Neutral Theory in Cancer Cell Population Genetics

Neutral Theory in Cancer Cell Population Genetics Abstract Kimura’s neutral theory provides the whole theoretical basis of the behavior of mutations in a Wright–Fisher population. We here discuss how it can be applied to a cancer cell population, in which there is an increasing interest in genetic variation within a tumor. We explain a couple of fundamental differences between cancer cell populations and asexual organismal populations. Once these differences are taken into account, a number of powerful theoretical tools developed for a Wright–Fisher population could be readily contribute to our deeper understanding of the evolutionary dynamics of cancer cell population. neutral theory, cancer, selection Introduction We here consider how Kimura’s neutral theory works in cancer cell populations. A tumor evolves from a single cells, grows rapidly, and divide into multiple cell lineages by accumulating numerous mutations. Such variations produce a tumor that consists of heterogeneous subclones rather than a single type of homogeneous clonal cells (Marusyk et al. 2012; Almassalha et al. 2016; Andor et al. 2016; McGranahan and Swanton 2017). There is an increasing interest in such genetic variation within a tumor, or ITH (Intratumor Heterogeneity). ITH could be a significant obstacle to cancer screening and treatment, and understanding how tumors proliferate and accumulate mutations is essential for early detection and treatment decisions (Yates and Campbell 2012; Burrell et al. 2013; Vogelstein et al. 2013; McGranahan and Swanton 2015). Although a cancer cell population can be considered as an asexual population, the evolutionary process may not follow the standard population genetic theory that were mainly built for Wright–Fisher organismal populations. We here describes how Kimura’s neutral theory works in a cancer cell population, and then explain a couple of fundamental differences between cancer cell populations and asexual organismal populations. Kimura’s Contribution to Organismal Population Genetics In addition to the proposal of the “neutral mutation hypothesis” (Kimura 1968a,b; Kimura and Ohta 1971), Kimura’s major contribution to population genetics is the development of fundamental theoretical framework of the behavior of the frequency of neutral (and also nonneutral) alleles mainly in a Wright–Fisher population (Kimura 1955a,b, 1964, 1968b). Neutral theory provides beautiful mathematical expressions on the amounts and patterns of polymorphism and divergence, which are still very useful in the modern population genetics at the molecular level. The analytic expression under neutrality is particularly simple when assuming a single nucleotide polymorphism (SNP) (Kimura 1969). As the mutation rate per site is very low, a biallelic single-locus model with no recurrent mutation can be reasonably applied, where the probability density distribution of the frequency of the derived allele is approximately given by φ(x)∝1/x in a Wright–Fisher population. φ(x) is directly transformed to the infinite-site model that assumes that any new mutation creates a novel polymorphic site: S(i), the expected number of sites with derived allele frequency i/N, is given by   S(i)=Nμ1i (1) in an asexual population with N haploids where μ is the mutation rate per region (Kimura 1969). In molecular population genetics, the neutral theory provides quantitative predictions that can be applied to actual observed data. A genome-wide deviation of allele frequency spectrum (AFS) from equation (1) implies that the population size has not been constant over time, and theoretical predictions of neutral alleles in various population models provide an opportunity to infer demographic models that explain the data well. A commonly used alternative model is an exponentially growing population (fig. 1A), in which excess of rare alleles is expected in comparison with a constant-size population (fig. 1B). Based on this theoretical prediction, it is possible to estimate the growth rate, r, by likelihood-based statistical methods (e.g., Wooding and Rogers 2002). In addition to the past demography, natural selection is another important factor to affect AFS, and the action of natural selection could be documented by testing neutral predictions. Thus, the neutral theory provides the whole theoretical basis of molecular population genetics for living organisms. Fig. 1. View largeDownload slide (A) Exponentially growing populations with different growth rates. (B) Expected allele frequency spectra in exponentially growing populations. The colors are identical to those in (A). Fig. 1. View largeDownload slide (A) Exponentially growing populations with different growth rates. (B) Expected allele frequency spectra in exponentially growing populations. The colors are identical to those in (A). Evolution of Cancer Cell Population In 1976, Nowell proposed that cancer originates from a single normal cell, and cell clones acquiring somatic mutations are subject to natural selection in a stepwise manner (Nowell 1976). In other words, cancer evolution can be regarded as Darwinian evolution of a unicellular organism that undergo asexual reproduction. Since this clonal evolution model was proposed, a number of proto-oncogenes and tumor suppressor genes have been discovered. Several dozens of common cancer driver genes were repeatedly identified, whereas the current list of cancer genes exhibits a long tail of genes mutated at very low frequency (Garraway and Lander 2013; Vogelstein et al. 2013; Rubio-Perez et al. 2015). In 1990, Fearon and Vogelstein (1990) integrated these discoveries with the clonal evolution model, thereby proposing the multistep tumorigenesis model; in colorectal tumorigenesis, while accumulating strong driver mutations in APC, KRAS, TP53 etc., a normal epithelial cell linearly transforms through a benign tumor to a malignant tumor (fig. 2A). Since then, the view that linear Darwinian evolution shapes a uniform population of malignant cells has been well accepted. Fig. 2. View largeDownload slide Illustration of possible models for the evolution of cancer cell population and ITH. (A) Linear Darwinian model, in which driver mutations (red stars) repeatedly fix through evolution. A low level of ITH is expected. (B) Nonneutral model for ITH, where extensive ITH arises through driver mutations (red stars). (C) Neutral model for ITH, where extensive ITH arises through the accumulation of a number of neutral mutations. Fig. 2. View largeDownload slide Illustration of possible models for the evolution of cancer cell population and ITH. (A) Linear Darwinian model, in which driver mutations (red stars) repeatedly fix through evolution. A low level of ITH is expected. (B) Nonneutral model for ITH, where extensive ITH arises through driver mutations (red stars). (C) Neutral model for ITH, where extensive ITH arises through the accumulation of a number of neutral mutations. However, genomic studies employing next-generation sequencers have recently demonstrated that evolution of cancer cell populations is not that simple. A cell population in a single tumor may not be genetically uniform, rather there could be many types of subclones, exhibiting extensive ITH. The evolutionary process is more complicated when metastasis is involved so that multiple tumors develop within a single patient. Our focus is on ITH in a single tumor, and see Hong et al. (2015) and Zhao et al. (2016) for cancer evolution involving metastasis. A fundamental question here is whether ITH consists of subclones with different physiological properties (e.g., growth rate; fig. 2B) or merely neutral mutations (fig. 2C). This problem could be clinically relevant. ITH presumably contributes to many difficulties in cancer treatments; it is expected that the different characteristics of ITH provide clues for developing new therapeutic and diagnostic strategies (Yates and Campbell 2012; Burrell et al. 2013; Vogelstein et al. 2013). Cancer genomics usually relies on next-generation sequencing of a piece from a tumor tissue (usually consists of ∼106 cells [Chen et al. 2015]) compared with the sequence of a normal tissue as a control. Deep sequencing is a very powerful means for studying ITH because it allows us to evaluate the level of genetic variation within a sampled tissue. Deep sequencing produces sequence reads at a much higher coverage than usually (typically from hundreds to thousands), from which we could estimate the frequencies of somatic mutations in the cancer cell population. Although some somatic mutations that occurred in the early phases of cancer progression are nearly fixed, some new mutations could be “polymorphic” in the cell population, which constitute ITH. Applying deep sequencing to multiple regions in a single tumor is even more powerful to understand the entire picture of ITH, and in the recent few years, this multiregion sequencing has been applied to many types of solid cancer (Yap et al. 2012; Martinez et al. 2013; McGranahan and Swanton 2017). It is found that the observed patterns are quite heterogeneous depending on cancer type (Gomez et al. 2018). In some types of cancer exemplified by renal cancer (Gerlinger et al. 2012, 2014) and low-grade glioma (Suzuki et al. 2015), a number of subclonal (i.e., polymorphic) mutations were found in well known cancer driver genes, indicating loss or altered function of these genes could cause fitness differences between subclones (fig. 2B). More interestingly, multiple subclonal mutations existed in the same gene or genes that work together in the same molecular pathway, consistent with the pattern of convergent evolution. These observations would fit to the scenario of nonneutral ITH (fig. 2B). In contrast, recent genomic analyses of multiregion sequencing data together with mathematical modeling have established a novel view: for some types of cancer such as colorectal cancer, the majority of ITH may be shaped by “neutral evolution” (fig. 2C) (Ling et al. 2015; Sottoriva et al. 2015; Uchi et al. 2016). For example, extensive ITH was unveiled by multiregion sequencing of colorectal cancer, but very few subclonal mutations were identified in known driver genes (Uchi et al. 2016). Sottoriva et al. (2015) also found extensive ITH in colorectal tumors that uniformly spread over the entire tumors. Moreover, the authors found that mutations identified in a region of tumor were mixed with spatially separated regions in the same tumor as subclonal mutations of low allele frequencies. Based on this observation, the authors proposed the Big-Bang model, which states that a large number of subclones are generated in the early phases of cancer evolution, after which they expand without selective competition, while partially mixing, to eventually shape uniformly high ITH in the entire region of a tumor. Sottoriva et al. (2015) and Uchi et al. (2016) independently demonstrated that certain simulation models could produce similar patterns to their observations (see Iwasaki and Innan [2017] for detailed arguments on the conditions under which this holds). A new approach for distinguishing the neutral and nonneutral scenarios of ITH is to incorporate information of allele frequency distribution (Bozic et al. 2016; Williams et al. 2016; Sun et al. 2017). Williams et al. (2016) showed that, under neutrality, the expected number of mutations has a linear correlation with the inverse of their frequencies. Therefore, the authors claimed that deviation from this prediction could be considered as a signature of selection. It is technically possible to apply this approach to deep sequencing data from a single region in a tumor. Provided that a large amount of single-profile data are available through public databases such as TCGA (The Cancer Genome Atlas; https://cancergenome.nih.gov, last accessed May 3, 2018), this approach is attractive in that we can compare various cancer types (i.e., pan-cancer analyses). Williams et al. (2016) applied this method to multiple cancer types and found that the neutral prediction well holds for many cases, although the power and accuracy of this analysis to detect selection is under debate (Noorbakhsh and Chuang 2017; Gomez et al. 2018). Bozic et al. (2016) provided a more mathematical treatment and also showed that their neutral prediction was consistent with observed data. Sun et al. (2017) proposed a method to compare allele frequencies between multiple regions. In summary, ITH has been receiving extensive attention particularly in recent few years, but statistical analyses are uninformed by relevant knowledge in theoretical population genetics and do not take full advantage of the highly sophisticated statistical methodologies developed for organismal populations (see refs in this issue). To solve this problem, it is crucial to understand the difference between cancer cell populations and organismal populations, which is described in the next section. Neutral Theory for a Cancer Cell Population As a cancer cell population evolves through proliferation from a single origin cell, one might think that population genetic theory for an asexual Wright–Fisher population may be applicable. However, this is not strictly true because there is no concept of “generation” in a cancer cell population (Sidow and Spies 2015). Whereas all individuals die and replaced by new offspring every generation in a Wright–Fisher population, cancer cells proliferate very quickly, and as a consequence, the population size in general increases dramatically. The reproductive process of a cell population is better described by the Moran model (Moran 1958, 1962), and when a rapid population growth is involved, the branching process provides useful theory that is specified by the birth and death rates (b and d, respectively), where the population growth rate is given by r=b−d (e.g., Durrett 2015). Through the following arguments, we assume b≫d, which should be reasonable in a rapidly growing cancer cell population. Then, under neutrality (i.e., the birth and death rates are uniform for all cells), S(i) is given by   S(i)=Nμ∑k=0∞1(i+k)(i+k+1)(db)k (2) (Ohtsuki and Innan 2017). This equation works fairly well for point mutations with a relatively small mutation rate, but overestimates very rare mutations when mutation rate is highly variable so that recurrent mutation occurs at a certain proportion of sites (Ohtsuki and Innan 2017). We thus can obtain the neutral prediction of allele frequency spectrum for a cancer cell population, which could provide powerful statistical basis for estimating past demography and testing neutrality (or detecting selection) as we commonly do with SNP data from organismal populations. However, there are a couple of caveats in doing so. First is the use of the shape of AFS for inferring the past demographic model (assuming neutrality). AFS should work also in a cancer cell population, but the shape of spectrum may not be very informative because equation (2) converges to   S(i)→Nμ1i(i−1), (3) as r increases (or φ(x)→1/x2), which is independent of the growth rate. Therefore, similar AFS are predicted for large r in figure 1B, indicating that it is very difficult to obtain a reliable estimate of r from the shape of AFS for a very rapidly growing population. Second is on inferring the role of selection. It is a common sense in analyzing SNP data from organismal populations that the absolute number of SNPs is informative to detect recent selection (e.g., Nielsen and Slatkin 2013). This idea should also work in a cancer cell population, but we need more careful consideration. As an example, let us consider a simple exponentially growing population (shown in green) as illustrated in figure 3A (we do not assume neutrality here). Quite recently, an advantageous variant arose within the original population (shown in pink), and it has grown more rapidly than the original type. Then, at present, it is assumed that the sizes of the original and derived types are the same, say N, from which seven individuals are sampled. The true genealogical relationship is shown by the solid line in figure 3A. In such a situation in an organismal population, we expect a reduced amount of variation within the derived type in comparison with the original types as illustrated in figure 3B, which shows the absolute allele frequent distributions of SNPs. Fig. 3. View largeDownload slide (A) An exponentially growing population (green), within which a strong advantageous mutation occurred (the red star) and this subclone has spread more rapidly (pink). (B, C) Suppose the scenario of (A) is applied to an asexual species, where the generation time is constant over the population. (B) AFS for derived SNPs within the original and derived subclones and (C) a gene tree reconstructed from the observed pattern of SNPs. (D, E) Suppose the scenario of (A) is applied to a cancer cell population, where the longevity of cells is very long (i.e., b≫d). AFS and a gene tree are shown in (B) and (C). Fig. 3. View largeDownload slide (A) An exponentially growing population (green), within which a strong advantageous mutation occurred (the red star) and this subclone has spread more rapidly (pink). (B, C) Suppose the scenario of (A) is applied to an asexual species, where the generation time is constant over the population. (B) AFS for derived SNPs within the original and derived subclones and (C) a gene tree reconstructed from the observed pattern of SNPs. (D, E) Suppose the scenario of (A) is applied to a cancer cell population, where the longevity of cells is very long (i.e., b≫d). AFS and a gene tree are shown in (B) and (C). Suppose the same situation is applied to a cancer cell population, within which nucleotide variation is referred to as SNVs (single nucleotide variants), rather than SNPs. From equation (2), we expect that the allele frequency distributions of SNVs for the original and derived types are very similar in both the absolute amount and the shape assuming r for both types are very large (see fig. 3C). It should be noted that equation (2) can be applied to ITH within a subclone in nonneutral model as long as all mutations within the subclone are neutral. It is indicated that comparing AFS between subclone may not work well as a test for selection. This difference comes from the assumption that mutations arise through cell division, and the speed of cell division differs between the two cases: the speed of cell division is accelerated in the advantageous subclones in a cancer cell population, but not in a Wright–Fisher population. In other words, the “molecular clock” works along regular time (or generation) in a Wright–Fisher population, whereas the clock proceeds as the number of cell division increases in a cancer cell population, and the speed of cell division may differ between lineages. Noted that the latter situation is also applicable to an asexual population with heterogenous birth rates (Moran 1958, 1962). Thus, in a rapidly growing cancer cell population, signature of selection might appear differently from typical positive selection in organismal populations. Our description above applies to a very simplified situation with many assumptions. Obviously, the evolution of cancer is not that simple, and more detailed theoretical work under various conditions is needed to capture the general pattern of cancer evolution. Perspectives There is no doubt that growing medical interests are in ITH in cancer cell populations, and the fundamental goals should overlap with those in population genetics. Through a century of the history of population genetics, a number of powerful statistical tools were developed for understanding evolutionary mechanisms behind various kinds of polymorphism data, from simple SNPs, presence/absence polymorphism with recurrent mutations, to complex copy number variations (Crow and Kimura 1970). This population genetic framework should readily contribute to cancer genetics once the differences between cancer cell populations and organismal populations are taken into account. In addition, there are many other factors that potentially important in considering cancer cell evolutions. A tumor consists of different kinds of cells, for example, cancer stem cells and differentiated cells. Cancer stem cells can maintain themselves by self-renewal, but their differentiated daughters are allowed to divide only limited times. Cell differentiation and hierarchical age structure may also affect the formation of ITH through the development of a solid tumor. Cells with different sets of driver mutations should have different physiological properties, including birth and death rates and how they migrate and invade. Environmental factors should also affect these properties because cells compete with each other for various resources, such as space, oxygen, and other nutrients, which are heterogeneous across a tumor. Mutation rate could change over the evolutionary process, particularly due to exposure to stress (e.g., mutagens and replication stress) and deficiency in DNA repair machinery. Furthermore, typical evolutionary behavior and physiological environment differ heavily between cancer types. We have recently developed a simulator named “tumopp” to explore the joint effects of these factors on the formation of ITH (Iwasaki and Innan 2017). As is illustrated in figure 4, the spacial structure dramatically varies depending on parameters. Figure 4A and B both show typical outcomes for neutral cases, but differ in the rate and mode of cell migration (for details, see Iwasaki and Innan 2017). Each simulation run starts from a single cell, and the first four cells after two rounds of cell division are in four different colors. With a setting of less frequent migration, cells with the same color make a local cluster (fig. 4A), whereas cells are well mixed when extensive migration is assumed (fig. 4B). Figure 4C shows a case with driver mutations, where the accumulation of drivers is exhibited through the change in color, from blue to red. It is demonstrated that the tumor shows a uni-directional growth along with acquisition of new drivers. Further detailed analyses on the joint effect of a number of factors on the spatial and temporal evolution of cancer would contribute to our deeper understanding of the the population genetics of cancer cells. Fig. 4. View largeDownload slide 3D structures of simulated tumors by tumopp. (A, B) Neutral cases with less (A) and frequent cell migration (B). Descendants from the first four cells in each simulation run are shown in blue, green, yellow, and red. (C) A case with selection for driver mutations. The colors of cells represent their cell division rates which increased along with the accumulation of driver mutations, scaled from blue to red. Fig. 4. View largeDownload slide 3D structures of simulated tumors by tumopp. (A, B) Neutral cases with less (A) and frequent cell migration (B). Descendants from the first four cells in each simulation run are shown in blue, green, yellow, and red. (C) A case with selection for driver mutations. The colors of cells represent their cell division rates which increased along with the accumulation of driver mutations, scaled from blue to red. References Almassalha LM, Bauer GM, Chandler JE, Gladstein S, Szleifer I, Roy HK, Backman V. 2016. 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Neutral Theory in Cancer Cell Population Genetics

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Abstract

Abstract Kimura’s neutral theory provides the whole theoretical basis of the behavior of mutations in a Wright–Fisher population. We here discuss how it can be applied to a cancer cell population, in which there is an increasing interest in genetic variation within a tumor. We explain a couple of fundamental differences between cancer cell populations and asexual organismal populations. Once these differences are taken into account, a number of powerful theoretical tools developed for a Wright–Fisher population could be readily contribute to our deeper understanding of the evolutionary dynamics of cancer cell population. neutral theory, cancer, selection Introduction We here consider how Kimura’s neutral theory works in cancer cell populations. A tumor evolves from a single cells, grows rapidly, and divide into multiple cell lineages by accumulating numerous mutations. Such variations produce a tumor that consists of heterogeneous subclones rather than a single type of homogeneous clonal cells (Marusyk et al. 2012; Almassalha et al. 2016; Andor et al. 2016; McGranahan and Swanton 2017). There is an increasing interest in such genetic variation within a tumor, or ITH (Intratumor Heterogeneity). ITH could be a significant obstacle to cancer screening and treatment, and understanding how tumors proliferate and accumulate mutations is essential for early detection and treatment decisions (Yates and Campbell 2012; Burrell et al. 2013; Vogelstein et al. 2013; McGranahan and Swanton 2015). Although a cancer cell population can be considered as an asexual population, the evolutionary process may not follow the standard population genetic theory that were mainly built for Wright–Fisher organismal populations. We here describes how Kimura’s neutral theory works in a cancer cell population, and then explain a couple of fundamental differences between cancer cell populations and asexual organismal populations. Kimura’s Contribution to Organismal Population Genetics In addition to the proposal of the “neutral mutation hypothesis” (Kimura 1968a,b; Kimura and Ohta 1971), Kimura’s major contribution to population genetics is the development of fundamental theoretical framework of the behavior of the frequency of neutral (and also nonneutral) alleles mainly in a Wright–Fisher population (Kimura 1955a,b, 1964, 1968b). Neutral theory provides beautiful mathematical expressions on the amounts and patterns of polymorphism and divergence, which are still very useful in the modern population genetics at the molecular level. The analytic expression under neutrality is particularly simple when assuming a single nucleotide polymorphism (SNP) (Kimura 1969). As the mutation rate per site is very low, a biallelic single-locus model with no recurrent mutation can be reasonably applied, where the probability density distribution of the frequency of the derived allele is approximately given by φ(x)∝1/x in a Wright–Fisher population. φ(x) is directly transformed to the infinite-site model that assumes that any new mutation creates a novel polymorphic site: S(i), the expected number of sites with derived allele frequency i/N, is given by   S(i)=Nμ1i (1) in an asexual population with N haploids where μ is the mutation rate per region (Kimura 1969). In molecular population genetics, the neutral theory provides quantitative predictions that can be applied to actual observed data. A genome-wide deviation of allele frequency spectrum (AFS) from equation (1) implies that the population size has not been constant over time, and theoretical predictions of neutral alleles in various population models provide an opportunity to infer demographic models that explain the data well. A commonly used alternative model is an exponentially growing population (fig. 1A), in which excess of rare alleles is expected in comparison with a constant-size population (fig. 1B). Based on this theoretical prediction, it is possible to estimate the growth rate, r, by likelihood-based statistical methods (e.g., Wooding and Rogers 2002). In addition to the past demography, natural selection is another important factor to affect AFS, and the action of natural selection could be documented by testing neutral predictions. Thus, the neutral theory provides the whole theoretical basis of molecular population genetics for living organisms. Fig. 1. View largeDownload slide (A) Exponentially growing populations with different growth rates. (B) Expected allele frequency spectra in exponentially growing populations. The colors are identical to those in (A). Fig. 1. View largeDownload slide (A) Exponentially growing populations with different growth rates. (B) Expected allele frequency spectra in exponentially growing populations. The colors are identical to those in (A). Evolution of Cancer Cell Population In 1976, Nowell proposed that cancer originates from a single normal cell, and cell clones acquiring somatic mutations are subject to natural selection in a stepwise manner (Nowell 1976). In other words, cancer evolution can be regarded as Darwinian evolution of a unicellular organism that undergo asexual reproduction. Since this clonal evolution model was proposed, a number of proto-oncogenes and tumor suppressor genes have been discovered. Several dozens of common cancer driver genes were repeatedly identified, whereas the current list of cancer genes exhibits a long tail of genes mutated at very low frequency (Garraway and Lander 2013; Vogelstein et al. 2013; Rubio-Perez et al. 2015). In 1990, Fearon and Vogelstein (1990) integrated these discoveries with the clonal evolution model, thereby proposing the multistep tumorigenesis model; in colorectal tumorigenesis, while accumulating strong driver mutations in APC, KRAS, TP53 etc., a normal epithelial cell linearly transforms through a benign tumor to a malignant tumor (fig. 2A). Since then, the view that linear Darwinian evolution shapes a uniform population of malignant cells has been well accepted. Fig. 2. View largeDownload slide Illustration of possible models for the evolution of cancer cell population and ITH. (A) Linear Darwinian model, in which driver mutations (red stars) repeatedly fix through evolution. A low level of ITH is expected. (B) Nonneutral model for ITH, where extensive ITH arises through driver mutations (red stars). (C) Neutral model for ITH, where extensive ITH arises through the accumulation of a number of neutral mutations. Fig. 2. View largeDownload slide Illustration of possible models for the evolution of cancer cell population and ITH. (A) Linear Darwinian model, in which driver mutations (red stars) repeatedly fix through evolution. A low level of ITH is expected. (B) Nonneutral model for ITH, where extensive ITH arises through driver mutations (red stars). (C) Neutral model for ITH, where extensive ITH arises through the accumulation of a number of neutral mutations. However, genomic studies employing next-generation sequencers have recently demonstrated that evolution of cancer cell populations is not that simple. A cell population in a single tumor may not be genetically uniform, rather there could be many types of subclones, exhibiting extensive ITH. The evolutionary process is more complicated when metastasis is involved so that multiple tumors develop within a single patient. Our focus is on ITH in a single tumor, and see Hong et al. (2015) and Zhao et al. (2016) for cancer evolution involving metastasis. A fundamental question here is whether ITH consists of subclones with different physiological properties (e.g., growth rate; fig. 2B) or merely neutral mutations (fig. 2C). This problem could be clinically relevant. ITH presumably contributes to many difficulties in cancer treatments; it is expected that the different characteristics of ITH provide clues for developing new therapeutic and diagnostic strategies (Yates and Campbell 2012; Burrell et al. 2013; Vogelstein et al. 2013). Cancer genomics usually relies on next-generation sequencing of a piece from a tumor tissue (usually consists of ∼106 cells [Chen et al. 2015]) compared with the sequence of a normal tissue as a control. Deep sequencing is a very powerful means for studying ITH because it allows us to evaluate the level of genetic variation within a sampled tissue. Deep sequencing produces sequence reads at a much higher coverage than usually (typically from hundreds to thousands), from which we could estimate the frequencies of somatic mutations in the cancer cell population. Although some somatic mutations that occurred in the early phases of cancer progression are nearly fixed, some new mutations could be “polymorphic” in the cell population, which constitute ITH. Applying deep sequencing to multiple regions in a single tumor is even more powerful to understand the entire picture of ITH, and in the recent few years, this multiregion sequencing has been applied to many types of solid cancer (Yap et al. 2012; Martinez et al. 2013; McGranahan and Swanton 2017). It is found that the observed patterns are quite heterogeneous depending on cancer type (Gomez et al. 2018). In some types of cancer exemplified by renal cancer (Gerlinger et al. 2012, 2014) and low-grade glioma (Suzuki et al. 2015), a number of subclonal (i.e., polymorphic) mutations were found in well known cancer driver genes, indicating loss or altered function of these genes could cause fitness differences between subclones (fig. 2B). More interestingly, multiple subclonal mutations existed in the same gene or genes that work together in the same molecular pathway, consistent with the pattern of convergent evolution. These observations would fit to the scenario of nonneutral ITH (fig. 2B). In contrast, recent genomic analyses of multiregion sequencing data together with mathematical modeling have established a novel view: for some types of cancer such as colorectal cancer, the majority of ITH may be shaped by “neutral evolution” (fig. 2C) (Ling et al. 2015; Sottoriva et al. 2015; Uchi et al. 2016). For example, extensive ITH was unveiled by multiregion sequencing of colorectal cancer, but very few subclonal mutations were identified in known driver genes (Uchi et al. 2016). Sottoriva et al. (2015) also found extensive ITH in colorectal tumors that uniformly spread over the entire tumors. Moreover, the authors found that mutations identified in a region of tumor were mixed with spatially separated regions in the same tumor as subclonal mutations of low allele frequencies. Based on this observation, the authors proposed the Big-Bang model, which states that a large number of subclones are generated in the early phases of cancer evolution, after which they expand without selective competition, while partially mixing, to eventually shape uniformly high ITH in the entire region of a tumor. Sottoriva et al. (2015) and Uchi et al. (2016) independently demonstrated that certain simulation models could produce similar patterns to their observations (see Iwasaki and Innan [2017] for detailed arguments on the conditions under which this holds). A new approach for distinguishing the neutral and nonneutral scenarios of ITH is to incorporate information of allele frequency distribution (Bozic et al. 2016; Williams et al. 2016; Sun et al. 2017). Williams et al. (2016) showed that, under neutrality, the expected number of mutations has a linear correlation with the inverse of their frequencies. Therefore, the authors claimed that deviation from this prediction could be considered as a signature of selection. It is technically possible to apply this approach to deep sequencing data from a single region in a tumor. Provided that a large amount of single-profile data are available through public databases such as TCGA (The Cancer Genome Atlas; https://cancergenome.nih.gov, last accessed May 3, 2018), this approach is attractive in that we can compare various cancer types (i.e., pan-cancer analyses). Williams et al. (2016) applied this method to multiple cancer types and found that the neutral prediction well holds for many cases, although the power and accuracy of this analysis to detect selection is under debate (Noorbakhsh and Chuang 2017; Gomez et al. 2018). Bozic et al. (2016) provided a more mathematical treatment and also showed that their neutral prediction was consistent with observed data. Sun et al. (2017) proposed a method to compare allele frequencies between multiple regions. In summary, ITH has been receiving extensive attention particularly in recent few years, but statistical analyses are uninformed by relevant knowledge in theoretical population genetics and do not take full advantage of the highly sophisticated statistical methodologies developed for organismal populations (see refs in this issue). To solve this problem, it is crucial to understand the difference between cancer cell populations and organismal populations, which is described in the next section. Neutral Theory for a Cancer Cell Population As a cancer cell population evolves through proliferation from a single origin cell, one might think that population genetic theory for an asexual Wright–Fisher population may be applicable. However, this is not strictly true because there is no concept of “generation” in a cancer cell population (Sidow and Spies 2015). Whereas all individuals die and replaced by new offspring every generation in a Wright–Fisher population, cancer cells proliferate very quickly, and as a consequence, the population size in general increases dramatically. The reproductive process of a cell population is better described by the Moran model (Moran 1958, 1962), and when a rapid population growth is involved, the branching process provides useful theory that is specified by the birth and death rates (b and d, respectively), where the population growth rate is given by r=b−d (e.g., Durrett 2015). Through the following arguments, we assume b≫d, which should be reasonable in a rapidly growing cancer cell population. Then, under neutrality (i.e., the birth and death rates are uniform for all cells), S(i) is given by   S(i)=Nμ∑k=0∞1(i+k)(i+k+1)(db)k (2) (Ohtsuki and Innan 2017). This equation works fairly well for point mutations with a relatively small mutation rate, but overestimates very rare mutations when mutation rate is highly variable so that recurrent mutation occurs at a certain proportion of sites (Ohtsuki and Innan 2017). We thus can obtain the neutral prediction of allele frequency spectrum for a cancer cell population, which could provide powerful statistical basis for estimating past demography and testing neutrality (or detecting selection) as we commonly do with SNP data from organismal populations. However, there are a couple of caveats in doing so. First is the use of the shape of AFS for inferring the past demographic model (assuming neutrality). AFS should work also in a cancer cell population, but the shape of spectrum may not be very informative because equation (2) converges to   S(i)→Nμ1i(i−1), (3) as r increases (or φ(x)→1/x2), which is independent of the growth rate. Therefore, similar AFS are predicted for large r in figure 1B, indicating that it is very difficult to obtain a reliable estimate of r from the shape of AFS for a very rapidly growing population. Second is on inferring the role of selection. It is a common sense in analyzing SNP data from organismal populations that the absolute number of SNPs is informative to detect recent selection (e.g., Nielsen and Slatkin 2013). This idea should also work in a cancer cell population, but we need more careful consideration. As an example, let us consider a simple exponentially growing population (shown in green) as illustrated in figure 3A (we do not assume neutrality here). Quite recently, an advantageous variant arose within the original population (shown in pink), and it has grown more rapidly than the original type. Then, at present, it is assumed that the sizes of the original and derived types are the same, say N, from which seven individuals are sampled. The true genealogical relationship is shown by the solid line in figure 3A. In such a situation in an organismal population, we expect a reduced amount of variation within the derived type in comparison with the original types as illustrated in figure 3B, which shows the absolute allele frequent distributions of SNPs. Fig. 3. View largeDownload slide (A) An exponentially growing population (green), within which a strong advantageous mutation occurred (the red star) and this subclone has spread more rapidly (pink). (B, C) Suppose the scenario of (A) is applied to an asexual species, where the generation time is constant over the population. (B) AFS for derived SNPs within the original and derived subclones and (C) a gene tree reconstructed from the observed pattern of SNPs. (D, E) Suppose the scenario of (A) is applied to a cancer cell population, where the longevity of cells is very long (i.e., b≫d). AFS and a gene tree are shown in (B) and (C). Fig. 3. View largeDownload slide (A) An exponentially growing population (green), within which a strong advantageous mutation occurred (the red star) and this subclone has spread more rapidly (pink). (B, C) Suppose the scenario of (A) is applied to an asexual species, where the generation time is constant over the population. (B) AFS for derived SNPs within the original and derived subclones and (C) a gene tree reconstructed from the observed pattern of SNPs. (D, E) Suppose the scenario of (A) is applied to a cancer cell population, where the longevity of cells is very long (i.e., b≫d). AFS and a gene tree are shown in (B) and (C). Suppose the same situation is applied to a cancer cell population, within which nucleotide variation is referred to as SNVs (single nucleotide variants), rather than SNPs. From equation (2), we expect that the allele frequency distributions of SNVs for the original and derived types are very similar in both the absolute amount and the shape assuming r for both types are very large (see fig. 3C). It should be noted that equation (2) can be applied to ITH within a subclone in nonneutral model as long as all mutations within the subclone are neutral. It is indicated that comparing AFS between subclone may not work well as a test for selection. This difference comes from the assumption that mutations arise through cell division, and the speed of cell division differs between the two cases: the speed of cell division is accelerated in the advantageous subclones in a cancer cell population, but not in a Wright–Fisher population. In other words, the “molecular clock” works along regular time (or generation) in a Wright–Fisher population, whereas the clock proceeds as the number of cell division increases in a cancer cell population, and the speed of cell division may differ between lineages. Noted that the latter situation is also applicable to an asexual population with heterogenous birth rates (Moran 1958, 1962). Thus, in a rapidly growing cancer cell population, signature of selection might appear differently from typical positive selection in organismal populations. Our description above applies to a very simplified situation with many assumptions. Obviously, the evolution of cancer is not that simple, and more detailed theoretical work under various conditions is needed to capture the general pattern of cancer evolution. Perspectives There is no doubt that growing medical interests are in ITH in cancer cell populations, and the fundamental goals should overlap with those in population genetics. Through a century of the history of population genetics, a number of powerful statistical tools were developed for understanding evolutionary mechanisms behind various kinds of polymorphism data, from simple SNPs, presence/absence polymorphism with recurrent mutations, to complex copy number variations (Crow and Kimura 1970). This population genetic framework should readily contribute to cancer genetics once the differences between cancer cell populations and organismal populations are taken into account. In addition, there are many other factors that potentially important in considering cancer cell evolutions. A tumor consists of different kinds of cells, for example, cancer stem cells and differentiated cells. Cancer stem cells can maintain themselves by self-renewal, but their differentiated daughters are allowed to divide only limited times. Cell differentiation and hierarchical age structure may also affect the formation of ITH through the development of a solid tumor. Cells with different sets of driver mutations should have different physiological properties, including birth and death rates and how they migrate and invade. Environmental factors should also affect these properties because cells compete with each other for various resources, such as space, oxygen, and other nutrients, which are heterogeneous across a tumor. Mutation rate could change over the evolutionary process, particularly due to exposure to stress (e.g., mutagens and replication stress) and deficiency in DNA repair machinery. Furthermore, typical evolutionary behavior and physiological environment differ heavily between cancer types. We have recently developed a simulator named “tumopp” to explore the joint effects of these factors on the formation of ITH (Iwasaki and Innan 2017). As is illustrated in figure 4, the spacial structure dramatically varies depending on parameters. Figure 4A and B both show typical outcomes for neutral cases, but differ in the rate and mode of cell migration (for details, see Iwasaki and Innan 2017). Each simulation run starts from a single cell, and the first four cells after two rounds of cell division are in four different colors. With a setting of less frequent migration, cells with the same color make a local cluster (fig. 4A), whereas cells are well mixed when extensive migration is assumed (fig. 4B). Figure 4C shows a case with driver mutations, where the accumulation of drivers is exhibited through the change in color, from blue to red. It is demonstrated that the tumor shows a uni-directional growth along with acquisition of new drivers. Further detailed analyses on the joint effect of a number of factors on the spatial and temporal evolution of cancer would contribute to our deeper understanding of the the population genetics of cancer cells. Fig. 4. View largeDownload slide 3D structures of simulated tumors by tumopp. (A, B) Neutral cases with less (A) and frequent cell migration (B). Descendants from the first four cells in each simulation run are shown in blue, green, yellow, and red. (C) A case with selection for driver mutations. The colors of cells represent their cell division rates which increased along with the accumulation of driver mutations, scaled from blue to red. Fig. 4. View largeDownload slide 3D structures of simulated tumors by tumopp. (A, B) Neutral cases with less (A) and frequent cell migration (B). Descendants from the first four cells in each simulation run are shown in blue, green, yellow, and red. (C) A case with selection for driver mutations. The colors of cells represent their cell division rates which increased along with the accumulation of driver mutations, scaled from blue to red. References Almassalha LM, Bauer GM, Chandler JE, Gladstein S, Szleifer I, Roy HK, Backman V. 2016. 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Published: Apr 27, 2018

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