It is a short sequence, 16-24bp

It binds spontaneously to the

*target*It does not bind easily to other things

It works well with the other primer

It works well with the polymerase

It has to be

*thermodynamically stable*It has to be

*taxonomically specific*

These two conditions imply that the sequence must be *short, but
not too short*

should not form

*hairpins*should not form

*homodimers*should not form

*heterodimers*should be stable in the 3’ end

- so the polymerase can extend

should match the target organism only once

- even if there are some mismatches

should not match other organisms

- even if there are some mismatches

When we use a primer to detect a target, 4 things can happen.

The primer either

binds to the target (True Positive)

binds to something else (False Positive)

does not bind to the target, even if the target is there (False Negative)

does not bind to anything, and the target was not there (True Negative)

When we test a possible primer, the four outcomes may happen

We measure the *number* of each case, using a database of all
possible sequences

\(TP\): True Positive number

\(TN\): True Negative number

\(FP\): False Positive number

\(FN\): False negative number

\[ \begin{aligned} \text{Sensitivity}&=\frac{TP}{TP+FN}=\frac{\text{Detected}}{\text{All targets}}\\ \text{Specificity}&=\frac{TN}{TN+FP}=\frac{\text{Not targets}}{\text{Not detected}} \end{aligned} \]

“Say the truth, all the truth, nothing but the truth”

The Polymerase Chain Reaction (PCR) is a method used to synthesize millions of copies of a given DNA sequence.

A typical PCR reaction consists of series of cycles:

- template DNA denaturation,
- primer annealing, and
- extension of the annealed primers by DNA polymerase.

This loop is repeated between 25 and 30 times

Why do you want to use qPCR?

- Presence of a specific mRNA molecule in the sample
- Concentration of that mRNA in the sample
- Change in concentration of that mRNA between two samples

(you can replace “mRNA” for other keywords)

Questions are more important than answers

Use the following ingredients

- Template: the thing we want to study
- Primers for that template, in high concentration
- dNTP, in high concentration
- Taq polymerase
- Some marker, such as a fluorophore
- Salt & pepper (Na++, K+, Mg++)
- Cooking machine: thermocycler with sensors

The thermocycler is a computer.

You program it to cook your cake

We simplify and we forget about the polymerase and the dNTP

They both will be represented by *primers*

The system is represented by this diagram:

Question:

Does final DNA depends on

initial DNA concentration?

initial primer concentration?

PCR reaction rate?

The curve depends on the initial DNA concentration

We will call \(X(C)\) to the signal measured when DNA concentration is \(C\)

We are mainly interested in the initial concentration \(X(0)\)

We care only about the *exponential* phase

The signal increases 2 times on every cycle

\[X(C) = X(0)⋅2^C\]

So we can find the initial concentration

\[X(0) = X(C)⋅2^{-C}\]

DNA concentration crosses 50% at 13.73 cycles

Start with a large concentration of template, and dilute it several times. Measure the CT of each dilution

If everything is right, we get

\[X(0) = X(CT)⋅2^{-CT}\]

But sometimes we do not know \(X(CT)\)

because the *Signal* is not 100% *DNA
concentration*

It depends on the fluorophore assimilation

We still can measure *change* of concentration

Let’s say we extract mRNA *before* and *after* a
shock

\[ \begin{aligned} X_B(0) & = X_B(CT_B)⋅2^{-CT_B}\\ X_A(0) &= X_A(CT_A)⋅2^{-CT_A}\end{aligned} \]

therefore the fold change of expression is

\[\frac{X_B(0)}{X_A(0)} = \frac{X_B(CT_B)⋅2^{-CT_B}}{X_A(CT_A)⋅2^{-CT_A}}\]

If we assume that the DNA concentrations are the same when the signal crosses the threshold, i.e.

\[X_B(CT_B)=X_A(CT_A)\] then \[\frac{X_B(0)}{X_A(0)} = 2^{-(CT_B-CT_A)} = 2^{-ΔCT}\]

\(Δ CT\) is the change in CT for one gene between two conditions

This change of concentration has two components

- The real biological change
- The variability of the RNA extraction protocol

To avoid the second component, we use an *endogenous
reference*

(typically, a housekeeping gene)

A single pipet, at the same time

We normalize each sample

\[ \begin{aligned} \frac{X_B(0)}{R_B(0)} &= \frac{X_B(CT_{XB})⋅2^{-CT_{XB}}}{R_B(CT_{RB})⋅2^{-CT_{RB}}}= K_B⋅ 2^{-(CT_{XB}-CT_{RB})}\\ \frac{X_A(0)}{R_A(0)} &= \frac{X_A(CT_{XA})⋅2^{-CT_{XA}}}{R_A(CT_{RA})⋅2^{-CT_{RA}}}= K_A⋅ 2^{-(CT_{XA}-CT_{RA})} \end{aligned} \]

\(K_A\) and \(K_B\) are constants that depend on the
target and reference genes, and how each *Signal* changes with
concentration

\[ \frac{X_B(0)}{R_B(0)}÷\frac{X_A(0)}{R_A(0)} = \frac{K_B}{K_A} ⋅ 2^{-(\Delta CT_B-Δ CT_A)} \]

We can assume that \(K_B=K_A,\) because we are comparing the same pair of genes every time

In that case the change in relative expression is

\[\frac{X_B(0)}{R_B(0)}÷\frac{X_A(0)}{R_A(0)} = 2^{-(\Delta CT_B-Δ CT_A)}\]

It is usually a good idea to take logarithms

Using \(\log_2\) we get log fold change, which can be written as

\[ \begin{aligned} Δ CT_A - Δ CT_B &= \overbrace{(CT_{BX}-CT_{BR})}^{\text{before}} - \overbrace{(CT_{AX}-CT_{AR})}^{\text{after}}\\ &=\underbrace{(CT_{BX}-CT_{AX})}_{\text{target}} - \underbrace{(CT_{BR}-CT_{AR})}_{\text{reference}} \end{aligned} \]

This works very well with a *linear model*

The reordering of deltas is equivalent to

\[ \frac{X_B(0)}{R_B(0)}÷\frac{X_A(0)}{R_A(0)} = \frac{X_B(0)}{R_B(0)}⋅ \frac{R_A(0)}{X_A(0)} = \frac{X_B(0)}{X_A(0)}÷\frac{R_B(0)}{R_A(0)} \]

In other words, the ratio of normalized values is also the ratio between ratios of change

(I think the formula is more clear than the text)

In that case the *standard curve slope* changes

If the primer efficiency is \(E_X,\) the correct formula is

\[ \begin{aligned} \frac{X_B(0)}{X_A(0)} &= \frac{X_B(CT_{BX})⋅(1+E_X)^{-CT_{BX}}}{X_A(CT_{AX})⋅(1+E_X)^{-CT_{AX}}}\\ &=(1+E_X)^{-(CT_{BX}-CT_{AX})} \end{aligned} \]

\[ \begin{aligned} \frac{R_B(0)}{R_A(0)} &= \frac{R_B(CT_{BR})⋅(1+E_R)^{-CT_{BR}}}{R_A(CT_{AR})⋅(1+E_R)^{-CT_{AR}}}\\ &=(1+E_R)^{-(CT_{BR}-CT_{AR})} \end{aligned} \]

\[ \frac{X_B(0)}{X_A(0)}÷\frac{R_B(0)}{R_A(0)}= \frac{(1+E_X)^{-(CT_{BX}-CT_{AX})}}{(1+E_R)^{-(CT_{BR}-CT_{AR})}} \]

The log ratio is \[-F_X(CT_{BX}-CT_{AX})+
F_R(CT_{BR}-CT_{AR})\] where \(F_X=-\log_2 (1+E_X)\) and \(F_R=-\log_2 (1+E_R)\) are the slopes of the
corresponding *standard curves*

We know the dilutions, the CT values, and their relationship \[X(0) = X(C)⋅(1+E)^{-C}\] We can connect several initial concentrations with their \(CT\)

\[ \text{initial}_i = \text{threshold}⋅(1+E)^{-CT_i} \]

Notice that threshold concentration should be the same for all \(i\)

Now we have

\[ \log(\text{initial}_i) = \log(\text{threshold}) + \log(1+E)⋅ -CT_i \]

This is an equation of a straight line, like \[y=a+b\,x\]

We can find \(E\) using a linear model like \[\log_2(\text{initial}_i) = β_0 + β_1 ⋅ CT_i + e_i\]

Fitting the linear models gives us \[β_1 = -log(1+E)\] and from there we can find the primers efficiency \(E\)

For the first set of primers, we have

```
2.5 % 97.5 %
(Intercept) 0.288 0.332
CT -0.694 -0.692
```

We get intervals for \(\beta_0\) and \(\beta_1.\) The efficiency is

```
2.5 % 97.5 %
1.000 0.997
```

In this case the real value is 100%

For the second set of primers, we have

The efficiency is in the interval

```
2.5 % 97.5 %
0.833 0.790
```

The real value is 80%

It is recommended to always do the standard curve by triplicate

**All** our measuring devices have a margin of
error.

There may be a small error measuring the signal.

That will affect the resulting CT value

When the initial concentration is small, the signal may not reach 50% in 30 or 40 cycles

Then we will use a lower threshold, and pay the price

Since the curve is nearly flat in the lower signals, a small error in the signal has a large impact in the concentration

Thr | CT | Concentration | Error ratio |
---|---|---|---|

0.05 | 10.00 | 1025.1 | |

0.06 | 10.21 | 1186.4 | 0.1573 |

0.50 | 13.73 | 13624.5 | |

0.51 | 13.78 | 14054.3 | 0.0315 |

An error between signal 0.05 and 0.06 results in an apparent increase of 16% in concentration. An error between signal 0.5 and 0.51 results in an error of 3%

Livak KJ, Schmittgen TD. “Analysis of relative gene expression data
using real-time quantitative PCR and the 2(-Delta Delta C(T)) Method”.
*Methods*. 2001 Dec; 25(4):402-8. doi: 10.1006/meth.2001.1262

Pfaffl, Michael W. “Relative Quantification.” In Real-Time PCR, Published by International University Line (Editor: T. Dorak), 64–82, 2007.