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Evolutionary amplifiers: Difference between revisions
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The simplest example of an evolutionary amplifier is the [https://en.wikipedia.org/wiki/Star_(graph_theory) star graph], which connects a central hub to a reservoir of leaf vertices through undirected (bi-directional) links. The hub represents a bottleneck for the evolutionary progression because if one leaf vertex is occupied by a mutant it needs to conquer the hub before another leaf vertex can be taken over. Most of the time, the ‘hot’ hub is repeatedly replaced by reproducing leaf vertices and only occasionally the hub itself reproduces and replaces a leaf vertex. For an advantageous mutant in a leaf vertex this means that compared to a resident leaf vertex it has a relative advantage of \(r\) to occupy the hub and similarly the mutant hub has again a relative reproductive advantage of \(r\). Thus, the overall relative advantage of a mutant leaf vertex to proliferate and occupy another leaf vertex is \(r^2\). Note that there is no other way for a mutant to spread through the population. As a consequence, a mutant with fitness \(r^2\) on the star graph has (approximately) the same fixation probability as a mutant with fitness \(r^2\) on a circulation graph. Thus, the fixation probability is approximately | The simplest example of an evolutionary amplifier is the [https://en.wikipedia.org/wiki/Star_(graph_theory) star graph], which connects a central hub to a reservoir of leaf vertices through undirected (bi-directional) links. The hub represents a bottleneck for the evolutionary progression because if one leaf vertex is occupied by a mutant it needs to conquer the hub before another leaf vertex can be taken over. Most of the time, the ‘hot’ hub is repeatedly replaced by reproducing leaf vertices and only occasionally the hub itself reproduces and replaces a leaf vertex. For an advantageous mutant in a leaf vertex this means that compared to a resident leaf vertex it has a relative advantage of \(r\) to occupy the hub and similarly the mutant hub has again a relative reproductive advantage of \(r\). Thus, the overall relative advantage of a mutant leaf vertex to proliferate and occupy another leaf vertex is \(r^2\). Note that there is no other way for a mutant to spread through the population. As a consequence, a mutant with fitness \(r^2\) on the star graph has (approximately) the same fixation probability as a mutant with fitness \(r^2\) on a circulation graph. Thus, the fixation probability is approximately | ||
\begin{align} | \begin{align} | ||
\rho_2 = \frac{\displaystyle 1-\frac1{r^2}}{\displaystyle 1-\frac1{r^{2N}}} | \rho_2 = \frac{\displaystyle 1-\frac1{r^2}}{\displaystyle 1-\frac1{r^{2N}}} | ||
\end{align} | \end{align} | ||
for a star graph of size \(N\). The approximation improves for large \(N\) and becomes exact in the limit \(N\to\infty\). | for a star graph of size \(N\). The approximation improves for large \(N\) and becomes exact in the limit \(N\to\infty\). | ||
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The intriguing dynamical features of evolutionary amplifiers have first been reported by [[#Publications|Lieberman et al. (2005)]]. However, the formula for the fixation probabilities reported therein has resulted in a controversy that took time and effort to resolve. Lieberman et al. (2005) originally derived the fixation probability of a single mutant with fitness \(r\) on a superstar graph as | The intriguing dynamical features of evolutionary amplifiers have first been reported by [[#Publications|Lieberman et al. (2005)]]. However, the formula for the fixation probabilities reported therein has resulted in a controversy that took time and effort to resolve. Lieberman et al. (2005) originally derived the fixation probability of a single mutant with fitness \(r\) on a superstar graph as | ||
\begin{align}\label{eq:nature05} | \begin{align}\label{eq:nature05} | ||
\rho_k = \dfrac{1-\dfrac1{r^k}}{1-\dfrac1{r^{kN}}}. | \rho_k = \dfrac{1-\dfrac1{r^k}}{1-\dfrac1{r^{kN}}}. | ||
\end{align} | \end{align} | ||
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\begin{align} | \begin{align} | ||
\label{eq:jtb05} | \label{eq:jtb05} | ||
1-\dfrac1{r^4(k-1)\left(1-\frac1r\right)^2} \leq \rho_k \leq 1-\dfrac1{1+r^4(k-1)} | 1-\dfrac1{r^4(k-1)\left(1-\frac1r\right)^2} \leq \rho_k \leq 1-\dfrac1{1+r^4(k-1)} | ||
\end{align} | \end{align} | ||
although the bounds are not as tight anymore. At the same time, the lower bound in this idealized limit is consistently too high. Quite surprisingly, this indicates that even for \(B=M=200\), which translates to a population size of \(N\approx40'000\), finite size effects remain significant. Although, the primary cause for the finite size effects is easily accounted for because it simply relates to the chance that a randomly placed mutant ends up in one of the chains (or the hub) and hence almost certainly goes extinct. | although the bounds are not as tight anymore. At the same time, the lower bound in this idealized limit is consistently too high. Quite surprisingly, this indicates that even for \(B=M=200\), which translates to a population size of \(N\approx40'000\), finite size effects remain significant. Although, the primary cause for the finite size effects is easily accounted for because it simply relates to the chance that a randomly placed mutant ends up in one of the chains (or the hub) and hence almost certainly goes extinct. | ||
