Fig 1.
Birth-death (Bd) updating with mutation.
Here we show an example of the single time step of the Moran Bd updating rule with mutation. First an individual is selected with probability proportional to its fitness to give birth to an offspring. The offspring resembles the parent with probability 1 − μ, or mutates with probability μ. If the mutation occurs, the offspring fitness f′ is then sampled from the distribution ζ(f′, f) with f being the parent’s fitness. In the figure, we have shown the case when mutation takes place. The mutant offspring will then replaces one of the individuals neighboring the parent individual, or the parent individual itself via the self-loop. The choice is made at random with probability proportional to the outgoing weight from the parent node. Here, we have shown the case when the parent individual is replaced by the offspring via the self-loop. The stronger the self-loop, more likely it is for the parent to be replaced by its offspring.
Fig 2.
(A) We mostly work with these five graphs throughout the manuscript. (B) The steady-state average fitnesses obtained using the Moran Birth-death mutation-selection dynamics simulations for the self-looped (weighted) star graph, an amplifier of selection, the star graph, a suppressor of fixation, the self-looped directed line, a suppressor of selection, the cycle graph, an isothermal graph, and the complete graph are shown via circles as a function of mutation rates. Solid horizontal lines represent steady-state average fitnesses for different graphs obtained under the low mutation rate approximation, Eq 1. The arrows mark the mutation rates beyond which the low mutation rate approximation is violated for respective graphs. The graphs with higher average fixation time is expected to deviate earlier, see Eq 4. (C) The average fixation time scaling with N at neutrality is shown for different graphs. Solid lines are the analytical results whereas circles represent Moran Bd simulations. For larger N, it gets computationally expensive to work with microscopic Moran Bd simulations, in such cases we use a Gillespie algorithm, shown via triangles. For details on the Gillespie algorithm, refer to App. 5.3.3. (D) The scaling of the average fixation time with population size N for the different graphs. (Parameters: (B) population size, N = 10, uniform mutant fitness distribution, i.e., , (B,C) with 2000 total number of independent realisations used for averaging, fmin = 0.1 and fmax = 10).
Fig 3.
Reference graph: Complete graph with self-loops.
Here, the mutation-selection dynamics is studied for the self-looped complete graph with μ → 1. We find a very good agreement for the steady-state statistics between the analytics and the simulations. The thick line represents the analytical average fitness, while the shaded grey area represents the standard deviation around the average. Symbols and error bars show simulations. In the steady-state, on average the self-looped complete graph attains the midpoint of the fitness domain, as the fitness dynamics for each individual node of the population becomes uncorrelated in the fitness space and time. The steady-state average fitness is also independent of the population size. The fluctuations in the steady-state however depends on the population size and decreases with the increase in population size as (Parameters: fmin = 0.1, fmax = 10, number of independent realisations is equal to 2000, mutant fitness distribution, ζ(f′, f) = 1/(fmax − fmin)).
Fig 4.
Nodewise analysis of the star graph with self-loops and the directed line with self-loops.
Here, the average fitness trajectories for each node of the self-looped star graph (shown in panel A) and the self-looped directed line (shown in panel B) are shown. Thick lines represent average fitness trajectories at the population level, whereas, thin lines represent average fitness trajectories for the nodes. The effect of self-loops on a node’s fitness depends on the incoming and outgoing weight flowing out of that node. In panel A, self-loops have the least effect on the central node because of relatively higher incoming and outgoing weight. As a result, the central node attains higher average steady-state fitness than the leaf nodes. In panel B, the root node of the directed line has the lowest steady-state average fitness because of the absence of an incoming link to the root node. (Parameters: N = 10, μ = 1, fmin = 0.1, fmax = 10, number of independent realisations is equal to 2000, mutant fitness distribution, . For the directed line with self-loops, every outgoing link from a node (including the self-loop) has the same weight. For the self-looped star graphs, the weights of the links follows Eq (23), such that λ = 1/(N − 1) and δ = 1/(N − 1)2).
Fig 5.
Sampling mutant’s fitness from the Gaussian.
(A) When mutant fitness is sampled from the (truncated) Gaussian distribution with σ = 0.1, we find that adding self-loops decreases the population fitness in all the graphs. (B) Increasing the σ from 0.1 to 1, the average fitness in the steady-state goes down for many graphs. The effect of increasing the σ is largest in the heterogeneous star graphs and smallest in the more homogeneous structure like the complete graph. (C) We recover the uniform mutant fitness distribution case for very large σ, here σ = 10. In this case, all the non-self looped graphs attain the same steady-state. All self-looped graphs have lower average steady-state fitness than a non-self looped graph and the self-looped complete graph (Parameters: N = 10, μ = 1, fmin = 0.1, fmax = 10, 2000 independent realisations).
Fig 6.
Average extinction and fixation time for the self-looped star graph and the standard star graph.
Here, we plot the average extinction and fixation time of a mutant for the self-looped (weighted) star graph (panel A) and the star graph (panel B) as a function of mutant’s relative fitness. Solid lines corresponds to the analytic results, Eqs 31 and 39. The circles represent Moran Bd simulations. Firstly, we observe that for both the graphs, the average fixation time of a mutant is higher than its extinction time, regardless of the mutant’s relative fitness. Secondly, the average fixation time peaks near neutrality for both of the graphs. Therefore, according to Eq 4, μth for the star graphs scales as the inverse average fixation time at neutrality. Because the fixation of a mutant takes longer on the self-looped star graph, the weak mutation rate approximation is more restrictive for the self-looped star graph than the star graph. (Parameters: N = 10, wild-type fitness, f = 1, and the number of independent realisations conditioned on mutant’s fixation or extinction are 2000).
Fig 7.
Average extinction and fixation time for the isothermal graphs.
The average fixation (via solid lines) and average extinction (via dashed lines) times for the two isothermal graphs, namely, the complete graph and the cycle graph. To plot the analytical results, we have used Eqs 66 and 69. Open circles represent microscopic Moran Bd simulations. Although, the probability for a mutant to fix on any of these structures is the same due to isothermal theorem, the times it takes to reach fixation are different. Fixation on the cycle graph is slower than on the complete graph. As a result, the cycle graph is more restrictive to the weak mutation approximation. The parameters are same as in Fig 6.
Fig 8.
Average extinction and fixation time for the self-looped directed line.
The average extinction time (dashed line) and the average fixation time (solid line) are shown for the self-looped directed line. Circles represent the total average time of the trajectories that lead to the fixation of mutants, whereas diamonds represent the average time spanned by the trajectories where mutants get extinct. We see a good agreement between analytical results and the corresponding simulations. The approximated formula for the average extinction time, Eq 82, works well in the regime of high relative fitness, as the dashed line starts coinciding with the simulations. Note that the average extinction time for a mutant can exceed the average fixation time. This is different from what we have observed in Figs 6 and 7. Also, for a given fitness domain, the average extinction and fixation time peaks away from the neutrality. Therefore, to decide the validity of the weak mutation rate approximation, fitness regions different from neutrality must be considered.
Fig 9.
Here, the possible mutant extinction routes are shown for the self-looped directed line when the initial mutant appears on a non-root node. For purpose of illustrations, we have chosen N = 4. Broadly speaking, there are three categories of extinction trajectories. (i) The case where the initial mutant goes extinct without spreading in the population. This would be a one time step extinction process, shown by arrow leading from the boxed initial state to the wild-type state, highlighted in grey. (ii) The second category corresponds to the case, where the initial mutant spreads, but the mutant goes extinct before the terminal node is ever occupied by a mutant. This would contain all the paths that go from the boxed state via two mutants to the grey highlighted state without going through the states highlighted in red. (iii) The third category refers to the case, where the initial mutant spreads and reach the state highlighted in red. After the terminal node is occupied by the mutant type, the number of mutants then starts to decrease from the left (shown via the trajectory marked with blue arrows). This third category is especially relevant when the mutant’s relative fitness is very high. We make use of this argument to approximate the extinction time for the self-looped directed line by computing the time covered by the blue arrowed trajectory.
Fig 10.
Universal steady-state fitness among non self-looped graphs.
(A) In the steady-state, the complete graph, the cycle graph, and the star graph, attain the same average fitness in the mutation-selection balance. The steady-state average fitness obtained by these graphs is higher than that of the self-looped complete graph, indicating that the dynamics on these graphs—unlike the self-looped complete graph—is not entirely uncorrelated in the fitness space and time. One common thing in these three graphs is that every node has at least one incoming link. (B) The molded directed line is constructed by adding a link directed from node 1 to node 0, the root node, so that every node has non-zero incoming links. In the steady-state, not only the molded directed line attains the same average fitness as the complete graph (non self-looped), but every single node becomes indistinguishable. (C) The same observation is made for the modified burst graph. (D) These observations remain valid for different population sizes. The difference between the steady-state average fitness of the non self-looped and the self-looped complete graph decreases with increasing N, indicating that the evolutionary dynamics becomes more random with increasing N (Parameters: same as Fig 3).