Abstract
Evolutionary graph theory models the effects of natural selection and random drift on structured populations of mutant and non-mutant individuals. Recent studies have shown that fixation times, which determine the rate of evolution, often have right-skewed distributions. Little is known, however, about how these distributions and their skew depend on mutant fitness. Here we calculate the fitness dependence of the fixation-time distribution for the Moran Birth-death process in populations modeled by two extreme networks: the complete graph and the one-dimensional ring lattice, each of which admits an exact solution in the limit of large network size. We find that with non-neutral fitness, the Moran process on the ring has normally distributed fixation times, independent of the relative fitness of mutants and non-mutants. In contrast, on the complete graph, the fixation-time distribution is a weighted convolution of two Gumbel distributions, with a weight depending on the relative fitness. When fitness is neutral, however, the Moran process has a highly skewed fixation-time distribution on both the complete graph and the ring. In this sense, the case of neutral fitness is singular. Even on these simple network structures, the fixation-time distribution exhibits rich fitness dependence, with discontinuities and regions of universality. Applications of our methods to a multi-fitness Moran model, times to partial fixation, and evolution on random networks are discussed.
Significance Statement Evolutionary graph theory studies the interplay among natural selection, random drift, and the network structure of a population. Fitter mutants tend to reproduce at the expense of wild-type individuals that reproduce less quickly. After a certain amount of time their lineage can take over the entire population. We study a simple stochastic model for the spread of an advantageous mutation and calculate the distribution of times required for the mutants to take over. We find the distribution is often skewed, and establish analytically how the skew depends on the relative fitness of mutants and wild-types, for large networks of individuals connected in a one-dimensional ring or a complete graph. In both cases, the skew jumps discontinuously when fitness is neutral.
Reproducing populations undergo evolutionary dynamics. Mutations can endow individuals with a fitness advantage, allowing them to reproduce more quickly and outcompete non-mutant individuals [1]. Two natural questions arise: If a single mutant individual is introduced into a population, what is the probability that the mutant lineage will spread and ultimately take over the population (an outcome known as fixation)? And if fixation occurs, how much time does it take?
These questions have been addressed, in part, by evolutionary graph theory, which has advanced our understanding of evolutionary dynamics in structured populations. In particular, fixation probabilities have been studied for various models on various networks [2–12]. The results show that the inuence of a network’s structure on its fixation probability can be subtle. For instance, under constant selection, a large class of networks known as isothermal graphs have a fixation probability identical to that of a structureless, well-mixed population [2–5], whereas other networks either suppress or amplify selection relative to the well-mixed case [5, 7, 10, 12–14]. When mutations are rare, the fixation probability determines the rate of evolution [1, 15] and can be used to formulate a thermodynamic description of evolution [16, 17].
Other intriguing theoretical studies have focused on constructing optimal structural amplifiers or suppressors to selection [14, 18], or on analyzing more sophisticated dynamics governed by evolutionary games, where selection is frequency dependent rather than constant [6, 11, 19–22]. Parallel efforts have addressed evolutionary dynamics in real-world systems involving cancer growth [23] and response to treatment [24], CRISPR gene drives, [25], and HIV drug resistance [26].
Along with the fixation probability, the fixation time is also of interest, as it too impacts the rate of evolution [15]. Given a model of evolutionary dynamics, one would like to predict the mean, variance, and ideally the full distribution of its fixation times. Of these three quantities, the mean is the best understood. Numerical and analytical results exist for mean fixation times on both deterministic [4, 6, 11, 12, 20, 27–29] and random [28–31] networks. The results reveal curious trade-offs between fixation probability and mean fixation time: network structures that increase the likelihood of fixation may cause the system to take longer to achieve it [12, 15].
Although mean fixation times are important to understand, the information they provide can be misleading, because fixation-time distributions tend to be broad and skewed and hence are not well characterized by their means alone [11, 32–35]. Initial analytical results have determined the asymptotic fixation-time distribution for several simple networks, but only in the analytically convenient limit of infinite mutant fitness [36–38]. For other values of the relative fitness, almost nothing is known. Numerical evidence and preliminary theoretical results suggest that at neutral fitness (when mutants and nonmutants are equally fit), the fixation-time distribution becomes particularly right-skewed [38].
In this paper we investigate the full fitness dependence of fixation times for a simple model of evolutionary dynamics known as the Moran process [39, 40]. In the limit of large network size, we derive asymptotically exact results for the fixation-time distribution and its skew for two network structures at opposite ends of the connectivity spectrum: the complete graph, in which every individual interacts with every other individual; and the one-dimensional ring lattice, in which each individual interacts only with its nearest neighbors on a ring. The resulting fixation-time distributions depend on fitness in rich and contrasting ways.
The specific model we consider is the Moran Birth-death (Bd) process on a network, defined as follows. On each node of the network there is an individual, either mutant or non-mutant. The mutants have a fitness level r designating their relative reproduction rate compared to non-mutants. When r > 1, the mutants have a fitness advantage, whereas when r = 1 they have neutral fitness. At each time step we choose a node at random, with probability proportional to its fitness, and choose one of its neighbors with uniform probability. The first individual gives birth to an offspring of the same type. That offspring replaces the neighbor, which dies. The model population is updated until either the mutant lineage takes over (in which case fixation occurs) or until the mutant lineage goes extinct (a case we do not consider here).
The quantity of interest is the fixation time, T. Given a single initial mutant on a network of N nodes, and assuming that fixation occurs, how long does it take for the mutants to replace all the non-mutants? As mentioned above, the distribution of fixation times often turns out to be skewed. The skew merges from the stochastic competition between mutants and non-mutants through multiple mechanisms. For instance, when the mutants have neutral fitness, the process resembles an unbiased random walk. Successful runs to fixation will sometimes take long recurrent excursions before the mutants finally sweep through the entire non-mutant population, giving rise to long takeover times and hence a skewed distribution.
Since the neighbor is chosen uniformly, the individuals do not discriminate between mutants and non-mutants during the replacement step of the Moran process. Thus, on certain networks, when there are very few remaining non-mutants, the mutants can waste time replacing each other. This effect produces characteristic slowdowns near the end of the Moran process. These are reminiscent of similar slowdowns seen in a classic problem from probability theory, the coupon collector’s problem, which asks: How long does it take to complete a collection of N distinct coupons if one coupon is received randomly (with replacement) at each time step? The problem was first solved by Erdős and Rényi, who proved that for large N, the time to complete the collection has a Gumbel distribution [41]. The intuition for the long slowdowns is clear: When nearly all the coupons have been collected, it can take an exasperatingly long time to collect the final few, because one keeps acquiring coupons that one already has. In fact, an exact mapping onto coupon collection exists for evolutionary processes with infinite fitness [37, 38]. Remarkably, while this correspondence breaks down for finite fitness, the coupon collection heuristic still allows us to write down correct asymptotic fixation-time distributions for non-neutral fitness.
In the following sections we show that for N ≫ 1, the neutral fitness Moran process on the complete graph and the one-dimensional ring lattice has highly skewed fixation-time distributions, and we solve for their cumulants exactly. For non-neutral fitness the fixation-time distribution is normal on the lattice and a weighted convolution of Gumbel distributions on the complete graph. We begin by developing a general framework for computing fixation-time distributions and cumulants of birth-death Markov chains, and then apply it to the Moran process to prove the results above. We also consider the effects of truncation on the process and examine how long it takes to reach partial, rather than complete, fixation. The fixation-time distributions have rich dependence on fitness level and the degree of truncation, with both discontinuities and regions of universality. To conclude, we discuss extensions of our results to a multi-fitness Moran model and to more complicated network topologies.
GENERAL THEORY OF BIRTH-DEATH PROCESSES
For simplicity, we restrict attention to network topologies on which the probability of adding or removing a mutant in a given time step depends only on the number of existing mutants, not on where the mutants are located on the network. The state of the system can therefore be defined in terms of the number of mutants, m = 0,1,…, N, where N is the total number of nodes on the network. The Moran process is then a birth-death Markov chain with N + 1 states, transition probabilities bm and dm determined by the network structure, and absorbing boundaries at m = 0 and m = N. In this section we review and develop general analytical results on the fixation-time distributions for these Markov chains.
On more complicated networks, the probability of adding or removing a mutant depends on the configuration of existing mutants. For some of these networks, however, the transition probabilities can be estimated using a mean-field approximation [31, 37, 38]. Then, to a good approximation, the results below apply to such networks as well.
Eigen-decomposition of the birth-death process
Assuming a continuous-time process, the state of the Markov chain described above evolves according to the master equation. If p(t) is the probability of occupying each state of the system at time t, then p evolves according to where Ω is the transition rate matrix, with columns summing to zero. In terms of the transition probabilities bm and dm, the entries of Ω can be written as where m and n run from 0 to N, δm,n is the Kronecker delta, and b0 = d0 = bN = dN = 0. The final condition guarantees the system has absorbing boundaries with stationary states pm = δm,0 and pm = δm,N when the population is homogeneous. Thus we can decompose the transition matrix into stationary and transient parts, defining the transient part Ωtr as in Eq. 2, but with m, n = 1,…, N − 1. The eigenvalues of Ωtr are real and strictly negative, since probability flows away from these states toward the absorbing boundaries. To ease notation in the following discussion and later applications, we shall refer to the positive eigenvalues of −Ωtr as the eigenvalues of the transition matrix, denoted λm, where m = 1,…, N − 1.
From the perspective of Markov chains, the fixation time T is the time required for first passage to state m = N, given i initial mutants, pm(0) = δm,i. At time t, the probability that state N has been reached (i.e., the cumulative distribution function for the first passage times) is simply , where φi is the fixation probability given i initial mutants. The distribution of first passage times is therefore . Since the columns of Ω sum to zero, the right hand side of this equation can be written as , where 1 is the row vector containing all ones.
The solutions to the transient master equation are given by the matrix exponential p(t) = exp(Ωtrt) · p(0), yielding a fixation-time distribution ·p(0). If we assume one initial mutant, pm(0) = δm,1 then this can be written as . Since we normalize by the fixation probability, this is exactly the fixation-time distribution conditioned on reaching N. The matrix exponential can be evaluated in terms of the eigenvalues λm by taking a Fourier (or Laplace) transform [for details, see Ref. [34]]. For a single initial mutant, the result is that the fixation time T has a distribution fT (t) given by
Generalizations of this result for arbitrarily many initial mutants have also recently been derived, in terms of eigenvalues of the transition matrix and certain submatrices [34].
The distribution in Eq. 3 is exactly that corresponding to a sum of exponential random variables with rate parameters λm. The corresponding cumulants equal . As our primary interest is the asymptotic shape of the distribution, we normalize T to zero mean and unit variance and study (T − μ)/σ, where μ and a denote the mean and standard deviation of T. The standardized distribution is then given by σfT (σt + μ). The rescaled fixation time has cumulants which, for many systems including those considered below, are finite as N → ∞. When the limit exists, we define the asymptotic cumulants by κn = limN→∞κn(N). In particular, because we have standardized our distribution, the third cumulant κ3 is the skew. Since λm > 0, it is clear from this expression that, for finite N, the skew and all higher order cumulants must be positive, in agreement with analysis on random walks with nonuniform bias [42]. As N → ∞ this is not necessarily true; in some cases the cumulants vanish.
This analytical theory gives the fixation-time distribution and cumulants in terms of the non-zero eigenvalues of the transition matrix. In general the eigenvalues must be found numerically, but in cases where they have a closed form expression the asymptotic form of the cumulants and distribution can often be obtained exactly.
Exact computation of skew via visit statistics
In this section we develop machinery to compute the cumulants of the fixation time analytically without relying on matrix eigenvalues. For this analysis, we specialize to cases where bm/dm = r for all m, relevant for the Moran processes considered below. These processes can be thought of as biased random walks overlaid with non-constant waiting times at each state.
It is helpful to consider the Markov chain conditioned on hitting N, with new transition probabilities and so that the fixation probability φi = 1. If Xt is the state of the system at time t, then with defined analogously. We derive explicit expressions for and in SI Abstract, S1. Conditioning is equivalent to a similarity transformation on the transient part of the transition matrix: , where S is diagonal with Smm = 1 − 1/rm. Furthermore, since bm/dm = r, we can decompose Ωtr = ΩRWD, where D is a diagonal matrix, Dmm = bm + dm, that encodes the time spent in each state and ΩRW is the transition matrix for a random walk with uniform bias,
Applying the results of the previous section, the fixationtime distribution of the conditioned Markov chain is . The corresponding characteristic function is
Taking derivatives of this expression gives the moments of T, in terms of . This inverse has a nice analytical form because S and D are diagonal and ΩRW is tridiagonal Toeplitz. We call this approach visit statistics because the elements Vij of encode the average number of visits to state i starting from state j.
Each power of in Eq. 7 produces products of (bi + di) that arise in linear combinations determined by the visit numbers Vij. Therefore, the cumulants of the fixation time have the general form where are weighting factors based on the visit statistics of the biased random walk. In SI Abstract, S2, we give a detailed derivation of Eq. 8 and compute explicit expressions for and . This expression for the cumulants is easier to handle asymptotically than the closed form recursive solution for fixationtime moments [33], and can be useful even without explicit expressions for . Estimating these sums allows us to compute the asymptotic fixation time cumulants exactly.
Recurrence relation for distribution moments
Evaluation of the eigenvalues of the transition matrix for large systems can be computationally expensive, with the best algorithms having run times quadratic in matrix size. Numerical evaluation of the cumulants Kn(N) given by Eq. 8 is even worse, as it requires summing elements. If only a finite number of fixation time cu-mulants (and not the full distribution) are desired, there are better numerical approaches. Using an approach well known in the probability theory literature [43], we derive a recurrence relation which allows for numerical moment computation with run time linear in system size N. For completeness we provide the full derivation of the reccurence for the fixation-time skew in SI Abstract, S3.
ONE-DIMENSIONAL LATTICE
We now specialize to Moran Birth-death (Bd) processes, starting with the one-dimensional (1D) lattice. We assume periodic boundary conditions, so that the N nodes form a ring. The mutants have relative fitness r, meaning they give birth r times faster, on average, than non-mutant individuals do.
Starting from one mutant, suppose that at some later time m of the N nodes are mutants. On the 1D lattice, the population of mutants always forms a connected arc, with two mutants at the endpoints of the arc. Therefore, the probability bm of increasing the mutant population by one in the next time step is the probability of choosing a mutant node at an endpoint to give birth, namely 2r/(rm + N – m), times the probability 1/2 that the neighboring node to be replaced is not itself a mutant. (The latter probability equals 1/2 because there are two neighbors to choose for replacement: a mutant neighbor on the interior of the arc and a non-mutant neighbor on the exterior. Only the second of these choices produces an increase in the number of mutants.) By multiplying these probabilities together we obtain where the probability dm of decreasing the mutant population by one is found by similar reasoning. These quantities play the role of transition probabilities in a Markov transition matrix. The next step is to find the eigenvalues of that matrix.
Neutral fitness
First we work out the eigenvalues for the case of neutral fitness, r = 1. In this case, the transition probabilities are equal, bm = dm = 1/N, and independent of m. Therefore, the Moran process is simply a random walk, with events occurring at a rate of 2/N per time step. The associated transition matrix is tridiagonal Toeplitz, which has known eigenvalues given by
Applying Eq. 4 and computing the leading asymptotic form of the sums (see SI Abstract, S4), we find that as N → ∞ the fixation-time distribution has cumulants where ζ denotes the Riemann zeta function. In particular, the skew , as calculated by Ottino-Löffler et al. [38] via martingale methods. The largeness of the skew stems from the recurrent propertyof the random walk. As N → ∞ to, long walks with large fixation times become common and the system revisits each state infinitely often [44].
Knowledge of the cumulants allows us to obtain the exact characteristic function of the fixation-time distribution:
Although we cannot find a simple expression for the distribution itself, we can efficiently evaluate it by taking the inverse Fourier transform of the characteristic function numerically. Figure 1(a) shows that the predicted fixation-time distribution agrees well with numerical experiments.
Non-neutral fitness
Next, consider r ≠1 with the transition probabilities given by Eq. 9. Then the eigenvalues of the transition matrix are no longer expressible in closed form. If r is not too large, however, the probabilities bm and dm do not vary dramatically with m, the number of mutants. In particular, bm ~ 1/N for all m when N is large. Therefore, as a first approximation we treat the Bd process on a 1D lattice as a biased random walk with bm = r/(1 + r) and dm = 1/(1+r). The eigenvalues of the corresponding transition matrix are
The cumulants again involve sums , which can be approximated in the limit N → ∞ to by an integral, Since the integral is independent of N and converges for r = 1, each of the sums scales linearly: Sn ~ N. Thus, using Eq. 4, we see that all cumulants past second order approach 0,
Hence the fixation-time distribution is asymptotically normal, independent of fitness level.
By evaluating the integrals in Eq. 14, we can more precisely compute the scaling of the cumulants. For the skew we find
This integral approximation becomes accurate when the first term in the sums Sn for n = 2 and 3 becomes close to the value of the integrand evaluated at the lower bound (x = 0). The fractional difference between these quantities is
Then we have Δ ≪ 1 when (assuming r is near 1). To compute the skew, we require the sums with n = 2 and 3, giving Nc ≈ 10/(r − 1).
The above calculation fails for r ≫ 1, because when r = ∞ the transition probabilities bm = 1/m have different asymptotic behavior as N → ∞. In particular, more time is spent waiting at states with large m. The process still has normally distributed fixation times, but the skew becomes for large N. Notice that the coefficient is different from that given by the infinite-r limit of Eq. 16, .
We conjecture that there is a smooth crossover between these two scaling laws with the true skew given approximately by for some exponent q, where κ3 is the skew given in Eq. 16. For small r this ansatz has skew similar to that of a random walk, but captures the correct large-r limit. We do not have precise theoretical motivation for this ansatz, but as discussed below, it appears to work quite well.
Numerical calculation of the skew for the 1D lattice was performed using the recurrence relation method discussed above and detailed in SI Abstract, S3. The results are shown in Fig. 2 for a few values of r. This calculation confirms our initial hypothesis, that near neutral fitness the waiting times are uniform enough that the process is well approximated by a biased random walk and the skew approaches 0, scaling in excellent agreement with Eq. 16. When N ≪ Nc, the bias is not sufficient to give the mutants a substantial advantage: the process behaves like an unbiased random walk and the fixationtime distribution has large skew κ3 ≈ 1.807, as found in the preceding section. For N ≫ Nc, the bias takes over, the cumulants approach 0, and the distribution become normal. For large fitness r ≫ 1, the crossover ansatz (Eq. 19) captures the scaling behavior quite well if we use an exponent q = 1/2. Direct numerical simulations of the process confirm that, for any r > 1, the fixation time on the 1D lattice has an asymptotically normal distribution [Fig. 1(b)].
The random walk approximation allows us to find the scaling law for the fixation-time cumulants, but ignores the heterogeneity of waiting times present in the Moran process. Using visit statistics we can compute the cumulants exactly with Eq. 8 and rigorously prove they vanish as N → ∞, verifying that the waiting times have no influence on the asymptotic form of the distribution. Details are provided in SI Abstract, S4.
COMPLETE GRAPH
We now turn to the Moran process on a complete graph, useful for modeling well-mixed populations in which all individuals interact. By similar reasoning to above, given m mutants the probability of adding a mutant in the next time step is while the probability of subtracting a mutant is while the probability of subtracting a mutant is
Interestingly, as we will see in this section, these transition probabilities give rise to a rich fitness dependence of the fixation-time distribution in stark contrast to the universality observed on the 1D lattice.
Neutral Fitness
Again we begin with neutral fitness r = 1. Now bm = dm = (Nm – m2)/(N2 — N). The eigenvalues of this transition matrix also have a nice analytical form:
The asymptotic form of the sums Sn (defined identically to above), can be found by taking the partial fraction decomposition of (λm)−n and evaluating each term individually. The resulting cumulants are
We have not managed to calculate the distribution of the fixation times or its characteristic function in closed form, but fortunately the full set of its cumulants provides equivalent information. By using the first ten cumulants given by Eq. 23, we approximate the true distribution with a fifth-order Pearson-style distribution [as described in Ref. [45], Chapter 5]. Figure 3 shows that the result agrees well with simulations.
The numerical value of the fixation-time skew for the Bd process on the complete graph is κ3 ≈ 1.671, slightly less than that for the 1D lattice. The decrease is the result of two competing effects contributing to skew. First, since the birth and death transition probabilities are the same, the process is a random walk, which has a highly skewed fixation-time distribution, as shown above. The average time spent in each state, however, varies with m. For instance, when m = 1 or N — 1, bm → 0 for large N. But if m = αN for some constant 0 < α < 1 independent of N, then bm approaches a constant. The beginning and end of the process are very slow because the transition probabilities are vanishingly small. To start, the single mutant must be chosen to give birth from the N available nodes. Near fixation, the reproducing mutant must find and replace one of the few remaining non-mutants.
The characteristic slowing down at certain states is reminiscent of “coupon collection”, as discussed earlier. Erdős and Réenyi proved that for large N, the normalized time to complete the coupon collection follows a Gumbel distribution [41], which we denote by Gumbel(α, β) with density
For the Moran process, each slow region is produced by long waits for the random selection of rare types of individuals: either mutants near the beginning of the process or non-mutants near the end. In fact, for infinite fitness there is an exact mapping between Bd on the complete graph and coupon collection [37, 38]. In the next section we show that the two coupon collection regions of the Bd process on a complete graph lead to fixation-time distributions that are convolutions of two Gumbel distributions. In the case of neutral fitness, these Gumbel distributions combine with the random walk to produce a new highly skewed distribution with cumulants given by Eq. 23.
Non-neutral fitness
We saw above that when the average time spent in each state is constant or slowly varying, as is the case for the biased random walk and Moran Bd on the 1D lattice respectively, the fixation-time distribution is normal. In contrast to these systems, Moran Bd on the complete graph exhibits coupon collection regions at the beginning and end of the process, where the transition probabilities are very small. We begin this section with a heuristic argument that correctly gives the asymptotic fixationtime distribution in terms of independent iterations of coupon collection.
Differentiating bm with respect to m, we find the slope near m = 0 is (r + 1)/N, while the slope near m = N has magnitude (r + 1)/(rN) for N ≫ 1. The transition rates approach zero at each of these points, so we expect behavior similar to coupon collection giving rise to two Gumbel distributions. Since the slope is greater for m near 0 than for m near N, the Moran process completes its coupon collection faster near the beginning of the process than near fixation.
This heuristic suggests that the asymptotic fixation time should be equal in distribution to the sum of two Gumbel random variables, one weighted by r, which is the ratio of the slopes in the coupon collection regions. Specifically, if T is the fixation time with mean u and variance σ2, we expect where means convergence in distribution for large N and G is a Gumbel random variable with zero mean and unit variance. It is easy to check that G has distribution Gumbel , where γ ≈ 0.5772 is the Euler-Mascheroni constant.
Let us make this argument more rigorous. Previous theoretical analysis showed that in the infinite fitness limit, the fixation time has an asymptotically Gumbel distribution [38]. This result can be recovered within our framework, since when r = to it follows that dm = 0, so the eigenvalues of the transition matrix are just bm = (N – m)/(N – 1) and the cumulants can be directly calculated using Eq. 4.
Now consider large, but not infinite, fitness. In this limit, the number of mutants is monotonically increasing, to good approximation, since the probability that the next change in state increases the mutant population is r/(1 + r) ≈ 1. The time spent waiting in each state, however, changes dramatically, especially near m = 1. Here, b1 → 0 for large N, in stark contrast to the infinite fitness system where b1 → 1. The time spent at each state, tm is an exponential random variable, ε(bm + dm). In this approximation each state is visited exactly once, so the total fixation time is a sum of these waiting times:
But this sum of exponential random variables has density given by Eq. 3, with the substitution λm → bm + dm. Thus, the cumulants of (T – μ)/σ are which are exactly the cumulants corresponding to the sum of Gumbel random variables given in Eq. 25. In the limit r → ∞, the first term in Eq. 27 becomes 1, and the cumulants are those for a single Gumbel distribution, in agreement with previous results.
For smaller networks, it is fascinating to see how the results converge to the asymptotic predictions as N grows. Figure 5 shows how the skew of the fixation-time distribution depends on r and N for the complete graph. Note the fixation-time distributions for these systems are invariant under r → 1/r. Therefore we show the skew for all r > 0, to emphasize the intriguing behavior near neutral fitness, where r = 1. We find that non-uniform convergence of the fixation-time skew leads to the discontinuity predicted at r = 1. For finite N, the skew is a non-monotonic function of r and has a minimum value at some fitness rmin(N). Furthermore, at fixed r, the convergence to the N = ∞ limit is itself non-monotone. Though beyond the scope of the current study, further investigation of this finite-N behavior would be worth pursuing.
PARTIAL FIXATION TIMES: TRUNCATING COUPON COLLECTION
In many applications, we may be interested in the time to partial fixation of the network. For instance, considering cancer progression [23, 24, 46] or the incubation of infectious diseases [38], symptoms can appear in a patient even when a relatively small proportion of cells are malignant or infected. We therefore consider Tα, the total time to first reach aN mutants on the network, where 0 < α < 1. The methods developed above apply to these processes as well. For the eigen-decomposition approach we instead use the sub-matrix of Ωtr containing the first αN rows and columns. In calculations involving the numerical recurrence relations or visit statistics, we simply cut the sums off at αN instead of N and for the latter, replace with .
Truncation of the Moran Bd process on the 1D lattice has no effect on the asymptotic shape of the fixation-time distributions. In both the neutral fitness system and the random walk approximation to the non-neutral fitness system, the transition matrix has no explicit dependence on the state or N (aside from proportionality factors that cancel in Eq. 4). Thus, the eigenvalues are identical to those calculated previously, but correspond to a smaller effective system size αN. Taking the limit N → ∞ therefore yields the same asymptotic distribution found above. This result is borne out numerically.
In contrast to the 1D lattice, the complete graph exhibits more interesting dependence on truncation. Since the transition probabilities have state dependence, the eigenvalues vary with truncation (they don’t exactly correspond to the full system with a smaller effective N). Our intuition from coupon collection, however, lets us predict the resulting distribution.
First suppose fitness is non-neutral. Then there are two coupon collection stages, one near the beginning and another near the end of the process, and together they generate a fixation-time distribution that is a weighted convolution of two Gumbel distributions. The effect of truncating the process near its end should now become clear: it simply removes the second coupon collection, since truncation means the process stops before the mutants have to laboriously find and replace the last remaining non-mutants. Therefore, we intuitively expect the fixation time for non-neutral fitness to be distributed according to just a single Gumbel distribution, regardless of fitness level.
The only exception occurs if r = ∞ then no coupon collection occurs at the beginning of the process either, because the lone mutant is guaranteed to be selected to give birth in the first time step, thanks to its infinite fitness advantage. Thus, when fitness is infinite and the process is truncated at the end, both coupon collection phases are deleted. In this case, the fixation times are normally distributed.
Similar reasoning applies to the Birth-death process with neutral fitness. It also has two coupon collection regions, one of which is removed by truncation. In this case, however, the random walk mechanism contributes to the skew of the overall fixation-time distribution, combining non-trivially with the coupon collectionlike process. We find that the skew of the fixation time depends on the truncation factor α, varying between when α = 1, and when α = 0. A derivation of the α → 0 limit of the skew is given in SI Abstract, S5.
The above results are summarized in Fig. 6, which shows the asymptotic fitness dependence of fixation-time skew for each network considered in this paper. As before, we show the skew for all r > 0 (not just r > 1) to emphasize the discontinuities at zero, neutral, and infinite fitness r. On the 1D lattice, independent of the truncation factor α, the Bd process has normally distributed fixation times, except at neutral fitness where the distribution is highly skewed. The complete graph fixation-time distributions are the weighted convolution of two Gumbel distributions for r ≠ 1, again with a highly skewed distribution at r = 1. With truncation by a factor α, the distribution for the complete graph is a Gumbel for 1 < r < ∞, and a normal for r = ∞. With neutral fitness the fixation distribution is again highly skewed, with skew dependent on the truncation factor α.
DISCUSSION
Multi-fitness Moran process: comparison to the Isotherm Theorem
So far, we have considered Moran processes with a single fitness level, designating the relative reproduction rates between mutants and non-mutants. It also makes sense to consider a second fitness level during the replacement step, measuring the resilience of mutants versus non-mutants [9]. Taking this into account, when a mutant or non-mutant is trying to replace its neighbors, mutants are replaced with probability proportional to . Taking returns to the model used throughout the preceding sections. The multi-fitness model may better capture the complexity of real-world evolutionary systems but, as we will see below, does not generally give rise to qualitatively different fixation-time distributions.
On the 1D lattice, the Moran process with fitness at both steps (birth and death), has new transition probabilities for 1 < m < N – 1. The probabilities are different when m =1 and m = N – 1 and there is only one mutant or non-mutant. In this case the cells on the population boundary don’t have one mutant and one non-mutant as neighbors, as is the case for all other m. In the limit N ≫ 1, however, changing these two probabilities does not affect the fixation-time distribution and we can use the probabilities given in Eq. 28.
The multi-fitness Moran model on the 1D lattice differs from the previously considered Moran Bd process in two ways. First, the transition probabilities have the same functional form as before, but are scaled by a factor . This factor determines the time-scale of the process but does not alter the shape of the fixation-time distribution because it drops out of the expression for the cumulants, Eq. 4. Second, the ratio shows that the process is still a random walk, but with new bias corresponding to an effective fitness level . With these observations, when reff ≠ 1, our preceding analysis applies and we predict normally distributed fixation times. If reff = 1, the random walk is unbiased, and we expect highly skewed fixation-time distributions.
On the complete graph, considering fitness during the replacement step leads to transition probabilities and
In this case, the ratio of transition probabilities is m-dependent, but as , again motivating the definition of the effective fitness level . If we take the large (but not infinite) fitness limit reff ≫ 1, so that the mutant population is monotonically increasing to good approximation, then the fixation time cumulants are again given by Eq. 4 with λm → bm + dm. As N → ∞, the cumulants become identical to the Moran Bd process on the complete graph, with . Simulations again indicate this expression for the cumulants holds for all r, not just in the high fitness limit. When reff = 1, we expect highly skewed fixation distributions arising from the unbiased random walk underlying the dynamics. This is indeed the case, though simulations show we get a whole family of distributions dependent on .
It is interesting to contrast the above observations with a result in evolutionary dynamics known as the isotherm theorem. The theorem states that for , the Moran process on a large class of networks, known as isothermal graphs, has fixation probability identical to the complete graph [5]. Recent work has shown that this breaks down if ; the fixation probability develops new network dependence [9]. In contrast, even isothermal graphs (including the complete graph and 1D lattice) have fixation-time distributions that depend on network structure. Likewise, adding a second fitness level to the model breaks the universality in fixation probabilities predicted by the isotherm theorem, but leads to the same family of fixation distributions that arise due to network structure.
Mean-Field Transition Probabilities: Random Networks, Stars, and Lattices
While the 1D lattice and complete graph provide illustrative exactly solvable models of the fitness dependence of fixation-time distributions, other networks may be more realistic. On more complicated networks many of the analytical tools developed here fail because the transition probabilities (the probability of adding or subtracting a mutant given the current state) depend on the full configuration of mutants, not just the number of mutants. In principle such a system could still be modeled as a Markov process, but the state space would become prohibitively large.
Fortunately, for certain networks the effect of different configurations can be averaged over, giving a mean-field approximation to the transition probabilities. This approach has been used on a variety of networks to calculate fixation times with reasonable success [31, 35]. For instance, the mean-field transition probabilities for the Erdos-Renyi random network were recently estimated in Ref. [31]. The result is identical to the complete graph probabilities (Eqs. 20-21) up to a constant factor 1 – 2/Np, dependent on the edge probability p for the network. This correction is important for computing the mean fixation times, but does not affect the shape of the fixation-time distribution, since proportionality factors cancel in the expression for the cumulants. Therefore we again expect a fixation-time distribution corresponding to the weighted sum of two Gumbel distributions. Previous work indicates this prediction holds for infinite fitness, where the fixation time on an Erdos-Renyi network has a Gumbel distribution [38].
Preliminary simulations show that the Erdős-Rényi network has the expected fixation-time distributions for p =1/4 and r = 2 (see Figure 7). Whether this result holds for the full range of fitness and edge probabilities requires further investigation. It may be that for some values of p and r the mean-field approximation is not sufficient to capture the higher-order moments of the distribution. In the same vein, what other networks admit accurate mean-field approximations to the transition probabilities? Do many complex networks have fixationtime distributions identical to the complete graph?
Another nice approximation maps the Moran process on a star graph, a simple amplifier of selection, onto a birth-death Markov chain [15]. The resulting transition probabilities exhibit coupon collection regions, similar to the complete graph. The ratio of slopes near these regions (few mutants or non-mutants), however, is r2. Our heuristic predicts the fixation-time distribution on the star is G + r2G. In addition to amplifying fixation probability, the star increases fixation-time skew. This raises a broader question: do evolutionary amplifiers also amplify fixation-time skew?
Similar mean-field arguments have also been applied to d-dimensional lattices in the infinite-fitness limit [37, 38]. In this limit the population of mutants grows in an approximately spherical shape near the beginning of the process and the population of non-mutants is approximately spherical near fixation. The surface area to volume ratio of the d-dimensional sphere gives the probability of adding a mutant. With finite fitness non-mutants can now replace their counterparts and the surface of the sphere of growing mutants roughens [46]. For nearneutral fitness, the configurations of mutants resembles the shape of real cancerous tumors. Perhaps, by extending our theoretical framework, connections can be made between the fitness-dependent roughness of a growing mutant population and fixation-time distributions for Moran processes on lattices.
Future Directions
Future studies could analyze random networks and lattices more deeply, as well as stars, the prototypical evolutionary amplifiers [5]. More sophisticated models involving evolutionary games are also of interest. These have skewed fixation-time distributions [34] whose asymptotic form remains unknown. Finally, we hope that methods developed here will prove useful in other areas, such as epidemiology [47], ecology [48], and protein folding [49], where stochastic dynamics may similarly give rise to skewed first-passage times.
S5: ASYMPTOTIC ANALYSIS FOR THE COMPLETE GRAPH
Non-neutral fitness
In the main text we predicted that the asymptotic fixation-time distribution for the Moran Birth-death process on the complete graph is a convolution of two Gumbel distributions by applying our intuition from coupon collection. Furthermore, our calculation of the fixation-time cumulants to arbitrary order in the large (but finite) fitness limit agrees with this prediction. Surprisingly, numerical calculations using the recurrence relation formulated above and direct simulations of the Moran process indicate that this result holds for all r > 1. In this section we prove, using the visit statistics formulation, that the asymptotic skew of the fixation time for r > 1 is identical to that of a convolution of Gumbel distributions. Based on our numerical evidence, we conjecture that an analogous calculation holds to all orders. The below calculation also shows why the coupon collection heuristic works: the asymptotically dominant terms come exclusively from the regions near fixation (m = N − 1) and near the beginning of the process when a single mutant is introduced into the system (m = 1).
As for the 1D lattice, we want to derive the asymptotic behavior of the sums where the transition probabilities bi and di are those for the Moran process on the complete graph,
To start, consider the sums Eq. 41, but with two indices i1 and i2 constrained to integers from αN to (1 − α)N for 1/2 > α > 0. This sum may be written as
Now we may apply the upper bound in Eq. 34, except for the sums restricted to αN < i,j < (1 − α)N the maximum of (bi + di)−1 can be restricted to this range,
In the second line we used the integral approximation from Eq. 38 and evaluated the maximum of (bi + di)−1 over the indicated intervals. Since we constructed to be symmetric, this upper bound holds for any permutation of the indices in Eq. 43.
We now consider the same sums but with 1 < i1 < αN or (1 − α)N < i1 < N − 1, and
These sums can be estimated using the same upper bound, but without extending the sum on to the entire domain. Specifically,
Note that the weighting factors fall off exponentially away from the diagonal elements. This is because the visit numbers in the biased random walk become only very weakly correlated if the states are far away. Thus, the sum in Eq. 47 over terms away from the diagonal elements converges to a constant as N → ∞. We have verified this explicitly for and . The series is similarly bounded, as are all sums of the form Eq. 45 or 46 with the indices permuted.
The remaining terms in Sn involve all indices in either [1, αN] or [(1 − α)N, N − 1]. If not all indices are in the same interval, the weighting factors are exponentially small: the visit numbers near m = 1 are uncorrelated with those near m = N − 1. Thus each term in the sum is exponentially suppressed and doesn’t contribute to Sn asymptotically. With this observation only two parts of the sum remain: those with bounds 1 ≤ i1, i2 … in ≤ αN or (1 − α)N < i1, i2 … in < N − 1. We call the sums with these bounds and respectively. As we will see below, the sums over these regions have leading order 𝒪(Nn). Since all the above terms are order 𝒪(Nn−1) or smaller, the asymptotic behavior of the cumulants is entirely determined by these regions near the beginning and end of the process, i.e. the coupon collection regions. The fact that we can restrict the sums to this region allows us to make approximations that do not change the asymptotic behavior, but make the sums easier to carry out. For instance, in , we can set rN − ri → rN and (N − i) → N, since the indices run only up to αN. This gives for N ≫ 1. A similar approximation shows . For the third order sums, we find for N ≫ 1. Again the other sum, with indices near N − 1, is identical up to a factor of r3, . Overall, we have that
Then the asymptotic skew is given by which is exactly the skew corresponding to the convolution of Gumbel distributions with relative weighting given by the fitness, G + rG. While evaluating the series to higher orders is increasingly difficult, our simulations and the large-fitness approximation suggest this result holds to all orders and that indeed, the asymptotic fixation-time distribution is a weighted convolution of Gumbel distributions.
Neutral fitness with truncation
As discussed in the main text, the neutral fitness Moran process on the complete graph has a fixation-time skew that depends on the level of truncation. That is, the time Tα it takes for the process to reach aN mutants, where 0 ≤ α ≤ 1, has a distribution whose skew depends on α. Here we show that the α → 0 limit of the fixation-time skew equals .
To start, we take the neutral fitness limit of the weighting factors to obtain for i ≥ j and for i ≥ j ≤ k, again with the expressions for other orderings obtained by permuting the indices accordingly. The neutral fitness Moran process on the complete graph has transition probabilities bi + di = 2(Ni − i2)/(N2 − N). Since we are computing the truncated fixation-time skew, we use Eq. 15, but cut the sums off at αN. In this case, these sums are dominated by the off-diagonal terms, so that where in the second line we approximated N − i and N − j by N. This approximation is exact in the limit a → 0 since the upper limit on the sum is αN, which is much smaller than N. Using analogous approximations, we find
The asymptotic fixation-time skew as α → 0 is therefore as claimed in the main text. This value agrees perfectly with our numerical calculations, which show the above approximation breaks down when α ≈ 1/2. Above this threshold, the random walk causes mixing between the two coupon collection regions, thereby lowering the overall skew of the fixation-time distribution toward the α = 1 value of .
ACKNOWLEDGMENTS
Research supported by an NSF Graduate Research fellowship grant DGE-1650441 to D. H. and by NSF grants DMS-1513179 and CCF-1522054 to S.H.S.