Efficiency of mutation in Darwinian evolution

Richard Dawkins has identified the framework of Darwinian evolution as a series of stages or cycles of random numbers generation and discriminate integer filtering. He has used a multiple dial lock to demonstrate Darwinian evolution, both in The God Delusion and in a video lecture.

In the video lecture, Richard Dawkins demonstrates the efficiency of mutation due to sub-staging in Darwinian evolution (http://www.youtube.com/watch?v=JW1rVGgFzWU). I noticed a discrepancy in the model of the 3-dial lock, which is used for illustration. The lock is verbally described as having six different mutation sites for each dial. However, pausing at 5:15 of the video reveals that the digits adjacent to the 6 on the dials are 5 and 0, rather than 5 and 1. There are 7 symbols in the range 0 to 6, whereas there are 6 symbols in the range 1 to 6. The number of mutations defined by 0 0 0 to 6 6 6 is 7^3 = 343. The number of mutations defined by 1 1 1 to 6 6 6 is 216, which is the number discussed in the audio of the lecture.

Nevertheless, the demonstration, as narrated, is an excellent illustration of efficiency in the number of mutations due to replacing a single cycle of non-random mutation and non-random selection with sub-cycles.

The video illustrates the evolution of the sequence 6 5 1, but based on non-random rather than random mutation. A better illustration of Darwinian mutational efficiency would have been the use of three dice, each of a different color. Rolling dice is conventionally recognized as random numbers generation. As an illustration of Darwinian random mutational generation, the rolling of three dice 479 times would generate a pool of random numbers to the base 216. Subjecting such a pool of random numbers to Darwinian non-random selection would yield a probability of 89.17% success of natural selection of the sequence 6 5 1. This can also be accomplished by subjecting each of the three mutation sites individually to Darwinian evolution, which is the theme of the video illustration. Generating three pools of 18 random numbers to the base 6 would yield a probability of 96.25% evolutionary success for each site and a probability of overall evolutionary success of 89.17%.

Although replacing a single stage of Darwinian random mutation and natural selection with a series of sub-stages does not affect the probability of Darwinian evolutionary success, it does increase the efficiency of Darwinian random mutation. For an 89.17% probability of the evolution of the sequence 6 5 1, the total of random numbers generated drops from 479 for a single stage to 54 for the series of three sub-stages. That is an efficiency factor of 479/54 = 8.87 for random mutations due to sub-staging. There is no change in the probability of success of 87.19% = 96.25^3.

Three dice would serve better to illustrate what Dawkins has dubbed ‘the smearing out of the luck’. In Darwinian evolution ‘the luck is smeared out’ due to sub-staging, but there is no change in ‘the luck’ overall. There is no change to the overall probability of Darwinian evolutionary success. However, there is an increase in the efficiency of random mutation.

Three dice, each of a different color would be an excellent model of three mutations sites of 6 mutations each, in contrast to the 3-dial lock of 6 mutations per dial. The reason is that the dice lend themselves very well to random mutation rather than serve simply to illustrate sequential, non-random mutation, to which Dawkins was constrained in the video by the model of the 3-dial lock. He could have spun the dials rather than turning them methodically, but rolling dice would teach the lesson more clearly. One drawback to spinning the dials is that in the video the dials were analog rather than digital. This difficulty could have been overcome by constructing the dials as prize wheels.

Of course, an illustration employing tetrahedral dice would be preferable to one employing cubic dice because each mutation site in a genome is of four, not six, mutations. The chimpanzee genome can be visualized as a map of 3 billion tetrahedral dice where each tetrahedron represents a site of 4 possible mutations.

One drawback to using tetrahedral dice is that the outcome of a roll is the face-down triangle. Another drawback is that they are hard to roll. These two difficulties may be avoided by using four sided rolling pins.

Whatever the model used for illustration, it should serve to elucidate and not to obscure the distinction between the efficiency of random mutation and the probability of success of natural selection in Darwinian evolution. In the video, Dawkins compares the 216 non-random mutations of a single stage with the total of 18 non-random mutations, i.e. 6 for each of three sub-stages. That is an efficiency factor of 216/18 = 12 for non-random mutations. The probability of the evolution of the sequence 6 5 1 is 100% in both cases. Yet, it is easy to get the impression that the lecture illustrates an increase in the probability of success of natural selection and not an increase in the efficiency of mutation. A synopsis of the video in Wikipedia gives such a false impression, “In Dawkins’ case we have 3 dials, with 6 positions each, so the probability that you open the lock by sheer luck is one to two hundred and sixteen. Dawkins then shows the mechanism of the lock with a big model: Each dial has to be in the correct position in order to open up the lock. The model is then adapted to demonstrate a staged or gradualist solution to finding the right combination to open the lock. The probability of unlocking the combination in three separate phases falls (sic) to one in eighteen.” ‘Falls’ is inadvertent. An increase to 1/18 from 1/216 would be a rise, not a fall. However, what matters is that the writer is under the false impression that the 3-dial lock serves to illustrate an increase in the probability of evolutionary success.
(http://en.wikipedia.org/wiki/Growing_Up_in_the_Universe#Part_3:_Climbing_Mount_Improbable)

Similarly, in The God Delusion, pages 121-122, Dawkins gives the impression that replacing a single cycle of Darwinian evolution with a series of sub-cycles increases the probability of success of natural selection. In fact such a replacement increases the efficiency of random mutation, while having no effect on the probability of evolutionary success. Dawkins states, “What is it that makes natural selection succeed as a solution to the problem of improbability, where chance and design both fail at the starting gate? The answer is that natural selection is a cumulative process, which breaks up the problem of improbability up into small pieces. Each of the small pieces is slightly improbable, but not prohibitively so.” The overall probability of a series is the product of multiplication. Dawkins’ explanation could only make sense, if the overall probability of a series was the sum of addition.

See also, “Too improbable to be due to chance” (https://theyhavenowine.wordpress.com/2013/12/30/too-improbable-to-be-due-to-chance/).

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