Regular structures and isothermal theorem
For networks where each node represents a single individual, the isothermal theorem of evolutionary graph theory shows that the fixation probability is the same as the fixation probability of a well-mixed population if the temperature distribution is homogeneous across the whole population1. The temperature of a node defined as the sum over all the weights leads to that node. This theorem extends to structured meta-populations for any migration probability (lambda ): If the underlying structure of the meta-population that connects the patches is a regular network and the local population size is identical in each patch, the temperature of all individuals is identical, regardless of the value of the migration probability. Therefore, the fixation probability in a population with such a structure is the same as the fixation probability in a well-mixed population of the same total population size (N=sum _{j=1}^M N_j), given by ( phi _{mathrm{wm}}^N(r)).
Small migration regime
If the migration probability is small enough such that the time between two subsequent migration events (( sim frac{1}{lambda } )) is much longer than the absorption time within any patch, then at the time of each migration event we may suppose that the meta-population is in a homogeneous configuration22,28. In other words, the low migration regime is an approximation in which we neglect the probability that the meta-population is not in a homogeneous configuration at the time of migration events. We define a homogeneous configuration of the meta-population as a configuration in which in all patches either all individuals are mutants, or all are wild-types.
Therefore, instead of having (2^N) states, where N is the population size, the system has only (2^M) states, where M is the number of patches. Thus, we can calculate the fixation probability exactly as in the case of a standard evolutionary graph model where each node represents a single individual but with a modified transition probabilities.
In a network with homogeneous patches, in order to increase the number of homogeneous mutant-patches one individual mutant needs to migrate to one of its neighbouring homogeneous wild-type-patches and reaches fixation there. For example if node j is occupied by mutants and one of its neighbouring patches, node k, is occupied by wild-types, the probability that one mutant individual from patch j migrates to patch k and reaches fixation there is (frac{lambda }{mathrm{deg} (j)}phi _{mathrm{wm}}^{N_{k}}(r) ), where (mathrm{deg} (j) ) is the degree of node j to take into account that the mutant can move to different patches. This is analogous to the probability that one mutant in node j replaces one wild-type in node k ,(T^{jrightarrow k}), in the network of individuals.
Similarly, if node j is occupied by wild-types and one of its neighbouring patches, node j, is occupied by mutants the probability that one wild-type individual from patch j migrates to patch k and reaches fixation there equals to (frac{lambda }{mathrm{deg} (j)}phi _{mathrm{wm}}^{N_{k}}(1/r) ) where (mathrm{deg} (j) ). Overall, we can move from network of individuals to the network of homogeneous patches by replacing the transition probabilities with the product of migration and fixation probabilities.
Two-patch meta-population
The simplest non-trivial case is the fixation probability in a two-patch meta-population with different local size for small migration probability (lambda ). If the migration probability (lambda ) is very small, a new mutant first needs to take over its own patch and only then the first migrant arrives in the second patch. To be more precise, the time between two migration events has to be much higher than the typical time that it takes for the migrant to take over the patch or go extinct again38. In this case, we can divide the dynamics into two phases: A first phase in which a mutant invades one patch and a second phase in which a homogeneous patch of mutants invades the whole meta-population. Assume a new mutation arises in patch 1. Only if this mutant reaches fixation in patch 1, it also has a chance to reach fixation in patch 2. When patch 1 consists of only mutants and patch 2 consists of only wild-types, there are two possibilities for the ultimate fate of the mutant:
- (i)
Eventually, the offspring of one mutant selected from patch 1 for reproduction will migrate to patch 2 and reach fixation there. The wild-type goes extinct. This happens with probability ( frac{N_1 r}{N_1 r+N_2} phi _{mathrm{wm}}^{N_2}(r)).
- (ii)
Eventually, the offspring of one wild-type selected from patch 2 for reproduction will migrate to patch 1 and the mutant goes extinct. This occurs with probability ( frac{N_2}{N_1r+N_2} phi _{mathrm{wm}}^{N_1}(tfrac{1}{r})).
Therefore, the probability that a single mutant arising in patch 1 reaches fixation in the entire population is
$$begin{aligned} phi _{mathrm{wm}}^{N_1}(r) frac{frac{N_1 r}{N_1 r+N_2} phi _{mathrm{wm}}^{N_2}(r)}{frac{N_1 r}{N_1 r+N_2} phi _{mathrm{wm}}^{N_2}(r)+frac{N_2}{N_1r+N_2} phi _{mathrm{wm}}^{N_1}left( tfrac{1}{r}right) }=phi _{mathrm{wm}}^{N_1}(r) phi _{mathrm{wm}}^{N_2}(r) frac{1 }{ phi _{mathrm{wm}}^{N_2}(r) +frac{N_2}{N_1} frac{1}{r}phi _{mathrm{wm}}^{N_1} left( tfrac{1}{r}right) }. end{aligned}$$
(3a)
Similarly the probability that a mutant arising in patch 2 takes over the whole population equals
$$begin{aligned} phi _{mathrm{wm}}^{N_2}(r) phi _{mathrm{wm}}^{N_1}(r) frac{1 }{phi _{mathrm{wm}}^{N_1}(r)+frac{N_1}{N_2} frac{1}{r} phi _{mathrm{wm}}^{N_2}left( tfrac{1}{r}right) }. end{aligned}$$
(3b)
If we assume that the mutant arises in a patch with a probability proportional to the patch size, the average fixation probability (phi _{bullet !!-!!bullet }) in a two patch population for small migration probability is the weighted sum of Eqs. (3a) and (3b),
$$begin{aligned} phi _{bullet !!-!!bullet }&= phi _{mathrm{wm}}^{N_1}(r) phi _{mathrm{wm}}^{N_2}(r) nonumber &quad times left( frac{frac{N_1}{N_1+N_2} }{ phi _{mathrm{wm}}^{N_2}(r) +frac{N_2}{N_1} frac{1}{r}phi _{mathrm{wm}}^{N_1}left( tfrac{1}{r}right) } +frac{frac{N_2}{N_1+N_2} }{ phi _{mathrm{wm}}^{N_1}(r) +frac{N_1}{N_2} frac{1}{r} phi _{mathrm{wm}}^{N_2}left( tfrac{1}{r}right) }right) . end{aligned}$$
(4)
In the case of neutrality, (r=1), we recover (phi _{bullet !!-!!bullet } = frac{1}{N_1+N_2})—the fixation probability in a population of the total size of the two patches. For identical patch sizes, ( N_1=N_2 ), Eq. (4) simplifies to
$$begin{aligned} phi _{bullet !!-!!bullet } = left( phi _{mathrm{wm}}^{N_1}(r)right) ^2 frac{1}{phi _{mathrm{wm}}^{N_1}(r)+frac{1}{r} phi _{mathrm{wm}}^{N_1}left( tfrac{1}{r}right) } = phi _{mathrm{wm}}^{2 N_1}(r), end{aligned}$$
(5)
where the simplification to the fixation probability within a single population of size (2N_1) reflects the validity of the isothermal theorem.
For (N_1 ne N_2), we approximate Eq. (4) for weak and strong selection. Let us first consider highly advantageous mutants, (r gg 1). In this case, we have (phi _{mathrm{wm}}^{N_1}(r) gg phi _{mathrm{wm}}^{N_1}(tfrac{1}{r})) and thus we can neglect the possibility that a wild-type takes over a mutant patch if patch sizes are sufficiently large. The probability (phi _{bullet !!-!!bullet } ) then becomes a weighted average reflecting patch sizes. For identical patch size (N_1=N_2 = N/2), it reduces to (phi _{bullet !!-!!bullet } approx phi _{mathrm{wm}}^{N_1}(r)=phi _{mathrm{wm}}^{N/2}(r)). In other words, taking over the first patch is sufficient to make fixation in the entire population certain. For patches of very different size, (N_1 gg N_2), we have (N approx N_1) and find (phi _{bullet !!-!! bullet } approx phi _{mathrm{wm}}^{N}(r), ) which implies that fixation is driven by the fixation process in the larger patch, regardless of where the mutant arises. Note that there is a difference between the case of identical patch size and very different patch size . The case of highly disadvantageous mutants, (r ll 1), can be handled in a very similar way.
Next, we consider weak selection, (r approx 1). We can approximate the fixation probability as (phi _{mathrm{wm}}^{N}(r^{pm 1}) approx frac{1}{N} pm frac{N-1}{2N} (r-1)). With this, we find
$$begin{aligned} phi _{bullet !!-!!bullet } approx frac{1}{N_1+N_2} +frac{1}{2} left( 1 – frac{1}{N_1+N_2} -frac{(N_1-N_2)^2}{(N_1^2+N_2^2)^2} N_1 N_2right) (r-1). end{aligned}$$
(6)
For identical patch size (N_1=N_2 = N/2), this reduces to
$$begin{aligned} phi _{bullet !!-!!bullet } approx tfrac{1}{N} +tfrac{N-1}{2N} (r-1), end{aligned}$$
(7)
which is the known result for a single population of size (N=N_1+N_2). When patches have very different size, (N_1 gg N_2) such that (N approx N_1), we recover the same result. Thus, the difference between the fixation probability of a two-patch meta-population with identical patch size and the fixation probability of a two-patch meta-population with very different patch size that we found for highly advantageous mutants is no longer observed for weak selection.
When migration probabilities become larger, our approximation is no longer valid and we need to rely on numerical approaches. Figure 2 illustrates the difference between the fixation probability of a two-patch structure meta-population and the equivalent well-mixed population of size (N_1+N_2 ) when migration is low using Eq. (4) and comparing with the numerical approach in Ref.39.
While the fixation probability of the two-patch meta-population is very close to the fixation probability of the well-mixed population40, a close inspection reveals an interesting property: For low migration probabilities and (N_1 ne N_2), the two patch structure is a suppressor of selection in the original sense of Lieberman et al.1: For advantageous mutations, (r>1), it decreases the fixation probability, whereas for disadvantageous mutations, (r<1), it increases the fixation probability compared to the well mixed case. For weak selection, we show this analytically: For (rapprox 1), we can write
$$begin{aligned} phi _{mathrm{wm}}^{N_1+N_2}&approx frac{1}{N_1+N_2} +frac{N_1+N_2-1}{2(N_1+N_2)}(r-1) end{aligned}$$
(8)
$$begin{aligned} phi _{bullet !!-!!bullet }&approx phi _{mathrm{wm}}^{N_1+N_2} -N_1 N_2 frac{(N_1-N_2)^2}{2(N_1^2+N_2^2)^2} (r-1). end{aligned}$$
(9)
While the difference to the well mixed case vanishes for (N_1=N_2) in first order in (r-1), the fixation probability of the two patch structure is larger for (r<1) and smaller for (r>1). Thus, under weak selection the two patch structure with (N_1 ne N_2) is a suppressor of selection.
While the structure remains a suppressor of selection for most values of the migration probability (lambda ), Fig. 2 reveals that for very large (lambda ) it becomes an amplifier of selection.
The difference between the fixation probability of a two-patch meta-population and a well-mixed population of the same size, (N=10). The two patch sizes are (N_1=3) and ( N_2=7 ). Lines show analytical results for low migration probabilities (Eq. 4) and migration probability (lambda =1) (Eq. 19) as a function of fitness. Symbols show numerical results based on a transition matrix approach39. The numerical result and analytical result for low migration probability and high migration match perfectly. In the low migration regime the two-patch meta-population is a suppressor of selection, indicated by the fact that the symbols are never in the area with a white background. However, in the high migration regime (( lambda =1 )), where simulations and analytical results again match, the two-patch meta-population is an amplifier of selection. The fixation probability for ( lambda =1 ) is obtained analytically using the Martingale approach discussed by Monk41.
The meta-star
Here, we approximate the fixation probability (phi _{bigstar }) of a meta-star with ( M-1 ) leaves for low migration probability. For simplicity, we assume that all patches are of the same size (N_1=frac{N}{M}) and omit the notation for patch size. As long as the migration probability is sufficiently low, such that before the next migration the immigrant gets fixed or lost, patches tend to be homogeneous. We denote the number of homogeneous mutant patches among the leaves by j and use a lower index to represent the state of the central patch, which is either occupied by wild-types ((circ )) or by mutants ((bullet )). The number of homogeneous mutant patches increases in two ways:
- (i)
the center is occupied by mutants and is selected for birth and its offspring migrates to one of the peripheral homogeneous wild-type patches and reaches fixation in that patch,
$$begin{aligned} T^{j+}_{bullet rightarrow bullet }=frac{r}{(j+1)r+M-1-j}frac{M-1-j}{M-1} lambda phi _{mathrm{wm}}^{N_1}(r), end{aligned}$$
(10)
where (phi _{mathrm{wm}}^{N_1}(r)) determines the fixation probability of a mutant in a local population,
- (ii)
the center is occupied by wild-types and one of the homogeneous mutant leaves is selected for birth and its offspring migrates to the center and gets fixed there,
$$begin{aligned} T^{j}_{circ rightarrow bullet }=frac{jr}{jr+M-j} lambda phi _{mathrm{wm}}^{N_1}(r). end{aligned}$$
(11)
Note that the number of homogeneous mutant leave nodes cannot increase if the center is occupied by wild-type individuals, i.e. (T^{j+}_{circ rightarrow circ } =0). Similarly, the number of homogeneous mutant leave nodes cannot decrease if the center is occupied by mutants, i.e. (T^{j-}_{bullet rightarrow bullet } =0). Thus, the number of homogeneous mutant patches can decrease in two ways,
- (i)
the center is occupied by wild-types and is selected for birth and its offspring migrates to one of the homogeneous mutant leaves and gets fixed there,
$$begin{aligned} T^{j-}_{circ rightarrow circ }=frac{1}{j r+M-j} frac{j}{M-1} lambda phi _{mathrm{wm}}^{N_1}left( tfrac{1}{r}right) , end{aligned}$$
(12)
where (phi _{mathrm{wm}}^{N_1}(tfrac{1}{r}) ) is the fixation probability of a wild-type in a local mutant population, or
- (ii)
the center is occupied by mutants and one of the leaves is selected for birth and its offspring migrates to the center and gets fixed there,
$$begin{aligned} T^{j}_{bullet rightarrow circ }=frac{M-1-j}{(j+1)r+M-1-j} lambda phi _{mathrm{wm}}^{N_1}left( tfrac{1}{r}right) . end{aligned}$$
(13)
For the star graph, depending on where the mutant emerges, two different fixation probabilities are defined: The fixation probability when a single mutant emerges in the center (phi _{bullet }^0) and the fixation probability when a single mutant emerges in one of the leaves, ( phi _{circ }^1 ). Following the same arguments as in Ref.42, we find
$$begin{aligned} phi _{bullet }^0= & {} dfrac{T_{bullet }^0}{1+(1-T_{bullet }^0) sum _{j=1}^{M-2}left( dfrac{T_{circ }^j}{T_{bullet }^j}right) ^j} end{aligned}$$
(14a)
$$begin{aligned} phi _{circ }^1= & {} dfrac{1-T_{circ }^j}{T_{bullet }^j}phi _{bullet }^0, end{aligned}$$
(14b)
where the probability to leave the state with a wild-type patch in the center and j mutant patch leaves is
$$begin{aligned} T_{circ }^j=frac{T^{j-}_{circ rightarrow circ }}{T^{j}_{circ rightarrow bullet } +T^{j-}_{circ rightarrow circ }} =frac{frac{1}{r} phi _{mathrm{wm}}^{N_1} left( frac{1}{r}right) }{frac{1}{r} phi _{mathrm{wm}}^{N_1} left( frac{1}{r}right) +(M-1) phi _{mathrm{wm}}^{N_1}(r) }, end{aligned}$$
(15)
and the probability to leave the state with a mutant patch in the center and j mutant patch leaves is
$$begin{aligned} T_{bullet }^j= frac{T^{j+}_{bullet rightarrow bullet }}{T^{j+}_{bullet rightarrow bullet }+T^{j}_{bullet rightarrow circ }} =frac{rphi _{mathrm{wm}}^{N_1}(r)}{rphi _{mathrm{wm}}^{N_1}(r)+(M-1) phi _{mathrm{wm}}^{N_1}left( frac{1}{r}right) }. end{aligned}$$
(16)
Note that the two probabilities ( T_{circ }^j) and (T_{bullet }^j) are independent of j in our particular case. Thus, also their ratio (Gamma = T_{circ }^j / T_{bullet }^j) is independent of j, which makes our calculation of (phi _{bullet }^0) easier.
Evaluating Eq. (14), we find the average fixation probability in the entire patch structured meta-population starting from a single homogeneous mutant patch,
$$begin{aligned} phi _{mathrm{patch}}=tfrac{M-1}{M} phi _{circ }^1+tfrac{1}{M} phi _{bullet }^0. end{aligned}$$
(17)
Therefore, the fixation probability of the whole population equals
$$begin{aligned} phi _{bigstar }=phi _{mathrm{wm}}^{N_1}(r) phi _{mathrm{patch}}. end{aligned}$$
(18)
Figure 3 illustrates the fixation probability of a meta-star as a function of fitness. Numerical solutions for low migration agree very well with the low migration approximation. According to this plot, the meta-star is an amplifier of selection in the low migration regime—similar to the star network of individuals1. A numerical investigation of Eq. (18) reveals that this result carries over to larger (N_1) as well. For any value of M and (N_1) between 1 and 100, we find that the star network of patches amplifies selection. However, as expected from earlier work40, the extent of amplification becomes smaller with growing population size.
Using the same approach as we used for the two patch meta-population, we find that the meta-star is an amplifier for small and high migration probability, but not in between. For intermediate migration probability, it is only a piecewise amplifier43,44 and does not fall into one of the originally defined categories, see Fig. 3.
The meta-star in low migration is equivalent to the “star of islands” discussed by Allen et al.45. In their study for death–Birth updating they found that the comparison of the size of the hub to the size of the leaves makes a determinative difference. When the leaves are larger, the structure amplifies under weak selection; when the hub is larger, it suppresses under weak selection. When the hub and leaves are the same size, the structure acts as a “reducer”, meaning that it lessens the fixation probability for all r not equal to 1 (termed “suppressor of fixation” elsewhere34). Doing the same comparison in the whole range of selection, we find that the meta-star under Birth–death is an amplifier when the hub has greater or equal size to the leaves, and a transient amplifier when the leaves are larger than the hub.
The difference between the fixation probability of a meta-star and the equivalent well-mixed population of size with ( M=4 ) patches and identical local size in each patch ( N_1=N/M=5 ) for different fitness values. Lines show analytical results for low migration probabilities (Eq. 18) and migration probability (lambda =1) (Eq. 19) symbols show numerical results based on a transition matrix approach39 for (lambda =10^{-6} ), (lambda =0.01), (lambda =0.1 ), and (lambda =0.5). For migration probability (lambda =10^{-6} ), we observe an almost perfect agreement, the low migration result serves as a good approximation. The fixation probability in ( lambda =1 ) is obtained analytically using the Martingale approach. For extremely high and low migration probability the meta-star acts as an amplifier of selection (such that the lines only pass through the white shaded area of the plot) while in the intermediate migration regime shows a very different behavior where it could be an amplifier or a suppressor of selection depending on the fitness value.
High migration probability
For moderate migration probabilities, it is challenging to calculate the fixation probability. However, in the case of the maximum possible migration probability, ( lambda =1 ), the two-patch meta-population and meta-star transform to complete bipartite graphs: In the two-patch meta-population, every offspring will be immediately moved to the other patch. In the meta-star, the offspring of individuals in the center node will be placed in a random leaf, whereas the offspring of the individuals in the leaf nodes will be placed in the center. Thus, the meta-star can be thought of as a bipartite graph in which one part is made out of all leaf nodes and the other part out of the center.
The fixation probability of complete bipartite graphs has been calculated analytically previously41,44,46 using the specific features of Martingales. The probability to reach fixation if the initial mutant arises either in patch 1, (phi ^1_{mathrm{bp}}) or patch 2, ( phi ^2 _{mathrm{bp}}) are
$$begin{aligned} phi ^1_{mathrm{bp}}=frac{h_1-1}{(h_1)^{N_1}(h_2)^{N_2}-1}, quad phi ^2_{mathrm{bp}}=frac{h_2-1}{(h_1)^{N_1}(h_2)^{N_2}-1}, end{aligned}$$
(19)
where ( h_1=frac{frac{N_2}{N_1}+frac{1}{r}}{frac{N_2}{N_1}+r} ) and ( h_2=frac{frac{N_1}{N_2}+frac{1}{r}}{frac{N_1}{N_2}+r} ). The average fixation probability is
$$begin{aligned} phi _{mathrm{bp}}=dfrac{N_1phi ^1_{mathrm{bp}} +N_2phi ^2_{mathrm{bp}}}{N_1+N_2}. end{aligned}$$
(20)
In the case where one of the patch size is much larger than the other, ( N_2 gg N_1 ), the fixation probability converges to the fixation probability of a star graph (frac{1-1/r^2}{1-1/r^{2N_2}}).
As discussed above, a star-structured meta-population in (lambda =1) can be reduced to a complete bipartite graph. As a result the fixation probability of a star meta-population with (M-1) leaves and population size N such that the population distributing homogeneously in all the patches is obtained by replacing (N_1) with ( frac{(M-1)N}{M} ) and ( N_2 ) with ( frac{N}{M} ) in Eqs. (19) and (20).
As shown in Figs. 2 and 3 both the two-patch meta-population and meta-star are amplifiers of selection for ( lambda =1 ). It has been proven in Ref.47 that a complete bipartite amplifies selection for weak selection. This can also be seen from a Taylor expansion of the difference between Eq. (20) and the corresponding result for the well mixed population at (r=1), which leads to
$$begin{aligned} phi _{mathrm{bp}} – phi _{mathrm{wm}}approx frac{1}{2} tfrac{(N_1 – N_2)^2}{(N_1 + N_2)^2} tfrac{ N_1 (N_1 – 1) +N_2 (N_2 – 1)}{N_1^2 + N_2^2}(r-1). end{aligned}$$
(21)
This quantity is positive for (r>1) and negative for (r<1), such that the structure is an amplifier of selection for weak selection. Eq. (20) reveals this fact holds for the whole range of selection strength. If we have a fixed population of size N on a complete bipartite graph, for fitness values ( r>1 ) the minimum fixation probability occurs when the two patch sizes are identical, ( N_1=N_2=N/2 ). Similarly, for fitness values ( r<1 ) the maximum fixation probability occurs when the two patch sizes are identical, ( N_1=N_2=N/2 ).
Since a complete bipartite graph with identical patch size is an isothermal graph and its fixation probability is the same as the fixation probability of a well-mixed population, we conclude that any complete bipartite graph is the amplifier of selection when (N_1ne N_2 ). This result is implicitly contained in Refs.41,46, but deserves special attention: It implies that a graph can be turned into the amplifier if we enforce a very large degree of exchange of individuals between patches. Combined with the observation that many graphs of individuals are amplifiers of selection34, it suggests that it may be easier to construct amplifiers of selection than suppressors of selection in undirected networks5,48.
Source: Ecology - nature.com
