Population Genetics Made Simple

The characteristics of organisms are determined by their genetic material (DNA), and random mutations (changes) in the DNA can result in slight changes in organisms. As these accumulate, there can be changes in organisms, resulting in evolution. Population genetics attempts to analyze this process mathematically.

About 90 percent of DNA is thought to be non-functional, and mutations there generally have no effect. The remaining 10 percent is functional, and has an influence on the properties of an organism, as it is used to direct the synthesis of proteins that guide the metabolism of the organism. Mutations to this 10 percent can be neutral, beneficial, or harmful. Probably less than half of the mutations to this 10 percent of DNA are neutral. Of the remainder, 999/1000 are harmful or fatal and the remainder may be beneficial. (Remine, The Biotic Message, page 221.) This model is actually not realistic, because it does not take into account the interactions between various mutations. Nor does it distinguish major mutations, which change the shape of proteins, from minor mutations, which do not. Furthermore, it does not consider that the beneficial mutations observed are generally only of a restricted kind that cannot explain evolution. However, we consider the model in some detail anyway, because it is so widely used. In addition, population genetics can help to explain the rapid adaptation of organisms to their environment by changes in frequency of existing genetic material (alleles) even without mutations.

Harmful mutations result in organisms less likely to survive, and so these mutations tend to be eliminated from the population (group of organisms in a species). Beneficial mutations also tend to be eliminated by chance, but less often, and tend to be preserved. As these accumulate, the species can gradually adapt to its environment. Neutral mutations are generally eliminated, curiously, but sometimes can spread to the whole population. We then say that the mutation has fixed in the population. The rate of evolution is the rate at which mutations fix in the population. These can be either beneficial or neutral mutations.

If the offspring have on the average one harmful mutation each, then the population will degenerate; this is called "error catastrophe." This puts a bound on how many non-neutral mutations can occur per generation. It cannot be much more than about one per generation, and in fact, it must be significantly less, since most non-neutral mutations are harmful.

Rates of evolution are computed using the likelihood that various mutations will be passed on to offspring. If for each individual having a mutation, two of its offspring also have it, then the mutation will rapidly spread through the population. If less than one offspring has it, the mutation will tend to be eliminated. Often one speaks in terms of the difference (ratio) between this number of offspring for individuals with or without the mutation. If this "selective advantage" is .01, then that means that the chance this mutation will be passed on to an offspring is about 1.01 times higher than the chance that DNA without this mutation will be passed on to an offspring. The value of .01 is considered to be high. Note that in this case, after k generations, the frequency of this mutation in the population should increase by a ratio of (1.01)^k. Different mutations in the same individual have a multiplier effect. So if an individual has two mutations with selective advantages of s and t, respectively, then this individual will have about (1+s)*(1+t) offspring per parent. Thus their combined effect is a selective advantage of (1+s)*(1+t) - 1. For small s and t, that is, much less than one, this expression is about equal to s + t, so we can often think of selective advantages as additive.

There seem to be many different uses of terminology in population genetics. What I call selective advantage, Spetner refers to as the selective value. Others call this the selective coefficient. ReMine uses the term selective value for something else.

For the sake of illustration, assume that the selective advantage of every beneficial mutation is 1, meaning that it doubles in frequency each generation. Suppose in some population, there are 1000 beneficial mutations. Each one will double its frequency each generation, so in about 10 generations each will multiply its frequency by about 1000 and in 30 generations, each will multiply its frequency by about a billion. Thus all these mutations can rapidly spread throughout the population, and the rate of evolution will be larger for a larger population, since there will be more beneficial mutations. Actually, after 30 generations, each individual will have only about 500 mutations, since the individuals originally in the population continue to reproduce themselves, too.

Those who are not interested in a detailed discussion of the technicalities of population genetics and Haldane's dilemma may want to skip down to the section More about Rates of Evolution now. The reason that we discuss Haldane's dilemma is that it is sometimes used by creationists to argue against evolution. However, we believe that only valid arguments should be used to support creationism, and so Haldane's dilemma is not appropriate. Furthermore, it is sometimes convenient from a creationist viewpoint to argue for rapid evolution, which Haldane's dilemma prohibits.

Problems with the Standard Analysis

Also of interest is the fact that that as the population evolves, the average fitness of the population will increase, so each individual will not have such an advantage over others as it might have had in the beginning. As competition increases, the effect of beneficial mutations will be less pronounced. We can assume that the number of offspring of all individuals will be reduced by the same factor, so that the average individual will have about one offspring per parent and a selective advantage of zero. All individuals will be penalized to the same extent by this, so it will not affect the changes in frequency of mutations with time. This is intended to keep the size of the population roughly constant, also.

Suppose there are 1000 beneficial mutations spreading through a population, doubling each generation, that is, having a selective advantage of 1. Suppose each has a frequency of 1/2, that is, each mutation occurs in half of the individuals in the species. Then the average number of mutations per individual is 500. Most individuals will have a few more or a few less than 500 mutations, but not many will have other numbers. An astronomically small number of individuals will have more than 750 mutations, for example. The curious fact is, however, that the astronomically small number of individuals having way more than 500 mutations contribute a considerable fraction to the speed of evolution, since an individual with k beneficial mutations has proportional to 2^k copies of its DNA passed on to the next generation, on the average. So if we restrict these individuals with many beneficial mutations to have at most two offspring, then the speed of evolution will be drastically reduced.

It is useful to have some background in probability theory in order to discuss population genetics. Suppose in a population there are n mutations altogether spreading through the population and on the average m mutations per individual. These mutations will be distributed more or less randomly, because of the effect of recombination (crossing over) during sexual reproduction. Later we will consider the case of asexual reproduction. The number of mutations per individual for organisms with sexual reproduction will be distributed as a binomial, or, normal, distribution, due to the randomizing effect of crossing over. The standard deviation of the number of mutations per individual will be at most 1/2 sqrt(m). This means that only about .31 of the population will have a number of mutations differing from m by more than 1/2 sqrt(m). Only about .046 of the population will have a number of mutations differing from m by more than sqrt(m). The normal distribution with mean zero and standard deviation one is proportional to e ^{- x * x / 2} , where e is about 2.718.

For the case of 1000 mutations with a selective advantage of 1 each, assume that each individual has on the average 500 of them. The square root of 500 is about 23, so the standard deviation is about 11.5. This means that .31 of the population will have more than 511.5 mutations or less than 488.5 mutations, so about .16 of the population will have more than 511.5 mutations. About .023 of the population will have more than 523 mutations. Recall our assumption that competition means the average individual (with 500 beneficial mutations) will have on the average one child per parent surviving (and thus a selective advantage of zero). The individuals with 511.5 mutations will have 2^11.5 times this many children, or, over 2,000 children per parent! The individuals with 523 mutations will have over 8 million children per parent! Clearly these numbers are ridiculous and something is seriously wrong. These are not just extreme cases, because over two percent of the population will have 523 mutations or more. However, it turns out that the largest contribution to the speed of evolution comes from individuals having 667 mutations! Though there are very few of them, each has an astronomical number of children to make up for it.

For an average of 500 mutations per individual, since the standard deviation is about 11.5, an individual with 667 mutations is about 14.5 standard deviations away from the average. The number of such individuals relative to the number of individuals with 500 mutations is e ^{- 14.5 * 14.5 / 2} , or, e ^{- 105} , which is about 2.5 * 10 ^{- 46}. Even though less than one in 10 ⁴⁵ individuals has this many mutations for every individual with 500 mutations, each such individual has 2 ¹⁶⁷ or about 10 ⁵⁰ times as many offspring per parent to make up for it. In any reasonable sized population, it is doubtful that even one such prolific individual would be found. On the other hand, if each of these mutations had a selective advantage of .01, the greatest contribution to evolution would come from individuals with 502 or 503 mutations each.

One obvious problem with 1000 mutations with selective advantages of 1 is that the size of the population is increasing, since one individual has 8 million offspring, and nearly half of the population has many more than two offspring per parent. This is so even though the average individual has one offspring per parent (thus two altogether). The next generation would have (3/2)¹⁰⁰⁰ or about 10 ¹⁷⁶ times as many individuals. However, we are assuming that competition reduces the absolute number of offspring so that the size of the population is maintained. This has the curious effect that an individual with an average number of beneficial mutations may at times have a very small number of offspring (much less than one) and a negative selective advantage. In our case, an individual with 500 beneficial mutations would have 10¹⁵⁰ offspring per parent in the absence of competition, but with competition it would be reduced to 10^-26 offspring per parent.

We now look at this in another way. Suppose that there are 1000 beneficial mutations spreading through a population. Let's give each mutation a selective advantage of 1/1000. This means that even an individual with all 1000 beneficial mutations will have a selective advantage of (1 + 1/1000)¹⁰⁰⁰ - 1, which is 2.718 - 1, not unreasonable. (It's useful to know that for large n, (1 + 1/n)ⁿ is about 2.718 and (1 - 1/n)ⁿ is about 1/2.718.) Then each mutation will increase its frequency by a factor of 2.718 in 1000 generations, and will increase its frequency by a factor of about a million in 14,000 generations. Thus if the population is about a million, we get 1000 beneficial mutations fixing in 14,000 generations, or, about one per 14 generations. Now, if we just had one beneficial mutation spreading through the population with a selective advantage of 2.718 - 1, then it would spread through the population in 14 generations, giving the same rate of evolution. So the end result is that we do not get any gain by having mutations spreading through the population in parallel, and we cannot obtain rates of evolution faster than would be obtained by single mutations with selective advantages of about 1 (assuming that no individual has a selective advantage of more than about 1.) This means that large populations cannot lead to significantly increased rates of evolution, unless some individuals have tremendously high selective advantages. The only advantage of a large population is that if there are say 100 mutations with selective advantages of .01, they can spread through the population in parallel, leading to a rate of evolution similar to that of one mutation with a selective advantage of about 1.

Now, if we assume 20 beneficial mutations spreading through the population, each with a selective advantage of 1, and on the average 10 mutations per individual, then we obtain that the largest contribution to the speed of evolution comes from those individuals having 13 beneficial mutations. Each such individual has 8 times as many children per parent as the average. For 12 beneficial mutations, with an average of 6 per individual, the largest contribution to evolution comes from individuals having 4 times as many surviving children as the average individual. Even this seems implausible.

These results do not depend on the large selective advantages of 1. If we assume 10,000,000 mutations spreading through a population, each with a selective advantage of .01, and an average of 5,000,000 mutations per individual, then we still obtain that over 2 percent of the population has 8 million times as many children as average. For 200,000 mutations with a selective advantage of .01, and an average of 100,000 per individual, half of the speed of evolution comes from individuals with 8 or more times as many children as average. For 120,000 mutations and similar assumptions, half of the speed comes from individuals with 4 or more times as many (surviving) children as average. However, it is true that smaller selective advantages per mutation make the number of mutations to get a given selective advantage larger and tend to reduce the selective advantage of typical individuals with respect to the average.

Rates of Evolution under Reasonable Assumptions

We can get some increase in the rate of evolution by "pipelining" mutations. Suppose there is a population of 1,000,000. Suppose there are 50 beneficial mutations per generation and therefore 50,000 harmful ones per generation. We assume that these beneficial mutations have selective advantages of .01 at most, which seems typical. Then all but one of them will be eliminated from the population, on the average. This remaining mutation will double its frequency in about 70 generations (since 1.01⁷⁰ is about 2) and will spread through the population in about 1400 generations (since 2 ²⁰ is about a million). At this time each individual will have an average of half of a mutation, and it will take 2800 generations total to complete the spread. During this time, other mutations will be entering the population and also spreading. Thus we will obtain one mutation fixing in the population per generation, in the long run. Each individual will have on the average one mutation that has spread to .99 of the population, one that has spread to (.99)² of the population, et cetera, for an average of about 100 mutations altogether. This will give the average individual a selective advantage of about 1, which is not unreasonable. The standard deviation in the number of mutations will be about 5, so most individuals will have only a small selective advantage over the average individual. This shows that we can reach a speed of about one mutation per generation under reasonable assumptions. The actual rate is about half a mutation per generation, because we also need to consider the original individuals in the population that continue to reproduce, slowing down the process.

We now perform the analysis in another way, which permits a faster rate of evolution than one mutation every two generations. The preceding analysis assumes that the selective advantage of an individual is not larger than about 1. Instead, we can assume that the selective advantage of an individual relative to the average individual is not larger than about 1. That is, we can assume that no individual has more than about twice as many offpsring as the typical individual. Since the typical individual probably has one offspring per parent, this is reasonable. Even with this reasonable assumption, we can get faster rates of evolution. Suppose, for example, that 100 mutations fix per generation and have a selective advantage of .01. These mutations increase their frequency in the popululation by 1.01 each generation. The number of mutations in a typical individual at equilibrium that have not yet fixed will be 100(.99 + .99² + .99³ + ... ) which is (summing the geometric series) 100*100 or 10,000. The standard deviation is 50, so the selective advantage of a typical individual with 50 extra beneficial mutations is only .5 over an average individual, consistent with our assumption. A typical individual with a selective advantage of .5 would have only 1.5 times as many children per parent as an average individual. But we obtain a rate of evolution of 100 mutations per generation. The population size to permit this must have 5000 beneficial mutations per generation (since only 1/50 of them fix). With 1000 harmful mutations to a beneficial one, this is 5 million harmful mutations per generation. The population size would probably have to be substantially larger than 5 million to prevent error catastrophe.

Suppose now that 1000 mutations fix per generation and have a selective advantage of .002. (We cannot use .01 again for the selective advantage, or else the selective advantages of typical individuals would be too high.) The number of mutations in a typical individual at equilibrium that have not yet fixed will be 1000(1/(1+s) + 1/(1+s)² + 1/(1+s)³+ ...) for s = .002, which is (summing the geometric series) 1000*500 or 500,000. The standard deviation is about 350, so the selective advantage of a typical individual with 350 extra beneficial mutations is only 350*.002 or .7 over an average individual. This is consistent with our hypothesis that a typical individual should not have too great an advantage over an average individual. This time we obtain a rate of evolution of 1000 mutations per generation (though the fitness advantage of each mutation is less). The population size to permit this must have 250,000 beneficial mutations per generation (since only 2*s or 1/250 of the beneficial mutations fix). With 1000 harmful mutations to a beneficial one, this is 250,000,000 harmful mutations per generation. The population size would probably have to be substantially larger than 250 million to prevent error catastrophe.

Haldane's Dilemma and Problems with it

Haldane's Dilemma seems to contradict our previous result about obtaining 100 (or even 1000) mutations fixing per generation under reasonable assumptions. We are not sure where the disagreement is, but perhaps our assumptions are different. In any event, there does not seem to be anything preventing high rates of evolution, even assuming the maximum relative selective advantage is about 1.

We now present another seeming paradox, in which we obtain a rate of evolution apparently faster than Haldane's Dilemma allows. This material is fairly technical, and may not be of interest to the average reader. Let us assume the maximum selective advantage of any individual in a population is 1. The "cost" of substitution is paid by individuals having more than one descendent per parent. We placed an upper bound of 1 on the selective advantage of any individual, and used this as an upper bound on how these costs can be paid. However, many individuals will have a selective advantage of less than 1, meaning that the rate at which costs can be paid will be much less than one per individual. We recall that the sum of the selective advantages of the mutations in an average individual in the population cannot be larger than 1. We also assume that mutations have selective advantages of at most .01. Let m be the average number of beneficial mutations in an individual in the population, and suppose for illustration that m is about 10,000. The standard deviation of the number of beneficial mutations would be at most on the order of one-half the square root of m, or, 50, since we have something approximating a binomial or a normal distribution. Therefore, very few individuals would have more than m+100 mutations. A typical individual would have mutations with a combined selective advantage of at most 1. This means that the average mutation would have a selective advantage of at most 1/10,000. Thus 50 mutations would have a combined selective advantage of at most about .005. The selective advantage such individuals would have over an average individual would be very small (about .005). As a result, it would appear that very few individuals would be able to pay the maximum cost corresponding to a selective advantage of 1. The astronomically small number of individuals with many beneficial mutations would not be able to pay the cost, either, because of the upper bound of 1 on selective advantage. So it would appear that we could get only about .005 substitutions per individual in this generation, by Haldane's result.

However, of the half of the population with at least 10,000 beneficial mutations, a typical individual would have on the average about 50 extra beneficial ones. Individuals in this half of the population will produce about 1/200 extra offspring every generation, since they have a selective advantage of about .005 over an average individual. Since we are dealing with half of the population, we obtain 1/400 of the population produced from such individuals in one generation. A typical new offspring will have an extra 50 beneficial mutations, and will replace an individual having typically 50 mutations less than average, a net gain of 100 beneficial mutations per offspring. Thus we obtain a number of substitutions equal to 100 * 1/400 or 1/4 of the population in one generation, even though only 1/200 of the population is being generated each generation. This contradicts the idea that each substitution requires one birth in the population.

Even if there are 100 mutations with combined selective advantages of .01 in an average individual, we obtain a standard deviation of about 5 at most, giving a selective advantage of only .05 over an average individual. Reasoning as above, we generate 1/40 of the population from such individuals, and each such birth replaces an individual having about 10 less beneficial mutations, resulting in about 1/4 of the population receiving an additional beneficial mutation in one generation. Thus one can get more than one substitution per offspring, a seeming contradiction to Haldane's dilemma.

Recall also that we obtained a maximum rate of evolution of one mutation per two generations under reasonable assumptions in our above analysis using pipelining. This is much higher than one mutation every 30 generations, which is the maximum possible according to Haldane's analysis under the assumption that the maximum selective advantage of any individual is at most 1. It could be that our assumtions are different, or maybe there is a flaw in Haldane's analysis. We are not considering the extra cost of recessive mutations, but this should not make much difference, according to ReMine.

There is a way, however, that we can get a bound very similar to Haldane's, and that is if we assume that the most fit individual in a population has only a limited advantage over an average individual. Suppose we assume that the most fit individual (the one with all the beneficial mutations currently spreading through the population) has a selective advantage of at most 1 over an average individual. Then we can get at most about one mutation every 20 generations. For example, if there is just one mutation spreading with a selective advantage of 1, then in 20 generations it can increase its frequency by a factor of about a million, and thus spread through a reasonable sized population. This gives one mutation per 20 generations. If we assume there are 10 mutations with selective advantages of 1/10 each, then an individual with all of them has a selective advantage of about 1.718. In 10 generations, these mutations can increase their frequency by a factor of about 2.718, and in 14 such increases, they can increase their frequency by about a million. In this way we obtain 10 mutations in 140 generations, or about 14 mutations per generation. The result is about the same for say 100 mutations with a selective advantage of .01 each. Of course, 1.718 is larger than our bound of 1, explaining the increased speed. If we assume, as Haldane did, that the maximum selective advantage of an individual with all beneficial mutations is about 1/10, then we obtain a maximum of one mutation every 200 generations, not far from Haldane's bound.

A problem with this analysis is that different mutations do not combine in a multiplicative way. If we assume that a tall monkey has a selective advantage of 1, a handsome monkey has a selective advantage of 1, a muscular monkey has a selective advantage of 1, and a smart monkey has a selective advantage of 1, then it is not correct to assume that a tall, handsome, muscular, smart monkey has a selective advantage of 15. Probably it only means that all of his children will survive, and so if half of the average monkey's children die, the exceptional monkey will still only have a selective advantage of 1. If we have a population in which most of the evolution is occurring due to individuals with a selective advantage of 1 or less, then we still get more rapid rates of evolution than Haldane allows if we bound the selective advantage in this way to 1.

More about Rates of Evolution

On the other hand, there are a number of other applications of population genetics that cause severe problems for the theory of evolution. We now consider these, beginning with a discussion of small populations. We then discuss error catastrophe and the fact that the high rate of mutation for humans should lead to rapid degeneration of the human genetic material. We also show that the human race must be young, since such deterioration is not yet observed. Finally, we discuss the ape-human gap, and show that if the human race is not degenerating, then this gap would take at least 100 million years to cross, and more probably a billion years, just counting numbers of mutations needed and not considering the problems of creating new structures for speech and thought.

We first discuss small populations, taking some material from ReMine's book, The Biotic Message. We note that the maximum possible rate of non-neutral mutation is less than one per generation, due to error catastrophe. This rate leads to at most about 1/1000 beneficial mutations per individual per generation. These mutations will probably have a selective advantage of at most .01, implying that at most 1/50 of them fix in the population. Thus we can get at most 1/50,000 mutations fixing in the population per generation per individual. Thus in a population of 100,000 individuals, we can expect at most two beneficial mutations to fix in the population per generation. In a population of a million, we might expect 20 beneficial mutations to fix per generation. However, Haldane's analysis implies that this rate does not continue to increase as the population becomes larger and larger. In fact, a more reasonable estimate is about .015 lethal mutations per generation, leading to .000015 beneficial mutations per individual per generation at most. At most 1/50 of these would fix, leading to at most about .0000003 fixing per individual per generation. This would be 0.3 mutations per generation fixing in a population of a million. We see right away that we cannot expect rapid evolution in small populations, as assumed in the theory of punctuated equilibrium. Haldane's reasoning (if it were true) would imply that even this rate is too high, and moreover, we cannot expect bursts of evolution, even in large populations.

We now discuss the subject of error catastrophe in humans. In humans, it is estimated that there are about 30 mutations per individual per generation, thus three in the functional part of the DNA. This implies that on the average there are about 3/2000 beneficial mutations per individual per generation and about 1.5 harmful mutations. ReMine notes that this rate of mutation should lead to error catastrophe in the human race, which is a puzzle for the theory of evolution.

To be precise about the human situation, there are about 3.5 * 10 ⁹ base pairs in the human genome, and each person has two copies of these, one from the father and one from the mother. The rate of mutation in humans is believed to be about 1 * 10 ^-8 per base pair per generation, or, 35 mutations per generation per individual. We can assume that 9/10 of these occur in the non-functional part of the DNA, and that of the remainder, possibly half are neutral, leading to about 1.75 harmful or fatal mutations per generation. Now, the typical individual will have mutations from both parents, and therefore will have about 3.5 harmful mutations per generation. Let's divide a generation into n small time intervals; then the chance of a harmful mutation during 1/n of a generation is 3.5/n, and the chance to avoid a harmful mutation during this small time interval is 1 - 3.5/n. Thus the chance to avoid any harmful mutations during the entire generation is (1 - 3.5/n)ⁿ, which is about (1/2.718)^3.5, or, 0.0302084. This means one child free of defects per 33.1 children. In order to have two children free of defects on the average, to avoid error catastrophe, the typical female would need 66 conceptions! ReMine obtains a figure of 16.3 conceptions assuming only 3 percent of the DNA is functional. Both are unrealistically large. The consequence is that humans are experiencing error catastrophe, that is, instead of evolving, we are degenerating. This implies that at one time humans were much more advanced than we now are. This is consistent with a creationist view that the human race was created with so much vitality that it could endure the degeration resulting from harmful mutations.

It is important in this respect to understand what a harmful mutation is. It must be a mutation that hinders the organism in a way that the organism is not able to compensate for. Since populations are generally stable over long time periods, this means that an organism with a harmful mutation will typically have less than one surviving offspring.

Equilibrium is the state at which the frequencies of occurrences of various mutations are fixed over time. After sufficient time, any population with constant rates of mutation should reach equilibrium. When the population reaches equilibrium, harmful mutations will be entering and leaving the population at the same rate. This means that a mutation that is only slightly harmful will spread to much of the population, but a mutation that is very harmful will spread to a small percent of the population. But both mutations will cause a number of deaths per generation equal to their frequency of occurrence in the population. So the chance that a fertilized egg will survive is less than or equal to the chance that it has no new harmful mutations, when the population is at equilibrium. Thus if there is a small chance that an offspring will be free of new defects, then there is a small chance that it will survive, when the population is at equilibrium. So at equilibrium we should expect that only about one in 33 fertilized human eggs will survive, based on the above calculations.

It is also important to understand that only essential genes will persist in a population as functioning genes for long periods of time. Inessential genes will suffer mutations that destroy their function, and since the organism is able to compensate, such mutations will accumulate over time until the gene is non functional in the whole population. This means that evolution eventually eliminates all redundancy, and only the genes that an organism really needs to survive will be maintained. So all species will eventually be living on the edge of extinction, in a sense.

One way to understand how the human race can endure a high rate of mutation is to realize that many mutations are recessive. They are only expressed (fully) when both copies of the gene have the mutation. Mutations to enzymes tend to be recessive, because an organism with one gene generating functioning enzymes can still survive, though possibly with some degradation. Mutations to structures of an organism tend to be dominant. Now, recessive mutations tend to accumulate until there are enough of them to be expressed. Assuming 10^-8 point mutations per base pair per generation, a recessive mutation would accumulate until it reaches equilibrium, when the numbers of mutations entering and leaving the population are equal. Assuming the mutation were fatal, this would happen when about 1/10,000 of the individuals had a mutation at this base pair, since the probability of expression would be the square of 1/10,000, or 1/100,000,000, the same as the probability of the mutation occurring. It would take on the order of 10,000 generations to reach equilibrium (actually, somewhat longer). Now, there may be about 3*10⁸ functional base pairs in the human genome. It would be reasonable to assume that half of these were for enzymes, which might mean about 10⁸ recessive fatal mutations. When each reached equilibrium, a typical individual would have about 10,000 recessive mutations. Each child would then on the average have one of them expressed. This would be essentially the same as if there were one fatal mutation per generation at the start; the cost to the population is the same, but recessive mutations take longer to reach the full cost.

The fact that we have not reached the point where 66 conceptions per couple are needed means that the cost of the harmful mutations has not yet been realized. This seems to imply that many of these mutations are recessive, and also that the human race has not been around for very long (under 10,000 generations). However, these recessive harmful mutations, even when not fully expressed, may render the human organism more inefficient in various ways, and may make us increasingly susceptible to disease and degeneration of various kinds. It is possible that a better life style would reduce the rate of harmful mutations to an acceptable amount.

Actually, it is more realistic to look at the picture in terms of genes rather than base pairs. If both copies of a gene in an individual have a harmful mutation, then this will probably be expressed in the individual, even if these mutations are different. There are estimated to be about 100,000 genes in human beings, and typical rates of mutation are about one in 25,000 as observed in the population. The square root of this is about one in 160, meaning that equilibrium should be reached for recessive mutations in about 160 generations. The fact that we have not yet reached equilibrium suggests that the human race is not much more than about 200 generations (6000 years) old.

Another possibility is that a very small portion of the genome comprises the functional DNA, perhaps under one percent. ReMine mentions that some biologists have come to this conclusion due to reasoning such as the above. This would seem surprising, but there is no way we can absolutely rule it out at present. This seems to contradict Maynard Smith (cited in ReMine, The Biotic Message, page 250), who estimates that 9 to 27 percent of the human genome codes for protein. It will be interesting to see what the human genome project reveals.

In general, one can compute the time it takes for a population to reach equilibrium in the following way, assuming that the population begins perfect (with no or very few harmful mutations). Suppose f is the frequency with which a mutation occurs per generation, and m is the fraction of the population that has the mutation. Suppose s is its selective disadvantage (which will be a negative number). Now, if the mutation is dominant, then it will reach equilibrium when the probability of expression equals f, so that -ms = f. Thus if its selective disadvantage is -1/1000, it will reach equilibrium when its frequency of appearance in the population is about 1000 times its frequency of occurrence. This will happen in slightly over 1000 generations, since the frequency of appearance in the population will initially increase by f per generation, but will slow down as m approaches -f/s. In general, it will take somewhat over -1/s generations to reach equilibrium. If the mutation is recessive, the probability of expression is m², and the fraction of the population that will die from this is -s*m². Equilibrium will occur when -s*m² = f. Thus m = sqrt(-f/s) at equilibrium. The frequency of occurrence will increase by f each generation and converge to sqrt(-f/s), which will take somewhat over sqrt(-f/s)/f generations, or, sqrt(-1/fs) generations.

A detailed calculation shows that if a mutation is recessive and fatal (when expressed), and occurs on the average once per generation, as assumed above, in sqrt(1/f) generations, its frequency in the population will be over .75, which implies that over half of all fertilized eggs must die due to this mutation. Even this close of an approach to equilibrium would imply too high a death rate for humans in 160 generations. It could be that the mutation rate has increased recently, or else many harmful mutations are only slightly harmful. These will take longer to reach equilibrium.

For example, harmful mutations with f = -.01 will take 10 times longer to reach equilibrium and harmful mutations with f = -.0001 will take 100 times longer to reach equilibrium. If most harmful mutations have f about -0.1, then they will take about 3 times as long to reach equilibrium. If we assume that the high mutation rate began about the time of Abraham, 4000 years ago, then there would be about 130 generations since then. Equilibrium would take 3 * 160 or 480 generations, so we would be about a fourth of the way to equilibrium, meaning that for recessive mutations only about 1/16 of the equilibrium rate would be expressed today. This might be tolerable. But in a few thousand years, the death rate would become unacceptably high.

We also note that mutations on small genes take longer to reach equilibrium. A gene having half the average size will probably have half the average rate of mutation. This means that if the gene is recessive, the frequency of mutated alleles of this gene at equilibrium will be about .707 of the average for all genes. It will also take about 1.414 times as long to reach equilibrium as an average gene. So in the early stages of approach to equilibrium, genetic disease from this gene will be about a fourth the frequency of an average gene. For a gene 1/4 average size, the frequency of genetic disease will be about 1/16 that of a gene of average size in these early stages, and mutations to this gene will take twice as long as mutations to an average size gene to reach equilibrium. The fact that these smaller genes take so much longer to reach equilibrium could also explain why the rate of genetic disease at present is relatively small.

Another possibility is that the human genome has many more than 100,000 genes and a lot of redundancy. In this case, equilibrium would take a lot longer to reach, which could explain why genetic defects are so rare today. However, at equilibrium, degeneration would occur even faster due to the larger number of genes.

Redundant genes take longer to reach equilibrium. Assuming a mutation rate per gene of 1/25,000 and assuming triple redundancy, at equilibrium each gene would have a frequency of about 1/30 in the population (the cube root of 1/25,000). This is assuming a mutation which is fatal when expressed. To reach equilibrium would require about 25,000/30 or about 900 generations. After 900 generations, the population would be at least .75 of the way to equilibrium (probably significantly more), and after 1800 generations, almost all the way. 900 generations is about 27000 years. If there are triple redundant genes and they have not reached equilibrium, then the age of the human race must be less than about 27,000 years and more than 160 generations (48,000 years) to account for those recessive genes that have reached equilibrium.

There is an interesting anomaly having to do with cystic fibrosis which is recessive and has a frequency of about one in 2500 births among Caucasians. This seems to imply that the frequency of the mutations in the population is about 1/50. This illness is caused by a number of mutations, including the loss of a single amino acid, but this is still a huge frequency, difficult to explain on the basis of frequency of occurrence of mutations. About 70 percent of the cases of CF are due to the deletion of a single amino acid in the gene. The frequency of this mutation might then be (7/10)*(1/50) or about 1/70. One explanation offered by biologists for the large frequency of these mutations is that this disease may have a heterozygote advantage. Another possibility from a creationist viewpoint is that there was once a severe population bottleneck and one of the humans at that time had this mutation. So it could be that one of the first Caucasians had this gene. Then we can compute how long it would take to come to its current frequency in the population, which is about one in 70. Starting out with a frequency of one in 8, how long would it take to reduce to one in 9? The chance of expression would be about 1/8^2 or 1/9^2, so we will take 1/(8*9) as a reasonable average. The number of generations required is then (1/8 - 1/9) divided by 1/(8*9), which is one. In general, it will take about one generation to reduce its frequency from 1/n to 1/(n+1). Thus about 70 generations is required to reach the present level among Caucasians. That is only 2100 years, which is too small.

We can explain this as follows: A couple that has a child with CF may then have another child to make up for it. Two parents with the recessive CF gene will on the average have 1/4 of their offspring with CF. Their offspring without CF will have on the average 2/3 of a CF gene per offspring. If they have an average of 2 children then on the average 1.5 will survive, and thus one CF gene will propogate to the next generation. If for each offspring with CF, they continue having children until there are two survivors, they will propogate 4/3 of a CF gene to the next generation instead of one. This means that the loss of CF genes is reduced from 1 to 2/3, so it takes 3/2 as long to obtain a given reduction. This would increase the time to about 105 generations, or 3150 years. This is still to small, but more reasonable.

Now, we can also guess that some couples will want to have an extra child just to make sure they have enough normal ones. So parents with a CF child may actually have more children than average. If they have an average of 3 children without CF, they are passing on 2 CF genes to the next generation, so the frequency of CF in the population will hardly decrease at all. If we assume half of the parents have 3 normal children and half have 2, then each such couple passes on an average of 5/3 CF genes, which will increase the time needed by a factor of three. This gives 210 generations, or, 6300 years, possibly less with a 20 year generation time. If we assume that the stress of having a CF child leads to divorce and remarriages, then the number of children could actually be even larger. So we see that it is possible to explain just about any age one wants to. But the creationist viewpoint does explain why CF should be so high in Caucasians and low in other races, if the father or mother of the race (at the time of a severe population bottleneck) happened to have a recessive CF gene among them.

The effect of harmful mutations seems to be misunderstood frequently. Spetner (Not by Chance, page 81) states that if ten percent of the offspring suffer a harmful mutation, then the genome of most of the population will be ruined. This is based on the assumption that harmful mutations have selective advantages of about -0.1 and therefore at equilibrium will spread to the whole population. However, just because the whole population has a harmful mutation does not mean that the genome is ruined; it only means that some number of conceptions do not lead to viable individuals, so there must be an excess of births to make up for it.

Suppose, for example, that one percent of the population per generation suffers a harmful mutation whose selective advantage is between 0 and -0.001. Mutations with a selective advantage of -0.001 result in 1000 individuals having on the average only 999 descendents, assuming that a typical individual has one descendent per parent. Such mutations will eventually increase their frequency in the population a thousandfold or more, so that at equilibrium each individual will have 10 such mutations per parent. Counting both parents, there will be 20 such mutations per individual. The chance of an individual being free of such mutations is then about one in 2.718 ¹⁰ , which is much less than one in a million. Now, if those individuals with harmful mutations have on the average less than one offspring per individual, this means that the one in a million individuals without harmful mutations have to have enough offspring to make up for the loss in population, which is unrealistic. In fact, it suffices for a defect-free individual to have about 101 offspring per parent to make up for the loss due to genetic defects, and so this average of 20 harmful mutations per individual is not necessarily a serious problem.

We now derive an absolute bound on rates of evolution. It is interesting that by considering rates of recombination (crossovers), one can get absolute bounds on mutations fixing per generation. This analysis applies both to beneficial and neutral mutations. There are some subtleties that I am ignoring, but I believe that the following analysis is valid. The human genome has about 3.5 * 10 ⁹ base pairs, and crossovers occur on the average about once every 10⁸ base pairs, for an average of 35 per generation. If a mutation spreads to the entire population, it will generally take a region of its chromosome with it, this region being larger the faster it spreads. In 1000 generations, this region would have about 10⁵ base pairs. This region would likely have no other mutations in it that had not yet fixed. Thus we could have probably at most 35000 mutations spreading in this way at one time, in 1000 generations, for 35 per generation. If the mutations spread more slowly, more of them can spread in parallel, but the bound of 35 per generation still holds. This bound does not depend on the selective advantages of the mutations or the rate of mutation. This bound is actually reduced by the following factors:

Smaller genomes would permit fewer mutations per generation

If the genes are clustered on chromosomes, the rate of evolution would be less.

There may not be enough beneficial mutations to attain this maximum rate of 35 per generation.

Mutations tend to compete against each other and only a small fraction of them may spread to the whole population. For example, a rapidly spreading mutation can eliminate many slowly spreading ones near it on the chromosome from the population.

This maximum rate is interesting in view of the ape-human gap which is about 2 percent of the genome (I've also read one percent). This would be about 70 million base pairs. Assuming both humans and apes diverged at the same rate, this would be 35 million base pairs for each. If we can have at most 35 mutations fixing per generation, this would mean at least a million generations, which would be at least 20 million years. (ReMine gives evolutionary sources on page 208 (The Biotic Message) which say the 20 year value is the right one to use.) If the bound is much less than 35, then it would take much longer than 20 million years.

The maximum rate of neutral mutations (in the non functional DNA) cannot be very high due to error catastrophe. In functional DNA, probably 2/3 of the mutations are harmful; mutations to the first two codon positions almost always change an amino acid, and mutations to the third position sometimes do. When an amino acid is changed, the mutation is almost always harmful. Kimura (cited in ReMine, The Biotic Message, page 246) estimates that a mutation which alters an amino acid is ten times more likely to be harmful than neutral or beneficial. In fact, one can derive something close to this figure from the fact that replacement sites in DNA evolve about 10 times slower than neutral sites (fibrinogen peptide) according to evolutionists. Assuming 1/10 of the DNA is functional, we can have at most about 7 neutral mutations per generation; this would be 7/10 of a mutation per generation in the functional DNA, and about half a harmful mutation per generation. With two parents, this would be one harmful mutation per generation, and bordering on error catastrophe. Thus most of these 35 mutations per generation would have to be beneficial mutations. In a million generations we could have at most 7 million neutral mutations, which is a fifth of a percent of the genome. To get a one percent change in the non-functional DNA would require probably 5 million generations and 100 million years.

This bound of 7 neutral mutations per generation applies to any organism having one-tenth of its DNA functional. In fact, since the mutation rate is somewhat affected by chance factors, it would seem highly improbable that this mutation rate should be so close to its maximum possible value, so a much smaller value seems more likely. In addition, this rate of mutation probably fluctuates to some extent, and if it were so near to error catastrophe, when the rate increased, the species would suffer. Therefore a much smaller rate, possibly 3 or 4 neutral mutations per generation, seems more reasonable. This would put the ape-man divergence at at least 200 million years ago and might have other difficult consequences for the theory of evolution.

Spetner (Not by Chance, page 81) says that species can tolerate at most 1/10 of a mutation per generation in the functional DNA. This would increase the time required by a factor of about 5, to a billion years, assuming the mutation rate happened to be at the maximum acceptable value. Remine (The Biotic Message, page 221) states that a typical rate of lethal mutations for a species is 1.5 percent per generation. Assuming at least half of the harmful mutations are lethal, this leads to a 3.0 percent harmful mutation rate per generation. This would increase the time required for the ape-man gap by a factor of about 30 over the 100 million year estimate, to 3 billion years. To change half of the human DNA would require 7.5 billion generations at this rate.

One might attempt to get around this problem by noting that some mutations affect more than one base pair of DNA. Suppose, for example, that on the average, a mutation affects five base pairs. Then the times for the ape-human divergence could be reduced by a factor of five, from a billion to say 200 million years. However, the age estimate for the human race computed in How Non-Functional DNA Testifies Against Evolution, and Shared Errors between Humans and Apes would also be reduced by a factor of five, to 40,000 years or less. This is because of the quote "Silent sites evolved at an average rate of 4.61 nucleotide substitution per 10⁹ years" found there. Silent sites could only change by mutations that affect a single base pair (i.e., substitutions). This indicates that the rate of base pair substitution is about one per 200 million years if evolution is true. Thus the rate of all mutations is at least this large. So if each mutation changes an average of 5 base pairs of non-functional DNA, then the rate of change of non-functional DNA would be five times as large, or, one base pair per 40 million years. This would mean that mitochondrial Eve lived 40,000 years ago or less, meaning the mitochondrial clock runs 5 times faster than assumed. Since the ape-human split is also dated in terms of the mitochondrial clock, this would reduce the ape-human split to 2 million years ago from 10 million years. We would still have a disagreement between 2 million years and 200 million years for the ape-human split. So there would still be a discrepancy in dating.

We now consider the maximum rate of evolution for asexual organisms. For these, each mutation can only spread from the descendents of one individual. After a mutation has spread to 1000 individuals, then there might be a chance of another favorable mutation. Error catastrophe prevents us from using a smaller number. The maximum selective advantage of a mutation is generally .01 or less. This would require 70 generations to double its frequency, and about 700 generations to multiply by a thousand. So we cannot expect much more than one beneficial mutation to fix in the population per 700 generations.

The subject of population genetics has been difficult for me to grasp, and, apparently, for population geneticists as well, judging from the history of the subject. I hope that this short discussion will do something to remedy this problem for users of the world-wide web.

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