Turned up to eleven: Fair and Balanced

Tuesday, July 23, 2002


One more time through the thicket

This will be my last post on the subject of mathematics, modeling (not the fun kind, either!), and biology, for a while. I had a thought last night on the subject, that was just too interesting (to me!) to let go of. Here's the thought; Wolfram is right!. Well, maybe not about the whole thing (I haven't read the book, so I really don't know), but there is a certain principle here that many people ignore, for good reason; because it has no meaning to them. That principle is this...

The universe is discrete rather than continuous in structure

I don't think this is a wild ass conjecture on my part. The fundamental physics of quantum mechanics, which clearly does describe the fundamental nature of the universe, although we don't really understand it all that well (see the comments from the last post for a nice Feynman quote on the subject), shows that 1)particles energy state is quantized (Einstein), and 2) the physical structure of the Universe is described by "bits" of the Planck scale variety. These bits have structure in space and time, that is much, much smaller than is possible to examine, even theoretically, at the moment. The Planck length is 1.6x10^-36 meters. By comparison, the typical atomic unit length of 1 angstrom is 1x10^-10 meters. So a "Planck length" is incredibly, incredibly small. The Planck time is 5.4x10^-44 sec. These quantities are so unimaginably small, that even things that occur on scales that are ultra-microscopic (i.e., only detected using atomic force microscopes and scanning, tunneling electron microscopes), are unimaginably larger than this scale. Here is an article in Scientific American by Steven Weinberg that descibes this area of physics. I am really not qualified to discuss either the merits or shortcomings of this theoretical frontier, but what is interesting is that quantum mechanics, as the name implies, describes a Universe that is discontinuous.

There are ramifications to this notion, the biggest one being that there is a phase transition (analogous to those in chemistry and physics), between the discontinuous world of the very small, and the world we inhabit, that appears to all intents and purposes to be made of continuous events and a "stream of life" that flows ever by. Phase transitions are very important things, because they are utterly counterintuitive. Suppose you take a material, and you heat it by 1 degree Celcius. Your expectation, when you then test the material to find out about it, is that the atoms in the material are now moving around a bit faster. If it is a solid, they are vibrating more vigorously, if it is a liquid or a solid, they are moving a bit faster. At phase transition boundaries, however, something very different happens. The atoms in the material undergo a tremendous change in the way they behave, all at once (more or less). A solid block of ice becomes liquid (the shape of the solid is important too, for reasons that are far too technical to discuss here; for our purposes, consider what happens in a thin section near the surface, rather than the entire solid or liquid. The phase transitions can be mathematically and graphically described by a phase diagram, which I expect most people saw in chemistry classes in high school and college. Phase transitions are governed by two components, temperature and pressure, both of which really amount to the input of energy into the system, in different ways. Phase transitions are very well described mathematically, in terms of using differential equations to describe their progression (I'm a little rusty on this stuff), but it seems to me that the molecular basis for the change is sort of ignored (anyone who knows better than me should definitely comment, and enlighten me!). For example, you can make supercooled liquids, at standard temperature and pressure, by using tricks to avoid ice nucleation. So the process requires a "catalyst" of some kind. In any event, I really like the analogy of a phase transition to the difference between quantum mechanical scales and classical physics scales.

What is interesting about this is the notion of scaling problems is that it is not limited to these huge differences, like the difference between a quantum mechanical system and classical physics. Relatively small differences in magnitude can show these sorts of patterns as well. I would like to start a "gedanken experiment", and see where it takes us.

Orwin's Demon: A "Gedanken Experiment"

Suppose you are an observer of molecular processes at the level of a nucleotide. In other words, suppose we shrink down to the level where we leap from base to base, like kids jumping across a river on the exposed rocks. Now, suppose we watch, waiting for events to occur in the life of the organism. If the gene is getting used, we will see the approach of the RNA polymerase holoenzyme (better get out of the way!!). The enzyme complex binds to the DNA, and starts the transcription process. If we hang around this operator/promoter region, we can watch as this occurs, and over time we can even see patterns in the events. For example, every 5 seconds, the promoter might suddenly get deluged with RNA polymerases, which all bind and transcribe the gene in rapid succession. We might even write an equation to describe this behavior.

Now, if we make ourselves a bit bigger, we can stand astride the gene like a Colossus. From this vantage point, we can see the binding of cis-acting transcription factors, which bind to the DNA near the gene of interest, usually having some mechanical effect on the DNA itself, allowing for the RNA polymerase to work its magic. Here, still, we can tabulate the ways that all of these factors bind, and we can see how they bind in particular orders, or at the same time, to make the whole machinery work. This is the level at which much of standard molecular biology is done. We might still be able to write equations to describe this, but they have a lot of independent variables, and become less tractable.

Moving "one level up the ladder", we can see an entire "regulatory pathway". This is essentially the level of biochemical analysis. We may still be able to observe the activity of the gene of interest itself, but we often lose the fine control and "texture" of the interactions at that level, in order to gain access to the overall heirarchy of events. This is largely a measurement problem. It takes a lot of effort and skill to dissect the workings of a gene, and still more to understand a regulatory or biochemical pathway. To continue our "Orwin's Demon" (there's some hubris for you!) analogy, we can imagine our hypothetical observer is about the size of an organelle in side a cell, able to reach out and touch any particular part of the genome, or hold a protein in his hands (maybe juggling a few at at time). This observer has probably lost the ability to see and count the particular instances of each transcription factor or polymerase touching down on the genome at any particular time, and is now seeing transcription and translation as a flow of information from the genome to the cytoplasm (the part of the cell where the business of biology gets done).

Finally, our observer can get so large that he(or she) no longer fits into the cell. This is essentially what we are. Although we can make very fine tuned observations, we are limited to either very close but destructive observation, which can involve microscopy, but often involves molecular techniques to tease out information about proteins and genes, but destroys the cell in the process, or coarser but non-destructive observation, which doesn't destroy the subject, but can only tell us so much. As an aside, there are absolutely amazing techniques for watching the inner workings of cells using fluorescent labels, but again, they are limited by this tradeoff. The more information you want to get, the more destructive your technique has to be.

As we got larger in our observations, our perspective became more continuous. As we widen the scope of inquiry, we become less concerned with what is going on a point (x,y,z) in the cell, and more concerned with how things interact. I submit that this "phase transition" is at the heart of the challenge to modern biology. I further submit that this core problem will not be trivial to solve. Finally, I think it will be solved. Simply put, I never want to be known as a naysayer, but I do want to be known as a realist. My point to "Godless", and all others, was not that it is impossible to understand things, but that it is farther away than we think. It is also clear, based on the experiences in mathematics (Godel) and physics (Schroedinger, Heisenberg, et al), that limits exist, and sometimes we bump up against them in rather unexpected places. It may be that we can explain how things work, in some useful sense, but still not be able to use that information to the ends that "Godless" would like.

Finally, let's all remember that this is a blog!, not the final word handed down from on high. These are my (sometimes considered) opinions, and subject to revision when better information comes to light. I write this stuff because the ideas intrigue me, not because I know for a fact that everything I write is the unvarnished truth.

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