A two-type model

McDonald, Chakrabarti, and Michor [8] consider two alternative evolutionary models in order to argue that other underlying models can produce a linear relationship between 1/f and the cumulative number of mutations with frequency ≥ f. Their second model is an infinite alleles branching process model previously studied by McDonald and Kimmel [11]. We will ignore this model, since in studying DNA sequence data the appropriate mutation scheme is the infinite sites model.

In their first model, clonal expansion begins with a single cell of the original tumor-initiating type (type 0). To make it easier to connect with previous mathematical work, we will describe their model using the notation used in [10] and [12]. We suppose that type 0 individuals give birth at rate a₀ and die at rate b₀, so the exponential growth rate is λ₀ = a₀ − b₀. For simplicity, we will suppose that neutral mutations accumulate during the individual’s life time at rate ν, instead of only at birth.

Type 0 individuals mutate to type 1 at rate u₁. Type 1 individuals give birth at rate a₁ and die at rate b₁. Their exponential growth rate is λ₁ = a₁ − b₁ where λ₁ > λ₀. In [8], different type 1 families have different increases in their growth rates that follow a normal distribution. In this section, we will assume all type 1 mutations have the same growth rate. Later, we will consider the implications of random fitness changes for the behavior of the model.

The reader will see many complicated formulas in this paper, so it will be useful to have a concrete set of parameters to plug into these formulas. Borrowing an example from [10], we will set (1) We do not pretend that these parameters apply to any specific cancer, but for a mental picture, you can imagine that type 0s are colon cancer cells in which both copies of APC have been knocked out, while type 1 cells in addition have a KRAS mutation.

Site frequency spectrum.

There are three classes of mutations in the two-phase model

• type 0: Neutral mutations that occur to type 0 individuals.
• type 1A: Advantageous mutations that turn type 0 individuals into type 1.
• type 1: Neutral mutations that occur to type 1 individuals.

By the argument in Methods, the type 0 mutations will have a 1/f site frequency. The argument can also be used to prove the next result so the details are hidden away in Methods.

Theorem 2 The number of type 1 mutations with frequency ≥ f with in the type 1 population will be asymptotically ν/(λ₁ f).

The points in the Poisson process in Theorem 1 indicate the contributions of the various type one families to the limit , so if we let x₁ > x₂ > x₃… be the points, then the jth largest family makes up a fraction of the population. Intuitively, this implies that the number of type 1A mutations with frequency ≥ f will be asymptotically Cf^−α where α = λ₀/λ₁. However, the fact that the sum of the points in the Poisson process is random makes this difficult to study. Fortunately for us, the work has already been done in 1997 by Pitman and Yor [14], who proved that the points in the Poisson process divided by their sum follow the Poisson-Dirichlet distribution PD(α, 0). See the remark after Theorem 5 in [12]. This gives us that when 0 < α < 1 the site frequency spectrum of 1A mutations is: (4) When α = 1/2, the constant is 2/π = 0.6366.

Including type 0 passenger mutations in type 1A families does not significantly change the f^−α shape in (4). This is because all important 1A mutations happen soon after the first mutation, which implies that all important 1A mutations have roughly the same number of passengers. See Methods.

To illustrate the results proved above, we turn to simulations seen in Figs 1 and 2.

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Fig 1. Site frequency spectrum in the type 1 population.

The figure shows the contribution of the different mutation types to the site frequency spectrum. The simulation was performed with parameters ν = 0.02, u₁ = 2 × 10⁻⁴, λ₀ = 0.02, λ₁ = 0.04 and a₀ = a₁ = 1 and is the average site frequency spectrum of 1000 runs. We simulated the 1A families and type 0 passenger mutations on their founders. Then, we obtained type 1 mutations for each 1A family by applying (8) in Methods. We only consider mutations present in the type 1 population because, as t → ∞, the proportion of the population that is type 0 cells approaches 0. As suggested from Theorem 2, the type 1 site frequency spectrum is linear when plotted against 1/f. The 1A + 0 line looks similar to a power law, as suggested by (4).

https://doi.org/10.1371/journal.pcbi.1008701.g001

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Fig 2. Distribution of 1A family sizes in the type 1 population.

To better understand the distribution of 1A family sizes, we used the Poisson-Dirichlet(α, 0) distribution to generate the six largest families. The plot gives the probability that the number of individuals in the top i families are greater than a fraction x of the total type 1 population.

https://doi.org/10.1371/journal.pcbi.1008701.g002

Random fitness increases

McDonald, Chakrabarti, and Michor [8] considered the case in which type 1 individuals have growth rates that are normal with mean m and standard deviation d. Early work on models with random fitness increases in the two-type model led to very unusual behavior in the limit t → ∞, see [15]. Results in that paper show

• If the fitness distribution was bounded then, as t → ∞, individuals with fitnesses that were close to the upper limit dominated the population.
• If the distribution was unbounded, then the population could grow faster than exponential.

In this section, we will modify our example from Fig 1 so that type 1 individuals have growth rates drawn from the normal distribution with mean m = 0.04 and standard deviation d = 0.005. We will see that in contrast to the limiting results just mentioned, random fitnesses do not substantially change the behavior.

To find the distribution of the growth rates of the mutations with the largest family sizes, we note that a mutant that occurs at time s_i and has growth rate λ_1,i will grow to size W₁exp(λ_1,i(1000 − s_i)) at time 1000. The number of i that are successful and have λ_1,i(1000 − s_i) > x is Poisson with mean given by the following integral (5) where ϕ and Φ are the density function and distribution function, of a normal distribution with mean m = 0.04 and standard deviation d = 0.005. The equality follows from substituting u = (λ − 0.04)² for the inner integral. Fig 3 graphs (5).

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Fig 3. Size of 1A families with random fitness.

The graph indicates the expected number of 1A families with λ_1,i(1000 − s_i) > x. The parameters are almost the same as in (1); rather than a single λ₁ for all type 1 families, we have a different λ_1,i for each type 1A family. Each λ_1,i is normally distributed with mean 0.04 and standard deviation 0.005. 500 runs were done up until time t = 1000. The graph shows that on average there is one family with e^x > 10¹⁰. If the λ_1,i of the largest family is within 2 standard deviations, then multiplying e^x by 1/λ_1,i implies a family of magnitude around 2 × 10¹¹ or greater.

https://doi.org/10.1371/journal.pcbi.1008701.g003

The random fitnesses cause the relative sizes of the contributions of mutations to the final population to change, but as Fig 4 shows, the site frequency still has the form C/f^β, where β ≤ α and achieves equality in the case of non-random changes, i.e. d = 0.

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Fig 4. Site frequency spectrum with random fitnesses.

(A) shows the site frequency spectrum for multiple values of d. The other parameters are the same as in Fig 3. As the contribution from neutral mutations is negligible, we will only show the contribution from 1A families. The line for constant, i.e., d = 0, is plotted from theory; the others are plotted from simulations with 200 runs. As d increases, the expected size of the frequency of the largest mutation increases. Also, fewer mutations reach above the 0.05 frequency threshold. (B) displays the same data on a log-log plot. The slopes β of the linear fits indicate that the site frequency spectrum takes the form C/f^β, with β decreasing as d increases.

https://doi.org/10.1371/journal.pcbi.1008701.g004

The authors of [8] claim that the site frequency spectrum in the two-type model is 1/f. However, their simulation methods take the very crude approach of considering the binary split process until 1,000 or 1,000,000 cells are produced. This corresponds to 10 and 20 generations respectively. To make it possible for something to happen in this short amount of time the mutation rate for advantageous mutations is set to be 0.1 in the 1000 cell scenario, and to 0.03 when there are 1,000,000 cells. At birth, each cell acquires a Poisson mean 100 number of mutations. In contrast our simulations run for approximately 1000 generations, leading to populations of order 10⁹ cells, and neutral mutations occur slowly, leading to genealogical relationships that are more like those found in growing cancer tumors.

Subclonal mutation frequencies

Bozic, Paterson, and Waclaw [6] argue that “the fact that no subclonal driver is present at intermediate frequencies cannot be taken as proof of neutral or effectively neutral evolution. It can be a consequence of population dynamics which create only a short window during which the driver mutation can be detected but not fixed in the population.” In this section we will describe their results and give a simple analytic derivation.

To argue for this viewpoint, they use the two-type model but with different notation In addition they define c = r₁/r > 1, and g = c − 1. They assume that the mutation to type 1 occurs at time 0 and run the process until the time t at which the total population size is M. Let X₀ be the population of type 0’s when the mutation occurs. Since X₀ is large, X_t ≈ X₀ e^rt. The type 1 population at time t is Y_t ≈ W₁ e^rct, where W₁ is an exponentially distributed random variable with rate cr/b₁. Note that as in Bozic et al [16] the possibility of subsequent driver mutations is ignored. As Fig 5 shows, that change does not lead to a substantial error.

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Fig 5. Driver frequencies.

This graph gives the probability of having a driver with frequency greater than y once the tumor reaches size 10⁹. The parameters used are a₀ = a₁ = 1, λ₀ = 0.02, λ₁ = .035 and u₁ = 10⁻⁵ and the data was generated from 1000 runs. Single 1A refers to approach taken by Bozic et al. where there is only 1 selective mutation. Multiple 1A is our approach. The theory curve comes using a Riemann sum with interval size 500 to evaluate the integral in Eq (6).

https://doi.org/10.1371/journal.pcbi.1008701.g005

Writing f_sub = Y_t/(X_t + Y_t) they prove that when the total tumor size is M = X_t + Y_t the subclonal mutation frequency has (6) which is (1) in [6]. From this they can compute the probability of a subclonal driver being detectable, that is, P(0.2 ≤ f_sub ≤ 0.8).

To see what this complicated formula implies, the authors turn to simulation. The mutation rate to produce an additional driver is u = 10⁻⁵. Their Fig 2A shows a moderately growing tumor b = 0.14, r = 0.01, 2B a fast growing tumor b = 0.25, r = 0.07, and 2C a slowly growing tumor b = 0.33, r = 0.0013. For moderate values of selection, e.g. g = 30%, the probability that a driver mutation is in the detectable range [0.2, 0.8] is < 15% for population sizes up to M = 10⁹ cells and remain below 1/3 for M ≤ 10¹¹. For other cases considered there (g = 70% and 100%) the chance of detecting the subclonal driver is always < 60% and for a broad range of sizes is less than 30%. Panels d,e,f in their Fig 2 show the frequency of a subclonal driver in the case of moderate growth when the size M_d = 10⁷, M_e = 5 ⋅ 10¹⁰ and M_f = 2 ⋅ 10⁸. In the three cases the frequency is near 0, near 1, and almost uniformly distributed on [0, 1].

Rather than study the tumor when it reaches a fixed size, we will derive results at a fixed time by using Theorem 1. Recall that we have set and have shown Combining the last two results, we see that Inserting the values of the λ_i so goes from 0.2/0.8 = 1/4 to 0.8/0.2 = 4 in time ln(16)/0.015 = 184, confirming that the window in which competing subclones coexist is short.

Simple derivations of the 1/f spectrum

Sottoriva and Graham say in their original paper [2] that “the power law signature is common to multiple tumor types and is a consequence of the effectively-neutral evolutionary dynamics that underpin the evolution of a large proportion of cancers.” To explain the source of the 1/f curve in an exponentially growing tumor, we give the derivation of the 1/f frequency distribution from [3]. They assumed that cells divide at rate λ and use N(t) to be the number of cells at time t. If we assume that the mutation rate is μ (which we assume takes into account their ploidy parameter π), then the expected number of new mutations before time t, M(t), satisfies Solving gives Since N(s) = e^λs (we have set β in [3] to be 1 for simplicity), we observe that a mutation that occurs at time s will have frequency e^−λs in the population. Evaluating the integral in the previous formula, we have Ignoring the −1, if we set t_f = −(1/λ)log f to make N(t_f) = 1/f so that mutations before time t_f will have frequency ≥ f, then

Theorem 3 The number of mutations with frequency ≥ f is (7) Note that in this derivation, mutations occur only at birth. If we instead let mutations happen continuously throughout a cell’s lifetime and call the mutation rate ν, then Durrett [12] has shown (8)

From the derivation given above, we see that the 1/f site frequency spectrum comes from the fact that mutations occur at a rate proportional to the size of the population and the fact that the population is growing exponentially fast.

Proof of Theorem 2

Proof. We follow the derivation of Theorem 3. If we let , then the number of type 1 mutations by time t satisfies where we have again dropped the −1 that comes from the lower limit. A mutation that occurs at a time , when there are individuals, will occur in a fraction of ≥ f of the population, so computing M(t_f) gives the desired result.

Passengers do not change the shape of the SFS

To show that the important 1A mutations happen soon after the first, and that therefore all important 1A mutations have roughly the same number of passengers, consider two successful mutations at times s₀ and s₁ which have sizes W₀e^{λ₁(t−s₀)} and W₁e^{λ₁(t−s₁)}. For the second mutation to be larger, we’d need W₀/W₁ ≤ e^{λ₁(s₀−s₁)}. Since the cdf of the quotient of two exponentials with the same rate is P(W₀/W₁ ≤ x) = x/(x+ 1), we find that If s₁ = s₀ + 4/λ₁ = s₀ + 200, then the probability that the second mutation is larger is (1 + e⁴)⁻¹ = 0.018. Thus, in our concrete example the most significant mutants occur within 200 time units of the first successful mutation. The mean number of mutations in 200 units of time is 200ν.