Random Selection is a Rate, but not a Temporal Rate

I must begin with the disclaimer that I do not agree with the proponents of the Intelligent Design argument that it is a scientific argument or that it is even valid. However, I also do not agree with the common critiques of the ID argument especially in the contention that such critiques are themselves scientific.

This essay is to argue against the claim of John Derbyshire in the American Spectator (http://spectator.org/articles/57159/does-intelligent-design-provide-plausible-account-lifes-origins?page=3) that Michael Behe has not tried to get his scientific views expressed in the ‘scientific arena’. Derbyshire states, “You should at least try, as ID-ers like Behe obviously haven’t.”

Also, the animosity toward Michael Behe is so intense that he cannot state that the time interval between unexpected genetic mutations cannot be determined a priori but must be experimentally determined, without his opponents labelling his view as ‘naïve’ in a scientific journal.

My previous post on this blog noted the importance of distinguishing between probability and density. This post is another example. I composed the following essay some time ago, but it is timely in view of Derbyshire’s current essay:

Evolution in the Darwinian lexicon is a mathematical process which initially characterizes a particular numerical fraction as a random selection ratio and finally as a non-random selection ratio. Darwinian evolution itself has nothing to do with rates with respect to time. However, in some material applications the question of temporal rates will arise. It is therefore important to distinguish on the one hand, the two selection rates, which are essential to the concept of Darwinian evolution and are numerically equal to each other and, on the other hand, any temporal rates, which refer to particular material contexts.

Darwinian evolution consists of one or more stages of two selection processes applied essentially to the same logical set of n unique elements. They are random mutation and natural selection.

In each stage, the first selection process is random selection, at a ratio of 1/n. The selection is repeated multiple times randomly forming a pool of elements in which the probability of the pool’s containing each of the n elements is close to 1, due to the multiplicity of random selection. Randomness means that there is no rationale governing selection, so that the probability of selection, i.e. the rate of selection, is the fractional concentration of any element in the initial set, namely, 1/n. Note that random selection, which does not discriminate among the n different elements of the initial set, is a rate of selection with respect to the number of elements in the set. It is not the rate of selection with respect to time. Time is irrelevant.

In each stage, the second selection process is non-random (or if you prefer, ‘natural’) at a ratio of 1/n. This selection process is applied to each of the elements in the randomly formed pool. Since it follows a rationale, which recognizes each element in the initial set, it is the discrimination ratio of 1/n in its processing, by selection, of each of the elements in the random pool. Note that the non-random selection, which discriminates among the n different elements of the initial set, is a rate of selection with respect to the number of elements in the set. It is not the rate of selection with respect to time. Time is irrelevant.

These two mathematical selection processes constitute the mathematical transformation known as Darwinian evolution. In each stage, the process involves the initial labeling of a given numerical selection ratio as random and the final labeling of this same numerical selection ratio as non-random. Time and rates of selection with respect to time are intrinsically irrelevant.

Selection rates with respect to time can only be introduced alongside the completely distinct mathematical selection rates of Darwinian evolution by means of the analogical application of Darwinian evolution to material phenomena. The temporal rates of material changes can even lead to mistaking a non-Darwinian temporal material process for the purely logical process of Darwinian random mutation.

Random mutation is the random selection of an element from a logical set. Population genetics is concerned with the inheritance of a genetic marker based on its material density in a breeding population.

Consider a non-Darwinian scheme affecting two nucleotide sites in a genomic map. There are four different nucleotide pairs and consequently four possible variants for any site in a genomic map. Let some fraction of a population possess at one site a different specific base pair from the specific base pair of the rest of the population. Let the material density in the population for this inheritable marker be m1. For a second site, let the material density in the population for a second marker be m2. These material densities, m1 and m2 may be referred to as the respective probabilities of possessing these inheritable markers by a member of the population selected at random.

Contrast these probabilities with the probability of selecting a number at random to the base four. The probability is 1/4, which is the probability of the random selection from the set of four variant genomes identified by random mutation at a single site of a genomic map. Random mutation, of course, is random selection from a logical set. If the four variant genomes were a set of material elements, a random mutation would exist before the random mutation occurred.

Consider a population in which there are two distinct subsets, each identified by an inheritable marker. Over generations a subset of individuals in the population will form in which both of these markers are present. Durrett and Schmidt (Genetics 181: 821-822, 2009 http://www.genetics.org/content/181/2/821.full) showed that the mean waiting time for two inheritable markers to be inherited by the same individual of a population, one marker with a material density of m1 in the population and the other with a material density of m2 (when the first marker is neutral with respect to reproductive efficiency) is a function of m1 and a function of the square root of m2.

Contrast the mean generational waiting time for occurrence within a population with Darwinian random mutation of numbers to the base 16. Sixteen is the number of different genomes defined by random mutation at two genomic sites. Given a mere 72 Darwinian generators, at a probability of 99%, at least one copy of any of the 16 genomes, defined by variance at two nucleotide sites, will be generated in the length of time to generate one individual genome.

To label a change as random, where randomness implies the random selection of an element from a set of n unique elements, is to identify the change as the product of a random numbers generator to the base n. To label such a change as a random mutation, without adverting to it as the product of a random numbers generator, can lead to misunderstanding. Abandoning the concept of Darwinian random numbers generation at nucleotide sites in genomes, yet persisting with the terminology of random mutation at these sites, can be misleading.

Durrett and Schmidt, whose field of interest is population genetics, characterized the mathematics of Darwinian random mutation, i.e. the mathematics of random numbers generation, as naïve. Yet, they implied that Darwinian random mutation was their topic by the phrase, ‘the mean waiting time for two mutations to occur in the same individual’. Their actual topic was one of population genetics, which would have been evident, if they had used the phrase, ‘the mean generational time for two genetic markers to be inherited by the same individual, given their initial densities in the population’. Similarly, they alluded to the size of a subpopulation using the misleading phrase, ‘the probability of a mutation’, rather than the more clearly descriptive, ‘the material density or fractional concentration of an inheritable genetic marker within the breeding population’. The density of an inheritable marker in a population is not the probability of a mutation.


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