To perform the above analyses, we proceeded as follows. Similarly to the analysis of O2A/OPCs, we distinguished cells of first and of subsequent generations, and refer to them as type-1 and type-2 cells, respectively. Let Y_i = (Y_i1, Y_i2)’ denote the number of type-1 and the number of type-2 cells counted in the ith clone at time t_i. Let Z₁(t) and Z₂(t) be stochastic processes denoting the number of type-1 and the number of type-2 cells at any time t, and let m₁(t, θ) and m₂(t, θ) denote their expectation. Write Z(t) = {Z₁(t), Z₂(t)} and m(t, θ) = {m₁(t, θ), m₂(t, θ)}’. We have that m₁(t, θ) = 1 − G₁(t), and . Because the model contains infinitely many types, the variance-covariance matrix of the numbers of cells takes a complicated form, and we did not use it to fit the models. We estimated θ = (μ₁, σ₁, μ₂, γ, ρ)’ using the GEE approach, with each Ω_Y (t_i) taken as the empirical variance-covariance matrix of Y_i computed from clones observed at time t_i (the experiment provided approximately 100 replicates for each time point). The expression for m₁(t, θ) is explicitely known, and the value of m₂(t, θ) only needs to be approximated. The cumulant generating function for G₁ ★⋯★ G_g(t) is given by The saddlepoint is the root to the equation . We deduced from equation (6) where we set k = 1; for g ≥ 3, the equation is solved numerically. The saddlepoint approximations to the expectation follows from equations (3-5). Figure 3 shows the experimental and fitted (Models 0-2) mean number of type-1 and type-2 cells versus time. Simulations were used to assess the accuracy of the saddlepoint approximations to the expected number of cells.

To illustrate the proposed methods we analyzed a set of clonal data on the generation of oligodendrocytes from O2A/OPCs cultured in vitro. In this experiment, O2A/OPCs isolated from the optical nerve of 7-day old rats were plated in cell culture in the presence of thyroid hormone. Cells were fed every other day with fresh PDGF. Clones were observed 1, 2, 3, 4, 5, 7 and 8 days after plating. At each time of observation, the composition of either 50, 100 or 150 clones (depending on the time point; total sample size n = 850) was individually examined to obtain the number of O2A/OPCs and the number of oligodendrocytes in each of them (as illustrated in Figure 1). The clones were discarded after examination. The mean number of O2A/OPCs and the mean number of oligodendrocytes versus time are showed in the left and right panels (respectively) of Figure 2. Each clone started with a single O2A/OPC and zero oligodendrocyte. The mean number of O2A/OPCs appears to decrease at days 1 and 2, increase at day 3, and decrease again after day 3. In contrast, the mean number of oligodendrocytes increases steadily in an almost linear fashion from the start of the experiment. On day 8, any clone would contain on average 0.7 O2A/OPC and 2.3 oligodendrocytes. The largest clone contained a total of 12 cells.

One cellular system that has been extensively studied using age-dependent branching processes is that of oligodendrocytes (the cells of the central nervous system (CNS) that enwrap axons with myelin sheaths) generated from their immediate ancestral progenitor cells (Raff et al, 1983). The oligodendrocyte-type 2 astrocyte progenitor cell (also referred to as an oligodendrocyte precursor cell and here abbreviated as O2A/OPC) can be isolated from multiple regions of the CNS and studied in vitro. The O2A/OPC lineage has proven to be exceptionally suitable for studying differentiation at the clonal level. Dividing O2A/OPCs can be readily distinguished from oligodendrocytes on the basis of cell morphology (the former being bipolar with small elliptical cell bodies and the latter being multipolar with large round cell bodies), enabling unambiguous identification of every cell in a clone (e.g, Noble et al, 1988). Moreover, O2A/OPCs readily establish clones from single founder cells, and the subsequent development of these clones can be followed over time. (see Figure 1 for an illustration of the development of a clone of O2A/OPCs). Clonal analyses have been of enormous value in investigating the control of differentiation at the single cell level. Despite the power of clonal analyses in the study of differentiation, they also have an important limitation. The complete histories of the cellular trees are not fully observed, but instead the composition of the clones at the times of observation are used to provide information on the proliferative patterns of the cells. The quantitative analysis of such experiments calls for specific estimation methods. One objective of this article is to propose new methods for such data built on the proposed saddlepoint approximations.

We tested this approach using a two-type process that begins at time 0 with a large number of type-1 cells. Upon completion of its lifespan, every cell of generation g (g = 1, 2 ⋯) either divides into two cells of generation (g + 1) with probability p, or it dies (that is, disappears from the population) with probability q = 1 − p. The duration of the lifespan of any cell of first generation is represented by the sum of two r.v.s τ₀ + τ₁, where τ₀ and τ₁ follow Inverse Gaussian distributions G₀ and G with respective means μ₀ and μ, variances and σ², and c.d.f.s G₀ and G, while, for every g = 2, 3 ⋯, the duration of the lifespan of any cell of generation g is described by a r.v. τ_g that follows the Inverse Gaussian distribution with c.d.f. G. This model extends a model considered by Hyrien and Zand (2008) to describe the activation and proliferation of lymphocytes. The r.v. τ₀ models the time that is needed for lymphocytes to become activated, while τ_g, g = 1, 2 ⋯, models the time that is needed for any activated lymphocytes to die or divide. It can be shown that the mean number g of cells in any given generation takes the form m_g(t) = {1 − G₀(t)}1_{g=1} + (2p)^g {G₀ ★ G^★(g-1)(t) − G₀ ★ G^★g(t)}. The cumulant generating function for G₀★ G^★g is , the rth derivative of which is given by where the double factorial r!! is defined as The saddlepoint which solves the equation is approximated numerically. Plugging the above expressions into equations (3-5) yields the desired quantities. Depicted in Figure 4 are the saddlepoint approximations to p_g(t), g = 1, 2 ⋯, for three time points (left panels), and its expectation as a function of time (right panels). Each row corresponds to a particular set of parameter values (see figure legend for the parameter values). The saddlepoint approximations were compared to simulation-based approximations (computed using direct simulations of the branching process). As can be seen, the saddlepoint approximations are in excellent agreement with their simulated counterparts, and capture remarkably well transient features of the expected generation number. The only exception is for p₁(45) in Case 2, where we convoluted two distributions with very dissimilar variances.