Differentiation of leukemic blasts is not completely blocked in acute myeloid leukemia

Significance

A long-held tenet in acute myeloid leukemia (AML) is that multiple genetic events produce a block in the differentiation of primitive myeloblast cells. Using mathematical modeling, we show that, rather than a complete block, even a slight reduction or skewing of the differentiation rate in these cells produces an AML phenotype. We experimentally validated that mature myeloid cells have the same genotype as leukemic blasts, demonstrating that there is not a complete block in differentiation. While targeted differentiation therapies have shown success in isolated subtypes of AML, our findings suggest that maximizing the potential impact of these agents will require discernment of the specific actionable mechanisms underlying reduced or skewed differentiation in genetic subtypes of the disease.

Abstract

Hematopoiesis, the formation of blood cells, involves the hierarchical differentiation of immature blast cells into mature, functional cell types and lineages of the immune system. Hematopoietic stem cells precisely regulate self-renewal versus differentiation to balance the production of blood cells and maintenance of the stem cell pool. The canonical view of acute myeloid leukemia (AML) is that it results from a combination of molecular events in a hematopoietic stem cell that block differentiation and drive proliferation. These events result in the accumulation of primitive hematopoietic blast cells in the blood and bone marrow. We used mathematical modeling to determine the impact of varying differentiation rates on myeloblastic accumulation. Our model shows that, instead of the commonly held belief that AML results from a complete block of differentiation of the hematopoietic stem cell, even a slight skewing of the fraction of cells that differentiate would produce an accumulation of blasts. We confirmed this model by interphase fluorescent in situ hybridization (FISH) and sequencing of purified cell populations from patients with AML, which showed that different leukemia-causing molecular abnormalities typically thought to block differentiation were consistently present in mature myeloid cells such as neutrophils and monocytes at similar levels to those in immature myeloid cells. These findings suggest reduced or skewed, rather than blocked, differentiation is responsible for the development of AML. Approaches that restore normal regulation of hematopoiesis could be effective treatment strategies.

In healthy individuals, most mature blood cells live only days to months. These cells must be replaced, on average, at the same rate at which they die, which requires the production of millions of new blood cells per second (1, 2). The blood system has a small population of hematopoietic stem cells (HSCs) that have the unique property of self-renewal—they reproduce enough to maintain their population—and the ability to differentiate to generate all blood cell lineages. During differentiation, these HSCs yield blood precursors devoted to unilineage differentiation and production of mature blood cells, including red blood cells, megakaryocytes, myeloid cells (monocytes/macrophages and granulocytes including neutrophils), and lymphocytes that do not divide (Fig. 1) (3–5). This process is tightly regulated to maintain the proper balance of cell types, and the system is able to respond to external perturbations like infection or blood loss.

Fig. 1.

Conventionally hypothesized phylogeny of blood cell types in hematopoiesis. Adapted by permission of ref. 22, Springer Nature: Springer Science+Business Media, LLC, copyright 2009. Recent evidence supports a continuous differentiation process for hematopoiesis (21). Our model is independent of the particulars of the tree. Solid lines represent well-accepted differentiation paths. The dotted line represents a suspected path. LT-HSC is long-term hematopoietic stem cell. ST-HSC is short-term hematopoietic stem cell. MPP is multipotent progenitor. CMP is common myeloid progenitor. MEP is megakaryocyte-erythrocyte progenitor. GMP is granulocyte-macrophage progenitor. CLP is common lymphoid progenitor. DN is double negative thymocyte. DP is double positive thymocyte. NK is natural killer cell.

Cells at intermediate hematopoietic hierarchy levels have many functions, one of which is to act as an amplifier: The rate at which they create more-differentiated cells is larger than the rate at which they are created by less differentiated cells. Because there are very few stem cells that divide infrequently, while millions of new mature blood cells need to be created per second, the aggregate ratio of creation rates between long-term stem cells and mature cells is roughly 10⁹ (Datasets S1 and S2).

In the context of hematologic malignancies, this delicate balance of type and abundance of blood cells is dysregulated. The long-held dogma in acute myeloid leukemia (AML) is that multiple genetic events produce a differentiation block phenotype at the myeloblast stage, coupled with increased proliferation of these cells (6, 7). This putative loss in the normal ability of hematopoietic cells to differentiate is common to many genetic subtypes of AML, often due to recurrent chromosomal translocations and mutations in AML that affect transcription factors involved in regulation of myeloid differentiation. Further, the aberrant expression of these transcription factors interferes with differentiation events and plays a role in AML pathogenesis through activation or repression of genes regulating proliferation and differentiation or by interference with assembly of the transcription complex for these genes (7). It has been previously recognized that leukemia cells that harbor certain translocations/mutations such as t(8, 21) can differentiate. Our experiments show that this is true not only of this one subtype but also in many, possibly most, subtypes of AML (8).

Targeted therapy in AML proposed in the late twentieth century focused on forcing cancer cells to differentiate. This was a result of several analyses made in the 1970s and 1980s that showed that various compounds have antiproliferative and prodifferentiating properties on some AML cells. For example, the granulocytic differentiation of certain leukemic cell lines and in acute promyelocytic leukemia (APL) patients was possible after all-transretinoic acid (ATRA) treatment (9), which resulted in the rapid development of differentiation therapies (10–12). However, ATRA is not effective in AML outside of APL, suggesting that differentiation blocks do not have a uniform underlying mechanism and could be a reflection of the disease complexity in AML. The more recent advent of isocitrate dehydrogenase (IDH) inhibitors that are selectively effective only in IDH1 and IDH2 mutant tumors by inducing differentiation further reinforces this point (13).

Results and Discussion

To examine the potential of varying degrees of perturbed differentiation to contribute to leukemic cell architecture, we generated a mathematical model using functional and hierarchical features of hematopoietic differentiation. Specifically, for each type of cell $\left(\sigma \right)$, the model accounts for total cell number $\left(N_{\sigma }\right)$, single-cell death rate $\left({\delta }_{\sigma }\right)$, single-cell division rate $\left({\rho }_{\sigma }\right)$, and the probability $\left(p_{\sigma ,\tau }\right)$ that each daughter cell differentiates into a particular type of cell $\left(\tau \right)$. The probability that the cell does not differentiate is $p_{\sigma ,\sigma }$; that is, the differentiation rate is $1-p_{\sigma ,\sigma }$.

Suppose cell type $\sigma$ differentiates into cell type $\tau$, which is itself not a mature cell type (it divides and differentiates). The total rate at which cells of type $\sigma$ produce cells of type $\tau$ is $N_{\sigma }{\rho }_{\sigma }2p_{\sigma ,\tau }$. Cells of type $\tau$ also divide and either differentiate or produce more cells of type $\tau$, which decreases the population of cells of type $\tau$ at total rate $N_{\tau }{\rho }_{\tau }\left(1-2p_{\tau ,\tau }\right)$. Healthy blood is in homeostasis, in which these 2 rates are equal, which leads to the equation

$$\frac{N_{\tau }}{N_{\sigma }}=\frac{{\rho }_{\sigma }2p_{\sigma ,\tau }}{{\rho }_{\tau }\left(1-2p_{\tau ,\tau }\right)}.$$

Here the key fact is that $N_{\tau }/N_{\sigma }$ is proportional to $1/\left(1-2p_{\tau ,\tau }\right)$. Since the more mature cell types are much more numerous than their progenitors, that is, $N_{\tau }/N_{\sigma }$ is very large, this implies that the differentiation rate is very close to 0.5 (Fig. 2A). Since the aggregate amplification ratio is roughly 10⁹, we expect 0.501 to be a typical value of the differentiation rate in healthy hematopoiesis. If regulation fails and the differentiation rate becomes even slightly less than 0.5, then the number of intermediate cells grows without bound, suggesting normal blood cell production must be on a knife’s edge of being leukemic to work as it does. (How quickly the intermediate cell population grows depends on the particulars of regulation and how it fails [Fig. 2 B–E].) The complete model and its derivation are provided in Methods.

Fig. 2.

Model for reduced or skewed, rather than completely blocked, differentiation in AML. For each type of cell (σ), the model accounts for total cell number (N_σ), single-cell death rate (δ_σ), single-cell division rate (ρ_σ), and the probability (p_σ,τ) that each daughter cell differentiates into a particular type of cell (τ). The probability that the cell does not differentiate is p_σ,σ. (A) The ratio N_τ/N_σ can be effectively controlled by the (non)differentiation rate p_τ,τ. Since N_τ ≫ N_σ, we expect p_τ,τ $\underset{\approx }{<}$ 1/2. If p_τ,τ ≥ 1/2 and ρ_τ > 0, the system is no longer stable, and N_τ grows without bound. (B) In healthy hematopoiesis, differentiation rates in the hierarchy of cells are tightly regulated. This figure is a greatly simplified example with only 4 cell types and one feedback point. Here, A are stem cells, B and C are intermediate progenitors, and D is a mature cell type that does not divide or differentiate. (C–E) A mutation may perturb the differentiation rate of one of the cell types τ, so that p_τ,τ $\underset{\approx }{>}$ 1/2 rather than p_τ,τ $\underset{\approx }{<}$ 1/2, as shown in the panels.

Hematopoiesis is quite complicated, with many cell types and regulatory feedback loops. Therefore, its dynamics are not yet completely understood. While our model describes features of the complete system, we chose a greatly simplified setting in which to illustrate it. We show several examples with only 4 cell types: stem cells (A), 2 stages of intermediate progenitors (B and C), and terminally differentiated mature cells (D). These cell types differentiate into one another in a chain. Our examples include 2 different versions of a feedback loop between the mature cells (D) and intermediate progenitors (B) (Figs. 2 and 3). The details of these examples are provided in Methods.

Fig. 3.

Evolution over time with impaired differentiation in one compartment and regulation affecting either the growth or differentiation rate of compartment B representing intermediate progenitors. Here, A are stem cells, B and C are intermediate progenitors, and D is a mature cell type that does not divide or differentiate. (A) The growth in compartment A with impaired differentiation as p_A,A varies; all except p_A,A = 0.5 are exponential, but with different bases. (B) The healthy system. In Left, cell counts stay at their base levels, and in Right, the control of ρ_B (the growth rate of compartment B) is regulated and corrects for random up to ±25% changes in compartment D every cell cycle. (C) The ρ_B (the growth rate of B) is controlled and differentiation is impaired in (Left) compartment A, (Middle) B, or (Right) C. When differentiation in compartment B is impaired, the counts grow linearly, while all others are exponential. (D) The system with p_B,B (the differentiation rate of B) controlled and differentiation impaired in compartments A or C. The examples in D have a higher base growth rate than in C, which is why N_A grows differently in the 2 cases where its differentiation is impaired.

Our example systems illustrate how dysregulation of differentiation at different stages of hematopoiesis can affect cell growth in different ways. They demonstrate exponential or linear growth in one or more hematopoietic compartments, roughly corresponding to acute or chronic leukemias. If the growth rate of intermediate progenitors (compartment B) is controlled, some of the examples begin to show oscillation of cell counts over time (Fig. 3C). This is, at least in part, because the only input to the growth rate is the size of the mature cells’ compartment (D). As intermediate progenitors (compartment B) grow over time, the same percentage change in growth rate results in more cell growth leading to an overreaction of the system (see Methods for details). The differentiation-rate controlled examples (Fig. 3D) do not have oscillations and result in somewhat different growth patterns but are still unable to compensate for too-small differentiation rates in other compartments. While these examples are simple, they still show a number of very different behaviors in cell growth, depending on where differentiation is impaired and how the regulation works.

Given the large numbers of intermediate cells characteristic of leukemia, our model argues that the leukemic phenotype requires only a small reduction rather than a complete block in the delicate balance of differentiation. If this model of skewed differentiation as opposed to blocked differentiation is correct, that would mean that mature cell types would have a leukemic genotype.

We tested this by collecting peripheral blood samples from 12 patients with AML with abnormal cytogenetics such as t(8, 21), inv 16, trisomy 8, 13q34, and Mixed-Lineage Leukemia (MLL) rearrangements or mutations such as FLT3-ITD and NPM1 and sorting the cells into several subpopulations, including immature leukemic blasts, monocytes, neutrophils, and lymphocytes. We analyzed the isolated cell populations using interphase fluorescent in situ hybridization (FISH) (Fig. 4 A–C) or by sequencing (Fig. 4D) to determine the identity and relative abundance of the leukemia-associated molecular aberrations. We found that all samples for which we were able to isolate neutrophils and myeloid blasts had (usually many) cells with abnormal cytogenetics or mutations at levels similar to those in the blast cell population, while, as a control, most of the purified T cell subpopulations had either no or few abnormal cells detected (Fig. 4). These findings are in contrast to the existing assumption that many of these abnormalities will lead to a full block in differentiation (14). Werner et al. (15) modeled a system with some differentiation in APL, but at a very low rate that is inconsistent with our observations of roughly equal mutant fractions in blasts and fully differentiated cells.

Fig. 4.

Leukemic genotypes are present at similar levels in both blast and mature, differentiated AML cells. Proportion of cells with (A−C) leukemic karyotype or (D) mutation, from 12 AML patients. Error bars represent a 95% binomial confidence interval, so smaller error bars in these measured proportions indicate that more cells of the given type were tested for the leukemic genotype. N for each point ranged from 2 to 112 cells; exact counts are provided in Dataset S2. T cell distributions were compared to each sorted leukemic cell population by 2-tailed Mann−Whitney U test: neutrophils (P = 1.3 × 10⁻⁴), monocytes (P = 3.9 × 10⁻⁴), and myeloid blasts (P = 1.3 × 10⁻⁵). Neutrophils and monocytes were not significantly different from myeloid blasts (P = 0.29 and 0.79, respectively), or from each other (P = 0.20).

Our model is similar to a number of previously published models of hematopoiesis and of more general multicompartment tissue development (16–19). Like ours, these models describe hierarchical tissues rooted in stem cells (including blood), and consider replication and differentiation of various compartments. Those models explore questions such as the dynamics of mutations in cell populations, and optimal tissue architectures for minimizing the possibility of tumorigenic mutations. Our model differs from a number of them by explicitly considering feedback between compartments, a branching rather than linear compartment structure, and (compared with some) a much shallower hierarchy, which results in much lower differentiation rates and corresponds more closely with morphologically defined compartments.

Our mathematical model shows that a complete differentiation block is not necessary for leukemia, and very small changes in differentiation rate could be the difference between healthy hematopoiesis and leukemia. Our experimental evidence confirms this model by showing that an incomplete block in differentiation is common in AML across different genetic driver lesions, thus changing an existing paradigm of AML biology. This, in turn, suggests that therapies aimed at increasing the differentiation fraction may be fruitful, but will require discernment of the specific actionable mechanisms underlying reduced or skewed differentiation in genetic subtypes of the disease. Our model will also be helpful in understanding the effect of therapeutic perturbation on hematopoiesis. Together, these findings support a model of AML wherein disease is driven, in part, by a failure of regulation to generate a high enough differentiation fraction of myeloid blasts, and that this fraction is commonly more than zero. This also warrants potential application to other hematologic malignancies characterized by different cells of origin and kinetics of disease progression. Further, AML is a heterogeneous disease harboring multiple mutations and multiple clones. In future studies, it will be valuable to perform such mathematical modeling at the single-cell level (20) and with differentiation along a continuum rather than in discrete states to capture clonal and genetic heterogeneity and to understand how the differences in degree of differentiation of individual cells could be the result of a different set of cooperating mutations and regulatory feedback. Such modeling should also consider how the various growth and differentiation rates among healthy cells and various tumor subclones affect the clonal heterogeneity and system behavior.

Methods

Mathematical Model of Hematopoietic Differentiation.

For each cell type $\sigma$ (e.g., $\sigma =\mathrm{CMP}$), let $N_{\sigma }$ denote the number of cells of type $\sigma$, ${\delta }_{\sigma }$ be the rate at which a single cell of type $\sigma$ dies, ${\rho }_{\sigma }$ denote the rate at which a cell of type $\sigma$ divides, and $p_{\sigma ,\tau }$ be the probability that a new cell with parent of type $\sigma$ differentiates into type $\tau$. Let $r_{\sigma ,\tau }$ denote the number of cells of type $\tau$ created by cells of type $\sigma$ per unit time. Then

$$N_{\sigma }{\rho }_{\sigma }2p_{\sigma ,\tau }=r_{\sigma ,\tau },$$ [1]

and, for cell types $\tau$ with only one parent $\sigma$, the rate of change of $N_{\tau }$ is given by

$$d{N}_{\tau }/dt=r_{\sigma ,\tau }+N_{\tau }{\rho }_{\tau }\left(2p_{\tau ,\tau }-1\right)-{\delta }_{\tau }{N}_{\tau }.$$ [2]

In steady state, $N_{\tau }$ does not change (i.e., $d{N}_{\tau }/dt=0$), so

$$N_{\sigma }{\rho }_{\sigma }2p_{\sigma ,\tau }=N_{\tau }{\rho }_{\tau }\left(1-2p_{\tau ,\tau }\right)+{\delta }_{\tau }{N}_{\tau }.$$ [3]

The death rate of nonmature cell types $\tau$ (such as stem or progenitor cells) is negligible (i.e., ${\delta }_{\tau }\approx 0$), so

$$\frac{N_{\tau }}{N_{\sigma }}=\frac{{\rho }_{\sigma }2p_{\sigma ,\tau }}{{\rho }_{\tau }\left(1-2p_{\tau ,\tau }\right)}.$$ [4]

For fixed values of the cell division rates ${\rho }_{\sigma }$ and ${\rho }_{\tau }$ and differentiation rate $p_{\sigma ,\tau }$, this equation implies $p_{\tau ,\tau }\mathrm{\lesssim }1/2$ if and only if $N_{\tau }\gg N_{\sigma }$. (The symbol $\mathrm{\lesssim }$ means “less than but nearly equal to.”)

For mature cell types $\tau$, ${\rho }_{\tau }=0$, so

$$\frac{N_{\tau }}{N_{\sigma }}=\frac{{\rho }_{\sigma }2p_{\sigma ,\tau }}{{\delta }_{\tau }}.$$ [5]

Consider what would happen if there was a mutation that perturbed the differentiation rates from cells of type $\tau$, so that, instead of $p_{\tau ,\tau }\mathrm{\lesssim }1/2$, the mutant lineage had $p_{\tau ,\tau }\ge 1/2$, and there was no regulation of cell growth or differentiation (i.e., hematopoiesis was completely open loop). If $p_{\tau ,\tau }>1/2$, then $N_{\tau }$ will grow exponentially, and, furthermore, all descendant cell types will grow exponentially. If $p_{\tau ,\tau }=1/2$, then $N_{\tau }$ will grow linearly, and all descendant cell types will grow linearly. Fig. 3A illustrates this for different values of $p_{A,A}$. Other than $p_{A,A}=1/2$ (which is constant because of cells differentiating out), they all have exponential growth, but the base of the exponent, and hence the growth rate, depends on the specific value of $p_{A,A}$.

We present several examples of systems with simple feedback control and show how they misbehave as nondifferentiation probabilities become too high. The examples all use a sequence of 4 cell types, $A\to B\to C\to D$, where A is defined as stem cells, B and C are defined as intermediate progenitors, and D is defined as terminally differentiated mature cells with only a death process, not growth or differentiation. In this model, each type differentiates into the next type, and the number of cells of type D affects the behavior of the cells of type B, as shown schematically in Fig. 2B. The examples differ in which cell type differentiates too little and in what is controlled in compartment B: either the growth rate ${\rho }_B$ or the nondifferentiation probability $p_{B,B}$. In each of the examples, in the healthy model, there are 1,000 cells of type A, 3,000 of type B, 9,000 of type C, and 27,000 of type D; $\rho =1$ is the maximum rate at which cells can divide, corresponding to one division per cell cycle duration. In the first 5 examples (2 in Fig. 3B without dysregulated differentiation and 3 in Fig. 3C with one compartment insufficiently differentiating), compartment B controls its growth rate, ${\rho }_B$, with the following control law:

$${\rho }_{Bn+1}=\text{max}\left(0,\text{min}\left(1,{\rho }_{Bn}+0.15\frac{N_{D\hspace{0.17em}target}-N_D}{N_{D\hspace{0.17em}target}}\right)\right).$$ [6]

This control law looks only at how far away from the target $N_D$ is, so its responsiveness in terms of cells differentiating into compartment C varies with $N_B$; effectively, the response size increases as the B compartment grows, which can lead to some overcontrolling and oscillations when $N_B$is much larger than its target value. A different control law that took into account $N_B$wouldn’t do this but seems less biologically plausible, because it would require a second signal from cells of type B to themselves. The 0.15 factor damps the response to avoid dynamic instability when the cell counts are close to normal.

The other parameters are fixed as ${\rho }_A=0.1$, ${\rho }_C=1.0$, ${\delta }_D=106/270$ (about 0.39), $p_{A,A}=0.5$, $p_{B,B}=14/30$ (about 0.47), $p_{C,C}=74/180$ (about 0.41) and $N_{D\hspace{0.17em}target}=\mathrm{27,000}$. We chose these values so that the system doesn’t vary when ${\rho }_B=0.5$, which gives it maximum flexibility to increase or decrease growth around the stable point. We discuss what happens when there are mutations that perturb the differentiation rates of cells of type $A$, $B$, or $C$, as illustrated in Fig. 2 C–E.

Fig. 3B shows 2 examples where differentiation is not dysregulated. The first one is the system with no perturbations (Fig. 3 B, Left). The cell counts never change. The second one shows that control law [6] is able to maintain stability even when presented with very large perturbations (Fig. 3 B, Right). In it, $N_D$ is randomly adjusted in either direction by as much as 25% of its value at every time step. Even with this large perturbation, the cell counts vary, but do not drift over time.

The case where $p_{A,A}=0.55$ is illustrated in Fig. 2C and Fig. 3 C, Left. All cell types grow exponentially, but types C and D oscillate as the feedback at B tries to control $N_D$. A different control law that considered $N_B$ could eliminate the growth in compartments C and D at the expense of more growth in compartment B, but could not avoid exponential growth in total cell count.

Fig. 2D and Fig. 3 C, Middle illustrate $p_{B,B}=0.55$. In this case, it’s never possible for $N_B$ to decrease, but it can stay steady for a while when the feedback suppresses all growth, that is, sets ${\rho }_B=0$. The growth is superlinear but less than exponential, because the periods between nonzero growth increase. $N_C$ and $N_D$ oscillate between somewhat less than their target values and a peak that increases superlinearly. Again, a control law that took into account $N_B$ could remove these oscillations and keep $N_C$ and $N_D$ near their target values in exchange for somewhat faster (but still nonexponential) growth of $N_B$. With a slightly smaller value of $p_{B,B}$, this might behave akin to a chronic leukemia.

The final case is when $p_{C,C}=0.55$. In this case, because ${\rho }_C$ is fixed, both compartments C and D grow exponentially, while the feedback shuts down all division in compartment B. Compartment B then grows linearly at the rate of cells coming from compartment A. This is illustrated in Fig. 2E and Fig. 3 C, Right.

The final 2 examples control $p_{B,B}$, the nondifferentiation rate at B, rather than the growth rate ρ_B (Fig. 3D). The control law is

$$p_{B,B}=\{\begin{array}{cc}{p}_{B,B}+\left(1-p_{B,B}\right)\min \left(1,\frac{N_D}{targe{t}_D}-1\right)0.15& \text{if}\hspace{0.17em}{N}_D>targe{t}_D\\ p_{B,B}+\left(0.5-p_{B,B}\right)\left(\frac{N_D}{targe{t}_D}-1\right)0.15& \text{otherwise}\end{array}.$$ [7]

This adjusts $p_{B,B}$ as a fraction of the distance between it and its limit (1 or 0.5) as the fraction that $N_D$ varies from its target. For these examples, the other parameters are set as ${\rho }_A={\rho }_B={\rho }_C=1$ (all cells are dividing as fast as they can) and $p_{A,A}=1/2$, $p_{C,C}=5/18$ (about 0.28), and ${\delta }_D=13/27$ (about 0.48). These parameters are selected so that, in steady state, $p_{B,B}=1/3$. Again, the factor of 0.15 is to avoid overresponsiveness and oscillations.

When $p_{A,A}=0.55$, all lineages grow exponentially. Compartment B decreases $p_{B,B}$ until it gets to zero, which is able to reduce the growth in compartments C and D for a short time, but, eventually, even with no amplification in compartment B, the rate of growth from compartment A drives the growth of all cell types. This is illustrated in Figs. 2C and 3 D, Left.

In our final example, $p_{C,C}=0.55$, and is illustrated in Fig. 2E and 3 D, Right. The regulatory mechanism increases the differentiation rate in compartment B to 1, which reduces growth in compartments C and D for a time. Eventually, $N_B$ gets to 1,000, the number of cells that differentiate from compartment A, and cannot reduce farther. At that point, compartments C and D grow exponentially.

These examples all use a constant death rate for compartment D $\left({\delta }_D\right)$, which causes it to decline on a negative exponential if no new cells differentiate into it. It is possible to use other death processes, such as having the cells die after a fixed number of cell cycles (i.e., unactivated neutrophils live for 3 d). While this will change the number of cells in D $\left(N_D\right)$, it will not greatly affect the overall behavior or growth of the system, unless the death rate in D depends on the size of D (i.e., there is another feedback loop).

In order for the population of some cell type $\tau$ to grow exponentially, it is sufficient to have ${\rho }_{\tau }>0$ and $p_{\tau ,\tau }>1/2$; in particular, it is not necessary to have $p_{\tau ,\tau }=1$. If a mutant lineage of type $\tau$ has $1/2<p_{\tau ,\tau }<1$ (and ${\rho }_{\tau }>0$), then $N_{\tau }$ will grow exponentially, and the mutant genotype will appear in descendant cell types. Blast crisis is consistent with $p_{\tau ,\tau }\mathrm{\gtrsim }1/2$, as shown in Fig. 3A. This conclusion does not depend on the details of the hematopoiesis cell-type phylogeny, and, in particular, it holds for the conventional model illustrated in Fig. 1 and the more recent continuous differentiation model (21).

Real hematopoiesis has a much more complicated regulatory mechanism than these simple examples and, correspondingly, will have more complex and varied behavior, and more ways in which the system can fail due to mutations. Modeling its control is beyond the scope of this work. Our model describes properties that are true regardless of the regulation, and so can be used as a basis of future work that fills in the details of the biological control laws.

An estimate of the aggregate amplification factor of the neutrophil path.

The product of the amplification factors on the path from long-term HSCs (LT-HSCs) to any mature blood cell type must equal the ratio of the creation rate of the mature blood cell to that of LT-HSCs. We derive an order-of-magnitude estimate of this ratio here, realizing that there will be large variation between people and over time. This is relevant to our result only in that the ratio is quite large, which, in turn, implies large amplification factors and, as is shown in our model, differentiation fractions only slightly above 0.5. Therefore, this order-of-magnitude approximation shows that a tiny change in differentiation fraction is sufficient to convert healthy blood into a leukemic phenotype.

Adults have about 10⁴ LT-HSCs (21, 22), each of which divide roughly once per 40 wk (7) (∼4 × 10⁻⁸/s) for an overall rate of ∼4 × 10⁻⁴/s. In healthy individuals, white blood cell count is roughly 4 × 10⁹ to 11 × 10⁹/L, about half of which are neutrophils. Given an average of ∼5 L of blood per adult individual, this works out to ∼10¹⁰ to 3 × 10¹⁰ neutrophils per person. Unactivated neutrophils live on the order of hours to a few days. Using a day (roughly 10⁵ s) as an average suggests about 10⁵ to 3 × 10⁵ new neutrophils per second are required to replace the ones that die and maintain steady state. This gives a ratio of LT-HSC division to neutrophil creation of 2.5 × 10⁸ to 7.5 × 10⁸. At least half of all LT-HSC divisions go to maintain the LT-HSC population (their differentiation fraction must be exactly 0.5 to maintain population, assuming no LT-HSCs ever die), so we can double the ratio. Furthermore, the LT-HSC divisions need to support all blood cell types, not just neutrophils, which, in turn, will multiply the factor by that fraction. Therefore, a very rough estimate of the aggregate amplification factor from LT-HSCs to mature blood cells is 10⁹.

Primary AML Samples.

AML samples were obtained following informed consent from patients evaluated at Oregon Health & Science University (OHSU) (Dataset S1). This study is approved by the OHSU Institutional Review Board.

Flow cytometry-based subpopulation sorting.

Peripheral blood derived from primary leukemia samples was stained with CD3-FITC, CD117-PE, CD13-PE-Cy7, CD34-APC, CD14-APC-H7, CD16-Pacific Blue, CD33-PerCP-Cy5.5, and CD45-Pacific Orange following standard protocol, and populations were sorted using a FACSAria flow cytometer (Becton Dickinson) using surface markers as follows: myeloid blasts (CD34+CD117+ or CD34+CD117+CD13+ or CD117+CD33+ cells), promyelocytes or myelocytes (CD34−CD13+CD16− cells), monocytes (CD14+CD34− cells), neutrophils (CD34−CD13+CD16+ cells), and lymphocytes (bright CD45+ cells or CD3+ cells).

FISH analysis.

Sorted subpopulations from relevant samples were analyzed by interphase FISH standard protocols for detecting recurrent AML abnormalities. Vysis (Abbott) FISH Probes utilized were RUNX1T1 (8q21.3)/CEP 8 to detect trisomy 8, D13S319 (13q14.3)/13q34 to detect deleted 13q, MLL (11q23) to detect trisomy 11 in hyperdiploid clone—no rearrangement, MLL (11q23) to detect MLL rearrangement, RUNX1T1/RUNX1 to detect RUNX1T1/RUNX1 fusion [t (8, 21)], and CBFB (16q22) to detect inverted 16.

DNA sequencing.

Genomic DNA was isolated from sorted cell subpopulations for relevant samples and subjected to targeted next-generation sequencing at OHSU Knight Diagnostic Laboratory.

Statistical analysis and considerations.

We presented 6 P values, 3 of which are significant at the 0.01 level, without correcting them for multiple hypothesis testing. We did this because we wanted the P values for the nonsignificant results to be in the range of [0,1]. Had we applied a Bonferroni correction of 6, each of the significant results would still have been significant at the 0.01 level, with the largest having a value of 0.002.

Data availability statement.

All materials, data, and associated protocols are included in text and in Datasets S1 and S2. If any additional information is needed, it will be available upon request from the corresponding authors.