The stem cell concept revisited: Self-renewal capacity is a dynamic property of hemopoietic cells

Leukemia Research Vol. 10. No. 8. pp. 937-950, 1986, Printed in Great Britain.

0145-2126/86 $3.00 + .00 Pergamon Journals Ltd.

OPEN FORUM THE STEM CELL CONCEPT REVISITED: SELFR E N E W A L CAPACITY IS A DYNAMIC PROPERTY OF HEMOPOIETIC CELLS ZvI GROSSMAN Tel Aviv University, Tel Aviv, Israel and Pittsburgh Cancer Institute and Departments of Medicine and Mathematics, University of Pittsburgh (Received 7 December 1985. Revision accepted 6 January 1986) Abstract--A rigid developmental program of stem cell division and progressive maturation into blood cells is challenged. It is proposed that the capacity for self-renewal is not limited to pluripotent stem cells but is shared by committed progenitors and even cells of later compartments. The relative probability of self-replication vs maturation in mitotic cells is controlled by extracellular influences. At the cell population level, the balance between proliferation and maturation and between compartments is regulated by feedback interactions. Inducibility of maturation in response to regulatory signals is smaller at earlier stages; consequently, at steady state primitive cells self-renew while their more differentiated progeny are forced to be transitory. The proposed dynamic linkage between compartments can be destabilized in a number of ways, resulting in defective hemopoiesis or leukemia. At all stages hemopoietic cells are able to change their patterns of gene expression, in an inheritable manner, in response to changes in their microenvironment. In particular, the capacity for self-renewal itself can vary even within a conventionally-defined compartment. On this basis of adaptive differentiation and self-renewal it is possible to account for the progression of chronic myelocytic leukemia and its "blastic conversion"; to analyse the hemopoietic system's response to various physiological and experimental perturbations; and to reinterpret the excessive phenotypic plasticity and apparent "lineage infidelity" manifested by leukemic cells and cell lines. Key words: Stem cells, self-renewal, steady states, feedback, leukemia, adaptive differentiation.

INTRODUCTION HEMOPOIETIC cells are generally assigned to one of these compartments or stages of differentiation: (a) terminally-differentiated cells that perform a specific function; (b) committed progenitors or precursors; or (c) pluripotent stem cells. Pluripotent cells can either self-renew or start a program of differentiation, while committed progenitors apparently cannot self-renew-they only divided a limited number of times, maturing as they head for the predetermined terminal state [43, 83]. Recent experimental evidence is at variance with this rigid distinction between pluripotent and committed cells on the basis of self-renewal capacity. "The relative

Abbreviations: CML, chronic myelocytic leukemia; AML, acute myelocytic leukemia; CFU-S, colony forming unitsspleen; CFU-C, colony forming units-culture. Correspondence to: Dr Z. Grossman, Pittsburgh Cancer Institute, 230 Lothrop Street, Pittsburgh, PA 15213-2592, U.S.A.

contributions to normal hemopoieis of self-renewal of committed progenitor cells versus an influx of committed progenitors derived from pluripotential stem cells are undergoing some reassessment" [72]. This state-of-the-art review goes on to say that, in light of current evidence, the following are now open to question: the self-renewal capacity of various cells, the branching patterns of different lineages, and their internal programming or external regulation. Available data indicate that pluripotent stem cells are a heterogeneous population, and that they as well as committed cells exhibit variable regeneration [90, 38, 64, 71, 4, 75, 48, 14]. Committed cells maturing in the bone marrow can add cell generation cycles in response to increased peripheral demand [12]. Also, mature T cells and other committed progenitor cells self-renew in vitro for long periods of time [55, 22, 15, 67, 68, 88]. Based on such observations of in-vitro growth, Salahuddin et al. [67] concluded, "the demonstration that committed normal hematopoietic cells, e.g., T lymphocytes, monocytes, and myeloid cells, can proliferate for extended periods indicates that readily recognizable

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cells can have long, self-replicative capacities, a property often attributed only to an unidentifiable, so-called "stem cell". Wendling et al. [88] state, "restricted erythroid progenitor cells exist which are capable of extensive self-renewal." Similar observations had been made by Dexter [14]. Thus, committed cells--and possibly pluripotent cells---exhibit flexibility by adjusting the ratio of selfrenewal vs maturation in response to external conditions both in vitro and in vivo. The observations of flexibility and adjustment suggest feedback regulation [27, 35, 34, 28, 30, 29]. A model will be presented and biological implications discussed. The model assumes that the feedback interactions adjust the relative probabilities of maturation and replication of the committed as well as the pluripotent cells. Besides self-renewal and its regulation, another issue is that of "branching patterns of different lineages, and their internal programming or external regulation" [72]. Conventionally, hemopoiesis is depicted in the form of lineage diagrams based on colony assays. Recently, McCulloch [52,51] has proposed a concept of "differentiation programs" assembled of many components or stages each representing gene activation or inactivation events modified by epigenetic or environmental influences. While more flexible than the concept of differentiation expressed as a lineage diagram, the programmatic concept still places great significance on "determination" or "commitment"--the hypothetical genetic events leading to "the transition from pluripotent stem cells to progenitors capable only of maturation" [52]. For normal hemopoietie cells "lineage fidelity" is implied: once a cell has been committed to a specific myelopoietic pathway, its descendants follow that pathway faithfully. The capacity to deviate from normally occurring differentiation programs and to manifest lineage infidelity is attributed only to genetically transformed, leukemic cells and not to normal hemopoietic cells. In addition, although it is stressed that gene activation and inactivation events are subject to epigenetic and environmental influences, the possibility of extracellular or feedback regulation of cell differentiation is not considered (perhaps in light of the failure of several previous attempts to demonstrate the operation of particular feedback mechanisms which had been proposed). The view taken here is that normal mitotic hemopoietic cells at all stages of differentiation possess phenotypic adaptability: the capacity to change their patterns of gene expression, in an inheritable manner, in response to changes in their microenvironment. However this capacity is manifest in self-renewing populations to a much larger extent than in populations that are transitory. It is proposed to link the pluripotency of normal stem cells to the excessive phenotypic plasticity manifested by tumor cells and long-term cell lines and in particular to the observation of lineage infidelity in AML blasts (reviewed in [52]). Through adaptation, replicating cells may lose as well as gain differentiative capacities, to which the capacity for self-renewal is inversely related. Implications of this view for normal and abnormal hemopoiesis will be discussed.

A MODEL

Definitions and concepts Despite recurrent attempts to provide a glossary of terms commonly in use for describing functional properties of hemopoietic cells, these terms are often used in an ambiguous way, sometimes reflecting preconceptions with regard to the biologic issues. It is useful to introduce the definitions in a way that is devoid of such preconceptions. Maturation. Cell transition from one state to another, requiring gene activation or inactivation events. The transition can occur either with or without amplificative divisions (see below). Self-renewal. For a single cell, a cell division in which the daughter cells are identical--within biologic variability-to the parental cell (implying identical patterns of gene expression). When referring to a cell population it describes an overall activity whereby the fraction of self-renewal divisions, as compared to maturation events, is sufficient to support a constant (for strict self-renewal) or increasing number of cells within the population even in the absence of inflow of cells from other cell populations. (A division in which only one daughter cell is identical to the parental cell would also be a strictly self-renewal division.) Self-renewal capacity. The meaning of this term is ambiguous. Sometimes it is used as a qualitative trait, as in "committed cells do not possess self-renewal capacity"--meaning that these cells are incapable of performing self-renewal divisions. It is also used as a quantitative property; for example, "progenitors with a limited (or extensive) capacity of self-renewal"--in this case it usually means that the cells and their progeny will undergo a small (or large) series of divisions (relative to other cells) under given conditions. Compartment. Cell population defined on the basis of a set of markers--morphological, biochemical and functional--which can be assayed experimentally. Compartments so defined are usually ordered in terms of lineage and/or stages of maturation. Transitory compartment. A cell population that is constantly replaced through maturation of cells from other (generally less differentiated) compartments. Amplification. This usually describes increase through division in the number of cells originating from a single cell within a compartment or across more than one transitory compartment. Amplificative (or constitutive) division. Cell division which is an obligatory part of a maturation process. Stochastic cellular event. Such an event can be predicted, for an individual cell, only in terms of probability-given all the information about the cell and its environment that can be measured. Pluripotency (vs commitment). The ability of a cell to give rise, through proliferation and maturation (differentiation), to functional cells belonging to more than one hemopoietic lineage. Pluripotency is assayed by the generation of mixed clones, in vitro, from single cells. Committed cells lack pluripotency under a range of conditions tested. Stem cells. Pluripotent, self-renewing cells.

The stem cell concept revisited With the help of these definitions, it is now possible to re-evaluate the data, distinguish between assumptions and facts and consider alternative assumptions. It is obvious that as a cell population the hemopoietic system is self-renewing. Moreover, considering normal hemopoiesis as a steady state, any subpopulation that includes the most primitive (least differentiated) hemopoietic cells must be self-renewing; whereas any other compartment must be transitory: cells that reside in such compartment at any given time and their progeny will eventually be "washed out" by the influx of maturing cells originating in earlier compartments. However, this in itself by no means implies that cells within transitory populations lack a capacity for self-renewal, or that they do not actually undergo self-renewal divisions at some rate; it only implies that such self-renewal activity, if it exists, must be over-balanced by cell maturation (and cell death). By the same token, amplification at the compartmental and intercompartmental level does not imply the existence of obligatory amplificative (as opposed to self-renewal) divisions at the single cell level. Observation of the behavior of recognizable precursor cells appears to suggest that these cells divide and mature "simultaneously". However, this does not imply a constitutive relationship. The classification by and ordering of compartments, though useful in documenting data and in modelling, is artificial and substantially underestimates the full heterogeneity of hemopoietic cells [52]. Cells representing a range of phenotypes are lumped into a single compartment. Thus, precursor cells may be undergoing a sequence of quick maturation steps, without division, intercalated by a smaller number of self-renewal divisions. If the time between divisions is relatively short as compared to the cell cycle time, the maturation events will appear indistinguishable from and associated with the cell cycle---especially as post-transcriptional processes could overlap each other and the cell cycle. The stochastic nature of cellular events is often misinterpreted in the literature to imply independence of external regulation. There is evidence for the stochastic nature of differentiation and self-renewal of stem cells [84, 61]. This can be explained theoretically on the grounds that the available information only partially characterizes individual cells and their environment. In fact, when considering cell populations the transition probabilities may be translated into cell flow rates and the description of the system in terms of cell numbers and average trends becomes fully deterministic. The question whether the transition probabilities associated with a given type of cell are subject to external influence and regulation is a major open issue. Finally, the description of cells as committed, pluripotent or stem cells is solely operational by the above definitions; no intrinsic or immutable cellular character can be associated with this description without making further assumptions. Assumptions A model based on the notion of dynamic regulation of self-renewal should explain the kinetic hierarchy

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exhibited by the hemopoietic system [43, 4] in terms of inter-cellular interactions rather than as resulting from a faithful followup of a differentiation program. The basic assumptions are [28, 30, 31] (1) The relative probabilities of maturation and selfrenewal of hemopoietic cells in any mitotic stage are subject to external regulation (namely, by other cells and factors). (2) The relative probabilities of maturation in all mitotic compartments in the bone marrow increase with the size of the mature cell compartment. This mode of feedback regulation was termed "balance of growth". (3) Beyond a certain limit, increased cell density has a negative effect on hemopoietic cell growth. This relatively non-specific feedback suppression mode is normally secondary to the balance of growth hut becomes potentially significant in hyperplasia (cell crowding). A mathematical representation of assumptions (1) and (2) (see below) suggests a hierarchy of cell maturation compartments, such that the earlier (more primitive or less differentiated) cells possess a lower sensitivity, or responsiveness, to external regulation. This hierarchy, in terms of a quantitative trait (responsiveness), rather than a genetically pre-determined, absolute restriction, is sufficient to account for the transitory nature of maturing cells (at steady state) in vivo. Similarly, the difficulty of demonstrating long-term selfrenewal in culture may reflect predominantly a consequence of culture conditions rather than an absolute inherent limitation. Assumption (2) does not specify the biologic mechanism(s) underlying the postulated feedback effects. Although there are candidates for the mature marrow cells that directly affect the ratio of self-renewal vs maturation in progenitors, an indirect effect would suffice: Mature (or maturing) cells could modulate signals exchanged between progenitors or delivered to them by stromal cells. For instance, it has been proposed that the stroma maintains proliferation and "protects" CFU-S and CFU-C from maturation [94, 93]; accumulating mature hemopoietic cells could interfere with this protection. Preliminary studies support this suggestion (D. Zipori, personal communication). Alternatively, mature cells could interfere with autocatalytic proliferative signals exchanged among physically adjacent stem cells and other progenitors. More direct feedback mechanisms have also been proposed. Experimental evidence implicates T cells, monocytes and granulocytes (for recent reviews see [72, 87]). The involvement of granulocytes, in particular, is pertinent to the discussion below of myelocytic leukemia. Sachs' MGI-2 can be produced by mature granulocytes and induce maturation in normal and leukemic myeloblasts [66]. An in-vioo role of mature granulocytes in controlling the growth of pluripotent cells by promoting differentiation was proposed [2]. An inhibitory effect of prostaglandin was suggested to be due to the induction of differentiation [5], and prostaglandins can be produced by mature marrow granulocytes. Recent data suggest that a single factor may influence an entire range of hemopoietic cells, from the

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pluripotent stem cell down to mature neutrophils and macrophages [40]. Mathematical representation [30, 31] This requires some additional assumptions and specifications. In each compartment the commitment of a cell either for self-renewal or for maturation has been postulated to require a preceding activation event (see Fig. 1). In the figure, S represents stimulation, or modulation, which is mediated in part in a feedback fashion by post-mitotic marrow cells (Z), and in part constitutively by intracellular or microenvironmental sources. The feedback component need not be the main driving force, but it provides the "steering". S acts with different interaction strengths, and probably via distinct biochemical pathways, on cells belonging to different compartments. It is convenient to think of S as representing a factor or a set of factors; however, as stressed earlier, the influence of Z cells could be exerted indirectly, i is the maturation compartment number (n compartments altogether), X denotes resting (Go) cells, Y denotes activated (G1) cells, f f and ~ indicate state transitions with and without division, respectively. In the model, maturation (Yi---~ Xi+ 1, or Y, --* Z) depends on S more than self-renewal (Y/--~ Xi) does (assumption 2). The transitions are stochastic at the single cell level, as proposed for stem cells [84], but are described in terms of (average) rates at the cell population level. The heterogeneity associated with different cell lineages is not considered explicitly in this schematic model.

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Finally, Fig. 1 indicates the possibility of incorporation in the model of a "source" of stem cells, supplying X1 cells at a fixed rate. Such a source might represent a fixed number of primitive cells dividing within "stem cell niches" [70]. A simple mathematical representation of the model facilitates a systematic analysis and derivation of several quantitative relationships (see Appendix) which display the implications of the hypothesis. These results can now be traced back directly to the underlying assumptions.

BIOLOGICAL

IMPLICATIONS

"Balance of growth" Consider a small population of regenerating cells. The observed sequence of events generally consists of (a) an extensive proliferation of precursors (with a limited degree of maturation), (b) the diminution of precursor multiplication when the total number has increased "sufficiently", and (c) terminal differentiation into functional non-dividing cells. The model explains precisely these events and the maintenance of the proper balance at equilibrium. At first, the process of proliferation and maturation is constitutive and auto-catalytic, with S mediating a positive feedback effect. When the number of terminally-differentiated cells approaches the "required" level, cells shift progressively toward maturation, and maturation in turn slows and finally stops further expansion. Accelerated maturation is also the signal for growth inhibition and by-products of growth provide the drive for maturation. In other schemes (e.g. [66]) the sequence is controlled in each individual cell by an independent genetic program.

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Fio 1. Balance of growth, xi, y~ and z denote resting, active and non-mitotic cells, respectively, s stands for stimulation. Double lines indicate amplification.

In each compartment there are two pathways: selfrenewal (xi--~ Yi ~ Xi) and maturation (xi ~ Yi - - ~ x i + 1). This structure is consistent with a body of experimental evidence. Mackey and Dormer ([46]; see also [80]) observed that erythroid and neutrophilic precursors may undergo transitions between compartments during an active phase of the cell cycle. Considering other types of cells in culture, Brooks et al. [6] suggested that cell cycles of proliferating mammalian cells possess two random transitions. These transitions resemble xi to yi and Yi into mitosis in Fig. 1. Models with distinct requirements for activation, proliferation, and differentiation of lymphocytes [33, 34] have recently found experimental support [39, 45, 47].

Adjustment to demand In the steady state the number of mitotic cells is proportional to the per-cell rate at which mature cells leave the marrow. This rate presumably depends upon peripheral demand for functional cells, as expressed by the levels of endotoxin, erythropoietin and thrombopoietin. One direct effect of these agents is indeed to promote (or in the case of erythropoietin, facilitate) terminal differentiation and release, shortening the transit time in the post-mitotic compartment [11]. Thus, increased demand--especially when prolonged or chronic--will trigger in mitotic cells a proliferative response extending beyond the actual target cells of the inducer. The mechanism by which lineage-specific external factor, e.g. erythropoietin, positively affects the production of cells belonging to that lineage is out of the scope of the present model. However, two possible ways are readily envisioned: (a) Pulling out mature cells of a given lineage may reduce the feedback differentiation pressure, locally, triggering a proliferative response in microenvironmental niches [85] populated mainly by cells of that lineage. Accordingly, only the peripheral signal is strictly lineage-specific; the apparent specificity of the tissue response is a consequence of its localized nature. (b) The factor may be involved directly

The stem cell concept revisited in the induction of commitment [24], thus channelling smaller or larger portions of the total influx into a particular lineage. The number of Z cells at steady state does not depend on the per-cell rate of release. With a chronically increased demand, for example, the larger outflow of mature cells can be maintained by an increased inflow from expanded progenitor compartments.

Compartmental hierarchy For a steady state to exist, the model requires that the first compartment, the one with no inflow, have the lowest per-cell maturation rate (this is equivalent to saying it is the least responsive to S). The steady state is reached when this first compartment's cells have equal probabilities of self-renewal and maturation; in the other compartments, the probability of maturation is then larger. As pointed out earlier, there is no contradiction between the capacity of differentiated progeny of stem cells for self-renewal and their transitory kinetics. The real question is of the configuration's stability: why don't these cells overrun the more primitive, slowly dividing pluripotent cells, replacing them as "stem cells"? Within the proposed balance of growth mode of regulation, it is their lower responsiveness to maturation pressures that endows the primitive cells with a growth advantage and stably couples them to the rest of the clone, in spite of their slower cycling rate. By inference, the hierarchy in which responsiveness to maturation pressures generally increases with maturation endows extra stability to the coupling of each compartment to the next one in the series. Nevertheless, the possibility of cell population uncoupling is visualized--under some intensive manipulations of the marrow, or in certain mutant animal strains, or in leukemia.

Defective hemopoiesis and cell population uncoupling It has been shown elsewhere ([30]; see also Appendix) that in the presence of certain growth-suppressive effects (other than maturation pressures) a stable configuration that does not include the most primitive cells is feasible. If the first compartment is inactivated, missing or suppressed, the feedback pressure will be readjusted so that the earliest available cell subpopulation will be strictly self-renewing. Although the hemopoietic system does not absolutely depend on specific primitive cells for self-renewal, as is commonly asserted, its performance is affected by the kinetic characteristics of the cells in the earliest compartment. The number of mature functional cells (for a given release rate) depends inversely on the maturationrate constant of these early cells: the more sensitive the stem cells are to the feedback, the less the system is allowed to expand. Irradiation or chemical agents may annihilate all or most primary stem cells [70]. The truncated clone supports a smaller number of terminally-differentiated cells. The turnover of these cells must then increase to meet peripheral demand, but the transit time through the mature compartment cannot be shortened beyond a certain limit. Progressive and selective loss of the least

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differentiated cells upon repeated transplantations to irradiated recipients will lead eventually to death [4, 37]. In addition the shift toward more differentiated and more sensitive progenitors is assayed as "reduced quality" or "ageing" in the spleen colony method [90, 64]. The macrocytic anemia seen in W / W v mice is caused by a genetically-determined defect in the stem cell population [53]. Virtually no CFU-S can be detected in these mice which nevertheless produce a near-optimal supply of erythrocytes and a normal supply of other blood elements. A conventional interpretation is that hemopoietic stem cells must exist but, for some reason, they do not form spleen colonies. The present model suggests that W / W v mice represent an example of hemopoiesis maintained by cells equivalent in their kinetic characteristics to relatively mature pluripotent cells or even normal committed progenitors. Typical stem cells are really absent. The inverse relationship between the sensitivity of the operating stem cells and the size of the mature compartment explains the tendency to anemia, largely but not entirely compensated by the readjustment of proliferation at all stages to the demand. W / W ~mice can be cured by the successful engrafment of a single normal stem cell [49]. Normal donor hemopoietic cells become competitively established in nonirradiated recipients. Obviously, the normal clone possesses a growth advantage over the defective clone(s). In the present model this advantage is attributed to the relative primitiveness--the higher resistance to maturation pressures--of the engrafted stem cells rather than a faster growth rate: as the normal engrafted clone approaches steady state, all other cells---including the host's original stem cells--are forced to become transitory. If the hypoplastic tissue of an anemic animal is seeded by donor cells that are not superior to the recipient's stem cells in the hierarchy of feedback resistance, but slightly inferior, a transient partial replacement of the recipient's cells can occur. The feedback pressure builds up gradually as the engrafted colonies expand, and only when these merge with the host's colonies can the combined pressure lead to the "out-washing" of the engrafted clonogenic cells. This scenario is consistent with the observations of Fleischman et al. [19] who engrafted fetal liver cells in a series of anemic fetal recipients and noticed high incidence of "transient replacement". It explains the late manifestation of dominance by the superior clones as well as the sequential expression and loss of self-renewal activity by some totipotent hemopoietic stem cells. In more recent experiments [54] some mice displayed a regular and complementary rise and fall in proportions of cells of different phenotypes, supporting the hypothesis of clonal succession proposed by Kay [41]. Analysis of this phenomenon is beyond the scope of this presentation but a few comments are due: (a) the present model is essentially a single clone model. (b) More than one clone may co-exist due to spacio-structural constraints that limit cross-reactivity among the clones. (c) Regular dynamic fluctuations can arise in such a system of dynamically balanced, inter-connected populations, leading to "complementary rise and fall" [28].

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Leukemia The idea that cells at various stages of maturation are not devoid of self-renewal potential but that this potential is tightly controlled by other cells and factors suggests a different conception of the origin of leukemia: a modification of normal cellular characteristics rather than immortalization of intrinsically short-lived cells. In particular, chronic myelocytic leukemia (CML) can arise and evolve into the blastic crisis from a progressive decline in a single kinetic characteristic-responsivess to maturation signals [30-32]. In a first, crude approximation for the pathogenesis of the disease, the blastic crisis results simply from the effects of cell crowding: breakdown of the balance of growth configuration through uncoupling of cell populations. If responsiveness to feedback is reduced in some hemopoietic clone, the clone will expand, since a larger number of mature cells is required to maintain a steady state. Normal clones, including normal stem cells, will be suppressed primarily as a result of over differentiation induced by the excessive number of these mature cells in the leukemic clone. In addition, since the carrying capacity of the marrow is limited, cell crowding arises. Keeping in mind the potential for selfrenewal in several compartments, intra-clonal competition among cells at different stages of maturation is predicted. As inductivity is further reduced, the competition for "space" results in the suppression of mature cells in the leukemic clone and concomitantly the emergence of a subpopulation of"blast" cells that grows faster than any others under these conditions. When blast cells are separated from their precursors and progeny as a result of selection, they exhibit an ability to selfrenew that is strongly restricted in the normal steady state. The whole process could take place in one of the small "regulatory volumes" into which the marrow is divided [49]. Subsequently, the growing uncoupled malignant population may spread out into other parts of the hemopoietic tissue. In this case the generation of acute leukemia would not be preceded by a detectable phase of (macroscopic) chronic leukemia. The hypothesis as formulated above departs in two significant (and inter-related) ways from most currently held hypotheses: (i) it requires only one type of quantitative change in the function of individual cells to explain both CML and the blastic crisis; and (ii) it attributes a more dynamic role to cell crowding. Crowding is usually envisioned at most as a factor in the growth-limiting microenvironment, imposing competition among normal and transformed clones and leading to successive selection of more competitive newly emerging clones. In the present approach cell crowding interferes with the stability of the dynamic coupling between cell compartments and facilitates intra-clonal competition. This certainly is not the whole story. No biological mechanism is offered by the model as to what causes the progressive decline in responsiveness. Several other phenotypes are also known to change during the course of evolution of the disease. In the next section an

elaboration of the dynamic model is proposed in which phenotypic plasticity is an important ingredient.

Response to perturbations A consequence of the requirement of a separately regulated cell activation step prior to self-renewal or maturation is that the relative probability of self-renewal does not depend on the activation rate. This explains some empirical observations and leads to predictions. The hierarchy of increasing responsiveness with maturation to maturation signals extends also to activation: the more differentiated hemopoietic progenitors and precursors are likely to be in an active phase of division or maturation and not in Go. The relative contributions to population self-renewal of the earlier and the later cell compartments are predicted to undergo dramatic changes during transient, non-equilibrium phases when the feedback pressure is reduced, e.g. following partial depletion of the marrow or a sudden increase in peripheral demand. The regenerative activity then can be shown to be greater in the more active compartments [30]. This prediction is contrary to standard dogma. Thus, enhanced proliferation is not generated necessarily by shortening large G0-times but rather by diverting small Go cells from maturation into cycling. The response to a sudden change of demand should propagate from the end of the line backwards, not in the direction of lineage progression as is commonly asserted [43]. Enhanced activation rate alone will simultaneously increase the rates of division and maturation, not changing the ratio between them and therefore not affecting hemopoiesis substantially. Indeed, stimulation of D N A synthesis by stem cells (CFU-S) in culture [14] or in vivo [2] in some cases did not lead to a parallel increase in the rates Of mature cell production. Inducers that affect activation rates but do not induce differentiation directly would have a "mitogenic effect" [23]. If the decrease in the maturation rate constant in CML is not accompanied with a parallel decrease in the activation rate constant, the predicted result is increased positive feedback at the level of activation, so that more progenitors will be in cycle in CML as compared to their normal counterparts. This, however, is a cell population effect, and the ability of individual leukemic progenitors to proliferate in a culture assay would not be elevated. This prediction is compatible with the observed behavior [87].

Cyclic hemopoiesis A simple mathematical representation of the present model is capable of manifesting periodic solutions for a range of the parameters within the chronic stage of CML [30]. The emergence of periodic ("cyclic") hemopoiesis in CML is documented and is probably quite common [56], lending support to the concept of feedback regulation [57]. Interestingly, periodic solutions can appear also when under-production is simulated by increased maturation rate constants [30], corresponding, perhaps, to the observed cases of cyclic neutropenia.

The stem cell concept revisited ADAPTIVE

DIFFERENTIATION LEUKEMIA

AND

Phenotypic adaptability can be defined as the capacity of somatic cells to change their patterns of gene expression, in an inheritable manner, in response to changes in their microenvironment. Extensive evidence in favor of this concept and its significance in normal development and in cancer has recently been reviewed by Rubin [65]. "Tissue interactions are required even into adult life to maintain the identity of cells which can be changed in a highly specific way to a closely related but easily distinguished cell type by switching the interacting mesenchyme" [65]. The orderly, predictable pattern in which hemopoietic cells differentiate and mature in the marrow or form colonies in a culture plate is in no contradiction to the concept of adaptability. Like the sequence of developmental events in embryogenesis, it owes its regularity to a network of causal relations and not just to the unfolding of a genetic program [79].

Regulation of commitment Phenotypic adaptability is consistent with the proposition that hemopoietic cell differentiation and maturation is guided to some extent by interactions with the microenvironment [85, 82, 24], including feedback interactions within the hemopoietic cell population. Models based on this proposition are often referred to as "instructive" but this is somewhat misleading: the cell itself participates in selecting the "instructions" for changing its pattern of gene expression (gene activation, inactivation or quantitative modulation of the level of expression) in accordance with its state at that time. Thus, a cell that is strongly biased in favor of a given signal, e.g. through the expression of the appropriate distribution of receptors, is effectively "committed" for the consequent change even if the environment is just "permissive" (rather than "instructive") with respect to this signal. This commitment, however, is not necessarily absolute and may be non-operational under different conditions, in a different microenvironment. It is also limited to a section of the differentiation pathway. Transitory hemopoietic cells are effectively committed. They are undergoing a sequence of changes which reflect both the structure of the microenvironment (hemopoietic tissue, inducer factors, and stroma) and their own dynamic bias. They can hardly be expected to manifest their greater inherent potential for phenotypic adaptation under these circumstances. In contrast, selfrenewing cells are sufficiently insensitive and unbiased in relation to their environment [82] to allow for the continuous maintenance of their genomes in multiple copies. (Actually, as was discussed, the feedback mechanism adjusts "the environment" to the sensitivity of these cells.) There is sufficient opportunity for the competing inductive forces to pick up (in a probabilistic fashion) different copies and induce inheritable changes, requiring perhaps repeated interactions. It may be proposed that pluripotency is dynamically linked to self-renewal. This is a two-way linkage: The same conditions that facilitate one also permit the other to be

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displayed. Moreover, some normal progenitor cells may be effectively committed, but potentially pluripotent.

Phenotypic plasticity A corollary of this conclusion is that cells which normally are transitory or inactive in the physiological context will be able to express a variety of phenotypes when allowed to self-renew for extended periods of time. There is ample documentation of the excessive phenotypic plasticity manifested by tumor cells and by long-term cell lines in culture (recently reviewed and discussed by Rubin [65]). Some examples involving hemopoietic cells are particularly pertinent. Lineage infidelity in leukemic blasts [52] has already been mentioned. Similar infidelity or "switch" across lineage phenotypes was reported in Friend Leukemia cells, which could be induced to express both erythropoietic and granulopoietic lineage markers [18]; and in a human myeloid leukemia cell line (K562) which apparently "underwent red shift" [36, 50]. Cytotoxic T-cell clones grown in culture using IL-2 regularly undergo reproducible phenotypic shifts, acquiring some of the characteristics of natural killer cells [7, 74]. Both normal and transformed T- and B-cell clones in culture undergo quantitative and qualitative variations in the expression of cell-surface antigens (e.g. [91, 92, 13]). The issue is to what extent this observed plasticity reflects normal potential for phenotypic adaptability-under abnormal conditions---as suggested here. The prevailing view at present is that phenotypic shifts and lineage infidelity are caused by genetic aberrations or mutations, i.e. abnormal changes in the genetic information units. There is no doubt that such aberrations do occur at high frequency in established tumors and cell lines. Three general explanations, not mutually exclusive, have been offerred: (a) An initial alteration of cellular DNA renders the cell intrinsically unstable [1]. (b) The genomic mechanisms of D N A repair cannot cope with errors generated throughout repeated division--in cells designed to be transitory or resting most of the time [43]. (c) Substantially altered microenvironment induces chromosomal changes in the cells, either actively or by interfering with the physiologic constraints that normally inhibit such changes. This has been termed "abnormal adaptation" [65]. On the other hand, there are many examples in which unusual phenotypic switches appear to result not only from mutations or aberrations but from inheritable epigenetic effects [69, 65, 42, 20, 21, 17, 77]. A role for D N A methylation in stabilizing epigenetically induced changes in gene expression has been proposed [62, 8]. The quantitative nature, flexibility and partial reversibility of D N A methylation patterns are compatible with the concept of dynamic regulation, whereby both normal differentiation and cellular evolution in tumors and cell lines are seen as adaptive processes.

Extended model of CML and the blastic crisis The major phenotype of interest here is that of selfrenewal. In the model presented above, a set of transition rate parameters define the growth characteristics

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of each cell. A fixed set of constants have been associated with each compartment. This implies a rigid linkage between the set of growth characteristics on one hand and a full set of other markers on the other: the first set can change only in conjunction with the second. The concept of adaptive differentiation, or even the more conservative programmatic concept [52], suggests more flexibility. When populations identified by their capacity to form colonies under certain conditions are examined by other means, striking heterogeneities are uncovered [51]. It is assumed, in particular, that the maturation rate constant, or "inductivity", is variable even within a conventionally defined compartment. In addition to the feedback influence mediated by Z cells (Fig. 1) which acts to increase the inductivity by enhancing the expression of a set of genes--a process called maturation-there is an opposite force, of constitutive and/ or extra-cellular origin, which tends to reduce the inductivity and equivalently increase the self-renewal capacity [31, 32]. Three lines of evidence indicate the plausibility of the new assumption: First, cell lines and clones maintained in culture tend gradually to lose their capacity to differentiate in response to various inducers [65, 7, 74]. Second, some growth factors can regulate the expression and affinity of their own receptors, perhaps by affecting the ratio between m R N A transcripts of the receptor's gene [76, 63]. Third, several observations indicate that the capacity of hemopoietic cells for self-renewal can be changed in a quantitative way by external influences [78, 9]. In particular, there is indirect evidence for a role for D N A methylation in the regulation of blast cell self-renewal [58]. It is further assumed that the rate of change in inductivity within compartments is slow relative to the transit rates through compartments. Therefore, the magnitude of the decline is normally limited. The rate of decline could depend on the density of cells, as suggested by evidence on autocatalytic growth effects [60, 23, 24]. Note, that while the self-renewing stem cells are in equilibrium with respect to their environment and therefore phenotypically adapted to it, transitory cells are not: a just-differentiated cell is like a newcomer, or new-born, in this environment. The assumption implies that if normal committed progenitors could somehow be freed from external maturation pressures, not only would they switch into enhanced self-renewal activity, but their growth characteristics will consequently change towards a higher degree of maturation "arrest" and greater capacity for self-renewal. The force that stimulates self-renewal would then be driving a positive feedback process. The transformation toward cellular "immortality" under these conditions is conceived of as independent initially of D N A aberrations, although the latter could eventually contribute to it. Normally, however, the negative balance-of-growth feedback mechanisms ensure the transitory nature of the transformable cells, and this transitory nature in turn makes it impossible for some changes in inductivity to accumulate significantly. The key question at this point is what kind of perturbation of the normal tissue can neutralize the nega-

tive feedback mechanism and start a positive feedback loop leading to "immortalization". Mathematical representation of the extended set of assumptions has been used to demonstrate a number of possible scenarios [31, 32]. Two of these will be briefly outlined. In both cases it is cell crowding that leads to destabilization of the normal functional balance among cell subpopulations, resulting in a selection-adaptation process terminating in the blastic crisis. Both scenarios begin with an initial, heritable event in an early hemopoietic cell, generating a clone with a reduced inductivity. This leads to CML and, depending on the level of hyperplasia, to a distorted population balance in the dominant leukemic clone in favor of immature cells. The modified cellular environment feeds back onto the individual members of the clone, inducing further decline in inductivity and consequently more crowding and distortion (selection). This cascade of dynamic changes leads to blast cell dominance. Scenario I: All compartments remain stably intercoupled (in CML) unless a threshold cell crowding is exceeded. Beyond this point any further decrease in inductivity will lead to a decrease, rather than an increase, in the number of Z-cells. This is the threshold beyond which the negative feedback fails and the inherent tendency for adaptive increase in self-renewal capacity causes a progressive change in inductivity. This progressive change drives the selection process that culminates in blastic crisis as described in the previous section. The process has been analysed and simulated numerically [311. Scenario II: Moderate cell crowding leads to the uncoupling of a population of primitive stem cells. These cells are suppressed because of their relative slow rate of replication. Self-renewal is thereafter assumed by more differentiated progenitors, which gradually undergo an adaptive overshooting increase in selfrenewal capacity; this leads to more crowding and uncoupling. Eventually the threshold of Scenario I is exceeded. As mentioned earlier, acute myelocytic leukemia may be essentially a localized version of a blastic crisis. The model also suggests a mechanism by which acute leukemia may arise from a pre-leukemic state of hypoplasia, rather than from the more prevalent hyperplasia. Such a hypoplastic state could result from a number of defects, e.g. inefficient supply of growth factor. The steady state numbers of all hemopoietic cells would tend to be small, and in particular the differentiated cells will generate only a weak feedback. Under this weaklyregulated condition (weak intercompartment coupling) blast cells could gain a growth advantage. Finally, we return briefly to the questions of heterogeneity, lineage infidelity and phenotype switches in the leukemic blast cell population (associated mainly with AML and lymphocytic leukemias [26, 52]. These questions are secondary to the issue of self-renewal capacity within the purview of the paper). Phenotypic heterogeneity reflects the various constitutive differentiative pressures in the environment on the partially responsive blast cells, possibly in combination with peripheral demand pressures. Lineage infidelity and

The stem cell concept revisited alterations in phenotypes during leukemia progression and relapse do not necessarily indicate the original stemcell nature of these blast cells. Even if the clonogenic cells in the blast population are the direct counterparts of normal committed progenitors, as proposed here, they may manifest latent (though limited) pluripotency that is normally unexpressed. With the change of their transit cell character into self-renewal--the expression of a normal, latent capacity--these cells may have gained enhanced opportunity to respond and adapt phenotypically to a range of differentiative pressures. Inter-lineage shifts however could also be explained by selection among the progeny of residual cells from the original (transformed) pluripotent stem cell population [26].

DISCUSSION Several elements of the present approach are testable. As pointed out elsewhere [30], models describing biological systems of interacting cells can seldom be expected to predict numbers or to provide unequivocal critical tests, unlike some theories in physics. A model is valuable here if it provides new interpretations and insights and if it can motivate experimental strategies. The present approach is an attempt to construct a theoretical framework within which to image quantitative descriptions of cell behavior at the level of the intact functioning cells, as recommended by Rubin [65]. The problem is that full quantitative descriptions are lacking as yet. Perhaps the most direct way to invalidate the basic conjecture of the present approach would be to identify the "self-renewal genes" (among cellular oncogenes? See [52]) and to show that they are irreversibly inactivated in committed progenitor cells. Some published hypotheses do not adopt a purelygenetic interpretation of "stemness". The "positional information" theory of Wolpert [89] as well as Schofield's "stem cell niches" hypothesis [70] suggested that a cell may be a stem cell only by virtue of its position. In the present scheme, primitive stem cells are dynamically protected from over-maturation--as well as from aberrant growth--by their very "primitiveness" and by the feedback-type coupling to differentiated cells. The hypothetical existence of "protective niches" is neither the primary cause for continuous self-renewal nor an obligatory requirement. A fixed number of primitive dividing cells confined to such niches would appear as a fixed source in the model of Fig. 1. As such their potential effect on the dynamics of the system is insignificant, except perhaps for one aspect: by forcing the cells in the next compartment to be transitory they could reduce the risk of phenotypic destabilization in these more active cells and the development of leukemia. The model reinterprets several in-oioo observations and predicts certain kinetic patterns to occur in response to experimental manipulations of the hemopoietic system. Critical evaluation of the basic assumptions on the basis of such predictions must await a better characterization of cell subpopulations, including early

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progenitors, and of the factors which affect branching of the different lineages. Some kinetic characteristics are largely model-independent. For instance, several different theoretical models describe cyclic hemopoiesis. As mentioned earlier, this model predicts the switching on and off of periodic patterns, with the correct frequency ranges to match those observed in cyclic neutropenia and in CML, by varying a single kinetic parameter [30]. Although consistent with the notion of feedback regulation, this behavior was shown to be a characteristic of a general class of feedback systems with respect to a variable time delay [10]. The development of cell colonies in culture is a highly non-equilibrium process and any potential feedback effects are likely to be masked by external factors. In addition, cell densities might be too low for effective cell-cell interactions [24]. Nevertheless, some observations made recently by Ogawa et al. are intriguing [59, 81]. They observed that colonies with many pluripotent cells may have unexpressed differentiation capabilities. For instance, one colony of 3000 cells, which contained 13.5% potentially pluripotent ceils and no erythroid cells, revealed erythroid lineage upon replating [81]. Such observations challenge the strictly stochastic model of differentiation [61] in which qualitative variability in the composition of mixed colonies, developed from single cells under identical culture conditions, can arise only from early events in the colony development. In the above example, the loss of the capability to differentiate into the erythroid lineage in that subpopulation which gave rise to the mature cells must have occurred in one of the earliest cycles of division. Otherwise, there would be too many erythropotent cells, and the probability of all of them losing their potency would be negligible. If it can be shown consistently that the variability in colonies arising from an assembly of cells that includes initially many pluripotent cells is comparable to the variability in mixed colonies originating in a single cell, this would strongly suggest the involvement of cell population effects in commitment. In particular it would suggest that selective or inductive feedback influences may be generated late in colony development, when large numbers of mature cells are exponentially generated. It would also be of interest to reassess the fate of erythroid progenitors developing in oivo in hyper-transfused animals. When these cells reach the stage of erythropoietin dependence, are they all subject to maturation arrest and/or death, or do some of these cells manifest under these conditions a capacity to be diverted into different lineages? Definite answers would require characterization of stable lineage markers on erythroid progenitors. The "Dexter culture" [14] is potentially a tool to demonstrate the hypothesized effects of cell crowding and other microenvironmental changes on the interpopulation balance and growth characteristics of the hemopoietic cells. Such studies would optimally consist of a combination of multivariate flow cytometry and functional assays. Recently, infection of long-term marrow cultures with a virus, src (MoMuLV), led to a dramatic change in the relative numbers of stem cells,

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progenitor cells and mature cells [3]. Furthermore, the stem cells from the src-infected cultures, even when transferred to src-free cultures, showed a remarkably increased capacity for self-renewal and the self-renewal/ differentiation ratio could be modified by culture conditions [78]. Most interestingly, these stem cells themselves did not appear to be infected by the virus--only the stromal cells were infected. These findings not only support assumption 1 of the present model but are also compatible with the interpretation of inheritable adaptive change that apparently affected the growth characteristics of early hemopoietic cells subject to the src-modified microenvironment. It has been suggested that chemotherapy should focus on the "clonogenic" leukemic cells [44, 26]. A corollary of the present interpretation is that the "clonogenic" cells in the blastic phase are not necessarily selected by any particular genetic alteration [26]; they emerge as a dominant subpopulation in a systemic process. This would make these cells a rather elusive, "moving target". In the model, the differentiative capacity of blastic crisis myeloblasts is reduced but not necessarily blocked. This seems to be supported by observations on the onset of the blastic crisis in a guinea pig model of transplantable CML [16], which suggest that blastic crisis cells transplanted into healthy recipients undergo differentiation in the host. As the model would predict, a CML-like phase which develops in the host precedes the re-emergence of blasts as the dominant population. It would be of great interest to study whether chronic leukemia in this animal model can be cured at an early stage by manipulations (chemical and biological response modifications) aimed at enhancing the feedback pressure exerted by the normal clones. In the human, this approach would have a better chance of success in A M L patients rather than in CML, according to the respective scenarios offered previously for the pathogenesis of these diseases: in CML depletion of normal stem cells is affected mainly by excessive differentiation pressures, while in A M L they are suppressed through a direct competition "for space" against colonizing blast cells; the first mode of suppression is presumably more exhaustive. "Uncoupling" or asynchrony of the intra-cellular controls which coordinate division and differentiation in the normal cell were proposed by Greaves [25] and by Sachs [66] to be the "lesion" at the root of the leukemia. Sachs defines a hierarchy of distinct cellular changes that give rise to different phases of malignancy (see also [73]). In contrast, the present model associates the development of imbalance between proliferation and differentiation with uncoupling of cell sub-populations, and a progressive, quantitative maturation arrest with changes in the tissue. The present dynamic approach underscores the limitations of a clonal analysis of tumors in assessing early events in carcinogenesis. A multiclonal tumor is logically inconsistent with the premise that a rare genetic event was responsible for the formation of the tumor [86]. However, the inverse statement is not necessarily correct: clonal dominance can ensue late in the devel-

opment of an originally multiclonal tumorogenic cell population following a selection process. Lajtha [44] emphasized that the target cell for leukemogenic transformation is most likely a self-renewing stem cell, in which genetic errors can accumulate. The present analysis suggests that (a) the most significant cellular changes in the early evolution of malignancy do not have to be rare genetic events, e.g. D N A mutations or chromosomal rearrangements, and (b) such changes can buildup even in a transitory cell population through their accumulative impact on the global structure of the tissue. To discriminate at any phase of the leukemogenic process between the two possibilities---rare genetic events or adaptive variation involving many cells simultaneously--a strategy based on analysis of polymorphisms must be used repeatedly during the process: for instance, introducing a traceable set of well-defined polymorphic changes, unrelated to growth characteristics, into the guinea pig transplantable cell line [16] could facilitate assessment of the nature of the cellular events involved in the transformation from the chronic to the blastic phase in the recipient animals.

CONCLUSION With the acquisition of a better knowledge of the multitude of ways in which the pattern of gene expression can be affected and controlled, the rigid classification of hemopoietic cells into qualitatively distinct compartments may be realized to be not only artificial but also misleading. In fact--starting in the fertilized egg--cells are on a continuum of differentiation and commitment to an ever-narrowing spectrum of functions. Hemopoietic stem cells and progenitors alike are embedded in this continuum. Progenitors are certainly not equivalent with respect to "stemness", i.e. the capacity for self-renewal. The unifying concept is the association of a given inductivity with each mitotic cell, so that its self-renewal probability can vary within a certain range. When this probability is equal to or greater than 50% then the cell behaves as a "'stem cell". The larger inductivity renders the more differentiated hemopoietic cells susceptible to constitutive and external maturation pressures so that they cannot act as stem cells under most conditions. They may do so in vivo temporarily, or under special conditions in culture or if their inductivity is reduced by some "transformation". The hierarchy of inductivities embeds in it the functional heterogeneity of stem cells that led to the notion of "age structure" [4]. Primitive stem cells are important for the optimal functioning of the system not because they are singularly characterized by immortality, but because they (a) maintain large populations; (b) force inherently faster populations to be transitory, establishing a flexible hierarchy of amplification steps; (c) impose stability against maturation pressures: and (d) associate self-renewal activity with long G0-times, as required for "genetic housekeeping" of stem cells [43], rendering them karyotypically and phenotypically stable. Breakdown of stability of the normal configuration can follow from a

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Physical separation of hemopoietic stem cells differing in their capacity for self-renewal. J. exp. Med. 130, 91. Zaguri D., Bernard J., Morgan D. A., Fouchard M. & Feldman M. (1983) Phenotypic diversity within clones of human normal T cells. Int. J. Cancer 31,705. Zaguri D., Morgan D., Lenoir G., Fouchard M. & Feldman M. (1983) Human normal CTL clones: generation and properties. Int. J. Cancer 31,427. Zipori D. (1981) Cell interactions in the bone marrow microenvironment: role of endogenous colony-stimulating activity. Supermol. Struct. Cell Biochem. 17, 347. Zipori D. & Sasson T. (1980) Adherent cells from mouse bone marrow inhibit the formation of colony stimulating factor (CSF) induced myeloid colonies. Expl. Hemat. 8, 816.

APPENDIX: M A T H E M A T I C A L M O D E L Since the purpose of this mathematical model is to illustrate and support theoretical arguments in evaluating possible modes of regulation, and not to mimic the natural phenomena in great detail, a highly simplified description is justified. Hence the choice of linear and bilinear functions of the variables, x, y, z and s now represent relative densities of cells and inducers in the equations below. For simplicity, s(t) is replaced by di+ aiz(t) in the terms representing cell activation rates from Go to G1 (superposition of constitutive and feedback parts) and by a'z(t) for maturation rates (neglecting constitutive maturation), d is the inflow rate of cells from the (optional) stem cell source into the first compartment. The per-cell rate of transition from G~ to the S-phase (towards mitosis) is assumed to be the same constant b for all compartments, c is the per-cell rate of release of mature cells from the marrow into the circulation, presumably geared to peripheral demand for functional cells. The factor 2 corresponds 1o doubling in the number of resting cells by mitosis, and p~are (constitutive) amplification factors associated with the maturation steps. It is not clear whether constitutive replication really occurs, but the factors p~ may be required anyway if the heterogeneity of the clone is underestimated, to account (partially) for compartments which are not included in the model explicitly. Nonspecific density-dependent suppression is represented by a positive, monotonically increasing per-cell rate function, f(N), of the total cell density N, where N = ~'~ (xi + y i ) + z. i

In numerical simulations, this function is chosen as f ( N ) = (N/Iqr) p, where ~/is a measure of the capacity of the bone marrow and p reflects the "steepness" of its resistance to increasing cellularity (p > 1). Finally, r and r[ are time delays associated with cell division and with cell maturation, respectively. (Notations: t is time; ~(t) is dx/dt, etc; and dt,~= I for i = 1, 6,1 = 0 for i > 1,). The equations are: Yci(t) = -(d, + aiz(t))xi(t) + 2byi(t - r) + 6,1d

+ (1 - 6n)pi la; l Z ( t - r~ a)Yi l ( t - 7:; ~) - f ( N ) x i ( t ) ,

(la)

)~i(t) = (a, + a,z(t))xi(t) - a[z(t)y,(t) - by,(t) - f ( N ) y , ( t ) ,

(lb)

2(0 = p,a',z(t - ~'~)y,(t - r') - cz(t) -f(N)z(t);i=

l . . . . n.

(lc)

950

ZvI GROSSMAN

It is assumed that under normal conditions f ( N ) is negligible. There are several constant steady state solutions characterized by different numbers of non-varnishing compartments. These can be classified by a n u m b e r j denoting the first non-varnishing population, such that x~0 = y~ = 0 for 1 ~< i < j. For j = 1 (a fully populated clone) we have, if f(N) = 0 and d = 0: z,, = b/al, y~,, = c/p~a~,, y, 1,~,= [(a; - al)/p~ la; I]Y,,/,

(2)

xio = [b(a; + al)/(ba~ + a'16,)]yio. We note a number of significant consequences. (a)x~0 and Y~0 are proportional to c, for any i. Thus, the system responds to increased peripheral demand by proliferation. (b) The number of mature ceils depends on the maturation-rate constant of the earliest compartment in the series. (c) Feasibility requires Y,- 1.0 > 0, and consequently a[ > al for i > 1. This is consistent with the observed hierarchy within hemopoietic clones [43]: The general observation is that both a[ and a~ of progenitors increase with i (shorter Go times, faster maturation). (d) If a[ > a~ > al, for i > 2, then an uncoupled steady state of the type j = 2 is feasible, namely, one in which xl0 = Yl0 = 0, and

xa~, Y,0, z0 are positive for i > 1. It has been shown [30] that this uncoupled state is unstable if f = 0. Thus, within the balance of growth mode of regulation, it is their lower responsiveness that stably couples the primitive progenitors to the rest of the clone, in spite of their slower cycling rate. By inference, the hierarchy in which a ' l > a[ for all (or most) values of/, i > 1, endows extra stability to the coupling of each compartment to the next one in the series (compartments are of course coupled to the preceding ones through intra-lineage relationships). It has also been shown [30] that if f > 0, the state of uncoupling can be locally stable for a certain range of parameters. Certain systemic or parametric perturbations can lead to uncoupling and reorganization of the hemopoietic population. (e) From Equation (lb), the probabilities of maturation and regenerative replication for an activated cell are, respectively, P,(M) = a~z/(a~z + b) and e i ( R ) = b/(a~z + b). At steady state, P I ( M ) = P j ( R ) = 0 . 5 , and P ~ ( M ) > P i ( R ) for i > 1 . Hence, the most primitive cells are dynamically adjusted to be self-renewing while cells in more differentiated compartments are forced to be transitory. (f) The relative probability of selfrenewal does not depend on the activation rate.