# Genotype frequencies and fitness values in haploid population with clonal growth

I found a small problem, and I am not sure at all about my answers. If someone could correct it, that would be awesome.

Consider a heterogeneous population with the following genotype frequencies and fitness values (attention: haploid individuals with clonal growth!) :

ab : $P_{ab} = 0.8, \quad \lambda_{ab} = 1$

Ab : $P_{Ab} = 0.1, \quad \lambda_{Ab} = 1.2$

aB : $P_{aB} = 0.1, \quad \lambda_{aB} = 1.1$

1. Determine the average fitness of the population

2. After one generation, what are the new genotype frequencies and the average fitness of the population ?

3. What are their values in the long-time limit ?

4. Explain clonal interference in this context.

1. Relative fitnesses values - we take the highest as reference (here: Ab): $$\omega_{ab} = \dfrac{1}{1.2} = 0.83$$ $$\omega_{Ab} = \dfrac{1.2}{1.2} = 1$$ $$\omega_{aB} = \dfrac{1.1}{1.2} = 0.916$$

Average fitness of the population : $$\overline{\omega} = p_{ab}.\omega_{ab} + p_{Ab}.\omega_{Ab} + p_{aB}.\omega_{aB} = 0.8556$$

1. $$p_{ab}(t+1) = \dfrac{p_{ab}(t).\omega_{ab}}{\overline{\omega}} = \dfrac{0.8\times 0.83}{0.8556} = 0.776$$

same for the others frequencies : $$p_{Ab}(t+1) = \dfrac{0.1}{0.8556} = 0.117$$ $$p_{aB}(t+1) = \dfrac{0.1 \times 0.916}{0.8556} = 0.107$$

1. I know that the fraction of the fitter genotype will grow exponentially compared to the genotype of lower fitness. But how do you demonstrate that ?

2. I learnt that due to the haploid structure of the population, the two new alleles A and B are in competition under natural selection : this will lead to the loss of one of them in the long term (even if they are both advantageous for the organisme). Then how do I explain that we those data ? Does this mean allele B will disappear ?

Any help would be welcomed :)