# phi(x):Mutation & Selection

Joe Felsenstein joe at evolution.genetics.washington.edu
Sat Feb 10 22:22:59 EST 1996

```In article <4fj7cl\$akn at mark.ucdavis.edu>,
Daniel Mcgoldrick <ez005139 at chip.ucdavis.edu> wrote:
>I can't seem to integrate this expression:
>
>phi(x)=exp(4Nesx^2 +4Neshx(1-x)) x^(4Nev-1) (1-x)^(4Neu-1)dx over the domain
>[0,1]
>
>(its the factor exp(f(x)) that throws me (the rest would just be a beta))
>
>The answer is a constant "C" that is required to normalize the frequency
>distribution of a deleterious gene under mutation-selection balance
>(Wright's stationary distribution) its equation 9.3.4 in Crow and Kimura's
>Introduction to population genetics theory ( a very excellent book) , but
>the authors don't mention whether or not the integral exists or what the
>value of C is (they do graph the function - I can do that, but the absence
>of the analytical expression for C is a bummer). I have Likelihood
>estimation in mind and want to do some tests using this distribution.

I've never seen any solution for that integral.  You would just have to
do it numerically, which is easy enough.  You can still do your
likelihood estimation numerically, but a closed-form expression is not
going to be available for your estimates.  That looks less elegant in the
published paper but is not actually going to make the resulting program
substantially harder to run.

There has been a tradition in evolutionary genetics of getting a nice
closed-form formula, publishing it, and then going home.  But to get anyone
to use your method, you'll have to write a program.  In which case the
absence of it as a closed-form expression won't hurt that much.

--
Joe Felsenstein         joe at genetics.washington.edu     (IP No. 128.95.12.41)
Dept. of Genetics, Univ. of Washington, Box 357360, Seattle, WA 98195-7360 USA

```