genetics - Easy derivation of Kimura's approximation for the probability of fixation of a mutation

Kimura's approximation for the probability of fixation of a mutation under selection finds recurrent use in population genetics models till date. I am trying to understand the mathematical basis of this equation but none of the textbooks or online resources that I have checked, provide an easy derivation of this approximation but rather simply cite Kimura's 1962 paper.

$$P_\text{fix} \approx \frac{1-e^{-4Nsp} }{1-e^{-4Ns}} \qquad (1)$$

So, I was reading the original paper but the provided derivation doesn't appear clear to me.

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Details

Kimura starts with definition of probability of change in allele frequency as:

$$u(p,t+\delta t) = \int f(p+\delta p; \delta t)\ u(p+\delta p,t)\ d(\delta p) \qquad (2)$$

where (quoted exactly)

• $u(p,t)$ is the probability that an allele will be fixed in a time interval $t$ given that it's initial frequency is $p$.


• $f(p+\delta p; \delta t)$ is the probability density of the change from $p$ to $p+\delta p$

Then he uses Taylor series approximation to obtain an equation of this form:

$$\frac{\partial u(p,t)}{\partial t}= \frac{V}{2}\frac{\partial ^2u}{\partial p^2}+M\frac{\partial u}{\partial p} \qquad (3)$$

He defines $M$ and $V$ as mean and variance of change of $p$ per generation. These are formally defined as:

$$M=\lim_{\delta t \to 0} \frac{1}{\delta t}\int (\delta p).\ f(p+\delta p; \delta t).\ d(\delta p)$$

$$V=\lim_{\delta t \to 0} \frac{1}{\delta t}\int (\delta p)^2.\ f(p+\delta p; \delta t).\ d(\delta p)$$

($V$ actually should be just the second moment as per the mathematical definition and not variance)

Then he solves equation 3 at steady state with boundary conditions $u(0,t)=0$ and $u(1,t)=1$ to obtain this:

$$u(p)=\frac{\displaystyle\int_0 ^p G(x) dx}{\displaystyle\int_0 ^1 G(x) dx} \qquad (4)$$

where:

$$G(x)=\exp\left(-\int \frac{2M}{V}dx\right)$$

I understood the derivation till this point.

Then he just puts:

$$M=sx(1-x)$$ $$V=x(1-x)/2N$$

and obtains equation 1.

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In short

Is there an easy derivation for equation 1?
If not, can someone explain me how M and V were approximated as above?