A mathematical model for scanning and catalysis on single-stranded DNA, illustrated with activation-induced deoxycytidine deaminase.

We formulated a master equation-based mathematical model to analyze random scanning and catalysis for enzymes that act on single-stranded DNA (ssDNA) substrates. Catalytic efficiencies and intrinsic scanning distances are deduced from the distribution of positions and gap lengths between a series of catalytic events occurring over time, which are detected as point mutations in a lacZα-based reporter sequence containing enzyme target motifs. Mathematical analysis of the model shows how scanning motions become separable from the catalysis when the proper statistical properties of the mutation pattern are used to interpret the readouts. Two-point correlations between all catalytic events determine intrinsic scanning distances, whereas gap statistics between mutations determine their catalytic efficiencies. Applying this model to activation-induced deoxycytidine deaminase (AID), which catalyzes C→U deaminations processively on ssDNA, we have established that deaminations of AGC hot motifs occur at a low rate, ∼0.03 s(-1), and low efficiency, ∼3%. AID performs random bidirectional movements for an average distance of 6.2 motifs, at a rate of about 15 nucleotides per second, and "dwells" at a motif site for 2.7 s while bound >4 min to the same DNA molecule. These results provide new and important insights on how AID may be optimized for generating mutational diversity in Ig genes, and we discuss how the properties of AID acting freely on a "naked" ssDNA relate to the constrained action of AID during transcription-dependent somatic hypermutation and class-switch recombination.