Inherent randomness of evolving populations

The entropy rates of the Wright-Fisher process, the Moran process, and generalizations are computed and used to compare these processes and their dependence on standard evolutionary parameters. Entropy rates are measures of the variation dependent on both short-run and long-run behaviors and allow the relationships between mutation, selection, and population size to be examined. Bounds for the entropy rate are given for the Moran process (independent of population size) and for the Wright-Fisher process (bounded for fixed population size). A generational Moran process is also presented for comparison to the Wright-Fisher Process. Results include analytic results and computational extensions.

Figure 1

(Top) Transition probabilities for $N=100$, ${\mu }_{AB}=0.001$, ${\mu }_{BA}=0.01$, uniform mutations, and game matrix given by $a=2$, $b=4$, $c=3$, and $d=2$. Bars top to bottom correspond to $T_{i\to i-1},T_{i\to i},T_{i\to i+1}$ for each state $i$. (Middle) Transition entropies. (Bottom) stationary distribution. The entropy rate in this example is approximately $H=0.9472$. Stationary distributions are not generally Gaussian [11].

Figure 2

Entropy rate vs population size $N$ for ${\mu }_{AB}={\mu }_{BA}\in \{0.5,0.1,0.05,0.01,0.005,0.001,0.0001\}$ (top to bottom) with a neutral fitness landscape. (Left) Mutations only at the boundary states (boundary regime); the entropy rate eventually approaches zero as $N\to \infty$. (Right) Mutations for all states (uniform regime); the entropy rate approaches $3/2ln2$ as $N\to \infty$.

Figure 3

Relative entropy [Eq. (5)] for the stationary distributions of Moran processes for the neutral fitness landscape computed with the boundary regime and the uniform regime, for each combination of $N$ and $\mu$. For small values of $\mu$ (horizontal axis) relative to $N$ (vertical axis), i.e., $N\mu \ll 1$, the distance between the stationary distributions is small. For larger population sizes and fixed $\mu$, the stationary distributions diverge substantially.

Figure 4

Scaled entropy rate vs population size $N$ for ${\mu }_{AB}={\mu }_{BA}\in \{0.5,0.1,0.05,0.01,0.005,0.001,0.0001\}$ (top to bottom) with a neutral fitness landscape for the Wright-Fisher process. Entropy rates are divided by $ln(N+1)$. (Left) Mutations only at the boundary states (boundary regime); the entropy rate approaches a value less than 1/2 as $N\to \infty$. (Right) Mutations for all states (uniform regime); the entropy rates approach $1/2$ as $N\to \infty$. Compare to Fig. 2.

Figure 5

Entropy rate heat maps for the Moran process, $r\in [0.6,1.4]$ in the horizontal axis, $N\in [3,150]$ on the vertical axis. (Top row) Boundary mutation regime. (Bottom Row) Uniform mutation regime. Though the plots look symmetric horizontally about $r=1$, they are not precisely so. Color bars are not consistent across plots (for additional resolution). Entropy rates are generally smaller as $\mu$ decreases (left to right). For the top row, the entropy rate is eventually decreasing as $N$ increases; for the bottom row, the entropy rate increases as $N$ increases. The apparent near symmetry about $r=1$ is due to the $r\leftrightarrow 1/r$ symmetry and the Taylor series $1/(1-r)=1+r+\cdots$.

Figure 6

Mutation-drift balance. Population size $N$ for which the entropy rate is maximal versus the relative fitness $r$ for various fixed values of $\mu$ (boundary mutations). From bottom to top: $\mu =0.05$, 0.01, 0.005, 0.003, 0.001. The maximum for $r=1$ appears to occur when $N\mu \approx 1$. The maximum entropy rate may occur for a large population size even though the entropy rate tends to zero as the population size gets large. These curves correspond to the maximum values in Fig. 2.

Figure 7

Scaled entropy rate heat maps for the Wright-Fisher process: $r\in [0.6,1.4]$ in the horizontal axis, $N\in [2,100]$ on the vertical axis. (Top row) Boundary mutation regime. (Bottom row) Uniform mutation regime. Color bars are not consistent across plots (for additional resolution). Entropy rates are generally smaller as $\mu$ decreases (left to right). For the top row, the entropy rate is eventually decreasing as $N$ increases; for the bottom row, the entropy rate increases as $N$ increases.

Figure 8

Scaled entropy rate vs population size $N$ for ${\mu }_{AB}={\mu }_{BA}\in \{0.5,0.1,0.05,0.01,0.005,0.001,0.0001\}$ (top to bottom) with a neutral fitness landscape for the $n$-fold Moran process. Entropy rates are divided by $lnN$. (Left) Mutations only at the boundary states (boundary regime). (Right) Mutations for all states (uniform regime). Compare to Figs. 2 and 4.

Figure 9

Scaled entropy rate vs population size $N$ for the $N$-fold Moran process with ${\mu }_{AB}={\mu }_{BA}\in \{0.5,0.1,0.05,0.01,0.005,0.001,0.0001\}$ (top to bottom) and a neutral fitness landscape. Entropy rates are divided by $ln(N+1)$. (Left) Mutations only at the boundary states (boundary regime). (Right) Mutations for all states (uniform regime). Compare to Figs. 2 and 4.