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Frank SA. Dynamics of Cancer: Incidence, Inheritance, and Evolution. Princeton (NJ): Princeton University Press; 2007.
Heritable changes in populations of cells drive cancer progression. In this chapter, I discuss three topics concerning population-level aspects of cellular genetics.
The first section shows that mutations during development may contribute significantly to cancer risk. In development, cell lineages expand exponentially to produce the cells that initially seed a tissue. A single mutation in an expanding population carries forward to many descendant cells. By contrast, once the tissue has developed, each new mutation usually remains confined to the localized area of the tissue that descends directly from the mutated cell. Because mutations during development carry forward to many more cells than mutations during renewal, a significant fraction of cancer risk may be determined in the short period of development early in life.
The second section analyzes the distinction between stem lineages and transit lineages. To renew a tissue, cells must be continuously produced to balance the equal number of cells that die. Cell death prunes certain cell lineages—the transit lineages—and requires that other lineages continue to provide future renewal—the stem lineages. Renewal imposes a constraint on the shape of stem and transit lineages. Within this constraint, if the mutation rate is relatively lower in stem cells, then relatively longer stem lineages and shorter transit lineages reduce cancer risk.
The third section contrasts symmetric and asymmetric mitoses in stem cells. Each stem cell may divide asymmetrically, every division giving rise to one daughter stem cell and one daughter transit cell. Alternatively, each stem cell may divide symmetrically, giving rise to two daughters that retain the potential to continue in the stem lineage; random selection among the pool of excess potential stem cells reduces the stem pool back to its constant size. With asymmetric division, any heritable change remains confined to the independent lineage in which it arose. With symmetric division, the random selection process causes each heritable change eventually to disappear or to become fixed in the stem pool; only one lineage survives over time.
13.1 Mutations during Development
Renewing tissues typically have two distinct phases in the history of their cellular lineages. Early in life, cellular lineages expand exponentially to form the tissue. For the remainder of life, stem cells renew the tissue by dividing to form a nearly linear cellular history. Figure 13.1 shows a schematic diagram of the exponential and linear phases of cellular division.
Figure 13.1
Lineage history of cells in renewing tissues. All cells trace their ancestry back to the zygote. Each tissue, or subset of tissue, derives from a precursor cell; np rounds of cell division separate the precursor cell from the zygote. From a precursor (more...)
Mutations accumulate differently in the exponential and linear phases of cellular division (Frank and Nowak 2003). During the exponential phase of development, a mutation carries forward to many descendant cells. The initial stem cells derive from the exponential, developmental phase: one mutational event during development can cause many of the initial stem cells to carry and transmit that mutation. During the renewal phase, a mutation transmits only to the localized line of descent in that tissue compartment: one mutational event has limited consequences.
Development occurs over a relatively short fraction of the human lifespan. However, a significant fraction of cancer risk may arise from mutations during development, because the shape of cell lineage history differs during development from that in later periods of tissue renewal (Frank and Nowak 2003).
Mutational Events versus the Number of Mutated Cells
Individuals begin life with one cell. At the end of development, a renewing tissue may have millions of stem cells. To go from one precursor cell to N initial stem cells requires at least N − 1 cell divisions, because each cell division increases the number of cells by one.
If the mutation rate per locus in each cellular generation is u, then how many of the initial N stem cells carry a mutation at a particular locus? This general kind of problem was first studied in microbial populations by Luria and Delbrück (Luria and Delbrück 1943; Zheng 1999, 2005). They wanted to estimate the mutation rate, u, in microbial populations by observing the fraction of the final N cells that carry a mutation.
The Luria-Delbrück problem plays a central role in the study of cancer, because progression depends on how heritable changes accumulate in cell lineages. The Luria-Delbrück analysis focuses on one aspect of mutation accumulation in cell lineages: the distribution of mutations in an exponentially expanding clone of cells.
To study the Luria-Delbrück problem, we must distinguish between mutational events and the number of cells that carry a mutation. Figure 13.2 shows an example in which one cell divides through three cellular generations to yield N = 23 = 8 descendants. This exponential growth requires a total of N − 1 = 7 cell divisions. Each cell division causes one cell to branch into two descendants, so there are 2(N − 1) = 14 branches in which DNA is copied and a mutational event may occur. If one mutational event occurs among those 14 replications, then how many of the final 8 cells carry the mutation?
Figure 13.2
Probability of the number of mutated cells for a single mutational event. Each of the three sequences starts with a single cell that then proceeds through three generations of cell division, yielding N = 23 = 8 descendants. In each sequence, there are (more...)
Figure 13.2 enumerates the possible outcomes for the simple example in which there is exactly one mutational event and a single cell divides regularly to produce 8 descendants. We can gain an intuitive understanding of the problem by generalizing the example in Figure 13.2.
Suppose we begin with one precursor cell, which then divides n times to yield N = 2n descendants. Assume that exactly one mutational event occurs, and that the mutational event happens with equal probability on any of the 2(N − 1) branches. If the mutation occurs on one branch in the first division, then 2−1 = 1/2 of the descendants carry the mutation; if the mutation occurs on one branch in the second division, then 2−2 = 1/4 of the descendants carry the mutation. In general, a fraction 2−i of the descendants carries the mutation with probability 2i/[2(N − 1)] for i = 1,...,n (Frank 2003b).
My simple calculations in the previous paragraph do not provide a full description of the Luria-Delbrück distribution, because I assumed exactly one mutational event over the entire population growth period. In reality, mutational events arise stochastically, so a full analysis must consider how the stochastic process of mutational events translates into the number of final cells that carry a mutation at a particular locus (Zheng 1999; Frank 2003b; Zheng 2005). For example, how do mutations during development translate into the number of initially mutated stem cells at the end of development?
Number of Initially Mutated Stem Cells
A small number of somatic mutations during development can lead to a significant fraction of stem cells carrying a mutation that predisposes to cancer. How much of the risk of cancer can be attributed to mutations that arise in development?
No one has tried to measure developmental risk. But a few simple calculations based on standard assumptions about cell division and mutation rate show that developmental risk may be important (Frank and Nowak 2003).
Suppose that N = 108 stem cells must be produced during development to seed the colon. Exponential growth of one cell into N cells requires about ln(N) cellular generations in the absence of cell death. In this case, ln(108) ≈ 18. If the mutation rate per locus per cell division during exponential growth is ue, then the probability that any final stem cell carries a mutation at a particular locus, x, is roughly the mutation rate per cell division multiplied by the number of cell divisions, x = ue ln(N). This probability is usually small: for example, if ue = 10−6, then x is of the order of 10−5.
The frequency of initially mutated stem cells may be small, but the number may be significant. The average number of mutated cells at a particular locus is the number of cells, N, multiplied by the probability of mutation per cell, x. In this example, Nx ≈ 103, or about one thousand.
I have focused on mutations at a single locus. Mutations at many different loci may predispose to cancer. Suppose mutations at L different loci can contribute to predisposition. We can get a rough idea of how multiple loci affect the process by simply adjusting the mutation rate per cell division to be a genome-wide rate of predisposing mutations, equal to ueL. The number of loci that may affect predisposition may reasonably be around L ≈ 102 and perhaps higher. Following the calculation in the previous paragraph, with L ≥ 102, the number of initial stem cells carrying a predisposing mutation would on average be at least 105. Some individuals might have two predisposing mutations in a single initial stem cell.
The average number of initially mutated cells may be misleading, because the distribution for the number of mutants is highly skewed. A few rare individuals have a great excess; in those individuals, the mutation arises early in development, and most of the stem cells would carry the mutation. Those individuals would have the same risk as one who inherited the mutation.
Figure 13.3 shows the distribution for the number of initially mutated stem cells at the end of development. For example, in the right panel, with a mutation probability per cell division of 10−6, a y value of 2 means that approximately 10−2, or 1%, of the population has more than 104 initially mutated stem cells at a particular locus (L = 1). Similarly, with a mutation probability per cell division of 10−5, a y value of 3 means that approximately 10−3, or 0.1%, of the population has more than 104 initially mutated stem cells.
Figure 13.3
Number of initially mutated stem cells at the end of development. The total number of initial stem cells, N, derive by exponential growth from a single precursor cell. Each plot shows the cumulative probability, p, for the number of mutated initial stem (more...)
Excess Risk from Developmental Predisposition
A significant fraction of adult-onset cancers may arise from mutations that occur during the short period of development early in life (Frank and Nowak 2003). In this section, I briefly summarize Meza et al.'s (2005) thorough quantitative analysis of this problem.
Meza et al. (2005) evaluated the role of developmental mutations in the context of colorectal cancer. They began with a model of progression and incidence that they had previously studied (Luebeck and Moolgavkar 2002). In that model, carcinogenesis progresses through four stages: two initial transitions, followed by a third transition that triggers clonal expansion, and then a final transition to the malignant stage.
In their new study, Meza et al. (2005) began with the same four-stage model. They then added a Luria-Delbrück process to obtain the probability distribution for the number of stem cells mutated at the end of development. The stochasticity in the Luria-Delbrück process causes a wide variation between individuals in the number of mutated stem cells. Meza et al. (2005) first calculated the probability that an individual carries Nx initially mutated stem cells at the end of development. To obtain overall population incidence, they summed the probability for each Nx multiplied by the incidence for individuals with Nx mutations.
Meza et al. (2005) summed incidence in their four-stage model over the number of initially mutated stem cells to fit the model's predicted incidence curve to the observed incidence of colorectal cancer in the USA. From their fitted model, they then estimated the proportion of cancers attributable to mutations that arise during development. Figure 13.4 shows that a high proportion of cancers may arise from mutations during the earliest stage of life.
Figure 13.4
The proportion of cancers that arise from cells mutated during development. These plots show calculations based on a specific four-stage model of colorectal cancer progression (Meza et al. 2005). The parameters of the progression model were estimated (more...)
Cancers at unusually young ages are often attributed to inheritance. However, Figure 13.4 suggests that early-onset cancers may often arise from developmental mutations. Developmental mutations act similarly to inherited mutations: if the developmental mutation happens in one of the first rounds of post-zygotic cell division, then many stem cells start life with the mutation. Inheritance is, in effect, a mutation that happened before the first zygotic division.
Cell Generations to a Common Precursor Cell
When will cases with early onset and multiple tumors be caused by developmental mutations rather than inherited mutations? The answer depends on the pattern of cellular lineages that produce a tissue.
All cells in a tissue trace their ancestry back to a precursor cell. That common precursor would be the zygote if both cells from the first zygotic division contribute descendants to the tissue. Alternatively, several cell divisions derived from the zygote may occur before a precursor cell begins to differentiate into a particular tissue.
Figure 13.1 shows the different phases in the ancestry of a tissue. In that figure, np rounds of cell division happen between the zygote and the common precursor cell for the tissue. The precursor then seeds an exponentially growing clone through ne cell generations. Once the tissue is formed, the stem cells renew the tissue by proceeding through ns cell divisions, where ns increases with age.
Consider an example to illustrate the potential importance of the number of cell generations to a common precursor for a tissue. Suppose a particular cancer syndrome has the characteristics of inherited disease—early onset and multiple independent tumors. Assume that the syndrome causes such severe early-onset disease that individuals who suffer the disease rarely reproduce. Then each case must arise from a new mutation.
The new mutation could occur in the parent: either in the germline or in a precursor to the germline that does not give rise to the affected tissue. A parental mutation would give rise to an inherited case, in which the offspring carries the mutation in all somatic cells. Suppose the number of cell generations between the parental germline precursor and the gamete is ng. Alternatively, the new mutation could occur in the off-spring. The number of cell generations between the zygote and the common precursor for the tissue is np.
The probability that an observed case arose from a developmental mutation rather than an inherited mutation would be approximately np/(np + ng). We could refine this approximation by adjusting for the mutation rates in the maternal and paternal germlines and the somatic precursor lineage and for the frequency of mutations carried by parents that derived from an earlier generation. For example, if the mutation rate per cell division is u, and the frequency of mutations carried by parents from earlier generations is f, then the approximation expands to unp/(unp + ung + f). My point here is simply that, as long as f is small, a significant fraction of important de novo mutations may happen developmentally rather than be inherited from parents.
Few estimates exist for np, the number of precursor cell generations. The little bit known about retinal development and the inherited cancer syndrome retinoblastoma raises some interesting issues. Retino-blastoma usually occurs before the age of five. Without modern medical treatment, the disease would often be fatal, so the affected individual would not reproduce. The inherited syndrome includes early onset and multiple independent tumors, usually with tumors in both eyes. According to the analysis here, the inherited syndrome would derive from developmental mutations approximately in a proportion of cases np/(np + ng).
The number of retinal precursor generations, np, remains unknown. Zaghloul et al. (2005) recently reviewed the subject of retinal development and concluded that, based on the Xenopus model, the left and right retina diverge rather late in development. Thus, there may be a significant number of precursor generations, np, before divergence of the common retinal precursors into the left and right eye. A developmental mutation before left-right divergence could predispose to bilateral retinoblastoma, a symptom usually attributed to an inherited mutation.
13.2 Stem-Transit Design
Mutations in transit cells usually get washed out as the transit cells slough at the surface (Cairns 1975). Most cell divisions occur in the transit lineages, and those divisions pose relatively little cancer risk. Because of the mutational washout advantage of transit lineages, it would seem that natural selection would favor a stem-transit separation with short-lived transit lineages. But adaptation may be more subtle.
Figure 13.5 shows the possibilities for design of a stem-transit architecture (Frank et al. 2003). Suppose a tissue requires k new cells over a certain period to renew itself. For now, assume that no other constraints exist with regard to renewal. To make k cells starting from one cell, the tissue may use n1 stem cell divisions leading to n1 transit lineages, each transit lineage dividing n2 times to produce 2n2 final cells, for a total of k = n12n2 cells.
Figure 13.5
The pattern of cell division giving rise to a total of k cells. The single, initial cell divides to produce a stem cell and a transit lineage. Each transit lineage divides n2 times, yielding 2n2 cells. The stem lineage divides n1 times, producing a total (more...)
Given the need to make k cells, consider how natural selection might increase benefit. Suppose short-lived transit lineages pose little risk. An improved design would add more cell divisions to those low-risk transit lineages and reduce the number of divisions in the long-lived stem lineage, that is, decrease n1 and increase n2.
In general, suppose we may choose to add one additional cell division to any lineage, with the goal to minimize cancer risk (Frank et al. 2003). If cancer requires n rate-limiting steps, and each step happens only during cell division, the risk rises with dn, where d is the number of cell divisions. Risk increases exponentially with number of cell divisions in a lineage, thus natural selection favors prevention of long lineages. It is always most advantageous to add any new cell division to the shortest extant lineage. This optimal design maintains equal length among cell lineages.
In terms of tissue architecture, if we start with one cell, then the best design follows perfect binary cell division with all lineages remaining the same length, such that k = 2n2, where n2 is the number of rounds of cell division. No stem divisions would occur except the first to seed the transit lineages.
This optimal design, with long transit lineages and no stem lineage, assumes that all k cells survive to the end of the required period, with no sloughing of cells. However, the requirement for continual cell death at epithelial surfaces imposes an additional requirement. But for now, I am just asking about the best design in the absence of the constraint imposed by renewal, to understand how much of tissue architecture may be explained by natural selection among alternative designs versus how much may be explained by the unavoidable constraints of renewal.
This first analysis suggests that natural selection favors long transit lineages and no stem lineage. If so, then the stem-transit design may be the consequence solely of continual cell death at the tissue surface, which imposes a stem-transit separation by shortening the cell lineages that lead to the sloughing of surface cells. But we should consider two additional factors.
First, the stem lineage may have a lower mutation rate than the transit lineage. Cairns (1975) proposed that immortal stranding and high sensitivity to DNA damage lower the stem-line mutation rate (See Section 12.4). If the stem lineage does have a lower mutation rate than the transit lineage, then natural selection would favor adding more cell divisions to the lower-risk stem line. In terms of design, this benefit of stem divisions would lengthen the stem lineage, that is, increase n1 in Figure 13.5, and would shorten the higher-risk transit lineages, that is, decrease n2.
Second, the transit lineage may be partially protected, because a transit cell that gets the required n carcinogenic changes may still slough off. This benefit would favor lengthening the transit lineages, because natural selection always tends to allocate additional divisions to those lineages with the lowest relative risk. This particular benefit for transit lineages works against the maintenance of a distinct, long-lived stem line.
In summary, two factors appear to favor a stem-transit design. A renewing tissue necessarily has continual cell death that prunes cell lineages and creates a dichotomy between short and long cell lineages. That constraint of tissue renewal may be sufficient to explain the stem-transit design, even though, with regard to cancer risk, natural selection often favors a more even distribution of cell lineage length. Alternatively, if the stem line accrues mutations at a lower rate than the transit lines, then natural selection favors short transit lineages and long stem lineages.
13.3 Symmetric versus Asymmetric Mitoses
Suppose a tissue compartment, such as an intestinal crypt, maintains N stem cells. To maintain a constant stem pool size, each stem cell may divide asymmetrically, every division giving rise to one daughter stem cell and one daughter transit cell. Alternatively, each stem cell may divide symmetrically, giving rise to two daughters that retain the potential to continue in the stem lineage; random selection among the pool of excess potential stem cells reduces the stem pool back to N.
With asymmetric division, the stem pool maintains N independent cell lineages. Any heritable change remains confined to the particular lineage in which it arose. The N distinct lineages form N parallel lines of evolution.
With symmetric division, the random selection process causes each heritable change eventually to disappear or to become fixed in the stem pool. In effect, only one lineage survives over many generation.
Figure 13.6 introduces a rough guide to the sorting of lineages under symmetric division. That figure shows a stem pool with N = 2, and the probability that the pool maintains two distinct lineages or coalesces into one lineage after a single round of cell division. Figure 13.7 calculates the probability of lineage diversity versus coalescence through two rounds of symmetric cell division.
Figure 13.6
Stem-transit design to renew a tissue based on symmetric stem cell division and regulation of the stem pool to a constant size. Each alternative begins with two stem cells at the left. The two stem cells differ genetically. Each stem cell divides to produce (more...)
Figure 13.7
Symmetric stem cell division and regulation of the stem pool to a constant size by random selection of daughter cells. The three patterns in each generation—polymorphism, fixation for the light type, or fixation for the dark type—are shown (more...)
Asymmetric and symmetric division have different consequences for the evolution of stem cell compartments. With asymmetric division, mutations remain in the stem pool but do not spread, unless those mutations break the asymmetry and force competition between lineages. With symmetric division, a mutation may be lost by chance or may take over the entire compartment. If a mutation takes over the compartment, any subsequent mutation in the compartment adds a second hit.
13.4 Summary
This chapter described the population genetics of somatic cell lineages, with an emphasis on stem cells. The theory of population genetics provides analytical tools to calculate how mutation, competition (selection), and random sorting of lineages (drift) influence the rate at which mutations accumulate in cell lineages. Several recent papers have applied population genetic theory to analyze how the demography of the stem cell compartment influences the accumulation of mutations and the progression of cancer (e.g., Komarova et al. 2003; Michor et al. 2003; Frank 2003c; Michor et al. 2004). The next chapter begins with empirical studies of stem cell population genetics, and follows with a more general review of cell lineage evolution and somatic mosaicism.
- Stem Cells: Population Genetics - Dynamics of CancerStem Cells: Population Genetics - Dynamics of Cancer
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