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It's simplest to think about a haploid population with size $N$. Say there are two alleles, $A$ and $a$, with $A$ having frequency $p$ in this generation. Each $A$ individual will have some realized fitness (i.e., number of offspring) in this generation; let's call this number $w_A^{(i)}$ for the $i^\text{th}$ individual. The total number of $A$ individuals ...


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If the fitness of a heterozygote is $(1+s/2)$ and of a homozygote is $(1-s/2)$ then why is the probability for a given state $(1+s/2)^j(1-s/2)^{m-k}$ $$\binom mj (1/2)^j(1/2)^{m-j}= \binom mj (1/2)^m~~ ?$$ As you pointed out earlier, in the general case it need not be true that $p = q = 1/2$ but that is what the form of the probability above implies. So ...


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Is this the exact text from the book? The left side seems to represent the probability for "No coalescence in $k$ lines in $t$ generations (i.e. the $Pr(k)^t$ term), and at least one coalescence among those lines in generation $t+1$ (the $1-Pr(k)$ term)" which is the same event as "First coalescence event in $k$ lines is exactly in generation ...


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Not in general -- there can be linkage disequilibrium among the loci. For instance, say that there are two di-allelic loci, $A/a$ and $B/b$, and that the frequencies of the $A$ and $B$ alleles are both $1/2$ and that they have the same effect on the trait, with no dominance. If all haplotypes in the population are either $Ab$ or $aB$ (with no $AB$ or $ab$ ...


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To derive it, first use that $E[x(1-x)]= E[x-x^2]=E[x]-E[x^2]$ and that $E[x^2]=\text{Var}[x]+E[x]^2$ to rewrite the left-hand side: $$E\left[x_{t+1}(1-x_{t+1})\right] = E\left[x_{t+1}\right](1-E\left[x_{t+1}\right])-\text{Var}\left[x_{t+1}\right].$$ The equation for $p_{ij}$ is just saying that $2Nx_{t+1}$ is binomially distributed with $2N$ trials with ...


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The notation at this site resembles that in your question but preserves the $\frac{x_t}{2N}$ notation for probability of selecting an allele. $$E[\frac{x_{t+1}}{2N})(1 - \frac{x_{t+1}}{2N} )|x_t] = (\frac{x_{t}}{2N})(1 - \frac{x_{t}}{2N}) (1 - \frac{1}{2N}) $$ The expression $(\frac{x_{t}}{2N})(1 - \frac{x_{t}}{2N}) $ is the probability of heterozygosity ...


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Well, the total genetic variance is just, by the definition of the variance, $$ \sigma^2 =\sum_{i,j} f_i f_j (w_{ij}-\bar{w})^2 $$ (using $f_i$ and $w_{ij}$ for frequency and fitness, respectively), and $$\bar{w} = \sum_{i,j} f_i f_j w_{ij}$$ is just the average fitness. You can calculate the additive genetic variance for different loci by simply assuming ...


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To a good first approximation $\overline{\Delta f} = 0$. Where $\overline{\Delta f}$ is the mean change in fitness down to any point or indel mutation. The reasons for this are as follows: In the genome of higher organisms, most of the genome is non-functional ("junk") so most mutations will not have any effect regardless of the change made. A substantial ...


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I don't believe you can produce a general function for this. It will depend on the exact gene and organism you are considering. From a molecular point of view, the vast majority of recessive mutations result from a change producing either a non-functional protein product or a truncated product that is cleaned up by the cell. We can reasonably assume that ...


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[This is purely speculative] Assumptions: impact on fitness is measured by survival chance impact is because of protein coding genes Probability of a mutation at position $i$ $P(m=i\ |\ g)$ where $g$ is the genome with its annotations. Probability that activity of some protein changes by X-fold given mutation at $i^{th}$ position(s) in the genome: ...


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An easy way to visualize the mistake in your thought experiment is to consider a bottleneck event, when the ancestral population was very small, maybe just a few individuals. This would mean that the entire current population is descending from just a few individuals. Your thought experiment is assuming that the "pyramid" of your ancentors is expanding all ...


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Your calculations are the following. Assuming non-overlapping generations, the number of ancestors you have in the last $t$ generation is given by: $$\sum_{i=1}^t 2^t$$ This sounds correct. But there are some very strong assumptions: Generations are non-overlapping. A more realistic model would need to consider $t$ as a continuous variable a give a ...



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