Elements of Physical Biology

to certain staples of agricultural production, our population has advanced in a succession of moving equilibria; yet the progress of modern industrial civilization on the whole is essentially the very antithesis of a moving equilibrium conditioned by and following upon the changes of a slowly varying parameter. Quite on the contrary, the development of this age is rather of the nature of a rocket-like ascent with a speed altogether unparalleled in all previous history of organic evolution, and at the cost of rapid depletion of capital resources. Certain aspects of this phenomenon are reserved for consideration in a later chapter. Here it will be sufficiently to the point to draw attention to table 31, reproduced from R. Pearl’s essay on The Population Problem," which shows the altogether disproportionate increase in the growth of our material accessories in recent years, as compared with that of the population itself.

The human species, considered in broad perspective, as aunit including its economic and industrial accessories, has swiftly and radically changed its character during the epoch in which our life has been laid. In this sense we are far removed from equilibrium—a fact which is of the highest practical significance, since it implies that a period of adjustment to equilibrium conditions lies before us, and he would be an extreme optimist who should expect that such adjustment can be reached without labor and travail. We can only hope that our race may be spared a decline as precipitous as is the upward slope along which we have been carried, heedless, for the most part, both of our privileges and of the threatened privation ahead. While such sudden decline might, from a detached standpoint, appear as in accord with the eternal equities, since previous gains would in cold terms balance the losses, yet it would be felt as a superlative catastrophy. Our descendants, if such as this should be their fate, will see poor compensation for ‘their ills in the fact that we did live in abundance and

Die Physik wird aus dem Studium des Organischen an sich noch sehr viel neue EHinsichten schépfen miissen, bevor sie auch das Organische bewaltigen kann.—H. Mach. In preceding pages we have passed in review some of the principal features of interest presented by systems maintained constantly at or near equilibrium, while one of more of the parameters determining such equilibrium were slowly changing, thus engendering a moving equilibrium. One might proceed to a consideration, on a more general basis, of the changes brought about in an evolving system through changes of any kind, including rapid ones, in the parameters. In themost general case this would amount to the discussion of a system of differential equations of the form.

It is not proposed to take up the study of this perfectly general case ; it must suffice to point to the mathematical literature regarding equations of this form.! But there is another special phase of the general problem which, like the case of slow changes, yields with comparative ease to analytical treatment; namely, that special phase which enquires only into the ultimate effect, upon equilibrium, of a given total change in a parameter, leaving aside all questions relating to the path by which the displacement of equilibrium takes place. Such a separate consideration of this special and restricted phase of the general problem is rendered possible by the fact that, in certain cases at any rate, the displacement of the equilibrium is independent of the path of the change, and depends only on the given initial and

final values of the parameters whose modification provokes or is associated with the change. So, in physico-chemical transformations (‘‘changes of state”) the principle of Le Chatelier enables us to predicate, within certain limits, the sign of the displacement of equilibrium conditioned by a change in certain of the parameters upon which the equilibrium depends. The principle of Le Chatelier is best illustrated by a simple example. Consider the simple chemical reaction

At high temperatures this reaction is reversible; that is to say, it takes place to some extent in the direction of the upper arrow, but also to some extent in the direction of the lower arrow, and an equilibrium is finally established between these two opposing reactions. Now this is what the Le Chatelier principle tells us: If we add either H alone or O alone to the system, the equilibrium is shifted in the direction of the upper arrow, that is to say, in such direction as to absorb some of the added constituent.

Similarly, if we heat the system, the equilibrium is shifted in the direction of the lower arrow, that is to say, in the direction of the reaction which absorbs heat. The principle, as enunciated by Le Chatelier? himself, is: Every system in chemical equilibrium, under the influence of a change of any single one of the factors of equilibrium,’ undergoes a transformation in such direction that, if this transformation took place alone, it would produce a change in the opposite direction of the factor in question.

The factors of equilibrium are temperature, pressure, and electromotive force, corresponding to three forms of energy—heat, electricity and mechanical energy. The second paragraph of the principle as quoted above, requires specialemphasis. It is often omitted, even by authors of the highest 3 It appears that some French writers employ the term ‘‘facteur d’équilibre”’ as synonymous with ‘‘intensity factor of an energy.” (Cf. F. Michaud, Ann. de Phys., vol. 16, 1921, p. 132.)

repute, with the result that a vagueness is introduced for which Le Chatelier himself cannot justly be made responsible. This vagueness is then often rendered still worse by departures from the original wording, aimed at an extension of the scope of the law to all conceivable systems and ‘‘factors,”’ an extension which is gained with a total sacrifice of all validity of the principle. So, for example, if we seek to apply the principle as quoted above, but omitting the restriction of the second paragraph, to the water equilibrium already mentioned, and if we select as “factor” of equilibrium not pressure but volume, the principle would lead us to reason as follows: On diminishing the volume of the system, that transformation will take place which, did it take place alone (i.e., at constant pressure), would be accompanied by increase in volume; a conclusion which is false. As has been shown by Ehrenfest,® the error arises through failure to discriminate, in the application of the principle, between the intensity factor (e.g., pressure) and the capacity factor (e.g., volume) of an energy.

It must appear singular that so obvious a defect of the principle, as commonly quoted, should so generally have escaped attention, and should for example, have passed unnoted through seven editions of so excellent a work as Nernst’s Theoretische Chemie. Ehrenfest points out that the explanation lies in the very vagueness of the principles, which permits it to be construed in each case to suit circumstances. The principle is commonly applied ex post facto, and its competence to predict thus escapes any serious test. This, however, is only a partial explanation. After all, the fundamental reason for the tardy recognition, and the still more tardy admission in the general literature, of the weakness of the principle, as commonly quoted, must be sought in an inherent weakness of the human mind: by a curious inversion of what might be expected in logical sequence, the last things to receive critical scrutiny are always the fundamental premises of our arguments. This is true both as regards the judgment of the average individual, of the people at large, and often even of the man of very superior intellect. One recalls, in this connection, MacAuley’s remarks regarding Dr. Johnson: “How it chanced that a man who reasoned upon his

premises so ably should assume his premises so foolishly is one of the great mysteries of human nature.” If such an outwardly slight departure from Le Chatelier’s original enunciation as the omission of his second “explanatory” paragraph, thus completely destroys the validity of his principle, what is to be said of such sweepingly vague settings as in the following examples: The broadest definition of the principle of Le Chatelier is that a system tends to change so as to minimize an external disturbance (W. D. Bancroft, Journal of the American Chemical Society, 1911, p. 92).

Every external action produces in a body or system changes in such direction, that in consequence of this change the resistance of the body or system against the external action ts increased. If we regard the faculty of adaptation of animals and plants from the point of view that the organisms undergo, under the influence of external actions, changes which render them more resistant to those actions, then the property of non-living matter which is expressed by the principle of Le Chatelier-Braun may be regarded as a sort of adaptation of such non-living matter (Chwolson, Traité de Physique, 1909, vol. 3, p. 547).

If the equilibrium of a natural complex (system of masses, organism, system of ideas) is disturbed, it adapts itself to the stimulus (Reiz) which causes the disturbance, in such manner that the said stimulus continually diminishes until finally the original or a new equilibrium is again established (J. Lowy, Kosmos, 1911, p. 331). The last two examples are of particular interest to us here as suggesting application of the principle to biological systems. Asa matter of fact, such application of the vaguely formulated principle (in a form in which it would be injustice to link it with the name of Le Chatelier) antedates by many years its enunciation by the French physicist. The following passages in Herbert Spencer’s First Principles are pertinent:

Among the involved rhythmical changes constituting organic life, any disturbing force that works an excess of change in some direction is gradually diminished and finally neutralized by antagonistic forces, which thereupon work a compensating change in the opposite direction, and so, after more or less of oscillation, restore the medium condition. And this process it is which constitutes what physicians call the vis medicatrix naturae. This is a conclusion which we may safely draw without knowing the special re-arrangements that effect the equilibration: If we see that a different mode of life is followed after a period of functional derangement by some altered condition of the system—if we see that this altered condition, becoming by

and by established, continues without further change, we have no alternative but to say that the new forces brought to bear on the system have been compensated by the opposing forces they have evoked (First Principles, Chapter XXII, Equilibriation, 173). Almost simultaneous with Le Chatelier’s publication (1884) is the following pronouncement. L’étre vivant est agencé de telle maniére que chaque influence perturbatrice provoque d’elle méme la mise en activité de l’appareil compensateur qui doit neutraliser et reparer le dommage (Léon Frédéricq, Archives de Zoologie Exp. et Gén., ser. 2, vol. 3, 1885, p. xxxv).

Now it is not denied that such expressions as this have a certain utility, as describing with fair accuracy a goodly proportion of a class of phenomena to which they relate. But to designate such statements, as ‘‘Le Chatelier’s Principle,’’ is wholly misleading. That principle, in its exact and narrower formulation is rigorously true, as much so as the laws of thermodynamics from which it can be deduced; it has no exceptions, any more than there is any exception to the law that heat flows by simple conduction from the hotter of two bodies to the colder.

The alleged ‘‘principle,’”’ as applied to biological systems, lacks the sureness which the true Le Chatelier principle possesses, in its stricter formulations, in physical chemistry. An organism may, by exposure to a certain influence A, become more resistant to the influence, as in the case of acquired immunity after an attack of infectious disease, or after habituation to such a poison as arsenic. But, by exposure to another influence B it may become less resistant to B, as in the case of cumulative poisons, or of anaphylaxis. The Le Chatelier principle does not enable us here to predict in which direction the effect will take place in a new and untried case of some influence C.

Conditions of Validity of Le Chatelier’s Principle. The question arises why the principle thus breaks down in its application to biological cases of the kind cited. The answer is found by examining the basis on which the proof of the principle rests. Such an examination brings out the fact that one of the necessary conditions for the applicability of the principle is stability of the equilibrium to which application is made. Now~ the equilibria commonly contemplated in physical chemistry are stable, so that

this condition is satisfied. But it is not always satisfied in the equilibrium of the living organism. The organism is, indeed, stable with regard to many of the commonly occurring attacks of its environment. But it is of little consequence to the species whether, for example, the individual organism is stable with regard to the ingestion of a large dose of strychnine, for in nature such ingestion will occur so rarely, if at all, as to influence in no appreciable degree the life of the species. It is not necessary for the stability of the species, that the individual be stable at all times.é In point of fact, we know perfectly well that sooner or later each individual finds itself in a condition of instability, by “accident?’ or sickness, and dies. An analysis’ of the basis of the principle of Le Chatelier reveals the fact, among others, that all demonstrations of this principle postulate, as a fundamental characteristic of the systems to which it applies, that they be in stable equilibrium. The principle can, therefore be applied at best only with cautious reservation to living organisms, reservation such as, for example, Le Dantec® makes: ‘In studying as closely as possible the consequences of disease in living organisms, when they survive such diseases, I have drawn attention . . . . to the fact that all these consequences, such as acquired immunity and the production of antitoxic sera, can be summarized in the principle of Le Chatelier.”” But with such reservation the principle loses its chief utility, which consists in its power to predict the course of events. Indeed, it might be accused, in such restricted form, of being little more than a tautological platitude, which tells us that if the system or unit in question is stable, then it is stable.

This is not quite such a damning accusation as may at first sight appear, for the same can be brought against the principle of the survival of the fittest, which nevertheless has proved supremely fertile in biological research. In point of fact there is a close relationship between 6 Compare what has been said in the discussion of chemical equilibrium regarding the stability of aggregates composed of individuals, themselves of limited stability, of limited life period (Chapter XII).

7 Such an analysis, carried out in considerable detail, has been given by the writer in Proc. Am. Acad. Arts and Sci., 1922, vol. 57, pp. 21-37. The importance of the restriction to stable equilibria, in connection with biological systems, has also been pointed out by C. Benedicks, Zeitschr. f. phys. Chemic, 1922, vol. 100, pp. 42-51. A.J. Lotka, Am. Jour. Hygiene, 1923, p. 375. the two principles. But it is important to note that the principle of the survival of the fittest is avowedly statistical in character, and is to be applied to organisms in the gross. This is true, also, of the principle of Le Chatelier in physico-chemical systems; its field of application is to aggregates of molecules, not to the individual. But the applications that have been essayed in biology have been made to the individual; such application can at the best yield a judgment of probabilities. In physical chemistry we deal for the most part with stable equilibria. But in biology, as has already been pointed out, though the races that come under our observation possess stability as races (else they would not have survived to be our contemporaries), it does not follow at all, that each and every individual is at all times in a state of steady equilibrium.

Aside from the limitation in the applicability of the principle to stable systems, other limitations appear in such an analysis of its foundations as has been referred to above. So, for example, loose analogy to the physico-chemical equilibrium, as affected by the addition of a quantity of one of the reacting substances, might lead one to draw the erroneous inference that in a community infected with malaria, the introduction of additional malaria parasites would shift the equilibrium in the direction of a higher malaria rate. But there is every reason to expect, on the contrary, that the equilibrium remains unchanged by such addition. For a close analysis of the reason for this divergence in the two cases the reader must be referred to the original paper already cited. It must suffice here to state briefly that this reason is to be found in the existence in the physico-chemical case of equations of constraint, relations between certain variables, and in the absence of analogous relations in the case of malaria.

Extension of Scope of Rigorous Applicability. While the prime result of a searching analysis of the foundations of the Le Chatelier principle is to emphasize rather the restrictions of its scope, yet in certain respects such analysis does furnish a rigorous basis for a certain generalization of its applicability beyond those bounds where its warrant rests on the firm ground of thermodynamics, And this extension of the strict applicability of the principle takes place essentially in two directions. On the one hand the thermodynamic justification, at any rate in the form commonly presented, covers only true equilibria, and does not extend to

steady states maintained with constant dissipation of energy. This restriction does not appear in the demonstration of the principle on the broadest grounds that suffice for its establishment,? Le Chatelier’s principle applies, in certain cases, to steady states of the more general type, as well as to true equilibria. The second direction in which the analysis, on general grounds, of the principle, enlarges its field of warrant, is in the matter of the kinds of “factors” to which it is properly applicable. It has already been pointed out that, in its physico-chemical application, it must be used with proper discrimination as to the distinction between the capacity and the intensity factor of an energy, as, for example, volume and pressure. It is found, upon analysis, that the applicability of the principle to the effect of a change in pressure, for example rests upon the following fundamental property of the pressure and volume of a system in stable equilibrium.

1. For every value of v, the volume of the system, there is a definite value of p;, the pressure which zt exerts, the internal pressure, aS we may term it. 2. The volume v increases or decreases according as the internal pressure 7; is greater or less than the external pressure p, upon the enclosure, that is to say, 3. It can be shown that, given (1) and (2), stability demands that the curves representing the relative between p (ordinates) and v (abscissae) must slope from left to right downwards. For if such a curve slopes in the opposite direction, then the slightest displacement from equilibrium will immediately cause the system to travel with cumulative effect, avalanche-like, along the pv curve further

9 For justification of this and other statements made in these paragraphs the reader is referred to the author’s paper already cited. It may be remarked that Ehrenfest (loc. cit.) expresses the belief that such broader scope belongs to the principle, but he does not support his impression with proof. 10 [t is interesting to note that an upward slope, from left to right, occurs in the middle limb of the van der Waals’ pv curve of a gas. But this limb represents an unstable state which is never realized, the gas, instead of following this part of the curve, partially condenses and traces a horizontal straight

Now these fundamental properties (1), (2) and (3), of a capacity and an intensity factor of an energy" are shared by certain parameters that have no direct or simple relation to energy whatsoever; and since the applicability of the principle depends upon these properties, it will extend to such other parameters possessing them. As an example may be mentioned the relation between area a occupied by a population, and the rent per unit area R; that an (average) individual is willing to pay. If R; is greater than R,, the rent at market rate, the individual will move into a more spacious apartment, and a will increase, and vice versa; so that

= =0 according as Rj = Re On the other hand the curves representing, in rectangular coordinates, the relation between rent and area available per head, necessarily slope from left to right downward. If it were true, as sometimes stated, that the more a man has, the more he wants, economic equilibrium would be an unstable condition. This example is presented here with reservation. There may be various complications in practice that may form obstacles to the simple application of the principle indicated. But it will serve to show how a perfectly rigorous justification may exist for the application of the principle of Le Chatelier outside the field of plain energetics and thermodynamics. Where, and only where such justification can be clearly shown to exist, there it will be permissible and useful to apply the principle. Applications made broadcast, without prior examination of the parameters involved, perhaps without any thought at all of reasonable parameters, are of little if any worth.

One other word of caution must be said, for which the example of area and rent will furnish a suitable illustration. Before we apply 4 Owing to the custom of counting heat absorbed by a system as positive, but work done upon it as negative, the relation analogous to that of (2) takes the form, in the case of heat energy, dQ dt where Q is the quantity of heat absorbed by the system at a temperature from a source at the temperature 6. Here the Qé curves slope upward from left to right. Cf. A. J. Lotka, loc. cit., p. 36.

the principle to any particular parameter, we must be sure that the contemplated change will modify this parameter alone, and not also at the same time others that are in principle, if not in physical fact, to be regarded as independent. So, for example, one reason why the example of area and rent was presented above with express reservation is that, ordinarily at any rate, it may be difficult or impossible to modify the area of a population without modifying at the same time certain other features, such as the supply of nutriments furnished in the soil, ete.