  Detection of extremely rare variant alleles, such as tumour DNA, within a
complex mixture of DNA molecules is experimentally challenging due to
sequencing errors. Barcoding of target DNA molecules in library construction
for next-generation sequencing provides a way to identify and bioinformatically
remove polymerase induced errors. During the barcoding procedure involving $t$
consecutive PCR cycles, the DNA molecules become barcoded by unique molecular
identifiers (UMI). Different library construction protocols utilise different
values of $t$. The effect of a larger $t$ and imperfect PCR amplifications is
poorly described.
  This paper proposes a branching process with growing immigration as a model
describing the random outcome of $t$ cycles of PCR barcoding. Our model
discriminates between five different amplification rates $r_1$, $r_2$, $r_3$,
$r_4$, $r$ for different types of molecules associated with the PCR barcoding
procedure. We study this model by focussing on $C_t$, the number of clusters of
molecules sharing the same UMI, as well as $C_t(m)$, the number of UMI clusters
of size $m$. Our main finding is a remarkable asymptotic pattern valid for
moderately large $t$. It turns out that $E(C_t(m))/E(C_t)\approx 2^{-m}$ for
$m=1,2,\ldots$, regardless of the underlying parameters $(r_1,r_2,r_3,r_4,r)$.
The knowledge of the quantities $C_t$ and $C_t(m)$ as functions of the
experimental parameters $t$ and $(r_1,r_2,r_3,r_4,r)$ will help the users to
draw more adequate conclusions from the outcomes of different sequencing
protocols.
