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 . 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 % . 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 , 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 ; 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 . 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 , 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 , 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  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 . The findings of that paper have been challenged by Axe . 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  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  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 , and Axe , 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  and deleterious mutations  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.