  A common approach to quantifying DNA involves repeated cycles of DNA
amplification. This approach, employed by the polymerase chain reaction (PCR),
produces outputs that are corrupted by amplification noise, making it
challenging to accurately back-calculate the amount of input DNA. Standard
mathematical solutions to this back-calculation problem do not take adequate
account of such noise and are error-prone. Here, we develop a parsimonious
mathematical model of the stochastic mapping of input DNA onto experimental
outputs that accounts, in a natural way, for amplification noise. We use the
model to derive the probability density of the quantification cycle, a
frequently reported experimental output, which can be fit to data to estimate
input DNA. Strikingly, the model predicts that a sample with only one input DNA
molecule has a $<$4% chance of testing positive, which is $>$25-fold lower than
assumed by a standard method of interpreting PCR data. We provide formulae for
calculating both the limit of detection and the limit of quantification, two
important operating characteristics of DNA quantification methods that are
frequently assessed by using ad-hoc mathematical techniques. Our results
provide a mathematical foundation for the rigorous analysis of DNA
quantification.
