Real-Time PCR is a technology that detects Polymerase Chain Reaction (PCR) amplification of a specific gene target automatically each cycle. The real-time data enables the original gene target quantity in the PCR reaction to be deduced mathematically.

PCR can be divided into 3 phases (see figure 1). The first phase may be termed geometric, exponential or logarithmic. In this phase, PCR reagents are in excess, fueling consistent amplification efficiency.

**Figure 1:**The 3 phases of PCR as seen on a log y-axis amplification plot.

The remarkable consistency of geometric amplification maintains the original quantitative relationships of the target gene across samples. The geometric phase is also remarkable in that efficiency does not change with original gene quantity over a wide range. For example, using highly concentrated target, such as plasmid, 20ml PCR reactions can produce a high quality standard curve spanning 9 logs or more.

During PCR, the DNA target may accumulate to a high enough level that one of the PCR reagents will no longer be sufficient to support geometric amplification. At this point, geometric phase transitions to linear phase. Linear phase is the second phase in PCR in which the efficiency declines cycle-to-cycle. Changes in efficiency during linear phase become less and less consistent with increasing cycle number, so the data becomes less and less quantitative.

Eventually, PCR efficiency may become so low that there is no appreciable target amplification. This final phase of PCR is called plateau. Plateau phase data is not considered quantitative, unless special techniques are employed, such as those used for “digital PCR.”

Quantitative data can be acquired from the geometric phase using a variety of methods, such as baseline-threshold or relative-threshold. Consequently, these geometric data points may have slightly different abbreviations, such as Ct, Crt or Cq, but they are all treated the same way in subsequent calculations. For simplicity, the term “Ct” will be used in this document.

The original gene amount or “quantity” in the PCR reaction can be deduced from Ct values due to a mathematical relationship that exists between Ct and quantity (see equation 1).

Two methods exist to transform Ct values into quantities, called the Standard Curve Method and the DDCt Method.

Two additional mathematical steps may be involved in the quantification process. The data may be normalized by sample mass, especially for biological samples. A gene, referred to as a normalizer, endogenous control or house-keeping gene, is often used to measure sample mass. In addition, the data may be calibrated to a sample referred to as a calibrator or reference sample. Calibrators may be relative or absolute.

**Equation 1**

Quantity ~ e^{–Ct}

Where:

Quantity = original gene amount in the PCR reaction

e = geometric efficiency (1< e __<__ 2)

Ct = geometric data point (threshold cycle number)

## Geometric efficiency

PCR efficiency can be defined as the ratio of the number of target gene molecules at the end of a PCR cycle divided by the number of target molecules at the start of the same PCR cycle. In the geometric phase, the efficiency is constant cycle-to-cycle.

Efficiency can be represented as a ratio or a percentage. At maximum, a target sequence can double each cycle, because DNA only has two strands. That efficiency can be represented as 2 or 100%.

In equation 1, efficiency (e) is a base in an exponential function, which means a change in the value of e can have a significant impact on the resulting quantity. For example, for a Ct of 20, the quantities resulting from 100% vs. 80% efficiency differ by 8.2-fold. Therefore, assigning accurate efficiency values is important for producing accurate quantity results.

In the 1990s, Perkin Elmer Corporation scientists developed a universal system for real-time PCR that consistently produced a geometric efficiency of 100%. The system integrates universal cycling conditions, chemistry and assay design. The proof of the system was demonstrated by a study of the efficiencies of 750 randomly selected TaqMan Gene Expression Assays (Amplification Efficiency of TaqMan Gene Expression Assays application note).

Real-time PCR is best performed when all assays have 100% geometric efficiency. While it is possible to successfully run assays with less than 100% efficiency, lowered efficiency creates a list of issues that can add complexity and may cause problems.

## How to design assays with 100% efficiency

Thermo Fisher Scientific offers a number of options for real-time PCR assay design that conform to the universal system. TaqMan Assays are off-the-shelf assays that have already been designed and tested. Custom assays may be designed for gene targets not covered by TaqMan Assays. Primer Express is desktop software that can be used to design custom assays and the Custom TaqMan Assay Design Tool is a web-based tool found on the Thermo Fisher Scientific website.

## Assessing geometric efficiency

To help maximize accuracy, the safest recommendation is to perform a geometric efficiency assessment on all involved assays. Note that the geometric efficiency for TaqMan Assays is guaranteed to be 100% and it is extremely likely that Custom TaqMan Assays designed with either Primer Express or the Custom TaqMan Assay Design Tool will have 100% efficiency. However, assays not designed in accordance with the universal system are at risk for lower efficiency.

One method to assess geometric efficiency is to calculate it from the slopes of standard curves. A theoretical relationship exists between a standard curve slope and efficiency.

In equation 2, theoretically a slope of -3.32 corresponds to an efficiency of 100%. Slopes steeper than -3.32 (e.g., -3.5) imply lower efficiency. Slopes shallower than -3.32 (e.g., -3.2) imply greater than 100% efficiency, even though geometric efficiency cannot actually be greater than 100%.

**Equation 2**

e = 10^{–1/slope}

Where:

e = theoretical efficiency

Slope = the slope of the stanard curve, plotted with

the y axis as Ct and the x axis as log(quantity)

## Disadvantages of assessing efficiency using standard curves

Using standard curves to assess efficiency entails additional cost and labor.

The ideal structure for a standard curve intended to assess efficiency is 7-points with a 10-fold series (see Amplification Efficiency of TaqMan Gene Expression Assays application note). Such a dilution series would require highly concentrated target, which may not be available naturally in samples.

Errors in standard curve slopes are very common. Many factors can cause such errors, including inhibitors, contamination, pipet precision error, pipet calibration error and dilution point mixing problems. These errors explain how slopes that theoretically correspond to greater than 100% efficiency could occur, even though geometric efficiency cannot exceed 100%. Therefore, any efficiency assessments based on equation 2 should be taken with a great deal of caution.

## User Bulletin #2 method for assessing 100% efficiency

One factor that has a predictable, mathematical impact on standard curve slopes is pipet calibration error. A method to correct for potential pipet calibration error is to subtract the slopes of two standard curves that were generated from the same dilution series. This method is shown in a Perkin Elmer Corporation document called User Bulletin #2. Theoretically, if two assays have the same geometric efficiency, the difference in the two standard curve slopes should be zero. If the efficiencies are equal, they are most likely 100%, because an assay that is not 100% efficient will have a random efficiency, making it highly unlikely the two assays will have equal efficiencies.

However, subtracting slopes does not correct for the other types of errors, so the User Bulletin #2 method is still error prone.

## Visual assessment method

The basis of real-time quantitative PCR is constant geometric efficiency, which means that in an amplification plot with a log y-axis scale, all the geometric slopes should be parallel within a given assay. If multiple assays have 100% geometric efficiency, the efficiencies are equal, which means geometric slopes should be parallel inter-assay as well. Such parallelism is in fact observed (see figure 2).

**Figure 2: **Four different TaqMan Gene Expression Assays.

Geometric amplification slopes that are not parallel indicate less than 100% efficiency (see figure 3).

Comparisons of this kind are most easily done when the assays are all on the same plate.

A related method is comparing geometric slopes of assays to those of an assay known to have 100% geometric efficiency, such as the RNase P assay, which is used in the instrument performance verification test.

The visual method offers the advantages that standard curves are not needed to make the assessment, the assessment does not involve any equations and the method is not impacted by contamination, pipet precision error, pipet calibration error and dilution point mixing problems. However, this method does not produce a mathematically determined number.

**Figure 3:**Two SYBR Green assays that did not conform to universal assay design guidelines.

## Standard curve quantification and efficiency

The original quantification method for real-time PCR involved standard curves. A standard curve was run for each assay on the plate, best fit line equations were calculated for each standard curve and Ct values were transformed into quantities based on those line equations.

In the standard curve quantification method, the slope designates the geometric efficiency and the data is calibrated from the y-intercept. In this method, Ct values are transformed into quantities as a first step. Additional steps, such as normalizing to a normalizer gene, are done by dividing quantities.

**Equation 3**

Standard curve line equation

y = mx + b

Where:

y = Ct value

m = slope

x = log(quantity)

b = y-intercept

ΔΔCt method

The ΔΔCt method is a method to quantify using real-time PCR data that does not require standard curves. The ΔΔCt method has multiple variants. In the traditional version, transformation of Ct values into quantities occurs as the final step. Steps such as normalizing to a normalizer gene or a calibrator sample are done earlier by subtracting Ct values, producing ΔCt or ΔΔCt values (see equation 4).

The ΔΔCt method offers reduced cost, lower labor, higher throughput and greater accuracy compared to the standard curve method. Note that standard curves are valuable for purposes of dynamic range testing, which would be done prior to quantification using the ΔΔCt method.

**Equation 4**

ΔΔCt equation

Quantity = 2^{–}ΔΔCt

Where:

2 = 100% geometric efficiency

ΔΔCt = Ct values normalized to the normalizer gene and calibrator sample

In the original version of the ΔΔCt method, geometric efficiency is set at 2, which means both the target and normalizer assays are 100% efficient. However, DDCt is not limited to assays with 100% efficiency. A version of ΔΔCt method easily allows for differing efficiencies (see equation 5).

Another way to perform the ΔΔCt method with differing efficiencies is to use “adjusted” Ct values, which are what the Ct values would have been had the efficiency been 100%.

A number of Thermo Fisher Scientific software programs offer the ΔΔCt method with adjustable efficiencies. However, best practice is to only use assays with 100% efficiency. If an assay has lower efficiency, a new assay should be selected or designed.

**Equation 5**

Modified ΔΔCt Equations

Uncalibrated Quantity = (e_target ^{–Ct_target} )/(e_norm ^{–Ct_norm})

Calibrated Quantity = Uncalibrated Quantity/Calibrator Quantity

Where:

e = geometric efficiency of either the target or normalizer assay

Ct = geometric data point

Norm = normalizer assay

Calibrator = sample used to calibrate the data set

## Can inhibitors change geometric efficiency?

The answer is no. PCR related inhibition falls into 3 categories: reverse transcription inhibition, solubility inhibition and *Taq* DNA polymerase inhibition. For those chemistries involving a traditional retrovirus-derived reverse transcriptase, inhibiting that enzyme increases any resulting Ct value, but has no impact on *Taq* DNA polymerase activity. Solubility inhibition reduces the amount of DNA target available for amplification, resulting in a Ct increase, but the geometric phase is unaffected. *Taq* inhibition reduces the concentration of active enzyme the reaction, which may cause the geometric phase to prematurely transition to linear phase. In worst case scenario, *Taq* inhibition may make detection of the geometric phase very difficult, but geometric phase efficiency remains the same.

For Research Use Only. Not for use in diagnostic procedures.