Proliferating parasites in dividing cells: Kimmel’s branching model revisited

Abstract

We consider a branching model introduced by Kimmel for cell division with parasite infection. Cells contain proliferating parasites which are shared randomly between the two daughter cells when they divide. We determine the probability that the organism recovers, meaning that the asymptotic proportion of contaminated cells vanishes. We study the tree of contaminated cells, give the asymptotic number of contaminated cells and the asymptotic proportions of contaminated cells with a given number of parasites. This depends on domains inherited from the behavior of branching processes in random environment (BPRE) and given by the bivariate value of the means of parasite offsprings. In one of these domains, the convergence of proportions holds in probability, the limit is deterministic and given by the Yaglom quasistationary distribution. Moreover, we get an interpretation of the limit of the Q-process as the size-biased quasistationary distribution.

A Host-Parasite Model for a Two-Type Cell Population

“…We continue with the introduction of some necessary notation which is similar to the one in [5]. Making the usual assumption of starting from one ancestor cell, denoted as ∅, we put G 0 := {∅}, G n := {0, 1} n for n ≥ 1, and let…”

“…It has transition probabilities P(T v0 = x, T v1 = y |T v = A) = p x y , (x, y) ∈ {(A, A), (A, B), (B, B)}, P(T v0 = B, T v1 = B |T v = B) = 1, and we denote by p 0 := p AA + p AB = 1 − p BB and p 1 := p AA the probabilities that the first and the second daughter cell are of type A, respectively. In order to rule out total segregation of type-A and type-B cells, which would just lead back to the model studied in [5], it will be assumed throughout that…”

“…For the definition of a random A-cell line, a little more care than in [5] is needed because cells occur in two types and parasitic reproduction may depend on the types of the host and both its daughter cells. On the other hand, we will show in Section 2 that a random A-cell line still behaves like a branching process in iid random environment (BPRE) which has been a fundamental observation in [5] for a random cell line in the single-type situation.…”

“…The random mechanism that describes how parasites within a cell multiply and are then shared into the daughter cells is allowed to depend on the hosting mother cell as well as its daughter cells. Focusing on the subpopulation of A-cells and its parasites, our model differs from the single-type model recently studied by Bansaye [5] in that the sharing mechanism may be biased towards one of the two types. Our main results are concerned with the nonextinctive case and provide information on the behavior, as n → ∞, of the number A-parasites in generation n and the relative proportion of A-and B-cells in this generation which host a given number of parasites.…”

“…Our main results are concerned with the nonextinctive case and provide information on the behavior, as n → ∞, of the number A-parasites in generation n and the relative proportion of A-and B-cells in this generation which host a given number of parasites. As in [5], proofs will make use of a so-called random cell line which, when conditioned to be of type A, behaves like a branching process in random environment. …”