Understanding the informational content of Evolutionary Processes (part 3)
2. Evolution Driven by Recombination (Sex)
Now, let's contrast this to evolution driven by recombination, or sexual reproduction. We assume the simplest recombination model with no linkage:
- Each child genotype is formed by randomly selecting bits from two parents' genotypes with equal probability (1/2 per bit).
Consider parents of identical fitness \(F = fG\). Under recombination, the expected fitness of the children remains the same as the parents (\(F\)), but the variation increases significantly. The standard deviation of fitness for children generated by recombination scales as:
\( \sqrt{G f (1 - f)} \)
After selection, the mean fitness at the next generation is increased proportionally to this standard deviation. Thus, the rate of fitness increase under sexual reproduction approximately follows:
\( \frac{dF}{dt} \approx \eta \sqrt{f(1 - f) G} \)
where \( \eta \approx \sqrt{\frac{2}{\pi + 2}}\).
This differential equation can be solved explicitly to find how fitness evolves in a sexual population:
\( f(t) = \frac{1}{2}\left[ 1 + \sin\left(\eta \frac{t + c}{\sqrt{G}}\right)\right] \)
where \( c = \sin^{-1}(2f(0) - 1) \). Thus, under recombination, the population reaches optimal fitness in approximately \( (\pi / \eta) \sqrt{G} \) generations, significantly faster than mutation alone.
Comparing Mutation vs. Recombination (Sex)
Key insight:
- Mutation-driven evolution: fitness increases slowly at a rate of roughly \(O(1)\) bits per generation at optimal conditions.
- Recombination-driven evolution: fitness increases at a rate proportional to \(\sqrt{G}\), allowing much faster adaptation and optimization.
Thus, from an informational and evolutionary perspective, sex (recombination) is a powerful evolutionary strategy, significantly accelerating adaptation and overcoming local fitness optima more efficiently than mutation alone.
Informational interpretation
From an information-theoretic standpoint, we can interpret sexual reproduction as parallel information acquisition and inference:
- Each individual can be viewed as holding a "genetic hypothesis" about the environment.
- Recombination rapidly combines these hypotheses to discover optimal fitness configurations.
- Mutation, by contrast, performs incremental and isolated searches, yielding slower information gain.
The evolutionary advantage of sex thus reflects the benefit of efficient parallel processing and inference.
In conclusion, from rigorous mathematical analysis, we clearly see that sexual reproduction (recombination) significantly enhances the informational content of evolutionary processes, justifying its evolutionary cost. This perspective aligns beautifully with the principles of information theory and Bayesian inference.