Elements of Physical Biology

It will be convenient, in the development of the subject, to follow, in the main, the schedule thus set forth. Kinetic, Dynamic, and Energetic Conceptions of Equilbrium. While we shall, in this section, conceive a stationary state from the standpoint of kinetics, defining it as a state in which certain velocities vanish, it must be noted that there are also other conceptions of equilibrium. Etymologically the word equilibrium is tied, in stricter usage, to a dynamic conception: Aequa libra, the poised balance, is symbolic of a state in which forces are balanced, in which

A third conception of equilibrium, differing from the second, the dynamic, only in point of view, not in scope, is derived from a consideration of energy relations. A system in dynamic equilibrium is found to be characterized by the attainment of a minimum (or sometimes a maximum) of certain functions having the dimensions of energy; a state in which the virtual work done in any very small displacement compatible with the constraints vanishes.1 So, for example, a ball placed in a hemispherical cup, is in equilibrium when its potential energy is a minimum compatible with the geometry of the system. More generally, equilibrium is, according to this view, defined as a state in which certain potentials have a minimum (or a maximum).

Pedantic usage would demand that the term equilibrium be reserved for states satisfying the dynamic and energetic conditions of rest or invariability in time. It would deny the appellation equilibrium to certain states commonly so designated. Metabolic equilibrium, population equilibrium, and the like, are not true equilibria, in this narrower sense, but are steady states maintained with a constant expenditure, a constant dissipation, of energy. Itis not necessary, however, at present, to lay any stress on this distinction. The occasional use of the word equilibrium in speaking of what is merely a steady state maintained with a continuous expenditure of free energy is not likely to cause any serious confusion; and we may as well take the usual liberties in the matter, whenever this course is dictated by convenience and does not offend against essential principles. Where express distinction becomes necessary, we may speak in specific terms of true equilibrium and quasi-equilibrium, respectively, to denote the two separate types included in the generic term “stationary state” or “steady state.”

A complete treatment of the entire field of the statics of evolving systems should, to be entirely systematic, cover both types of stationary states. There are, however, two reasons for departing somewhat from such strictly systematic arrangement. The first is that the statics of true equilibria have been developed to a high degree in the discipline of thermodynamics, so that an exposition of the pertinent principles and conclusions would be little more than

1 Stability of equilibrium demands, further, that the work done on the system in any small, but finite, displacement, be positive, that the potential energy be a minimum (maximum being, in this case, excluded). a transcription into these pages of what can be found abundantly set forth elsewhere in the standard literature. However much one might be tempted, in the interest of a well rounded presentation, to sketch at this point at least an outline of the relevant chapters of thermodynamics, economy of space dictates the briefer expedient of referring the reader to the existing literature, so abundant that it seems superfluous to mention titles.

A second reason for passing lightly over true equilibria at this point, is that the steady states with which we are most frequently and most closely concerned in the field of organic evolution (our main topic here), are of the second class; not true equilibria in the dynamic sense, equilibria in which all forces are balanced; but what we have termed above quasi-equilibria, states maintained constant or approximately so with a continual expenditure, a continual dissipation or degradation of available energy. To such as these we shall give our chief attention, though in part our discussion will be framed broadly to cover indifferently either type of steady state. For the sake of example, too, reference will be made, on occasion, to systems evolving toward a true equilibrium; systems for which the law of evolution is capable of direct expression in comparatively simple thermodynamical terms; systems which, by that very fact, are peculiarly adapted to serve as paradigms exhibiting the characteristic form of a law of evolution.

General Equilibrium Condition. As has already been noted incidentally, the general condition for equilibrium, or, to be more precise, for a stationary state, is obtained by equating to zero the velocity of growth of each component of the system, thus This condition, in general, furnishes n independent equations, which determine one or more sets of values of the variables X, thus If the values C thus determined are real and positive? they evidently define an equilibrium or a steady state, the character (stability, mode of approach) of which depends upon the nature of the roots X of a certain characteristic equation, as has been indicated on an earlier occasion.

Different Types of Equilibrium. Graphic Representation. A particularly graphic representation of the different types of equilibrium is obtained if, instead of seeking solutions of the fundamental equations (1) expressing X,, X.. . . Xx, in terms of ¢, we on the contrary eliminate ¢ from this system of equations. This is very readily effected by division, which leads to the new system This system of equations defines a family of curves passing through the equilibrium points, which here appear as singular points. The situation is particularly transparent in the case of two variables X1, Xe, since this readily permits of plotting the integral curves in rectangular codrdinates in the plane of the paper. We have already had occasion incidentally to employ this method of treatment in an example in Chapter VIII, in which the conflict between a host species and a parasite species was examined analytically. Without going into extensive technical details it is advisable now at least to enumerate and briefly describe the several types of equilibria and the topography, characteristic of each type, presented by the integral curves in and about a singular point. These types are somewhat numerous, even if we restrict ourselves to the case of two variables, and brevity is therefore imperative.

Type 1. Roots \; and » real and negative. Equilibrium is stable; integral curves run directly into singular point as in figure 27, A. Type 2. Roots A; and, real and positive. The topography is similar to that of type 1, but integral curves are traversed outward from singular point. Unstable equilibrium, figure 27, B. Type 3. Roots di and 2: real and of opposite sign. Integral curves in general do not pass through singular points, but curve away from it. Unstable equilibrium, figure 27, C.

* Masses cannot assume negative or imaginary values. Hence negative roots may fail to define equilibria; a similar statement holds regarding complex roots. Type 4. Roots \1 and d2 complex, real parts negative. The integral curves are spirals winding into the origin, forever approaching it without ever reaching it. Stable equilibrium, figure 27, D. Type 5. Roots 1 and A: complex, real parts positive. The topography is similar to that of type 4, but integral curves are traversed outward from singular point. Unstable equilibrium. figure 27, EZ.

Type 6. Roots A; and d2 pure imaginaries. This gives rise to several distinct subtypes. Subtype F. Integral curves are closed loops enclosing the origin. Process is purely periodic. Figure 27, F. Subtype G. Integral curves are spirals winding inward. Stable equilibrium. This is the case treated in Chapter VIII, where a representative diagram willbe found. Figure 27,G. Another subtype is similar to G but spiral winds outward. Unstable equilibrium.

Subtype H. Integral curves are spirals winding about a closed loop. As an example illustrating the occurrence of two types of equilibrium, two types of singular points, the topographic chart of the integral curves defined by the Ross equations for the spread of malaria under certain conditions is shown in figure 28. It will be seen that there are two singular points, one at the origin O, unstable, of type C; the other at 7, stable, of type A. This chart obviously suggests “stream lines’? and a three dimensional model. Such a model (purely qualitative) is shown in figure 29. The feature of interest is that a singular point like O, of type C, is represented by a col (“notch”) in the landscape; whereas the stable equilibrium of type A is represented by a pit, as at the point T.

While this model refers to a very particular case, it serves to bring out a noteworthy fact, namely, that there are necessarily certain regularities in the occurrence of the various types of equilibria. So, for example, it is clear that two pits of the character of the point T cannot occur without some other type of singular point between them, just as it is physically impossible for two mountains to rise from a landscape without some kind of a valley between. For a detailed study of this phase of the subject the reader must be referred to the mathematical literature.’

Fia, 27, Some FunpAMENTAL TYPES OF EQUILIBRIUM, IN A SYSTEM WITH Two Fic. 28. Map or INTEGRAL CurRVES FOR THE Ross MALARIA EQuaTIONs, AS AN ExaMpLe Exuisitinc Two SINGuLAR Points, or TyPp Figure 27, A anp C) The heavy lines are integral curves; the lighter lines are auxiliaries (isoclines) employed in constructing the graphic solution of the differential equations. (Reproduced from A. J. Lotka, Am. Jour. Hygiene, Januar y Supplement, 1923.) ‘A[UO VAIVBYITBNH Si GZ puvw SZ Soinsy us0M4oq 9oUepUOdsedI00 OY T, *(8% “BY ul Q) wMIAGITIMbe sjqvysun jo yurod oy} 0} spuodseii09 (19U109 JUOJJ oy} WOIF SuUUNOD OAINO [eATOZUT PUODS OY} PUB BAIND [BUOZEIP oY} JO UOTJDeSIO}UT JO JuTOd 9Y}) Yo}OU ay} Jo Joqyued oy} {(8z “Sy ul 7) UMIAqI{IMbs 9,qQVq4s Jo yuI0d oY} 07 spuodsoi109 JopOUr oY} JO I9}U99 94 4B yid oy

(HOVAUNY OIHdVUNOdOT, V NO LNGOSAC 1 NIT) MOTW] dO SANIT SV ( INT @ Metastable Equilibrium. The graphic representations of the malaria equilibrium furnish the occasion for another remark of general character regarding certain equilibria. The circumstance that gives rise to the first malaria equilibrium, the one in which the malaria rate is zero, (point O in figs. 28 and 29) is the autocatakinetic character of the growth of amalariaendemic. This is a common characteristic of the growth of living systems; growth is initiated by a nucleus of the same species of matter that is added by the growth. Conversely, in the entire absence of any nucleus of a particular species of living matter, growth of that species cannot take place, even though all other conditions for such growth may be satisfied, even though the system may be, as it were, supersaturated with regard to that species of matter. In these circumstances an equilibrium may be presented which is unstable in the sense that, upon the introduction of a suitable nucleus, growth immediately sets in. Equilibria of this type, which are stable in the absence of a suitable ‘‘nucleus”’ but in which change is immediately initiated upon introduction of such a nucleus, have been termed ‘metastable’ equilibria.

Exceptional Cases: A brief reference must suffice regarding certain exceptional cases that may arise. So it may happen that one of the roots \ of the characteristic equation vanishes. An example of this was encountered in dealing with the Ross malaria equations. It was found that as the number of mosquitoes per head of the human population approaches a certain critical value, two singular points approach each other, and finally fuse, giving one “double” point.®

Another special case that may arise, and whose mention must here suffice, is that of multiple roots of the characteristic equation, the case in which two or more of the roots are equal.® 4In inorganic systems an analogous state of affairs is observed in supersaturated solutions or vapors which are brought to crystallization or to condensation by the introduction of a suitable nucleus. Dynamically the characteristic of a metastable equilibrium is that the thermodynamic potential of the system, though a minimum, is not an absolute minimum.

I wanted to remind the biologists that in the early stages of life what they are accustomed to speak of as natural selection passes over into what might be described as a mere physical selection of stabler compounds.—K. Pearson. One of the simplest examples of equilibria in systems of the type that interests us here—systems composed of several groups each consisting of numerous similar individuals as units—is the equilibrium resulting from a pair of balanced or opposing chemical reactions.

This case illustrates so well, in their simplest form, a number of typical traits of the phenomena here under discussion, that it will pay to give it brief consideration. We shall select for this purpose the simplest possible type of balanced chemical reaction at constant volume and temperature, namely a reaction which is monomolecular in both directions. A substance S; undergoes a transformation into S», and S, in turn is converted back into S;, one molecule alone taking part, in each case, in the transformation. If x; and x2 are the respective concentrations of S, and S:, we have, at a given temperature, by the law of mass action, the rate of decomposition of S, and S: respectively.

Or, since at constant volume concentrations x are proportional to numbers n of molecules where ky, kz are coefficients (functions of the temperature) characteristic of the reaction. The rate of increase of the substance S is the excess of its rate of formation b over its rate of decomposition, in strict analogy to the birth rate and death rate in a human population In a population of living organisms the material for the formation of new individuals must ultimately be derived from the bodies of those that have died. But the connection is a complicated one involving many steps. In the population of molecules here under consideration the relation between birth rate and death rate is of the simplest possible form. Each molecule of S; that ‘dies’ becomes a molecule of S2, and vice versa. Thus equations (5), (6) assume the form

If we fix our attention upon m molecules of S; at the moment of their formation, we can apply to these particular molecules the equation (3), from which we have, by integration But — a A is the probability p:(a), at the moment of its formation, ny that a molecule of S, picked at random at such moment, will reach age a. ‘The life curve for the molecules of S, is thus defined by the molecules are present in amounts proportional to their respective mean lengths of life, although, they are “‘born’”’ in equal numbers, since kynz = kyn:. The significance of this is brought out in the diagram figure 30, in which the population of molecules is plotted ‘in age groups,” for the substance S; and S: separately. Since the birth rate is the same for both populations, they begin at a common ordinate; but the curve for S., the substance with greater k, greater force of mortality, lies entirely below that for S,; the areas of the two curves are, in fact, proportional to the mean lengths of life of the molecules of the corresponding substances. Thus, in the struggle for existence the stabler (fitter) molecules of S; have the advantage, being, on an average, longer-lived.

There is thus an obvious analogy between the course of events in such a population of different species of molecules, on the one hand, and a mixed population of different species of organism on the other, an analogy which extends into details for the exposition of which space is lacking here.!_ The analogy is not a meaningless accidental circumstance, but depends on identity of type in the two cases. It can be said quite generally (so as to apply to either case), that in a material system in which physical conditions vary from instant to instant and from point to point, certain individual constituents (molecules, organisms) may have a transitory existence as such, each lasting just so long as its conditions and those of its neighborhood continue within certain limits. Although the “life period” of each individual constituent may be thus limited, an aggregate of a number

of such individuals may nevertheless have prolonged existence, provided that the fluctuations in the conditions of the system, from point to point and from instant to instant, do not exceed certain limits, and that by some process or other new individuals are formed as the old are eliminated. Of the character of the fluctuations, and their relation to the “length of life” in the case of living organisms, more will be said in a later chapter. As to the circumstances, the fluctuations, that lead to the translation of a molecule in chemical reaction from one state into another, we may with advantage adopt a view-

Fic. 30. Ace Disrripution In Popunation or Moxuecutses or Two Sus- The birth rate per head, of both species, is the same as indicated by their common zero ordinate; but the species having the lesser force of mortality (reaction constant) predominates, as shown by the greater area under the corresponding curve. (Reproduced from A. J. Lotka, Am. Jour. Science, point set forth in some detail by the writer on an earlier occasion,? and expressed more recently by Professor Baly* in these terms:

Every complete reaction consists of three separate stages, with each of which is associated its characteristic energy change. In general, molecules in the free state exist in a phase which is non-reactive, and in order to carry out any reaction it is first of all necessary to bring them into a reactive phase. This, which is the first stage of the reaction, requires that a definite amount of energy should be supplied to each molecule, the amount necessary being the difference in energy contents of the initial phase and the particular phase necessary for the reaction in question.

The second stage of the reaction is the atomic rearrangement whereby new molecules are produced, and it is this stage, and this stage alone, which is represented by the equation of the reaction. The third and final stage is the change in phase of the newly synthesised molecules, whereby they pass into their normal and non-reactive phases. These last two stages are both accompanied by an escape of energy. If the sum of the amounts of energy evolved in the second and third stages is greater than that absorbed in the first stage, the reaction is exothermic; whilst an endothermic reaction is one in which the energy necessary for the first stage is greater than the total amount evolved in the second and third stages.

It should be remarked that the second stage is, apparently, passed through in an exceedingly brief space of time, so that at any instant only an imperceptibly small amount of substance exists in the transitional state. We are, in fact, almost wholly devoid of any information regarding matter in this state, and the words of Schénbein‘ hold true in almost their full force today: “‘Presumably, between the state in which two portions of matter exist after completion of chemical combination, and the state in which they previously existed separately, there is a series of transition states of which the chemistry of today knows nothing.’ Probably the only positive and direct experimental evidence we have of matter in this intermediate state between two compounds is furnished by the superlatively refined methods of Sir J. J. Thomson and Dr. F. W. Aston, which not only reveal but actually weigh such decapitated molecules as CH3, whose length of life is measured in ten-millionths of a second.

As to the agencies, the “fluctuations” that provide, every now and again, the requisite energy to carry a transforming molecule “‘over the crest of the hill,” there is first the thermal agitation of the molecules, second the influence of incident light in photochemical reactions, and third the influence of catalysts, whose action probably depends on a flattening of the path over the hill crest, the point of departure and the final state remaining unchanged. For discussions of these technical details the reader must be referred to the literature, a few of the more recent publications being noted in a footnote below.®

While the details of the manner of the “birth” and “death” of the molecules in chemical transformation are, as yet beyond the range of the observation of the physicist, the fundamental laws of energetics, which hold true generally, and independently of particular features of mechanism, are competent to give substantial information as to the end product, at any rate, of the evolution of such a system as considered in the simple example above. The final equilibrium must accord, as regards its dependence on temperature, pressure and other factors, with the second law of thermodynamics, which may thus be said to function as a law of evolution for a system of this kind. This is a point worth dwelling on a little at length, inasmuch as our knowledge of the form and character of the law of evolution for this special type of system may be expected to serve as a guide in the search for the laws of evolution in the more complicated systems, belonging to an essentially different type, which confront us in the study of organic evolution. The second law of thermodynamics can be expressed in various ways, but the form in which it serves our present purpose best is that which states that the system evolves toward a state in which certain functions (thermodynamic potentials) of the variables defining its condition are at a minimum, somewhat as a ball placed in a hemispherical bowl ultimately comes to rest in the position in which its (gravitational) potential is a minimum, namely, at the lowest point of the bowl. Many laws of nature are conveniently’ expressed in this form, as minimum (or maximum)

7 Fundamentally this is a matter of convenience, and does not predicate anything narrowly characteristic of natural laws. The fact that the course of events is wniquely determined implies that the laws which determine that course can be expressed in the manner referred to. For a discussion of this question see J. Petzoldt, Maxima and Minima und Okonomie, Altenburg, laws, and it is to be expected that the law of evolution in life-bearing systems also, (where, as we shall see later, mechanism cannot be lightly waved aside into the convenient catch-all of the laws of thermodynamics), will be found to receive its most convenient expression in this form. In another respect the case of chemical evolution may confidently be expected to be found a good model in the treatment of the broader problem of evolution. It is to be noted that the law of chemical evolution is expressed in terms of the system asa whole. It is the thermodynamic potential of the entire system that approaches a minimum. Biologists have rather been in the habit of reflecting upon the evolution of individual species. This point of view does not bear the promise of success, if our aim is to find expression for the fundamental law of evolution. We shall probably fare better if we constantly recall that the physical object before us is an undivided system, that the divisions we make therein are more or less arbitrary importations, psychological rather than physical, and as such, are likely to introduce complications into the expression of natural laws operating upon the system as a whole.