Novel type of coagulation equation and its application to DNA repeat expansion process

DNA molecules containing repetitive motifs are prone to expand in their lengths. Once there appear a head to tail tandem of two identical DNA sequences in the system, they can propagate indefinitely by the mechanism involving cycles of staggered annealing of complementary DNA strands of variable lengths and polymerase mediated filling-in of the generated overhangs. Microgene Polymerization Reaction (MPR) is an experimental model for expansion of short repetitive DNA to longer lengths. The testable kinetic model of (MPR) was formulated and solved numerically by Itsko et al. in Kinetics of Repeat Propagation in the Microgene Polymerization Reaction (2009). Here, the simple cases of MPR were solved analytically using modified Smoluchowski coagulation equation. It was found that the repeats propagate according to Gumbel probability density function when the distribution of lengths of obtained polymers follows inverted Gumbel probability density function.