mutation Fixation Model

This model employs binomial distributions to represent mutation and repair probabilities and hypergeometric distributions to calculate distribution into the daughter cells. By utilizing mutation matrices M, repair matrix R, and distribution matrix D, the approach enables efficient tracking of mutation and repair frequencies across generations. The iterative matrix multiplication (DRM)^g effectively captures generational propagation, allowing for the observation of the landscape of mutant copy number distributions over time. The outline of this model is shown schematically. 1 represents the stochastic mutation production step, 2 represents the stochastic mutation repair step, 3 represents the deterministic duplication process, and 4 represents the stochastic distribution process. The binomial distribution is employed in the mutation generation and repair steps, while the hypergeometric distribution is used to describe the partitioning step.  The model assumed that each episome duplicated once to sustain a constant copy number per cell. As an example, the first generation starts from the non-mutated state (no mutation: 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 a total of 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 (resulting in mutation count i = 2 and total copy number n = 6 in each daughter cell).  Thick arrows indicate the stochastic processes, and the thin lines represent the deterministic processes.