Haldane’s Dilemma

A satisfactory theory of natural selection must be quantitative. In order to establish the view that natural selection is capable of accounting for the known facts of evolution we must show not only that it can cause a species to change, but that it can cause it to change at a rate which will account for present and past transmutations.
— J. B. S. Haldane¹

Finding a suitable lover can be arduous. The fewer demands you make the better—introduce standards, and the solitary road may stretch long and wide before you. Do you wish for a member of a particular sex? Your potential candidates are halved. They must enjoy swing dancing? One percent remains. Soon your list of suitors reaches, for all practical purposes, zero. If these criteria are necessary for your emotional survival, you’re out of luck.

Nature also has trouble selecting several traits simultaneously. In 1957, J. B. S. Haldane recognized the difficulty encountered by breeders selecting for multiple characteristics at once.² For example, a breeder might successfully increase milk production, meat yield, or reproductive output, in a herd of dairy cattle; achieving all three at the same time is difficult. The probability is small that the individual with the most desirable milk production will also happen to be the meatiest, even ignoring the possibility that one of the traits may correlate negatively with the other. In general, the number of individuals combining several specific characteristics decreases exponentially with each additional requirement.

The improbability of obtaining an organism combining multiple beneficial characteristics might be more practically formulated as a limit on the rate of change produced by natural selection. Imagine that a breeder prioritizes high milk production over other characteristics. Once the desired milk production is achieved, the breeder may then wish to increase meat yield. Unfortunately, since the previous selection for milk may have resulted in less meat, the breeder must hope that mutation will reintroduce new beneficial variation. This waiting time is roughly equivalent to the improbability of obtaining an individual combining both desired traits at the start.³ Thus, whether the breeder waits for the improbable occurrence of an individual with all desired traits, or else decides to select for the traits one by one, the population cannot be transformed instantly. Once the necessary variations arise, selection still requires time to do its work.

Why is this so? The short answer is that reproduction is finite. Evolution involves the genetic change of whole populations of organisms, and this change occurs because some organisms leave more offspring than others.⁴ Whether differential survival is thought of as excess death of the suboptimal trait, or as excess reproduction of the advantageous trait, the underlying principles are the same.⁵ Imagine a population that has trouble digesting the only food resource in its environment. Over time, by chance, a mutation in some digestive enzyme gene confers an enhanced ability to digest this food. From birth, the mutant individual has an extreme advantage in terms of survival and reproduction, and it might leave twice as many offspring in the next generation as the other members of the species. As generations pass, the population as a whole will change; an increasing proportion of its members will carry the advantageous mutation. Eventually, the mutation will be fixed, reaching a frequency of 100 percent. Every individual now carries the new mutation.

Evolution by natural selection has occurred.

This scenario is too vague to be testable. Suppose we add two pieces of information: (1) our hypothetical population consists of ten thousand individuals; and (2) each produces ten offspring during its lifetime, just one of which will successfully grow up to produce its own offspring. For convenience, let us further suppose that the population size does not fluctuate wildly. Could evolution occur at such a rate that the new mutation is fixed in a single generation? Obviously not. This would require the mutant individual to have ten thousand children—a mechanically impossible feat for a member of this species. Would two generations be sufficient? Again, the answer is no. Given our information, the first mutant individual might have a maximum of ten successful offspring. Her offspring might have the same number, while individuals carrying the old (pre-mutation) gene would leave many fewer offspring on average. By generation two, under the most favorable conditions, 10 × 10 = 100 individuals carrying the new mutation might exist. They replace a similar number of non-mutant individuals who would in theory have been born had the new mutation not appeared.

This is the simplest possible presentation of the mathematical basis for Haldane’s cost of natural selection. Evolution by natural selection requires excess reproduction of a new (advantageous) mutant, implying excess death of an old (suboptimal) gene, and a species’ reproductive capacity limits the rate at which this can proceed.⁶

Haldane, one of the founders (along with Ronald Fisher and Sewall Wright) of mathematical population genetics, was the first to quantify such a limit on the speed of adaptive evolution.⁷ He concluded that the cost of selection “defines one of the factors, perhaps the main one, determining the speed of evolution.”⁸ Cost was the main reason Motoo Kimura proposed the neutral theory of molecular evolution.⁹ Many others cite its importance.¹⁰

The implications for mammalian evolution were considered so severe that the issue became known as Haldane’s dilemma.¹¹

Despite Haldane’s work, a massive body of literature has accumulated asserting the primary role of natural selection in evolutionary change, often implying rates of adaptive evolution that exceed plausible limits.¹² I maintain that cost, though often neglected in contemporary studies, remains as important as Haldane’s mathematical theory of selection. By setting a limit on the number of selective evolutionary changes, Haldane provided a simple way to test the plausibility of many evolutionary scenarios.

If cost is the primary limitation on natural selection, it has wide-ranging implications for assessing the validity of competing theories of molecular evolution.¹³ Once the maximum number of evolutionary changes is determined, it may be compared to the observed number of changes. Where positive selection fails, other mechanisms must take over.

The Population Genetics of Selection

Population genetics provides a mathematical tool for modeling evolutionary change, making hypotheses more exact and testable. Before covering some of the mathematics, I will mention a few fundamental concepts. New genetic variation is introduced by random mutations. These occur when cellular machinery makes copying errors during DNA replication or repair. Most often, mutations simply substitute one of the four nucleotide monomers—A, C, G, or T—for another.¹⁴ When a mutation occurs in a gene, it produces a variant form called an allele at a specific site, or locus. We will consider alleles to be two stretches of DNA encoding the same gene at a locus, but differing in sequence or length by one or more nucleotides. Mutations occur at the level of the genetic material, or genotype. These changes may or may not affect the physical or physiological characteristics of an organism, its phenotype. If a mutation alters the reproductive success of an organism, it is said to affect its fitness. Virtually all experimental and observational studies agree that the majority of mutations decrease an organism’s fitness, and these mutations are characterized as deleterious or slightly deleterious.¹⁵ When selection acts to remove such mutations from a population, it is considered negative or purifying. On the other hand, occasional mutations may be beneficial, providing an advantage to the organism.¹⁶ When selection acts to increase their frequency, it is called positive or Darwinian. Although well-documented cases of Darwinian selection exist, purifying selection is much more common.¹⁷

Substitutions occur when a mutation increases in frequency, typically via excess reproduction, until it reaches fixation in a population. I will use the term substitution primarily to refer to the process of increasing frequency, whereas fixation will refer to the final state in which the frequency reaches 100 percent. Substitutions need not occur via positive selection; the neutral theory of molecular evolution implies that the majority of substitutions over the course of evolution occur via genetic drift, or random changes in allele frequencies.¹⁸ Because natural populations are not infinitely large, such random changes can accumulate over time, causing alleles to be fixed by chance. Allelic variants that do not cause dramatic effects on the organism’s phenotype (and even those that do) can be subject to substantial amounts of random genetic drift.

Positive selection occurs when individuals carrying beneficial mutations leave more offspring than others.¹⁹ If such mutations occur, the new allele will begin to replace the old allele at that genetic locus. Let A represent the new allele, having a frequency of p, and a represent the old allele, having a frequency of q. Denote their relative fitness as w_A and w_a. In this case, w_A > w_a, and the frequency of A will increase via positive selection. By convention, the superior fitness is defined as 1, with the inferior fitness expressed as a fraction of 1. Because positive selection typically requires beneficial alleles to increase dramatically in number, achieving substitutions by this means can be relatively difficult. New mutations that are deleterious can easily be removed by purifying selection, since only one or a few copies need be eliminated.

For simplicity, we begin by considering a population of size N with non-overlapping generations. For the time being, we also assume haploidy, i.e., one copy of the genome per adult cell. All individuals carry the old a allele, such that p = 0 and q = 1, until a beneficial mutation produces A. If mutation produces a single copy of A, it will be present at an initial frequency of p₀ = 1/N, since there are N copies of the genome in the population. We can quantify the fitness advantage of A by specifying its selection coefficient, s. This measures the fertility difference between a and A individuals. Individuals carrying a leave an average of only 1 – s offspring for every one left by individuals carrying A, and the fitness of the two genotypes are defined as w_A = 1 and w_a = 1 – s.

From this simple case, Haldane calculated the relative number of individuals that must die in each generation as a result of selection, choosing to focus on mortality rather than reproduction.²⁰ For mathematical simplicity, let us assume that a population remains approximately constant. This assumption is not necessary for cost, and to allow its relaxation later on, it can be formalized as follows: let R be a reproduction rate, with R_i being the reproduction rate of individuals carrying allele i. Further, let $\stackrel{-}{R}$ be the average reproduction rate of the population over an interval lasting multiple generations. Thus, if $\stackrel{-}{R}$ = 1, members of the population leave an average of 1 offspring per generation, and the population size is maintained. If $\stackrel{-}{R}$ > 1, the population expands; if $\stackrel{-}{R}$ < 1, the population might eventually become extinct. Thus, $\stackrel{-}{R}$ ≠ 1 leads to an unsustainable ecological situation if the deviation is great for an extended period. In our case, individuals of the A genotype have R_A > 1, while individuals of the a genotype have R_a < 1, such that long-term $\stackrel{-}{R}$ ≈ 1. A substitution may occur, during which the relative frequencies of the two alleles change enormously. Still, N remains approximately constant due to population regulating mechanisms that define its carrying capacity, such as food, space, and competitors.²¹

I have now established a simple context for an advantageous mutation A. Haldane chose to focus on the selective deaths that occur during this process, calling this the cost. It is important to note that cost is not a burden to the population; it is simply a measure of the differential survival required should a substitution take place. In order for A to replace a, individuals carrying a must leave fewer successful offspring than can fully replace themselves in the next generation. This empty space in the environment may then be filled by the excess offspring of genotype A.

Let us begin with a single generation. If 1 – s individuals of genotype a (with frequency q) reproduce for every one of type A, then sq is the fraction of the total population that fails to reproduce. For example, suppose A is a new mutation in a population of size N = 1000, such that its initial frequency is p₀ = 1/N = 0.001. The old allele will then have a frequency of q = 1 – p = 0.999. If the selective disadvantage of a is s = 0.01, then a individuals leave only 1 – s = 0.99 offspring for every one left by A individuals. Before selection p + q = 1. If selection occurs during growth to adulthood, then the total remaining after selection is 1 – sq, which is less than 1. Thus the total fraction of the initial population that failed to reproduce is sq. Given our example, this fraction is sq = 0.01 × 0.999 = 0.00999. For a population of size N = 1000, this amounts to about 10 individuals per generation.

For simplicity, Haldane approximated 1 – sq as 1, and formulated the cost of selection as the fraction of the total population that dies (or fails to reproduce) as a result of carrying out a substitution.²² C_j = sq is the cost of a substitution during generation j. In the long-term $\stackrel{-}{R}$ ≈ 1. A individuals can provide an excess where a left a deficit, allowing us to re-normalize the total by 1 – sq. James Crow introduced this improvement.²³ In generation j, cost is measured as the ratio of individuals eliminated by selection to those that survived

$$C_j=\frac{sq}{1-sq}.$$

In our example, C_j is 0.0101, or about 10.1 individuals—very similar to the previous estimate of 10.

Because this measure of C focuses only on mortality, it is not at all determined by the advantageous allele’s reproduction rate, R_A, or the length of the substitution in generations, t. Regarding R, it is tacitly assumed that the species is able to reproduce at a rate consistent with the speed of the substitution, whatever that may be. Secondly, since s is inversely proportional to t—because advantageous mutations proceed more quickly to fixation—t is not a factor in these calculations.²⁴

Having established the cost C for a single generation of selection, we may now ask how C accumulates as A increases from p₀ to 1 (fixation). If the change in the frequency of A is sufficiently slow²⁵, then

$$C=\sum _{j=1}^t{C}_j\approx {\int }_0^{\infty }\frac{sq}{1-sq}dt={\int }_{p_0}^1\frac{dp}{p}=-\ln p_0.$$

In our example, p₀ = 0.001, yielding C = 6.9.

How is this measure interpreted? Assuming that the number of individuals, N, remains approximately constant at 1000, it follows that CN = 6.9 × 1000 = 6900 selective deaths occurred over the course of the substitution. Accumulated cost is simply multiplied by the size of the population to yield the number of deaths that have occurred. N individuals have lived during each of the t generations of the substitution, for a total of tN. Of these, CN were eliminated by the selection required to carry out a substitution. Selective death can work via any of the components of fitness: actual death of the organism (viability), failure to produce viable gametes (fecundity), or failure to mate.

Thus far we have considered the relatively simple case of a haploid population. Because each genetic locus has only one allele in a haploid individual, a mutation’s effects are not influenced by other alleles. This perfect exposure of mutations in the phenotype allows selection to act with maximum efficacy. However, diploid organisms such as humans have two copies of the genome in each cell, so that beneficial alleles may exist in either the homozygous (AA) or heterozygous (Aa) state. When a mutation first occurs, it will necessarily exist in the heterozygous state. Of the population’s 2N copies of the genome, only one contains the beneficial mutation. As p increases from 1/(2N), the number of possible heterozygotes increases.

The way in which a mutation affects fitness in the heterozygous state is specified by its dominance, h. Just as s is customarily applied to the inferior allele, h refers to its degree of dominance. If h = 0, the advantageous allele A is fully dominant; if h = 1, the suboptimal allele a is fully dominant. In the case where a is fully dominant, it will mask the beneficial effects of A in the phenotype, and selection cannot act. When h = 0.5, there is no dominance, such that the fitness of the heterozygote (Aa) is exactly intermediate between the two homozygotes (AA and aa).

Solutions for C in a diploid population are²⁶

$$C=\frac{-1}{1-h}\left[\ln p_0+h\ln \frac{h}{1-h-(1-2h)p_0}\right]$$

when the substituting allele is partially dominant (h ≠ 1); and otherwise

$$C=-\ln p_0+\frac{1-p_0}{p_0}$$

when the substituting allele is fully recessive (h = 1).

Several conclusions can be drawn from these results. First, the total cost C incurred over an evolutionary substitution is higher when the starting frequency of the beneficial allele (p₀) is small. This makes sense, because there will be more organisms with the old genotype, which must be replaced. Total cost might be lowered, it is obvious, by an increase in p₀. Second, cost is identical for haploids and diploids with a completely dominant beneficial allele. A dominant allele will always be expressed in the phenotype of a diploid. And finally, as the dominance of the advantageous allele decreases, its cost increases exponentially. While the new allele is still rare, it will generally pair with an old allele in the genotype (i.e., Aa) of a diploid organism. If the new allele is not expressed, then any differential survival that occurs is wasted. Selection can only act effectively once the new allele becomes common enough to stand a reasonable chance of pairing up with another copy of itself.

The haploid case represents the best possible scenario (with lowest cost) for a diploid organism.

The Dilemma

Haldane was interested in developing a way to test natural selection in the evolution of mammals.²⁷ One way to do this would be to specify the maximum amount of genetic death that a population can tolerate, and then determine the number of generations required by a substitution. Imagine a diploid mammalian population in which a new beneficial mutation has frequency p₀ = 10^-4, and specify that selection may cull 10 percent of the population in each generation. C might be any value from 9.2 to 10,000 when p₀ = 10^-4, depending on the dominance of the allele. Haldane reasoned that the typical range might be between 10 and 100, and took C = 30 as a mean value. If only 10 percent of the population may be selectively eliminated in any given generation j, then the substitution must occur slowly enough for C_j = 0.1. This condition is met if the substitution takes 300 generations, since 30/300 = 0.1. Of the tN = 300N organisms that live over t generations, 30N of them must be culled by selection to result in a selection intensity I, of 10 percent. In general

$$I=\frac{CN}{tN},$$

where I measures the fraction of the population that selection must cull every generation in order to accomplish a substitution in t generations. In our case, I = (30N)/(300N) = 0.1. Haldane concluded that over the course of mammalian evolution, substitutions must occur only once every 300 generations on average.²⁸ These might occur sequentially or concurrently; Haldane thought that the latter was more probable.²⁹ If selection acts on multiple mutations at once, the cost per substitution remains the same. Once we specify the intensity of selection that the species can tolerate, we know what fraction of the population is available for selective culling. If two mutations are selected simultaneously, they will impose a cost of 2C, requiring 600 generations in the present case, and the length of time will increase for each additional substitution.

The number of simultaneous substitutions, George Williams has argued, cannot be too large.³⁰ Imagine that numerous dominant advantageous mutations exist in a population, each with selective advantage s = 0.01. The diploid genotype aa is only 99 percent as fit as the others; it leaves 1 percent fewer offspring. Suppose that such beneficial mutations exist at n = 10,000 loci, each undergoing substitution. An average individual might have the less favorable genotype aa at 1000 of these loci; its fitness would be only 0.99¹⁰⁰⁰ = 0.00432% of an individual with the superior genotypes. This difference in fitness is so great that, if the typical individual were to leave one offspring per generation, individuals with the superior genotype at all such loci would leave over 23,000 offspring. Such a drastic range in fitness is biophysically implausible for mammals. Even if s is redefined as a positive number, as Bruce Wallace has suggested, the same variation exists and demands explanation.³¹ Indeed, using a positive value for s still implies a deficiency in the old allele, such that the only change brought about by switching signs is cosmetic.³²

This discussion has assumed that the two advantageous mutations occupy distinct loci in the genome, and that they influence fitness independently. I discuss the first assumption later. The second seems a very reasonable approximation. Most of the heritable variation in mammalian fitness results from largely independent genetic differences.³³ If mutations did not influence fitness independently, each would contribute a much smaller fraction of an individual’s total fitness advantage, rendering positive selection for any particular beneficial mutation less effective.

It is easy to see why Haldane’s conclusions posed a dilemma for biologists interested in mammalian evolution. Human and chimpanzee species diverged from a common ancestor approximately 4.5 to 13 million years ago.³⁴ Humans currently have an average generation time of 30 years, chimpanzees 20 years.³⁵ At most, 500,000 generations have elapsed. Given Haldane’s limit, this makes for 3333.3 adaptive differences.³⁶

Can roughly 3000 changes explain all of the complex adaptive differences between humans and chimpanzees?

This is Haldane’s dilemma.

It is a dilemma that has been exacerbated by genome sequencing. Humans and chimpanzees both have genome sizes of roughly 3 billion nucleotides. Yet these species differ by some 30 million fixed nucleotide differences.³⁷ If these differences were fixed individually by positive selection, then the substitution rate would have been 1.5 substitutions per year in each line of descent, or 30 per generation—a biological impossibility.

Haldane’s conclusion has not stopped evolutionary biologists from searching for evidence of positive selection in the nucleotides of the human genome, an endeavor Austin Hughes has described as a “misguided quest.”³⁸ This “outpouring of pseudo-Darwinian hype,” he says, “has been genuinely harmful to the credibility of evolutionary biology as a science.”³⁹

Part of the problem has certainly been a failure to address Haldane’s work. With no reference to cost, Fay et al. claim that adaptive substitutions have occurred at a rate of one per every ~200 years for the last 30 million years.⁴⁰

John Hawks et al. claim an incredible rate of ~0.53 per year.⁴¹

Charles Lumsden and Edward Wilson define a thousand-year rule, or approximately 50 generations per substitution in Homo sapiens.⁴²

Pierre Luisi et al. report 554 positively selected genes, each presumably requiring numerous mutational innovations, while Ralph Haygood et al. claim positive selection in certain promoter regions.⁴³

These studies ignore Haldane’s dilemma.

The troubling implications of Haldane’s results earned him great criticism.⁴⁴ Some contentions were largely semantic, Alice Brues argued thus that:

[T]here is no risk to species survival inherent in the selective process itself which will restrict the number of [adaptive] substitutions which may take place in a given population during any particular period of time.⁴⁵

But this is only partially correct. On the assumption that substitution occurs, Haldane was providing a way to measure the differential survival necessary to carry it out. It is true that there is no risk inherent in the selective process, if by this we mean the mere coexistence of two alleles in a population. However, if we stipulate not only the coexistence of two alleles, but require that one substitutes for the other over a fixed period, then there is clearly a restriction that depends on differential survival.⁴⁶

Cost as Reproduction

We have established that substitution occurs due to differential survival. The processes involved are reproduction and death. When formulating cost, Haldane chose to quantify death. However, death is not biologically restrictive: any organism in a population can die at any time. The alternative is reproduction. Indeed, it makes good sense that the biological process underlying the evolutionary process of substitution would be excess reproduction of the beneficial allele, and reproduction rates are measurable characteristics of a species.

Soon after the concept of cost was proposed, it was recognized that reproduction was somehow central to the problem—indeed, that it should replace mortality. Haldane himself often referred to reproductive capacity in relation to cost, even though genetic death was his focus.⁴⁷ Numerous authors have expanded on this, Crow and Joseph Felsenstein drawing attention to the reproductive excess involved in substitution.⁴⁸ Crow even equated reproductive excess with cost, writing that “[t]he cost is the excess in survival and fertility that the favored genotype must have in order to carry out the gene substitution at a specified rate.”⁴⁹ Masatoshi Nei also equated cost with the “fertility excess necessary for gene substitution,” noting that “the number of possible gene substitutions per unit length of time is limited by the fertility of the species concerned,” and that an overwhelming cost must imply either a slower gene substitution, or else extinction.⁵⁰

It was Walter ReMine, expanding on the work of Felsenstein and Nei, who finally gave the concept of reproductive cost an intuitive foundation.⁵¹ His formulation focuses on absolute rather than relative frequencies. Imagine a new mutation arises: one copy is now present in the population. Suppose that it must reach fixation in 300 generations. It is then simple to calculate the minimum uniform reproduction rate R necessary to accomplish the required increase. For a new mutation in a population of size N that starts at n₀ copies and must be fixed in t generations

$$R=\sqrt[t]{\frac{N}{n_0}}.$$

For example, if a new (n₀ = 1) mutation occurs with N = 10,000 and t = 300, the minimum R required is $\sqrt[300]{\mathrm{10,000}}$ = 1.03. Each member of the species carrying the new mutation must leave an average of 1.03 offspring in each generation in order to reach N = 10,000 at the end of 300 generations. Some generations might leave fewer, but this would then require that other generations leave more. Assuming that R remains constant, the mutation would amplify from 1 to an average of 1 × 1.03 = 1.03 copies in the first generation, 1.03 × 1.03 = 1.06 in the second generation, and so on.

This approach to cost illuminates three important points.

First, death is irrelevant. Whether the members of the population carrying the old allele die instantly when the new mutation arises, or gradually over the course of the substitution, it is only the absolute number of copies of the new allele that matters.

Second, an increase in population size or carrying capacity actually increases cost. If a beneficial mutation increases carrying capacity from 1000 to 1500, perhaps by enabling a more efficient use of food resources, then the new mutant must increase from one to N = 1500 copies.

Finally, cost is only severely limiting for species with a limited reproductive output. Excess offspring are the payment spent for the cost of substitution. Some species have enough cash flow to pay for faster substitutions than others. Biological parameters having been specified, it then becomes possible to determine the rate of substitution a species can afford. Given a reproduction rate R, the total cost for a substitution lasting t generations is the reproductive excess, t(R – 1). Summing over t generations, C = ln(N/n₀). Substituting 1/(2N) for p₀ in Haldane’s formulation, –ln(p₀) = ln(2N), confirming our intuition that the absolute number of allele copies in the population is the relevant parameter of cost.

I prefer to define cost as the required reproduction rate R for one generation, because this may be compared to the actual reproductive potential of a member of the species. This formulation has several desirable features. First, it is intuitive. Second, it equates cost with a measurable biological parameter: reproduction rate. Third, the cost is spread evenly throughout. Fourth, because s is nowhere used in its derivation, this way of measuring cost is not limited to slow substitutions. Finally, it uses absolute rather than relative frequencies.

The last point deserves elaboration. When using relative frequencies (e.g., p₀ = 10^-4) and ignoring the actual population size N, the mathematics of population genetics generally involves an assumption of infinite population size. This can mask important processes.⁵² Consider two populations in which a beneficial allele A arises. The populations are the same size, N = 6, such that the initial frequencies of A in each are identical: p₀ = 1/N = 0.167. After t generations, the frequencies of the new mutation are p = 1.0 in each; in both populations mutations have been fixed. Yet A is fixed in population (A) due to excess reproduction, whereas it is fixed in population (B) due to chance survival in a shrinking population on the brink of extinction. In the case of (A), R = $\sqrt[t]{6}$. In (B), R is only 1. Death of the old allele is the sole reason for fixation.

This is just one example of how relative frequencies and the focus on death of the old allele can create confusion.

ReMine makes the point another way.⁵³ In the derivation of costs, the numerator (sq) is the relative number of selective deaths, and the denominator (1 – sq), the relative number of survivors. If there are no survivors, the cost is mathematically undefined. There must be survivors for mortality-based cost to make sense.

Can this formulation of cost be used to calculate an achievable number of substitutions over the course of human evolution? If we assume a total fertility rate of five per female, R = 2.5 per individual.⁵⁴ If the historic population size N = 10,000, there are then roughly t ≈ 10 generations per substitution, implying a maximum of 50,000 substitutions.⁵⁵

The dilemma remains.

The Neutral Theory

Haldane’s results suggest that roughly one substitution occurs in every 300 generations. This calculation came to Kimura’s attention, who found it at odds with the rates of substitution that must have occurred during mammalian evolution.⁵⁶ Using sequence data, Kimura estimated a mammalian substitution rate of about one amino acid substitution every 28 million years in a polypeptide 100 residues in length. Given a mean mammalian genome size of 4 × 10⁹ base pairs, and accounting for the finding that ~20 percent of nucleotide mutations do not alter amino acids, he estimated that one amino acid substitution corresponds to about 1.2 nucleotides. Thus there must have been one nucleotide substitution every 1.8 years in mammalian evolution—far more than one every 300 generations.⁵⁷ A modified version of his derivation follows

$$\frac{2.8 \times {10}^{7}yr}{\left(\frac{{sub}_{aa}}{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{i}\mathrm{n}}\right) }\times \frac{1 {sub}_{aa}}{1.2 {sub}_{nt}}\times \frac{100 aa}{1 \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{i}\mathrm{n}}\times \frac{3 nt}{1 aa}\times \frac{\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{m}\mathrm{e}}{4 \times {10}^{9}nt}=\frac{1.75 yr}{\left(\frac{{sub}_{nt}}{\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{m}\mathrm{e}}\right)}$$

where aa represents amino acid, nt represents nucleotide, and sub indicates substitution. Jack King and Thomas Jukes independently derived similar results.⁵⁸ Because of this discrepancy, Kimura concluded that the majority of substitutions must occur by random genetic drift.

This claim is the central tenet of the neutral theory of molecular evolution.⁵⁹

How does neutrality address Haldane’s dilemma? In a diploid population, there are 2N copies of the genome, each of which receives µ mutations per generation. Thus, µ is the genomic mutation rate, and the total number of mutations entering the population in each generation is 2Nµ. The vast majority of these will be approximately neutral.⁶⁰ If f_n is the proportion of neutral mutations, then 2Nµf_n neutral mutations enter the population in each generation.

What are reasonable values for µ and N in Homo sapiens? Estimates of the human mutation rate range from ~70 mutations to ~150 mutations per diploid genome per generation.⁶¹ Assume a mutation rate of ~100, or µ = 50 per haploid copy of the genome. Estimating N is complicated. Population genetics assumes that populations exhibit several idealized features, such as random mating. Yet the human population does not mate randomly, and its size has fluctuated quite drastically over time. These considerations must be taken into account in order to define an effective N for mathematical use.⁶² Because deviations from assumptions introduce random errors, the effective measure of N is almost always drastically smaller than the actual N, approximately 10,000 for recent human history.⁶³

The total number of mutations entering the human population in each generation is thus 2Nµ = 2 × 10⁴ × 50 = 10⁶. If f_n = 0.99 of these are neutral, then 2Nµf_n = 9.9 × 10⁵ neutral mutations occur each generation.

What does this imply about the rate of substitution? Let δ_n be the substitution rate of neutral mutations per generation. If we know P_fix, the probability that a given neutral mutation will reach fixation, then the rate of substitution is

$${\delta }_n=2N\mu f_n{P}_{fix}.$$

For a new neutral mutation, p₀ = 1/2N. Its probability of fixation is simply its frequency. Each copy of the genome has an equal probability of fixation. Thus 2Nµf_n × 1/2N = µf_n.

This is a remarkable conclusion of the neutral theory.

It now becomes apparent how the neutral theory solves the quantitative problem of a high substitution rate. Imagine that all mutations are neutral. There are many neutral mutations in a population and these may be substituted together for the cost of one.⁶⁴ Over the 500,000 generations since divergence, this amounts to 25 million substitutions, which is very close to the actual number of substituted nucleotides observed. In this way, the neutral theory allows a faster rate of evolution.

Still, the question of beneficial mutations remains. According to Kimura’s analysis, they enter the population at a rate of 2Nµf_b in each generation. The rate of substitution, δ_b, is obtained by multiplying the number of mutations by their probability of fixation,

$${\delta }_b=2N\mu f_b{P}_{fix}.$$

Kimura showed that

$$P_{fix}\approx \frac{1-e^{-4Ns{p}_0}}{1-e^{-4Ns}}\approx \frac{1-e^{-2s}}{1-e^{-4Ns}},$$

for a new beneficial mutation with no dominance (h = 0.5); or approximately 2s if Ns ≥ 2.⁶⁵

What fraction f_b of all mutations are beneficial? It turns out that beneficial mutations are so rare that they cannot be reliably measured, with estimates up to 15 percent for various organisms.⁶⁶ The best opportunity for observing beneficial mutations comes from the long-term evolution experiment by Richard Lenski et al. with E. coli, because large effective N, and consistent selection pressures, provided an ideal environment for adaptive substitutions.⁶⁷ Lenski et al. esimated f_b to be approximately 1 in a million.⁶⁸ The value is probably similar for eukaryotic organisms.⁶⁹ An independent line of reasoning agrees: if a mutation in a protein-coding region is to be beneficial, then it must probably affect the amino acid composition of its protein product. Such changes usually occur with frequencies significantly less than those which do not change amino acid composition. This suggests that the proportion of such mutations is very low.⁷⁰ It is reasonable to use one in a million (0.000001 = 10^-6) to estimate f_b, yielding a rate of one beneficial mutation entering the population in each generation.

Calculating P_fix for a beneficial mutation requires that we specify s. Theoretical and in silico results agree that most beneficial mutations have small effects, with the frequency of mutations decreasing exponentially with increasing effects.⁷¹ Suppose that s = 10^-4 on average. If so, then P_fix = 2.04 × 10^-4. This is over four times higher than P_fix for a neutral mutation. While beneficial mutations are more likely than neutral mutations to be fixed, beneficial mutations are also much rarer than neutral mutations.⁷² More than 99.9% of beneficial mutations appearing in the human population are lost by random genetic drift.

Under various parametric estimates, δ_b = 2.04 × 10^-4 per generation for the human species. Approximately 102 beneficial mutations might be substituted over 500,000 generations. Depending on s and f_b, this number may take on drastically different values.

These results represent an independent theoretical estimate of the number of adaptive substitutions that took place in human evolution. They are in rough agreement with Haldane. Given the assumption of an infinite population size, the cost of selection still functions as an independent limit.

Kimura’s last contribution to this subject was published posthumously and entitled “Limitations of Darwinian selection in a finite population.”⁷³ He showed that the number of loci n at which beneficial mutations can be simultaneously substituted is such that

$$n<-\left(\frac{N}{2}\right)\ln \left(1-I\right),$$

where I is again the reproductive excess available for positive natural selection. Notice that this limit does not depend on s, but on N and I. If N increases, the actual number of reproducing adults increases, and some of these will be available to carry and propagate advantageous mutations. Likewise, if I increases, so does the actual number of excess offspring, providing the next generation with more individual carriers of the advantageous mutations. Assume I = 0.1. If N = 10⁴, n must be < 527. Thus, positive natural selection cannot act effectively if more than 527 advantageous mutations are present in the population.

With respect to neutral substitutions, it follows that there is no real dilemma. They can easily account for the approximately 30 million nucleotide differences observed between humans and chimpanzees. But neutral mutations have no effect on fitness, and so by definition do not contribute to complex adaptations. If we consider the case of beneficial mutations, then positive natural selection appears limited to driving the substitution of fewer than 10,000 nucleotides.

We are faced with a different formulation of the same dilemma. Can so few differences account for the complex adaptive changes and innovations that shaped humans over the 10 million years since their divergence from the chimpanzees?

Either Haldane’s limit on adaptive changes can be modified while still assuming positive natural selection, or natural selection must be supplemented by a different adaptive mechanism.

Epicycles

If a substantial number of beneficial mutations could coexist in the same population, this might make for an effective means of evading Haldane’s limit.⁷⁴ Multiple mutations linked together in the same genome might behave as one large beneficial multi-nucleotide mutation. Haldane published one of the first documented cases of mammalian linkage in 1915.⁷⁵ He apparently felt that beneficial mutations were too rare for this mechanism to have an impact.

Nonetheless, we might imagine how this could occur. Begin with the average number of beneficial mutations expected in one individual. If there are n loci containing advantageous alleles in the population, each at a frequency p, then we would expect an average of np advantageous alleles per individual, with a standard deviation of $\sqrt{np(1-p)}$.⁷⁶ If n = 100 loci and p = 0.001, this amounts to an average of 0.01 ± 0.316 beneficial alleles per individual. Yet, in a population of size N = 10⁴, only 0.464%, or 46 individuals, contain ≥ 2 loci with these alleles; only 1.5 individuals contain ≥ 3 such loci. Thus, the number of loci containing beneficial alleles, or the frequency of alleles at those loci, would have to be unexpectedly large for this mechanism to greatly increase the number of substituted nucleotides.

In terms of cost, the unlikelihood of combining two rare mutations will offset the gain they would achieve by combination. Two mutations with frequency p that substitute one at a time each a have cost –ln(p), for a total of –2ln(p). The frequency of genotypes that contain both beneficial mutations by chance would be p × p = p². In this case, there is one cost, –ln(p²), which simplifies to –2ln(p).⁷⁷

Perhaps beneficial alleles at separate loci need not rely solely on chance association? Mutations might arise in separate individuals. Their descendants might then mate, affording an opportunity for genetic recombination to place the two mutations in the same genome. Although recombination relies on chance, the fitness advantage of the recombinant offspring might increase, raising P_fix for the multi-mutation complex as a whole.

There are several obstacles to this scenario. First, recombination is random, and has no tendency to link beneficial alleles more often than to separate them. Unless the two mutations interacted to boost their collective s by several orders of magnitude, random extinction would remain their most likely fate. One might surmise that two mutations which reside close to one another in the genome might have a higher probability of interaction; however, the closer they are, the lower the probability of recombination. This mechanism is also biologically limited insofar as the amount of recombination is usually limited to about one event per chromosome arm in meiosis.⁷⁸ For humans, the average rate of recombination is such that a given stretch of 10,000 nucleotides must wait 3,846 generations to stand an even chance of one recombination event.⁷⁹

I have used p = 0.001 to account for a selection-driven increase in the number of beneficial alleles. Perhaps beneficial mutations might increase further in frequency, followed by the occurrence of a second beneficial mutation. Unfortunately, the probability is 1 – p that the second mutation would occur in a non-advantageous genotype. Imagine two beneficial mutations, A and B. If the frequency of A had risen to p = 0.01, then B would occur in a non-A genotype 99 percent of the time. Should this happen, B would compete with A, resulting in selective interference. One of the two alleles would eventually drive the other to extinction, and any reproductive excess that the failed allele had enjoyed would be wasted. If recombination happened to link A and B, the initial frequency of the new genotype would be 1/2N. This would involve replacing even the advantageous A (but no B) and B (but no A) genotypes. The wasted reproductive excess involved in failed substitutions adds to the total cost.

John Maynard Smith argued that multiple linked mutations can increase the rate of adaptive substitution if the amount of selective death is 50 percent, and if these deaths involve only those individuals carrying the fewest beneficial mutations.⁸⁰ This type of selection is called truncation selection, because it effectively lines up every individual in a population by its value for some characteristic and truncates those below some critical value.⁸¹ Under this scheme, assuming p ≥ 0.1 and s = 0.01, selection might act at ≥ 25,000 loci at once. J. A. Sved has made a similar argument, claiming that no obvious upper limit exists for the rate of substitution if only one assumes an infinite number of simultaneous beneficial substitutions.⁸² These are interesting theoretical points. But real populations have finite genome sizes, and leave a finite number of offspring.

Considerations of cost remain.

Could beneficial mutations begin as neutral mutations and drift to relatively high frequencies? If an environmental change causes a common neutral variant to become beneficial, then p₀ > 1/2N, and the cost of substitution is lowered. This is precisely the reasoning that Haldane used to estimate C = 30.

Unless p₀ greatly exceeds expectations, the dilemma remains unaltered.⁸³

In order, therefore, to allow for an increase in p₀, Ron Woodruff et al. have recently suggested that since mutations need not arise in single gametes but may instead occur before meiosis, multiple gametes might contain them.⁸⁴ Let this number of gametes be g. Woodruff et al. argued that in this case, p₀ is not 1/2N but rather g/2N.

This reasoning is flawed.

Under the standard model of selection, each diploid individual in a population contains two copies of the genome—one maternal and one paternal. When mating occurs, every individual contributes an equal fraction 1/N of the gametes present in the population’s gene pool. Mutations occur immediately upon zygote formation, and this must happen in either the maternal or paternal copy of the genome. Suppose it is the maternal copy. A beneficial mutation will be present in the maternal genome copy in all adult cells derived from the zygote, including those that generate gametes. When gametes are formed, half will contain the beneficial mutation; when this organism contributes its share 1/N to the gene pool, half will contain the beneficial mutation with frequency ½ × 1/N = 1/2N. Under the best of circumstances, a lucky gene conversion event might occur early in development, using the maternal copy of the genome to overwrite the paternal copy. The frequency of the mutation in the gene pool would then rise to a maximum of 2/2 × 1/N = 1/N.

Perhaps Woodruff et al. meant that the individual containing the beneficial mutation contributes more than its fair share of gametes, 1/N, to the gene pool? But this requires excess reproduction—another way of saying higher cost. As it stands, then, premeiotic clusters of mutations do not seem to alter cost considerations. They might actually make the dilemma worse. What if a beneficial mutation does not arise immediately upon zygote formation, but later in development? In that case, some of the cells generating gametes in the adult might not contain the mutation, causing the fraction of gametes carrying it to fall well under 50 percent. This situation can be treated more generally if we define G to be the total number of gametes contributed to the gene pool by an individual, g of which contain the beneficial mutation. In that case, g/G of the gametes will contain the mutation, and its initial frequency in the gene pool becomes

$$p_0=\frac{1}{N} \times \frac{g}{G},$$

where 0 ≤ g ≤ G. It has historically been assumed that g/G = ½.

Although some factors might reduce cost, others might worsen it. A divergence time in human evolution of 5 million, rather than 10 million years, doubles the problem.⁸⁵ What of the assumption that beneficial mutations are readily available and that they have a selective advantage s that remains constant? Neither assumption appears true. Few beneficial mutations have been observed in eukaryotes.⁸⁶ Even prokaryotes display waiting times between sequential advantageous substitutions.⁸⁷

Selection itself may not remain consistent. In the case of the Galápagos finches, strong selective pressures can reverse themselves over very short periods.⁸⁸ Experiments on Drosophila suggest that, even when selection pressure is constant, s may fluctuate, perhaps due to the accumulation of mutations elsewhere in the genome.⁸⁹ Worse, the substitution of one beneficial mutation might negate the effects of others, either by changing the genetic background or exhausting the adaptive potential of a gene. If balancing selection were to occur often, it would act to prevent the substitution of any linked beneficial alleles. Finally, if many beneficial mutations are recessive, which is more likely if they are high in impact, their substitution rates will be disastrously slow.⁹⁰

Substitutes for Natural Selection

It is reasonable to ask at this point whether any other mechanisms can compensate for natural selection. Small changes might go a long way. Some evidence suggests that mutations in regulatory regions can have dramatic effects.⁹¹ In The Origins of Genome Architecture, Michael Lynch has argued that random genetic drift is, by itself, quite sufficient to explain many of the features of complex genomes.⁹² Neutral mutations that fix over the course of evolution might provide novel genetic contexts that can be converted into beneficial alleles by subsequent mutations. As few as one in 10⁷⁷ protein domain-sized polypeptides may be able to form functional folds.⁹³ This suggests that the number of neutral mutations needed to result in a functional gene product is vast. Lynch and Adam Abegg have observed that a mutation involving three nucleotide sites and two neutral steps would take ~760 million years to be fixed.⁹⁴

But processes of this sort do not explain functional innovations in structural genes, let alone genes that might be unique to the human species.⁹⁵

Some evidence suggests an important role for whole-genome duplication in the evolutionary history of teleosts.⁹⁶ Duplication provides extra genetic material for evolutionary experimentation, and decreases the constraints imposed by purifying selection. But proteins that are not initially multifunctional show little promise of evolving distinct functions; new functions still require either drift or substitution.⁹⁷

Austin Hughes recently proposed an intriguing new mechanism by which adaptive changes might become fixed.⁹⁸ A population exhibits phenotypic plasticity as a trait. A species might then be able to change its behavior in the presence of two different predators. Any mutation reducing the ability of an organism to adapt to either predator would be subject to purifying selection. Suppose one predator disappears. Mutations continue to occur in the underlying gene, but purifying selection against them is relaxed. Upon relaxation, drift commences, leading to interspecies differences and neutral substitutions.

Whether this scheme explains the origin of genes encoding novel functions is a question for another day.