Abstract
Cancer progression is an example of a rapid adaptive process where evolving new traits is essential for survival and requires a high mutation rate. Precancerous cells acquire a few key mutations that drive rapid population growth and carcinogenesis. Cancer genomics demonstrates that these few ‘driver’ mutations occur alongside thousands of random ‘passenger’ mutations—a natural consequence of cancer’s elevated mutation rate. Some passengers can be deleterious to cancer cells, yet have been largely ignored in cancer research. In population genetics, however, the accumulation of mildly deleterious mutations has been shown to cause population meltdown. Here we develop a stochastic population model where beneficial drivers engage in a tug-of-war with frequent mildly deleterious passengers. These passengers present a barrier to cancer progression that is described by a critical population size, below which most lesions fail to progress, and a critical mutation rate, above which cancers meltdown. We find support for the model in cancer age-incidence and cancer genomics data that also allow us to estimate the fitness advantage of drivers and fitness costs of passengers. We identify two regimes of adaptive evolutionary dynamics and use these regimes to rationalize successes and failures of different treatment strategies. We find that a tumor’s load of deleterious passengers can explain previously paradoxical treatment outcomes and suggest that it could potentially serve as a biomarker of response to mutagenic therapies. The collective deleterious effect of passengers is currently an unexploited therapeutic target. We discuss how their effects might be exacerbated by both current and future therapies.
Introduction
While many populations evolve new traits via a gradual accumulation of changes, some adapt very rapidly. Examples include viral adaptation during infection (1); the emergence of antibiotic resistance (2); artificial selection in biotechnology (3); and cancer (4). Rapid adaptation is characterize by three key features: (i) the availability of strongly advantageous traits accessible by a few mutations, (ii) an elevated mutation rate (5, 6), and (iii) a dynamic population size (7). Traditional theories of gradual adaptation are not applicable under these conditions, and new approaches are needed to address pressing problems in medicine and biotechnology.
Cancer progression is an example of a rapidly adapting population: cancers develop as many as ten new traits (8), often have a high mutation rate (8–10), and a population size that is rapidly changing over time. This process is driven by a handful of mutations and chromosomal abnormalities in cancer-related genes (oncogenes and tumor suppressors), collectively called drivers. From an evolutionary point of view, drivers are mutations that are beneficial to cancer cells because their phenotypes increase the cell proliferation rate or eliminate brakes on proliferation (8). Drivers, however, arise alongside thousands of other mutations and alterations dispersed through the genome that have no immediate beneficial effect, collectively called passengers.
Passengers have been previously assumed to be neutral and largely ignored in cancer research, yet growing evidence suggests that they may sometimes be deleterious to cancer cells and, thus, play an important role in both neoplastic progression and clinical outcomes. In an earlier study, we showed that deleterious passengers can readily accumulate during tumor progression and found that many passengers present in cancer genomes exhibit signatures of damaging mutations (11). Additionally, chromosomal gains and losses that are pervasive in cancer can be passengers, and have been shown to be highly damaging to cancer cells (12). Lastly, cancers with high levels of chromosomal alterations exhibit better clinical outcomes in breast, ovarian, gastric, and non-small cell lung cancer (13). Passenger mutations and chromosomal abnormalities can be deleterious via a variety of mechanisms: direct loss-of-function (14), proteotoxic cytotoxicity from protein disbalance and aggregation (15), or by inciting an immune response (16).
While the role of deleterious mutations in cancer is largely unknown, their effects on natural populations has been extensively studied in genetics (5, 17–19). The accumulations of deleterious mutations can cause the extinction of a population via processes known as Muller’s ratchet and mutational meltdown (17, 20, 21) It was recently proposed that inevitable accumulation of deleterious mutations in natural populations should be offset by new beneficial mutations, leading to long-term population stability (19). Here we consider a rapid adaptation of a population with a variable size and subject of a high mutation rate. A rapidly adapting population faces a double bind: it must quickly acquire, often exceeding rare, adaptive mutations and yet avoid mutational meltdown. As a result, adaptive processes frequently fail. Indeed, less than 0.1% of species on earth have adapted fast enough to avoid extinction (22) and, similarly, only about 0.1% of precancerous lesions ever advance to cancer (23). To control cancer or pathogens, we should understand the constraints that evolution imposes on their rapid adaptation.
Here we investigate how asexual populations such as tumors rapidly evolve new traits while avoiding mutational meltdown. Unlike classical theories of gradual adaptation, the evolutionary model we develop has three key features: (i) rare, strongly advantageous driver mutations, (ii) a high mutation rate that makes moderately deleterious passengers relevant, and (iii) a population size that varies with the fitness of individual cells. We found that a tug-of-war between beneficial drivers and deleterious passengers creates two major regimes of population dynamics: an adaptive regime, where the probability of adaptation (cancer) is high; and a non-adaptive regime, where adaptation (cancer) is exceedingly rare.
Adaptive and non-adaptive regimes are separated by a critical population size or barrier to cancer progression that most lesions fail to overcome, and a critical mutation rate that leads to mutational meltdown. We found strong evidence of these phenomena in age-incidence curves and recent cancer genomics data. Agreement of the model with these data allows us to estimate the selective advantages of drivers as 10-50%, a range consistent with recent direct experimental measurements (24). Genomic data also show that deleterious passengers are approximately 100 times weaker. Our model offers a new interpretation of cancer treatment strategies and explains a previously paradoxical relationship between cancer mutation rates and therapeutic outcomes. Most importantly, it suggests that deleterious passengers offer a new, unexploited avenue of cancer therapy.
Results
Model
We consider a dynamic population of cells that can divide, mutate (in a general sense, i.e. including alterations, epigenetic changes, etc), and die stochastically. Mutations occur during cell divisions with a per-locus rate μ. The number of driver loci in the genome, i.e. a driver target size, is Td, while the target size for deleterious passengers is Tp. Hence, the genome-wide driver and passenger mutation rates are μd = μTd and μp = μΤp respectively. A driver increases an individual’s growth rate by sd ~ 0.1, while a new passenger decreases the growth rate by sp ~ 10−4 – 10−1. Here we only consider fixed values of sd and sp because previous work showed that drivers and passengers sampled from various fitness distributions (exponential, normal, and Gamma) exhibit essentially the same dynamics (11). The net effect of multiple mutations on cell fitness w is given by w = (1 + sd)nd (1 + sp)np, where nd and np are the total number of drivers and passengers in a cell.
The birth and death rates of a cell in our model depend not only on fitness, but also on the population size N via a Gompertzian growth function often used to describe cancerous populations (25) (see SI for details). At large N, deaths exceed births and tumors must adapt (or innovate) via new drivers to progress to larger population sizes. Thus, populations in our model expand and shrink in two ways: on a short time-scale due to stochastic cell divisions and deaths, and on a long time-scale due to the accumulation of advantageous and deleterious mutations. Previous models of advantageous and deleterious mutations have not considered a varying population size (26, 27).
In cancer and other adapting populations the target size for advantageous mutations (drivers) is much smaller than the target size for deleterious mutations (Td ≪ Tp). If driver loci include a few specific sites (~ 10 per gene) in all cancer-associated genes (approximately 100, (28)), then collectively drivers will constitute less than one one-millionth of the genome. Conversely, as much as 10% of the human genome is well-conserved and likely deleterious when mutated (29, 30). In natural populations, Tp should still remain much greater than Td simply because natural selection optimizes genomes to their environment, implying that most changes will be neutral or damaging. Indeed, most protein coding mutations and alterations were deleterious or neutral when investigated in fly (31), yeast (32), and bacterial genomes (33). We consider only moderately deleterious loci here (sp ≈ 10−4 – 10−1)—which account for most nonsynonymous mutations (34, 35). Deleterious mutations outside of this range either do not fixate or negligibly alter progression (11). Hence, we used a conservative size of Tp ≈ 105 – 107 loci to account for passengers with fitness effects outside of this range that we are neglecting (see SI and Table S1 for details of parameters estimation). This quantity is still much greater than Td. Finally, we explored a variety of driver fitness advantages, as estimates in the literature ranged from 0.0001 (36) to 1 (24).
A critical population size
Figure 1A shows the dynamics N(t) of individual populations starting at different initial sizes N0, which correspond to different potential hyperplasia sizes (we begin trajectories immediately after a stem cell acquires its first driver, see SI for a discussion of dynamics before this time point). Populations exhibit two ultimate outcomes: growth to macroscopic size (i.e. cancer progression), or extinction. The prevalence of either outcome is determined by a critical population size N*, about which larger populations (N > N*) generally commit to rapid growth and smaller populations (N < N*) generally commit to extinction.
To understand the cause of this critical population size N*, we looked at the short-term dynamics of populations. All trajectories show a reversed saw-toothed pattern (Fig. 1B), which result from a tug-of-war between drivers and passengers (11). When a new driver arises and takes over the population, the population size increases to a new stationary value. In between these rare driver events, the population size gradually decreases due to the accumulation of deleterious passengers. The relative rate of these competing processes determines whether a population commits to rapid growth or goes extinct.
We can identify the location of N* by considering the average change in population size over time (< dN/dt >), which is simply the average population growth due to driver accumulation (vd) minus the population decline due to passenger accumulation (vp). Fixation of a new driver causes an immediate jump in population size ΔN = Nsd. These jumps occur randomly at a nearly constant rate f = μdNsd, given by the driver occurrence rate μdN, multiplied by a driver’s fixation probability sd/(1 + sd) ≈ sd. Hence, the velocity due to drivers is . Similarly, passengers’ velocity (vp = μpNsp) is a product of their rate of occurrence (μpΝ); their effect on population size (Nsp); and their probability of fixation (~ 1/N, a more accurate measure of this probability is used below and provided in the SI). Thus, we obtain:
where N* is the critical population size.
Because the population velocity is negative below N* and positive above N* there is an effective barrier for cancer (Fig. 1C) : smaller populations tend to shrink, while larger populations tend to expand. Simulations support our conclusion that the probability of cancer increases with N0 and sharply transitions at N* (Fig. 1D). Indeed, drastically different probability curves collapse onto a single curve once N0 is rescaled by N* (computed from equation 2). Since N* captures only the average, or mean-field, dynamics, it misses the variability of outcomes in rapidly adapting populations. Figure 1E illustrates that the variability of outcomes depends upon the strength of drivers sd. Higher values of sd lead to larger stochastic jumps, which leads to larger deviations from mean behavior and more gradual changes in the probability of cancer across N0. Thus, we formulated and analytically solved a stochastic generalization of equation 1 that incorporates this variability (SI). Our solution provides an excellent fit to simulations (Fig. 1E) and indicates that N* and sd fully describe population dynamics (SI).
We can understand how N* and sd control cancer progression using a simple random-walk analogy. The population size experiences random jumps, resulting from driver fixation events, which are described by equation 1. These random jumps and declines are effectively a random walk in a one-dimensional effective potential , Fig. 1C and SI) with stochastic jumps of frequency f and size ΔN. Similar to chemical reactions activated by thermal energy, cancer progression is a rare event triggered by a quick succession of driver fixations. Below, we show that human tissues operate in a regime where progression is rare and successful lesions are the rare lesions that happen to acquire drivers faster than average. We found that population dynamics depend entirely on two dimensionless parameters: a deterministic mean velocity, dependent only upon N/N*, and a stochastic step-size that is approximately proportional to sd. By reducing the complexity of our evolutionary system to two parameters, we were next able to infer their values for real cancers without over-fitting.
Model validation using cancer incidence and genomic data
Our model of cancer progression predicts the presence of an effective barrier to cancer where small lesions are very unlikely to ever progress to cancer. It also predicts a specific distribution in the number of passenger mutations and a specific relationship between drivers and passengers in individual cancer samples. We looked for evidences of these phenomena in age-incidence data (37) and cancer genomics data (28, 38–40). These comparisons also allowed us to estimate some critical parameters of the model: N0, sd, and sp.
Figure 2A presents the incidence rate of breast cancer versus age (37) alongside the predictions from a classic driver-only model (SI) and our model. The incidence rate was calculated by considering a process where precancerous lesions arise with a constant rate r beginning at birth. These lesions then progress to cancer in time τ with a probability P(τ) that we determined from simulations (Fig. S1). By convoluting this distribution P (τ) with the lesion initiation rate r, we can predict the age-incidence rate I(t). Because many lesions go extinct in our model and never progress to cancer, the predicted incidence rate saturates at old-age: where P∞ is the probability that a lesion will ever progresses to cancer, determined above.
Both the observed population incidence rates and our driver-passenger model saturate with age. This is a direct result of the probability of progression from a lesion to cancer being low. We estimate a lower bound for the rate of lesion formation r in breast cancer of at least 10 lesions per year that can be arrived at through two separate considerations: first, by considering the quantity of breast epithelial stem cells and the rate at which they can mutate into lesions (SI), and second, by considering the number of lesions observed within the breast tissue of normal cadavers (23). By comparing this limit (~ 10 lesions · year−1) to the maximum observed breast cancer incidence rate Imax ≈ 10−2 cancers · year−1, we find that P∞ ≈ 10−3, or only about 1 in 1,000 lesions ever progress. This finding is consistent with a number of clinical studies that have observed that very few lesions ever progress to cancer, while many more regress to undetectable size (41, 42)—another property seen in our model. Thus, good agreement between age-incidence data and our model is obtained when sd ≈ 0.1 – 0.2 and N0/N* is chosen such that P∞ = 10−3. This suggests that cancer begins at a population size far below N*, where drivers are most often overpowered by passengers. Indeed, 21 of the 25 most prevalent cancers plateau at old-age suggesting that progression is inefficient in most tumor types (Fig. S1). In a driver-only model (see SI for details), every lesion progresses to cancer after sufficient time (i.e. P∞ = 1), therefore a plateau in incidence rate can only result from a very low lesion formation rate (0.01 per year), which is inconsistent with abundant pathology data (23, 43).
Recent cancer genomics data offer a new opportunity to validate our model. Specifically, we looked at Somatic Nonsynonymous Mutations (SNMs) and Somatic Copy-Number Alterations (SCNAs) derived from over 700 individual cancer-normal sample pairs obtain from the breast (38), colon (28), lung (40), and skin (39) (Table S2). We found similar results when analyzing SNMs and SCNAs both separately (Fig. S2, Table S3) or in aggregation (Fig. 2BC). Figure 2B shows a wide and asymmetric distribution of the total number of mutations, which is consistent with our model under realistic parameters. A driver-only model yields a narrower and more symmetric distribution that is inconsistent with the data (Fig. 2B). The driver-only model can fit the observed distribution only if it assumes that just 1-2 drivers are needed for cancer (Fig. S3, Table S4)—a value inconsistent with both the extent of recurrent mutations seen in cancer (10) and known biology. This large variance in mutation totals further supports our model and suggests that driver mutations and alterations have a large effect size: sd ≈ 0.4. Our estimate of sd ≈ 0.1 – 0.4 obtained from age-incidence and mutation histograms is in excellent agreement with experimentally measured changes in the growth rate of mouse intestinal stem cells upon induction of p53, APC or k-RAS mutations where measured values ranged from of 0.16 to 0.58 (24).
We then used cancer genomics data to compare the number of drivers and passengers observed in individual cancer samples to our model’s predicted relationship. In our model, additional passengers must be counterbalanced by additional drivers for the population to succeed. If a lesion lingers around N* for a long time, then it must have acquired both many passengers and many counterbalancing drivers; while lesions that quickly progress through the barrier at N* acquire fewer of each. As a result, we expect a positive linear relationship between the number of drivers and passengers: nd · sd – np · sp = constant; this result follows directly from the definition of fitness in our model (SI). Our predicted positive linear relationship between drivers and passengers is indeed observed in all tumor types that we studied (Fig. 2C, Table S3, p < 0.08 – 10−6). The slope of this regression line is predicted to be sp/sd, which ranged from 1/21 to 1/193 (Table S3) for the various subtypes. While there is considerable variation and large margins of error in these numbers, these slopes (sp/sd ≈ 5 · (10−2 – 10−3)) correspond to an sp of 5 · (10−3 – 10−4) when sd = 0.1. These rough values are similar to germ-line SNMs in humans of European descent, where 64% of all mutations exhibit an sp between 10−5 and 10−2 (35).
We considered and refuted several alternative explanations for the observed positive linear relationship between drivers and passengers. First, that the strength of SCNAs may differ from SNMs. Hence, we investigated each alteration-type separately and found positive linear relationships in both cases (Fig. S2, Table S3). Second, that the number of driver alterations might be explained by variation in the tumor stage, or the rate and/or mechanism of mutagenesis. In Table S3 we show that these factors cannot suppress the correlation between drivers and passengers. Lastly, we considered and refuted the possibility that this relationship between drivers and passengers is non-linear (Fig. 2C insert). Because the data disagrees with all of these alternate hypotheses, we believe that it supports our conclusion that cancer progression is a tug-of-war between drivers and passengers.
A critical mutation rate
We next used simulations to investigate the probability of cancer over a broad range of evolutionary parameters (Fig. S4) and found that there is a critical mutation rate above which the probability of cancer is exceedingly low (Fig. 3A). To explain this phenomenon and to find the parameters that determine this critical mutation rate μ*, we modified our analytical framework to consider selection against passengers and the effects of unfixed passengers on the accumulation of drivers. The modified framework, described in the SI, explains observed dynamics well (Fig. 3AB, S4). Previous theoretical work has shown that the number of unfixed passengers per cell is Poisson distributed with mean μp/sp [first described in (44)]. This result assumes an approximate balance between the mutation rate of passengers and the selection against them, otherwise known as mutation-selection balance. The average fitness reduction of a cell due to this mutational load (i.e. the reduction in fitness relative to the fittest cells in the population) is μp. A new driver arises in one of these cells at random and must carry the load of passengers residing in its cell along with it to fixation (18) (Fig. 3C); this process is often referred to as hitchhiking, so we describe these passengers as ‘hitchhikers’. If the reduction in fitness due to the load of passengers (μp = μTp exceeds the benefit of a new driver (sd), then the driver will not fixate (Fig. 3C). Hence, cancer is extremely rare when μp > sd. This suggests a critical mutation rate:
The critical mutation rate suggests a new mode by which mutational meltdown operates. Prior models of mutational meltdown consider deleterious mutations in isolation (17), whereas our model points at the ability of deleterious mutations to inhibit the accumulation of advantageous mutations as a mechanism of meltdown. While it has been previously shown that deleterious mutations interfere with the fixation of beneficial alleles (18, 26, 45), this phenomenon has never been studied in the context of population survival. We discuss some important implications of this critical mutation rate for cancer treatment below.
We found support for the critical mutation rate in both cancer age-incidence and cancer genomics data. If we constraint cancer progression to develop within the typical timeframe for cancer progression (i.e. when we begin to see a plateau in incidence: ~ 60 years or 10,000 generations), the probability of cancer exhibits an optimum across mutation rates (Fig. 3D). Above μ* population meltdown is very common, while at very low mutation rates progression is too slow. The optimal mutation rate (10−9 – 10−8 mutations · nucleotide−1 · generation−1) is similar to mutation rates observed in cancer cell lines with a mutator phenotype (9) and the inferred mutation rate derived from the median number of mutations observed in a pan-cancer study of >3,000 tumors (10). Because μ* depends only on sd and Tp, and is independent of other variables, we believe the maximal mutation rate should be the same across tumor subtypes (SI, Fig. S4). Indeed, a maximum of approximately 100 somatic mutations per Mb (99.4th percentile) was observed in the pan-cancer study mentioned above, which corresponds to our theoretical estimate (if we assume that the most mutagenic cancers still require 1,000 generations to progress).
Two regimes of dynamics
Taken together, our results demonstrate that the tug-of-war between advantageous drivers and deleterious passengers creates two major regimes of population dynamics: an adaptive regime where the probability of progression (cancer) is high (~ 100 %) and a non-adaptive regime where cancer progression is exceedingly rare (Fig. 3, S4). Evolving populations that fail to adapt and go extinct may do so because they reside in a non-adaptive region of the phase space. Similarly, normal tissues that avoid cancers may present a tumor microenvironment that is in this non-adaptive regime. By keeping N* sufficiently high, a tissue or clinician could keep cancerous populations outside of the adaptive regime. This critical population size depends on the evolutionary parameters of the system. For example, if sp were increased by tuning the response of the immune system to mutation-harboring cells, or if Td were decreased via a driver-targeted therapy, adaptation would become less likely. Below we demonstrate that a successful treatment must push a cancer back to the non-adaptive regime.
The adaptive barrier and critical mutation rate explain cancer treatment outcomes
We simulated cancer growths and treatments and then monitored the long-term dynamics of these populations. Most treatments used today attempt to reduce tumor size, e.g. by specifically inhibiting key drivers (46) or by simply killing rapidly dividing cells (chemotherapy and radiation). Chemotherapy and radiation also elevate the mutation rate, thus affecting evolutionary dynamics. Previous work on the evolution of resistance to therapy has not considered the barriers to adaptation that we observe, so we re-investigated evolutionary outcomes from standard therapies and identified new potential ways to treat cancer. While real cancers have a varied evolutionary history, our analytical formalism predicts that cancer’s future dynamics depend only on their current state, not their history (i.e. cancer dynamics are approximately path-independent, Fig. S5). We show below that this assumption can accurately predict outcomes in simulations and the clinic.
In Figure 4, we present the evolutionary paths of cancer—from hyperplasia, to cancer, to treatment, and relapse or remission—on top of the phase diagrams described earlier. Our analysis demonstrates that a treatment is successful if it pushes a cancer into the non-adaptive regime of evolutionary dynamics where the probability of adaptation is low. Conversely, therapies fail, and populations re-adapt and remiss, when the therapy does not move cancer far enough to place it in the non-adaptive regime.
Our model suggests that chemotherapies succeed, in part, because they move cancers across the mutational threshold μ*. Beyond this threshold, the probability that a driver is strong enough to overpower a load of passengers becomes small (see above, Fig. 2C), making it hard for cancer to readapt. Increasing the mutation rate has little effect on the critical population size N* (see above).
Thus, our model suggest that cancers with a very high load of mutations/alterations are close to the critical mutation rate and should be more susceptible to mutagenic treatments, such as chemotherapy. Several recent studies (13, 47) have noticed that patients survive breast and ovarian cancer most often when their tumors exhibited exceptional high levels of chromosomal alterations. This phenomenon was robust within and between subtypes of breast cancer (47). This finding is paradoxical for all previous models of cancer, where a greater mutation rate always accelerates cancer evolution and adaptation; yet is fully consistent with our model (Fig. 4B).
Treatments exploiting the mutational load of cancers (i.e. their accumulated passengers) remain largely unexplored. We show that increasing the deleterious effect of passengers sp causes tumors to enter remission. Increasing sp is doubly effective because it exacerbates the deleteriousness of accumulated passengers and also slows down future adaptation. When we simulate such treatment by a 3-5 fold increase of sp (Fig. 4C), we observe an immediate decline in the population size followed by a low probability of replace due to an increased N*. The phase diagram shows that a mild increase in sp is sufficient to push a population into an extinction regime and thus induce remission. Below we discuss possible treatment strategies that would increase sp.
Given the large number of treatment options, finding therapies that work synergistically is a very important problem in cancer research (reviewed in (48)). While synergism is often discussed in the context of pharmacology, our phase diagrams identify evolutionarily synergistic treatments. We found that remission was most likely to occur when the mutation rate and the fitness cost of passengers were increased simultaneously, more so than would be expected from simply adding together the effects of the individual therapies (Fig. S6). Hence, combinations of mutagenic chemotherapy along with treatments that elevate the cost of a mutational load may be most effective. According to our model, these therapies should also be compatible and complementary to driver-targeted therapies.
Discussion
We present an evolutionary model of rapid adaptation that incorporates rare, strongly advantageous driver mutations and frequent, mildly deleterious passenger mutations. In this process, a population can either succeed and adapt, or fail and go extinct. We found theoretically, and confirmed by simulations, two regimes of dynamics: one where a population almost always adapts, and another where it almost never adapts. Complex stochastic dynamics, which emerge due to a tug-of-war between drivers and passengers, can be faithfully described as diffusion over a potential barrier that separates these two regimes. The potential barrier is located at a critical population size that a population must overcome to adapt. We also found a critical mutation rate, above which populations quickly meltdown. This general framework for adaptive asexual populations appears to be perfectly suited to characterize the dynamics of cancer progression and responses to therapy.
Progression to cancer is an adaptive process, driven by a few mutations in oncogenes and tumor suppressors. During this process, however, cells acquire tens of thousands of random mutations many of which may be deleterious to cancer cells. While strongly deleterious passenger mutations are weeded out by selection, mildly deleterious can fixate and even accumulate in a cancer by hitchhiking on drivers, as we have shown earlier (11). Passengers may be deleterious by inducing loss-of-function in critical proteins (14), gain-of-function toxicity via proteotoxic/misfolding stress (15, 49), or by triggering an immune response by a mutated epitope (16, 50). While we looked at passengers in cancer exomes and in SCNAs, passengers may also constitute epigenetic modifications, or karyotypic imbalances (15). Hence the number of deleterious passengers may be more than currently observed by genome-wide assays.
Our framework suggests that most normal tissues reside in a regime where cancer progression is exceedingly rare; i.e. most lesions fail to grow above the critical population size and, thus, fail to overcome the adaptive barrier. Clinical cancers, on the contrary, reside above the adaptive barrier in a rapidly adapting state. Therapies must push a cancer below this adaptive barrier to succeed. In our framework, this entails moving the population below N* or increasing the mutation rate above μ*. The availability of a broad range of data for cancer allowed us to thoroughly test our framework’s applicability.
We tested out model and estimated its parameters using cancer age-incidence curves, cancer exome sequences from almost 1,000 tumors in four cancer subtypes, and data on clinical outcomes. Age-incidence curves support the notion that the vast majority of lesions fail to progress and allow us to estimate the fitness benefit of a driver as sd ~ 0.1 – 0.4. Genomics data suggests that passengers are indeed deleterious and that their deleterious effect is approximately one hundred times weaker than driver’s beneficial effect. Moreover, the fitness benefit of a driver estimated from genomic data is roughly sd ~ 0.4. Our range of values are consistent with recently measured 16-58% increases in the mouse intestinal stem cell proliferation rate upon mutations in APC, k-RAS or p53 (24). Taken together these data support the notion of a tug-of-war between rare and large-effect drivers and frequent, but mildly deleterious passengers sp ~ 5 · (10−4 – 10−3), which nevertheless have a large collective effect.
Results of our analysis have direct clinical implications. Available clinical data (13, 47, 51) show that cancers with a higher load of chromosomal alterations, i.e. close to μ*, respond better to treatments. Our study suggests two potentially synergistic therapeutic strategies: to increase the mutation rate above μ*, and/or to increase the deleterious effect of accumulated passengers. An increase in the fitness cost of passengers would not only magnify the effects of an already accumulated mutational load, but also reduce future adaptation. This may be accomplished by (i) targeting unfolding protein response (UPR) pathways and/or the proteasome (15), (ii) hyperthermia that may further destabilize mutated proteins or clog UPR pathways (52), or (iii) by eliciting an immune response (16). Intriguingly, all these strategies are in clinical trials, yet they are often believed to work for reasons other than by exacerbating passengers’ deleterious effects. In contrast, we predict that these therapies will be most effective in cancers with more passengers and an elevated mutation rate. Thus, characterizing the load of mutations/alterations in tumors may offer a new biomarker for predicting treatment outcomes and identify the best candidates for mutational chemotherapies.
While this study focused on asexual innovative evolution in cancer, our model may be generally applicable to other innovating populations. Consider a population in a new environment. The population is often initially small and fluctuating in size and often goes extinct, yet occasionally it expands to a larger stationary size by rapidly acquiring several new traits that are highly advantageous in the new environment. Both the evolutionary parameters (53) and observed phenomena (54) match our model well. Our mathematical framework may explain why these populations sometimes adapt, yet often fail.
Materials and Methods
All simulations were run using a previously-described first-order Gillespie algorithm (11). Driver genes were identified using MutSig (10) and GISTIC 2.0 (55) for potential NSM and SCNA drivers, respectively; requiring a Bonferroni-corrected enrichment p-value ≤ 5 · 10−3 for classification of a gene as driver-harboring. All other genes were classified as ‘passenger-harboring’.
The probability of progression, determined from the outcome of 3,000 simulations propagated until extinction or rapid growth, across the parameter range of our model. In simulations, we observe parameters where progression occurs, fails, and is rare. A sophisticated analytical framework incorporating selection against passengers, hitchhiking of passengers onto driver mutations, and stochasticity predicts phase patterns well (black lines). This sophisticated analytical model uses two solutions for Muller’s Ratchet in various parameter regimes (see Selection against passengers.), an estimate of the quantity of hitchhiking passengers (see Effects of passengers on driver fixation), and a stochastic differential equation to estimate probabilities of progression (see Estimating the probability of cancer). A simplified framework, which offers a closed-form solution, is possible and works reasonably well (magenta). This solution differs from the more precise solution in two ways: (1) a novel estimate of Muller’s ratchet is used (Eq. 9), and (2) hitchhikers that accumulate after the new driver clone arises δpS are neglected. A-A‴ Pcancer increases as the relative target size of drivers Td verses passengers Tp increases, for all parameters. B-B‴, A‴ Pcancer exhibits a local minimum with respect to the selection against passengers sp. When the selection against passengers is very weak, passengers are effectively neutral. When the selection against passengers is too strong, natural selection weeds out passengers and they never accumulate. Deleterious passengers are most effective at preventing cancer when at moderate fitness effects. This local minimum suggests that there may be two types of cancers: those that exist in an environment/genetic context where passengers are weak and buffered by, for example, an activated UPR; and those that succeed by exacerbating passengers’ deleterious effects, perhaps by minimizing the mutation rate. C, C′, A″, B″ The probability of cancer always increases with sd. This parameter has a profound affect on cancer progression, as can be seen by how little the boundary between success and failure appears to be almost independent of the other parameters. Thus, this parameter is easy to estimate from epidemiological patterns. D, A′ An increasing mutation rate affects the probability of cancer very little at first; however, once it exceeds a critical value (μ*), the probability of cancer drops precipitously.
We explored our evolutionary model incorporating driver and passenger mutations across a broad range of parameters. The ranges were motivated by literature estimates that we discussed previously [28]. Note that in simulations μd = μTd and μp = μTp), hence we can explore the entire phase space by only altering μ and Td/Tp; altering all three parameters would be redundant. In Figure 2 we compare our model to epidemiological and genomic data and find that the best-fitting parameters agree well with these prior published estimates.
The total number of identified Somatic Nonsynonymous Mutations (SNMs) and Somatic Copy Number Alterations (SCNAs) for various tumor-normal paired sequences from various tissues of origin: 100 breast [34], 183 lung [13], 159 Colon without Micro-satellite INstability (MIN−), 64 Colon with Micro-Satellite Instability (MIN+) [10], and 121 melanomas [5].
Negative values are in gray. We observe a linear relationship between drivers and passengers that was predicted by our model. Above the thick black line are relationships that appear to robustly covary, while the bottom half contains relationships that we believe are insignificant. In our model, driver’s and passenger’s linear relationship results from their competing effect: additional deleterious passengers must be overcome by additional drivers. However, this relationship could conceivably be explained by alternate factors; in particular, we were concerned that the mutation type, mutation rate, or aggressiveness of the tumor could also explain the observed relationship. The data above suggest that these possibilities are unlikely, thus supporting our conclusion that drivers compete with passengers. Our rational for the other competing hypotheses and why we reject them: (a) SCNAs and SNMs might have drastically different effects on cancer progression and undermine our model. The slope and y-intercept between drivers and passengers is approximately equal in SCNAs and SNMs, suggesting the relative fitness effects of these mutations is similar. (b) Some cancers might progress via CIN, while others progress via an elevated point mutation rate. If so, a negative correlation between SCNAs and SNMs within tumor subtypes would be expected, which has been observed previously in a pan-cancer study [11] and within the aggregate colorectal dataset. However, this does not appear to be so in other tumor types, nor in colorectal cancer after segregation according to MIN phenotype. Thus, the observed patterns are not explicable by varying mutational mechanisms. (c) The relationship between drivers and passengers might be a result of variation in mutation rate. Variation in the mutation rate should only alter the waiting time to cancer in the neutral-passenger model, and not alter mutation totals. Nevertheless, if variation in the mutation rate could explain the correlation between drivers and passengers, then stratifying tumors by their mutation rate should reduce the correlation. Because the relationship between drivers and passengers is persistent and strong within the MIN+ and MIN−subtypes—expected to have and not-have a mutator phenotype—we reject this hypothesis. (d) Tumors with more drivers and passengers might simply be more evolutionarily advance. Suppose some cancers are detected and sequenced later than others. These late cancers would not only possess additional drivers, but also additional passengers, even if passengers were neutral; thus, retaining the correlation between drivers and passengers. However, late-detected tumors with additional drivers should also be more advanced and more aggressive. We find that a tumor’s pathological grade is uncorrelated with the number of drivers, refuting this possibility. Pathological grade was quantified by converting roman numerals into a linear scale (i.e. A Stage IV tumor corresponds to an aggressiveness of 4). Many tumors had intermediate grades that were given corresponding fractional values (e.g. a Stage IIIa tumor was translated into a 3.0, a Stage IIIb was given 3.3, and a Stage IIIc was given 3.7). Because this quantification of tumor grade may distort the scale of aggressiveness, Spearman’s Rank correlations are provided. For completeness, we have also show the relationship between SNMs and Pathological Grade and SCNAs and Pathological Grade.
A classical model of cancer progression with neutral passengers predicts that the distribution of passengers will follow a Negative Binomial distribution (see Fig. S3 and A traditional model of cancer progression with drivers and neutral passengers. for details). Tissues are ordered according to mean number of mutations and tissues with >100 samples are in bold. The MLE fits to observed distributions predict that only 1-3 drivers are needed for carcinogenesis. This is far fewer than expected from age-incidence curves, known biology, and driver classification algorithms. Therefore, we believe that the neutral-passenger model of cancer progression can not explain the observed spectrum of mutation totals.
Acknowledgements
CM and LM were supported by the National Cancer Institute-supported Physical Sciences in Oncology Center at MIT. KK was supported by Pappalardo Fellowship in Physics at MIT. We are grateful to Shamil Sunyaev, Gregory Krykov, Geoffrey Fudenberg, Maxim Imakaev, Anton Goloborodko for many productive discussions.
Footnotes
↵1 Depending upon the number of cells and genomes in the population, it may be more efficient to model cancer as a set genotypes that can gain and lose cells (rather than a set of cells that can gain mutations—as we have done). This alternate design is best when the number of genomes in the population is significantly less than the number of cells. However, the efficiency of a Gillespie algorithm depends very weakly on the number of chemical species n (Specifically, it affects computational speed by O(log n) in the Next Reaction Method [17]). More importantly however, the other steps in the simulation: creating mutations, and calculating birth/death rates is faster for individual cells than for genomes. Thus, using cells as the basic element of simulations, rather than genomes, is faster even under circumstances where the number of cells outnumber the number of genomes by a few orders of magnitude. This design choice also afforded more plasticity in model design and allowed us to create and investigate coalescences trees, which were informative. In fact, the genetics community at large tends to simulate genomes (rather than individuals) for what appears to be ill-conceived speed considerations, and may want to consider adopting our style.
↵2 While we assume that drivers arise at random time intervals, this assumption is not always true. Because unfixed passengers can interfere with the fixation of drivers, a driver is more likely to fixate immediately following a previous driver fixation event [23]. Ignoring this caveat does not significantly alter dynamics in the parameter space explored here.