Counting unique molecular identifiers in sequencing using a multi-type branching process with immigration.

Detection of extremely rare variant alleles, such as tumor 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 (UMIs). Different library construction protocols utilize different values of t. The effect of a larger t and imperfect PCR amplifications in relation to UMI cluster sizes 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, r, r, r, r for different types of molecules associated with the PCR barcoding procedure. We study this model by focussing on C, the number of clusters of molecules sharing the same UMI, as well as C(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(m))/E(C)≈2 for m=1,2,…, regardless of the underlying parameters (r,r,r,r,r). The knowledge of the quantities C and C(m) as functions of the experimental parameters t and (r,r,r,r,r) will help the users to draw more adequate conclusions from the outcomes of different sequencing protocols.