Dawkins’ Equation of Missing Mutations

Richard Dawkins’ most important contribution to our understanding of Darwinian evolution is identifying the role of gradualism as increasing the efficiency of mutation while having no effect on the probability of success of natural selection. He outlined that role in The God Delusion (p. 122), proposing two analogies of gradualism:

(1) “On the summit sits a complex device such as an eye or a bacterial flagellar motor. The absurd notion that such complexity could self-assemble is symbolized by leaping from the foot of the cliff to the summit in one bound. Evolution by contrast goes around the back of the mountain and creeps up the gentle slope to the summit.”
(2) “Theoretically a bank robber could get lucky and hit upon the right combination of numbers by chance. In practice, . . . this (is) tantamount to impossible. . . . Suppose that when each one of the dials approaches a correct setting, the vault door opens another chink, and a dribble of money trickles out. The burglar would hone in on the jackpot in no time.”

On page 121 of The God Delusion, Dawkins labels a large single stage of Darwinian evolution as ‘a single, one-off event’ in contrast to the gradualism of a series of sub-stages, which he describes as incorporating ‘the power of accumulation”.

In 1991 in a lecture on Climbing Mount Improbable at the 4:30 minute, Dawkins gave a numerical illustration of the replacement of a single large Darwinian stage of mutation and natural selection. The large stage was replaced by the gradualism of an equivalent series of Darwinian sub-stages. The single large stage generated mutations based on three mutations sites of six mutations each, thereby defining 216 different mutations. The replacement series consisted of three independent sub-stages. In each sub-stage one mutation site was subjected to Darwinian evolution independently of the other two sites. Each sub-stage defined six mutations. However, the beginning mutation of the second sub-stage was a duplicate of the ending mutation of the first sub-stage and similarly the beginning mutation of the third sub-stage was the ending mutation of the second sub-stage. Consequently, there were 3 x 6 = 18 total mutations defined by the three sub-stages, but only 16 different mutations.

In the single large stage, there were 216 different mutations liable to generation. In the equivalent, series of sub-stages, there were only 16 different mutations liable to generation. That amounts to 200 different mutations of the initial spectrum, which were missing due to the role of gradualism of sub-stages in Darwinian evolution.

Dawkins demonstrated that the role of gradualism in Darwinian evolution is to reduce the spectrum of mutations. The missing mutations cannot be subject to generation due to the gradualism of the series of evolutionary sub-stages. Dawkins’ numerical illustration implies his algebraic equation for the number of mutations missing from the spectrum of evolutionary mutations, which would be liable to generation in a single large, or one-off, stage of Darwinian evolution.

Dawkins’ Equation of Missing Mutations

Let M be the number of different mutations defined by a single, one-off Darwinian event, i.e. a large stage of Darwinian evolution. In Dawkins’ illustration, M = 216.

Let M be factored into n sub-stages, each defining the same number of mutations. The number of mutations per sub-stage would be M^(1/n). In Dawkins’ illustration, n = 3, while M^(1/n) = 216^(1/3) = 6.

The total number of mutations defined by the series of sub-stages is n[M^(1/n)]. In Dawkins’ illustration n[M^(1/n)] = 3[6] = 18. The number of different mutations is n-1 less than the total number. In Dawkins’ illustration the number of different mutations is 18 – (3-1) = 16.

The number of mutations in the single, one-off stage, which are missing from the series of sub-stages is readily calculated using Dawkins’ Equation of Missing Mutations, m:

m = M – n[M^(1/n)] – (n-1)               Dawkins’ Equation of Missing Mutations

In Dawkins’ illustration, m = 216 – 18 – 2 = 200.

In Dawkins’ illustration, only 16 of the mutations defined in the spectrum of 216 mutations are liable to be generated due to the gradualism of the series of sub-stages of Darwinian evolution. This is the basis of the mutational efficiency of Darwinian gradualism, the understanding of which we owe to Richard Dawkins.

Two Other Numerical Examples of Dawkins’ Equation

The human genome has roughly 3 billion base-pair sites, while the fruit fly genome has roughly 165 million. Each base pair site represents four possible mutations.

If only 8 base-pair sites of a genome were subject to mutation in a single, one-off stage of Darwinian evolution, these 8 base-pairs would define a spectrum of M = 4^8 = 65536 different mutations. Replacing the one-off stage with n = 4 sub-stages of equal size results in each sub-stage defining M^(1/n) = 16 mutations. However, n – 1 of these are duplicates. We then have the information to calculate the number of missing mutations, m, from the spectrum of 65536, due to the series of sub-stages:

m = M – n[M^(1/n)] – (n-1)                Dawkins’ Equation of Missing Mutations

m = 65536 – 4 [16] – 3 = 65475

There are 65536 different mutations liable to be generated in the spectrum of the one-off stage of Darwinian evolution, affecting 8 base-pair mutation sites of a genome. However, employing Dawkins’ Equation for missing mutations, 65475 of these are missing from the total different mutations which are liable to generation in the equivalent series of four sub-stages of Darwinian evolution. That leaves only 61 of the full spectrum of 65536 as liable to generation in the series of sub-stages. This is the basis of the increased mutational efficiency, which is the role of sub-stage gradualism in Darwinian evolution, as Richard Dawkins has elucidated it.

As another example, let 20 base-pair sites be subject to mutation in a one-off Darwinian stage. Let this be replaced with a series of 4 equal sub-stages. Then,

M = 4^20 = 1,099,511,627,776 different mutations liable to generation in the one-off stage.

Each sub-stage defines M^(1/4) = 1024 mutations, and a total of 4096 for the series.

The total of different mutations liable to generation in the series of sub-stages is 4,093.

The number of different, but missing mutations using Dawkins’ Equation is:

m = 1,099,511,627,776 – 4[1024] – (4-1) = 1,099,511,623,683

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