After Cas9 finds a PAM (5′-NGG on the non-target strand), the guide RNA invades the duplex and hybridizes to the target strand one base pair at a time, starting immediately next to the PAM — the "seed" — and zipping outward. This directional R-loop propagation is modeled as a biased 1-D random walk over R-loop extension x ∈ {0…20}: at each step the next base pair forms with rate kf or the R-loop retracts with rate kb. Cleavage (HNH/RuvC activation) requires reaching x = 20; retracting to x = 0 means dissociation.
matched bp: k_f = k0, k_b = kb0
mismatched bp: k_f = k0·e^(−B(i)), k_b = kb0·e^(B(i))
B(i) = Bmax·[0.15 + 0.85·e^(−(i−1)/6)] (i = distance from PAM)
B(i) is the mismatch energy penalty in the zipper model: it is large for i near the PAM (seed, positions 1–8) and decays exponentially toward the PAM-distal end, matching single-molecule and structural studies of Cas9 (Sternberg et al. 2014; Boyle et al. 2017) — a seed mismatch stalls the walk almost immediately, while the same mismatch 18 nt from the PAM barely slows propagation. That asymmetry is the physical basis of off-target cleavage: sites differing from the guide only in PAM-distal positions are cut nearly as efficiently as a perfect match.
The exact absorption probability P(reach 20 before 0 | start at bp 1) is solved analytically from the per-site forward/backward branching probabilities (a birth–death chain), giving the "Model P(cleave)" readout. "Simulate Binding Event" runs one stochastic realization of the same walk in real time; "Run 100 Fast Trials" repeats it instantly to build an empirical success rate that converges to the theoretical value.
- Mismatch position — where along the 20-nt protospacer (counted from the PAM) the mismatch cluster starts.
- Mismatch cluster length — how many consecutive mismatched bases sit there (0 = perfect match, positive control).
- PAM quality — a suboptimal NAG PAM lowers the initial engagement probability, scaling predicted off-target frequency down independently of the R-loop kinetics.