  Real time, or quantitative, PCR typically starts from a very low
concentration of initial DNA strands. During iterations the numbers increase,
first essentially by doubling, later predominantly in a linear way. Observation
of the number of DNA molecules in the experiment becomes possible only when it
is substantially larger than initial numbers, and then possibly affected by the
randomness in individual replication. Can the initial copy number still be
determined?
  This is a classical problem and, indeed, a concrete special case of the
general problem of determining the number of ancestors, mutants or invaders, of
a population observed only later. We approach it through a generalised version
of the branching process model introduced by Jagers and Klebaner, 2003 and
based on Michaelis-Menten type enzyme kinetical considerations from Schnell and
Mendoza, 1997. A crucial role is played by the Michaelis-Menten constant being
large, as compared to initial copy numbers. In a strange way, determination of
the initial number turns out to be completely possible if the initial rate $v$
is one, i.e all DNA strands replicate, but only partly so when $v<1$, and thus
the initial rate or probability of successful replication is lower than one.
Then, the starting molecule number becomes hidden behind a "veil of
uncertainty". This is a special case, of a hitherto unobserved general
phenomenon in population growth processes, which will be addressed elsewhere.

Probability