The Discrimination Ratio of Natural Selection

In understanding Darwinian evolution an important distinction is that between the discrimination ratio of natural selection and the probability of success of natural selection, which is also a ratio. The discrimination ratio of natural selection is the ratio of the one mutation, which can survive natural selection, to the total number of different mutations, which are defined by a cycle of random mutation and natural selection. The probability of success of natural selection is the probability that the mutation, which can survive natural selection, is present in at least one copy in the set of mutations randomly generated and subjected to natural selection.

There is a third ratio, which is also characteristic of Darwinian evolution. It is the probability of the random generation of any one mutation, which also applies to the one mutation, which can survive natural selection. This probability is numerically equal to the discrimination ratio of natural selection.

Natural selection screens the set of mutations randomly generated. It permits one mutation to survive and culls the others. Natural selection discriminates among the set of different mutations defined by the cycle of random mutation and natural selection. If a single base pair site of a genome alone were subject to random mutation and natural selection, that cycle would define four mutations. Natural selection would operate at a discrimination ratio of 1/4. If a cycle involved three base pair sites, the cycle would define 64 mutations (4 × 4 × 4). The discrimination ratio of natural selection would be 1/64.

If only one random mutation were generated in a cycle of random mutation and natural selection, then the probability of success of natural selection would equal the probability of the generation of any one random mutation. This would also equal the value of the discrimination ratio of natural selection. In a cycle affecting one base pair, the three ratios would equal 1/4. In a cycle affecting three base pairs, the three ratios would equal 1/64. Although equal, each of the three ratios has a unique definition.

A pool of generated mutations, consisting of a total of one mutation, is irrelevant because many mutations, not just one, are randomly generated in a cycle of Darwinian evolution. No matter how many random mutations are generated, the discrimination ratio remains constant as does the probability of the generation of any one random mutation. Also they are equal to each other. In contrast, the ratio, which is the probability of success of natural selection, increases with the total number of random mutations that are generated in a cycle. The probability of success of natural selection begins at the value of the discrimination ratio. With the increasing number of mutations randomly generated, the probability of success increases toward 100%, but never reaches it.

Before considering the probability of success of natural selection in greater detail, let us familiarize ourselves with probability using the common prop, a deck of 52 playing cards.

In selecting one card at random from the deck, the probability is 1/52 that it is the Seven of Diamonds. That is because probability is the fractional concentration of an element in a logical set. The logically defined set is that of 52 playing cards. The fractional concentration of Sevens of Diamonds is 1/52. Similarly, the probability of a diamond is the fractional concentration of diamonds, which is 13/52 = 1/4. The probability of a seven in the set of playing cards is the fractional concentration of sevens, namely 4/52 = 1/13.

That which is random is that which lacks order; that which lacks any rationale. When we select a card at random from a deck we equate the lack of order or the lack of rationale with our human ignorance of the ID of the card. We call the selection random, while not denying that the cards are in some order of location. We cannot draw a card without selecting it in the exact, ordered location in which it exists. Thus we call random selection, the selecting of a card in ignorance of its ID in simulation of the logical concept of a random selection. The logical concept of random selection is the formation of a new logical set of elements based solely on the fractional concentration of the elements in the logical source set.

Having selected a card at random from a deck, we can then subject it (the single element in the newly formed set) to natural selection. In the above example the discrimination ratio of natural selection is 1/52. Natural selection culls the card, if it is any of 51 and permits the card to survive, if it is the Seven of Diamonds. The probability of success of natural selection is 1/52. Instead of drawing one card at random from the deck, let us draw two, subjecting them to natural selection. The discrimination ratio of natural selection is 1/52, but the probability of success of natural selection is now 2/52. If the set of cards randomly generated and subjected to natural selection is 26, then the discrimination ratio is still 1/52, but the probability of success of natural selection is now 26/52 = 1/2 = 50% . If we choose all 52 cards one by one randomly from a deck, the discrimination ratio of natural selection is still 1/52, but the probability of success of natural selection is 100%.

Because we have selected the entire deck, which is one copy of each of the possible 52 mutations of a playing card, we would not call the selected set of 52 cards a random set. But, didn’t we draw them at random one by one? If the resultant set is a non-random set, then to what does the obvious simulation of random selection refer, once the entire deck has been selected? The answer is that the randomness refers to the sequence of the deck of 52 cards, rather than to the full set of 52 as a newly formed set. We can look at the resultant deck logically in either of two ways: as a defined set of 52 elements, i.e. as non-random, or as a random sequence of 52 elements. When we look at the resultant set as a random sequence, then drawing the cards randomly one by one until all 52 cards were drawn was simply one method of shuffling the deck.

Recognizing the random selection of all 52 cards one by one as a method of shuffling the deck, resulting in a random sequence of 52 elements, brings up a fascinating aspect of materially simulating the logical concepts of random selection and probability. When we drew one card ‘randomly’ from the deck in a material simulation of the logical concepts of random selection and probability, three material analogs prevailed. The source set was materially simulated as a deck of 52 cards. Random selection was materially simulated by the draw in ignorance of the ID of the card. The new set formed was materially simulated by the drawn card. In contrast, these three material analogs cannot obtain in the random selection and probability of a 52-card sequence. Working in reverse order, the shuffled deck is the material analog of an element in the new set. Shuffling is the material analog of random selection. However, there can be no material analog of the source set of all possible 52-card sequences from which random selection is made. The reason is the magnitude of the logical source set. It consists of 52 factorial or 8.0 x 10 ^ 67 elements. [As homework you may calculate how many times the mass of the earth would be the equivalent to the mass of that many material elements (decks of cards), if each of the elements (each deck of cards) had the mass of a hydrogen atom.] Nevertheless, the lack of a material analog for the source set does not invalidate the material simulation of the purely logical concepts of random selection and probability. Shuffling a deck of cards is by simulation the materializing of an element out of a purely logical set of elements. The purely logical source set, from which one element is materialized by shuffling, is that of all possible sequences of a deck of playing cards. Shuffling a deck of cards is the simulation of materially drawing a material rabbit out of a purely logical hat.

Getting back to Darwinian evolution, it is to be noted that random mutation generates a set of random elements which is subjected to natural selection. In Darwinian evolution random mutation does not deplete the source set. Rather, each random mutation is a random selection from a full source set (the source set is logical). The newly formed set is a set of random elements and not simply a non-random set of elements forming a random sequence, which would be the result of depleting the source set. Consequently, the probability of evolutionary success can never achieve a value of 100%. In a cycle of Darwinian evolution affecting 3 base pairs, the discrimination ratio of natural selection is always 1/64. The probability of success of natural selection is 1/64 after one random mutation. When the set of random mutations reaches 2, the probability is not 2/64 = 1/32 = 3.125% as it would be, if the source set were depleted by random mutation. What then is the probability of success of natural selection?

Probability and improbability are complements of 1. That is: the sum of a probability and its improbability equals one. The probability of drawing the Seven of Diamonds is 1/52. The improbability of drawing the Seven of Diamonds is 51/52 (1/52 + 51/52 = 52/52 = 1). The improbability of drawing the Seven of Diamonds is the probability of drawing any card but the Seven of Diamonds.

The probability of success of natural selection is the probability that the one mutation, which can survive natural selection, is present in at least one copy in the pool of mutations randomly generated. The probability of success of natural selection equals 1 minus its improbability. That improbability, conveniently, is the probability of generating any non-survivable mutation. In 1 random selection, the probability of generating any non-survivable mutation is 63/64. The probability of non-generation of the survivable mutation in 2 selections is (63/64) ^ 2. In general, the probability of not randomly selecting the survivable mutation is ((n-1)/n) ^ x, where n is the number of different mutations defined by the cycle and x is the number of random mutations of whatever kind that have been randomly generated. Consequently, the probability, P, of success of natural selection is: P = 1- ((n-1)/n) ^ x. For n = 64 and x = 2, the probability, P, of success of natural selection is 3.10%. That is not much smaller than the probability of success in the case of depleting the source set. However, when x = 64, then P = 63.5%. In contrast, if the source set were depleted by mutation, then for 64 mutations the probability of success of natural selection would be 64/64 = 100%.

In 1991 Richard Dawkins demonstrated that the efficiency of random mutation increases due to the gradualism of Darwinian evolution (http://www.youtube.com/watch?v=JW1rVGgFzWU, minute 4:25). He didn’t realize that he had demonstrated an increase in the efficiency of random mutation. He thought he had demonstrated an increase in the probability of success of natural selection. For illustration he chose 3 mutation sites of 6 mutations each. That defines 216 different mutations (6 × 6 × 6 = 216). For one random mutation, the probability of success of natural selection equals the discrimination ratio of natural selection, which is 1/216 = 0.46%. When the size of the pool of random mutations subjected to natural selection equals the number of mutations defined by a single cycle, n = 216 and x = 216. The discrimination ratio of natural selection is still 1/216 but the probability of evolutionary success is: P = 1 – (215/216) ^ 216 = 63.3%.

For this probability of success (63.3%) when this single cycle is replaced by three sub-cycles, each affecting a single mutation site, the probability of success of natural selection for each mutation site must be (0.633) ^ (1/3), which, within each sub-cycle, equals 85.9%. At this probability of success for each sub-cycle, the overall probability of success is 63.3% (0.859 × 0.859 × 0.859 = 0.633). In each of the three sub-cycles, the discrimination ratio of natural selection is 1/6. The probability of success of natural selection is P = 1 – (5/6) ^ x, where x is the number of randomly generated mutations subjected to natural selection within a sub-cycle.

For each sub-cycle, when P = 85.9%, x = 10.7. Rounding this up to the integer 11, then x = 11 and P = 86.5%. For the series of three sub-cycles, the overall probability is then 64.8% for 11 random mutations per sub-cycle (0.865 × 0.865 × 0.865 = 0.648 = 64.8%). When for the overall single cycle, P = 64.8%, then x = 225. In this illustration, at a probability of success of natural selection of 64.8%, the gradualism of three sub-cycles reduces the total number of random mutations form 225 to 33. That is an efficiency factor of 225/33 = 6.8 for random mutations due to gradualism.

In contrast, Dawkins’ illustration was one of non-random mutation, where the probability of evolutionary success was 100% for both the overall single cycle and the series of three sub-cycles. The efficiency factor for non-random mutation was 216/18 = 12. For either non-random mutation or random mutation the discrimination ratio of natural selection is 1/216 for the overall single cycle and 1/6 for each of the sub-cycles in the series of three sub-cycles of gradualism.

It is obvious from his lecture in 1991 that when he compared the total of 216 non-random mutations to the total of 18 non-random mutations, Dawkins thought he was demonstrating an increase in the probability of success of natural selection from 1/216 to 1/6 (or possibly 1/18).  In fact he was illustrating an increase in the efficiency of mutation from a total of 216 mutations to a total of 18 mutations due to gradualism.  Because he was illustrating non-random mutation at a probability of success of natural selection at 100%, there was no change in that probability from the single cycle compared to any one of its sub-cycles.  In contrast when a probability is less than 100%, each of its factors (the value in a sub-cycle) is greater than the overall probability, but the product of the factors equals the overall probability.

To understand Darwinian evolution you’ve got to keep the distinctions among the three defining ratios clearly in mind as well as any other pertinent ratios such as that of an efficiency factor.

Gradualism in Darwinian evolution increases the efficiency of random mutation.  However, gradualism has no effect on the overall discrimination ratio of natural selection, the overall probability of success of natural selection or the overall probability of any one random mutation.  Because these ratios are less than one, each of their factors (their values in sub-cycles) is greater than the ratio of the single main cycle.  However, their product, i.e. their overall value, equals the ratio of the main cycle.

The efficiency of mutation due to gradualism is most easily appreciated from the perspective of the discrimination ratio of natural selection.

Natural selection is by definition non-random. In any cycle, only a specific mutation can survive. However, because the set of elements processed by natural selection in Darwinian evolution is a set of randomly generated mutations, the probability of success of natural selection is always less than 100%. Consequently the success of natural selection or, synonymously the outcome of Darwinian evolution, is always random. Thus, Darwinian evolution is a random process. It is a matter of mathematical probability. For an astonishingly unembarrassed claim to the contrary, while acknowledging that genetic variation in Darwinian evolution is random, see minute 38:20 to minute 40:10 of the YouTube video, http://www.youtube.com/watch?v=ibLo-Cxg1L4.

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: