The Probable Cell of Origin of NF1- and PDGF-Driven Glioblastomas

Primary glioblastomas are subdivided into several molecular subtypes. There is an ongoing debate over the cell of origin for these tumor types where some suggest a progenitor while others argue for a stem cell origin. Even within the same molecular subgroup, and using lineage tracing in mouse models, different groups have reached different conclusions. We addressed this problem from a combined mathematical modeling and experimental standpoint. We designed a novel mathematical framework to identify the most likely cells of origin of two glioma subtypes. Our mathematical model of the unperturbed in vivo system predicts that if a genetic event contributing to tumor initiation imparts symmetric self-renewing cell division (such as PDGF overexpression), then the cell of origin is a transit amplifier. Otherwise, the initiating mutations arise in stem cells. The mathematical framework was validated with the RCAS/tv-a system of somatic gene transfer in mice. We demonstrated that PDGF-induced gliomas can be derived from GFAP-expressing cells of the subventricular zone or the cortex (reactive astrocytes), thus validating the predictions of our mathematical model. This interdisciplinary approach allowed us to determine the likelihood that individual cell types serve as the cells of origin of gliomas in an unperturbed system.


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Supplemental Experimental Procedures
We designed a stochastic mathematical model of gliomagenesis to identify the most likely cell of origin of PDGF-and NF1-induced gliomas. This model is part of a growing body of literature devoted to the mathematical modeling of glioma growth [1,2,3,4,5,6,7,8,9,10,11,12].
We performed exact stochastic computer simulations of the system (for a full description of the system, see the main text) and derived analytical approximations of the probabilities of cancer initiation from the different cell types. These analytical approximations are useful to analyze parameter regimes for which stochastic simulations would be computationally too expensive. We considered multiple independent cell clusters or niches of neural stem and progenitor cells within the brain [13]. We investigated the dynamics of one niche within the brain since the total probability of cancer initiation is given by the probability per niche times the number of niches; hence, a consideration of all niches does not alter the most likely cell of origin of brain cancer.
We derived equations for the PDGF and NF1 cases individually. Let us first discuss the case of PDGF-driven gliomas; we will consider NF1-driven gliomas in a later section.
In the PDGF-driven case, we consider a population of N self-renewing (SR) cells, one of which divides each time step. This division can be self-renewing symmetric with probability α, in which case another SR cell is chosen to die to maintain homeostasis in the tissue, or differentiating with probability 1 -α. With probability λ, the division resulting in differentiation is asymmetric, in which case one of the daughter cells remains a SR cell while the other becomes a transit amplifying (TA) cell. The remaining fraction 1λ differentiate symmetrically, producing two TA cells and causing a second SR cell to divide symmetrically to maintain homeostasis. Since the inclusion of symmetric differentiation did not change the relative ordering of the probabilities of cancer initiation between cell types (Table S4), we set λ = 1 and neglected it in the differential equation system for mathematical simplicity.
Each TA cell divides symmetrically, producing more differentiated cells with each division, for a total of z times before terminal differentiation and loss from the system. Additional accidental cell deaths can also occur, in which case a cell of equal maturity will divide to replace it. Genetic alterations can arise in one of the daughter cells of any division. We denote the mutation rate per allele per division by µ ARF and µ PDGF for alterations leading to INK4A/ARF inactivation and those leading to PDGF overexpression, respectively. Bi-allelic inactivation of INK4A/ARF results in an increased growth rate (i.e. relative fitness) of R ARF > 1 in SR cells and an additional β ARF cell divisions in TA cells. PDGF overexpression in 100% of cells in the SR cell population results in expansion by a factor of C of that population and also C SR cell divisions per time step. In TA cells, PDGF overexpression confers a probability of regaining self-renewal capabilities γ i , which is dependent on the differentiation level i of the cell which overexpresses PDGF. When the first PDGF-overexpressing TA cell becomes self-renewing, a new population of self-renewing transit amplifying (SRTA) cells of size CN is created. Cancer is initiated when a self-renewing cell with loss of both INK4A/AR alleles and overexpression of PDGF has emerged. This cell can arise from an SR or SRTA cell, or alternatively from a TA cell which has undergone a gamma event.
Let us first consider the dynamics of SR and SRTA cells. Thus, we use a state space consisting of all possible combinations of mutations in SR and SRTA cell populations. Since we consider the number of SR cells, N, in each niche to be small [13], the mean waiting time until the appearance of a mutated cell is much longer than the average time for fixation of its lineage.
Hence we can describe the evolutionary dynamics as a simple Markovian jump between populations consisting only of a single type of SR cells; the dynamics of SRTA cells are treated similarly. There are five states of SR cells and three states of SRTA cells before the generation of the first cancer-initiating cell. Thus, we have fifteen states before the emergence of a cancer-initiating clone. For each cell type, there also exists a state in which the cancer-initiating cell has already emerged. The states are enumerated as in Table S1.
Denote by X ij the probability that the SR and SRTA cell populations are in states i and j, and by X S , X T , and X R the probabilities that a cancer-initiating clone arises from an SR, TA, and SRTA cell, respectively. Let us first consider only the dynamics within the SR cell population, given by Here α represents the probability of a symmetric SR cell division. When an asymmetric SR cell division occurs and a mutation emerges, this mutation is retained in the SR cell population with In the derivation of the analytical approximation, we ignore the cases in which TA cells need to accumulate more than one mutation before terminally differentiating since the probability of these cases occurring is very small. The cell of origin is then defined as the cell that accumulates the last mutation necessary for cancer initiation.
Let us consider the case in which a lineage of TA cells needs to accumulate only one mutation and a gamma event to produce a cancer-initiating cell. These lineages consist of the populations of TA cells arising from states 3 and 4 in the SR cell compartment (see Table S1). The probabilities that a cancer-initiating clone is produced from these lineages per time unit are respectively given by The chance that a TA clone accumulates two events before terminally differentiating was derived in Haeno et al. [14]. These rates of cancer-initiation from a TA compartment are used to derive the dynamics of TA cells, These quantities contain a factor of N since time is measured in units of N SR cell divisions.
Let us now consider the probability of cancer initiation from TA cells that have acquired selfrenewing propensities before accumulating the final mutation (SRTA cells). The difference between the dynamics of the SR and SRTA cell populations is that (i) the latter cells always undergo symmetric self-renewing cell divisions, (ii) at the beginning, there is no population of SRTA cells, and (iii) the SRTA population never contains unmutated cells. The latter situation arises since we assume that PDGF overexpression is required for cells to experience a gamma event.
The dynamics within the SRTA population ignoring gamma events is given by Denote by G p the rate at which TA cells overexpressing PDGF experience a gamma event; it is given by A special scenario arises when a mutation leading to PDGF overexpression arises in the TA cell population, and such a cell acquires self-renewal before the clone terminally differentiates. The rates at which such a gamma event occurs is given by Similarly, a PDGF-overexpressing TA cell may evolve a mutation in INK4A/ARF, after which one of its offspring experiences a gamma event before the clone terminally differentiates. The rate at which such a gamma event occurs is given by Then the dynamics of SR and SRTA cells due to gamma events is given by Here the factor C -1 N -1 represents the fixation probability of the newly produced SRTA cells.
Note that the rate at which the first gamma event arises in PDGF-overexpressing TA cells is Finally, let us consider the scenario in which a PDGF-overexpressing SR cell produces a TA cell clone before dying out or reaching fixation, and a TA cell from this clone acquires self-renewal.
The rate at which a PDGF-overexpressing SR cell is chosen to differentiate but does not take over the SR population is given by ) ) ) ) µ Here 2µ(α + (1 − α)/2 + Nd) represents the probability that a mutation arises during any time step; the parameter µ can be either µ ARF or µ PDGF . The expression T is denoted by T ARF for cells that need to accumulate an INK4A/ARF mutation and by T PDGF for those requiring PDGF- # denotes the expected number of Moran (symmetric) divisions before a mutation is either fixed or lost, as derived by Rick Durrett [15]. Division by α + Nd, the expected number of symmetric divisions per time step, leads to the expected number of time steps before fixation or loss of the mutation. We then determine the probability that in those time steps, a mutant cell is selected to asymmetrically divide and become a TA cell based on the expected time that j cells are mutated, N/j; the probability that an asymmetric division occurs while in this state, (1α)/(α + Nd); and the probability of selecting a mutant cell as the asymmetrically dividing cell, j/N, summed over the number of cells j that could possibly be mutant in the SR compartment. The rate at which a gamma event occurs in such cells is given by Here we assume that the PDGF-overexpressing SR cells will eventually be chosen to differentiate before dying out. Then the dynamics of such events that result in the cell becoming an SRTA cell is given by In this case, the evolution of a cancer-initiating cell from a TA cell is very unlikely; it would require accumulating a mutation in the SR compartment, one such mutation differentiating before fixation or loss, gaining an additional mutation, and finally undergoing a gamma event, each of which has a low probability of occurring. Thus, we ignore such events.
Finally, we combine all separate scenarios into one system of differential equations to determine the cell of origin of PDGF-driven gliomas: We determined the equations for the NF1-driven case in a manner similar to the PDGF-driven case. The basic system remains the same; all that changes is the effect of each mutation. We consider per allele per division mutation rates of µ NF1 and µ TP53 for NF1 loss and TP53 mutation. We denote by Y ij the probability that the SR and SRTA cell populations are in states i and j, and by X S , X T , and X R the probabilities that a cancer-initiating clone arises from an SR, TA, and SRTA cell, respectively. We enumerate the states in Table S2. Let us first consider only the dynamics within the SR cell population, given by The factors ρ(R NF1,wt ) and ρ(R TP53 ) are defined in the same manner as ρ(R ARF ).
Next, we consider the SRTA compartment alone. The dynamics of this compartment are given In order for TA cells to either become cancerous or transfer to the SRTA compartment, some event that confers self-renewal must occur; this event arises at rate . In the NF1 case, there is no possibility for such event. Thus, the following equations all result in a change of zero:  (Figs. S1-S4). The analytical approximations of all three probabilities demonstrate a good fit with the results of the exact stochastic computer simulations ( Fig. S1 and S2). Note the divergence in results between the NF1-and PDGF-driven cases. In the NF1-driven case, the only cell type capable of initiating cancer is the SR cell type, regardless of the parameters used (Fig.   S2). This is a result of the need for some form of self-renewal capability in order to maintain a cancerous population. However, in the PDGF-driven case, SRTA cells almost always have the highest probability of cancer initiation. Only in the case where the probability of regaining selfrenewal capabilities is sufficiently low (below ~2-5×10 -5 ) does any other cell type have a higher probability of cancer initiation. Therefore, in the PDGF-driven case, SR cells are most likely to initiate gliomagenesis. Overall, the probability of cancer initiation from TA cells, X T , is almost always lowest. In only a few cases, namely when z or β ARF is large, is X T not the lowest probability. However, in all those cases, X R is always larger.
Since the probability of cancer initiating from non-SR cells in the NF1 case is zero, due to the assumption of no gamma effect in this case, we consider the effects of each parameter in the PDGF case only. The rate of symmetric SR cell division, α, has little effect on the probabilities (Fig. S3A). Also, X T increases with the number of additional cell divisions that INK4A/ARF -/-TA cells can undergo, β ARF , while the other probabilities show little dependence on that parameter (Fig. S3B). The probability of cancer initiation from the self-renewing TA cell population increases with the expansion factor resulting from PDGF overexpression, C (Fig. S3C). The probability of an individual cell dying per time step, d, results in an increase in the probabilities of cancer initiation from all three cell types, although not equally. However, even at very high cell death rates (1 in 10 cells dying every time step), the probability of cancer initiation from SRTA cells is still considerably larger than the probability of cancer initiation from any other cell type (Fig. S3D). The rate at which PDGF-overexpressing TA cells gain self-renewal capabilities, γ, and rate of decrease of that probability with increasing differentiation, γ step , enhances X R and, to a lesser extent, X T as γ increases and γ step decreases (Fig. S3E, F, G). Higher mutation rates increase the probabilities of cancer initiation from all three cell types (Fig. S3I-K).
Interestingly, when the number of SR cells (N) is large, the total probability of cancer initiation is small because both the time until fixation and the risk of extinction of mutated SR cells increase (Fig. S3H). The three probabilities are not significantly influenced by the relative fitness of INK4A/ARF -/-SR cells, R ARF (Fig. S3L). The number of cell divisions TA cells can undergo, z, increases primarily X R but also to a lesser extent X T since there is an increased chance for acquiring mutations and self-renewal capabilities when z is large (Fig. S3M). These results hold true even when we vary the ratio of symmetric to asymmetric differentiation divisions, λ, (Fig.   S4).
The predictions of our models are robust with regard to changes in parameters (Fig. 5B).
However, it is possible that other combinations of genetic alterations within the same subtype may result in gliomagenesis. Indeed, many mutations in the same pathway may be functionally equivalent [16,17,18,19,20,21]. Any such alterations would be expected to result in a similar phenotype; thus, they would act functionally as if one of the genes already included in the model were altered. In the model, this would be equivalent to increasing the mutation rate by a factor equal to the number of potential replacement mutations. As we have shown (Fig. S3I-K), increases in mutation rate even by orders of magnitude do not significantly change our results; therefore, our conclusions will hold when mutations with similar phenotype to already included alterations are added to the model. Nonetheless, should further mutational effects be discovered, new extensions to this model would need to be developed. Since the mathematical model described here represents only one possible approach to modeling the evolutionary dynamics leading to glioma formation, we also investigated the robustness of our conclusions to the model assumptions. If there are multiple independent TA cell populations originating from each SR population, then the identity of the cell of origin does not change appreciably (data not shown). If the SVZ is not subdivided into independent cell clusters but consists of a single well-mixed population of stem cells, then the conclusions of the model are robust as well (Fig. S3H). If the PDGF-driven SRTA cell population divides asymmetrically as well as symmetrically, thus producing its own non-self-renewing TA cells, it is possible that the final event before cancer initiation is a gamma event in these cells. However, these cells would In the presence of such a possibility, TA cells may be the most likely cells of origin. However, since there is no evidence for such an alteration arising in NF1-driven cases, we disregard this possibility in our model.

Enderling H, Hahnfeldt P (2011) Cancer stem cells in solid tumors: Is 'evading apoptosis' a hallmark of cancer? Prog Biophys Mol Biol.
Supplemental Tables   Table S1.   when fitting the approximation to simulation (Fig. S1), but the default mutation rate is decreased to µ ARF = µ PDGF = 10 -7 .  shows the probability of cancer initiation from self-renewing (SR) cells, the blue curve the probability of cancer initiation from transit-amplifying (TA) cells, and the green curve the probability of cancer initiation from self-renewing transit-amplifying (SRTA) cells. All parameters are kept the same as in Figure S1 except for death rates.

Probabilities of cancer initiation
Probability of symmetric SR cell division, α

Probabilities of cancer initiation
Expansion factor due to PDGF overexpression, C

Probabilities of cancer initiation
Number of additional cell divisions of ink4a/arf-null TA cells, β ARF

Probabilities of cancer initiation
Probability of accidental cell death,d

Probabilities of cancer initiation
Probability of acquisition of self-renewal, γ, with γ step = 0

Probabilities of cancer initiation
Probability of acquisition of self-renewal, γ, with γ step =γ/10

Probabilities of cancer initiation
Probability of acquisition of self-renewal, γ, with γ step = γ/5 Probabilities of cancer initiation

Fig. S1
Mutation rate per ink4a/arf or pdgf allele, μ Probabilities of cancer initiation I) J)

Probabilities of cancer initiation
Mutation rate per ink4a/arf allele, μ ARF Mutation rate per pdgf allele, μ PDGF Probabilities of cancer initiation

Probabilities of cancer initiation
Fitness of ink4a/arf-null SR cells, R ARF

Probabilities of cancer initiation
Probability of symmetric SR cell division, α

Probabilities of cancer initiation
Number of additional cell divisions of tp53-dom neg TA cells, β TP53

Probabilities of cancer initiation
Number of additional cell divisions of nf1-null TA cells, β NF1

Probabilities of cancer initiation
Probability of accidental cell death,d

Fig. S2
Mutation rate per nf1 and tp53 allele, μ Probabilities of cancer initiation E) F)

Probabilities of cancer initiation
Mutation rate per nf1 allele, μ NF1 Mutation rate per tp53 allele, μ TP53 Probabilities of cancer initiation

Probabilities of cancer initiation
Probability of symmetric SR cell division, α

Probabilities of cancer initiation
Expansion factor due to PDGF overexpression, C

Probabilities of cancer initiation
Number of additional cell divisions of ink4a/arf-null TA cells, β ARF

Probabilities of cancer initiation
Probability of accidental cell death,d

Probabilities of cancer initiation
Probability of acquisition of self-renewal, γ, with γ step = 0

Probabilities of cancer initiation
Probability of acquisition of self-renewal, γ, with γ step =γ/10

Probabilities of cancer initiation
Probability of acquisition of self-renewal, γ, with γ step = γ/5 Probabilities of cancer initiation

Fig. S3
Mutation rate per ink4a/arf or pdgf allele, μ Probabilities of cancer initiation I) J)

Probabilities of cancer initiation
Mutation rate per ink4a/arf allele, μ ARF Mutation rate per pdgf allele, μ PDGF Probabilities of cancer initiation

Probabilities of cancer initiation
Fitness of ink4a/arf-null SR cells, R ARF

Probabilities of cancer initiation
Probability of symmetric SR cell division, α

Probabilities of cancer initiation
Expansion factor due to PDGF overexpression, C

Probabilities of cancer initiation
Number of additional cell divisions of ink4a/arf-null TA cells, β ARF

Probabilities of cancer initiation
Probability of accidental cell death,d

Probabilities of cancer initiation
Probability of acquisition of self-renewal, γ, with γ step = 0

Probabilities of cancer initiation
Probability of acquisition of self-renewal, γ, with γ step =γ/10

Probabilities of cancer initiation
Probability of acquisition of self-renewal, γ, with γ step = γ/5 Probabilities of cancer initiation

Fig. S4
Mutation rate per ink4a/arf or pdgf allele, μ Probabilities of cancer initiation I) J)

Probabilities of cancer initiation
Mutation rate per ink4a/arf allele, μ ARF Mutation rate per pdgf allele, μ PDGF Probabilities of cancer initiation

Probabilities of cancer initiation
Fitness of ink4a/arf-null SR cells, R ARF

Probabilities of cancer initiation
Probability of accidental death in SR cells, d S

Probabilities of cancer initiation
Probability of accidental death in SRTA cells, d R

Probabilities of cancer initiation
Probability of accidental death in TA cells, d T

Probabilities of cancer initiation
Probability of accidental death in SR and TA cells, d S = d T