## Mathematical models

The relative expression of a GOI in relation to another gene, mostly to an adequate reference gene, can be calculated on the basis of 'delta Cp' (ACp, 24) or 'delta delta Ct' (AACt) values (Livak and Schmittgen, 2001). Today various mathematical models are established to calculate the relative expression ratio (R), based on the comparison of the distinct cycle differences. The CP value can be determined by various algorithms, e.g. CP at a constant level of fluorescence or CP acquisition according to the established mathematic algorithm (see Section 3.6).

Three general procedures of calculation of the relative quantification ratio are established:

1. The so-called 'delta Ct' (eqs. 1-2 using ACP) or 'delta-delta Ct' method (eqs. 3-4 using AACP) without efficiency correction. Here an optimal doubling of the target DNA during each performed real-time PCR cycle is assumed (Livak, 1997, 2001; Livak and Schmittgen, 2001). Such expression differences on basis of ACP values are shown in Figure 3.1.

2. The efficiency corrected calculation models, based on ONE sample (eqs. 5-6) (Souaze et al., 1996; LightCycler® Relative Quantification Software, 2001) and the efficiency corrected calculation models, based on MULTIPLE samples (eqs. 7) (Pfaffl, 2004).

(ERef) Cp sample (ERef) Cp calibrat°r

-■P calibrator ratio = —----* —--¡c--(eq. 6)

target target

80.075.070.065.060.055.050.0 45.040.035.030.0-1 25.020.015.0-1 10.0.5.0 0.0 -5.0-

-GAPDH (treatment)

-TNFa (treatment)

-analysis line

Cycle number

### Figure 3.1

Effect of LPS treatment of TNFa target gene expression and on GAPDH reference gene expression in bovine white blood cells. Expression differences are shown by ACP values.

/"c \ ACp tareet (MEAN control - MEAN sample)

(E )ACpRef (MEANcontrol-MEANsample) "t-' '

3. An efficiency corrected calculation models, based on MULTIPLE sample and on MULTIPLE reference genes, so-called REF index, consisting at least of three reference genes (eq. 8) (Pfaffl, 2004).

(E ) ACP target (MEAN control - MEAN sample)

(E ) ACP Ref index (MEAN control - MEAN sample)

(ERef index)

In these models, the target-gene expression is normalized by one or more non-regulated reference gene (REF) expression, e.g., derived from classical and frequently described reference genes (Bustin, 2000; Vandesompele et al., 2002; Pfaffl et al., 2005). The crucial problem in this approach is that the most common reference-gene transcripts from so-called stable expressed housekeeping gene are influenced by the applied treatment. The detected mRNA expressions can be regulated and these levels vary significantly during treatment, between tissues and/or individuals (Pfaffl, 2004; Schmittgen and Zakrajsek, 2000).

Thus always one question appears: which is the right reference to normalize with and which one(s) is (are) the best housekeeping- or reference gene(s) for my mRNA quantification assay? Up to now no general answer can be given. Each researcher has to search and validate each tissue and treatment analyzed for its own stable expressed reference genes. Further, each primer and probe combination, detection chemistry, tubes and the real-time cycler platform interfere with the test performance. However, qRT-PCR is influenced by numerous variables and appears as a multifactorial reaction. Thus, relative quantification must be highly validated to generate useful and biologically relevant information.

The main disadvantage of using reference genes as external standards is the lack of internal control for RT and PCR inhibitors. All quantitative PCR methods assume that the target and the sample amplify with similar efficiency (Wittwer et al., 2001; Livak and Schmittgen, 2001). The risk with external references is that some analyzed samples may contain substances that significantly influence the real-time PCR amplification efficiency of the PCR reaction. As discussed earlier (Pfaffl, 2004), sporadic RT and PCR inhibitors or enhancers can occur.

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