We have just submitted a manuscript. The abstract is here.

**The Genetic Equidistance Phenomenon at the Whole Proteomic Level**

Denghui
Luo and Shi Huang

Abstract

The
field of molecular evolution got started with the alignment of a few protein
sequences in the early 1960s. Among the first results found at the time, the
genetic equidistance result, has turned out to be also the most astonishing and
unexpected by any evolutionary theory of the time. It hence directly inspired
the

*ad hoc*universal molecular clock hypothesis that in turn inspired the neutral theory. Unfortunately and unknown to most, however, what is only a maximum distance phenomenon was mistakenly transformed into a mutation rate phenomenon and became known as such. Previous studies have suggested the universality of this phenomenon based on results from a small set of selected proteins. We have now confirmed this by whole proteome wide studies of 7 different sets of proteomes involving a total of 15 species. All 7 sets showed that within each set of 3 species the least complex one is approximately equidistant in average proteome wide identity to the two more complex ones. Thus, the genetic equidistance result is a universal phenomenon of maximum distance. There is a reality of constant but stepwise increase in complexity during evolution, the rate of which is what the original universal molecular clock is really about. These results provide additional lines of evidence for the recently proposed maximum genetic diversity (MGD) hypothesis.**Figure 3. The constant rate of complexity increase.**The fraction of identical residues between human and a lower complex species is equivalent to the fraction of non-changeable sites in the lower complexity species. The fraction of identical residues in cytochrome C (identity divided by length) between human and each of the species listed in the figure is plotted against the separation time between human and each of the listed species. Data for plots were obtained using homo cytochrome C to BLASTP Genbank.

**Figure 4. The prime number staircase.**The graph counts the cumulative number of primes up to 100.

The molecular clock interpretation of the maximum genetic
equidistance result is really about the constant rate of complexity increases.
People since Aristotle have long appreciated the direction of evolution towards
higher complexity. Darwin’s theory has long denied this but only by ignoring inconvenient
facts including the genetic equidistance phenomenon. The evidence for
complexity increase is commonplace and easy to notice by common sense. The
first molecular evidence for it is the maximum genetic equidistance phenomenon.
What is most striking is the nearly constant rate as measured in years of the complexity
increase, which could be quantitatively studied by the fraction of non-changeable
positions in a protein or the fraction of identical residues between human and
a lower complexity species (Fig. 3).

As nature is written in the language of mathematics,
it would be most unusual if the most fundamental natural phenomenon, i.e., the
constant rate of evolution towards higher complexity as measured in years, has
no counterpart in mathematics. The most relevant mathematics that we could find
is the pattern of prime numbers (du Sautoy 2003). The cumulative increase in prime numbers along
the progression in natural numbers is well known to follow a nearly constant
rate (Fig. 4). Here the progression in natural numbers is like a time clock,
rigid and predictable. The appearance of prime numbers is discontinuous or
staircase and unpredictable but follows nonetheless a well defined function
Li(N) as shown by the Riemann hypothesis, widely known as the most important
unproved problem in mathematics (du Sautoy 2003). Each new appearance of a more complex species is
like a new prime number, unpredictable, discontinuous, and yet constant. Individual
species are well known to appear in the fossil record abruptly as evidence for
the punctuated equilibrium model of macroevolution has shown (Gould, Eldredge 1993). However, the discontinuous appearance of higher
and higher complexity species still follows a very smooth and regular pattern
as shown by the equidistance phenomenon. We speculate that the mystery behind
the constant rate of complexity increase in nature might well turn out to be
the same as that behind the constant appearance of prime numbers. Indeed, the
common speculative and unproven answer to both mysteries has long been random
forces.