(A2) in Otto and Whitton (2000) contains typos and should read:
	
	(((Subscript[σ, c]) (Subscript[σ, c]^(')))/(ln[c N ((e^((Subscript[σ, c])/(2 c N v (Subscript[σ, c]^(')))))-1)((Subscript[σ, c]^('))/((Subscript[σ, c]^('))+(Subscript[σ, c])))]))

Note the c in the denominator and the (Subscript[σ, c]^(')) in the exponential.  These errors were in the manuscript only and did not affect subsequent derivations for equations (3).

Derivation of A2 of Otto and Whitton (2000)

Letting σ1 = (Subscript[σ, c]) and σ2 = (Subscript[σ, c])', the following derives equation A2, using the fixation probability given by equation A1:

Simplify[p[t] c N v fixation[t]/.fixation[t]->2 σ2(σ1+σ2)/(σ2+p[t] σ1)/.p[t]->p0/(p0+(1-p0)Exp[-σ1 t])]

The number of fixation events is given by the integral with respect to time:

Integrate[%,t]

Evaluating the definite integral from time = 0 to g:

(%/.t->g)-(%/.t->0)

The “Log” here is the natural log.

Setting this to one (i.e., one fixation event) and solving for g gives the number of generations that must pass, on average,  between the appearance of two successful beneficial mutations:

g/.Simplify[Solve[%==1,g]]

The rate of fixation is the inverse of this quantity.  Also setting the initial frequency to 1/(c*N) in an organism with ploidy level “c” gives:

Simplify[1/%/.p0->1/(c*N)]

Assuming that the population size is large (N large), we drop σ2 relative to c N σ2 in the above to get:

((σ1)/(Log[c N ((E^((σ1)/(2 c N v σ2)))-1)((σ2)/(σ1+σ2))]));

This gives the rate at which second beneficial alleles are expected to fix, which when multiplied by their beneficial effect σ2 gives the rate of increase in fitness:

RateFitnessIncrease=((σ1 σ2)/(Log[c N ((E^((σ1)/(2 c N v σ2)))-1)((σ2)/(σ1+σ2))]));

Thus (A2) contains two typos:  The “c” is missing in the denominator, and it should be σ2 in the denominator of the exponent term.

Confirming that equations (3) are unaffected

Assuming that the effects of the mutations are the same (σ1= σ2), the effect of ploidy on rate of fitness increase is:

hap=RateFitnessIncrease/.c->1/.σ2->σ/.σ1->σ

dip=RateFitnessIncrease/.c->2/.σ2->h σ/.σ1->h σ

tetra=RateFitnessIncrease/.c->4/.σ2->h1 σ/.σ1->h1 σ

For (3a):

dip/hap

We take the limit in two cases.

When N v is assumed very large, we obtain a limit of (h^2), using the following approximations (recall that -1+(e^x) ~ x for x small):

Log[(1/2) (-1+(e^(1/(2 N v)))) N] ~  Log[(1/2) ((1/(2 N v))) N] = Log[((1/(4 N v))) N]
Log[(-1+(e^(1/(4 N v)))) N] ~  Log[((1/(4 N v))) N] , which is the same as above.

When N v is assumed very small, we obtain a limit of 2(h^2), using the following approximations:

Log[(1/2) (-1+(e^(1/(2 N v)))) N] ~  Log[(1/2) ((e^(1/(2 N v)))) N] = Log[(1/2) ((e^(1/(2 N v)))) N] = Log[(1/2) N]+(1/(2 N v)) ~ (1/(2 N v))
Log[(-1+(e^(1/(4 N v)))) N] ~  Log[((e^(1/(4 N v)))) N] =  Log[N]+(1/(4 N v)) ~ (1/(4 N v))

Similarly, for (3b):

tetra/dip

We take the limit in two cases.

When N v is assumed very large, we obtain a limit of ((h1^2)/(h^2)), using the following approximations (recall that -1+(e^x) ~ x for x small):

Log[(-1+(e^(1/(4 N v)))) N] ~  Log[((1/(4 N v))) N] 
Log[2 (-1+(e^(1/(8 N v)))) N] ~  Log[2((1/(8 N v))) N] , which is the same as above.

When N v is assumed very small, we obtain a limit of 2((h1^2)/(h^2)), using the following approximations:

Log[(-1+(e^(1/(4 N v)))) N] ~  Log[((e^(1/(4 N v)))) N] =  Log[N]+(1/(4 N v)) ~ (1/(4 N v))
Log[2 (-1+(e^(1/(8 N v)))) N] ~  Log[2 ((e^(1/(8 N v)))) N] = Log[2((e^(1/(8 N v)))) N] = Log[2 N]+(1/(8 N v)) ~ (1/(8 N v))