pcr: an R package for quality assessment, analysis and testing of qPCR data

The comparative C_T methods

The comparative C_T methods assume that the cDNA templates of the gene/s of interest as well as the control/reference gene have similar amplification efficiency, and also that this amplification efficiency is near perfect. This means that, at a certain threshold during the linear portion of the PCR reaction, the amount of the gene of the interest and the control double each cycle. Another assumption is that the expression difference between two genes or two samples can be captured by subtracting one (gene or sample of interest) from another (reference). The final assumption is that the reference doesn’t change with the treatment or the course in question. The formal derivation of the double delta C_T model is described here Livak & Schmittgen (2001). Briefly, the ΔΔC_T is given by: (1)${\displaystyle \Delta \Delta C_T=\Delta C_{T,q}-\Delta C_{T,cb}}$

And the relative expression by: (2)${\displaystyle 2^{-\Delta \Delta C_T}}$ where ΔC_T,q is the difference in the C_T (or their average) of a gene of interest and a reference gene in a group of interest. ΔC_T,cb is the the difference in the C_T (or their average) of a gene of interest and a reference gene in a reference group. The error term is given by: (3)${\displaystyle s=\sqrt{s_1^2+s_2^2}}$ where s₁ is the standard deviation of a gene of interest and s₂ is the standard deviation of a reference gene.

Standard curve

In comparison, this model doesn’t assume perfect amplification but rather actively use the amplification in calculating the relative expression. Therefore, when the amplification efficiency of all genes are 100% both methods should give similar results. The standard curve method is applied using two steps. First, serial dilutions of the mRNAs from the samples of interest are used as input to the PCR reaction. The linear trend of the log input amount and the resulting C_T values for each gene are used to calculate an intercept and a slope. Secondly, these intercepts and slopes are used to calculate the amounts of mRNA of the genes of interest and the control/reference in the samples of interest and the control sample/reference. These amounts are finally used to calculate the relative expression in a manner similar to the later method, just using division instead of subtraction. The formal derivation of the model is described here (Yuan et al., 2006). Briefly, the amount of RNA in a sample is given by: (4)${\displaystyle log\mathrm{amount}=\frac{C_T-b}{m}.}$

And the relative expression is given by: (5)${\displaystyle 10^{log\mathrm{amount}}}$ where C_T is the cycle threshold of a gene. b is the intercept of C_T log10 input amount. m is the slope of C_T. And the error term is given by: (6)${\displaystyle s=\left(cv\right)\left(\bar{X}\right)}$ where: (7)${\displaystyle cv=\sqrt{c{v}_1^2+c{v}_2^2}}$ where s is the standard deviation. $\bar{X}$ is the average. cv is the coefficient of variation or relative standard deviation.

Statistical significance tests

Assuming that the assumptions of the first methods are holding true, the simple t-test can be used to test the significance of the difference between two conditions (ΔC_T). t-test assumes, in addition, that the input C_T values are normally distributed and the variance between conditions are comparable. Wilcoxon test can be used if sample size is small, and those two last assumptions are hard to achieve.

Two use the linear regression here. A null hypothesis is formulated as following, (8)${\displaystyle C_{T,\mathrm{target,treatment}}-C_{T,\mathrm{control,treatment}}=C_{T,\mathrm{target,control}}-C_{T,\mathrm{control,control}}.}$

This is exactly the ΔΔC_T value as explained earlier. So the ΔΔC_T is estimated and the null is rejected when ΔΔC_T ≠ 0.

Quality assessment

Fortunately, regardless of the method used in the analysis of qPCR data, The quality assessment can be done in a similar way. It requires an experiment similar to that of calculating the standard curve. Serial dilutions of the genes of interest and controls are used as input to the reaction and different calculations are made. The amplification efficiency is approximated be the linear trend between the difference between the C_T value of a gene of interest and a control/reference (ΔC_T) and the log input amount. This piece of information is required when using the ΔΔC_T model. Typically, the slope of the curve should be very small and the R² value should be very close to one. A value of the amplification efficiency itself is given by 10^−1∕slope and should be close to 2. Other analysis methods are recommended when this is not the case. Similar curves are required for each gene using the C_T value instead of the difference for applying the standard curve method. In this case, a separate slope and intercept for each gene are required for the calculation of the relative expression.

Quality assessment

pcr_assess provides two methods for assessing the quality of qPCR data. These are ‘efficiency’ and ‘standard_curve’ to calculate the amplification efficiency and gene standard curves as described in the methods section. The following code block applies both methods to the dataset ct3, shown in Table 1. Using the argument plot as TRUE in the pcr_assess function provides the a graphic presentation of the amplification and the standard curves as shown in Fig. 1.

 
 
# load  required  libraries 
library(pcr) 
library(ggplot2) 
library(cowplot) 
library(dplyr) 
library(xtable) 
library(readr)    

 
 
#  pcr_assess 
##  locate  and  read  data 
fl <− system.file('extdata', 'ct3.csv', package = 'pcr') 
ct3 <− read_csv(fl) 
 
## make a vector  of RNA  amounts 
amount <− rep(c(1, .5, .2, .1, .05, .02, .01),  each = 3) 
 
##  calculate  amplification  efficiency 
res1 <− pcr_assess(ct3, 
                      amount = amount, 
                      reference_gene  = 'GAPDH', 
                      method = 'efficiency') 
 
##  calculate  standard  curves 
res2 <− pcr_assess(ct3, 
                      amount = amount, 
                      method = 'standard_curve ') 
 
##  retain  curve  information 
intercept <− res2$intercept 
slope <− res2$slope    

Analysis models

Similarly, pcr_analyze provides two methods to model the C_T values and calculates the relative expression of target genes. ’delta_delta_ct’ performs the ΔΔC_T method described previously. The average relative expression of the target gene in the condition of interest is given by the Eqs. (1) and (2) and the standard deviation by Eq. (3). The calculations are applied to the dataset ‘ct1’, shown in Table 2 and Fig. 2A. ‘relative_curve’ performs the relative standard curve quantification, average relative expression/amount of the target gene in the condition of interest is given by Eqs. (4) and (5) and the standard deviation by Eqs. (6) and (7). The calculation is applied to the same dataset ‘ct1’ and is shown in Table 3 and Fig. 2B.

  
 
#  pcr_analyze 
##  locate  and  read  raw ct data 
fl <− system.file('extdata', 'ct1.csv', package = 'pcr') 
ct1 <− read_csv(fl) 
 
## add  grouping  variable 
group_var <− rep(c('brain', 'kidney'), each = 6) 
 
# calculate  all  values  and  errors  in one  step 
## mode == 'separate_tube' default 
res1 <− pcr_analyze(ct1, 
                        group_var  = group_var, 
                        reference_gene  = 'GAPDH', 
                        reference_group  = 'brain') 
 
##  calculate  standard  amounts  and  error 
res2 <− pcr_analyze(ct1, 
                        group_var  = group_var, 
                        reference_gene  = 'GAPDH', 
                        reference_group  = 'brain', 
                        intercept = intercept, 
                        slope = slope, 
                        method = 'relative_curve ')    

Significance testing

Finally, pcr_test can be used to calculate useful statistics, p-values and confidence intervals on the previous models. Two tests are available of the two-group comparisons; ‘t.test’ and ‘wilcox.test’ to test for the difference of the normalized target gene expression (ΔC_T) in one condition to another. Linear regression, ‘lm’, can be applied to estimate these differences between multiple conditions and a reference (Eq. (8)). The following code applies different testing methods to the dataset ‘ct4’. The dataset was published original in (Yuan et al., 2006), along with results of different testing method (Table 4). Table 5 shows the results of the three different tests as implemented in pcr_test.

 
 
#  pcr_test 
# locate  and  read  data 
fl <− system.file('extdata', 'ct4.csv', package = 'pcr') 
ct4 <− read_csv(fl) 
 
# make  group  variable 
group <− rep(c('control', 'treatment'), each = 12) 
 
# test  using t −test 
tst1 <− pcr_test(ct4, 
                     group_var  = group, 
                     reference_gene  = 'ref', 
                     reference_group  = 'control', 
                     test = 't.test') 
 
# test  using  wilcox.test 
tst2 <− pcr_test(ct4, 
                     group_var  = group, 
                     reference_gene  = 'ref', 
                     reference_group  = 'control', 
                     test = 'wilcox.test') 
# testing  using  lm 
tst3 <− pcr_test(ct4, 
                     group_var  = group, 
                     reference_gene  = 'ref', 
                     reference_group  = 'control', 
                     test = 'lm')