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Title: Quantifying Clonal and Subclonal Passenger Mutations in Cancer Evolution}
Article:
Abstract:
The vast majority of mutations in the exome of cancer cells are passengers, which do not affect the reproductive rate of the cell. Passengers can provide important information about the evolutionary history of an individual cancer, and serve as a molecular clock. Passengers can also become targets for immunotherapy or confer resistance to treatment. We study the stochastic expansion of a population of cancer cells describing the growth of primary tumors or metastatic lesions. We first analyze the process by looking forward in time and calculate the fixation probabilities and frequencies of successive passenger mutations ordered by their time of appearance. We compute the likelihood of specific evolutionary trees, thereby informing the phylogenetic reconstruction of cancer evolution in individual patients. Next, we derive results looking backward in time: for a given subclonal mutation we estimate the number of cancer cells that were present at the time when that mutation arose. We derive exact formulas for the expected numbers of subclonal mutations of any frequency. Fitting this formula to cancer sequencing data leads to an estimate for the ratio of birth and death rates of cancer cells during the early stages of clonal expansion.
Introduction:
In healthy tissues, cell division and cell death are tightly controlled processes, which enable a precise balance assuring that the number of cells in the body remains approximately constant. However, during each cell division mistakes in DNA replication can occur, leading to accumulation of mutations in individual cells[1, 2]. The majority of such mutations are effectively neutral( passengers), but some of them( drivers) can provide selective advantage to the cell, by tipping the balance of cell division and death slightly in favor of increased proliferation[3–5]. This unwanted evolution[6–8] of somatic cells can lead to a clonal expansion of cells with driver mutations, which can ultimately result in the formation of tumors and seeding of new lesions in distant tissues[9–11]. Sequencing efforts over the past decades have resulted in a compendium of genetic alterations in the exomes of common human cancers and revealed that adult cancers harbor dozens( leukemias) to hundreds( lung cancer and melanoma) of somatic mutations. A typical tumor is thought to contain 2-8 driver gene mutations, with the rest being neutral passengers[9]. Unlike driver mutations, passengers cannot be attacked by conventional targeted therapy, but some of them can become targets for immunotherapy or induce resistance to treatment[12–14]. In addition, passenger mutations can provide information about the timing of cancer evolution in individual patients by acting as a molecular clock[15]. Recent studies found that the evolution of metastases in colorectal[16] and pancreatic cancer[15], and even the evolution of primary colorectal cancer follows largely neutral evolution[17], in which the founding cell starts a clonal expansion during which cells accumulate neutral mutations( passengers) rather than drivers. Here we study neutral evolution during clonal expansion and show that passenger mutations can be used to infer the parameters of the tumorigenic process. Our model is a generalization of the famous Luria-Delbrück model for studying resistance mutations in bacteria[18, 19]. In contrast to the original Luria-Delbrück model, in which wild type and mutant populations grew deterministically and mutation occurred stochastically, in our model all cell types grow stochastically; our model also includes cell death. Many authors have studied the fully stochastic version of the Luria-Delbrück model in which all mutants are treated as a single population[20–25]. In these models there are only two cell populations: wild type and mutant. In contrast, here we study populations started by individual passenger mutations separately.
Discussion:
In summary, we have shown that the frequencies of the first and thus most abundant passenger mutations are influenced not only by the mutation rate, but also by the death-birth ratio, δ = d/b, of the cancer cells. If δ is close to 1, which is the relevant case for slow overall growth, then several clonal passengers may not have been present in the first tumor cell, but were collected during clonal expansion. The ratio δ also determines the shape of the cancer’s phylogenetic tree, which is star-shaped for fast growth and linear for slow growth( Fig 3). Additionally, if we consider a mutation that has a certain observed frequency in the population of cancer cells, we can ask: how many cells were present when that mutation arose? The answer varies hundred-fold as the death-birth ratio, δ, changes from 0 to 0. 99. This is particularly relevant as high levels of cell death are reported in both malignant and premalignant tissues[36, 37]. In this work, we derive a simple form Eq( 7) for the cumulative distribution of mutant allele frequencies under a neutral model of cancer evolution. Fitting the cumulative distribution of allele fractions in a sample to this functional form and analyzing the goodness of fit provides information about the nature of the process underlying the generation of mutant alleles. We computed this distribution with data from each patient by first dividing all allele fractions in a single patient by the sample purity and then restricting our view to allele fractions in the interval[. 12,. 25]. The high end of interval was chosen to minimize the chance clonal mutations appeared in it and the low end was chosen to ensure that sequencing was powered to detect such mutations. The cumulative distribution of allele fractions in this interval was computed at each mutant allele fraction as the number of mutations greater than that fraction but less than or equal to the cutoff at 0. 25. Finally, we performed a least-squares fit to the functional form described using MATLAB and report fits with R2 values at least 0. 9. Our model is applicable to individual cancers in which there are no subclonal drivers at observable cell frequencies. This includes both liquid and solid tumors. It has recently been shown that colorectal tumors fit this model often[17, 34]. In contrast, some liquid cancers such as chronic lymphocytic leukemia usually harbor subclonal drivers, and are thus not good candidates for the application of our model[38]. In a previous paper[14], we have studied the accumulation of individual resistance mutations in cancer using a fully stochastic Luria-Delbrück model. For targeted therapies, resistance mutations are estimated to be rare: only about a hundred positions in the genome can give rise to resistance if mutated[29, 39, 40]. Therefore all results in Ref.[14] were derived in the limit of very small mutation rates, about ∼10−7 per cell division. In that scenario, two resistance mutations are virtually never present in the same cell, and the fixation of mutations in the population is extremely unlikely. In contrast, here we are interested in the accumulation of passenger mutations in the whole exome, which leads to a large number of passenger mutations in the population. Even individual cells contain many different mutations. Therefore, new questions arise and a new mathematical approach is needed. Our model is a continuous time version of the infinite alleles branching process introduced by Griffiths and Pakes[41], and is a special case( birth-death process) of the model studied recently by Wu and Kimmel[42]. These works were mostly interested in the limiting frequency spectrum of the process, namely the number of mutations present in j individuals as time t → ∞; for example, Wu and Kimmel provide an explicit expression for the mean limiting frequency spectrum for the birth-death process in terms of the hypergeometric function. We study the same process with respect to tumor size and derive explicit expressions for the frequencies of mutations according to their order of appearance. We also study the expected number of mutations above a certain frequency, which has connections to the frequency spectrum. Sottoriva and Graham[34] estimate the number of cells, including the new cell, that were present when a mutation with observed frequency α appeared by 1/α, using a deterministic model with no cell death. A deterministic model is always useful as it provides the simplest approach to study evolutionary dynamics. Our formula( 4) provides the stochastic correction to their prediction. Recently, Durrett[43] derived formulas for the expected number of passenger mutations present at a frequency larger than α. His process differs from ours as in his model mutations occur independent of cell division. Consequently, the founding cell can collect mutations prior to the first cell division, and even for d = 0 there could be passenger mutations that are clonal. In contrast, in our model for d = 0 all mutations are strictly subclonal. In this paper we do not consider loss of heterozygosity( LOH), which implies that mutations can be lost from the cell during cell division. The rate of LOH is on the order of 10−6 or lower per cell division in tumors that do not have chromosomal instability( CIN)[44]. For such small LOH rate, if LOH events are neutral or deleterious, the fraction of cells that gained a mutation but then lost it will be very small, and our results would still hold; our results would also hold for LOH events that occur at higher rates but are deleterious. In a subsequent paper we will study the effect of LOH events on the evolutionary dynamics of passenger mutations, and focus on situations where they occur with high frequency( CIN)[45, 46]. Similarly, here we do not consider the effect of new driver mutations that may appear during clonal expansion of cancer[5, 47–49]. The addition of drivers may change both the observed mutational frequencies and the phylogenies of the occurring mutations. Instead we focus only on the accumulation of neutral mutations, which is the relevant case for studying the growth of metastases and even some primary tumors[15–17]. While cancer spends much time in clonal expansion, plateau stages are also common. The effects of plateau stages after clonal expansion on the frequency of passenger mutations can be studied using a density-dependent branching process used previously in the context of resistance to cancer therapy[50]. This density-dependent model can be approximated analytically with a two phase model: a branching process with constant birth and death rates b and d corresponding to the growth phase, and a plateau phase of length T in which birth rate of all cells is approximately equal to d. In this model, mutations present at the end of the plateau phase can be either “old” mutations that were present at the end of the growth phase, or “new” mutations that appeared during the plateau phase and were not lost. In the large time T limit, all old mutations will either reach fixation or be lost in the population, but for shorter times T frequencies of old mutations can be approximated by their frequencies at the end of the exponential growth phase. On the other hand, the number of new mutations that are present above a certain frequency can be studied analytically using techniques from[50] to show that new mutations will in general not be present at observable frequencies if the population size at the plateau is on the order of millions of cells or higher. Rodrigues-Brenes et al. recently studied inhibited cancer growth in the context of stem-cell driven cancers[51]. In our model we do not explicitly take into account the possible existence of a differentiation hierarchy within the cancer population: namely, the existence of cancer stem cells, which are able to propagate the tumor population indefinitely, and differentiated cells, which have limited life-spans. However, our results can also inform the study of this more complicated situation, and our analysis in fact applies to cancer stem cells. To add differentiated cells, we can consider the model in which, in addition to our basic assumptions, stem cells can also divide to produce one stem cell and one differentiated cell of generation 1, differentiated cells of generation i divide to produce two differentiated cells of generation i + 1 and the lifespan of differentiated cells is n divisions( i. e. differentiated cells of generation n are lost from the population). A reasonable estimate for the number of divisions before mitotic arrest is n = 10( e. g. 4-6 divisions in colon[52] and 15-20 in hematopoietic system[53]), which means that each differentiated cell of generation 1 produces ∼1000 cells before they are lost from the population. Typical detectable tumors contain billions of cells; mutations that occur in the lineage of a single differentiated cell will remain confined to that lineage, which will contain no more than ∼1000 cells( or no more than ∼106 cells if n = 20), and will not reach a significant fraction in the population( less than 1/106 for n = 10 or less than 1/103 for n as high as 20). Hence mutations appearing in the lineages of differentiated cells will not be present at frequencies above 0. 1% or 1% that we are interested in—mutations above these frequencies will be only those in the stem cell population, which will behave as described in our model. The only adjustment that needs to be made to our results when referring to cancer stem cells is the adjustment of the mutation rate to account also for mutations that occur to stem cells during asymmetric divisions. We recently developed a spatial version of the model studied in this paper, which we mostly analyzed through computer simulations[54]. In this spatial model, tumor growth occurs on a 3-dimensional lattice and birth rate is reduced in the presence of many neighboring cancer cells. This results in the inside of the tumor being in the state of equilibrium between birth and death, while the surface of the tumor is able to expand. If the effective birth rate of cells in the surface of the spatial model is comparable to the birth rate in the well-mixed model, there will be more mutations present above a certain frequency in the spatial case, as the spatial tumor experienced more divisions to reach the same size. However, the addition of migration of tumor cells allows cells to explore less crowded spatial positions and in turn reduces the number of divisions needed to reach the same size as well as the number of mutations above a certain frequency, bringing this model closer to the model without space[54].
Cancer is the consequence of an evolutionary process , which lasts several decades , is impossible to observe during most of its time , and only becomes apparent in late stages . We use mathematical modeling to shed light on the evolutionary dynamics of cancer by studying the accumulation of passenger mutations . We show that the frequencies obtained by passenger mutations depend strongly on the ratio of death and birth rates of cancer cells . We use genetic data of colorectal cancer to estimate this important quantity in vivo . We estimate the size of the cancer cell population that was present when a specific mutation first emerged . Our theory informs the analysis of cancer sequencing data and the phylogenetic reconstruction of cancer evolution .