efficiency

In a perfect PCR reaction, the amount of product should double with each additional reaction cycle. In reality, primers often deviate considerably from perfect doubling. This deviation causes problems for qPCR analysis, as an assumption of comparing Ct values between primer sets is that the amplification curves have approximately the same shape during their logarithmic phases. As such, verifying that primers amplify approximately exponentially is an important part of qPCR experimentation. efficiency() allows a user to easily convert Ct values associated with a standard curve to efficiency values for tested primers.

The following section provides a detailed explanation of efficiency testing. To jump directly to the usage of efficiency(), jump to “Using namer to get a dataframe for efficiency”.

What is efficiency?

Efficiency is the most common metric used to determine whether primers are properly amplifying exponentially. It is calculated from a standard curve. For each two-fold dilution in the standard curve, the Ct value should increase by 1. In other words, when Ct values are plotted against \(log_2(dilution)\), the absolute value of the slope should be 1. Efficiency is calculated by converting the slope to the fraction or percentage of product amplified.

If E > 1, the amount of PCR product is more than doubling each PCR cycle. This is often due to primer-dimers or other issues with multiple amplicons. If E < 1, the PCR product is doubling more slowly than once per PCR cycle. This could be due to primer secondary structure, non-optimal melting temperature, primer saturation, or difficult templates. In general, E-values between 0.9 and 1.1 are acceptable.

To test the primers over a large enough concentration range, each dilution should be at least a factor of two, and three to five dilutions should be used. For certain targets, the reverse transcription reaction conditions and dilution factors may need to be adjusted to keep the highest Ct values under 30. Additionally, optimal primer concentrations may vary, but in general a final concentration between 0.5-1.0µM gives good results.

efficiency() uses linear regression to determine the equation of the line of best fit, then uses the following equation to calculate efficiency:

\[E = 2^{- 1/slope} - 1\]

To explain this, remember that the slope represents the change in \(Ct\) over the change in \(log_2(dilution)\). Note that dilution in this case represents fraction of stock concentration (e.g. 1 volume of template plus 3 volumes diluent yields a dilution of \(\frac{1}{4}\)). We’ll label dilution as \(d\):

\[slope = \frac{\Delta Ct}{\Delta log_2{d}}\]

Note that \(\Delta Ct\) in this equation is not the same as the \(\Delta Ct\) in differential gene expression. Substituting this into the efficiency equation yields:

\begin{align} E &= 2^{\frac{-1}{slope}} - 1 \\ &= 2^{\frac{-1}{\Delta Ct / \Delta log_2{d}}} - 1 \\ &= 2^{\frac{- \Delta log_2{d}}{\Delta Ct}} - 1 \\ &= \left( 2^{\Delta log_2{d}} \right)^{-1/\Delta Ct} - 1\\ &= d^{-1/ \Delta Ct} - 1 \end{align}

Let’s examine what happens if we have a perfect PCR reaction:

\begin{align} E &= 0.5^{-1 / 1} - 1 \\ &= 2 - 1 \\ & = 1 \\ \end{align}

As expected, this yields 100% efficiency or \(E = 1\). We can back-calculate to see how an efficiency value of 1.1 would affect Ct:

\begin{align} 1.1 &= 0.5^{-1 / \Delta Ct} - 1 \\ 2.1 &= 0.5^{-1 / \Delta Ct} \\ log_0.5{2.1} &= \frac{-1}{\Delta Ct} \\ \Delta Ct &= \frac{-1}{log_0.5{2.1}} \\ \Delta Ct &\approx 0.93 \\ \end{align}

We can see that although the template concentration dropped by half, the \(Ct\) value only increased by 0.93. This means that the amount of PCR product is more than doubling each cycle.

The mathematics underlying methods such as \(\Delta \Delta Ct\) differential gene expression analysis assume that the efficiencies of all primers involved are approximately 1. As such, validating primer efficiency is essential for accurately interpreting qPCR results.

Using namer to get a dataframe for efficiency

For this example, we will just look at data from a Lightcycler 480 formatted as namer() anticipates. The namer() documentation has details on how to use the function for other instruments and dilution arrangements.

import equipt

primers = ['Fus (112734868c1)',
         'Fus (15029724a1)',
         'Ewsr1 (6679715a1)',
         'Ewsr1 (88853580c2)',
         'Taf15 (141803447c1)',
         'Taf15 (141803447c2)',
         'Tsix exon4']

samples = ['mESC total cDNA']

reps = 3

config = 'line'

kwargs = {'with_dil':samples,
         'dil_series':[20,40,80,160],
         'dil_rest':None}

df = equipt.namer('data/22.11.22_PrimerCurve_Ct.csv',
            primers,
            samples,
            reps,
            config,
            **kwargs)

df.iloc[:6]

	Pos	Cp	Primer	Name	NamePrim
0	A1	17.51	Fus (112734868c1)	mESC total cDNA_20	mESC total cDNA_20Fus (112734868c1)
1	A2	17.54	Fus (112734868c1)	mESC total cDNA_20	mESC total cDNA_20Fus (112734868c1)
2	A3	17.55	Fus (112734868c1)	mESC total cDNA_20	mESC total cDNA_20Fus (112734868c1)
3	A4	18.49	Fus (112734868c1)	mESC total cDNA_40	mESC total cDNA_40Fus (112734868c1)
4	A5	18.52	Fus (112734868c1)	mESC total cDNA_40	mESC total cDNA_40Fus (112734868c1)
5	A6	18.54	Fus (112734868c1)	mESC total cDNA_40	mESC total cDNA_40Fus (112734868c1)

We can now see that the ‘Name’ column has the sample name and the dilution factor separated by an underscore. If a user plans to write their own script to label dilutions, it is critical that samples are labeled in this way.

Using efficiency

from bokeh.layouts import gridplot
import bokeh.io

bokeh.io.output_notebook()

Loading BokehJS ...

efficiency() takes three parameters and two optional keyword arguments. Their documentation is reproduced here:

Params
______

ct_data : a dataframe
    Output of namer().

with_dil : list of strings
    List of samples that have dilution series. Generally has only one value,
    but in specific cases multiple samples may have curves.

returnmodel : Bool
    Whether to return a dataframe containing the linear regression model.
    Default False

**kwargs : dictionary

    thresh : float
        The acceptable standard deviation of replicate well Ct values.
        averager() is used to identify wells to drop.

    reps : int
        Number of replicate wells loaded per sample.

It returns either two or three outputs depending on the value of returnmodel, as specified in the documentation:

Returns
_______

plot_dict : a dictionary
    A dictionary in which the keys are samples and the value is a list of
    Bokeh plots for each primer tested.

eff_df : a dataframe
    A dataframe containing the efficiency values and R^2 statistic
    calculated for each sample-primer pair.

model_df : a dataframe
    Only returned if returnmodel == True. A dataframe containing the
    intercept and slope of the linear model for each sample-primer pair.

Using the dataframe generated above, we can easily calculate efficiency values:

eff_kwargs = {'thresh':0.1,
             'reps':3}

plot_dict, eff_df = equipt.efficiency(df,
                               samples,
                               returnmodel=False,
                               **eff_kwargs)

eff_df

	Name	Primer	Efficiency	Rsquared
0	mESC total cDNA	Ewsr1 (6679715a1)	0.953	0.934628
1	mESC total cDNA	Ewsr1 (88853580c2)	0.976	0.934442
2	mESC total cDNA	Fus (112734868c1)	0.952	0.921493
3	mESC total cDNA	Fus (15029724a1)	0.997	0.932829
4	mESC total cDNA	Taf15 (141803447c1)	1.023	0.925023
5	mESC total cDNA	Taf15 (141803447c2)	1.000	0.933382
6	mESC total cDNA	Tsix exon4	1.108	0.929611

Immediately we can see most of the primers have acceptable efficiency values, but Tsix exon4 is a bit high.

Standard curve visualization with efficiency

Having calculated the efficiency values, we noticed that two of our primer sets appear to have unacceptable values. Looking at the plots of the standard curve can give insights on what might be going wrong with those primers. efficiency() returns a dictionary that uses the sample names as keys and a list of Bokeh plots as values for easy visualization. Here are all the plots laid out in a gridplot:

eff_gp = gridplot(plot_dict['mESC total cDNA'], ncols=2)

bokeh.io.show(eff_gp)

Just looking at Tsix exon4, we can see that the Ct values are reasonable, the replicate wells cluster closely together, and the threshold filtering applied during the calculation did not exclude any samples:

bokeh.io.show(plot_dict['mESC total cDNA'][-1])

Since we applied a standard deviation threshold, efficiency() called averager() to remove poorly replicating wells. These wells can be found in the droppedWells.txt log file:

with open('droppedWells.txt','r') as f:
    print(f.read())

Outlier Wells Dropped:
mESC total cDNA_160Fus (112734868c1) - 1

Samples Removed:

One of the Fus (112734868c1) wells was removed at the highest dilution level. This likely indicates a pipetting error and is not a serious cause for concern. No samples were removed entirely, so the full standard curve was analyzed for each primer set.

Returning the linear model

Occasionally, we may want to know the exact function that efficiency() fit for a given standard curve. This is possible by setting returnmodel to True.

plot_dict, eff_df, model = equipt.efficiency(df,
                               samples,
                               returnmodel=True,
                               **eff_kwargs)

model

	Name	Primer	Coefficient	Intercept
0	mESC total cDNA	Ewsr1 (6679715a1)	-1.035	14.411
1	mESC total cDNA	Ewsr1 (88853580c2)	-1.018	13.527
2	mESC total cDNA	Fus (112734868c1)	-1.036	13.032
3	mESC total cDNA	Fus (15029724a1)	-1.002	13.355
4	mESC total cDNA	Taf15 (141803447c1)	-0.984	14.827
5	mESC total cDNA	Taf15 (141803447c2)	-1.000	14.561
6	mESC total cDNA	Tsix exon4	-0.930	21.148

The model dataframe contains the sample-primer pair as well as the slope and y-intercept of the linear function.

%load_ext watermark
%watermark -v -p bokeh,equipt,jupyterlab

Python implementation: CPython
Python version       : 3.9.17
IPython version      : 8.12.0

bokeh     : 3.1.1
equipt    : 1.0.0
jupyterlab: 3.6.3