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 that this amplification efficiency is near perfect. Meaning, 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 assumptions 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). This final assumption requires also that these references 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 and Schmittgen 2001). Briefly,
The ΔΔC_T is given by:

ΔΔC_T = ΔC_T, q − ΔC_T, cb

And the relative expression by:

2^−ΔΔC_T

Where:

• ΔC_T, q is the difference in the C_T (or their ) 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 ) of a gene of interest and a reference gene in a reference group

And the error term is given by:

$$ s = \sqrt{s_1^2 + s_2^2} $$

Where:

• s₁ is the of a gene of interest
  
• s₂ is the of a reference gene

Standard curve methods

In comparison, this model doesn’t assume perfect amplification but rather actively use the amplification in calculating the relative expression. So 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 mRNA 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:

$$ \log amount = \dfrac{C_T - b}{m} $$

And the relative expression is given by:

10^log amount

where:

• C_T is the cycle threshold of a gene

• b is the of C_T ~ log10 input amount

• m is the of C_T ~ log10 input amount

And the error term is given by:

s = (cv)(X̄)

Where:

$$ cv = \sqrt{cv_1^2 + cv_2^2} $$

Where:

• s is the

• X̄ is the

• cv is the or relative standard deviation

Two-group 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 when sample size is small and those two last assumptions are hard to achieve.

Linear regression

Two use the linear regression here. A null hypothesis is formulated as following,

C_{T, target, treatment} − C_{T, control, treatment} = C_{T, target, control} − C_{T, control, control}  or  ΔΔC_T

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

Amplification efficiency pcr_efficiency

To calculate the amplification efficiency in a qPCR experiment, the main input is a data.frame with columns contain the C_T values for each gene and rows correspond to the different input amounts/dilutions (Table 5).

The following code apply the calculation on the data.frame, ct3. It has two columns c_myc and GAPDH and 3 rows for each of the input amounts coded in the variable amounts. A reference gene passed to the reference_gene argument, in this case, the column name GAPDH.

library(pcr)
library(ggplot2)
library(cowplot)

## 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
res <- pcr_assess(ct3,
                  amount = amount,
                  reference_gene = 'GAPDH',
                  method = 'efficiency')
knitr::kable(res, caption = 'Table 7: amplification efficiency of c-myc')

The output of pcr_assess is a data.frame of 4 columns and n rows equals the input genes except for the reference. For each gene an intercept, slope and R² is calculated for a the difference between it and the reference (ΔC_T) (Table 7).

When the argument plot is TRUE a graph is produced instead shows the average and standard deviation of of the ΔC_T at difference input amounts. In addition, a linear trend line is drawn (Fig 1).

gg <- pcr_assess(ct3,
           amount = amount,
           reference_gene = 'GAPDH',
           method = 'efficiency',
           plot = TRUE)
gg + 
  labs(x = 'log10 amount', y = 'Delta Ct') +
  theme(strip.background = element_blank(),
        strip.text = element_blank())

## Warning: Use of `df$x` is discouraged.
## ℹ Use `x` instead.

## Warning: Use of `df$y` is discouraged.
## ℹ Use `y` instead.

## Warning: Use of `df$x` is discouraged.
## ℹ Use `x` instead.

## Warning: Use of `df$y` is discouraged.
## ℹ Use `y` instead.

Figure 1: Amplification efficiency of c-myc

The relevant summaries in calculating efficiency is the slope. Typically, slope should be very low (less than 0.1). Means the ΔC_T are not changing much as a consequence of changing the input concentration. A value of the amplification efficiency itself is given by 10^−1/slope and should be close to 2.

Standard curve pcr_standard

To calculate the standard curve for individual genes, pcr_assess takes a data.frame similar to that described above as input ct3, and the same amount variable. The following code calculates the curves for the two columns/genes by fitting a line between their C_T values and the log input amounts.

## calculate standard curve
res <- pcr_assess(ct3,
                  amount = amount,
                  method = 'standard_curve')
knitr::kable(res, caption = 'Table 8: Standard curves of c-myc and GAPDH')

The output put is similar to the previous call, except when ‘standard_curve’ is passed to method curves are calculated for individual genes, column gene (Table 8).

The information of the standard curves are required when using the standard curve methods, so we retain the relevant ones in the variables intercept and slope. Typically, the r_squared should be close to 1.

intercept <- res$intercept
slope <- res$slope

When the argument plot is TRUE a graph is returned instead. A panel for each gene showing the raw C_T values and the log input amounts (Fig. 2).

gg <- pcr_assess(ct3,
           amount = amount,
           method = 'standard_curve',
           plot = TRUE)
gg + 
  labs(x = 'Log 10 amount', y = 'CT value')

## Warning: Use of `df$x` is discouraged.
## ℹ Use `x` instead.

## Warning: Use of `df$y` is discouraged.
## ℹ Use `y` instead.

Figure 2: Standard curve of c-myc and GAPDH

Doube delta C_T (ΔΔC_T) pcr_ddct

To apply the double delta C_T model, the default method of pcr_analyze, the main input is a data.frame with columns containing the genes and the rows the C_T values from different samples. In addition, a group variable group_var corresponding to these rows/samples is required. Finally, a reference_gene and a reference_group are entered.

The following code chunk applies this method to a data.frame of 12 samples from 2 groups ct1 (Table 1).

# default mode delta_delta_ct
## 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')

knitr::kable(res1, caption = 'Table 9: Double delta $C_T$ method (separate tubes)')

The output of pcr_analyze is 8 columns; contains the calculations of each gene in each group and the error terms (Table 9). This analysis uses the default mode, ‘separate_tube’ as the input dataset came for an experiment where the target c_myc and the control gene GAPDH were ran in separate tubes.

In contrast, the ct2 dataset, also shown in Table 3, came from an identical experiment except the samples were run in the same tube. So the following analysis invokes a different mode, ‘same_tube’.

# calculate all values and errors in one step
## mode == 'same_tube'
res2 <- pcr_analyze(ct2,
  group_var = group_var,
  reference_gene = 'GAPDH',
  reference_group = 'brain',
  mode = 'same_tube')

knitr::kable(res2, caption = 'Table 10: Double delta $C_T$ method (same tube)')

The only difference here is that the average of the C_T values for the target gene c_myc is calculated after normalizing by the reference gene GAPDH. The rest of the calculations are expected to be slightly different than the previous case (Table 10).

Figure 3 shows the results of these two analysis. Bars represent the average relative expression of c-myc in the kidney, normalized by GAPDH and calibrated by the brain. The error bars are the standard deviations.

gg1 <- pcr_analyze(ct1,
  group_var = group_var,
  reference_gene = 'GAPDH',
  reference_group = 'brain',
  plot = TRUE) +
  labs(x = '', y = 'Relative mRNA expression') +
  ggtitle(label = 'Separate tubes')

gg2 <- pcr_analyze(ct2,
  group_var = group_var,
  reference_gene = 'GAPDH',
  reference_group = 'brain',
  mode = 'same_tube',
  plot = TRUE) +
  labs(x = '', y = 'Relative mRNA expression') +
  ggtitle(label = 'Same tubes')

plot_grid(gg1, gg2)

Figure 3: Relative expression of c-myc using double delta C_T

Delta C_T (ΔC_T) pcr_dct

This method is a variation of the double delta C_T model. It can be used to calculate the fold change of in one sample relative to the others. For example, it can be used to compare and choosing a control/reference genes.

## example to check fold change of control gens
## locate and read file
fl <- system.file('extdata', 'ct1.csv', package = 'pcr')
ct1 <- read.csv(fl)

## make a data.frame of two identical columns
pcr_hk <- data.frame(
  GAPDH1 = ct1$GAPDH,
  GAPDH2 = ct1$GAPDH
  )

## add grouping variable
group_var <- rep(c('brain', 'kidney'), each = 6)

Here, we used the column GAPDH from the dataset ct1 to make a data.frame, pcr_hk of two identical columns GAPDH1 and GAPDH2 two show how such comparison can be done.

The input to pcr_analyze is identical to that of the previous call, only method is specified this time to ‘delta_ct’.

# delta_ct method
## calculate caliberation
res <- pcr_analyze(pcr_hk,
            group_var = group_var,
            reference_group = 'brain',
            method = 'delta_ct')

knitr::kable(res, caption = 'Table 11: Delta $C_T$ method')

Similarly, the output contains the calculated model and the error terms (Table 11). The difference here will be skipping the normalization step and calibrating the C_T values of all genes to a reference_group.

Figure 4 shows the average relative fold change of the identical housekeeping genes and there error terms in two tissue samples.

pcr_analyze(pcr_hk,
            group_var = group_var,
            reference_group = 'brain',
            method = 'delta_ct', 
            plot = TRUE) +
  theme(legend.position = 'top',
        legend.direction = 'horizontal') +
  labs(x = '', y = 'Relative fold change')

Figure 4: GAPDH relative fold change using delta C_T

Standard curve pcr_curve

The calculation of the standard curve method involves to steps as explained earlier. First, a standard curve is calculated for each gene to find the intercept and the slope. Then the relative expression is calculated.

To apply this method to the ct1 dataset (Table 2). We used the variables slope and intercept that were calculated earlier using the ct3 dataset (Table 8).

## calculate standard amounts and error
res1 <- pcr_analyze(ct1,
                   group_var = group_var,
                   reference_gene = 'GAPDH',
                   reference_group = 'brain',
                   intercept = intercept,
                   slope = slope,
                   method = 'relative_curve')

knitr::kable(res1, caption = 'Table 12: Standard curve method (separate tubes)')

The output of pcr_analyze is the same as explained before. The calculated averages and error term of target genes in each group relative to the reference gene and group (Table 12).

The argument mode can be can be used to change the way the C_T values are averaged when samples from different genes were ran in the same tube (Table 4).

## calculate standard amounts and error
res2 <- pcr_analyze(ct2,
                   group_var = group_var,
                   reference_gene = 'GAPDH',
                   reference_group = 'brain',
                   intercept = intercept,
                   slope = slope,
                   method = 'relative_curve',
                   mode = 'same_tube')

knitr::kable(res2, caption = 'Table 13: Standard curve method (same tube)')

The output is similar to that described earlier (Table 13).

Figure 5 shows the output of the standard curve method. Relative expression values of c-myc in the kidney normalized by GAPDH and calibrated to the brain are shown as bars, Averages ± standard deviations.

gg1 <- pcr_analyze(ct1,
                   group_var = group_var,
                   reference_gene = 'GAPDH',
                   reference_group = 'brain',
                   intercept = intercept,
                   slope = slope,
                   method = 'relative_curve',
                   plot = TRUE) +
  labs(x = '', y = 'Relative mRNA expression') +
  ggtitle(label = 'Separate tubes')

gg2 <- pcr_analyze(ct2,
                   group_var = group_var,
                   reference_gene = 'GAPDH',
                   reference_group = 'brain',
                   intercept = intercept,
                   slope = slope,
                   method = 'relative_curve',
                   mode = 'same_tube',
                   plot = TRUE) +
  labs(x = '', y = 'Relative mRNA expression') +
  ggtitle(label = 'Same tubes')

plot_grid(gg1, gg2)

Figure 5: Relative expression of c-myc using the standard curve