September 22, 2011 3 Comments
It got cold really, really quickly today… Winter is coming –GRRM
In my last two posts I wrote about holey adaptive landscapes and some criticisms I had for the existing model. In this post, I will motivate the hijacking of the model for my own purposes, for macroevolution :-) That’s the great thing about models, by relabeling the ingredients something else, as long as the relationship between elements of the model holds true, we may suddenly have an entirley fresh insight about the world!
As we might recall, the holey adaptive landscapes model proposes that phenotypes are points in n-dimensional space, where n is a very large number. If each phenotype was a node, then mutations that change one phenotype to another can be considered edges. In this space for large enough n, even nodes with very rare properties can percolate and form a giant connected cluster. Gavrilets, one of the original authors of the model and its main proponent, considered this rare property to be high fitness. This way, evolution never has to cross fitness valleys. However, I find this unlikely; high fitness within a particular environment is a very rare property. If the property is sufficiently rare, then even if the nodes form a giant connected cluster, if the connections between nodes are sufficiently tenuous, then there is not enough time and population size for their exploration.
Think of a blind fruitfly banging its head within a sphere. On the wall of the sphere is pricked a single tiny hole just large enough for the fruitfly to escape. How much time will it take for the fruitfly to escape? The question clearly depends on the size of the sphere. In n-dimensions, where n is large, the sphere is awfully big. Now consider the sphere to be a single highly fit phenotype, and a hole is an edge to another highly fit phenotype. The existence of the hole is not sufficient to guarantee that the fruitfly will find the exit in finite time. In fact, even a giant pack of fruitflies — all the fruitflies that ever existed — may not be able to find it, given all the time that life has evolved on Earth. That’s how incredibly large the sphere is — the exit must not only exist, it must be sufficiently common.
The goal of this post is to detail why I’m so interested in the holey adaptive landscapes model. I’m interested in its property of historicity, the capacity to be contingent and irreversible. I will define these terms more carefully later. Gavrilets has noted this in his book, but I can find no insightful exploration of this potential. I hope this model can formalize some intuitions gained in complex adaptive systems, particularly those of evo-devo and my personal favorite pseudo-philosophical theory, generative entrenchment (also here and here). Gould had a great instinct for this when he argued for the historical contingency of evolutionary process (consider his work on gastropods, for example, or his long rants on historicity in his magnus opus — although I despair to pick out a specific section).
Before I go on, I must also rant. Gould’s contingency means that evolution is sensitive to initial conditions, yes, this does not mean history is chaotic, in the mathematical sense of chaos. Chaos is not the only way for a system to be sensitive to initial conditions, in fact, mathematical chaos is preeminently ahistorical — just like equilibriating systems, chaotic systems forget history in a hurry, in total contrast to what Gould meant, which is that history should leave an indelible imprint on all of future. No matter what the initial condition, chaotic systems settle in the same strange attractor, the same distribution over phase space. The exact trajectory depends on initial condition, yes, but because the smallest possible difference in initial condition quickly translates to a completely different trajectory, it means that no matter how you begin, you future is… chaotic. Consider two trajectories that began with very differently, the future difference between those two trajectories is no greater than two trajectories that began with the slightest possible difference. The difference in the difference of initial conditions is quickly obliviated by time. Whatever the atheist humanists say, chaos gives no more hope to free will than quantum mechanics. Lots of people seem to have gone down this hopeless route, not least of which is Michael Shermer, who, among many different places, writes here:
And as chaos and complexity theory have shown, small changes early in a historical sequence can trigger enormous changes later… the question is: what type of change will be triggered by human actions, and in what direction will it go?
If he’s drawing this conclusion from chaos theory, then the answer to his question is… we don’t know, we can’t possibly have an idea, and it doesn’t matter what we do, since all trajectories are statistically the same. If he’s drawing this conclusion from “complexity theory” — not yet a single theory with core results of any sort, then it’s a theory entirely unknown to me.
No, interesting historical contingency is quite different, we will see if the holey landscapes model can more accurately capture its essence.
Things in the holey landscapes model get generally better if we consider the rare property to be viability, instead of high fitness. In fact, Gavrilets mixes use of “viable” and highly fit, although I suspect him to always mean the latter. By viable, I mean that the phenotype is capable of reproduction in some environment, but I don’t care how well it reproduces. For ease of discussion, let’s say that viable phenotypes also reproduce above the error threshold, and there exist an environment where it is able to reproduce with absolute fitness >1. Else, it’s doomed to extinction in all environments, and then it’s not very viable, is it?
It turns out that the resultant model contains an interesting form of irreversibility. I will give the flavor here, while spending the next post being more technical. Consider our poor blind fruitfly, banging its head against the sphere. Because we consider viability instead of “high fitness”, there are now lots of potential holes in the sphere. Each potential hole is a neighboring viable phenotype, but the hole is opened or closed by the environment, which dictates whether that neighboring viable phenotype is fit.
Aha, an astute reader might say, but this is no better than Gavrilets’ basic model. The number of open holes at any point must be very small, since it’s also subject to the double filter of viability and high fitness. How can we find the open holes?
The difference is that after an environmental change, the sphere we are currently in might be very unfit. Thus, the second filter — high fitness — is much less constrictive, since it merely has to be fitter than the current sphere, which might be on the verge of extinction. A large porportion of viable phenotypes may be “open” holes, as opposed to the basic model, where only the highly fit phenotypes are open. Among viable phenotypes, highly fit ones may be rare, but those that are somewhat more fit than an exceedingly unfit phenotype may be much more common — and it’s only a matter of time, often fairly short time on the geological scale, before any phenotype is rendered exceedingly unfit. So you see, in this model evolution also did not have to cross a fitness valley, but I’m using a much more classical mechanism — peak shifts due to environmental change, rather than a percolation cluster of highly-fit phenotypes.
Now that our happier fruitfly is in the neighboring sphere, what is the chance that it will return to its previous sphere, as opposed to choosing some other neighbor? The answer is… very low. The probability of finding any particular hole has not improved, and verges on impossibility; although the probability of finding some hole is much better. Moreover, the particular hole that the fruitfly went through dictate what holes it will have access to next — and if it can’t return to its previous sphere, then it can’t go back to remake the choice. This gives the possibility of much more interesting contingency than mere chaos.
This mechanism has much in common with Muller’s Ratchet, or Dollo’s Law, and is an attempt to generalize the ratchet mechanism while formalizing what we mean, exactly, by irreversibility. I will tighten the argument next Thursday.