Fig 1.
How spatial structure affects the molecular clock rate K.
Relative to the rate in a well-mixed population (K = u), spatial structure can either accelerate (K > u) or slow (K < u) the accumulation of neutral substitutions, depending on how birth rates bi and death rates di vary across sites. The rate is unchanged from that of a well-mixed population (K = u) if either death rates are uniform across sites (Result 1), or the birth and death rates are equal at each site (Result 2). Almost all previous studies of neutral drift in spatially structured populations fall into one of these two categories; thus the effects of spatial structure on the molecular clock rate are unappreciated. We show that, in general, K can take any non-negative value less than Nu (Result 4). If one adds the constraint that the birth rate is the same at each site, then the molecular clock rate cannot exceed that of a well-mixed population (K ≤ u; Result 3).
Fig 2.
For spatial structures with symmetry, K = u.
(a) For a well-mixed population, represented by a complete graph with uniform edge weights, a neutral mutation has a 1/N chance of fixation, where N is the population size. It follows that the rate K of neutral substitution in the population equals the rate u of neutral mutation in individuals. (b) The same result holds for spatial structures in which each site is a priori identical, such as the cycle with uniform edge weights.
Fig 3.
Asymmetric spatial structure affects the rate of neutral substitution.
This is because the frequency of mutations and the probability of fixation differ across sites. Turnover rates are indicated by coloration, with red corresponding to frequent turnover and consequently frequent mutation. (a) A star consists of a hub and n leaves, so that the population size is N = n+1. Edge weights are chosen so that the birth rates are uniform (bi = 1 for all i). Solving Eqs. (1)–(2), we obtain site-specific fixation probabilities of ρH = 1/(1+n2) and ρL = n/(1+n2) for the hub and each leaf, respectively. From Eq. (4), the molecular clock rate is , which equals u for n = 1 and is less than u for n ≥ 2. Thus the star structure slows down the rate of neutral substitution, in accordance with Result 3. Intuitively, the slowdown occurs because mutations are more likely to arise at the hub, where their chances of fixation are reduced. (b) A one-dimensional population with self-replacement only in site 1. Solving Eqs. (1)–(2) we find , , and . (The powers of two arise because there is twice as much gene flow in one direction as the other.) From Eq. (4), the molecular clock rate is , thus the molecular clock is accelerated in this case.
Fig 4.
Results 1 and 2 give conditions leading to K = u.
(a) Our Result 1 states that the molecular clock has the same rate as in a well-mixed population, K = u, if the rate of turnover di is uniform across sites, as in this example (di = 0.2 for all i). (b) Result 2 asserts that ρi = 1/N for all i—again implying K = u—if and only if each site has birth rate equal to death rate, bi = di for all i, as in this example. Nodes are colored according to their rates of turnover di.
Fig 5.
Result 4 shows that K can achieve any value 0 ≤ K < Nu.
This is proven by considering a population structure with unidirectional gene flow from a hub (H) to N−1 leaves (L). Fixation is guaranteed for mutations arising in the hub (ρH = 1) and impossible for those arising in leaves (ρL = 0). The overall fixation probability is equal by Eq. (3) to the rate of turnover at the hub: ρ = dH = 1−(N−1)a. The molecular clock rate is therefore K = Nuρ = N[1−(N−1)a]u. It follows that K > u if and only if a < 1/N. Intuitively, the molecular clock is accelerated if the hub experiences more turnover (and hence more mutations) than the other sites. Any value of ρ greater than or equal to 0 and less than 1 can be achieved through a corresponding positive choice of a less than or equal to 1/(N−1). For a = 1/(N−1) we have K = 0, because mutations arise only at the leaves where there is no chance of fixation. At the opposite extreme, in the limit a → 0, we have K → Nu.
Fig 6.
A model of a population divided into upstream and downstream subpopulations.
Each subpopulation is well-mixed. The replacement probability eij equals e↑ if sites i and j are both upstream, e↓ if i and j are both downstream, e→ if i is upstream and j is downstream, and e← if i is downstream and j is upstream. We suppose there is net gene flow downstream, so that e→ > e←. We find that the molecular clock is accelerated, relative to the well-mixed case, if and only if the upstream subpopulation experiences more turnover than the downstream subpopulation: K > u if and only if d↑ > d↓.
Fig 7.
A simple model of cell replacement structure in epithelial crypts of the small intestine, based on results of [41] and [45].
A small number of stem cells (N↑ ∼ 5) residing at the bottom of the intestinal crypt and are replaced at rate d↑ ∼ 0.1 per stem cell per day. Empirical results [41, 45] suggest a cycle structure for stem cells. To achieve the correct replacement rate we set eij = 0.05/day for each neighboring pair. Stem cells in an individual crypt replace a much larger number of progenitor and differentiated cells (∼ 250; [46]). These downstream progenitor and differentiated cells are replaced about every day [46]. The hierarchical organization of intestinal crypts, combined with the low turnover rate of stem cells, limits the rate of neutral genetic substitutions ( substitutions per day), since only mutations that arise in stem cells can fix.
Fig 8.
The rate of neutral substitution K on Twitter “ego networks”.
(a–e) Five of the 973 networks analyzed, including those with (a) the largest value of K, (b) the smallest value of K, and (c) the fewest nodes. (f) A scatter plot of K/u versus N reveals a weak negative correlation (slope ≈ −0.00164 with 95% confidence interval (−0.0023, −0.001) based on the bootstrap method; R ≈ −0.45). The colored dots on the scatter plot correspond to the networks shown in (a–e). The dashed line corresponds to K/u = 1, above which network topology accelerates neutral substitution.
Fig 9.
Illustration of idea generation and spreading on Twitter “ego networks”.
Network structure accelerates idea substitution (K > u) if and only if there is a positive spatial correlation between the generation of new ideas (which for our model occurs proportionally to the rate di of incoming ideas) and the probability of fixation ρi. Panels (a) and (b) show the networks with the slowest (K ≈ 0.667u) and fastest (K ≈ 1.085u) rates of idea substitution, respectively, among networks of size 13. The coloration of nodes corresponds to their rate of turnover di, with warmer colors indicating more rapid turnover. The size of nodes corresponds to their fixation probability ρi.
Fig 10.
Illustration of a replacement event.
In this case, the occupant of site 3 produced one offspring, which displaced the occupant of site 2. The occupant of site 4 produced two offspring, one remaining in site 4 and displacing the parent, and the other displacing the occupant of site 5. Thus the set of replaced positions is R = {2, 4, 5} and the offspring-to-parent map α is given by α(2) = 3, α(4) = 4, α(5) = 4. There is no mutation in the example illustrated here, so each offspring inherits the type of the parent. Thus the population transitions from state s = (M, R, R, M, R) to s′ = (M, R, R, M, M).