This seems like a rather unusual subject even to be considering, but it has been raised in a discussion with a biologist. Let me propose an answer from a creationist view. The Creator will, I suppose, choose the features of each organism to optimize its function, whatever that may be. So if there is some purpose P for a hypothetical organism to be created, there will be a function fP associated with it, measuring how closely a possible organism fulfils purpose P. Thus if X is a possible organism, fP(X) is a real number measuring how closely organism X fulfils purpose P. We can assume that fP is a function from organisms to real numbers such that the higher the value of the function fP(X), the more closely the organism X meets the purpose P. So if the Creator has a purpose P in mind, He will presumably create the organism X that maximizes the associated function fP(X). Since we are considering the DNA, we can consider fP as a function from the DNA sequence of the organism to the real numbers. For this, it is convenient to express the DNA sequence as a sequence of bits (zeroes and ones). We can express each base by two bit positions. One bit position will specify whether the base is a purine or a pyrimidine, and the other bit position will specify the base. Thus, with two bits, we get four possibilities, corresponding to the four DNA bases. We can then represent n bases (base pairs of DNA) by 2n bit positions, each bit being either a 0 or a 1.
Suppose there are two purposes P and P' which are very similar. Now, suppose the functionality fP(X) of this sequence for a purpose P can be represented by a linear function c1b1 + c2b2 + ... + cnbn, where the ci are real coefficients and the bi are either 0 or 1. This is an oversimplification, but it will illustrate the ideas. Suppose the functionality fP'(X) of this sequence for purpose P' can be represented by the linear function c'1b1 + c'2b2 + ... + c'nbn. We can assume that the Creator would choose the bi to optimize (maximize) this value. Thus if ci is positive, bi would be one, and if ci is negative, bi would be zero. If the purposes P and P' are very similar, ci and c'i would also be very close. Let A and A' be the organisms maximizing fP(A) and fP'(A'), respectively. Thus A is the organism that best fulfils purpose P, and A' is the organism that best fulfils purpose P'. Then we would expect A and A' to be similar organisms, because the purposes P and P' are similar.
Silent sites (typically third codon positions) may have some function, even though they do not affect the amino acid coded for. For third codon positions, therefore, the coefficients ci would be very small, corresponding to the fact that these positions typically do not have much of an effect. Thus the "noise" or random difference between organisms A and A' is likely to have a large effect on these ci, so we would expect many of the third codon positions to differ between A and A' because the signs of ci and c'i differ. However, for non-silent sites, we would expect most of the ci to have values much different from zero, so the optimum value of bi would usually be the same in both organisms. This explains why the silent sites would tend to differ more than the non-silent sites. However, sometimes we might find the silent sites nearly the same in organisms that are fairly similar.
This also explains the observed pattern of amino acid sequences in similar proteins between different organisms. Organisms that are more similar tend to have proteins (hemoglobin, cytochrome C, etc.) that are similar, and more distantly related organisms have greater differences in the proteins, as a rule. Some proteins, such as fibrinogen peptide, vary a lot between different organisms; this can be explained by the fact that the choice of amino acid has very little effect, and thus it is likely that an optimum choice will be very different for two different organisms.
There may also be some third codon positions that have no effect at all. For these, the choice may be determined by artifacts of the creation process, by esthetic considerations, or by other means (such as providing a type of "serial number").
Since the Creation, mutations have accumulated, so some of the observed patterns are not due to created differences, but have arisen more recently. In general, however, this latter contribution should be relatively small.
We hope that readers will not find this discussion to be too far out!
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