Gödel, Escher, Bach: An Eternal Golden Braid (97 page)

Henkin Sentences and Viruses

Now both of these contrasting types of self-reference in molecular biology have their counterparts in mathematical logic. We have already discussed the analogue of the self-defeating phages-namely, strings of the G6del type, which assert their own unproducibility within specific formal sstems. But one can also make a counterpart sentence to a real phage: the' phage asserts its own producibility in a specific cell, and the sentence asserts its own producibility in a specific formal system. Sentences of this type are called Henkin sentences, after the mathematical logician Leon Henkin. They can be constructed exactly along the lines of Godel sentences, the only difference being the omission of a negation. One begins with an

"uncle", of course:

3a:3a':

and then proceeds by the standard trick. Say the Godel number of the above "uncle" is h.

Now by arithmoquining this very uncle, you get a Henkin sentence: 3a:3a': M

(By the way, can you spot how this sentence differs from -G?) The reason I show it explicitly is to point out that a Henkin sentence does not give a full recipe for its own derivation; it just asserts that there exists one. You might well wonder whether its claim is justified. Do Henkin sentences indeed possess derivations? Are they, as they claim, theorems? It is useful to recall that one need not believe a politician who says. "I am honest"-

he may be honest, and yet he may not be. Are Henkin sentences any more trustworthy than politicians? Or do Henkin sentences, like politicians, lie in cast-iron sinks?

It turns out that these Henkin sentences are invariably truth tellers. Why this is so is not obvious; but we will accept this curious fact without proof.

Implicit vs. Explicit Henkin Sentences

I mentioned that a Henkin sentence tells nothing about its own derivation; it just asserts that one exists. Now it is possible to invent a variation on the theme of Henkin sentences-namely sentences which explicitly describe their own derivations. Such a sentence's high-level interpretation would not be "Some Sequence of Strings Exists Which is a Derivation of Me", but rather, "The Herein-described Sequence of Strings

Is a Derivation of Me". Let us

call the first type of sentence an implicit Henkin sentence. The new sentences will be called explicit Henkin sentences, since they explicitly describe their own derivations. Note that, unlike their implicit brethren, explicit Henkin sentences need not be theorems. In fact, it is quite easy to write a string which asserts that its own derivation consists of the single string 0=0-a false statement, since 0=0 is not a derivation of anything. However, it is also possible to write an explicit Henkin sentence which is a theorem-that is, a sentence which in fact gives a recipe for its own derivation.

Henkin Sentences and Self-Assembly

The reason I bring up this distinction between explicit and implicit Henkin sentences is that it corresponds very nicely to a significant distinction between types of virus. There are certain viruses, such as the so-called "tobacco mosaic virus", which are called self-assembling viruses; and then there are others, such as our favorite T-evens, which are non-self-assembling. Now what is this distinction? It is a direct analogue to the distinction between implicit and explicit Henkin sentences.

The DNA of a self-assembling virus codes only for the parts of a new virus, but not for any enzymes. Once the parts are produced, the sneaky virus relies upon them to link up to each other without help from any enzymes. Such a process depends on chemical affinities which the parts have for each other, when swimming in the rich chemical brew of a cell. Not only viruses, but also some organelles-such as ribosomes-assemble themselves. Somtiems enzymes may be needed – but in such cases, they are recruited from the host cell, and enslaved. This is what is meant by self-assembly.

By contrast, the DNA of more complex viruses, such as the T-evens, codes not only for the parts, but in addition for various enzymes which play special roles in the assembly of the parts into wholes. Since the assembly process is not spontaneous but requires "machines", such viruses are not considered to be self-assembling. The essence of the distinction, then, between self-assembling units and non-self-assembling units is that the former get away with self-reproduction without telling the cell anything about their construction, while the latter need to give instructions as to how to assemble themselves.

Now the parallel to Henkin sentences, implicit and explicit, ought to be quite clear.

Implicit Henkin sentences are self-proving but do not tell anything at all about their proofs-they are analogous to self-assembling viruses; explicit Henkin sentences direct the construction of their own proofs-they are analogous to more complex viruses which direct their host cells in putting copies of themselves together.

The concept of self-assembling biological structures as complex as viruses raises the possibility of complex self-assembling machines as well. Imagine a set of parts which, when placed in the proper supporting environment, spontaneously group themselves in such a way as to form a complex machine. It seems unlikely, yet this is quite an accurate way to describe the process of the tobacco mosaic virus' method of selfreproduction via self-assembly. The information for the total conformation of the organism (or machine) is spread about in its parts; it is not concentrated in some single place.

Now this concept can lead in some strange directions, as was shown in the Edifying Thoughts of a Tobacco Smoker. There, we saw how the Crab used the idea that information for self-assembly can be distributed around, instead of being concentrated in a single place.

His hope was that this would prevent his new phonographs from succumbing to the Tortoise's phonograph-crashing method. Unfortunately, just as with the most sophisticated axiom schemata, once the system is all built and packaged into a box, its well-definedness renders it vulnerable to a sufficiently clever "Godelizer"; and that was the sad tale related by the Crab.

Despite its apparent absurdity, the fantastic scenario of that Dialogue is not so far from reality, in the strange, surreal world of the cell.

Two Outstanding Problems:

Differentiation and Morphogenesis

Now self-assembly may be the trick whereby certain subunits of cells are constructed, and certain viruses-but what of the most complex macroscopic structures, such as the body of an elephant or a spider, or the shape of a Venus's-Hyt-ap? How are homing instincts built into the brain of" a

bird, or hunting instincts into the brain of a dog% In short, how is it that merely by dictating which proteins are to be produced in cells, DNA exercises such spectacularly precise control over the exact structure and function of macroscopic living objects? There are two major distinct problems here. One is that of cellular differentiation: how do different cells, sharing exactly the same DNA, perform different roles-such as a kidney cell, a bone marrow cell, and a brain cell? The other is that of morphogenesis ("birth of form"): how does intercellular communication on a local level give rise to large-scale, global structures and organizations-such as the various organs of the body, the shape of the face, the suborgans of the brain, and so on? Although both cellular differentiation and rnorphogenesis are poorly understood at present. the trick appears to reside in exquisitely fine-tuned feedback and "feedforward"

mechanisms within cells and between cells, which tell a cell when to "turn on" and when to

"turn off" production of various proteins.

Feedback and Feedforward

Feedback takes place when there is too much or too little of some desired substance in the cell: then the cell must somehow regulate the production line which is assembling that substance. Feedforward also involves the regulation of" an assembly line, but not according to the amount of end product present: rather, according to the amount of some precursor of the end product of that assembly line. There are two major devices for achieving negative feedforward or feedback. One way is to prevent the relevant enzymes from being able to perform-that is, to "clog up" their active sites. This is called inhibition. The other way is to prevent the relevant enzymes from ever being manufactured! This is called repression.

Conceptually, inhibition is simple: you just block up the active site of the first enzyme in the assembly line, and the whole process of synthesis gets stopped dead.

Repressors and Inducers

Repression is trickier. How does a cell stop a gene from being expressed? The answer is, it prevents it from ever getting transcribed. This means that it has to prevent RNA polymerase from doing its job. This can be accomplished by placing a huge obstacle in its path, along the DNA. precisely in front of that gene which the cell wants not to get transcribed. Such obstacles do exist, and are called repressors. They are themselves proteins, and they bind to special obstacle-holding sites on the DNA, called (I am not sure why) operators. An operator therefore is a site of control for the gene (or genes) which immediately follow it: those genes are called its operon. Because a series of enzymes often act in concert in carrying out a long chemical transformation, they are often coded for in sequence; and this is why operons often contain several genes, rather than just one. The effect of the successful repression of an operon is that a whole series of genes is

prevented from being transcribed, which means that a whole set of related enzymes remains unsynthesized.

What about positive feedback and feedforward? Here again, there are two options: (1) unclog the clogged enzymes, or (2) stop the repression of the relevant operon. (Notice how nature seems to love double-negations! Probably there is some very deep reason for this.) The mechanism by which repression is repressed involves a class of molecules called inducers. The role of an inducer is simple: it combines with a repressor protein before the latter has had a chance to bind to an operator on a DNA molecule; the resulting "repressor-inducer complex" is incapable of binding to an operator, and this leaves the door open for the associated operon to be transcribed into mRNA and subsequently translated into protein.

Often the end product or some precursor of the end product can act as an inducer.

Feedback and Strange Loops Compared

Incidentally, this is a good time to distinguish between simple kinds of feedback, as in the processes of inhibition and repression, and the looping-hack between different informational levels, shown in the Central Dogrnap. Both are "feedback" in some sense; but the latter is much deeper than the former. When an amino acid, such as tryptophan or isoleucine, acts as feedback (in the form of an inducer) by binding to its repressor so that more of it gets made, it is not telling how to construct itself; it is just telling enzymes to make more of it. This could be compared to a radio's volume, which, when fed through a listener's ears, may cause itself to be turned down or up. This is another thing entirely from the case in which the broadcast itself tells you explicitly to turn your radio on or off, or to tune to another wavelength-or even how to build another radio! The latter is much more like the looping-back between informational levels, for here, information inside the radio signal gets

"decoded" and translated into mental structures. The radio signal is composed of symbolic constituents whose symbolic meaning matters-a case of use, rather than mention. On the other hand, when the sound is just too loud, the symbols are not conveying meaning: they are merely being perceived as loud sounds, and might as well be devoid of meaning-a case of mention, rather than use. This case more resembles the feedback loops by which proteins regulate their own rates of synthesis.

It has been theorized that the difference between two neighboring cells which share the exact same genotype and yet have different functions is that different segments of their genome have been repressed, and therefore they have different working sets of proteins. A hypopothesis like this could account for the phenomenal differences between cells in different organs of the body of a human being.

Two Simple Examples of Differentiation

The process by which one initial cell replicates over and over, giving rise to a myriad of differentiated cells with specialized functions, can be likened to the spread of a chain letter from person to person, in which each new participant is asked to propagate the message faithfully, but also to add some extra personal touch. Eventually, there will be letters which are tremendously different from each other.

Another illustration of the ideas of differentiation is provided by this extremely simple computer analogue of a differentiating self-rep. Consider a very short program which is controlled by an up-down switch, and which has an internal parameter N-a natural number.

This program can run in two modes-the up-mode, and the down-mode. When it runs in the upmode, it self-replicates into an adjacent part of the computer's memoryexcept it makes the internal parameter N of its "daughter" one greater than in itself. When it runs in the down-mode, it does not self-rep, but instead calculates the number

(-1)'/(2N + 1)

and adds it to a running total.

Well, suppose that at the beginning, there is one copy of the program in memory, N =