Over at
http://creationwiki.org/Haldane, Walter Remine has a page describing what he calls "Haldane's Dilemma". I recommend reading it; it's not some stupid quote-mine job, and Remine has thought hard about it. For one thing, he actually links to the original paper he's based this on (read that too!). And you can read his paper on cost theory at
http://www.creationontheweb.com/images/journal_of_creation/vol19/costtheory.pdf to see that work has gone into this.
The essence of the idea is that there is a limit on how fast beneficial mutations can be substituted into the genome by natural selection, and that limit is too slow to add meaningful change to the human genome since the split from the chimp/human common ancestor.
A crucial part of this is that substituting multiple mutations at once does not speed this up; if one mutation takes n generations, 2 mutations takes 2n generations. This conclusion is far as I can tell based purely on Haldane's work, which again as far as I can tell comes down to this paragraph:
"Can this slowness be avoided by selecting several genes at a time? I doubt it, for the following reason. Consider clonally reproducing bacteria, in which a number of disadvantageous genes are present, kept in being by mutation, each with frequencies of the order of 10^-4. They become slightly advantageous through a change of environment or residual genotype. Among 10^12 bacteria there might be one which possessed three such mutants. But since the cost of selection is proportional to the negative logarithm of the initial frequency the mean cost of selecting its descendants would be the same as that of selection for the three mutants in series, though the process might be quicker. The same argument applies to mutants linked by an inversion. Once several favorable mutants are so linked the inversion may be quickly selected. But the rarity of inversions containing several rare and favorable mutants will leave the cost unaltered."
Many of you will have already spotted the key words here - "clonally reproducing". That means that if you don't have the beneficial allele in your DNA already, the only way your descendants will get it is by mutation. So although each mutation may be present at 10^-4, there is no way to combine them unless they randomly already are combined. The mutations must occur in a single line of descent.
But humans don't reproduce that way (in reality, neither do bacteria). Humans reproduce sexually, which allows beneficial mutations to spread individually and combine later. So how does this change Remine's numbers?
Essentially Remine's formula (and Haldane's) is that the minimum number of generations required for substitution is the log of the population size (in his paper he uses a population of 1,000,000) divided by the selection intensity (which can be roughly considered as the log of the maximum growth rate, and which Haldane estimates at 0.1 for typical species).
So, log (1,000,000) / 0.1 = 13.8 / 0.1 = 138 generations per mutation; Haldane considered that 300 generations would be a more typical figure. This represents a beneficial mutation starting out in one individual, and growing at the expense of the old-type genome, until it completely replaces it.
The beneficial mutation to be substituted will be found on one of the chromosomes in the genome. In a sexually reproducing species, you would expect the substitution to leave the variation in the other chromosomes untouched, as they would be thoroughly mixed by reproduction. Variation in the substituting chromosome would depend on the rate at which chromosomes recombine. If no other beneficial mutations were around, then the entire 138 generations would indeed be required for a single substitution.
What if there are other beneficial mutations, however? They would have considerable room to increase their own frequencies, just as the target mutation does. A mutation that grows exponentially from 1 to 1M will spend half its growth time at a frequency of less than 1 in 1000 - basically invisible to other mutations. So 1,000 other mutations might easily do half their substitution on other chromosomes, stay around unchanged as the second half of the substitution took place, and then fight it out to see who went next. Two substitutions in only 1.5 times the minimum for one.
Let's formalize this intuition by considering a population of constant size K, with a maximum growth rate for beneficial mutations of r (where selection intensity = log r). It would take n generations to substitute, where K = r^n (r is the nth root of K). So in generation i, there will be r^i individuals with a copy of the gene (completely removing all the old allele is roughly equivalent in cost to substituting into a population of 2K, which we'll ignore for now). And this will be the result of (r-1)r^i deaths of old-type individuals - and the extra births of the same number of new-type individuals.
So we expect that in each generation, (K-r^i) individuals don't have the new gene, but aren't killed off to substitute it, and are available as a population for the substitution of other genes. So our new gene is using up r^i/K of the species' ability to substitute new genes, in that generation.
Over the course of the entire substitution, it will use up 1/K(1 + r + r^2 + ... + r^n). Using the old compound interest formula, the total cost usage is then 1/K(r^(n+1) - 1)/(r - 1). Remembering that r^n = K, and assuming K >> 1, that simplifies to r/(r - 1), which for r = 1.1 means one substitution every 11 generations.
The limit described by Haldane is the latency, not the throughput. It will still take 138+ generations to subsitute a given mutation into a population of 1M. But others could start or finish during that time, using the leftover capacity described above.
Note that the throughput result, unlike the latency, is no longer dependent on the population size. A bigger population takes longer to subsitute individual mutations, but can be working on more things simultaneously (and actually has more capacity to generate benefical mutations). The only thing that matters is how much excess reproductive capacity the species has.
Sidenote: Remine spends most of a chapter of "The Biotic Message" wondering how evolution can explain the origin of sexual reproduction. Above we see that throughput of beneficial mutations is one every logK/logr generations for asexual species, and roughly one every 1/logr generations for sexual species. That is, sexual reproducers can adapt to new situations about logK times faster. This is *more* pronounced for large populations such as bacteria - and no wonder they still use sex, even if they don't tie it so directly to reproduction. Any multicellular species that failed to allow sex somewhere in its reproductive cycle would have been outcompeted by its rivals in a geological heartbeat.
How does this apply to humans? Remine claims that over the last 10M years, humans could have only substituted 500,000 / 300 = 1,667 mutations. The above shows that we are in fact capable of up to 500,000 / 11 = 45,454! And it's worth noting that the human race is currently doubling in size about every 50 years, even with very low birth rates in the developed world, and all the wars, famine, disease etc that we see. That would give a figure for r of 1.32 (assuming generations of 20 years), or one substitution every 4.125 generations. And that rate requires 2.64 children per family surviving to adulthood; clearly plausible.
The latest results from the human genome project (see
http://www.ensembl.org/Homo_sapiens/index.html) show that humans have 21,541 genes. If we split from chimps 5M years ago - 250,000 generations - we've had time to substitute 60K mutations at 4.125 generations each. That is, almost three important mutations per gene; and that doesn't include neutral differences like multiple ways of coding for the same amino acid, or amino acid changes that don't affect protein function, all of which can piggyback on beneficial mutations. And that's just our side; the chimps have had time to do the same thing on theirs.
Calculations like the above are fine things, but experiment trumps theory. Fortunately, Remine notes that these limits also apply to computer simulations - so I don't need to keep a few hundred generations of cows at home to test my results. But this post is already way too long, so I'll leave that for another post.
