The theory of branching stochastic processes has proved a powerful tool for cell kinetics in general and for analyzing clonal growth of cultured cells in particular. The ongoing development of mathematical aspects of this theory is frequently stimulated by or directed towards applied problems. A comprehensive account of the theory and some biological applications are given in books by Harris [1
], Sevastyanov [2
], Mode [3
], Athreya and Ney [4
], Jagers [5
], Assmussen and Hering [6
], Yakovlev and Yanev [7
], Guttorp [8
], Kimmel and Axelrod [9
] and Haccou et al. [10
Since the choice of a particular model is frequently determined by its tractability, the Bellman-Harris branching process and its modifications have been traditionally considered as a fairly general framework for cell kinetics studies. The multi-type version of this process is defined as follows. Let
= 1, ..., K
be the number of cells of the kth
type at time t
given that the clonal growth starts with a single (initiator) cell of type i
at time t
= 0. The vector Z(i)
(t) = (
) is said to be a Bellman-Harris branching stochastic process with K
types of cells if the following conditions are met. Each cell of type k
, 1 ≤ k
, transforms into j1
, ..., jK
, daughter cells of types 1, ..., K
, respectively, with probability pk
, ..., jK
). The time to transformation is a non-negative random variable (r.v.) with cumulative distribution function (c.d.f.) Fk
(.). The usual independence assumptions are adopted.
The problem of quantitative inference from clonal data on cell development in tissue culture has been addressed in our publications [11
]. These papers employ a multi-type Bellman-Harris branching process to model the proliferation of oligodendrocyte/type-2 astrocyte progenitor cells and their transformation into terminally differentiated oligodendrocytes. This model is widely applicable to other in vitro
cell systems. The precursor cell that gives rise directly to oligodendrocytes was first discovered by Raff, Miller and Noble in 1983 [22
], when it was named as an oligodendrocyte/type-2 astrocyte (O-2A) progenitor for the two cell types it could generate in vitro
. This cell is also known as an oligodendrocyte precursor cell (OPC), and will be referred to henceforth as an O-2A/OPC. Such cells appear to be present in various regions of the perinatal rat CNS, and cells with similar properties also have been isolated from the human CNS [23
The O2A/OPC-oligodendrocyte lineage has provided a remarkably useful system for studying general problems in cellular and developmental biology. In the context of our present studies, three advantages of this lineage are that it is possible to analyze progenitor cells grown at the clonal level, that progenitor cells and oligodendrocytes can be readily distinguished visually, and that the generation of oligodendrocytes is associated with exit from the cell cycle. In the culture system we use in our experiments, the dividing O-2A/OPCs only make either more progenitor cells or oligodendrocytes; no other branching in the process of their development is possible. This makes it possible to conduct well-controlled experiments that generate quantitative information on cell division and differentiation at the clonal level and at the level of individual cells.
The earlier proposed model was designed to describe the development of cell clones derived from O-2A/OPCs under in vitro conditions. Cells of this type are partially committed to further differentiation into oligodendrocytes but they retain the ability to proliferate. It is believed that the main function of progenitor cells in vivo is to provide a quick proliferative response to an increased demand for cells in the population. Terminally differentiated oligodendrocytes represent a final cell type; they are responsible for maintaining tissue-specific functions and they do not divide under normal conditions. Both cell types are susceptible, in variable degrees, to death.
A substantial amount of new biological knowledge has emerged from applications of our model to experimental data, with a particular focus on understanding the regulation of differentiation at the clonal level. As all differentiation processes require that cells make a decision between differentiating and not differentiating, it is important to understand how this process is controlled at the level of the individual dividing precursor cell. Early studies had indicated that individual O-2A/OPCs would divide a limited number of times before all clonally related cells differentiated synchronously and symmetrically under the control of a cell-intrinsic biological clock. Subsequent biological studies showed that the cell-intrinsic regulator of differentiation promoted asymmetric and asynchronous differentiation among clonally related cells unless promoters of oligodendrocyte generation were present. It was only through our modeling studies, however, that the popular clock model of oligodendrocyte generation in vitro
was disproved by testing a more general (hierarchical) model against experimental data [11
In the earliest version of our model [11
], it was assumed that the initial population of progenitors is a mixture of subpopulations with different numbers of "critical" cycles. In each of these subpopulations the probability of division is 1 until the critical number is reached and drops sharply to a fixed value p
< 0.5 afterwards. The number of critical cycles is not directly observed, and one can only verify this basic assumption by fitting the model to experimental data on the evolution (over time) of clones consisting of two distinct types of cells. However, if one considers the whole population of cells, there is a more gradual decline in the division probability from 1 to p
, suggesting that an alternative model is also plausible, in which there is a single population of progenitor cells with a gradually decreasing division probability [17
]. While both models are in almost equally good agreement with clonal data, the latter model has a more parsimonious structure, which is also perfectly consistent with the time-lapse data to be reported in the present paper.
The basic stochastic model of proliferation and differentiation of O-2A/OPCs was based on the following assumptions:
A1. The process starts with a single progenitor cell of type 1 at time 0.
A2. After completion of its mitotic cycle, every progenitor cell of type l ≥ 1 either divides to produce two new progenitor cells of age 0 and type l + 1 with probability pl, or transforms into a differentiated cell of type l = 0 (oligodendrocyte) with probability 1 - pl.
A3. The time to division of a progenitor cell of type l ≥ 1 is a non-negative r.v. Tl,1 with c.d.f. F1(x), while the time to differentiation of a progenitor cell of type l ≥ 1 is a non-negative r.v. Tl,2 with c.d.f. F2(x).
A4. Differentiated oligodendrocytes neither divide nor differentiate further, but they may die; their lifespan T0 has c.d.f. L(x) = Pr(T0 ≤ x).
A5. Whenever counts of dead oligodendrocytes are utilized for estimation purposes, the model needs to be extended further to include the following assumption: every dead oligodendrocyte disappears (disintegrates) from the field of observation after a random lapse of time T-1 distributed in accordance with c.d.f. H(x) = Pr(T-1 ≤ x). The time to the disintegration event is expected to be quite long, as there are no macrophages present in the culture to clear away cell debris.
A6. The cells do not migrate out of the field of observation.
A7. Of the two cell types, oligodendrocytes appear to be more susceptible to death. Therefore, it was assumed that progenitor cells do not die during the period of observation.
A8. The assumption of independence of cell evolutions is adopted. This assumption is critical for making the mathematical treatment of the resultant branching stochastic process tractable.
The probabilities pl
can be described by an arbitrary function of the mitotic cycle label l
that satisfies the natural constraints: 0 ≤ pl
≤ 1 for all l
≥ 1. In [17
], these probabilities are specified as pl
, 1}, where p
are free positive parameters with p
representing the limiting probability of division of progenitor cells as the number of cycles tends to infinity. In our analysis of the time lapse data in the next section we proceeded from this choice as well. All the distributions introduced above were specified by a two-parameter family of gamma distributions, which is the most popular choice in cell kinetics studies [7
Assumption A3 was introduced in [19
] to allow the mitotic cycle duration and the time to differentiation to follow dissimilar distribution functions. The authors proceeded from the following line of reasoning. In the classical Bellman-Harris process, either the event of division or the event of differentiation is allowed to occur upon completion of the mitotic cycle. Let the r.v.s X
represent the time to division and the time to differentiation, respectively. Then the postulates of the Bellman-Harris process imply that the joint distribution of X
is singular along the diagonal X
. A natural alternative is to assume that the r.v.s X
have dissimilar continuous distributions. This alternative is biologically plausible because the proliferation and differentiation of cells involve different molecular mechanisms. The analysis of clonal growth of cultured O-2A/OPCs has corroborated this hypothesis [19
], and the time-lapse data presented in the next section provide additional evidence in favor of its validity.
], the mitotic cycle duration and the time to differentiation of O-2A/OPCs were assumed to follow the same distribution, that is, F1
) = F2
) for all x
, but we allowed the distribution of the time to division and differentiation of initiator cells to be potentially different from that of cells in subsequent generations. Our time-lapse data provide the opportunity to look more closely at variations in the mitotic cycle duration across cell generations and their consistency with this basic model assumption. The design of our previous studies generated cell counts in independent cell clones at different times after plating. We also used longitudinal data on cell counts produced by observations of the same cell clone at different time points [20
]. However, much more information can be extracted from data yielded by time-lapse video recording of individual cell evolutions, and we take advantage of this experimental technique to verify the most basic elements of the earlier proposed model.