“Stochastic”—40 Years of Use and Abuse

Till et al3 4 showed that colonies of hematopoietic cells developed on the spleens of irradiated mice injected with normal bone marrow cells. When these colonies were excised individually and the cells retransplanted into a second group of irradiated recipients they were able to observe colonies on their spleens. The numbers of these secondary colonies were highly variable and fitted a skewed (gamma) distribution. This skewed distribution could be explained by stem cells having a certain probability of differentiating into one of the maturation lineages and, therefore, either losing their clonogenicity or, alternatively, remaining as colony-forming cells. Computer simulations were used, because mathematical solutions are intractable, to show that this skewed distribution could be explained by stem cells having a probability of 0.4 of differentiating into one of the maturation lineages, and therefore losing their clonogenicity, or remaining as colony-forming cells, giving a self-renewal probability of 0.6. Cell death as a possible contributary factor was not considered at the time.

Till et al explicitly pointed out in their publications that stochastic effects would only be observed if the spleen colonies originated in a single or a very few cells. In fact, the highly skewed distribution provided evidence not that the cells had some special property, as they claimed, but rather that the spleen colonies were initiated by single colony-forming cells. They also mention that stochastic processes are important in relation to the size of bird populations and even to cosmic ray events.

The conclusions of this work were that stem cells may differentiate or not (self-renew) when they divide, but that the outcome of any individual stem cell division cannot be predicted in advance. The erroneous perceptions were that the probability of self-renewal/differentiation is fixed, that it is a special property of stem cells and that it is not a property of more mature progenitors.

The so-called “stochastic” property of stem cells was challenged by the observation that spleen colonies forming in the splenic red pulp consisted predominantly of red cells, while colonies found just under the splenic capsule and along the trabeculae consisted mainly of white blood cells.5 This indicated the response of stem cells was subject to changes in the “hematopoietic inductive microenvironment” (HIM), and a fierce argument developed. This seems quite surprising because one side of the argument related to lineage selection and the other to stem cell self-renewal/differentiation, which should be distinguished from each other. Nevertheless, the implied meaning of “stochastic” (hematopoiesis engendered randomly [HER]) became even more firmly entrenched. The coining of these acronyms, which are remembered to this day, is witness of the intensity of the debate between the proponents of two views.

According to the Shorter Oxford Dictionary, stochastic processes are “randomly determined, that follows some probability distribution or pattern so that its behaviour may be analysed statistically but not predicted precisely.” The quintessential example of a stochastic process is radioactive decay where the probability of an atom decaying is known very precisely, but the time that any particular atom will decay is completely unpredictable. For example, for radioactive carbon (¹⁴C) with a half-life of 5,770 years, an atom may disintegrate within a microsecond, or not until tens of thousands of years later.

Taking a cue from Aesop, consider three people entering a consulting room at random: it will be completely unpredictable whether there will be 3 men, or 2, or 1, or all women, but the probability of each is precisely known based on the equal probability of 0.5 that the person entering the room is male or female. In the same way is the growth of a single cell in a colony-forming assay variable, and the very wide skewed distribution is the result of such a process being repeated many times as the cells divide.

This effect has been amply shown by experimental data: for example, by Humphries et al6 when cells from erythroid colonies grown in vitro were replated for the formation of secondary colonies, by Ogawa’s group7,8 in the replating of blast cell colonies, and by Kurnit et al9 who used subcolony number in individual erythroid bursts (BFU-E) as surrogates for secondary colony-forming cells. In all cases, the skewed distributions were reminiscent of those found by Till and McCulloch and confirmed the existence of stochastic events in colony formation in general.

That the misconceptions behind this controversy have remained until today is surprising because self-renewal/differentiation and the direction of differentiation (lineage commitment) are different biological processes, and do not necessarily occur in the same cell populations. Furthermore, in the 1960s and 1970s there was almost certainly a reluctance of others to take issue with the two “combatants” who were the leaders in this field, and there were few experimentalists with sufficient interest/expertise in the mathematical aspects of this process. It seems appropriate to take this opportunity to summarize briefly the importance of taking into account stochastic effects in interpreting results obtained with colony-forming assays, because they may be used to study the molecular biology of progenitor cells.

There are four principal characteristics of the growth of single cells from a homogenous population of colony-forming cells: (1) the large variation in number of secondary colonies, already mentioned; (2) a similar wide variation in the total number of cells in the colony; (3) differences in growth rate between individual colonies; (4) some colonies will be long-lived while in other colonies all the stem cells will differentiate leading to the demise (“extinction”) of the colony.10 These points are frequently challenged by the argument that such results are a reflection of the heterogeneity of the cultured cell population. However, even with a 100% pure population, stochastic events will be evident during colony formation whenever some cells self-renew and others differentiate.

The very wide variation in the size of colonies predicted to grow from a uniform population of colony-forming cells means that colony size (eg, above and below a particular scoring threshold) is not necessarily evidence that they come from a different population or a subpopulation of progenitor cells. The same reasoning applies to the number of secondary colonies. Furthermore, the timing of the appearance of colonies and their disappearance is not necessarily evidence that the colony-forming cells are heterogenous or represent different populations. Consequently, an apparent heterogeneity in the growth of colonies is not in itself sufficient evidence for the existence of heterogeneity within a given progenitor population or even of different populations, although there is ample evidence for a hierarchy in progenitor cell maturation from the physical separation of cells depending on a variety of different cell properties.

Mathematical predictions cannot provide proof of the correctness of a particular biological explanation, but they are useful in showing whether it is consistent with a set of experimental results. One perhaps extreme case is that a simple computer simulation shows that a uniform population of colony-forming cells can predict the time of appearance of the so-called “transient tiny erythroid” spleen colonies11 and the near constant number of colonies between 7 and 10 days, resulting from the concomitant appearance of colonies and the disappearance of other colonies when appropriate size thresholds are used for scoring.

The original work of Till and colleagues indicated that the probability of stem cell renewal was 0.6 and that it was intrinsically fixed. However, under normal steady-state conditions, self-renewal must be 0.5 so that on average one daughter cell differentiates and one daughter “replaces” the cell that has divided, so as to maintain the same number of colony-forming cells. The increase in self-renewal from the steady-state value of 0.5 to 0.6 in spleen colony-forming cells is consistent with the rapid regeneration of hematopoiesis in irradiated mice and should have been sufficient to refute the conclusion that self-renewal of spleen colony-forming cells is determined “intrinsically” and so not subject to change.

There is other ample evidence12,13 that spleen colony-forming cells are subject to external control by feedback regulation. There is some evidence that self-renewal can be regulated in vitro from the demonstration by Metcalf14 that granulocyte colony-stimulating factor reduces the secondary colony-forming ability of WEHI-3B cells, while Lewis et al15 16 have shown that for erythroid cells the BFU-E subcolony number distribution, and for granulopoiesis secondary colony formation, are altered by cytokines.

To conclude, stochastic effects are the consequence of the uncertainty of the response of individual cells, and are not a property of the cells themselves. However, the effects only become significant when the population comprises a single or a few cells, as in colony-forming assays, and do not provide any information as to whether or not cells respond to the physiological environment in which they are growing. Therefore, de facto, they are not relevant to discussions about whether or not cells respond to extrinsic regulators.

As molecular biology unravels the interior “workings” of progenitor cells, a better understanding of the stochastic effects inherent in the use of colony-forming assays is important. Computer simulations can provide a useful guide as to the magnitude of these effects for different values of the various cell kinetic parameters.

M.G. is supported by the Leukaemia Research Fund of Great Britain.