Inferring average generation via division-linked labeling

Abstract

For proliferating cells subject to both division and death, how can one estimate the average generation number of the living population without continuous observation or a division-diluting dye? In this paper we provide a method for cell systems such that at each division there is an unlikely, heritable one-way label change that has no impact other than to serve as a distinguishing marker. If the probability of label change per cell generation can be determined and the proportion of labeled cells at a given time point can be measured, we establish that the average generation number of living cells can be estimated. Crucially, the estimator does not depend on knowledge of the statistics of cell cycle, death rates or total cell numbers. We validate the estimator and illustrate its features through comparison with published data and physiologically parameterized stochastic simulations, using it to suggest new experimental designs.

Sample path properties of the average generation of a Bellman–Harris process

“…Consequently, a complementary result is also established in [58] within the context of the standard model of an asynchronously developing tree, the Bellman-Harris branching process. That is, a growing tree model where cells have i.i.d.…”

“…We recently proposed a new design for in vivo inference of average generation that relies on a DNA coded randomised algorithm [58]. For illustration, consider a single initial cell at time t = 0.…”

“…numbers of offspring numbers at the end of their lives. With Z(t) being number of cells alive at time t in a super-critical Bellman-Harris branching process, so long as the label-positive sub-population Z + (t) is super-critical, it is established in [58] that…”

Retracing the in vivo haematopoietic tree using single‐cell methods

“…By following histone-GFP retention and dilution over time [70][71][72] and the development of an inducible fluorescent tagging of HSCs followed by observation of the fluorescent flux across FACs defined cell types over time [46], population level inferences in this regard have been made. Alternate singlecell systems include the use of MS mutation state [73], as well as a proposal for the design of a genetic delabelling construct [74], to infer tree depth. If tree depth could be measured in parallel with any of the singlecell techniques covered in this review, it will inform the inference approaches and should lead to additional hypothesis discrimination on the tree's structure.…”

The variance of the average depth of a pure birth process converges to 7

“…In addition to the results in [16], this finding is potentially surprising because it is known that the two processes {Z(t)} and {G(t)} have different growth rates, e λt and te λt , respectively [7,17], from which one might anticipate that the variability of the average depth of a tree diverges to infinity as t 2 . Those suppositions are incorrect as it has recently been established that, for general continuous time branching processes, Z(t) and G(t) are strongly correlated at the level of sample paths [10], and that, for a pure birth process, lim t Z(t)/(tG(t)) = 2λ almost surely.…”