Monday, September 14, 2009

Homeotic Mutationism
The guest post a while back by Dr.  Filler brought up the issue of "adaptationism" vs.  "homeotic mutationism".  It seems to me that this issue is an example of the simplistic use of formulas in science (which I recently decried) where a more thoughtful approach would end up without the controversy.

Summarizing from most of the various papers I read (see Links), the foundation of "adaptationism" is the assumption that the genetic variation on which Darwinian selection operates involves such small increments of change that they appear continuous.  Mutation as a source of variation operates (in this model) almost independently of selection, adding new variation to the gene pool, while natural selection operates against the entire existing pool variation by altering the relative distribution of different alleles for genes with variation.  There also seems to me to be a tacit assumption that variation exists in any direction (within "morphospace", see here for a discussion of morphospace), so that anytime there is a selective incentive for evolution in a particular direction, it will happen.

(Afarensis pointed me to the 1996 book Adaptation edited by Michael Robertson Rose and George V. Lauder, as discussing the current "adaptationist" program.  Evidently, the "adaptationism" discussed by Dr. Filler, and the papers mentioned above, is what Rose and Lauder define as the "Old Adaptationism".  This is, by all appearances, dead.  The "new adaptationism" takes some account of the ability of mutations to be "large", as well as accepting the the reality of developmental limitations on variation.)

A Recently Generated Example of Homeotic Mutation

I'm going to (briefly) discuss a recent paper documenting an artificial homeotic mutation[15], in which an entire homeotic gene was rendered non-functional, and the impacts to various developmental systems was analyzed.  This is certainly not the first such experiment, however its recent provenance, and the widespread impacts to diverse systems including "thoracic, lumbar, and sacral vertebrae and in the pelvis, along with alterations in the bones and ligaments of the hindlimbs" and "a reduction in lumbar motor neurons and a change in locomotor behavior",[15] make it an excellent example of the sort of mutations that underlie the concept of "homeotic mutationism".

Before we start, however, let's take a look at genes and how they work.  Wiki defines a gene as "the basic unit of heredity in a living organism." This is fine as a functional definition, but the history of research into the subject has provided some excess baggage:  some deadwood that needs to be cleared away.  ...

I'm not going to even summarize the process by which DNA sequences are translated into proteins.  I've discussed this in past posts, and Wiki has a good description here (including the larger articles they link to).

Figure 1:  Summary level cartoon of the "central dogma" regarding transcription and translation from DNA sequences to proteins.  (From Wiki)

Specifically, the "central dogma" has often been taken to mean that a gene consists of the portion of DNA that codes for proteins.  This isn't actually true, according to Wiki: 
The central dogma of molecular biology was first enunciated by Francis Crick in 1958[ref] and re-stated in a Nature paper published in 1970:[ref]
The central dogma of molecular biology deals with the detailed residue-by-residue transfer of sequential information.  It states that information cannot be transferred back from protein to either protein or nucleic acid. 
As you can see, this "dogma" applies only to the translation of DNA sequences into amino acid sequences in proteins.  It says nothing about whether the gene is, by definition, limited to the protein-coding region.

A Local and Temporary Definition of "Gene"

For purposes here, I'm going to use a limited definition of gene:  only for protein-coding genes of Eukaryotes.A1  A gene is the DNA that codes for proteins, all the sequence that is transcribed with it, and all the control sequences that (in sum) determine when/whether that gene is going to be transcribed.

This is different from the more common (often unstated) definition of only the coding region.  It includes all the introns that are transcribed to RNA and then removed by editing prior to the beginning of translation.  It also includes all the control sequences by which its transcription is controlled.  From the point of studying mutations, adaptation, and selection, this allows most of the mutations that affect development to be included within the gene, by definition.

Technically, a gene also should include the DNA sequences that, while not affecting transcription, are close enough to do so if they should mutate from a neutral sequence to one that impacts transcription.

Now, many genes code for proteins that perform simple metabolic and/or "housekeeping" functions within the cell.  Others code for structural proteins, that are used to build the cellular skeleton.  But the most important genes, from the point of view of controlling development, are those whose proteins contribute to the activation of other genes.  The network of information handling these proteins are involved in constitutes a powerful and complex analog computer within the cell.

How do these proteins and their DNA interact? Basically, any protein that can interact on a sequence-specific basis with DNA is a Transcription Factor (TF).  (Technically, TFs are enzymes, as are all proteins that can catalyze a chemical process of any sort.  In addition, we should probably demand that the interaction with DNA have some significant effect.) TFs can work to enhance the rate of transcription, reduce it, suppress it entirely, or do any of the foregoing to the effects of other TFs.  They do this by spending part of their time connected to the sequence(s) they act on, and while they do other parts of the molecule interact with other enzymes, creating a complex analog logic.  (See my early post How Smart is the Cell? Part II:  The Gene Activation network as an Analog Computer for a more detailed discussion of this, and links to more technical and peer-reviewed discussions.)

All this depends on the interaction between the TF and the "Transcription Factor Binding Sites (TFBS or "binding site") that interact with" the TF.  Now, as I mention
A TF can interact with many binding sites, and its activity with each site will be independent of the others, except that when it's present in relatively small amounts, there will be competition among sites for TF activity.

The interaction between a TF and its binding site depends on the specific sequence of DNA in the binding site.  However, experiments with TF binding have shown that there are many sequences that will bind any particular TF, usually all very similar.  By comparing these sequences, it's usually possible to find a consensus sequence that is very similar to all of them.

The consensus sequence will generally have the highest binding energy, that is it will stick tightest to the TF.  However, other similar sequences may be able to bind to the TF, although with different behavior.  A few example consensus sequences are found in table 2 from The Evolution of Transcriptional Regulation in Eukaryotes.
[10] This means that when there are multiple binding sites with slightly different sequences, they will have different binding energies, and the activity will be different for the same concentration of TF.  Note also that there can be multiple TF's with similar (but non-identical) consensus sequences, so that different binding sites may bind to different TF's depending on the relative concentrations of the TF's.

The effect of each TF concentration on transcription rate will be generally analog.  Although a high enough concentration will saturate any particular binding site, producing full-bore transcription (assuming it's an enhancer), lower concentrations will cause each TF/binding site activity to perform an analog calculation.
The previous post was concentrating on how the gene activation network makes up a complex analog computer, but here I'm going to concentrate on the way the various types of mutation interact with this computer.

The Effects of Point Mutations

Let's start with a simple point mutation, the replacement of one base in the DNA sequence by another.  When this happens in a coding sequence, it may be "silent", coding for the same amino acid, or "neutral", coding for a different amino acid that, however, doesn't make much (or any) difference to the protein function, or it might have an important effect.  In any event, it affects every interaction of the protein, one way or another (and to some extent or another).  But a change to one of the binding sites only affects that specific interaction with the TF(s) involved.  Many of these changes have a very small effect on binding energy (which, in turn, has a small effect on the transcription rate).  Others have a larger effect, sometimes much larger, but only on the transcription rate of the one gene involved.

But what happens when there's a similar point mutation to the coding regions for a TF? Here, the mutation may have an effect on many, sometimes many hundreds of, interactions.  If every binding site had the same sequence, the effect of a point mutation to the TF would be the same for all the "downstream" genes it affects.  However, the actual sequences at the binding sites often vary, and a change to the TF could well increase the binding energy for some binding sites, while reducing it for others.  Moreover, the magnitude of the change can also vary, with some being minor and others major.  Finally, the results of these changes will (often) differ depending on how the specific transcription logic for each binding site fits into the overall computing network.

This, then, is the foundation for homeotic mutations.  Not every mutation to a TF constitutes a homeotic mutation, but there are some TFs that are involved in hundreds (or even thousands) of transcription control operations, and when a mutation occurs to such a gene, the effects can be widespread, unconnected, and apparently random.  This, specifically, is the type of mutation that Dr. Filler has hypothesized occurred when the ancestors of the Great Apes split off from other lineages around 20MYA.

Now, the number of bases involved in any TF/binding site interaction might be around 5-20.  There are many possible mutations that could occur either to the binding site or the TF, but the effect of any change in binding energy will usually be limited to a single (scalar) change.  The more different binding sites a TF interacts with, the more "dimensions" a mutation to the TF can make changes to.

Although the case is not a simple example, I should mention that several Hox proteins (the quintessential homeotic genes) act on the same consensus sequence, but with: 
subtle, but distinct, preferences to DNA sites that contained variations of the nucleotides within the consensus motif.  We further showed that Hox proteins varied in their relative affinities for DNA.  These data demonstrate that closely related Hox proteins exhibit subtle differences in DNA binding specificities and affinities.  These differences are likely to contribute to the selective interactions of Hox proteins with target DNA sites in vivo.[11]
These aren't recent mutations, the hox genes have been evolving separately for over half a billion years, but they are highly conserved, with many specific genes (coding sequences) showing almost identical activity when mouse coding sequences are grafted into fruit flies in place of the native version, and vice versa.

What this shows, then, is that changes to homeotic genes can alter the binding affinity to a large number of different sequences, in different ways.  A new homeotic mutation would probably be much "rougher" in its effects, at least until subsequent mutations to the binding sites "smoothed out" the effects, but such mutations could certainly occur even with a point mutation.

Effects of Other Types of Mutation on Homeotic Genes

Several other types of mutation need to be considered here.  One type is where a section of code from another, related, gene gets grafted into the coding sequence in place of an original of similar length.  This could well happen during DNA repair of a gene with many relatives, such as the Hox genes.  (Specifically, the repair of a double-strand break through homologous recombination, which could potentially use a related gene as a template rather than the other copy of the original.)

Here, we potentially have the recombination of the part of the protein that interacts with one end of the consensus sequence with the part from another, related but not identical, protein that interacts with the other end.  The result could be a gene with widely changed interactions with the various sequences it controls.  Moreover, the "grafted in" section has undergone its own long evolution, creating its own system of binding energies with different sequence fragments.  Although the mutation is "random" in the sense that a the new protein's affinities to the various sequences it controls will change in unpredictable ways, it's "random" within a very constrained space of possible changes.

Unlike point mutations, these mutations are capable of producing enormously organized suites of changes, although such suites will almost always be mal-adaptive.  But not always, just almost always.

Another type of mutation also involves DNA repair, but this time instead of grafting in a homologous sequence from a related functional gene, the replacement sequence comes from a pseudogene originally created as a non-functional copy of this gene, which has been sitting in the genome accumulating random mutations for some time (ranging from very recent to somewhat older).  The importance of this type of mutation is that several point mutations can accumulate without having any effect (because a pseudogene doesn't get expressed), until a short sequence is copied into the functional gene during DNA repair.  As with most mutations, the very large majority would be mal-adaptive, but those that are viable can cross deep valleys in the "fitness landscape" that a working gene can't cross because any of the individual point mutations would be lethal or very mal-adaptive.

One more type of mutation must be considered, and that's a change to the control sequences of the homeotic gene itself.  While such a change wouldn't affect the binding affinities between the homeotic gene and the downstream genes it affects, it could cause the gene to be expressed in places it wasn't previously, with all sorts of potential odd effects.  Like the other types of mutation I've discussed, the vast majority of such mutations would probably be mal-adaptive, but the occasional adaptive one could produce homeotic effects.

There are other types of mutation that could produce potentially viable homeotic mutations, but the ones discussed are sufficient for an explanatory example.

An Artificial Example of a Homeotic Mutation

Now that we've examined just what a homeotic gene is, and how mutations to their protein coding can have such widespread effects, let's return to the example I mentioned above.[15]  Axial and appendicular skeletal transformations, ligament alterations, and motor neuron loss in Hoxc10 mutants by Sirkka Liisa Hostikka, Jun Gong, and Ellen M. Carpenter.

The specific gene involved is named Hoxc10.  We must note that in mammals the hox genes are actually defined in four separate families, the result of "the ancestral vertebrate genome being twice duplicated in its entirety", or at least one complete duplication and one duplication of both hox families.[16]  Hoxc10 is part of the third ("c") family of hox genes, descended from the Hox10 gene of the original single string in the old pre-vertebrate chordates.

What Hostikka et al. did was create a version of this gene "producing a protein lacking the ability to bind to DNA."[15]  In mice where the allele was homozygous (where both chromosomes had this inactive allele), results included the following: 
The lack of a functional Hoxc10 homeobox causes several homeotic transformations in the axial skeleton (Table 1, Figures 3 and 4).  Wild-type C57Bl/6 mice typically have 30 precaudal vertebrae, with the more caudal vertebrae organized in T13L6S4 pattern, with thirteen thoracic vertebrae, six lumbar vertebrae and four sacral vertebrae [refs].  Eighty percent of the wild-type mice examined in this study exhibited this pattern, with the remainder of the animals showing some mild variation in the shape of the L1, L6, or S1 segments; these types of variations are within the range of normal [ref].  Hoxc10 mutant mice also have 30 precaudal vertebrae, but the patterning of the vertebral column is altered.  Most Hoxc10 mutant mice (33 of 34 examined) show a partial to complete transformation of the thirteenth thoracic vertebrae, precaudal vertebra 20 (PC20) into a lumbar identity, typified by the reduction or complete loss of the thirteenth rib.  By definition, thoracic vertebrae are those vertebrae with attached ribs; hence loss of the thirteenth rib suggests a posterior transformation of the most caudal thoracic vertebra into a lumbar identity.  The most caudal lumbar vertebra (PC25) often undergoes a similar transformation into a sacral (S1) identity (Figures 3 and 4).  The sacrum in wild-type animals is typically comprised of four vertebrae.  The first two vertebrae, S1 and S2, have butterfly-shaped, fused transverse processes.  The next two sacral vertebrae, S3 and S4, have transverse processes that are not fused, with S3 having butterfly-shaped transverse processes and S4 having club-shaped transverse processes similar to those seen on more caudal vertebrae (Figures 2 and 3).  In Hoxc10 mutant animals, the fourth sacral vertebra (S3* in Figures 3 and 4) exhibits an intermediate shape between a normal S3 and S4, and there are usually five sacral-like vertebrae, three fused and two free.  This suggests an anterior transformation of the first caudal vertebra into a sacral identity.  This transformation restores the register of the vertebral column and thus there is no overall loss of precaudal vertebrae.  In combination, these alterations produce a T12(T13/L1)L5S5 pattern in 94% of mutant mice.  Heterozygous mice show several intermediate patterns, most commonly T13L5S5 (31.5%) or T12(T13/L1)L5S5 (38.9%), suggesting dosage compensation.  The transitional vertebra, defined as the most anterior vertebra to show a lumbar rather than a thoracic articulation between the pre- and postzygapophyses [refs] is normally the tenth thoracic vertebra (T10), whereas in the homozygous Hoxc10 mutants the transition occurs at the ninth thoracic vertebra (T9) (Figure 4).[15]
In addition to the changes to vertebrae: 
The bones in the hip, the ilium, ischium and pubis, are constructed from independent condensations that grow together after birth.  All three bones meet at the acetabulum, but the ischium and pubis also meet at the ventral edge of the pelvis.  These two bones are angled 45° from each other on the anterior, acetabular end, and are connected by a cartilaginous bridge where they meet on the posterior side.  This bridge undergoes calcification during puberty, and the pelvic apparatus is normally fully fused by eight weeks of age.  In Hoxc10 mutants, the cartilaginous bridge forms normally, but the calcification process is delayed, leaving a prominent seam or indentation on the posterior edge of the pelvis, dyssymphysis ischio-pubica.  In all mutant adults the bones touch each other but even in the mildest cases, a seam is visible unlike in the controls (Figure 5A, B; Table 2).  The C57Bl/6 background strain has been reported to have high incidence of dyssymphysis, especially in females [refs], however, somewhat newer reports have claimed only 30% incidence which was lost when C57Bl/6 mice were crossed to another strain [ref].  To minimize differences attributable to genetic background, wild-type sibling controls were used for all experiments.[15]
The hindlimbs were also affected: 
There are several alterations in hindlimbs and hip joints in Hoxc10 mutants.  During embryonic and early postnatal development, there are no visible skeletal defects in the hindlimbs.  However, by four to six weeks postnatally, a bony ridge develops along the anterior longitudinal line of the shaft of the femur (Figure 5).  A likely cause for the formation of this femoral ridge is the presence of an ectopic branch of the iliofemoral ligament.  The iliofemoral ligament is a sheet-like structure on the anterior side of the femoral neck, acting as a part of the synovial capsule.  Normally, the ligament joins the ilium anterior to the acetabulum of the hip joint to the intertrochanteric line of the femur.  During development, this ligament expresses Hoxc10 (Figure 2C-F).  The ischiofemoral ligament connects the dorsal aspect of the ischium to the femoral neck.  In the mutant, the ischiofemoral ligament is visibly weaker than in the wild-type mice, and attaches more dorsally to the medial side of the rim.  The iliofemoral ligament, on the other hand, has two femoral connections in the mutants:  one normal connection that attaches as a part of the synovial capsule into the femoral neck and another that attaches onto the anterior shaft of the femur (Figures 5 and 6).  This second ectopic branch attaches to and is part of the anterior femoral ridge.  The point of attachment varies from just distal to the intertrochanteric line to halfway down the femur, likely affecting the extent of ridge calcification that varies between animals and progresses with age.  This phenotypic attachment site variation likely drives the differences in the shapes of the femoral ridges of the mutants from prominent to moderate, reflecting additional stresses on the bone.  The femoral ridges are found in all the mutants six weeks and older (Table 2).  Histological study of the extra ligament shows an organized fibrillar structure, indicating no aberrant pathology (Figure 6C).  Serial sections of newborns show the ligament attaching to the femoral shaft (Supplemental Figure 2), also seen in some newborn skeletal preparations as Alcian blue-stained material in the ridge area indicating cartilaginous material, likely sulfated proteoglycans, at the femoral end of the abnormal ligament (Figure 5F, G).[15]
There were also effects on the nervous system: 
Initial analysis of peripheral nerve projection patterns in Hoxc10 mutant embryos using anti-neurofilament antibody labeling suggested that there were no gross defects in the formation, appearance, and projection of spinal nerves in Hoxc10 mutant embryos.  Motor and sensory projections into the developing hindlimb bud appeared normal as well (Figure 2).  However, in light of alterations in motor neuron positioning in paralogous Hox10 gene mutants [refs] we decided to examine these neurons more carefully in serial histological sections collected from newborn animals.  Serial 10 µm sections were collected from three wild-type and three Hoxc10 mutant animals.  Lumbar segmental position was established as previously described [ref] and the number of motor neurons in the medial and lateral motor columns (MMC and LMC, respectively) in the L1-L4 spinal segments was counted.  All motor neuron populations showed substantial reductions in Hoxc10 mutants (Figure 8).  We further distinguished both medial and lateral components of the LMC; both components showed similar reductions in numbers of motor neurons.  Despite the reduction in numbers of motor neurons, the distribution of these neurons appeared relatively unchanged, with MMC and LMCM neurons showing a peak in their distribution in segment L2 and LMCL neurons peaking in segments L3/L4.  This suggests an absence of an anterior spinal segmental transformation, in contrast to results observed in Hoxd10 mutants and Hoxa10/Hoxd10 double mutants [refs].  These cell counts appear to reflect an absolute loss of motor neurons, as there is no evidence of compensation in one pool for losses in another.[15]
Note that these changes are the result of complete elimination of DNA-binding functionality.  If the mutation had made small changes to the affinity for all the possible binding sites (rather than setting them all to zero) any or all of the areas found in this study might have been affected, but in different ways.  Of course, as mentioned above, the hox genes (in mammals) are a complex family of multi-duplicated genes, with similarities in affinities between members of the same family.  Thus, mutations to Hoxc10 will interact with Hoxa10 and Hoxd10 (there's no hoxb10), with the possibility for much more complex results.

The Adaptive Fitness of Homeotic Mutations

The subject gets sort of sticky here.  As with most mutations, the ratio of "adaptive" to "mal-adaptive" mutations is usually very small.  However, the number of possible mutations is also extremely tiny compared to the "morphospace" within which such developmental changes operate.  This is especially true because any change to the protein coded by a homeotic gene will have a specific effect on each possible sequence used in a binding site.  OTOH, in the case of a normal homeotic gene (as opposed to the complex families of Hox genes), the only thing that really matters for any specific binding site is the affinity its sequence has for the TF.  There can potentially be a number of different sequences for the same binding site (multiple alleles of the same gene, using my definition of gene) with essentially identical affinities for the old TF, that would respond differently to the mutant version.  Thus, closely related species could have several, or even many, binding sites with different sequences that all act identically (between species) with the old TF, but react differently to the same mutation.

Indeed, such an effect doesn't even require different species.  Since a mutation to a specific binding site is "silent" if it doesn't affect the affinity for the old TF, even a small population could have several different versions of the same binding site, with different responses to the same mutation.  Since the gene for the TF will often be on a different chromosome from the "downstream" genes the TF controls (including their binding sites, which are usually near the coding region on the same strand of DNA), the mutant form will undergo the full "shuffling" activity of sexual recombination.

Mutationism and Adaptationism

Now, lets take a look at how all this mixes with "adaptationism".  Many point mutations to binding sites will have a very small effect, either because they have a naturally small effect on affinity, or because the logic of the control network causes even a large change to affinity to have a small effect on overall phenotype.  This is probably the most common source of "variation" found in typical animal morphology.  Mutations with larger effects will occur, but will have a much larger chance of being mal-adaptive and thus being selected out of the population.

An important aspect, however, is that the only variation that this type of mutation can produce in the phenotype is that which can be caused by changes to the affinity of one or more binding sites for their TF(s).  Thus, the morphospace within which the typical variation resides is much "flatter", ie. it has many fewer dimensions, than the changes that can be imagined for observed morphology.  This places limits on how evolution can adapt a population for changing conditions:  if a particular position in morphospace can't be reached via changes to binding site affinities, the form involved won't appear in the species no matter how strong the selective incentive.

What mutations can occur, sooner or later do, assuming a large enough population.  They are then tested for "fitness" within the population, with mal-adaptive mutations quickly disappearing (with the rare exception of one that happens to be linked to a highly adaptive one).  However, as the probability of a mutation decreases, the size of a "large enough" population increases.  When the critical size for a mutation to likely occur becomes enough orders of magnitude larger than the typical population integrated over the entire lifetime of the species,A2 the chance that that mutation will occur during the existence of the species in question becomes small, and the activity of evolution by natural selection becomes much less like the "adaptationist" model, and much more "mutationist", indeed, such mutations, many of which are homeotic, fit fairly cleanly into the old "hopeful monster" paradigm.

And it is here that we come to the point of the dispute.  It can't be denied that many mutations produce very small changes in one or a few phenotypic characteristics.  These, in the model I've discussed, will usually be changes to the control sequences rather than the amino acid sequence of the protein coded for.  This means that the adaptationist models are certainly valid for evolution, both within species and to create new species.  But the existence of homeotic mutations, and our modern understanding of how the various TFs and their binding sites combine with other aspects of cellular intelligence to drive the fetal development of the organism, make the "hopeful monster" paradigm just as valid, if in a substantially updated form.

When it comes to the many types of homeotic mutations that have such a low probability of occurring that there's, say, only a few percent chance of it happening during the lifetime of the species, we have pretty much left "adaptationism" behind, and are into the realm of "random creativity".  In this type of scenario, the rare adaptive homeotic mutation can cause the origin of a new species, or higher level clade, if it occurs, but that occurrence is a matter of random chance.  This is not to suggest that such mutations are in any way "directed", or that there are other forces than natural selection that determine the success of a particular homeotic mutation.A3  Rather, the "inevitability" of any particular homeotic mutation has been totally falsified (again, only in the case of the very low-probability ones), and the course of evolution is basically determined by the random chance of which such mutations arise when.

We see, then, that the "debate" between "adaptationism" and "mutationism" is either a matter of emphasis, or the denial of one or another valid mechanism by extremist proponents of the other.  Both are valid, and technically even the most "hopeful monster" oriented form of homeotic mutationism still depends on natural selection and adaptation to pick out the rare adaptive mutation from the much larger mass of mal-adaptive.

Hostikka SL, Gong J, & Carpenter EM (2009). Axial and appendicular skeletal transformations, ligament alterations, and motor neuron loss in Hoxc10 mutants. International journal of biological sciences, 5 (5), 397-410 PMID: 19623272


A1.  only for protein-coding genes of Eukaryotes:  The differences between Eukaryotes and Prokaryotes include major differences in how they use DNA:  in Eukaryotes the coding section of each gene is usually isolated along with its control sequences, so that different genes can reside on different chromosomes.  In prokaryotes, most coding sequences tend to be combined into operons, with many genes being controlled by the same (set of) control sequences.

In addition, there are, in both types of life, genes that code for ribozymes (strands of modified RNA that function as catalysts in the manner of enzymes).  These are often combined into operons even in Eukaryotes, so they don't fit into the logic of my argument here.

A2.  the entire lifetime of the species:  This makes sense at a high level, however when you dig into the details it gets a little iffy.  For the typical species that lasts a few million years, with a population of millions to billions, if the probability of occurrence of a mutation is less than 1/total number of members (over the entire lifetime of the species), there will be significant chance that it will never occur.  If it is much smaller, then the chance of its occurrence at all becomes small as well.

Where things get complicated is for species that have very large populations, especially divided into many distinct sub-populations, and that last a long time, say an order of magnitude longer than the typical few million years.  Here, "silent" changes to various binding site sequences will have enough time to accumulate and diverge among sub-populations and over time, so that the exact same mutation to a homeotic gene will have different effects at different times and in different populations.  From an adaptive standpoint, these would count as different mutations, although the change to the DNA of the homeotic gene itself is identical.

A3.  other forces than natural selection that determine the success of a particular homeotic mutation:  As it happens, there are a number of mechanisms by which one copy of a chromosome, with all the alleles it carries, can be preferentially selected during oogenesis or spermatogenesis.[14]  I'm not going to discuss these here, although they cannot be ignored.  See reference 14 for a more complete discussion.

Links:  I've only included the link called out in this leader.  Not all of these links are called out in the text.  Use the back key if you came by clicking a footnote. 

1.  Adaptation from Leaps in the Dark Open Access

2.  Mutationism and the dual causation of evolutionary change paywall

3.  Climbing Mount Probable:  Mutation as a Cause of Nonrandomness in Evolution paywall

4.  Mutation-Biased Adaptation in a Protein NK Model Open Access

5.  The concept of developmental reprogramming and the quest for an inclusive theory of evolutionary mechanisms paywall


7.  A framework for evolutionary systems biology Open Access

8.  Saltational evolution:  hopeful monsters are here to stay Open Access

9.  The new mutation theory of phenotypic evolution Open Access

10.  The Evolution of Transcriptional Regulation in Eukaryotes Open Access

11.  Hox proteins have different affinities for a consensus DNA site that correlate with the positions of their genes on the hox cluster Open Access

12.  Adaptationism by Peter Godfrey-Smith and Jon F. Wilkins (final draft:  for the Blackwell Companion to the Philosophy of Biology.  This draft has probably been peer-reviewed, but I'm not certain.)

13.  Three Kinds of Adaptationism by Peter Godfrey-Smith (In S. H. Orzack & E. Sober (eds.), Adaptationism and Optimality, Cambridge University Press, 2001, pp. 335-357.  This is not the published version, and may be prior to peer-review.)

14.  Genes in conflict:  the biology of selfish genetic elements By Austin Burt and Robert Trivers

15.  Axial and appendicular skeletal transformations, ligament alterations, and motor neuron loss in Hoxc10 mutants Open Access

16.  Extensive genomic duplication during early chordate evolution Open Access

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