branching process

Iwasa Y, Minchor F & Nowak MA 2004 Stochastic tunnels in evolutionary dynamics. Genetics 166:1571-1579.

  • cells reproduce asynchronously
  • we use a Moran process instead of the standard Wright-Fisher model
  • each elementary step of the stochastic process consists of a birth and a death event
  • for birth, one of the N cells is chosen at random proportional to fitness
  • it will give rise to an offspring subject to possible mutation
  • for death, one of the N cells is chosen at random
  • the total population size, N, is strictly constant
  • for a given trajectory Y(t), the probability of appearance and fixation of a type 2 mutant is given by 1 − P
  • P = exp[− ru2ρ(a) ∫0∞Y(t)dt] ... (2)
  • this expression is the zeroth term of a Poisson distribution
  • the exponent is given by the cumulative number of type 1 cells multiplied by the mutation rate from type 1 to type 2, ru2, and the probability of fixation, ρ(a), of one type 2 cell that arises in a population of type 0 cell
  • using ρ(a) is an approximation, because the type 2 cell arises in a population that contains a mixture of type 0 and type 1 cells
  • the rate of tunneling is the product of three factors:
  • the rate of producing type 1 mutants times the probability that the lineage of type 1 mutants will become extinct times the conditional expectation of initiating a successful type 2 lineage
  • the expectation is subject to the conditions that the initial number of type 1 cells is 1, Y(0) = 1, and that the lineage will eventually become extinct, Y(∞) = 0
  • in this way, we exclude the contribution of the spread of type 2 mutants after fixation of type 1 mutants
  • the lineages starting from different mutants of type 1 behave independently, which is implicit in the calculation of the doubly stochastic process
  • this assumption holds if mutations from type 0 to type 1 occur infrequently, Nu1 << 1
  • if this inequality does not hold, then a lineage starting from a single type 1 might not go extinct before the next mutation creating a type 1 mutant arises
  • the approximate formula for the deleterious tunnel, R = N(u1 / (1 − r)) ru2ρ(a), is a product of three factors
  • the first factor is the product of the population size N and the frequency of mutation-selection balance, u1 / (1 − r), indicating the expected number of type 1 mutants in the population
  • the second factor, ru2, denotes the mutation rate from type 1 to type 2
  • the third factor, ρ(a), is the probability of successful fixation of the second mutant once it appears in the population
  • such a simple and intuitive understanding is not available for the corresponding formula when the type 1 mutant is neutral, R = Nu1√(u2ρ(a))
  • why it is proportional to the square root of the second mutation rate is not easy to interpret
  • genetic recombination can be neglected in the somatic evolution of cancer, while it is important for sexually reproducing organisms