Last Friday, UWO hosted a Distinguished Colloquium talk by Gregory Chaitin, who was talking about a proposal for a new field he calls “metabiology”, which he defined in the talk (and on the website above) as “*a field parallel to biology, dealing with the random evolution of artificial software (computer programs) rather than natural software (DNA), and simple enough that it is possible to prove rigorous theorems or formulate heuristic arguments at the same high level of precision that is common in theoretical physics.*” This field doesn’t really exist to date, but his talk was intended to argue that it should, and to suggest some ideas as to what it might look like. It was a well-attended talk with an interdisciplinary audience including (at least) people from the departments of mathematics, computer science, and biology. As you might expect for such a talk, it was also fairly nontechnical.

A lot of the ideas presented in the talk overlapped with those in this outline, but to summarize… One of the motivating ideas that he put forth was that there is currently no rigorous proof that Darwin-style biological evolution can work – i.e. that operations of mutation and natural selection can produce systems of very high complexity. This is a fundamental notion in biology, summarized by the slogan, “Nothing in biology makes sense except in light of evolution”. This phrase, funnily, was coined as the title of a defense of a “theistic evolution” – not obviously a majority position among scientists, but also not to be confused with “intelligent design” which claims that evolution can’t account for observed features of organisms. This is a touchy political issue in some countries, and it’s not obvious that a formal proof that mutation and selection CAN produce highly complex forms would resolve it. Even so, as Chaitin said, it seems likely that such a proof could exist – but if there’s a rigorous proof of the contrary, that would be good to know also!

Of course, such a formal proof doesn’t exist because formal proof doesn’t play much role in biology, or any other empirical science – since living things are very complex, and incompletely understood. Thus the proposal of a different field, “metabiology”, which would study simpler formal objects: “artificial software” in the form of Turing machines or program code, as opposed to “natural software” like DNA. This abstracts away everything about an organism except its genes (which is a lot!), with the aim of simplifying enough to prove that mutation and selection in this toy world can generate arbitrarily high levels of complexity.

Actually stating this precisely enough to prove ties in to the work that Chaitin is better known for, namely the study of algorithmic complexity and theoretical computer science. The two theorems Chaitin stated (but didn’t prove in the talk) did not – he admitted – really meet that goal, but perhaps did point in that direction. One measure of complexity is computability – that is, the size of a Turing machine (for example, though a similar definition applies to other universal ways of describing algorithms) which is needed to generate a particular pattern. A standard example is the “Busy Beaver function“, and one way to define

it is to say that is the largest number printed out by an -state Turing machine which then halts. Since the halting problem is uncomputable (i.e. there’s no Turing machine which, given a description of another machine, can always decide whether or not it halts), for reasons analogous to Cantor’s diagonal argument or Godel’s incompleteness theorem, generating , or a sequence of the same order, is a good task to measure complexity.

So the first toy model involved a single organism, being replaced in each generation by a mutant form. The “organism” is a Turing machine (or a program in some language, etc. – one key result from complexity theory is that all these different ways to specify an algorithm can simulate each other, with the addition of at most a fixed-size prefix, which is the part of the algorithm describing how to do the simulation). In each generation, it is mutated. The mutant replaces the original organism if: (a) the new code halts, and (b) outputs a number which (c) is larger than the number produced by the original. Now, this decision procedure is uncomputable since it requires solving the halting problem – so in particular, there’s no way to simulate this process. But the theorem says that, in exponential time (i.e. ), this process will produce a machine which produces a number of order . That is, as long as the “environment” (the thing doing the selection) can recognize and reward complexity, mutation is sufficient to produce it. But these are pretty big assumptions, which is one reason this theorem isn’t quite what’s wanted.

Still, within it’s limited domain, he also stated a theorem to the effect that, for any given level of complexity (in the above sense), there is a path through the space of possible programs which reaches it, such that the “mutation distance” (roughly, the negative logarithm of the probability of a mutation occurring) at each step is bounded, and the complexity (therefore fitness, in this toy model) increases at each step. He indicated that one could prove this using the bits of the halting probability Omega – he didn’t specify how, and this isn’t something I’m very familiar with, but apparently (as describeded in the linked article), there are somewhat standard ways to do this kind of thing.

So anyway, this little toy model doesn’t really do the job Chaitin is saying ought to be done, but it illustrates what the kind of theorems he’s asking for might look like. My reaction is that it would be great to have theorems like this that could tell us something meaningful about real biology (so the toy model certainly is too simple), though I’m not totally convinced there needs to be a “new field” for such study. But certainly theoretical biology seems to be much less developed than, say, theoretical physics, and even if rigorous proofs aren’t going to be as prominent there, if some can be found, it probably couldn’t hurt.

After the talk, there was some interesting discussion about other things going on in theoretical biology and “systems biology“. Chaitin commented that a lot of the work in this field involves detailed simulations of models of real systems, made as accurate as possible – which, while important, is different from the kind of pursuit of basic theoretical principles he was talking about. So this would include things like: modeling protein folding; studying patterns in big databases of gene frequencies in populations and how they change in time; biophysical modeling of organs and the biochemical reactions in them; simulating the dynamics of individual cells, their membranes and the molecular machinery that makes them work; and so on. All of which has been moving rapidly in recent years, but is only tangentially related to fundamental principles about how life works.

On the other hand, as audience members pointed out, there is another thread, exemplified by the Santa Fe Institute, which is more focused on understanding the dynamics of complex systems. Some well-known names in this area would be Stuart Kauffman, John Holland and Per Bak, among others. I’ve only looked into this stuff at the popular level, but there are some interesting books about their work – Holland’s “Hidden Order”, Kauffman’s “The Origins of Order” (more technical) and “At Home in the Universe” (more popular), and Solé and Goodwin’s “Signs of Life” (a popular survey, but with equations, of various

aspects of mathematical approaches to biological complexity). Chaitin’s main comment on this stuff is that it has produced plenty of convincing heuristic arguments, simulations and models with suggestive behaviour, and so on – but not many rigorous theorems. So: it’s good, but not exactly what he meant by “metabiology”.

Summarizing this stuff would be a big task in itself, but it does connect to Chaitin’s point that it might be nice to know (rigorously) if Darwinian evolution by itself were NOT enough to explain the complexity of living things. Stuart Kauffman, for example, has suggested that certain kinds of complex order tend to arise through “self-organization”. Philosopher Daniel Dennett

commented on this in “Darwin’s Dangerous Idea”, saying that although this might be true, at most it tells us more detail about what kinds of things Darwinian selection has available to act on.

This all seems to tie into the question over which appeared first as life was first coming into being: self-replicating molecules like RNA (and later DNA), or cells with metabolic reactions occurring inside. Organisms obviously both reproduce and metabolize, but these are two quite different kinds of process, and there seems to be a “chicken-and-egg” problem with which came first. Kauffman, among others, has looked at the emergence of “autocatalytic networks” of chemical reactions: these are collections of chemical reactions, some or all of which needing a catalyst, such that all the catalysts needed to make them run are products of some reaction in the network. They’ve shown in simulation that such networks can arise spontaneously under certain conditions – suggesting that metabolism might have come into existence without DNA or similar molecules around (one also thinks of larger phenomena, like the nitrogen cycle). In any case, this is the kind of thing which people sometimes point to when suggesting that Darwinian selection isn’t enough to completely explain the structure of organisms actually existing today. Which is a different claim (mind you) than the claim that Darwinian evolution *could not possibly* produce complex organisms. Chaitin’s whole motivation was to suggest that it should be provable one way or the other (and, he presumes, in the affirmative) whether mutation and selection CAN do this job. If it could be proved that it can’t – at least there are some other ingredients to consider.

All in all, I found the talk thought-provoking, in spite (or because) of being partial and inconclusive. Biology may be less rigorous than physics, but this could just be a sign that there’s a lot to learn and do in the field – and a lot of it is being done!

September 28, 2009 at 11:28 pm

[...] Gregory Chaitin: “Mathematics, Biology and Metabiology” [...]

October 6, 2009 at 6:59 pm

I’m happy to have the field in which I’ve worked since 1973 given a good name by a genius. I’ve been able to tell Gregory Chaitin in person what a strong influence that he’s had on my life. Similarly, I’ve conversed at length with this topic with Stuart Kauffman (whom I’ve been extensively citing in a long recent draft monograph of QM and Consciousness). I beta-tested John Holland’s breakthrough book on the Genetic Algorithm while it was still in draft manuscript, while in Grad school 1973-1977. I contend that my PhD dissertation (many chapters subsequently published in refereed proceedings) is a landmark in Metabiology, but the History of Metabiology will be the judge. You see, Theories and Implementations of Metabiology compete by Meta-Natural Selection, according to the recursiveness of Meta-Metabiology.

October 8, 2009 at 3:17 pm

“The result of a mathematical development should be continuously checked against one’s own intuition about what constitutes reasonable biological behavior.

When such a check reveals disagreement, then the following possibilities must be considered:

(a) A mistake has been made in the formal mathematical development;

(b) The starting assumptions are incorrect and/or constitute a too drastic oversimplification;

(c) One’s own intuition about the biological field is inadequately developed;

(d) A penetrating new principle has been discovered.

[Harvey J. Gold, Mathematical Modeling of Biological Systems]