= Gel electrophoresis size calibration
{title2=$d=a-b\log_{10}(L/L_0)$}
A <gel electrophoresis size calibration> relates migration distance to known fragment lengths on the same gel. Here is an explicitly chosen model for linear double-stranded <DNA>, not measured data or a universal law. Let $L$ be the numerical length in <base pairs> and $d$ the numerical migration distance in millimetres from the wells:
$$
d=55-12\log_{10}(L/100).
$$
The <logarithm> has a dimensionless argument: $L/100$ is the length divided by 100 <base pairs>. The chosen <DNA ladder> fragments $L=100,250,500,1000$ give $d=55.00,50.22,46.61,43.00$, respectively. The <gel bands> therefore appear in the reverse order of length along the direction of migration.
\Image[/gel-electrophoresis-ladder-calibration.png]
{title=Constructed <DNA ladder>, unknown, mixture and blank lanes with a <logarithm> calibration. All positions follow the chosen model; the unknown is 375 <base pairs> at 48.11 mm, and darkness does not represent <DNA> quantity}
To invert the calibration, rearrange the <logarithm>:
$$
L=100\,10^{(55-d)/12}.
$$
Our chosen unknown $L=375$ gives $d=48.1116248\ldots$, between the 250-<base pair> and 500-<base pair> references. Inserting that distance into the inverse recovers 375; retaining only $d=48.11$ gives about $375.12$ <base pairs>. The extra digits reflect rounding in this model, not the resolving power of a real gel.
For <linear interpolation> between those two reference <gel bands>, use the <logarithm> of length. Here the fraction of the distance from the 500-<base pair> reference toward the 250-<base pair> reference is $t=\log_{10}(500/375)/\log_{10}2\approx0.4150$, giving $L=500(250/500)^t=375$. Direct <linear interpolation> of lengths instead gives approximately $396.24$ <base pairs>. This difference explains why the choice of calibration coordinates matters. A real calibration must be checked over its usable range; folded <RNA> or circular <DNA> does not automatically share the linear double-stranded <DNA> calibration.
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