in which the log input DNA was related to the log of *F*_{
0
}[18]. Subsequently, this equation was used for quantification with log transformation of fluorescence data to increase goodness-of-fit as described in Goll *et al*. 2006 [19].

### Description of the *Cy*_{
0
}method

The *Cy*_{
0
}value is the intersection point between the abscissa axis and tangent of the inflection point of the Richards curve obtained by the non-linear regression of raw data (Fig. 1).

The *Cy*_{
0
}method was performed by nonlinear regression fitting of the Richards function [25], an extension of logistic growth curve, in order to fit fluorescence readings to the 5-parameter Richards function:

{F}_{x}=\frac{{F}_{\mathrm{max}}}{{\left[1+{e}^{\left(-\frac{1}{b}\left(x-c\right)\right)}\right]}^{d}}+{F}_{b}

(3)

where *x* is the cycle number, *F*_{
x
}is the reaction fluorescence at cycle *x*, *F*_{
max
}is the maximal reaction fluorescence, *x* is the fractional cycle of the turning point of the curve, *d* represents the Richards coefficient, and *F*_{
b
}is the background reaction fluorescence. The inflection point coordinate (*Flex*) was calculated as follows (Additional file 2):

Flex=\left[c+b\mathrm{ln}d;{F}_{\mathrm{max}}{\left(\frac{d}{d+1}\right)}^{d}+{F}_{b}\right]

(4)

and the tangent slope (*m*) was estimated as:

m=\frac{{F}_{\mathrm{max}}}{b}{\left(\frac{d}{d+1}\right)}^{d+1}

(5)

When *d* = 1, the Richards equation becomes the logistic equation shown above. The five parameters that characterized each run were used to calculate the *Cy*_{
0
}value by the following equation:

{C}_{y0}=c+b\mathrm{ln}d-b\left(\frac{d+1}{d}\right)\left[1-\frac{{F}_{b}}{{F}_{\mathrm{max}}}{\left(\frac{d+1}{d}\right)}^{d}\right]

(6)

Although the *Cy*_{
0
}is a single quantitative entity, as is the *Ct* or *Cp* for threshold methodologies, it accounts for the reaction kinetic because it is calculated on the basis of the slope of the inflection point of fluorescence data.

### Statistical data analysis

Nonlinear regressions (for 4-parameter sigmoid and 5-parameter Richards functions) were performed determining unweighted least squares estimates of parameters using the Levenberg-Marquardt method. Accuracy was calculated using the following equation:

R{E}_{\left({n}_{Dna},{\%}_{mix}\right)}={\displaystyle \sum _{i=1}^{n}\left(\frac{{x}_{{i}_{obs}\left({n}_{Dna},{\%}_{mix}\right)}}{{x}_{{i}_{\mathrm{exp}}\left({n}_{Dna},{\%}_{mix}\right)}}-1\right)}, where R{E}_{\left({n}_{Dna},{\%}_{mix}\right)} was the relative error, while {x}_{{i}_{obs}\left({n}_{Dna},{\%}_{mix}\right)} and {x}_{{i}_{\mathrm{exp}}\left({n}_{Dna},{\%}_{mix}\right)} were the estimated and the true number of DNA molecules for each combination of input DNA (*n*_{
Dna
}) and amplification mix percentage (*%*_{
mix
}) used in the PCR. Precision was calculated as:

C{V}_{({n}_{Dna},{\%}_{mix})}=\frac{{s}_{{\overline{x}}_{obs\left({n}_{Dna},{\%}_{mix}\right)}}}{{\overline{x}}_{obs\left({n}_{Dna},{\%}_{mix}\right)}}, where C{V}_{({n}_{Dna},{\%}_{mix})} was the coefficient of variation, {\overline{x}}_{obs\left({n}_{Dna},{\%}_{mix}\right)} and {s}_{{\overline{x}}_{obs\left({n}_{Dna},{\%}_{mix}\right)}} were the mean and the standard deviation for each combination of n_{Dna} and %_{mix}. In order to verify that the Richards curves, obtained by nonlinear regression of fluorescence data, were not significantly different from the sigmoidal curves, the values of *d* parameter were compared to the expected value *d* = 1, using *t* test for one sample. For each combination of *n*_{
Dna
}, *%*_{
mix
}, the *t* values were calculated as follow:

{t}_{\left({n}_{Dna},{\%}_{mix}\right)}=\frac{{\overline{d}}_{\left({n}_{Dna},{\%}_{mix}\right)}-1}{S{E}_{{d}_{\left({n}_{Dna},{\%}_{mix}\right)}}}, where {\overline{d}}_{\left({n}_{Dna},{\%}_{mix}\right)} and S{E}_{{d}_{\left({n}_{Dna},{\%}_{mix}\right)}} were the mean and the standard error of *d* values for each combination of *n*_{
Dna
}and *%*_{
mix
}, with p(*t*) < 0.05 for significance level. R{E}_{\left({n}_{Dna},{\%}_{mix}\right)} values were reported using 3-d scatterplot graphic, a complete second order polinomial regression function was shown to estimate the trend of accuracy values. C{V}_{({n}_{Dna},{\%}_{mix})} where also reported using 3-d contour plots using third-order polynomials spline fitting. All elaborations and graphics were obtained using Excel (Microsoft), Statistica (Statsoft) and Sigmaplot 10 (Systat Software Inc.).