The waiting time problem in a model hominin population

At the heart of all information systems, including biological information systems, are strings of symbolic characters. Within the human realm, communication is generally based either upon strings of sounds, strings of letters, or strings of binary bits. In the biological realm, much information is transmitted within the cell based upon long strings of either nucleotides or amino acids. The information within an RNA or protein string almost always traces back to DNA strings. Within DNA, a primary unit of functional information is the gene, which is a very long text string that can range in size from about 1000 to more than one million nucleotides long. No geneticist believes that these very long nucleotide strings could ever arise directly, from scratch, via the mutation/selection process. Rather, a new gene is thought to arise from a previously existing gene, with the mutation/selection process establishing mutations within the long text string that is already established and functional. If sets of mutations (or short sub-strings) can be established within the preexisting longer string, then a new gene might arise, having a new biological function. Since a typical human gene is roughly 50,000 nucleotides long, to form a new gene with a new function should typically require the establishment of multiple mutations (or multiple sub-strings) within a pre-existent gene.

In this paper we address the question, “How long does it take for the simplest biological text strings to arise and be fixed, within a hominin population?” Given the unique capabilities of humans, an evolving hominin population would need to establish a great deal of new information, leading to new functionalities. Since all information is based upon strings of symbolic characters, this suggests the need to establish a multitude of new nucleotide sub-strings. Chimp and human genomes are minimally 5 % divergent [1], representing about 150 million nucleotide differences. This substantial genomic difference implies many new sub-strings within the human genome. In this paper, we only examine the average waiting time needed to generate and fix a single DNA sub-string of minimal length (2–8 nucleotides). This sort of minimal genomic modification would alter only one (or a few) specific amino acids, or might conceivably result in one new specific protein fold.

Numerous scientists have previously examined the problem of waiting time. Haldane was the first to address this problem [2]. He realized that even after a highly selectable beneficial mutation arose within a given population, considerable time was required for the amplification of that new mutation – such that all individuals in the population inherited the new mutation (“genetic fixation”), thus driving the original allele to extinction. Haldane realized that for long-lived organisms such as Homo sapiens, waiting time for fixation appeared to be problematic (only allowing about 1000 beneficial mutations to be fixed within the evolving pre-human population during the span of 6 million years). This problem came to be known as Haldane’s Dilemma. Kimura [3] agreed with Haldane’s assessment of the problem, and this motivated him to develop his neutral model of evolution. He reasoned that since selective amplification is too slow to fix very many mutations, most genetic fixations must result from random genetic drift.

Although significant, the waiting time associated with Haldane’s Dilemma is eclipsed by a much more profound waiting time problem. Haldane assumed a continuous supply of high-impact beneficial mutations, and did not realize that such mutations occur very rarely. Waiting for such mutations to arise is a much more significant factor affecting waiting time than the time required to amplify a given beneficial mutation after it has arisen. It is now generally recognized that beneficial mutations are rare, and that high-impact beneficial mutations are extremely rare [4–9]. Because of this, waiting for just the right mutation to arise in just the right location can be a rate-limiting factor in terms of the long-term evolution of any relatively small population. When a population faces a specific evolutionary challenge, a specific fix is needed, and it must arise in a timely fashion. Positive selection cannot generally begin to resolve an evolutionary challenge until just the right mutation (or mutations) happens at just the right position (or positions). Selection for the required trait can only begin after the mutation (or mutations) result in a substantial (selectable) improvement in total biological functionality. In higher life forms where population sizes are modest, the mutation rate per nucleotide per generation is normally extremely low (about 10⁻⁸) [10–12] – so that waiting for a particular mutation to arise can require relatively deep time. The waiting time required for the generation and fixation of multiple specific mutations needed to combine to create a new function can require inordinately long waiting times.

Behe and Snoke [13] and Behe [14] have made the argument that when more than two specific mutations are required to create a specific new biological function, the waiting time problem can become prohibitive. To illustrate this point, Behe [14] used the historical case of natural selection for malarial resistance to the drug chloroquine, which required two mutually-dependent mutations. He used some rough calculations based upon malarial incidence data, and then extrapolated those numbers to a hypothetical human population. Arguably, there may be problems with that analysis, as well as the earlier analysis by Behe and Snoke [15–19]. It is beyond the scope of this paper to defend or refute those earlier studies. Regardless of the validity of the initial mathematical formulations of Behe and Snoke, the basic question they raised remains interesting, and their work has led to considerable additional research by other investigators (this paper; [15–26]).

Virtually all of the papers subsequent to the work of Behe and Snoke have confirmed that waiting times can be prohibitive – depending upon the exact circumstances. Some of the subsequent papers have been critical [15–17, 25]. Yet even those papers show that establishing just two specific co-dependent mutations within a hominin population of 10,000 can require waiting times that exceed 100 million years (see discussion). So there is little debate that waiting time can be a serious problem, and can be a limiting factor in macroevolution. The real question is; “How long must the required string of nucleotides be, before the waiting time becomes prohibitive?” As we will see, the answer to that question depends on the specific biological circumstances.

The major point of disagreement seems to involve the question of exactly how to formulate the waiting time problem. For example, is waiting time measured in terms of years to first instance of the string, or years until the string has arisen repeatedly – to the point where the string persists and goes to fixation? What constitutes a valid functional target string? Is a perfect nucleotide match required to create a functional string? Can the target string arise anywhere within the genome? Does selection begin when the string is complete (fully functional), or when it is only partially complete (limited function)? Are all of the “stepping-stone” mutations that lead to a selectable string neutral, or are some deleterious? How large a fitness benefit does the completed string provide?

The various researchers who have examined the issue of waiting time have approached the problem from different directions, using many different starting assumptions. This means that comparing their results can often be like comparing apples and oranges. It is unlikely that there is a universal way to characterize the waiting time problem, as each circumstance requires special treatment. For example, the size and generation time of a population has a large effect on waiting time. Some authors like Behe/Snoke, Lynch, Axe, and Gauger focus almost exclusively on mega-populations with very short generation times (i.e., microbes). Authors like Durrett and Schmidt focus on smaller populations with longer generations. In this paper we focus exclusively on the classic hominin population, as would give rise to modern man. It is in this type of population where the waiting time problem is most acute, and this is where our expertise in comprehensive numerical simulation is most useful. We do not claim that our findings involving hominin-type populations can necessarily be extended to microbial systems.

In this paper we take a fresh approach to the waiting time problem, using comprehensive numerical simulations that are designed to simulate the mutation/selection process in a biologically realistic manner – taking into account all relevant factors simultaneously. We use biologically realistic numerical simulations to analyze waiting times for the generation and fixation of specific strings of nucleotides of various lengths, given different mutation rates, given different selection pressures, and given different population sizes.

Our goal is not to support the formulations of Behe, or his supporters, or his critics – but simply to bring greater clarity to the waiting time issue. In order to cut through some of the confusion, we use the most easily understood formulation possible, so that both specialists and non-specialists can assess our findings with a reasonable level of understanding. For the purpose of clarity we made our formulation very specific, and we do not claim it is universally applicable. The specific features of our formulation are as follows: 1) we use biologically realistic numerical simulations to study the average waiting time required for the generation and the fixation of a specific string of nucleotides within a specific location of the genome; 2) we design the starting sequence so that it differs from the target sequence at every nucleotide site; 3) we specify that as the starting string begins to mutate, all the subsequent evolving strings are neutral in their fitness effect until an instance of the specified target string arises; 4) we specify that when an instance of the target string arises, a specific fitness benefit (a reproductive advantage) is assigned to each individual carrying that target string, with the target string being fully dominant; 5) the mutation/selection process is faithfully played out according to the standard understanding of the neo-Darwinian process. The only deviation from the classic Darwinian process in our simulation experiments is that once an instance of the target string arises, we protect it from further mutations (as we discuss in the methods section). This practical consideration is conservative, as it can only shorten waiting time.

We expect dialog regarding the details of our formulation. Some may say the string should be able to arise anywhere (genetic context does not matter). Some may say that the rewarded sequence should not have to match the specified target sequence exactly. Some may say that in a specified location, the string might already be there – perhaps lacking only a single new nucleotide. Some may say that each nucleotide in the string should be independently rewarded. In all these special circumstances the waiting time will certainly be shorter than this study indicates. But we are modeling, for the sake of clarity, a simple and well-defined model that is generic in its nature, and is easily executed and understood. One reasonable way to characterize our approach is to describe our target string of nucleotides not as what is required to create a given new gene or function, but rather our target string is what is lacking, as needed to complete a new gene or function that is already serendipitously near completion and only lacks the specified string of mutations. With this definition of our target string we can reasonably require that every nucleotide in our initial string be mutated, that all point mutations be neutral until the string is complete, and every nucleotide must eventually match the corresponding target nucleotide before there is any reproductive benefit for the carrier individual.

When we consider the nature of the problem stated above, it might seem that analytical approximation would be sufficient for estimating the waiting time required for generating and fixing any specific target string. It is true that analytical approximation can be helpful in making very crude estimates of waiting times, but it requires gross over-simplification of the problem and ignores many important biological factors that can profoundly affect time to fixation. As we will show, comprehensive numerical simulation requires fewer simplifying assumptions, and therefore provides estimates of waiting times that are more reliable and more informative compared to analytical approximations.

To do these simulation experiments, we employed the numerical simulation program Mendel’s Accountant (Mendel version 2.4.2, now being released as 2.5). This new version was specifically designed to enable these types of waiting time experiments, and is freely available for download at SourceForge [27]. The current downloadable Mendel version is for Mac computers only, and is not compatible with all Mac operating systems. Alternative versions for use with other computer systems will be available at http://www.mendelsaccountant.info/ [28]. Detailed instructions on how to use Mendel to execute waiting time experiments are also available [28].

Mendel is arguably the first comprehensive, biologically realistic simulator of the mutation/selection process [29–36]. Mendel allows the user to specify all of the major biological variables that affect selection efficiency. Mendel faithfully models our present understanding of the neo-Darwinian mutation/selection process and maintains biological realism at a level that has not been previously possible. Mendel is user-friendly and enables the user to adjust over 25 relevant parameters. Mendel simulates a virtual population, creates new mutations and applies them to individuals, followed by computation of individual fitness, probability selection based on each individual’s fitness, random mating, and then progeny production. This process is repeated for a designated number of generations, with every mutation being tracked. Output is automatically tabulated and plotted, revealing the overall outcome of the mutation/selection process.

Our simulations track the fixation process through three phases. Stage 1 is the waiting time to first instance of the target string within the population. This first instance of the string is almost always a dead-end, being lost from the population shortly after it arises, due to genetic drift. The target string, even though it is beneficial and is subject to amplification via natural selection, must generally arise many times before it will “catch hold” within the population. Stage 2 is the waiting time to that special instance (the effective instance) of the string in the population that escapes extinction due to drift, and so will eventually go to fixation. Stage 3 is the waiting time for the effective instance to be amplified by selection to the point of fixation. Comprehensive numerical simulation is very valuable for fully understanding the waiting time associated with each stage of the generation/fixation process, because all three stages are strongly affected by interacting biological parameters.

Unless otherwise specified, we used the default settings of Mendel’s Accountant. Fertility rate was set at the default setting, such that 50 % of all progeny were selectively eliminated every generation – reflecting very intense selection. The human mutation rate per nucleotide per generation is known to be slightly higher than 10⁻⁸ [10–12]. The mutation rate we used (per string) was calculated by multiplying 10⁻⁸ times the number of nucleotides in the string, times two (because the string is diploid). As needed, the mutation rate was enhanced, as described in the results section.

Mendel was specially modified for this research, allowing us to: 1) define a starting nucleotide string of a given length which resides at a specific location within the virtual genome; 2) define a corresponding target nucleotide string of the same length which may or may not share nucleotides with the starting sequence; 3) cause the initial string to undergo random single nucleotide substitutions (point mutations) at a specified rate; and 4) whenever the target string arises, Mendel confers a reproductive benefit to any individual carrying the target string. In other words, whenever random mutations give rise to the target string, it results in a fitness benefit for all carrier individuals such that those individuals have an enhanced probability of reproduction. This initiates the selective amplification process.

Mendel was modified so that once a string mutates into the target string, that string would be protected from further mutations. Otherwise, when mutation rates are significantly enhanced to allow faster runs, the elevated mutation rates could result in significant back-mutation pressure. This artificial constraint was generous in terms of accelerating the time to fixation, because it prevented any mutational degradation of the target string after it arises.

When natural selection successfully amplified the target string to the point of fixation (i.e., when the allele frequency reached 99–100 %), the experiment was automatically halted and summary statistics were calculated and plotted. The summary statistics include: a) waiting time to first instance of the target string; b) waiting time to the effective instance which will eventually “take hold” and go to fixation; c) duration of the selective amplification phase; d) total fixation time, and e) total number of independent instances of the target string that arose before the final fixation event.

All experiments were replicated 25 times, to determine average waiting times and average number of instances required. The following biological variables were tested to see how they impacted waiting times: a) mutation rate; b) string length; c) degree of fitness benefit; and d) population size.

We have focused on the waiting time problem as it would apply to an evolving hominin population of 10,000-100,000. It is within this type of population where the waiting time problem is most acute, due to long generation time and limited population size. This type of mammalian population is small enough so that it is possible to use comprehensive numerical simulation experiments – simulating all the essential variables simultaneously and producing results that are biologically realistic. While we believe our simulation results are broadly informative, we do not assume our results are directly transferable to microbial organisms. Simulating mega-populations using comprehensive numerical simulators such as Mendel will be possible only when adequate computing resources become available. Therefore we limit all our discussion to a relatively small hominin population.

Our numerical simulations consistently reveal that in such a population the waiting time problem is profound. Even waiting for the fixation of a single point mutation that converts a specific nucleotide within the genome into a specific alternative nucleotide is problematic. The waiting time for the establishment of such a simple event requires on average over 1.5 million years (Table 2). This is a very long time to wait for such a tiny genetic modification of a pre-human genome. It causes us to ask; “Is such a long waiting time credible?”

The ballpark figure for establishing a specific point mutation is easily validated using very straightforward analytical approximation. The human mutation rate is approximately 10⁻⁸ per nucleotide per generation (see [10–12]). This means that the waiting time for a specific nucleotide within a single chromosomal lineage would be 100 million generations. In a diploid population of 10,000 there are 20,000 copies of each nucleotide, so in such a population the waiting time to first mutation would be 20,000 - fold less (5,000 generations). But to have that specific nucleotide mutate into a specific alternative nucleotide (there are three alternatives), takes three times more time – which is 15 thousand generations. But even if the mutation is highly beneficial, on average it has to happen very roughly ten times (given the 10 % fitness benefit), before it “catches hold”. This yields 150,000 generations. Assuming a generation time of 20 years, the waiting time is about three million years. So analytical approximation yields a waiting time of very roughly 3 million years for the establishment of a specific point mutation, given a very strong fitness effect of 10 %. Our numerical simulations yielded a comparable waiting time (1.53 million years), which is in the same ballpark (actually being slightly shorter). This makes it clear that our simulation methodology is not yielding unreasonably long waiting times and is in fact conservative. The problem is not in our numerical simulations - rather it is in the biology. As a general rule, small mammalian populations must wait a very long time for a specific point mutation to arise and be fixed.

There are two primary reasons for such a long waiting time, even for the fixation of a single specific substitution: 1) the mutation rate per nucleotide site is so extremely low; and 2) a particular mutation needs to occur many times before it can “catch hold” in the population. Any analysis that equates waiting time to first instance with total waiting time will be grossly in error. Because we used a very large fitness benefit of 10 % for most of the simulations in this study, we minimized the time required for the mutation to catch hold. Therefore, our reported waiting times are very conservative. Indeed, we have generally coupled our exaggerated fitness benefits with full dominance and zero selection interference (we ignore all the other parts of the genome that would be mutating and segregating, which would otherwise result in selection interference and longer waiting times). Therefore, our results generally represent best-case scenarios in terms of minimizing waiting time. When we use more realistic parameter settings for our simulations, we consistently get much longer waiting times.

When we have reduced the single point mutation’s fitness benefit to a more realistic level of 1 %, waiting time increases ten fold (15.9 million years, rather than 1.5 million years). Given an even more reasonable fitness benefit of 0.1 %, average waiting time was 145 million years. So allowing for a more realistic range of fitness effects, we should more accurately say that even given very substantial fitness effects, the waiting time for a specific point mutation ranges between 1.5 and 15.9 million years. This is consistent with comments by Durrett and Schmidt [16]. They only calculated waiting time to first instance, but at the end of their paper acknowledged that with a 1 % beneficial effect their waiting time to fixation would have been about 100 times longer – due to the need to wait for the effective instance. The need to wait 1.5 – 15.9 million years for the fixation of a particular point mutation is very sobering, since it is estimated that mankind evolved from a chimp-like creature in just 6 million years.

While total waiting time for a particular point mutation to arise and be fixed is surprisingly long in this type of population, the waiting time for any particular string of mutations is vastly longer. Waiting times increase dramatically as we increase the string length (Fig. 2). As shown in Table 2, if an eight-nucleotide string is required, waiting time exceeds the estimated age of the universe.

The severity of the waiting time problem as it applies to specific sets of mutations is due to four levels of constraint. The first level of constraint is that we still have the same low mutation rate per site as with a point mutation, but we need multiple point mutations to arise on the same short strand of DNA, which is very difficult and requires a certain mutation density (which larger population size fails to provide). The second level of constraint is that the average number of mutations that must arise on each short strand of DNA increases more or less exponentially with string length. While for a single point mutation there are three possible mutations (two of which are wrong), for a specific set of ‘n’ linked nucleotide positions there are 4ⁿ possible strings (all of which are wrong - except one). The third level of constraint is that it is the entire completed string that must arise afresh, many different times, to overcome the problem of early extinction due to drift. The fourth level of restraint is that while a population is waiting (through deep time) for the correct string to arise, genetic drift is systematically eliminating almost all the string variants – including most of the necessary intermediate strings between the starting string and the target string. Given a modest population size and such a low mutation rate, genetic drift ensures that there will be only a few string variants within the population in any generation. So almost all mutations must arise within the random string that is currently dominant in the population. Almost all of the time there will be zero or essentially zero strings anywhere in the population that are even close to the target string.

Given optimal settings, what is the longest nucleotide string that can arise within a reasonable waiting time within a hominin population of 10,000? Arguably, the waiting time for the fixation of a “string-of-one” is by itself problematic (Table 2). Waiting a minimum of 1.5 million years (realistically, much longer), for a single point mutation is not timely adaptation in the face of any type of pressing evolutionary challenge. This is especially problematic when we consider that it is estimated that it only took six million years for the chimp and human genomes to diverge by over 5 % [1]. This represents at least 75 million nucleotide changes in the human lineage, many of which must encode new information.

While fixing one point mutation is problematic, our simulations show that the fixation of two co-dependent mutations is extremely problematic – requiring at least 84 million years (Table 2). This is ten-fold longer than the estimated time required for ape-to-man evolution. In this light, we suggest that a string of two specific mutations is a reasonable upper limit, in terms of the longest string length that is likely to evolve within a hominin population (at least in a way that is either timely or meaningful). Certainly the creation and fixation of a string of three (requiring at least 380 million years) would be extremely untimely (and trivial in effect), in terms of the evolution of modern man.

It is widely thought that a larger population size can eliminate the waiting time problem. If that were true, then the waiting time problem would only be meaningful within small populations. While our simulations show that larger populations do help reduce waiting time, we see that the benefit of larger population size produces rapidly diminishing returns (Table 4 and Fig. 4). When we increase the hominin population from 10,000 to 1 million (our current upper limit for these types of experiments), the waiting time for creating a string of five is only reduced from two billion to 482 million years. When we extrapolate our data to a population size of ten million we still get a waiting time of 202 million years. Even when we extrapolate to a population size of one billion we still have a waiting time of 40 million years. This is consistent with Fig. 3 of Lynch [15], which for a string of just two specific mutations (when n = 2), suggests extremely long waiting times in smaller populations, and suggests significant waiting times even in a population of 1 billion. As mentioned in our results section, it is true that a larger population size will always result in more mutations in less time, but it does not result in higher mutation-densities (more mutations arising in the same small linkage block of DNA), which is a critical factor limiting formation of specific nucleotide strings.

The only way around the profound waiting time problem within a hominin-type population is to invoke special, atypical circumstances. Different authors have invoked a variety of special circumstances to reduce waiting times. These special circumstances include: a) assuming fixation of the first instance of a string [16]; b) special strings with reduced context-dependence such as a protein binding-site [16, 17]; c) strings that are largely already in place, and only require one or two new mutations [15–17]; d) incomplete strings that are still considered beneficial and can be rewarded with a significant fitness benefit [15–17]; and e) extremely large population sizes [17]. These special circumstances can sometimes be honestly invoked to enable instances with greatly reduced waiting times, but they are not generically applicable, and so cannot be used as a general resolution to the waiting time problem. The generic waiting time problem remains unresolved.

The results of other researchers [20–24] are generally consistent with our own results. Interestingly, even the previous studies that have strongly argued against the waiting time problem [15–17] still indicate that for a hominin population, the fixation of two co-dependent mutations is extremely problematic. For example, in the analysis of Durrett and Schmidt [16], they studied the waiting time to first appearance (first instance) of various string types within a hominin-type population (a population essentially identical to our own simulated population and with exactly the same mutation rate). However, a specific formulation of the problem was chosen, designed for the special case of a protein-binding (regulatory) site. Several special cases were examined involving either reduced context constraint (many possible genomic sites), or reduced specificity restraint (incomplete strings are beneficial and selectable), or cases where the target string was already nearly complete (lacking only 1–2 nucleotide changes). So those results are only marginally comparable to the third column in all our tables (our time to first instance). In all these special cases, one would naturally predict significantly shorter waiting times than we report here. Yet for a string of 8, when a perfect match was required, they still calculated a waiting time to first instance of 650 million years. For a beneficial effect of 1 % they estimate that the time to the effective instance (followed by final fixation), would be about 100-fold higher (this would be about 65 billion years). Their results, when adjusted as they prescribe, make our own findings for a string of 8 seem quite modest (just 18.5 billion years). The primary reason our waiting time was less than their corrected waiting time was apparently because we used an over-generous fitness benefit 10 times stronger than what they were assuming.

In a second paper by Durrett and Schmidt [17], they examined a more limited problem – how long does it take to create two co-dependent mutations within a hominin population. Again, their formulation was designed for a special circumstance – switching a protein-binding site to a relatively non-specific alternative location. Yet their calculations indicated the average waiting time for establishment of these two mutations was still 216 million years (their simulations suggested a somewhat shorter time – 162 million years). Their waiting times again appear to be substantially longer than our own average waiting time for two co-dependent mutations (84 million years – Table 2). This again appears to be primarily because we used a ten-fold stronger fitness benefit. Their data is in good agreement with our own waiting time for two co-dependent mutations when we reduced our fitness benefit to a more reasonable 1 %. We then observed a waiting time of 270 million years (Table 3), which is in the same ballpark as their findings.

A paper by Lynch [15] suggests that the waiting time problem is not particularly serious – at least not for microbial populations. Yet that same analysis indicates a very significant waiting time problem for smaller populations. Figure 3 of that paper plots the waiting times for “neofunctionalization” (creation of a new function) after a gene duplication. When two mutations were required (needing to arise at two specific locations, such that n = 2), and when population size was 10,000, the waiting times goes off scale – exceeding 1 billion generations (which for a hominin population would be 20 billion years). Even with a population size of 10 million, this same plot seems to suggest that when n = 2 the waiting time would be prohibitively long for a model Hominin population - in the range of 10–100 million generations (i.e., 0.2 to 2 billion years). That paper’s formulation of the problem which is closest to our own (where fixation of two specific mutations are required within a population of 10,000) seems to suggest waiting times much longer than what we see in the current study. This again appears to largely be due to our use of such a strong fitness benefit - reflecting our decision to primarily model best-case scenarios.

The most recent attempt to resolve the waiting time problem is the paper by Lynch and Abegg [25]. The findings of that paper have been challenged by Axe [21]. The paper by Lynch and Abegg begins by acknowledging that the waiting time problem should be of great interest to the evolutionary community (“A central problem in evolutionary theory concerns the mechanisms by which adaptations requiring multiple mutations emerge in natural populations.”) These authors then suggest they have largely resolved that problem. However, that paper again suggests that for a hominin-type population, waiting times are much longer than we report in this paper. Figure 1 in that paper suggests that to establish two specific co-dependent mutations in a population of 10,000 (given that the first mutation to arise is neutral) requires roughly 10–100 million generations. In a hominin population this would be roughly 0.2 to 2 billion years – just to fix two specific mutations. Figure 1 in that paper also suggests that for the same population of 10,000, when the intermediate mutation has a deleterious effect of 1 %, the waiting time is nearly 100 billion generations (2 trillion years).

Lynch and Abegg [25] go on to invoke 3 special atypical circumstances (such as hyper-mutation) that might reduce waiting times, none of which would be generally applicable in our case (i.e., a hominin population of 10,000). Even when these special circumstances were assumed, prohibitively long waiting times were indicated for a hominin-type population of 10,000 (see Fig. 3 in that paper), consistently exceeding the waiting times we report here.

Lastly, Lynch and Abegg [25] also analyzed the waiting time required for strings longer than 2 nucleotides. They argue that beyond two mutations, longer string length has only a marginal effect on waiting time, especially in small populations. This conclusion is very counterintuitive and is strongly contradicted by our own findings (see our Fig. 2). Similarly, the findings of Durrett and Schmidt [16], and Axe [21], seem to directly contradict that claim. Even if the claim by Lynch and Abegg regarding string length were valid, all of their waiting times for strings longer than two were extremely prohibitive for a population of 10,000 (see Fig. 4 of that paper – also note, that plot employed a heightened fitness benefit of 2 %).

In light of all this, it is clear that there are multiple lines of evidence that support our findings. Firstly, our own mathematical approximations make us very confident that our simulations are yielding waiting times that are in the right ballpark (see above). Secondly, numerous other researchers have come to similar conclusions [13, 14, 20, 21, 23, 24, 26]. Lastly, the long waiting times we report here are even supported indirectly by the papers that have argued against a serious waiting time problem [15–17, 25]. When examined carefully, even those papers indicate that for a hominin-type population, waiting times are as long or even longer than we report here. Therefore, the theoretical evidence is very strong that there is a significant waiting time problem in any small hominin-type population, and this problem does not necessarily go away as the population gets larger.

When we began this research, our preliminary calculations and simulation experiments revealed to us that our observed waiting times were going to be extremely long. For this reason we tried to be conservative in terms of all our parameters. Firstly, we chose to use proportional correction of our elevated mutation rates, rather than correcting for the observed longer waiting times associated with drift-induced string homogenization. This would have yielded 10-fold longer waiting times (Table 1, Fig. 1). Secondly, we chose to use an extremely high fitness benefit of 10 %, rather than more realistic fitness effects – which would have yielded 10–100 fold longer waiting times (Table 3). Thirdly, we chose to use full dominance, rather than partial dominance – which would have yielded several-fold longer waiting times (data not shown). Fourthly, we disregarded all genetic variants within all other parts of the genome – which would have resulted in significant selection interference and much longer waiting times. Competing beneficial mutations [32] and deleterious mutations [37] both cause serious selection interference because simultaneous selection at countless other sites in the genome confounds selection for the target string. Ignoring all the other segregating sites in the genome greatly simplifies the waiting time problem, and more to the point, current computation capabilities are simply not able to track all mutations arising within the genome through such deep time. However, based upon previous studies [7, 32, 36], we know that selection interference at the genomic level can be severely limiting, and should greatly increase waiting times. For all these reasons, we view the analyses reported here as being a series of best-case scenarios that yield waiting times which are very conservative.

Which method of analyzing the waiting time problem is better - analytical approximation or comprehensive numerical simulation? We conclude that both methods can, and in this case do, yield very similar results. Regarding the subject of this research, both methods clearly show prohibitively long waiting times for establishing even the shortest nucleotide strings within a model hominin population of 10,000–100,000 individuals.

Comprehensive numerical simulation is a new research and teaching tool, and is useful for enhancing our understanding of population dynamics. This tool can be used to study population scenarios that might be too specific or too complex for the traditional methodology of employing analytical calculations (which require, by necessity, many simplifying assumptions). In terms of increasing clarity, comprehensive numerical simulation allows us to directly observe the detailed unfolding of the mechanistic process of mutation/selection. In this way comprehensive numerical simulation can counterbalance the very high degree of abstraction inherent in attempting to reduce complex population dynamics to simple mathematical formulas.

Comprehensive numerical simulation entails genuine empirical experimentation. Mendel experiments produce outcomes that are neither programmed nor formulated, but are truly emergent in nature – being solely dependent on the interaction of the numerous biological factors. This can produce results that are both unanticipated and instructive. Comprehensive numerical simulation has the potential to bring us greater clarity of understanding of complex population dynamics, and may even prompt the re-evaluation of some long-held prior assumptions. For these reasons, we believe comprehensive numerical simulation can serve to expand upon and complement the traditional analytical approach. These two methods can be used to enhance each other, providing independent testing, correction, validation, and refinement.

We have used comprehensive numerical simulations to show that in populations of modest size (such as a hominin population), there is a serious waiting time problem that can constrain macroevolution. Our studies show that in such a population there is a significant waiting time problem even in terms of waiting for a specific point mutation to arise and be fixed (minimally, about 1.5 million years). We show that the waiting time problem becomes very severe when more than one mutation is required to establish a new function. On a practical level, the waiting time problem greatly inhibits the establishment of any new function that requires any string or set of specific linked co-dependent mutations. We show that the waiting time problem becomes more extreme as string length increases, as fitness benefit decreases, and as population size decreases. In a population of 10,000 the establishment of a string of just two specific co-dependent mutations tends to be extremely problematic (conservatively requiring an average waiting time of at least 84 million years). For nucleotide strings of moderate length (eight or above), waiting times will typically exceed the estimated age of the universe – even when using highly favorable settings. Many levels of evidence support our conclusions, including the results of virtually all the other researchers who have looked at the waiting time problem in the context of establishing specific sequences in specific genomic locations within a small hominin-type population. In small populations the waiting time problem appears to be profound, and deserves very careful examination. To the extent that waiting time is a serious problem for classic neo-Darwinian theory, it is only reasonable that we begin to examine alternative models [39, 40] regarding how biological information arises.