2.3. Markov Model of Polymerase Stuttering

To describe the process of polymerase stuttering, we implemented a Markov model (Figure 1) with three possible states: transcribe (T), next (N), and stutter (S). In normal conditions, the polymerase would proceed from the state T where it just transcribed a base to the state N, where it would transcribe the next base. In our model, this transition happens with probability 1−ps. The system goes in a stuttering state (S) with probability ps. This means that the same guanine base is transcribed once more, resulting in a single insertion. The polymerase can then stutter again with a different probability p_r or the polymerase might move to the next base, ending up in the absorbing state N (corresponding to the transcription of the rest of the sequences) with probability 1 − p_r. After the first time, the polymerase can stutter any number of times, each time with probability p_r.

Markov model of polymerase stuttering. In this model, three states are possible: Transcribe, Stutter, Next. The two rates in the figure can be inferred from deep sequencing data.

The probability of the polymerase not stuttering, resulting in no insertion at all, is

while the probability of a stutter of length l > 0 (i.e., insertion of a homopolymeric stretch of l guanines) is

A fraction γ of genomic RNA would change the observed distribution of the length of the insertion as

where δ_l,0 = 1 for l = 0 and 0 otherwise. This would also change the inferred fraction of V and W, by reducing both of them by a factor 1 − γ; however, their ratio is unchanged.

We consider the logarithm (in base 10) of p(l). Our model predicts a linear dependence on l for this quantity, except for l = 0. The slope of the linear part is log10(pr), while its intercept is log10(ps(1−pr)/pr), provided that the contribution from genomic RNA is negligible.