Stem Cell Proliferation and Differentiation

Our interest in modeling the process of cell growth and differentiation in cell culture systems was stimulated by in vitro experiments designed to study the growth and differentiation of bone marrow cells into mature macrophages and by attempts to maintain a line of macrophages in culture. When bone marrow or other not fully differentiated cells are grown on plates in the presence of nutrients and appropriate growth factors, some single cells grow into large groups of cells. Even when cells are grown under identical conditions, one finds great heterogeneity in the sizes of the resulting groups (see Fig. 1.1; also Fig. 1B of Sachs, 1987). As the length of time a plate is incubated increases, it is observed that the distribution of sizes of groups becomes bimodal (Stewart, 1980, 1984). Typically, groups are either relatively small (≤50 cells) and are called clusters, or are orders of magnitude larger than clusters and are called colonies. As the culturing time increases further, fewer groups fall between these size limits. Pharr et al. (1985) observed a similar bimodality in the colony size distribution of proliferating mast cells.

The model we develop is one of the simplest nontrivial models of cell growth in culture. We assume that there are three general classes of cells, which we label stem cells (S), macrophage progenitors (M), and end cells (E). Discrimination among the classes is on the basis of function of the member cells since typically the cell types cannot be distinguished morphologically. Stem cells are described as having the capacity for infinite self-renewal. By this, we mean that a stem cell can divide an infinite number of times, and at each division its offspring can be an exact replica of itself. A daughter cell that has differentiated from the parent stem cell is called a progenitor. In contrast to stem cells, progenitors do not have the capacity for infinite self-renewal. Eventually, a progenitor must produce a daughter cell that differs significantly from itself: this process is called terminal differentiation. The fully mature offspring cannot divide and are called end cells. Thus the three cell types of the model form a natural progression according to increasing maturity and consequent decreasing proliferative potential; S → M → E.

In the previous chapter, we alluded to the problem of iteration of the generating functions: Essentially it is computationally infeasible to evaluate Eq. (2.11) for large n. Hence, we cannot obtain the detailed probability distribution for colony composition with time, Pr(Z _n = (i, j, k)|Z₀). However, in this and the following three chapters we characterize other aspects of colony growth in finite time and obtain results in the limit n → ∞. Because of the recursive nature of many of the expressions, explicit analytical results are not obtainable except in rather trivial special cases. The implications of the expressions will therefore not become apparent until Chapter 7 where we provide numerical examples for selected values of the branching probabilities.

Two related quantities that describe a colony that completes its growth are the time (number of generations) taken to reach completion and the colony size at completion. One might expect that a colony that grows for many generations before its growth finally ceases would be much larger than a colony that ceases growth after only a few generations. We examine first the distribution of the number of generations to completion since the derivation of the distribution follows directly from the finite-time recursion of the modified model in Section 3.2. Colony size at completion is covered later.

From Theorem 2 we can predict that colonies containing cell types with branching probabilities greater than 0.5 (i.e., colonies having supercritical growth) will, with non-zero probability, never reach completion of growth. When growth is unbounded, calculating the number of cells in each of the three compartments as n → ∞ is a meaningless exercise, for these numbers will be infinite or zero. (For example, in a colony with an S-cell parent, if \( \frac{1}{2} < {p_{s}} < 1 \) and p _m < 1, then there is a non-zero probability that the S, M and E compartments will each contain an infinite number of cells.) However, we can calculate the proportion of cells in each of the compartments by appealing to the Kesten-Stigum limit theorems for decomposable Galton-Watson processes (cf. Kesten and Stigum, 1967; Mode, 1971). The essence of these theorems is that, under conditions in which a colony can grow without bound, the limiting proportion of each cell type is a constant that can be calculated as a function of the branching probabilities, p _s , p _m , and p _e . The theorems require calculation of the eigenvalues and eigenvectors of the matrix of means of the offspring distribution for parents of each type. We shall consider separately the cases of colonies with an M-cell parent and colonies with an S-cell parent. Since the former case is easier and is in some sense a subset of the latter case, we shall consider it first.

As described in Section 3.2, a threshold exists between colonies completing growth in a finite number of generations and colonies with a positive probability of growing without bound. The condition defining this threshold is that a colony contains cell types for which the maximum branching probability is 0.5 (see Theorem 1). The branching process is then defined as critical. We have already seen that a colony with p _m , p _s ≤ 0.5 has probability 1 of completion of growth. However, if any of the cell types in the colony have a branching probability equal to 0.5, then the expected time for the colony to complete growth is infinite. Thus, aspects of finite and infinite growth are observed in a critical process. One might expect that, biologically, the critical case is important, for it possibly approximates the equilibrium situation in vivo.

We now use the results of our mathematical analyses in the previous chapters to make predictions about the general behavior of colony growth in culture. We will emphasize contrasts between colonies originating from stem cells and colonies originating from more mature cells that lack the capability of self-renewal, which in our model are macrophage progenitors. We will also describe the pattern of growth with increasing time in a heterogeneous culture.

Since the demonstration of the existence of a pluripotent hemopoietic stem cell (cf. Till et al., 1964), there has been discussion about the nature of the commitment and differentiation processes that cause the undifferentiated stem cell to develop into a functionally mature hemopoietic cell. Two opposing views of the process have developed: one view proposes that the decision of when, and along what pathway, a stem cell differentiates is preprogrammed in the cell, i.e., that decisions are under strict genetic control. Hence, offspring of a stem cell will pass through a regulated sequence of transformations until the (predetermined) end point is reached. This view we call deterministic. The other view, which we call stochastic, allows for the commitment process to be random. In the stochastic view a decision to commit may be made in any cell cycle. However, not all pathways may be available to a progenitor at a given degree of maturity. Thus the cell is only able to commit, with certain (unknown) probabilities, to any one of the pathways available. Furthermore, the probability of commitment may not be fixed but rather may be a function of the cell’s environment and the cell’s biochemical state. In particular, one may envision the situation in which the local concentration of hemopoietic growth and differentiation factors (e.g., CSF-1, IL-3, IL-1) and the number of cell surface receptors for these growth factors affect the probability of commitment.