Hydrodynamic flow and concentration gradients in the gut enhance neutral bacterial diversity

The gut microbiota features important genetic diversity, and the specific spatial features of the gut may shape evolution within this environment. We investigate the fixation probability of neutral bacterial mutants within a minimal model of the gut that includes hydrodynamic flow and resulting gradients of food and bacterial concentrations. We find that this fixation probability is substantially increased, compared with an equivalent well-mixed system, in the regime where the profiles of food and bacterial concentration are strongly spatially dependent. Fixation probability then becomes independent of total population size. We show that our results can be rationalized by introducing an active population, which consists of those bacteria that are actively consuming food and dividing. The active population size yields an effective population size for neutral mutant fixation probability in the gut.

Haldane’s formula in Cannings models: the case of moderately strong selection

For a class of Cannings models we prove Haldane’s formula, $$\pi (s_N) \sim \frac{2s_N}{\rho ^2}$$ π ( s N ) ∼ 2 s N ρ 2 , for the fixation probability of a single beneficial mutant in the limit of large population size N and in the regime of moderately strong selection, i.e. for $$s_N \sim N^{-b}$$ s N ∼ N - b and $$0< b<1/2$$ 0 < b < 1 / 2 . Here, $$s_N$$ s N is the selective advantage of an individual carrying the beneficial type, and $$\rho ^2$$ ρ 2 is the (asymptotic) offspring variance. Our assumptions on the reproduction mechanism allow for a coupling of the beneficial allele’s frequency process with slightly supercritical Galton–Watson processes in the early phase of fixation.

The network structure affects the fixation probability when it couples to the birth-death dynamics in finite population

The study of evolutionary dynamics on graphs is an interesting topic for researchers in various fields of science and mathematics. In systems with finite population, different model dynamics are distinguished by their effects on two important quantities: fixation probability and fixation time. The isothermal theorem declares that the fixation probability is the same for a wide range of graphs and it only depends on the population size. This has also been proved for more complex graphs that are called complex networks. In this work, we propose a model that couples the population dynamics to the network structure and show that in this case, the isothermal theorem is being violated. In our model the death rate of a mutant depends on its number of neighbors, and neutral drift holds only in the average. We investigate the fixation probability behavior in terms of the complexity parameter, such as the scale-free exponent for the scale-free network and the rewiring probability for the small-world network.

Fixation probabilities in network structured meta-populations

The effect of population structure on evolutionary dynamics is a long-lasting research topic in evolutionary ecology and population genetics. Evolutionary graph theory is a popular approach to this problem, where individuals are located on the nodes of a network and can replace each other via the links. We study the effect of complex network structure on the fixation probability, but instead of networks of individuals, we model a network of sub-populations with a probability of migration between them. We ask how the structure of such a meta-population and the rate of migration affect the fixation probability. Many of the known results for networks of individuals carry over to meta-populations, in particular for regular networks or low symmetric migration probabilities. However, when patch sizes differ we find interesting deviations between structured meta-populations and networks of individuals. For example, a two patch structure with unequal population size suppresses selection for low migration probabilities.

Fixation probabilities in network structured meta-populations

The effect of population structure on evolutionary dynamics is a long-lasting research topic in evolutionary ecology and population genetics. Evolutionary graph theory is a popular approach to this problem, where individuals are located on the nodes of a network and can replace each other via the links. We study the effect of complex network structure on the fixation probability, but instead of networks of individuals, we model a network of sub-populations with a probability of migration between them. We ask how the structure of such a meta-population and the rate of migration affect the fixation probability. Many of the known results for networks of individuals carry over to meta-populations, in particular for regular networks or low symmetric migration probabilities. However, when patch sizes differ we find interesting deviations between structured meta-populations and networks of individuals. For example, a two-patch structure with unequal population size suppresses selection for low migration probabilities.

Fast and strong amplifiers of natural selection

Selection and random drift determine the probability that novel mutations fixate in a population. Population structure is known to affect the dynamics of the evolutionary process. Amplifiers of selection are population structures that increase the fixation probability of beneficial mutants compared to well-mixed populations. Over the past 15 years, extensive research has produced remarkable structures called strong amplifiers which guarantee that every beneficial mutation fixates with high probability. But strong amplification has come at the cost of considerably delaying the fixation event, which can slow down the overall rate of evolution. However, the precise relationship between fixation probability and time has remained elusive. Here we characterize the slowdown effect of strong amplification. First, we prove that all strong amplifiers must delay the fixation event at least to some extent. Second, we construct strong amplifiers that delay the fixation event only marginally as compared to the well-mixed populations. Our results thus establish a tight relationship between fixation probability and time: Strong amplification always comes at a cost of a slowdown, but more than a marginal slowdown is not needed.

Hydrodynamic flow and concentration gradients in the gut enhance neutral bacterial diversity

The gut microbiota features important genetic diversity, and the specific spatial features of the gut may shape evolution within this environment. We investigate the fixation probability of neutral bacterial mutants within a minimal model of the gut that includes hydrodynamic flow and resulting gradients of food and bacterial concentrations. We find that this fixation probability is substantially increased compared to an equivalent well-mixed system, in the regime where the profiles of food and bacterial concentration are strongly spatially-dependent. Fixation probability then becomes independent of total population size. We show that our results can be rationalized by introducing an active population, which consists of those bacteria that are actively consuming food and dividing. The active population size yields an effective population size for neutral mutant fixation probability in the gut.

Spectral analysis of transient amplifiers for death–birth updating constructed from regular graphs

A central question of evolutionary dynamics on graphs is whether or not a mutation introduced in a population of residents survives and eventually even spreads to the whole population, or becomes extinct. The outcome naturally depends on the fitness of the mutant and the rules by which mutants and residents may propagate on the network, but arguably the most determining factor is the network structure. Some structured networks are transient amplifiers. They increase for a certain fitness range the fixation probability of beneficial mutations as compared to a well-mixed population. We study a perturbation method for identifying transient amplifiers for death–birth updating. The method involves calculating the coalescence times of random walks on graphs and finding the vertex with the largest remeeting time. If the graph is perturbed by removing an edge from this vertex, there is a certain likelihood that the resulting perturbed graph is a transient amplifier. We test all pairwise nonisomorphic regular graphs up to a certain order and thus cover the whole structural range expressible by these graphs. For cubic and quartic regular graphs we find a sufficiently large number of transient amplifiers. For these networks we carry out a spectral analysis and show that the graphs from which transient amplifiers can be constructed share certain structural properties. Identifying spectral and structural properties may promote finding and designing such networks.

Evolutionary game dynamics of death-birth process with interval payoffs on graphs

In this paper, we study the evolutionary game dynamics of the death-birth process with interval payoffs on graphs. First of all, we derive the interval replication dynamic equation. Secondly, we derive the fixation probability of the B-C prisoner’s dilemma game based on the death-birth process under the condition of weak selection, analyze the condition of the strategy fixed in the population, that is the condition of strategy A being dominant is analyzed. So we can judge whether natural selection is beneficial to strategy A in the game process through this condition. Finally, the feasibility of this method is verified by several examples.

Pointing the Way to Additional Topics

This concluding chapter highlights many of the concepts that are important to understanding modern-day population genetics research and explains that while they may not have been covered in this book, they are built on the foundations laid out in the preceding chapters. A series of small sections are provided which briefly introduce important concepts for continued learning. These focus especially on the coalescent theory but also touch on tests of neutrality, linkage disequilibrium, deleterious alleles, fixation probability, selfish genetic elements, future directions, and R packages.