can anyone really point out the relationship between fold change, efficiency and threshold cycle number as a method in gene expression determination. thanks
There are three points: i) the exponential amplification in the PCR, ii) the generation of a (fluorescence) signal that is proportional to the amount of amplified molecules, and iii) the measure "ct" that is taken as the cycle number needed for the signal crossing a given threshold.
The exponential amplification of N0 starting molecules with efficiency E is given by
N(c) = N0*E^c
where c is the number of PCR cycles. The signal is proportional to this amount. Using p as proportionality factor the signal F at cycle c is
F(c) = p*N(c) = p*N0*E^c
The ct value is the cycle for which F = Ft (some arbitrary but assay-specific value):
Ft = F(ct) = p*N0*E^ct
This is it. Now you can ask yourself what a given ct-value tells you about N0:
Ft = p*N0*E^ct
logE(Ft) = logE(p*N0*E^ct) = logE(p)+logE(N0)+ct
logE(N0) = logE(Ft)-logE(p)-ct
The term logE(Ft)-logE(p) is an assay-specific constant.Let us denote this by u, so we get
logE(N0) = u-ct
Conclusion: the logarithm of N0 is proportional to -ct ("minus ct").
Now consider two genes A and B with initial concentrations A0 and B0 and measured with the assay-specific constants uA and uB. The ct-values are ctA and ctB, respectively. We are interested in the relative concentration A0/B0 (i.e. the concentration of gene A normalized to the conc. of B). We know:
logE(A0) = uA-ctA
logE(B0) = uB-ctB
If the amplificantion efficiencies for A and B are identical we can write
logE(A0) - logE(B0) = logE(A0/B0) = (uA-ctA) - (uB-ctB) =(ctB-ctA) - (uB-uA)
Conclusion: the log of the relative expression is proportional to the delta-ct value.
If you compare the relative expression between two contitions (trt, ctl), the fold-change (FC) of the normalized concentrations is
FC = ( A0[trt]/B0[trt] ) / ( A0[ctl]/B0[ctl] )
And again,
logEFC = logE( ( A0[trt]/B0[trt] ) / ( A0[ctl]/B0[ctl] ) )
= logE(A0[trt]/B0[trt]) - logE(A0[ctl]/B0[ctl])
= ((ctB-ctA)[trt] - (uB-uA)) - ((ctB-ctA)[ctl] - (uB-uA))
Now the assay-specific constants cancel out it simplifies to
logEFC = (ctB-ctA)[trt] - (ctB-ctA)[ctl]
Conclusion: the difference of the delta-ct values is the logarithm of the fold-change. The base of the logarithm is the amplification efficiency E.
There are three points: i) the exponential amplification in the PCR, ii) the generation of a (fluorescence) signal that is proportional to the amount of amplified molecules, and iii) the measure "ct" that is taken as the cycle number needed for the signal crossing a given threshold.
The exponential amplification of N0 starting molecules with efficiency E is given by
N(c) = N0*E^c
where c is the number of PCR cycles. The signal is proportional to this amount. Using p as proportionality factor the signal F at cycle c is
F(c) = p*N(c) = p*N0*E^c
The ct value is the cycle for which F = Ft (some arbitrary but assay-specific value):
Ft = F(ct) = p*N0*E^ct
This is it. Now you can ask yourself what a given ct-value tells you about N0:
Ft = p*N0*E^ct
logE(Ft) = logE(p*N0*E^ct) = logE(p)+logE(N0)+ct
logE(N0) = logE(Ft)-logE(p)-ct
The term logE(Ft)-logE(p) is an assay-specific constant.Let us denote this by u, so we get
logE(N0) = u-ct
Conclusion: the logarithm of N0 is proportional to -ct ("minus ct").
Now consider two genes A and B with initial concentrations A0 and B0 and measured with the assay-specific constants uA and uB. The ct-values are ctA and ctB, respectively. We are interested in the relative concentration A0/B0 (i.e. the concentration of gene A normalized to the conc. of B). We know:
logE(A0) = uA-ctA
logE(B0) = uB-ctB
If the amplificantion efficiencies for A and B are identical we can write
logE(A0) - logE(B0) = logE(A0/B0) = (uA-ctA) - (uB-ctB) =(ctB-ctA) - (uB-uA)
Conclusion: the log of the relative expression is proportional to the delta-ct value.
If you compare the relative expression between two contitions (trt, ctl), the fold-change (FC) of the normalized concentrations is
FC = ( A0[trt]/B0[trt] ) / ( A0[ctl]/B0[ctl] )
And again,
logEFC = logE( ( A0[trt]/B0[trt] ) / ( A0[ctl]/B0[ctl] ) )
= logE(A0[trt]/B0[trt]) - logE(A0[ctl]/B0[ctl])
= ((ctB-ctA)[trt] - (uB-uA)) - ((ctB-ctA)[ctl] - (uB-uA))
Now the assay-specific constants cancel out it simplifies to
logEFC = (ctB-ctA)[trt] - (ctB-ctA)[ctl]
Conclusion: the difference of the delta-ct values is the logarithm of the fold-change. The base of the logarithm is the amplification efficiency E.
See the attached for several different ways of looking at this...
In addition (in simple terms): slope of the standard curve = -1/log10(EAMP)
and EAMP = 10-1/slope and "Efficiency" (in terms of percent) = (10-1/slope) -1
Fold difference = EAMP(target)(Cq control - Cq treated) / EAMP(ref gene)(Cq control - Cq treated)
(Again - this is all in the most simplistic of terms).
And: logEamp(serial dilution factor) = expected difference between Cq values of the dilution curve. E.g., if the dilution curve (standard curve) demonstrates an EAMP of 1.8
(or 80% Efficiency), the slope is thus -1/log10(EAMP) = -1/log10(1.8) = -3.91738232676218, and, if you are using a serial dilution factor of 4, you would expect an average distance between Cq values on the curve to thus be: log1.8(4) = ~2.3585. These relationships can be practiced ad nauseum in Excel until one gets a feel for them. But, in reality, what Jochen has written above is the most responsible way to look at these relationships.
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