A multicompartmental model of hematopoiesis has been described in detail elsewhere [8]. Briefly, under steady state conditions, HSC in the active pool divide symmetrically with probability *x* and produce either two stem cells or two differentiated cells with equal probability. With probability 1 − *x*, they divide asymmetrically, giving rise to one cell for self-renewal and another cell that enters the pool of short-term repopulating cells. Subsequent rounds of division in downstream compartments *k* are coupled either with differentiation (and migration to compartment *k* + 1) or self-renewal with both daughter cells staying within a given compartment *k*. Under steady state conditions, each compartment accommodates a number of cells, *N*_{k}. We used the principle of cell flux conservation to map out the size and rate of replication of cells in the various compartments of hematopoiesis. We obtain a progressive increase of compartment size and of the intrinsic replication rate of cells as they become more committed toward a specific cell lineage [8]. Within each compartment, cells divide at a rate *r*_{k} specific for each compartment. This rate increases as the cells become more differentiated. On the other hand, the mutation rate μ is the same across compartments. Since on average each compartment has a fixed number of cells, we will model cell dynamics within each compartment in the form of a Moran process [24].

#### Two Mutations Within the Stem Cell Pool

First we address whether two independent mutations in the same gene (e.g., *PIG-A*) can occur at the level of the stem cell pool before the disease is detected. The process is illustrated in Figure 1A and 1B. In Figure 1A, a single HSC mutation (red cell) is responsible for one clone, whereas in Figure 1B independent mutations in different HSC (red and green cells) give rise to two clones. In the case of PNH, at least 20% of circulating neutrophils have to be deficient in GPI-anchored proteins for diagnosis [22]. We estimate the probability that a second mutation occurs in the stem cell pool while the first mutant clone increases to a size for which diagnosis is possible. Assuming Moran dynamics, the conditional average time until a mutant reaches the threshold *M* in a stem cell pool of size *N* can be calculated from the general equation for fixation times [25]. For *neutral* mutants, the average number of times each cell divides is given by

- ((1))

For *M* = *N*, this reduces to the fixation time of a neutral mutant in *N* − 1 generations (one generation amounts to *N* Moran steps). In order to estimate how many mutations arise during these *C* cell divisions, we note that there are at most *N* − 1 unaffected cells during the process. Hence, the maximum number of cell divisions occurring in this population cannot exceed *C* · *N*. If the mutation rate per gene per cell division is *u*, then the upper limit for the expected number of new second mutants during the time until the first mutant reaches the threshold *M* is given by

- ((2))

Assuming that each hematopoietic stem cell is equally represented in the circulation, with *N* = 400 [3], *M* = 0.2 · *N* = 80, *x* = 0.5, and *u* = 5 × 10^{−7} [13], then from equation (2)*F* < 0.017. Hence, the expected number of stem cells with a second mutation in the same gene arising independently is below 0.02, suggesting that in PNH, one clone probably does not arise at the level of the stem cell (see Discussion).

#### Simultaneous Mutations in the Stem Cell and Downstream Compartments

In this case, we have to consider cell dynamics within and between compartments. Given the size of the hematopoietic organ and the turnover of cells [8], it is more likely that additional mutations occur in compartments downstream from the HSC compartment. This is illustrated in Figure 1C, where one clone (green cells) originates in a downstream compartment, whereas another clone (red cells) originates in the HSC compartment. Within any compartment (*k*) downstream of the stem cell pool (*k* > 1) in each time step, a cell is selected at random for reproduction. It can divide in two ways. First, with probability ε, the selected cell gives rise to two differentiated cells that both move to the next compartment. This process decreases the size of the compartment by one (*N*_{k} − 1), and so the upstream compartment (*k* − 1) replenishes the number of cells in compartment *k* to the constant value *N*_{k} that is characteristic for that compartment. Second, the cell contributes to self-renewal with probability 1 − ε and, in this case, a random cell from compartment *k* is chosen for differentiation and moves to the next compartment.

For simplicity, we consider the neutral case in which mutated cells replicate at the same rate as normal cells (normal fitness). We calculate the average time a mutation persists in a downstream compartment starting from *i* mutated cells. The probability that one of the *i* mutated cells is selected for reproduction out of a total number of *N* cells is *i/N*. The probability that this cell differentiates and leaves the compartment is *T*^{−}(*i*) = ε · *i*/*N*. A new cell from the upstream compartment where no mutants are present replaces it. Thus, this step always reduces the number of mutants. On the other hand, the number of mutated cells may increase due to self-renewal, with probability *T*^{+}(i) = (1 − ε) · *i*/*N* · (*N* − *i*)/*N* < (1 − ε) · *i*/*N*.

Self-renewal occurs with probability 1 − ε and takes place in a mutated cell with a probability *i/N*. Moreover, in this case we have to choose one (normal) cell for differentiation in order to keep the compartment size constant. This occurs with probability (*N* − *i*)/*N*, as reflected by the last term. Interestingly, mutated cells are disadvantageous in this process, as

- ((3))

for ε > 0.5. Consequently, the fixation probability φ(*j*) for *j* mutants in a non-stem cell compartment cannot exceed that associated with neutral mutants

- ((4))

For large populations, the probability that a single mutant reaches fixation φ(1) becomes very small. The average number of cell divisions in each cell until a single neutral mutation is lost again is formally given by

- ((5))

For disadvantageous mutants φ(*j*) < *j*/*N* (as in our case) and large *N*, we can determine an upper limit for this time, which is given by

- ((6))

Here, *Ln*[*x*] is the natural logarithm. A detailed mathematical description of these derivations will be reported elsewhere.

Mutations in the downstream compartment will ultimately die out due to the underlying structure of hematopoiesis, although the time until their extinction occurs can be long (see below). For normal hematopoiesis, we have estimated that ε = 0.84 [8]. Using this value in equation (6), we find that a mutant, on average, is only present for 1.32 generations within a given compartment. The replication rate in compartment *k* is *r*_{k} = *r*_{0} · *r*^{k} [8], where *r*_{0}, the replication rate at the level of the stem cell, is once per year [3, 5], and *r* = 1.27 [8]. Thus, we can now calculate the average time a neutral mutant stays in each compartment. In the first non-stem cell compartment, this time is 380 days and decreases to 54 days in compartment 10 (Fig. 2B). We must emphasize that these are averages and, stochastically, a mutant cell that arises in one of these downstream compartments can persist for longer or shorter times. For larger compartments, however, the mutant population will quickly die out, and the size of the clone will be small enough to be undetectable with current technologies.

#### Does the Clone Always Arise from a Single Cell in an Initial Compartment?

Let us consider a case where we observe a circulating clone that comprises 1% of all cells in the blood. Is it legitimate to conclude that such a clone is due to a single mutated cell present in a single compartment? To address this issue, we deduce the probability that a single neutral mutant reaches a certain threshold in a population. The probability that a second cell with the same mutation is produced within the same compartment is smaller than 1 − ε, which is 0.16 for the value of ε = 0.84 associated with normal hematopoiesis. If we assume the compartment has 10^{3} cells, the probability that 10 (1%) cells of this type are produced in the compartment as a result of the original mutation is already smaller than 10^{−6}. Thus, we can conclude that, usually, the mutant population does not reach a significant fraction of the compartment size. Notice that the size of compartment *k* = 1 is already of the order of 10^{3} cells in normal hematopoiesis. Hence, circulating mutant cells are truly clonal and maintained by a very small pool of cells in the bone marrow associated with the smaller compartments.