mutation Fixation Model

A base editor is a genome editing tool that enables precise nucleotide changes without creating double-strand breaks. It consists of a catalytically impaired CRISPR-Cas enzyme fused to a deaminase, allowing targeted base conversions (e.g., C•G to T•A or A•T to G•C). When used for gene inactivation, base editors introduce mutations that disrupt gene function over successive generations. To better understand this process, we aim to determine the number of inactivated gene copies at a specified generation. This model provides an estimate of these key values.

The model employs binomial distributions to represent mutation and repair probabilities and hypergeometric distributions to describe the distribution of gene copies into daughter cells. By utilizing mutation matrices (M), a repair matrix (R), and a distribution matrix (D), this approach efficiently tracks mutation and repair frequencies across generations. The iterative matrix multiplication (DRM)^g effectively captures generational propagation, enabling the observation of mutant copy number distributions over generation number g.

A schematic outline of the model is presented as follows:

1. Stochastic mutation production process
   
2. Stochastic mutation repair step
3. Deterministic duplication process
4. Stochastic distribution process
5. Deterministic separation step

The binomial distribution is applied in the mutation and repair steps, while the hypergeometric distribution is used to model the partitioning of gene copies into daughter cells. The model assumes that each episome duplicates once per generation to maintain a constant copy number per cell.

As an example, the first generation starts from a non-mutated state (mutation count: i=0, total copy number: n=6). Three mutations (red dots) are introduced (i=3, n=6), followed by a binomial repair process that corrects one mutation, resulting in j=2 mutations and a total copy number of n=6. The episomes then duplicate (2j=4, 2n=12), and the copies are distributed into two daughter cells according to a hypergeometric distribution, leading to a mutation count of i=2 and a total copy number of n=6 in each daughter cell.

Thick arrows in the schematic indicate stochastic processes, while thin lines represent deterministic processes.