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Explosive Ground-Setting

by MikeGene

Okay, so I'm a little slow. In 2001, Takashi Miyata and Hiroshi Suga [1] analyzed eight different animal-specific genes involved in cell-cell communication and developmental control. Each of these gene families has spawned a myriad of subfamilies through gene duplication and divergence. But when did all this inventive evolution occur? Here is their summary:

These results have several evolutionary implications. (1) Frequent gene duplications that gave rise to different subfamilies with distinct domain organizations and functions occurred in the very early evolution of animals, and these subfamilies were formed before the divergence of parazoans and eumetazoans, the earliest divergence among extant animals. (2) These duplications are characterized by explosive occurrence within a limited period, instead of proceeding gradually. (3) These duplications occurred at approximately the same time (i.e., in a limited period before the parazoan-eumetazoan split) for different gene families examined to date. It is not known, of course, whether these duplications occurred piecemeal or involved genome duplications. (4) The explosive subfamily-generating duplication precedes the Cambrian explosion by about 200-300 myrs. This strongly suggests that there was no direct link between subfamily-generating duplications (i.e., gene diversification) and the Cambrian explosion (i.e., organismal diversification). Instead, these duplications might be related to the acquisition of multicellularity during animal evolution rather than to the Cambrian explosion.

They also note: "Almost all the present-day subfamilies were established within this period."

The researchers looked more closely at the domain organization of some of these gene products:

As Figure 3 shows, the domain organizations of the PTK sequences are virtually identical among human, Drosophila and sponge for each of the subfamilies, except for the ror/Musk subfamily, in which the domain organizations appear to differ for different species groups.(14) This result strongly suggests that most, if not all, of domain shufflings antedate the divergence of parazoans and eumetazoans. A similar result is also found in the PDE family,(19) which provides further supporting evidence for the ancient domain shufflings before the parazoan-eumetazoan split.

So there is this explosive, inventive bit of evolutionary change that may be associated with the origin of multi-cellularity and this just happens to perfectly set the groundwork for the subsequent evolution of all animal phyla. As far as these genes are concerned, the rest of evolution was about tinkering and tweaking what this burst of innovation handed it, as subsequent rounds of duplication and divergence were not needed to evolve the different body plans. All that work was done hundreds of millions of years earlier. In essence, this ancient, early round of gene duplications got it right from the start.

1. BioEssays 23:1018-1027.

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22 Responses to “Explosive Ground-Setting”

  1. solip Says:
    February 14th, 2008 at 4:29 am

    This is an unrelated comment. I just don't know where to post this.

    Anyway, I think the issue I am going to present below would be a good point of discussion here in TelicThoughts. Actually, I think it is the most fundamental question anyone should ask before hypothesizing about other things.

    Recently, I've been reading about solipsism. (Perhaps, you all know what it means but I'll define it anyway.) Solipsism is the highest form of skepticism which states that either: only I exists (metaphysical) or the only thing I could ever know (with 100% certainty) is that I exist (epistemological).

    This is logically coherent. It is actually possible that everything is just my illusion or that I am the sole conscious being who creates reality. Every dialogue I hear from other people, every information I collect actually came from me. Perhaps, I was just bored by being alone so I created a reality with people who seem to possess minds but the thing is, they really don't. Perhaps, I just intentionally forgot that I created everything so that I can live in a dream world where I am not alone.

    Of course, this is totally absurd if taken at face value. But the thing is, if one would think about it, there is a possibility that it is true. There is nothing that could refute solipsism because it is internally consistent. I am not saying that it is true. It's just that there is no way to verify whether it is true or not.

    Just like in a dream — when I dream, the pavement I walk on seems solid, the people I talk to seems to be alive, the people I talk to seem to tell me things that I did not consciously make up (e.g. I meet a guy in my dream and he says, "You know that guy is a crank, blah, blah, blah" whereas I am not consciously thinking about that particular sentence at that moment [where did it come from?]). I also do not doubt that the reality I experience in my dream until I wake up. I also create the reality in my dream by "mere thought."

    The questions now are:

    1. How can I verily say that this reality is not just my own concoction/imagination [just like in a dream]?

    [You might say that I cannot do anything I want here and now (unlike in a dream) but a solipsist can say that perhaps, he sort of programmed everything before he entered the dream world so that they will remain consistent and constant until the solipsist's imagined death wherein he'll come back to his existence as the sole being. A solipsist can intentionally lie to himself so that he'll feel the comfort that other minds exist.]

    The only refutation I can think of right now is that it is more logical for a solipsist to actually create "other minds" (to solve his problem of being alone) than to do an elaborate delusion that there are indeed minds where there is actually none.

    The problem with the solution is: Is it logically coherent to create other minds? I mean, is it logically coherent to detach from yourself a particular consciousness so that it can live independently? What if there is really just one conscious being (monistic) who just lives different lives by creating different realities (at one point or another)? What if, it is logically incoherent that "two minds exist" or that a "mind can create another mind" Surely I can create thoughts, I can perceive, I can think. But I cannot think of a logical way (in principle) of how to create another being who can have separate thoughts from me.

    I think this is a serious philosophical and metaphysical problem: What if you're the sole being and you just created everything, every bit of information, every relative, every friend, every plant and animal are all forgotten figments of your imagination that just continue to linger (pre-programmed) just to make your grand delusion believable?

  2. Comment by solip — February 14, 2008 @ 4:29 am

  3. MikeGene Says:
    February 14th, 2008 at 8:09 am

    Hi solip,

    For future reference, any thread tagged as Rabbit or Duck is also considered an open thread.

  4. Comment by MikeGene — February 14, 2008 @ 8:09 am

  5. Todd Berkebile Says:
    February 14th, 2008 at 11:31 am

    Solip: Faith and faith alone can help you, its not a scientific issue.

    Mike: In essence, this ancient, early round of gene duplications got it right from the start.

    I don't think this conclusion follows from the evidence. I had this same reaction when reading your book. Any event that requires specific conditions will obviously only occur while those conditions are true. The fact that this event does not continue to occur constantly and forever more doesn't mean that it occurred perfectly the first time, but might instead mean that the conditions for it to continue occurring are no longer present. So "got it right the first time" is only one possible explanation from among many. Since we have no direct data of the period in question its presumptuous to assume one specific conclusion over all the others. At best you are highlighting one possibility which itself merely hints at the conclusion you wish to reach.

  6. Comment by Todd Berkebile — February 14, 2008 @ 11:31 am

  7. Zachriel Says:
    February 14th, 2008 at 11:32 am

    Gene families involved in signal transduction and developmental control are characterized by two active periods of gene duplication and diversification. Most of the families were established before the eumetazoan split, but a second period of diversification occurred around the time of the origin of gnathostomes (1).

    Miyata and Suga: Instead, these duplications might be related to the acquisition of multicellularity during animal evolution rather than to the Cambrian explosion.

    So the Cambrian Explosion was not an event without precedent as might appear at first blush, but one that had substantial groundwork. Interestingly, the Cambrian Explosion occurred during the quiet period between these rapid diversifications of developmental genes (1).

    MikeGene: As far as these genes are concerned, the rest of evolution was about tinkering and tweaking what this burst of innovation handed it, as subsequent rounds of duplication and divergence were not needed to evolve the different body plans.

    After all, humans are 'just' elaborated deuterostomes. A tube with appendages to stuff food into one end. Microevolution.

    Miyata along with Iwabe (2) did some related studies on the diversity of eukaryotic genes in the metazoan lineage. They determined there were three periods of divergence, the first well before eukaryotes, then before eumetazoa, then again about the time of gnathostomes. Suga, Iwabe, Miyata, et al. (3) showed that additional gene duplications occurred within each family of developmental genes at the fish"“tetrapod split.

    The key point I think you are raising is that the evolutionary rates of the developmental genes decreased with increasing organismal complexity during animal evolution, as has been shown by Hoshiyama, Kuma and Miyata (4). They suggest that this is due to developmental constraints.

    "”

    (1) Divergence pattern of animal gene families and relationship with the Cambrian explosion, 2001

    (2) Kinesin-Related Genes from Diplomonad, Sponge, 2002

    (3) Sponge Pax cDNA Related to Pax-2/5/8 and Ancient Gene Duplications in the Pax Family, 2004

    (4) Extremely reduced evolutionary rate of TATA-box binding protein in higher vertebrates and its evolutionary implications, 2001

  8. Comment by Zachriel — February 14, 2008 @ 11:32 am

  9. Stephen Says:
    February 14th, 2008 at 11:45 am

    So it seems that these genes, and their protein products, are context dependent; meaning that function changes with context. The genes are two-sided: their function; and the teleological context (front-loaded) that supports the function. This would be the only way to grow gene families, and so one could argue that life is underwritten by this process of growth that is found even in the genes and proteins.

    This is unlike machines, our machines show only a contrivance (created by us humans); our machines to not grow themselves. Otherwise show me the toaster that feels itself backing bread? And show me the computer that feels itself playing chess?

    There is an emotional unwillingness to see intelligent design as a simple contradiction of Darwin's theory; and this unwillingness is apparent in both creationists and Darwinists.

    The innate equivocation is presented as a contrivance, an agenda that is full of dirty tricks, or what has come to be known as intelligent design with all the self-evident tension that design brings, where the covert middle term is hidden beyond the synthesis of the representation (blind function) and its recognition (teleological context).

    We would have nothing if we could not contrive our feelings and turn them into words that are successfully communicated. There is no machine that can do what we can do, and this is because contrivance has been a two-sided affair that comes with feelings. Dennett's machines (his cranes) can only be contrived to feel, and this particular contrivance slips away from Dennett's awareness because be lacks the emotionality (self awareness) to see it.

  10. Comment by Stephen — February 14, 2008 @ 11:45 am

  11. Thought Provoker Says:
    February 14th, 2008 at 12:41 pm

    Hi Mike,

    I would love to look into this further and make a thoughtful and encouraging comment. However, I am so far behind my real life work, I can't.

    So let me just point to my favorite pre-cambrian animal, the Vernanimalcula guizhouena. Here is a link where I talked about it.

    You don't have to be an ID proponent to see that quite a bit was already "front loaded" into this interesting creature (including paired external pits that were sensitive to light).

    Let me just add…

    Yeah, more science stuff! :mrgreen:

  12. Comment by Thought Provoker — February 14, 2008 @ 12:41 pm

  13. Zachriel Says:
    February 14th, 2008 at 1:22 pm

    Scale-invariance is a mathematical structure exhibiting a power-law distribution.

    An example of a scale-invariant structure is the archtypical airport traffic grid. It starts with a single airport in Chicago, followed perhaps by another in New York. Over time, new airports connect to Chicago and New York. Later still, newer airports will connect to these main cities as well as to established outliers. To travel from Sacramento to Albany might mean taking a flight to Los Angeles then transferring in Chicago to New York, then taking a commuter flight from New York to Albany.

    For illustration, this is a typical scale-invariant network. But this one, however, is the pattern of an epidemic!

    One of the characteristics of a scale-invariant network is that certain nodes, typically older ones, are critically important. If Albany gets snowed in, it disrupts air travel to upper New York State. But if Chicago gets snowed in, it disrupts travel all across the country. Is there something special about Chicago? Is it the best possible location for a major hub of air travel? Not necessarily. It may just be a happenstance of history, or best at the time, or just conveniently located for the people who were most involved, but not necessarily the best today.

    Another characteristic of scale-invariant networks is that newer connections are much easier to change than older ones. Meanwhile, we tend to see lots of small changes, a few large changes, and only very rare revolutions.

    Scale-invariance is a natural consequence of local, preferential attachment under global pressure. Genetic networks are scale-invariant.

  14. Comment by Zachriel — February 14, 2008 @ 1:22 pm

  15. chunkdz Says:
    February 14th, 2008 at 5:02 pm

  16. Comment by chunkdz — February 14, 2008 @ 5:02 pm

  17. Zachriel Says:
    February 15th, 2008 at 10:39 am

    Zachriel: Genetic networks are scale-invariant.

    Just as the mathematical structure of a nested hierarchy is the natural consequence of divergence along uncrossed lines, the mathematical structure of a scale-invariant network is the natural consequence of local, preferential attachment to a network. By examining such a network, we can distinguish contributions of random attachment, preferential attachment, and global design.

  18. Comment by Zachriel — February 15, 2008 @ 10:39 am

  19. Guts Says:
    February 15th, 2008 at 7:11 pm

    Zachriel:

    Another characteristic of scale-invariant networks is that newer connections are much easier to change than older ones.

    Actually, this is not an inherent property of sf networks– the network can be scale-free and not have this property.

  20. Comment by Guts — February 15, 2008 @ 7:11 pm

  21. Anton Says:
    February 18th, 2008 at 12:30 am

    Zach,

    This is the oddest explanation of scale invariance I have ever read. What in the world makes you think your example represents scale invariance?

  22. Comment by Anton — February 18, 2008 @ 12:30 am

  23. Zachriel Says:
    February 18th, 2008 at 8:53 am

    Thanks for the bump.

    Anton: What in the world makes you think your example represents scale invariance?

    Classes of small-world networks, Amaral, et al. PNAS, 2000

    The air traffic network is scale-free (Amaral et al.), which means that connectivity is unbalanced: a few nodes are highly connected with others, while most are only sparsely connected. Other examples include Web linkages, electric power grids, proteins in yeast, viral epidemics, social networks (Amaral et al.), ecosystems, and scientific papers.

    Consider an established social network in a archtypical high school. The cheerleaders and athletes are highly connected socially (high sociability). Nerds and geeks have low numbers of connections, but do form small networks of their own (low sociability). Some people have characteristics that make them more sociable, and connections and relationships may vary over time. However, older relationships tend to be more durable than newer relationships. You try not to spurn the elite for fear of becoming disconnected from the network. No more cool parties for you!

    Now consider what happens when a new student with random sociability is transferred to our high school. This person will initially have few social connections.

    If she is sociable, then she will attempt to make friends. Wanting friends, she is most likely going to gravitate to those who also have friends. In other words, the new student will preferentially attach to one of the established social networks. And whatever her initial contact, she will be closer to the elites than to any other person chosen at random. Once connected to the establishment, she will be one or two degrees from all sorts of other attachment nodes. Indeed, even if she doesn't fit in with the princes and princesses of the elite, she can find her place more readily by starting with these preferred connections. After all, the class president and head cheerleader know nearly everybody!

    If the person has low sociability, then she may be rebuffed or scorned by the well-connected, perhaps gravitating or even being directed to a suitable low connection web. Go hang out with the geeks over there! Or because the geeks are less likely to make friends (low sociability), may remain unconnected and estranged.

    Now consider what happens when a new cool kid joins (high sociability). Now everyone wants to make a social connection to the new kid. That includes the elite. As the elite have the best connections to others and are held in high esteem for that reason, again, they will still be the preferred connections. Let's say the cool kid is a closet geek. He will may still preferentially attach to the elite network in order to be introduced to the local and poorly connected geeks. The elite know everyone, or at least are a minimal number of steps away.

    By the way, this is not the same as random attachment weighted by sociability. It's a power-law distribution. So what do traffic patterns and yeast proteins have in common? They form scale-free networks because they are the result of local and preferential attachment under global pressure. This mathematical pattern is of paramount importance to understanding evolutionary processes, just as the nested hierarchy is of paramount importance to understanding common descent along uncrossed lines.

  24. Comment by Zachriel — February 18, 2008 @ 8:53 am

  25. Zachriel Says:
    February 18th, 2008 at 11:34 am

    Scale-free networks have a lot of interesting characteristics.

    For instance, they are 'robust'. If you knock out a random hub, you are unlikely to disrupt the entire network. That's because critical hubs are a very small proportion of all hubs. However, if you do knock out a critical hub, either by happenstance or design, then you can disrupt large sections of the network. This is important in disease control, building a better Internet, or even fighting terrorist networks which are, after all, a type of social network (a directed scale-free network where the degree of protection of each node depends on its importance within the network, and where messages come from the hubs, but not to the hubs).

    Early entrance is an advantage in networks that grow over time. The longer a node has been around, the greater the number of links on average. And a greater number of links in the past means being preferred for new links in the future.

    Here are a couple of the seminal papers on scale-free networks.

    D. J. Watts and S. H. Strogatz., Collective dynamics of 'small-world' networks, Nature (1998), "We call them 'small-world' networks, by analogy with the small-world phenomenon (popularly known as six degrees of separation)."

    Barabási, A.-L., and R. Albert, Statistical mechanics of complex networks, Reviews of Modern Physics, 2002 "Complex networks describe a wide range of systems in nature and society, much quoted examples including the cell, a network of chemicals linked by chemical reactions, or the Internet, a network of routers and computers connected by physical links."

  26. Comment by Zachriel — February 18, 2008 @ 11:34 am

  27. Rock Says:
    February 18th, 2008 at 2:03 pm

    Heuristically Optimized Trade-Offs: A New Paradigm for Power Laws in the Internet
    Alex Fabrikant7 , Elias Koutsoupias8 and Christos H. Papadimitriou7

    We propose a plausible explanation of the power law distributions of degrees observed in the graphs arising in the Internet topology [Faloutsos, Faloutsos, and Faloutsos, SIGCOMM 1999] based on a toy model of Internet growth in which two objectives are optimized simultaneously: "last mile" connection costs, and transmission delays measured in hops. We also point out a similar phenomenon, anticipated in [Carlson and Doyle, Physics Review E 1999], in the distribution of file sizes. Our results seem to suggest that power laws tend to arise as a result of complex, multi-objective optimization.
    http://www.springerlink.com/co...

  28. Comment by Rock — February 18, 2008 @ 2:03 pm

  29. Rock Says:
    February 18th, 2008 at 5:20 pm

    The Non-Darwinian (Paretian? Even doubly-Paretian!) design (evolution) of complex networks (social webs, the Internet, genomes) is fascinating to me.

    (Slip of the tongue, so to speak, I meant Panglossian, not Paretian. LOL)

  30. Comment by Rock — February 18, 2008 @ 5:20 pm

  31. Zachriel Says:
    February 18th, 2008 at 7:05 pm

    Fabrikant et al: Our results seem to suggest that power laws tend to arise as a result of complex, multi-objective optimization.

    Consider a case where there are no constraints and the only goal is to minimize hops. Then everyone could simply connect to a single hub, and all nodes would now be at most two steps away. However, add constraints, such as proximity, and now it may make more sense to connect to someone close to the central hub. In the other extreme, if proximity is much more important than hops, then we might not see a power-law distribution at all, but a minimum spanning tree.

    We don't want to oversimplify. We don't expect physical systems to be perfectly modeled, but scale-invariance is often a valid first-order approximation, and can act as a null-hypothesis, just as the nested hierarchy does for common descent. With the nested hierarchy, we have ambiguity when we have rapidly diverging lines, during the process of speciation, or outright violations due to horizontal mechanisms. Similarly, the mathematical pattern of scale-invariance may apply only in part or to limited scale domains, but still be a useful model.

    Scale-free networks in cell biology, Réka Albert, 2005

  32. Comment by Zachriel — February 18, 2008 @ 7:05 pm

  33. Rock Says:
    February 19th, 2008 at 8:19 pm

    If Mike gene allows, I have no clue as to how this is topical, but you did bring it up, Z) I will pursue the matter with you.

    The repeated discovery (particularly in biology) of a particular scale, for even wildly (seemingly) disparate phenomena, everything from the size of communities to the branching of my respiratory tract, is more than interesting for any number of reasons.

    Most obviously, the existence of an invariant, an invariant scale, is suggestive of an underlying common cause, general principle, or law. (Per Planck & Einstein, e.g.)

    But the very existence of "biological" laws is contested. (often in these discussions.) Both philosophically and scientifically. Bohr's thesis, that inspired so many physicists to enter biology (and win Nobel's), was that there may be discovered laws in biology unknown to physicists.

    Did they discover such laws? Despite the prizes the answer is generally No. But No was the philosophical answer long before Bohr was born.

    That "law" they were looking for, the physicists-cum-biologists (and biologists-cum-physicists) didn't discover"¦

    But engineers, intelligent designers, did.

  34. Comment by Rock — February 19, 2008 @ 8:19 pm

  35. Joy Says:
    February 20th, 2008 at 10:39 am

    Here's a nifty little ditty from ScienceDaily (UC-Berkeley) with some front-loading implications…

    Genome of Marine Organism Tells of Humans' Unicellular Ancestors

    One finding confirmed by the sequencing is that choanoflagellates have many genes that, in animals, produce proteins essential to cell-to-cell signaling and in determining which cells stick to one another. Since Monosiga does not form colonies as do some other choanoflagellates, these proteins' roles are a mystery, King said.

    "In animals, some of these proteins, called cadherins, evolved for linking cells together; they are the glue that prevents clumps of cells from falling apart," King said. "Choanoflagellates show no hint of multicellularity, but they have 23 genes for cadherin proteins, about the same as the fruit fly or the mouse."

    Hmmm… But it gets worse:

    The choanoflagellate genome, like the genomes of many seemingly simple organisms sequenced in recent years, shows a surprising degree of complexity, King said. Many genes involved in the central nervous system of higher organisms, for example, have been found in simple organisms that lack a centralized nervous system.

    Likewise, choanoflagellates have five immunoglobulin domains, though they have no immune system; collagen, integrin and cadherin domains, though they have no skeleton or matrix binding cells together; and proteins called tyrosine kinases that are a key part of signaling between cells, even though Monosiga is not known to communicate, or at least does not form colonies.

  36. Comment by Joy — February 20, 2008 @ 10:39 am

  37. Zachriel Says:
    February 20th, 2008 at 7:48 pm

    Rock: If Mike gene allows, I have no clue as to how this is topical, but you did bring it up, Z) I will pursue the matter with you.

    Because network theory helps explain why many genetic structures are established early with everything built on top of those.

    Law: A descriptive generalization about how some aspect of the natural world behaves under stated circumstances.

    Laws of Nature are a stated regularity in the relations or order of phenomena in the world that holds, under a stipulated set of conditions, either universally or in a stated proportion of instances.

    Rock: Most obviously, the existence of an invariant, an invariant scale, is suggestive of an underlying common cause, general principle, or law. (Per Planck & Einstein, e.g.)

    That's correct. It's a specific and non-random pattern. It can be shown that scale-free architecture results from preferential attachment. (Just like a nested hierarchy results from diverging descent along uncrossed lines.)

    Rock: Did they discover such laws? Despite the prizes the answer is generally No. But No was the philosophical answer long before Bohr was born.

    Common Descent (that any two organisms share a common ancestor) was established well before Bohr. However, biology is messy, so there are many exceptions to such generalizations. Physics is not immune to this messiness. For instance, Mercury's precession anomaly was an exception to Newton's Laws of Gravity. They studied it, couldn't resolve it, but because the Theory could otherwise explain so many things, they kept the law while shelving the anomaly for a few centuries. Call it Newton's Laws of Gravity, except Mercury and those other things it can't explain. Scientific laws work only within limited empirical domains.

    Rock: Did they discover such laws?

    The Theory of Evolution is a scientific theory that unifies observations in everything from genetics to geology. There are scientific laws concerning reproduction, variation, selection, population dynamics, competition, common descent, inheritance, etc.

  38. Comment by Zachriel — February 20, 2008 @ 7:48 pm

  39. Rock Says:
    February 22nd, 2008 at 5:19 pm

    I have been repeatedly reminded in these discussions that evolution is "directionless," so a digraph (directed graph or "tree") would seem to be an inappropriate model of biological evolution.

    Oddly enough, digraphs (since at least the days of Darwin) are the only models of evolution.

    I also thought that an exponential (rather than linear) growth process with a power law "cut off' (extinction of linkages), like a truncated or relative frequencey dependent selection regime would be a more approprite (biologically) model.

    But why should I think that selection coefficents are drawn (randomly, as they must) from a power law distribution?

    Other than that the global statistics (the scale-invariant statistics) of the genome disctates it. If true, the statistics, then I should expect the distribution of mutants sampled (selected) to follow a power law (and as I suggested in both tails)–not any Gaussian or exponential distribution.

    As usual in popgen (Neo-Darwinian) models.

  40. Comment by Rock — February 22, 2008 @ 5:19 pm

  41. Rock Says:
    February 23rd, 2008 at 12:01 pm

    I really can think of no reason to identify natural selection S with this "exotic" exponent [gamma], S:= P(k) ~ k^[negative gamma].

    Except to say that the scale-invariant factor is invariant also wrt selection. I.e., selection invariantly selects the scale-invariant architecture. Selection does not transform it because the SI-architecture cannot be improved upon. It is, in some sense, an ideal design; very ingeniously engineered and optimized to operate effectively in a virtually unlimited environment.

    (It was these issues about the limits of selection, based on many observations, biological and technological, btw, that made former friends, Neo-Darwinists Ronald Fisher and Sewall Wright, life long enemies… and even beyond life, for Fisher!)

    And as Barabasi & Albert show, natural selection does not explain the origin of the system either, which is purely a statistical mechanical process in their model. (And it was to render the model a bit more biologically relevant that inspired my original question.)

    Now what about that nested hierarchy and those trees of life?

  42. Comment by Rock — February 23, 2008 @ 12:01 pm

  43. One Brow Says:
    February 23rd, 2008 at 12:57 pm

    I have been repeatedly reminded in these discussions that evolution is "directionless," so a digraph (directed graph or "tree") would seem to be an inappropriate model of biological evolution.

    You don't think evolution has a chronological/generational direction?

    I also thought that an exponential (rather than linear) growth process with a power law "cut off' (extinction of linkages), like a truncated or relative frequencey dependent selection regime would be a more approprite (biologically) model.

    Most population growth models I have seen are exponential, before resource limitations are included.

  44. Comment by One Brow — February 23, 2008 @ 12:57 pm

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