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16. Calculating Up/Down Regulations using qPCR /RTPCRWerner Van Belle^{1}  werner@yellowcouch.org, werner.van.belle@gmail.com Abstract : In this article explain the finer points of the deltadeltaCT method to calculate up/down regulations. The following text is a writeup of a course I gave at a local highschool.
Reference:
Werner Van Belle; Calculating Up/Down Regulations using qPCR /RTPCR; Bioinformatics; YellowCouch; August 2010 
To explain the analysis process of a quantitative PCR (qPCR) we will first look into a mathematical simulation of a PCR. How would a concentration curve look like if we assumed a specific copying process. Later on, we focus on a standard up/down regulation analysis that includes the use of a houshold gene and dilution series.
To simulate a quantitative PCR, we start off with a specific volume of DNA material: amount. With each cycle we will 'more or less' double the amount of DNA. This is the copied_dna. After a number of cycles this exponential process explodes. (click on the images to see the full sized version)
Simulation without any restriction. The Xaxis contains the cycle count, the Yaxis plots the amount of DNA 
Of course the well volume is finite, which means that, as soon as there are no longer sufficent reagentia left, we should limit the DNA that is copied. Below we restrict the the amount of newly produced DNA by the maximum of the available DNA and the available reagents.
Simulation with hard volume limit. Xaxis: cyclecount; Yaxis: amount of DNA 
The above plot has a fairly hard limit because we assumed that anything that was not DNA was reagentia used to create new DNA. That is of course not true. In general, the usable reagents are only part of the remaining volume, which leads to the very nicely simulated expected output from a quantitative PCR, as pictured below.
Simulation with volume and reagentia limit. Xaxis: cyclecount; Yaxis: amount of DNA 
The trick in quantitative PCR is that the initial concentration affects the time it takes to reach a specific target volume. This we can reverse, and thus, based on the time it takes to reach a specific amount of DNA, we can calculate the initial concentration.
A multiplication of to the initial concentration will shift the qPCR curve to the left (we need less cycles to reach the same absolute amount), or right (we need more cycles to reach the same amount).
In practice,
The reason we observe a curve shift (as opposed to a curve scaling) when the initial concentration is multiplied, lies in the exponential nature of the PCR process. Mathematically, we can express the exponential growth as an 'amount of DNA at cycle c' (annotated a_{c}). This amount depends on the amount of the previous cycle a_{c1}, which in turn depends on the amount of its previous cycle a_{c2}, etc. This dependency chain goes whole the way back to cycle 0, where a_0 equals the initial concentration. This can be written as
This chain of multiplications results in an amount of DNA at cycle c which is 2^cycle multiplied with the initial amount.
If we now assume an initial concentration of a0 and a target amount (T), how many cycles (c) will it take to reach that amount ?
To understand the effect a concentration increase of a factor 10 has, we can simply write 10a_0 and see what it does to the cycle count.
Using a quantitative PCR we can measure the number of cycles it takes to reach a certain threshold value. This cycle count is often called a CT value, or sometimes also a CP value. The finer nomenclature doesn't matter here. What is important is that the cycle count can be calculated back to the initial amount of DNA, and thus to the concentration that was present in the well.
There are of course two problems with this approach
In this discussion it matters not how to obtain the CT value, and from my point of view too much attention is paid to this particular aspect of qPCR.
Just out of cheer curiosity I tested how a small error in the CT value would affect the calculated concentration.
Well, staggeringly, a 0.1 change in cycle count (CT value) affects the initial concentration with 7% !
Another factor quite relevant to the qPCR debacle is the efficiency of the copy process itself. There are two somewhat confusing terms around:
In general, copyrate efficiencies of 100% are seldom achieved. Efficiencies of 90% are somewhat more realistic; and of course, we would like to understand what effect this has on a) the CT values and b) the calculated concentrations.
Based on a simulation, in which we assumed a percycle copyefficiency between 95% and 100%, we found the following outcome:
With an initial concentration/amount of 0.001These errors are staggering and it should be clear by now that the mantra 'qPCR is so accurate due to its amplification process', is sheer nonsense. The amplification process will also copy errors for one, and secondly, if the amplification process has not a sufficiently stable and/or known efficiency, then any subsequent calculation will be seriously misguided (not to say wrong).
To deal with the problem of such unknown efficiencies we can fall back to our observation that the CT distance between two curves of different concentrations is around 3.3, if one concentration was 10 times smaller (or larger) than the other.
The expected distance (x) between two curves, given a concentration ratio of 10, and a copyrate of (r) would be
This leads to the following observed copy rates, given a certain curvedistance
Depending on the used dilution series, the above equation might need a different value for 10.
Based on the measurement within one curve, it is possible to observe how much material was copied from cycle to cycle and use that to estimate the efficiency of the process for that particular curve. Based on that again, we can directly calculate the initial concentration. That is however not how it is often done because such change calculations at the lower cycles is very error prone, due to a limited machine accuracy. Normally, people tend to continue working with the CTvalues.
If we want to use the deltadeltaCT method later on we must be able to normalize the CT values as if the copy process was 100% accurate. If we for instance measured a CT value of 10, at an efficiency of 95% then, expressing this at an efficiency of 100% would result in a CT value of around 9.63 cycles.
To get a grasp on this mathematically, it suffices to assume that we reach a specific volume V after x cycles, given an initial amount of i and a copyrate of r.
To reach this volume if the copyrate were 2, we would need y cycles:
Equating the volumes gives
which in turn leads to
Based on the last equation, we can calculate the CT under perfect conditions, based on a nonoptimal copyrate.
When measuring concentrations in real life, the measured expression is often difficult to qualify because the initial amount of material depends on a large number of biological factors that might lower or increase the obtained concentration. E.g: lysing residuals, siRNA efficiency, cell metabolism etc. To deal with such effects, one tends to rely on a household gene, which is a gene of which we known that it is fairly constantly expressed.
Instead of 1 measurement, two measurements are conducted. One is the amplification curve of the household gene, and one is the amplification curve of the gene under investigation.
In the above equation. CP_A is the CT value of the gene of interest. CP_H, is the CT value of the household gene. Dividing the two obtained concentrations gives
which is the first 'delta' in the deltadeltaCT method
The concentrations produced by this normalization will be called relative concentration
Often, one is interested in the up or downregulation from one cell type to the next. E.g: between a constitutive active cell line and an unmodified cell line. In that case, one must compare the relative concentrations between for instance the transgene and the wildtype cell type. This can be directly written down and expanded as follows
This means that to measure the up or down regulation between two cell types, thereby relying on a household gene, leads to 4 qPCR measurements (at the very least). If we are somewhat more scientific we need 2 measurements for the household genes, 2 measurements for the gene under investigation, 4 measurements for the different dilutions (1:1, 1:2, 1:5, 1:10 for instance), and three replicas, which altogether leads to 48 measurements.
A common layout to deal with a number of genes and housholdgenes, while using triplicates and a dilution series is the followong
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