Understanding the informational content of Evolutionary Processes (part 2)
We have defined the basic evolutionary model, including variation (mutation and recombination) and truncation selection. Now, let’s delve deeper into the mathematical details of how these variations influence the evolution of fitness.
Rate of Fitness Increase
We will consider the evolution under two separate scenarios:
- Mutation-Driven Evolution
- Recombination (Sex)-Driven Evolution
1. Evolution Driven by Mutation
Consider a population with discrete generations. Each individual produces exactly two offspring whose genomes undergo mutation, flipping each bit with independent and identically distributed (iid) probability ( m ). After mutation, natural selection (truncation selection) retains only the top ( N ) fittest offspring, and this defines the next generation.
Let's mathematically characterize this process.
Define the normalized fitness as:
\( f(x) = \frac{F(x)}{G} \)
where \( F(x) = \sum_{g=1}^{G} x_g \) and \( G \) is the genome size (total number of bits).
It is helpful to work with the excess normalized fitness:
\( \delta f \equiv f - \frac{1}{2} \)
Assuming small mutation rates \( m \), if a parent has excess normalized fitness \( \delta f \), its offspring will have an expected excess fitness of:
\( \mathbb{E}[\delta f_{\text{child}}] = (1 - 2m)\delta f \)
with variance approximately:
\( \text{Var}(\delta f_{\text{child}}) \approx \frac{m}{G} \)
Dynamics of Fitness Evolution under Mutation
At generation \( t \), let the parent population have a mean excess normalized fitness \(\delta f(t)) and variance \(\sigma^2(t)). After mutation, the child generation (before selection) has:
- Mean: \((1 - 2m)\delta f(t)\)
- Variance: \(\sigma^2(t) + \frac{m}{G}\)
Natural selection retains only the top half of this distribution. Assuming a Gaussian approximation, the fitness after selection is:
\( \delta f(t+1) = (1 - 2m)\delta f(t) + \alpha \sqrt{\sigma^2(t) + \frac{m}{G}} \)
and the variance after selection becomes:
\( \sigma^2(t+1) = \gamma \left(\sigma^2(t) + \frac{m}{G}\right) \)
where constants \(\alpha) and \(\gamma) are derived from the properties of truncation selection of Gaussian distributions. For Gaussian truncation selection, typically:
- \(\alpha \approx \sqrt{2/\pi} \approx 0.8\)
- \(\gamma \approx 1 - 2/\pi \approx 0.36\)
Assuming variance reaches a steady-state (\(\sigma^2(t+1) \approx \sigma^2(t)\)), we find:
\( \sigma^2(t) = \frac{\gamma m/G}{1-\gamma} \)
For large \( G \), the rate of fitness increase becomes approximately:
\( \frac{df}{dt} \approx -2m\delta f + \sqrt{\frac{m}{G}} \)
This rate is maximized when:
\( m_{\text{opt}} = \frac{1}{16G(\delta f)^2} \)
giving a maximal rate of fitness increase per generation:
\( \left(\frac{df}{dt}\right)_{\text{opt}} = \frac{1}{8G\delta f} \)
Thus, under mutation-driven evolution, as the fitness approaches optimal (\(\delta f) large), the optimal mutation rate becomes very small, and the fitness increases at a rate of roughly one bit per generation.