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Not only does the combination of improbable factors demonstrate the untenable nature of new species originating by beneficial mutation, but we can also use take the numbers generated by evolutionary scientists themselves to see whether or not the mechanism of beneficial mutation is valid. Specifically, we want to answer the question of whether or not beneficial mutation is capable of generating the existing number of species (kinds of organisms) within the timescale allotted by evolutionary theory.

As we begin, it is important to note for reference that genes come in complimentary pairs, one on each chromosome in a chromosome pair. These complimentary genes are called alleles. Conversely, an allele is simply another term for an individual gene. And each allele codes for either the same or a different version of the trait that both genes control, such as “smooth” or “wrinkled” surface texture in peas.

“Allele – 1: any of the alternative forms of a gene that may occur at a given locus

2: either of a pair of alternative Mendelian characters (as smooth and wrinkled seed in the pea).” – Merriam-Webster’s Collegiate Dictionary

According to evolutionary sources, for just 1 new, mutated allele, just 1 mutation, to become present in even 1 percent of a population would take between 2,000-10,000 generations depending upon how prominent the original gene was in the population before it mutated.

“Evolution, Dynamics of genetic change, Processes of gene frequency change

Mutation – The allelic variations that make evolution possible are generated by the process of mutation; but new mutations change gene frequencies very slowly, since mutation rates are low. Assume that the gene allele A 1 mutates to allele A 2 at a rate m per generation, and that at a given time the frequency of A 1 is p…If the mutation rate is 10 [to a power of -5] (1 in 100,000) per gene per generation, about 2,000 generations will be required to change the frequency of A 1 from 0.50 to 0.49 and about 10,000 generations to change it from 0.10 to 0.09.” – Encyclopaedia Britannica 2004 Deluxe Edition

Notice 5 points from the quote above.

First, notice that the term “frequency” refers to the amount or percentage of a population that carries a particular allele.

Second, notice that these estimates are based upon the lowest end of the range for mutation rates in gametes. The estimates use 1 out of 100,000 rather than the much slower 1 out of 1,000,000 or even the lower rate among unicellular organisms.

“Evolution, The process of evolution, Evolution as a genetic function, The origin of genetic variation: mutations, Gene mutations – Mutation rates have been measured in a great variety of organisms, mostly for mutants that exhibit conspicuous effects. Mutation rates are generally lower in bacteria and other microorganisms than in more complex species. In humans and other multicellular organisms, the rate typically ranges from about one per 100,000 to one per 1,000,000 gametes.” – Encyclopaedia Britannica 2004 Deluxe Edition

Third, notice that A1 is the original allele and A2 is the new, mutant form of that allele. If A1 was only present in one-half (0.5) of the population to begin with, then it will take 2,000 generations for A2 to cause a shift in the amount of A1 in the population. And if A1 was only present in one-tenth (0.1) of the population to begin with, then it will take 10,000 generations for A2 to cause a shift in the amount of A1 in the population.

Fourth, notice that the decreasing shift described in A1’s frequency among the population is only 1 percent. In the case where 50 percent of the population starts with A1, it will take 2,000 generations for A1 to decrease in frequency from 0.50 to 0.49, which is just 1 percent. In the case where 10 percent of the population starts with A1, it will take 10,000 generations for A1 to decrease from 0.10 to 0.09, which is also just 1 percent. Conversely, the loss of A1 in 1 percent of the population is caused as A2 mutates from A1 taking A1’s place. Consequently, the estimates of 2,000 and 10,000 generations are also the amount of time it would take for the mutation, A2, to go from zero to 1 percent of the population.

Fifth, notice that the estimate does not indicate how many generations before a new species is produced. Nor does it indicate how many years it will take before a new species is produced. The estimate above simply designates how many generations it will take for a new mutation of a single gene to reach 1 percent of the population depending on what percentage of the population already has the original allele before it mutates and assuming that the same mutation occurs consistently in every member of the population with the original allele in every generation.

In contrast, we want to use the numbers and formulas in Britannica’s estimate as a basis for determining whether or not beneficial mutation is capable of producing all the species that exist today within the timescale allotted by evolutionary theory. In order to accomplish this goal, we need to take a few preliminary steps.

First, we’ll need to compile a list of the factors that were asserted by these common reference sources as relevant to the rate of mutation and speciation. The first 2 factors can be found in Britannica’s estimate.

1) Mutation rate in the gametes.
2) Starting frequency of original allele.

However, Britannica was not attempting to estimate the number of species that would be produced by beneficial mutation within a certain number of years, but instead Britannica was only attempting to present how slowly a mutation circulates throughout a population in terms of generations. Consequently, Britannica’s estimate only incorporated 2 factors: 1) the rate of general mutations in the gametes and 2) the frequency of the original allele among the population before it mutates. But the sources we’ve already examined, including Britannica, have asserted a number of other factors that are directly relevant to the rate of speciation itself, not just the rate and circulation of mutations calculated by Britannica. In other words, once we begin to apply Britannica’s estimates toward the goal of identifying the rate of speciation, we need to include some factors that Britannica did not. The first considerations for our equation involve the 2 factors, which Britannica did include. Because of the way these 2 items factor into the equation, we will cover them in reverse order.

Factor A: The Initial Gene Frequency

Concerning Britannica’s second factor, the starting frequency of the original allele before it mutates, there are a few items worth noting.

First, whether Britannica chooses 50, 10, or any other percentage of the population to start with, that percent is set according to how many members of the population have the original allele that is experiencing mutation. In other words, it is the number of individuals which have the critical allele that determines what the size of the starting percentage is. Second, whether starting from 50 percent or 10 percent of the population, the end result in both cases is a 1 percent portion of the population with the new mutation. Third, because the original allele is the basis of the mutant allele, this 1 percent is necessarily within either the 50 or the 10 percent that had the original allele and, therefore, served as the base group for the equation. Fourth, although the calculation sought to determine how many generations would be required before the mutation reached 1 percent of the population, Britannica could just as easily have gone further to determine how many generations would be required for the mutation to reach the full 50 percent that had the original allele or the full 10 percent that had the original allele. But instead, the amount of time required to reach just 1 percent was chosen in order to emphasize Britannica’s point concerning how long it takes for a mutant gene to become distributed throughout a population. However, using Britannica’s essential formula, we could just as easily start from just 1 percent of the population instead of 10 or 50 percent. And we could likewise ask how many generations would be required for the mutation to circulate throughout the entire 1 percent.

Britannica can set their hypothetical starting group to any percentage in order to demonstrate how different starting percentages affect the number of generations. However, our calculation cannot be as versatile. This is due to the fact that we are attempting to inquire farther than Britannica. In order to demonstrate the slow rate at which mutations rise in a population, Britannica only needs to track how long it will take for just 1 mutation to accumulate. However, our equation will calculate the speed of speciation, a process which requires the accumulation of multiple mutations, not just 1. And therefore, our equation needs to have Britannica’s base process repeat over and over again to represent the accumulation of not just 1 but many mutations by the same members of a population.

The reason for this is simple. The addition of other mutations to members of the population outside that 1 percent, who do not possess the previous mutation, will not produce the accumulation of new genes necessary to bring about a new species. Just as we saw that Britannica’s starting percentage had to reflect the number of members with a particular critical allele, our calculation must do the same. However, since a species (or new kind of organism) is built by the accumulation of multiple new (mutant) alleles, our starting percentage is defined by the number of members that acquired the previous mutation. And in Britannica’s calculation, the number of individuals who have acquired the new mutation by the end of the equation is 1 percent. Consequently, it will be a base of 1 percent that we will have to start from, rather than the base of 50 or 10 percent used for the purposes of an alternate, more general demonstration by Britannica.

Here our equation will follow Britannica’s model directly. Just as the resulting 1 percent in Britannica’s equation was within the starting 50 or 10 percent, in the same way the resulting group that receives the next mutation will be within the starting 1 percent.

And just as Britannica could have calculated how many generations it would have taken the new mutant to reach the entire 50 percent or the entire 10 percent, rather than just reaching a level of 1 percent, our equation will track how many generations it will take for the new mutant to permeate that entire 1 percent. In this way, we will be using Britannica’s formula to track how many generations it will take a second mutation to accumulate in the 1 percent of the population that possesses the previous mutation.

And there is an additional reason to use 1 percent: a low target quantity, such as 1 percent, is favorable to evolutionary theory. Effectively, in the equation, this percentage represents the individual organisms from which the newly forming species (or kind of organism) will emerge. The larger the percentage that we require, the more generations it will take for mutations to be accumulated and the longer the process of evolution would require. By requiring only 1 percent, we not only start with the exact percentage that results from Britannica’s calculations already, but we give evolution the maximum amount of time to occur. And so choosing a low target helps evolution by requiring the least amount of time for a new species to emerge.

Lastly, it should be noted that 1 percent is simply a portion of the population, not an exact numerical amount. This is the case in Britannica’s original equation. In this way, it doesn’t matter what the original size of the population is in terms of actual numbers. No matter how large or how small the population is, according to Britannica, it will still take 2,000 generations for a normal allele in 50 percent of that population to change to a mutated allele in just 1 percent and 10,000 generations if the normal allele starts in only 10 percent of the population. As such, our use of 1 percent is shown once again to be a generous assumption in favor of evolution. Since no actual numeric amounts are specified by Britannica or our equation, 1 percent truly is the lowest amount of the population we can require a mutation to reach, which in turn will take the shortest amount of time for mutations to accumulate toward the formation of a new species.

With these critical explanations out of the way, the essential question arises. If Britannica calculates a 50 percent starting gene frequency will require 2,000 generations and a 10 percent starting gene frequency will require 10,000 generations, how do we know how many generations a 1 percent starting frequency will require?

The answer comes by noticing a simple correspondence within Britannica’s figures. It’s no secret that Britannica is trying to demonstrate that the starting percentage directly affects the number of generations required for the new mutation to accumulate to 1 percent. In fact, if we look at Britannica’s calculation, the number of generations increases and decreases in direct proportion to any increase or decrease of the starting percentage. If the starting percentage is divided by 5 from 50 percent down to 10 percent, the number of generations also multiplies by a factor of 5, from 2,000 up to 10,000 generations. Thus, it would seem very simply that by whatever number you decrease the size of the starting percentage, the number of generations multiplies by the same number. Consequently, by moving the starting percentage from 10 percent down to 1 percent, we would simply need to multiply the number of generations by a factor of 10 as well. And if we multiply Britannica’s 10,000 generations by a factor of 10, the result is 100,000 generations. In other words, according to Britannica’s basic formula, starting from 1 percent of the population, it would take 100,000 generations for the next mutation to fully accumulate in that same 1 percent.

Factor B: The General Mutation Rate

Concerning Britannica’s first factor, the mutation rate in the gametes, we noted earlier that Britannica stipulated mutation rates in multicellular organisms, such as animals, ranged from 1 out of 100,000 to 1 out of 1,000,000 gametes.

Then Britannica chose the fastest rate from this range, 1 out of 100,000, as the basis of its estimates, which has the effect of speeding up the timeframe for circulating a new, mutant gene among the population.

“Evolution, Dynamics of genetic change, Processes of gene frequency change Mutation – The allelic variations that make evolution possible are generated by the process of mutation; but new mutations change gene frequencies very slowly, since mutation rates are low. Assume that the gene allele A 1 mutates to allele A 2 at a rate m per generation, and that at a given time the frequency of A 1 is p…If the mutation rate is 10 [to a power of -5] (1 in 100,000) per gene per generation, about 2,000 generations will be required to change the frequency of A 1 from 0.50 to 0.49 and about 10,000 generations to change it from 0.10 to 0.09…Changes in gene frequencies due to mutation occur, therefore, at even slower rates than was suggested above, because forward and backward mutations counteract each other.” – Encyclopaedia Britannica 2004 Deluxe Edition

To be more in line with the full range of rates, we’ll select a rate that is in the middle of the two extremes. Instead of 1 out of 100,000 or 1 out of 1,000,000, we’ll use 1 out of 500,000, which is right in the middle of the two, effectively the average rate of mutation. This decrease is half of 1 order of magnitude. Ordinarily, a decrease of this nature would probably affect the rest of the calculation exponentially. Specifically, if mutation rates in the gametes were to change from 1 out of 100,000 to a slower rate of 1 out of 500,000, the result would probably be an exponential increase in number of generations required to circulate the new gene. But leaving aside the issue of exponential increase, one thing is certain. If the rate is increase from 100,000 to 500,000, which is an increase of 500 percent, the resulting number of generations simply cannot decrease or remain the same, but must also increase by at least 500 percent as well. And, in the effort to remain favorable to evolutionary theory and to keep things easy to follow, rather than adjusting for an exponential change, since we simply multiplied the rate by 5 from 100,000 to 500,000, we’ll simply multiply the original estimate by a factor of 5.

Factor A, described above, multiplied Britannica’s 10,000 generations (which was resulted from a starting based of 10 percent) by a factor of 10 to reflect a starting base of only 1 percent. The result was 100,000 generations. Factor B multiplies those 100,000 generations by a factor of 5 to reflect the moderate middle range of mutation rates rather than the fastest end of the range, which is used in Britannica’s calculations. The result is 500,000 generations.

This addresses the issues concerning the 2 factors utilized in Britannica’s equation to illustrate the slow speed at which even 1 mutant allele accumulates in a population. However, as noted earlier, the quotes in the previous segment outlines several other factors that were not included in Britannica’s estimates. We now turn our attention to those remaining factors.

Factor C: “Backward” Mutation Slow Down

As indicated previously, forward and backward mutations counteract each other. Taking note of this and commenting on its original calculations, Britannica states that “Changes in gene frequencies due to mutation occur at even slower rates than was suggested above.”

“Evolution, Dynamics of genetic change, Processes of gene frequency change

Mutation – The allelic variations that make evolution possible are generated by the process of mutation; but new mutations change gene frequencies very slowly, since mutation rates are low. Assume that the gene allele A 1 mutates to allele A 2 at a rate m per generation, and that at a given time the frequency of A 1 is p…If the mutation rate is 10 [to a power of -5] (1 in 100,000) per gene per generation, about 2,000 generations will be required to change the frequency of A 1 from 0.50 to 0.49 and about 10,000 generations to change it from 0.10 to 0.09…Changes in gene frequencies due to mutation occur, therefore, at even slower rates than was suggested above, because forward and backward mutations counteract each other.” – Encyclopaedia Britannica 2004 Deluxe Edition

Once again, we want to be more than fair and to, in fact, be favorable to evolution concerning this factor. Consequently, we will only assume a slight slow down in the process. Let’s do a few examples to illustrate how this factor will affect the equation.

If we were to assume that the frequency of “backward” mutation was so high that 1 out of every 2 mutations were itself a backward mutation, reversing a previous mutation, then each forward mutation would be restored to the original gene, and the process of evolution would never occur. Consequently, to make sure the calculation gives evolution a fair chance of working, we’ll assume a lower rate of “backward” mutations than 1 out of every 2 mutations.

For illustration purposes, suppose we were to assume that the frequency of “backward” mutations was 1 out of every 3 mutations. Thus, 2 of the 3 mutations would be normal “forward” mutations and the third would be “backward” and would reverse 1 of the other 2 mutations. If the mutation rate were 1 out of 100,000 gametes this would mean it would take 300,000 mutations to produce just 1 forward mutation, which could build toward the production of a new species. This would allow the process of change by mutation to continue forward but at a rate that is three times slower, effectively one third of the original rate. As we can see, factoring this into our equation would simply require multiplying the original ratio of mutation by 3.

We can learn more about the impact of this factor by decreasing the frequency of “backward” mutations to a ratio of 1 out of 4 rather than 1 out of 3. In this scenario, every 400,000 gametes would produce 4 mutations, of which 3 were “forward” and 1 was a counteracting “backward” mutation, which nullified one of the 3 “forward” mutations. This would effectively result in 2 “un-reversed” “forward” mutations every 400,000 gametes. This mathematically reduces to a ratio of 1 out of 200,000 instead of the 1 out of 100,000 rate that Britannica started with. Consequently, a 1 out of 4 “backward” mutation ratio would require multiplying our equation by a factor of 2. So, we can see that a “backward” mutation ratio of 1 to 3 reduces the rate to one-third, or 33 percent, of the original speed and a “backward” mutation ratio of 1 out of 4 reduces the rate to one-half, or 50 percent, of the original speed.

With these patterns in mind, we can see the following. A “backward” mutation ratio of 1 out of 6 results in 4 “un-reversed” forward mutations for every 600,000 gametes. This reduces to 2 out of 300,000 and eventually to 1 out of 150,000 gametes. Consequently, we would simply need to multiply the original 1 out of 100,000 rate by a factor of 1.5, effectively two-thirds or 67 percent, of the original speed. Likewise, a “backward” mutation ratio of 1 out of 8 results in 6 “un-reversed” forward mutations for every 800,000 gametes. This reduces to 3 out of 400,000 and eventually to 1 out of 133,333 gametes. Consequently, we would simply need to multiply the original 1 out of 100,000 rate by a factor of 1.33, effectively three-quarters, or 75 percent, of the original speed. And finally, if there were only 1 “backward” mutation in every 10 mutations that would result in 8 “un-reversed” forward mutations in every 1,000,000 gametes. This reduces to 4 out of 500,000 gametes or 1 out of 125,000. And consequently, we would simply need to multiply the original 1 out of 100,000 rate by a factor of 1.25, effectively four-fifths, or 80 percent, of the original speed.

But again, we want to be generous to evolutionary theory and to keep things simple. So, for the purposes of our equation, we will assume a “backward” mutation ratio of 1 out of every 22 mutations, which results in 20 “un-reversed” “forward” mutations for every 2,200,000 gametes. This reduces to 10 out of 1,100,000 gametes and eventually to 1 out of 110,000 gametes. Consequently, we simply need to multiply the original 1 out of 100,000 rate by a factor of 1.1, effectively ten-elevenths or 91 percent, of the original speed. This is the ratio we will assume for “backward” mutations. We will assume that 21 out of every 22 mutations are “forward” mutations and only 1 is a counteracting “backward” mutation. This slows down the process only by less than 10 percent and requires us to multiply our equation by a factor of 1.1.

Factor D: Beneficial Mutation Ratio

Britannica’s original estimates operated as though all mutations produced by the ratio of either 1 out of 100,000 or 1 out of 1,000,000 were beneficial mutations when in reality, those were simply mutation rates for mutations in general, which as we have seen are predominantly harmful, lethal, or negligible. We know that Britannica’s equation was assuming that all of the mutations were beneficial because Britannica’s equation operated on the assumption that the same mutation survived for 2,000 to 10,000 generations. Because they result in no survival advantage, mutations that are either harmful, lethal, or negligible are eliminated by natural selection and do not last for thousands of generations.

“Evolution, Causes of evolutionary change – Mutations occur regularly but are usually infrequent, and most of them produce unfavorable traits…In most cases, such mutant genes are eliminated by natural selection because most individuals that inherit them die before producing any offspring.” – Worldbook, Contributor: Alan R. Templeton, Ph.D., Rebstock Professor of Biology, Washington University.

Consequently, only beneficial mutations, mutations that actually provided a manifested advantage, survive natural selection to rise in frequency.

“Gene – The mutation generally has little or no effect; when it does alter an organism, the change is frequently lethal. A beneficial mutation will rise in frequency within a population until it becomes the norm.” – Encyclopaedia Britannica 2004 Deluxe Edition

“Heredity, Heredity and evolution, The gene in populations, The Hardy–Weinberg principle – In 1908, Godfrey Harold Hardy and Wilhelm Weinberg independently formulated a theorem that became the foundation of population genetics. According to the Hardy–Weinberg principle, two or more gene alleles will have the same frequency in the gene pool generation after generation, until some agent acts to change that frequency.” – Encyclopaedia Britannica 2004 Deluxe Edition

“Heredity and evolution, Selection as an agent of change, Natural selection and Darwinian fitness – Sexual reproduction under simple (Mendelian) inheritance is a conservative force that tends to maintain the genetic status quo in a population. If a gene frequency is 1 percent in a population, it tends to remain at 1 percent indefinitely unless some force acts to change it. Outside of the laboratory, the most powerful force for changing gene frequencies is natural selection.” – Encyclopaedia Britannica 2004 Deluxe Edition

“Species – Interbreeding only within the species is of great importance for evolution in that individuals of one species share a common gene pool that members of other species do not. Within a single pool there is always a certain amount of variation among individuals, and those whose genetic variations leave them at a disadvantage in a particular environment tend to be eliminated in favour of those with advantageous variations. This process of natural selection results in the gene pool's evolving in such a way that the advantageous variations become the norm.” – Encyclopaedia Britannica 2004 Deluxe Edition

Therefore, since Britannica’s equation operates on the assumption that all mutations are beneficial, in order to properly calculate the possibility of speciation occurring by beneficial mutation, we need to factor into our equation how rare beneficial mutations are in comparison to mutations in general. This is an only an assumption and so we will err on the side of being favorable to evolutionary theory. As a result, we’re going to generously assume that at least 1 out of every 20 mutations is beneficial. This in turn requires us to multiply our equation by a factor of 20. Incidentally, it should be noted that here again we’re favoring evolutionary theory by assuming beneficial mutations are more frequent than backward mutations, to which we assigned a ratio of 1 out of 22.

Factor E: Environmental Considerations

Microsoft Encarta denoted that mutated alleles are usually recessive, which means that their traits are not expressed unless the genes in the complimentary pair are both the mutant allele.

“Evolution, XI MUTATIONS, A Gene Mutation – Mutations are usually recessive, and their harmful effects are not expressed unless two of them are brought together into the homozygous condition. This is most likely to occur as a result of inbreeding, the mating of closely related organisms that may have inherited the same recessive mutant gene from a common ancestor.” – "Genetics," Microsoft® Encarta® Encyclopedia 99. © 1993-1998 Microsoft Corporation. All rights reserved.

This is important because, as stated earlier, if an organism only has 1 mutant allele and the trait will not be expressed, then there will be no resulting advantage and the trait will likely be removed by natural selection. Even in the immediately preceding segment describing Factor D, we noted that only mutations that actually produce a benefit will survive natural selection to rise in frequency from generation to generation. And even further information on this point will be presented under Factor F below.

However, Britannica’s calculation operates on the assumption that all mutations produced by the general mutation rates will automatically survive long enough to last for 2,000 to 10,000 generations without being removed by natural selection. Thus, Britannica’s calculation does not include this factor at all. Even as indicated by the quote immediately above, the only way that both complimentary alleles in an organism could be the same mutation is if that mutation was passed on in the individual gamete inherited by both parents during reproduction. This is especially true since in our equation we will focus on the animal kingdom as the focal group of our calculation. And animals predominantly reproduce by sexual reproduction, in which half of the alleles are received from each parent during fertilization.

“Animal – Primitive members of all major taxa of animals reproduced sexually, and virtually all animals still do at some time or another.” – Encyclopaedia Britannica 2004 Deluxe Edition

Consequently, if the same mutation has to be present in both gametes that participate in fertilization and the mutation rate in each parent is 1 out of 100,000 normally we would have to multiply 100,000 x 100,000 to create the combined ratio of occurrence. It is the same as if we had 2 coins and asked what the odds were of getting heads-up on both coins. The odds of getting heads-up on either coin individually are 1 out of 2. But when the 2 coins are used together, there is only 1 chance in 4 that the coins will both come up heads. Consequently, the ratio of 1 out of 2 is multiplied by itself to create a combined ratio of 1 out of 4. Similarly, the odds of getting a 1 using one six-sided dice are 1 out of 6. But the odds of getting ones on 2 separate six-sided die is 1 out of 36. The “6” in the ratio is multiplied by itself to create the new combined ratio. In the same way, if the odds of 1 mutation in the gametes are 1 out of 100,000, then the odds of getting the same mutation in 2 gametes is 1 out of 100,000 x 100,000, or 1 out of 10,000,000,000. Here we’re going to be so enormously favorable to evolutionary theory that if there is somehow any unfairness in any of the rest of our equation, this one factor is enough to more than make up for it.

Rather than assuming a ratio of 1 out of 10 billion for both gametes to have the same mutation, which would require squaring our current calculations, we’re instead simply going to assume a ratio of 1 out of 4. This simple ratio reflects the principles of a basic Punnett’s square. A Punnett’s square is a means of arranging the 2 alleles from each parent to display the ratio of the potential combinations of those 4 total alleles in the offspring. For illustration, Britannica provides the following example using flowers in their article on “Heredity” under the subheading “Basic features of heredity.”

(Caption) “Figure 1: Mendel's law of segregation. From T. Dobzhansky, "Evolution, Genetics, and Man (1955);" John Wiley and Sons, Inc.” – Encyclopaedia Britannica 2004 Deluxe Edition

Britannica provides the following explanation of this diagram.

“Heredity, Basic features of heredity, Early conceptions of heredity – An example of one of Mendel's experiments will illustrate how the genes are transmitted and in what particular ratios. Let R stand for the gene for purple flowers and r for the gene for white flowers (dominant genes are conventionally symbolized by capital letters and recessive genes by small letters).” – Encyclopaedia Britannica 2004 Deluxe Edition

For our study, there are 4 features from this chart that are important. First, notice that 2 separate generations are being depicted here as indicated by the designations “F1” and “F2.” Second, notice that although the plant is self-pollinating, both a male and the female “parent” is represented, which is designated by the male and female symbols in the upper left hand corner of the diagram. Thus, the situation is analogous to sexual reproduction involving one male and one female parent in animals. Third, notice, particularly from Britannica’s explanation that the capital letter R represents dominant allele and the lowercase “r” represents the recessive allele. And fourth, notice that the female component, represented vertically on the left-hand side by the designation “ovules” and the male component, represented horizontally by the designation “pollen” across the top, are both comprised of one dominant allele and one recessive allele, which they are potentially contributing to the offspring. As the results of the square demonstrate, particularly the white flower in the lower right-hand corner, when both parents contain one dominant and one recessive allele, there is precisely a 1 in 4 chance that the offspring will contain both recessive alleles.

This translates directly to the assumption used in our equations, except that in the case of the flowers, the dominant and recessive alleles represented the specific trait of coloration, whereas in our equation, the dominant allele represents any normal, pre-mutated allele and the recessive allele represents the recessive mutation. Consequently, applying this basic genetic principle, if each parent has 1 mutant allele and one normal allele, the Punnett’s square demonstrates a simple 1 in 4 chance of the offspring having both recessive, mutant alleles.

(The graphic below, Mutation Figure 1, is a Punnett’s square depicting the distribution of recessive mutant alleles. In the graphic, capital "A" designates the original, unmutated allele and lowercase "a" designates the mutated form of this allele.)

Consequently, in order to keep the process as simple as possible as well as remaining favorable to the prospects of evolutionary theory, we will simply assume a ratio of 1 out of 4 to account for the role that the recessive nature of mutations plays in the likelihood of speciation by beneficial mutation. This will also prevent discrediting our equation by including such an astronomical factor as multiplying a ratio of 1 out of 100,000 gametes times itself, which would create an insurmountable obstacle to evolutionary due to the resulting improbability of 1 out of 10,000,000,000 (10 billion). So in short, we will simply multiply our equation by a factor of 4.

Factor A multiplied Britannica’s 10,000 generations (which was resulted from a starting based of 10 percent) by a factor of 10 to reflect a starting base of only 1 percent. The result was 100,000 generations. Factor B then multiplied those 100,000 generations by a factor of 5 to reflect a mutation rate of “1 out of every 500,000 gametes,” which was right in the middle of the 1 out of 100,000 to 1 out of 1,000,000 range asserted by Britannica. The 1 out of 500,000 rate was much more moderate than the extremely favorable 1 out of 100,000 rate selected by Britannica from the absolute fastest end of the available range. The result was 500,000 generations. Factor C multiplied those 500,000 generations by 1.1 to reflect a very small “slow-down” caused by counteracting “backward” mutations. The result was 550,000 generations, which is subsequently a very small change to the overall product. Factor D multiplied those 550,000 generations by 20 to reflect a hypothetical ratio between mutations in general and the much rarer beneficial mutations needed for evolution. The result was 11,000,000 (11 million) generations. Factor E multiplies those 11,000,000 generations by a factor of 4 to reflect the necessity for recessive mutations to be present in both alleles rather than just 1 in order to avoid elimination by natural selection in less than 2,000-10,000 generations. The result is 44,000,000 (44 million) generations.

Factor F: Recessive Limitations

As we have seen earlier in this study, natural selection is a process by which particular traits become advantageous when there is a change in an organism’s environment. Evolutionary and genetic theory assert that gene frequencies will remain constant rather than shift unless there is a corresponding change in environment that causes one of the alleles to suddenly become more advantageous. Conversely, so long as alleles, even mutant alleles, do not provide an advantage, they will not rise in frequency at all.

And Microsoft Encarta specifically included the need for an accompanying change in environment as a requirement if a beneficial mutation was to actually promote fitness and, therefore, contribute to the eventual origin of a new species (or new kind of organism).

“VI SPECIATION – Because all the established genes in a population have been monitored for fitness by selection, newly arisen mutations are unlikely to enhance fitness unless the environment changes so as to favor the new gene activity, as in the gene for dark color in the peppered moth. Novel genes that cause large changes rarely promote fitness and are usually lethal.” – "Evolution," Microsoft® Encarta® Encyclopedia 99. © 1993-1998 Microsoft Corporation. All rights reserved.

Consequently, if there is no accompanying change in environment, gene mutations are forced by the constant work of natural selection into 1 of 2 categories. If there is no accompanying change in environment, either the mutation will escape removal by natural selection because it is a negligible mutation which does not produce any “novel” or significant change at all to the organism or if the mutation does produce a “novel” and significant change to the organism, it will be a novel change that does not promotes fitness, in which case natural selection will eliminate it.

However, changes in environment can include anything from a climatological or ecological change to the organism’s current environment to a portion of the population moving to a new location where the environment is slightly different. Therefore, the chances for a change of some kind are not as unlikely as one might think. Nevertheless, given that gene mutations occur at times that do not in any way consider or correspond to such environmental changes, we’re going to assume that such environmental changes only occur as often as half of the time when a related mutation occurs.

“Evolution, III DARWINIAN THEORY – The basic rules of inheritance became known to science only at the turn of the century, when the earlier genetic work of Gregor Mendel came to light…The discovery was then made that inheritable changes in genes, termed mutations, could occur spontaneously and randomly without regard to the environment.” – "Evolution," Microsoft® Encarta® Encyclopedia 99. © 1993-1998 Microsoft Corporation. All rights reserved.

“Evolution, The process of evolution, Evolution as a genetic function, The origin of genetic variation: mutations, Gene mutations – …mutations are random events with respect to adaptation; that is, their occurrence is independent of any possible consequences.” – Encyclopaedia Britannica 2004 Deluxe Edition

This is effectively a 50-50 chance, which is a generous assumption that favors evolution. And a fifty-fifty chance that a beneficial, rather than harmful mutation will also be accompanied by a change in the environment is quite favorable to evolution. The optimistic nature of the fifty-fifty chance becomes even more apparent when we consider the added necessity, not only for a beneficial mutation to be accompanied by a change in environment, but the need for there to be a correspondence between the mutation and the environmental change. After all, a beneficial mutation producing white fur in a usually snowy environment will not contribute to the development of a new species if the accompanying change in the environment is a change involving greater warmth and the loss of most of the snowfall. From this example, we can see that simply having a beneficial mutation and an accompanying change to the environment will not suffice for evolutionary purposes unless the mutation and the environmental change correspond to one another in some way. Such correspondence between the two further adds to the improbability of the required scenario. And in this light, we can see that a fifty-fifty probability for this factor is indeed quite generous.

Furthermore, anything greater than a 50-50 chance would start to require a relationship in which mutations actually do occur not randomly, spontaneously, or independently but with regard for the consequences in a particular environment. As indicated by the quotes above, evolutionary science rejects any non-random, non-spontaneous version of mutation, any version of mutation in which the mutation occurs with regard for the effect in a particular environment, because it inherently involves mutation that entails tailoring toward anticipated, beneficial consequences in the environment, which is synonymous with foresight and teleology. In other words, because of the improbabilities inherent to chance events such as mutations, the suggestion of mutations occurring suddenly at the most ideal and convenient times just when an environmental change also happens to occur and providing the exact mutation that is perfectly functional and is suitably adapted to that new environment has strong indications toward foresight and teleology. Consequently, because a chance greater than fifty-fifty starts to entail foresight and evolution must avoid foresight, we will assume simply a fifty-fifty, or 1 out of 2, probability for this factor.

While Britannica’s estimates do not take this factor into account at all, in order to account for the required accompanying change in environment, we will need to multiply our equation by a factor of 2, to reflect the favorable 50-50 chance we’ve assigned to this event.

Factor A multiplied Britannica’s 10,000 generations (which was resulted from a starting based of 10 percent) by a factor of 10 to reflect a starting base of only 1 percent. The result was 100,000 generations. Factor B then multiplied those 100,000 generations by a factor of 5 to reflect a mutation rate of “1 out of every 500,000 gametes,” which was right in the middle of the 1 out of 100,000 to 1 out of 1,000,000 range asserted by Britannica. The 1 out of 500,000 rate was much more moderate than the extremely favorable 1 out of 100,000 rate selected by Britannica from the absolute fastest end of the available range. The result was 500,000 generations. Factor C multiplied those 500,000 generations by 1.1 to reflect a very small “slow-down” caused by counteracting “backward” mutations. The result was 550,000 generations, which is subsequently a very small change to the overall product. Factor D multiplied those 550,000 generations by 20 to reflect a hypothetical ratio between mutations in general and the much rarer beneficial mutations needed for evolution. The result was 11,000,000 (11 million) generations. Factor E multiplied those 11,000,000 generations by a factor of 4 to reflect the necessity for recessive mutations to be present in both alleles rather than just 1 in order to avoid elimination by natural selection in less than 2,000-10,000 generations. The result was 44,000,000 (44 million) generations. Factor F multiplies those 44,000,000 generations by 2. And the result is 88,000,000 (88 million) generations.

Factor G: New Species Threshold

As indicated earlier, Britannica’s calculations are not designed to determine whether or not a sufficient number of beneficial mutations could occur in order to produce the current number of species on earth in the timeframe allotted by evolutionary theory. Consequently, it simply was not necessary for Britannica to include the following factor. However, in order to determine if beneficial mutation is a viable method for producing all the species on the planet today, our equation will need a factor that represents how many new genes are required before a new species is produced.

For a comparative reference for this factor, we notice that Britannica states human beings have 30,000-40,000 genes.

“Human Genome Project – February 2001 declared that the human genome actually contains only about 30,000 to 40,000 genes, much fewer than originally thought.” – Encyclopaedia Britannica 2004 Deluxe Edition

Similarly, in the quote below, Discover magazine asserts that both humans and chimps have between 20,000 and 30,000 genes. Since according to these 2 quotes, 20,000 is the low end and 40,000 is the high end of the range, to be fair we’ll go with the middle number of 30,000, which is asserted by both sources. Likewise, notice that both quotes below assert that the gene code of humans and chimps are 98 percent the same, leaving only a 2 percent difference.

“Now that scientists have decoded the chimpanzee genome, we know that 98 percent of our DNA is the same…Human and chimps each have somewhere between 20,000 and 30,000 genes, so there are likely to be nucleotide differences in every single gene.” – “The 2 Percent Difference,” by Robert Sapolsky, DISCOVER, April 2006

“Chimpanzee, Taxonomy – Genetic analysis suggest that humans and chimps diverged four million to eight million years ago and that at least 98 percent of the human and chimpanzee genomes are identical.” – Encyclopaedia Britannica 2004 Deluxe Edition

Notice also that according to the Britannica quote above, humans and chimps “diverged” four million years ago, a statement which indicates the common ancestry of the two groups according to evolutionary theory.

This is important because the question arises, if humans and chimps diverged as species and presently only have 2 percent genetic difference, how much genetic difference was required before the 2 groups actually constituted separate species? In other words, exactly how many new, different genes does it take to produce a new species? If, in the case of humans and chimpanzees, it requires 2 percent difference before a separate species can be said to “diverge” then 2 percent of 30,000 genes is 600 genes. The more new genes have to come about by mutation, more time that evolution is going to require. So again, to be more than favorable to evolutionary theory, we want to assume a relatively low number. Consequently, we’ll cut the number down to only 30 new alleles, which is just one-tenth of 1 percent of all the genes in the human and chimp genomes. In other words, we are assuming that a new species (or new kind of organism) arises when they acquire less than a 1 percent difference in genetic material from their originating species. Likewise, this small figure should also account for the variety of genome sizes present in the animal kingdom and hopefully represent a number that is reasonably average.

This consideration means that in order for a new species to emerge, 30 beneficial mutations will have to accumulate according to the processes described above. As such, we will need to multiply our equation by a factor of 30 to represent the recurrence of the entire process 30 times, once for each new allele acquired along the road to a new species.

Factor A multiplied Britannica’s 10,000 generations (which was resulted from a starting based of 10 percent) by a factor of 10 to reflect a starting base of only 1 percent. The result was 100,000 generations. Factor B then multiplied those 100,000 generations by a factor of 5 to reflect a mutation rate of “1 out of every 500,000 gametes,” which was right in the middle of the 1 out of 100,000 to 1 out of 1,000,000 range asserted by Britannica. The 1 out of 500,000 rate was much more moderate than the extremely favorable 1 out of 100,000 rate selected by Britannica from the absolute fastest end of the available range. The result was 500,000 generations. Factor C multiplied those 500,000 generations by 1.1 to reflect a very small “slow-down” caused by counteracting “backward” mutations. The result was 550,000 generations, which is subsequently a very small change to the overall product. Factor D multiplied those 550,000 generations by 20 to reflect a hypothetical ratio between mutations in general and the much rarer beneficial mutations needed for evolution. The result was 11,000,000 (11 million) generations. Factor E multiplied those 11,000,000 generations by a factor of 4 to reflect the necessity for recessive mutations to be present in both alleles rather than just 1 in order to avoid elimination by natural selection in less than 2,000-10,000 generations. The result was 44,000,000 (44 million) generations. Factor F multiplied those 44,000,000 generations by 2. The result was 88,000,000 (88 million) generations. Factor G multiplies those 88,000,000 generations by 30 to reflect the need for this process to repeat 1 time for every new allele acquired on the road toward the emergence of a new species. The result is 2,640,000,000 (2.640 billion) generations.

Factor H: Generations per Year

There is one last factor to include before our equation will be complete. This factor also was not necessary to Britannica’s equation since Britannica’s estimate was designed only to reflect the number of generations, not the number of years, required for 1 mutation to reach 1 percent of the population. In contrast, because our calculation is designed to reflect whether or not beneficial mutation could produce the current amount of species on earth within evolution’s allotted timeframe, we’ll need to convert generations to years. Once again, on this last point, we want to make sure to be generous and favorable to evolutionary theory. And we also to take into account the diversity in the animal kingdom. For these reasons, we will assume that the average time for a generation is roughly 1 week, or one-fiftieth of a year. In other words, we will assume that a generation passes and a new generation arises to replace it once a week. Thus, since we are assuming that 50 generations pass in 1 year, to convert the number of generations in the equation to years, we will simply need to divide our equation by 50.

Factor A multiplied Britannica’s 10,000 generations (which was resulted from a starting based of 10 percent) by a factor of 10 to reflect a starting base of only 1 percent. The result was 100,000 generations. Factor B then multiplied those 100,000 generations by a factor of 5 to reflect a mutation rate of “1 out of every 500,000 gametes,” which was right in the middle of the 1 out of 100,000 to 1 out of 1,000,000 range asserted by Britannica. The 1 out of 500,000 rate was much more moderate than the extremely favorable 1 out of 100,000 rate selected by Britannica from the absolute fastest end of the available range. The result was 500,000 generations. Factor C multiplied those 500,000 generations by 1.1 to reflect a very small “slow-down” caused by counteracting “backward” mutations. The result was 550,000 generations, which is subsequently a very small change to the overall product. Factor D multiplied those 550,000 generations by 20 to reflect a hypothetical ratio between mutations in general and the much rarer beneficial mutations needed for evolution. The result was 11,000,000 (11 million) generations. Factor E multiplied those 11,000,000 generations by a factor of 4 to reflect the necessity for recessive mutations to be present in both alleles rather than just 1 in order to avoid elimination by natural selection in less than 2,000-10,000 generations. The result was 44,000,000 (44 million) generations. Factor F multiplied those 44,000,000 generations by 2. The result was 88,000,000 (88 million) generations. Factor G multiplied those 88,000,000 generations by 30 to reflect the need for this process to repeat 1 time for every new allele acquired on the road toward the emergence of a new species. The result was 2,640,000,000 (2.640 billion) generations. Factor H divides those 2,640,000,000 generations by 50 to reflect a hypothetical average of 50 generations per year in the animal kingdom. And the final result of the equation is 52,800,000 years.

In other words, according to these favorable estimates for evolution, it takes a little over 53 million years to accumulate enough mutant alleles to produce a new species (or new kind of organism).

Just for comparison, it should be noted that this figure of 50,000,000 years for speciation to occur by beneficial mutation is not necessarily very far off from the number of years asserted by evolutionary theory itself by 1 order of magnitude (which is to say by a factor of 10), at least as far as gradual evolution is concerned. (The difference between gradualism and punctuated equilibrium in evolutionary theory will be discussed in a later segment.) Evolutionary theory itself asserts that it takes millions of years for speciation to occur, as indicated by the example of humans and chimps, which are said to have diverged 4 to 8 million years ago.

Now, we arrive at the point of comparison. We need to compare the calculation of 52,800,000 years per speciation cycle to the actual timeframe allotted for the origin of species by evolutionary theory. As stated earlier, to accomplish this comparison we will simply examine whether or not these rates would be sufficient to bring about just the organisms in one single kingdom, the animal kingdom. This is ideal because we know a great deal about the animal kingdom, including a count of how many species we’ve identified and the evolutionary timeframe for the approximate origin of the first animals. Here again, even in these figures, there will be room for additional favorable assumptions to be granted on behalf of evolutionary theory.

Consequently, the first step is to identify a target number of species and the allotted timeframe in which they would have to be produced by this process of beneficial mutation. According to Worldbook Encyclopedia, the animal kingdom is the largest of the 5 kingdoms of organisms and there are over 1 million species in the animal kingdom alone.

“Classification, Scientific, Groups in classification – The kingdom Animalia is the largest kingdom. It has more than 1 million named species. These species include the organisms that most people easily recognize as animals, such as human beings, deer, fish, insects, and snails.” – Worldbook, Contributor: Theodore J. Crovello, Ph.D., Professor of Biology and Dean, Graduate Studies and Research, California State University, Los Angeles.

To be favorable to evolutionary theory, we’re going to assume only 1 million species need to be produced. However, in reality, this number could be much higher since secular sources currently estimate that there are anywhere from 10 to 30 million species that currently exist and are yet to be identified or discovered.

“Evolution – More than 2,000,000 existing species of plants and animals have been named and described; many more remain to be discovered—from 10,000,000 to 30,000,000 according to some estimates…The virtually infinite variations on life are the fruit of the evolutionary process.” – Encyclopaedia Britannica 2004 Deluxe Edition

Of that 10 to 30 million, if even one-tenth to one-fifth were animals, then beneficial mutation would have to produce anywhere from 2-3 million to 6 million species in the allotted timeframe instead of just 1 million. To be even more realistic, since the animal kingdom is the largest of the kingdoms, we would expect that quite a large percent of the additional 10-30 million species to be animals, in which case beneficial mutation would have to account for perhaps 15 to 20 million species. And even this doesn’t factor in all of the extinct species and requisite intermediary or transitional forms that are not present today or in the fossil record. So, once again, the number of 1 million is extremely generous to evolutionary theory.

According to evolutionary theory, animals have only existed on earth since the Vendian time period of evolutionary history, which spans from about 670 to 570 million years ago.

“Animals, Evolution and paleontology, Appearance of animals – Animals first appeared in the Vendian, soft-bodied forms that left traces of their bodies in shallow-water sediments.” – Encyclopaedia Britannica 2004 Deluxe Edition

“Animal, Ecology and habitats, Evolution of ecological roles – This was probably more common in the Vendian (the last interval of the Precambrian, from 670 to 590 million years ago on certain geologic time scales).” – Encyclopaedia Britannica 2004 Deluxe Edition

“Period, III PERIODS OF THE PROTEROZOIC EON – The Sinian Era is divided into two informal geologic periods-the Sturtian Period (from 800 million to 610 million years before present) and the Vendian Period (610 million to 570 million years before present).” – "Period," Microsoft® Encarta® Encyclopedia 99. © 1993-1998 Microsoft Corporation. All rights reserved.

So, the question is this: Can beneficial mutations produce over 1 million species of animals since the start of the Vendian Period? Once again, here we’ll take the higher number of 670 million years to be favorable to evolutionary theory and allow the maximum amount of time for the 1 million species of animals to come into being through beneficial mutation. This leaves us with 1 final item to cover before performing our calculation.

The second step is to explain that the rate of speciation will assume an exponential, rather than strictly linear, increase in the number of species. In other words, this calculation could be performed to determine how long it would take one animal species to evolve and produce over 1 million animal species on earth today. However, this would not be fair to evolutionary theory. According to evolutionary theory, each new species that evolves would itself undergo a process of evolution. So, if you start with one species experiencing evolution, in time another species will develop and then you will have 2 species experiencing evolution simultaneously. Thus, in time, both species will in turn produce 2 more species, at which point there would be 4 species evolving, not just 1 species evolving in linear fashion. Consequently, the process is exponential according to evolutionary model and so in order to be fair our equation will also work exponentially.

(For an explanation and illustration contrasting linear to exponential speciation, please see Speciation Figure 1.)

This exponential effect should work dramatically in favor of evolutionary theory because, working exponentially, it only takes 21 cycles to go from 1 to 1 million. If you start with 1 and then multiply by 2, the product is 2. From here multiplication becomes exponential. The existing 2 each produce 1 more, resulting in 4. Then each of the 4 produces 1 more, resulting in 8. Consequently, the progression doubles every time, starting with 1 and then proceeding upward to 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288, and finally 1048576 (1,048,576). At the 21st “doubling” the number breaches the barrier of 1 million. In other words, in order to produce 1 million species from just 1 starting species, the duration required for speciation would only have to repeat 21 times. After 21 back to back speciation cycles, over 1 million species would be produced from just a single species.

At last, we are ready to begin our calculation to “check the numbers” and see if the origin of species by beneficial mutation is mathematically feasible according to the probabilities and factors outlined by evolutionary and secular sources.

In order for beneficial mutations to be a viable mechanism for the origin of species, it would need to be able to produce over 1 million animal species in 670 million years or less that according to evolutionary theory have occurred since the emergence of the first animal.

According to these favorable estimates, it would take 52,800,000 years for just one cycle of speciation to occur. In fact, it’s going to take 52,800,000 years for each new cycle of speciation. As we noted, factoring this equation so that the number of species undergoing speciation is multiplied exponentially every round, starting with 1 species as the first cycle, it would only take 21 cycles of speciation in order to go from that 1 lonely animal species to over a million. But at 52,800,000 years for each speciation cycle, 21 cycles simply will not occur in 670 million years.

In 670 million years, there would only be enough for 12.7 cycles of speciation. To be favorable to evolution, we’ll round up to 13. If we start with 1 animal species, after 670 million years and 13 cycles of speciation there would only be 8,192 species, which is a long way short of the required 1 million that we observe today. Even if we assume that there are as many as 8 original animal species at the start of the process and continue to calculate exponentially, after 670 million years, there would still only be 65,536 species. In fact, we could assume that there were as many as 64 original animal species at the start of the process but after 670 million years calculating exponentially we would only have 524,288 species, not the 1 million to perhaps tens of millions that are said to exist on earth today.

Assuming an average mutation rate of 1 out of 500,000, assuming only 1 in 22 mutations is a “backward” mutation, assuming 1 in 20 mutations is beneficial, assuming only 30 new genes are required for a new species, assuming the accumulation of mutations only has to occur in a portion as small as 1 percent or the population no matter what the size, assuming that generations pass in one week’s time, and compounding this process exponentially as new species are produced, there simply isn’t enough time for beneficial mutation to produce all the animal species that we observe today, let alone the potentially tens of millions more that have yet to be discovered or the intermediary forms and other species that have become extinct.

Perhaps, though, it might be suggested that these calculations and results are close enough or that maybe if we adjusted the numbers slightly the evolutionary model might be vindicated as plausible. But we must remember that the numbers and rates that we employed in our calculations were generously biased in favor of the evolutionary model. Provided such generous, numerical accommodations, we must conclude that if the evolutionary model doesn’t work under these terms, then it doesn’t work at all. And, if we were, in fact, to adjust our figures in order to more closely reflect the actual probabilities and rates, the result would be even more decisively damaging to the viability of evolutionary theory.

As a result of this information and these critical factors asserted by secular and evolutionist sources, we are left to conclude that beneficial mutation is simply not a tenable mechanism for causing the origination of new species. However, in addition to the inadequacy of the explanatory mechanism of evolutionary theory, we can also evaluate the validity of the evolutionary origin of species by comparing it to the evidence of biological history itself preserved in the form of the fossil record itself. This brings us to our next segment of our expanded commentary on evolutionary theory.