What Is Life (Canto Classics) Read online

  QUANTUM THEORY – DISCRETE STATES –

  QUANTUM JUMPS

  The great revelation of quantum theory was that features of discreteness were discovered in the Book of Nature, in a context in which anything other than continuity seemed to be absurd according to the views held until then.

  The first case of this kind concerned energy. A body on the large scale changes its energy continuously. A pendulum, for instance, that is set swinging is gradually slowed down by the resistance of the air. Strangely enough, it proves necessary to admit that a system of the order of the atomic scale behaves differently. On grounds upon which we cannot enter here, we have to assume that a small system can by its very nature possess only certain discrete amounts of energy, called its peculiar energy levels. The transition from one state to another is a rather mysterious event, which is usually called a ‘quantum jump’.

  But energy is not the only characteristic of a system. Take again our pendulum, but think of one that can perform different kinds of movement, a heavy ball suspended by a string from the ceiling. It can be made to swing in a north–south or east–west or any other direction or in a circle or in an ellipse. By gently blowing the ball with a bellows, it can be made to pass continuously from one state of motion to any other.

  For small-scale systems most of these or similar characteristics – we cannot enter into details – change discontinuously. They are ‘quantized’, just as the energy is.

  The result is that a number of atomic nuclei, including their bodyguards of electrons, when they find themselves close to each other, forming ‘a system’, are unable by their very nature to adopt any arbitrary configuration we might think of. Their very nature leaves them only a very numerous but discrete series of ’states’ to choose from.2 We usually call them levels or energy levels, because the energy is a very relevant part of the characteristic. But it must be understood that the complete description includes much more than just the energy. It is virtually correct to think of a state as meaning a definite configuration of all the corpuscles.

  The transition from one of these configurations to another is a quantum jump. If the second one has the greater energy (‘is a higher level’), the system must be supplied from outside with at least the difference of the two energies to make the transition possible. To a lower level it can change spontaneously, spending the surplus of energy in radiation.

  MOLECULES

  Among the discrete set of states of a given selection of atoms there need not necessarily but there may be a lowest level, implying a close approach of the nuclei to each other. Atoms in such a state form a molecule. The point to stress here is, that the molecule will of necessity have a certain stability; the configuration cannot change, unless at least the energy difference, necessary to ‘lift’ it to the next higher level, is supplied from outside. Hence this level difference, which is a well-defined quantity, determines quantitatively the degree of stability of the molecule. It will be observed how intimately this fact is linked with the very basis of quantum theory, viz. with the discreteness of the level scheme.

  I must beg the reader to take it for granted that this order of ideas has been thoroughly checked by chemical facts; and that it has proved successful in explaining the basic fact of chemical valency and many details about the structure of molecules, their binding-energies, their stabilities at different temperatures, and so on. I am speaking of the Heitler– London theory, which, as I said, cannot be examined in detail here.

  THEIR STABILITY DEPENDENT ON TEMPERATURE

  We must content ourselves with examining the point which is of paramount interest for our biological question, namely, the stability of a molecule at different temperatures. Take our system of atoms at first to be actually in its state of lowest energy. The physicist would call it a molecule at the absolute zero of temperature. To lift it to the next higher state or level a definite supply of energy is required. The simplest way of trying to supply it is to ‘heat up’ your molecule. You bring it into an environment of higher temperature (‘heat bath’), thus allowing other systems (atoms, molecules) to impinge upon it. Considering the entire irregularity of heat motion, there is no sharp temperature limit at which the ‘lift’ will be brought about with certainty and immediately. Rather, at any temperature (different from absolute zero) there is a certain smaller or greater chance for the lift to occur, the chance increasing of course with the temperature of the heat bath. The best way to express this chance is to indicate the average time you will have to wait until the lift takes place, the ‘time of expectation’.

  From an investigation, due to M. Polanyi and E. Wigner,3 the ‘time of expectation’ largely depends on the ratio of two energies, one being just the energy difference itself that is required to effect the lift (let us write W for it), the other one characterizing the intensity of the heat motion at the temperature in question (let us write T for the absolute temperature and kT for the characteristic energy).4 It stands to reason that the chance for effecting the lift is smaller, and hence that the time of expectation is longer, the higher the lift itself compared with the average heat energy, that is to say, the greater the ratio W:kT. What is amazing is how enormously the time of expectation depends on comparatively small changes of the ratio W:kT. To give an example (following Delbrück): for W 30 times kT the time of expectation might be as short as s., but would rise to 16 months when W is 50 times kT, and to 30,000 years when W is 60 times kT!

  MATHEMATICAL INTERLUDE

  It might be as well to point out in mathematical language – for those readers to whom it appeals – the reason for this enormous sensitivity to changes in the level step or temperature, and to add a few physical remarks of a similar kind. The reason is that the time of expectation, call it t, depends on the ratio W/kT by an exponential function, thus

  τ is a certain small constant of the order of 10−13 or 10−14s. Now, this particular exponential function is not an accidental feature. It recurs again and again in the statistical theory of heat, forming, as it were, its backbone. It is a measure of the improbability of an energy amount as large as W gathering accidentally in some particular part of the system, and it is this improbability which increases so enormously when a considerable multiple of the ‘average energy’ kT is required.

  Actually a W = 30kT (see the example quoted above) is already extremely rare. That it does not yet lead to an enormously long time of expectation (only s. in our example) is, of course, due to the smallness of the factor r. This factor has a physical meaning. It is of the order of the period of the vibrations which take place in the system all the time. You could, very broadly, describe this factor as meaning that the chance of accumulating the required amount W, though very small, recurs again and again ‘at every vibration’, that is to say, about 1013 or 1014 times during every second.

  FIRST AMENDMENT

  In offering these considerations as a theory of the stability of the molecule it has been tacitly assumed that the quantum jump which we called the ‘lift’ leads, if not to a complete disintegration, at least to an essentially different configuration of the same atoms – an isomeric molecule, as the chemist would say, that is, a molecule composed of the same atoms in a different arrangement (in the application to biology it is going to represent a different ‘allele’ in the same locus’ and the quantum jump will represent a mutation).

  To allow of this interpretation two points must be amended in our story, which I purposely simplified to make it at all intelligible. From the way I told it, it might be imagined that only in its very lowest state does our group of atoms form what we call a molecule and that already the next higher state is ‘something else’. That is not so. Actually the lowest level is followed by a crowded series of levels which do not involve any appreciable change in the configuration as a whole, but only correspond to those small vibrations among the atoms which we have mentioned above. They, too, are ‘quantized’, but with comparatively small steps from one level to the next. Hence the impacts of th
e particles of the ‘heat bath’ may suffice to set them up already at fairly low temperature. If the molecule is an extended structure, you may conceive these vibrations as high-frequency sound waves, crossing the molecule without doing it any harm.

  Fig. 11. The two isomers of propyl-alcohol.

  So the first amendment is not very serious: we have to disregard the ‘vibrational fine-structure’ of the level scheme. The term ‘next higher level’ has to be understood as meaning the next level that corresponds to a relevant change of configuration.

  SECOND AMENDMENT

  The second amendment is far more difficult to explain, because it is concerned with certain vital, but rather complicated, features of the scheme of relevantly different levels. The free passage between two of them may be obstructed, quite apart from the required energy supply; in fact, it may be obstructed even from the higher to the lower state.

  Let us start from the empirical facts. It is known to the chemist that the same group of atoms can unite in more than one way to form a molecule. Such molecules are called isomeric (‘consisting of the same parts’; = same, = part). Isomerism is not an exception, it is the rule. The larger the molecule, the more isomeric alternatives are offered. Fig. 11 shows one of the simplest cases, the two kinds of propyl-alcohol, both consisting of 3 carbons (C), 8 hydrogens (H), 1 oxygen (O).5 The latter can be interposed between any hydrogen and its carbon, but only the two cases shown in our figure are different substances. And they really are. All their physical and chemical constants are distinctly different. Also their energies are different, they represent ‘different levels’.

  Fig. 12. Energy threshold (3) between the isomeric levels (1) and (2). The arrows indicate the minimum energies required for transition.

  The remarkable fact is that both molecules are perfectly stable, both behave as though they were ‘lowest states’. There are no spontaneous transitions from either state towards the other.

  The reason is that the two configurations are not neighbouring configurations. The transition from one to the other can only take place over intermediate configurations which have a greater energy than either of them. To put it crudely, the oxygen has to be extracted from one position and has to be inserted into the other. There does not seem to be a way of doing that without passing through configurations of considerably higher energy. The state of affairs is sometimes figuratively pictured as in Fig. 12, in which 1 and 2 represent the two isomers, 3 the ‘threshold’ between them, and the two arrows indicate the ‘lifts’, that is to say, the energy supplies required to produce the transition from state 1 to state 2 or from state 2 to state 1, respectively.

  Now we can give our ‘second amendment’, which is that transitions of this ‘isomeric’ kind are the only ones in which we shall be interested in our biological application. It was these we had in mind when explaining ‘stability’ on pp. 49–51. The ‘quantum jump’ which we mean is the transition from one relatively stable molecular configuration to another. The energy supply required for the transition (the quantity denoted by W) is not the actual level difference, but the step from the initial level up to the threshold (see the arrows in Fig. 12).

  Transitions with no threshold interposed between the initial and the final state are entirely uninteresting, and that not only in our biological application. They have actually nothing to contribute to the chemical stability of the molecule. Why? They have no lasting effect, they remain unnoticed. For, when they occur, they are almost immediately followed by a relapse into the initial state, since nothing prevents their return.

  1 And thy spirit’s fiery flight of imagination acquiesces in an image, in a parable.

  2 I am adopting the version which is usually given in popular treatment and which suffices for our present purpose. But I have the bad conscience of one who perpetuates a convenient error. The true story is much more complicated, inasmuch as it includes the occasional indeterminateness with regard to the state the system is in.

  3 Zeitschrift für Physik, Chemie (A), Haber-Band (1928), p. 439.

  4 k is a numerically known constant, called Boltzmann’s constant; kT is the average kinetic energy of a gas atom at temperature T.

  5 Models, in which C, H and O were represented by black, white and red wooden balls respectively, were exhibited at the lecture. I have not reproduced them here, because their likeness to the actual molecules is not appreciably greater than that of Fig. 11.

  CHAPTER 5

  Delbrück’s Model Discussed and Tested

  Sane sicut lux seipsam et tenebras manifestat, sic veritas norma sui et falsi est.1

  SPINOZA, Ethics, Pt II, Prop. 43.

  THE GENERAL PICTURE OF THE HEREDITARY

  SUBSTANCE

  From these facts emerges a very simple answer to our question, namely: Are these structures, composed of comparatively few atoms, capable of withstanding for long periods the disturbing influence of heat motion to which the hereditary substance is continually exposed? We shall assume the structure of a gene to be that of a huge molecule, capable only of discontinuous change, which consists in a rearrangement of the atoms and leads to an isomeric2 molecule. The rearrangement may affect only a small region of the gene, and a vast number of different rearrangements may be possible. The energy thresholds, separating the actual configuration from any possible isomeric ones, have to be high enough (compared with the average heat energy of an atom) to make the change-over a rare event. These rare events we shall identify with spontaneous mutations.

  The later parts of this chapter will be devoted to putting this general picture of a gene and of mutation (due mainly to the German physicist M. Delbrück) to the test, by comparing it in detail with genetical facts. Before doing so, we may fittingly make some comment on the foundation and general nature of the theory.

  THE UNIQUENESS OF THE PICTURE

  Was it absolutely essential for the biological question to dig up the deepest roots and found the picture on quantum mechanics? The conjecture that a gene is a molecule is today, I dare say, a commonplace. Few biologists, whether familiar with quantum theory or not, would disagree with it. On p. 47 we ventured to put it into the mouth of a pre-quantum physicist, as the only reasonable explanation of the observed permanence. The subsequent considerations about isomerism, threshold energy, the paramount role of the ratio W:kT in determining the probability of an isomeric transition – all that could very well be introduced on a purely empirical basis, at any rate without drawing on quantum-theory. Why did I so strongly insist on the quantum-mechanical point of view, though I could not really make it clear in this little book and may well have bored many a reader?

  Quantum mechanics is the first theoretical aspect which accounts from first principles for all kinds of aggregates of atoms actually encountered in Nature. The Heitler–London bondage is a unique, singular feature of the theory, not invented for the purpose of explaining the chemical bond. It comes in quite by itself, in a highly interesting and puzzling manner, being forced upon us by entirely different considerations. It proves to correspond exactly with the observed chemical facts, and, as I said, it is a unique feature, well enough understood to tell with reasonable certainty that ‘such a thing could not happen again’ in the further development of quantum theory.

  Consequently, we may safely assert that there is no alternative to the molecular explanation of the hereditary substance. The physical aspect leaves no other possibility to account for its permanence. If the Delbrück picture should fail, we would have to give up further attempts. That is the first point I wish to make.

  SOME TRADITIONAL MISCONCEPTIONS

  But it may be asked: Are there really no other endurable structures composed of atoms except molecules? Does not a gold coin, for example, buried in a tomb for a couple of thousand years, preserve the traits of the portrait stamped on it? It is true that the coin consists of an enormous number of atoms, but surely we are in this case not inclined to attribute the mere preservation of shape to the statistics of large numbers.
The same remark applies to a neatly developed batch of crystals we find embedded in a rock, where it must have been for geological periods without changing.

  That leads us to the second point I want to elucidate. The cases of a molecule, a solid, a crystal are not really different. In the light of present knowledge they are virtually the same. Unfortunately, school teaching keeps up certain traditional views, which have been out of date for many years and which obscure the understanding of the actual state of affairs.

  Indeed, what we have learnt at school about molecules does not give the idea that they are more closely akin to the solid state than to the liquid or gaseous state. On the contrary, we have been taught to distinguish carefully between a physical change, such as melting or evaporation in which the molecules are preserved (so that, for example, alcohol, whether solid, liquid or a gas, always consists of the same molecules, C2H6O), and a chemical change, as, for example, the burning of alcohol,

  C2H6O + 3O2 = 2CO2 + 3H2O,

  where an alcohol molecule and three oxygen molecules undergo a rearrangement to form two molecules of carbon dioxide and three molecules of water.

  About crystals, we have been taught that they form threefold periodic lattices, in which the structure of the single molecule is sometimes recognizable, as in the case of alcohol and most organic compounds, while in other crystals, e.g. rock-salt (NaCl), NaCl molecules cannot be unequivocally delimited, because every Na atom is symmetrically surrounded by six Cl atoms, and vice versa, so that it is largely arbitrary what pairs, if any, are regarded as molecular partners.