Elements of Physical Biology

Now a biological species, however defined, is not a homogenous group. It comprises portions (individuals) varying more or less widely with regard to numerous features, such as stature, weight, etc. If our description of the eicbicution of matter in the system is to be at all exhaustive, we shall need to know, not only the extent (total mass) of each species, but also its constitution, as expressed by the frequency, the relative abundance, of each statistical type within the species. In the case of man, for example, we may wish to know the fraction F (56) of the total population whose height is comprised within the limits 56 and 57 inches at a given instant. And in observing the evolution of the system of which this population forms part, we shall be interested, not only in the growth (or decay) of the population as a whole, but also in the rate of change of the abundance (frequency) of each statistical type. This phase of the problem does not differ essentially, in character, from that first considered: it is essentially a question of distribution and changes in distribution of mass in the system among its several components; only now we have fixed our attention on a different set of components, components defined in a different way, on a finer scale. In the first instance we had taken in view the distribution and changes in distribution of the matter of the system among the several major groups or species; now we are considering the distribution within each such group. This division of the general problem of organic evolution into two aspects has certain practical advantages, and it will be convenient to have names to designate the two separate aspects or domains of evolution. We shall accordingly speak of inter-growp evolution on the one hand, when

referring to changes in the distribution of the matter of the system _ among several component groups; and we shall speak of intra-group evolution when referring to changes in the distribution of matter within the group, among its statistical types, however defined. It is possible to set up equations relating to intra-group evolution, similar in general character to those set forth above relative to inter-group evolution. However, this phase of the problem is probably treated more satisfactorily in other ways, of which some examples will arise in due course.°

It is nevertheless, desirable, to indicate in the system of equations (3) the incidence of intra-group evolution. Conveniently this may be done by writing the ith equation, for example, in more detailed form: where Qi, Q2, . . . Qx are parameters defining the character of several components (e.g., biological species) S; such definition may take the form of a set of characteristic frequency functions, or some other form. These parameters will, in general, be functions of the time, that is to say, each component may, in general, be variable in character through the occurrence of intra-group evolution. Whether this variability is limited, or constrained to follow a certain course (orthogenesis), whether variations take place in continuously graded series or per saltwm, or in any or all of these ways, are questions which will not be discussed at this point. At present all that need be said is that the origination of a new species in any of these ways falls within the scheme of our description of evolution as a change in the distribution of matter among the components of the system.®

It may be remarked, in passing, that in general a complete definition of the system may require an infinite number of parameters P and Q; this does not necessarily cause any undue complication in practise, since in many cases certain of the parameters P, Q either remain constant, or change so slowly that, in discussing changes in the variables X, we may treat these parameters as if they were constant. The parameters Q, defining the character of the species, are in general functions of the time, as has already been remarked. In this respect organic evolution exhibits a very important distincetion as compared with chemical evolution, i.e., the evolution of a system in the course of chemical transformation. In such a system the character of the components is usually fixed once for all. Water is H,O for all time, unlike a species of organism which is subject to change in character. One important result of this is that, so far as we know, organic evolution is a process without end, for there seems to be no limit to the variety of forms of living matter, as there is no limit to the variety of geometric configurations and mechanical systems that can be formed from a given portion F matter. Chemical evolution, on the contrary, terminates, under

constant condition, in a definite equilibrium, determined once for all, by those conditions. * This is not a new definition of evolution, it is a conception of evolution wholly compatible with the definition that has been laid down in preceding pages. It will be observed that the fundamental equations (3) resemble in form the equation (4) of Chapter II, which was given as a typical example of the equation for an inertia-free or completely OF ues , : damped system. The velocities “ag te single-valued functions F of the variables X. It is the single-valued character of the functions F that gives the system its stamp as an inertia-free or completely damped system; a system in the course of typically irreversible transformation.

It is not maintained that these equations cover all cases that may be brought within the purview of the present study; nor shall we, in all cases, be tied down to the scope of these equations. They are, however, of very broad scope, and, upon reflection, will be found to cover at least a large and significant portion of the field of our interests here. To one point, however, it may be well to draw attention. To read these equations in their broadest interpretation we must be prepared to consider cases in which the phenomenon of lag or lead enters. Perhaps the terms lag and lead require explanation. In some cases the course of events today depends on certain features in the state of the world at a previous date. So, for example, the number of persons of age 50 in the year 1924 depends (among other things) on the birthrate in the year 1874. Or, to quote another instance, the number of new cases of scarlet fever today depends on the number of infections a week ago. There is thus a lag in the appearance of the observable effect in the system. In other cases there may be a lead. The price of land on Church Street today may suffer an increase or perhaps decrease because it becomes known that in a year’s time a railway station is to be built nearby.

Since effects of this kind must be contemplated as a possibility, we must be prepared to read our equations in the following sense: The rate of increase of the mass of the component S; at the present instant, is a function of the masses Xi, Xo, etc., at some other instant of time, say X, at (t — pi), X2 at (t — po), ete., where some of the p’s may be negative (corresponding to a lead). We must be prepared to consider our equations in this interpretation. Illustrative examples will not be offered here, as the mathematical treatment of these cases is somewhat troublesome. The interested reader will find an illustration in the author’s monograph on the Ross Malaria equations, Am. Jour. Hygiene, vol. 3, January Supplement, 1923. Only a reflection of general character shall find its place here. It is characteristic of systems whose history is defined by equations thus involving a lag, that, in general, the course of events at a given instant is dependent upon the previous history of the system over a certain finite range of time. The consequences of this feature are somewhat singular. If the world’s events followed a system of equations of this kind, we might have two worlds, in every respect identical today, but each with a different past, and, in consequence each with a different future. And a similar reflection necessarily applies inthe case of lead.

This conclusion is perhaps not in harmony with a mechanistic conception of the universe. But the phenomena of memory and of will are of precisely such character as to introduce lags and leads into the world’s equation, and we may be well advised to keep our minds open as to the possible effects of this circumstance upon the course of events. It should be observed, that the appearance of a lag or lead in our equation may be spurious. It may be due toa species of mathematical shorthand. It is easier to describe a person as having become infected with scarlet fever three days ago, than to describe precisely his present state today ensuant upon that infection. Hence we may prefer to, or, for lack of detailed knowledge we may be

_ forced to write our equations in terms of (¢ — p), although, if all the facts were known, we could, were we so disposed, write them in terms of ¢. And the same reflection applies to the appearance of a lead in our equations. Whether, with complete knowledge of all | circumstances bearing on the situation, there should still remain _a residuum of influences that could find expression only in terms of a lag or lead, this, perhaps, is fundamentally the nature of the problem of the influence of consciousness upon the course of the

; Tt will be the function of this new branch of science to investigate biological phenomena as regards their physical aspects, just as Physical Chemistry has treated the physical aspects of chemical phenomena. Because this field has not yet been systematically explored . . . . the individual data of Physical Biology appear, as yet, as more or less disconnected facts, or as regularities for which no proper place is found in the existing scheme of present-day science; and the investigations of isolated problems in this field are as yet carried on as something of a scientific hobby by amateurs, with the result that they are guided by chance rather than by plan. . . . and are often totally lacking in any fundamental guiding principles or connecting theory. As results gathered in this disconnected fashion accumulate, the need of their unification into a harmonious whole, into a distinct discipline of science, becomes more and more acutely felt. Such unification necessarily involves the working out of a viewpoint that shall make the several facts and relations fall in line naturally in an orderly system; in other words, what is needed is a labor of organisation. In the course of this, new and unforeseen problems will inevitably arise, and a fruitful field of scientific endeavor should thus be opened for the investigator.—Porstmann.

A first use to which we may with advantage put the results of the preceding analysis, is to systematise the subject here treated; there will thus be gained a general plan of work and a division of the topic into natural sections, upon which the arrangement of the succeeding chapters will, in the main, be based. Physical Biology,! as here conceived and discussed, is essentially a branch of the greater discipline of the General Mechanics of Evolution, the mechanics of systems undergoing irreversible changes in the distribution of matter among the several components of such

1 In introducing the term Physical Biology the writer would suggest that the term Biophysics be employed (as hitherto) to denote that branch of science which treats of the physics of individual life processes, as exhibited in the individual organism (e.g., conduction of an impulse along nerve or muscle); and that the term Physical Biology be reserved to denote the broader field of the application of physical principles in the study of life-bearing systems as a whole. Physical biology would, in this terminology, include biophysics as a subordinate province.

For a summary statement of what might be termed the program of biophysics see A. Forbes, Science, 1920, vol. 52, p. 331. It so happens that many of the components that play an important réle in nature, both organic and inorganic, are built up of large numbers of individuals, themselves very small as compared with the aggregations which they form. Accordingly the study of systems of this kind can be taken up in two separate aspects, namely, first with the attention centered upon the phenomena displayed by the component aggregates in bulk; we may speak of this as the Bulk Mechanics or Macro-Mechanics of the evolving system. And, secondly, the study of such systems may be conducted with the attention centered primarily upon the phenomena displayed by the individuals of which the aggregates are composed. This branch of the subject may suitably be termed the Micro-Mechanics_of

the evolving system. It is evident that between these two branches or aspects of the general discipline there is an inherent relation, _ arising from the fact that the bulk effects observed are of the nature _ of a statistical manifestation or resultant of the detail working of the micro-individuals. The study of this inherent connection is, accordingly, the special concern of a separate branch which we may speak of as Statistical Mechanics. This terminology is in part coincident with accepted usage, but in part must be understood to refer to an expansion of the subject beyond the bounds hitherto covered, whereby its scope shall be extended so as to include the statistical treatment of the dynamical problems presented by aggregates of living organisms; that is to say, aggregates of energy transformers possessing certain significant special properties.

Each of the branches of the mechanics of evolution enumerated so far naturally splits up into two subdivisions, according as we devote our attention to the material changes or to the energy changes involved. By an extension of prior usage in physical chemistry we may employ the term Stoichiometry to denote that branch of the science which concerns itself with \the material_transformations, with the relations between the masses of the components. The discussion of this branch presents itself as the more elementary task, and will therefore be taken up first; after this has been disposed of we shall be better prepared to discuss the second aspect, the Energetics or Dynamics of Evolution.

Taking now a survey of the stoichiometry of systems in evolutionary transformation, we can hardly find a better guide, for the organization of this subject, than the fundamental equations which we may speak of as the fundamental equations of the Kinetics of Evolution, since they furnish expressions for the velocities of transformation and exhibit the relations between these velocities and the masses of the several components, as well as the parameters Pow),

The very form of these equations suggests, as the first and most elementary problem, the treatment of the case of evolution under constant conditions, as defined by constant P’s and Q’s. This will, accordingly be the course here adopted, treating first the general case of n variables X1, Xo, . . . Xn, and then some special cases in which the number of variables is restricted to 1, 2 and 3. The perfectly general case, of evolution under conditions of wholly unrestrained variability of the P’s and Q’s, is mathematically uninviting (though not wholly intractable), and is also of minor interest in practice. Little will therefore be said of this. Certain special types of changes in the P’s and Q’s (e.g., slow changes) will find a place in the next subdivision of the general subject, the Statics of evolving systems. This branch is, in a sense, a special division of the Kinetics of Evolution, namely that which concerns itself with systems in which the velocities of transformation are zero, so that there is Equilibrium, or, to be more exact, a Steady State. This, of course, implies, strictly speaking, constancy of the parameters P, Q. But something very much like equilibrium presents itself in certain cases when these parameters change slowly. There may then arise what has been termed a Moving Equilibriwm. In view of the important réle which such moving equilibria play in nature, their discussion must form a part of the program of Physical Biology.

A second special problem of evolution under changing conditions (changing parameters P) that lends itself to treatment with comparative ease, is that which concerns itself with initial and end states (equilibria), as influenced by changing conditions, without demanding any information regarding the intermediate steps passed through by the evolving system. So, in physical chemistry, we may enquire what will be the effect, upon equilibrium, of a change in pressure or in temperature. Similar questions may be raised regarding

equilibria or steady states in life-bearing systems, and the matter calls for at least passing notice. This leads to the consideration of the Principle of Le Chatelier and, as a natural sequel of the train of thought thus started, to the examination of relations which may exist between certain of the parameters P, somewhat as, in physical chemistry, significant relations exist between pressure and volume, for example. It is found that some, at any rate, of the parameters P present analogy to the intensity and the capacity factors of an energy; out of this have arisen in the past sporadic efforts to establish systems of social dynamics and the like, which, however, have been based upon an acceptance as identity of what is only analogy. Those who have followed this road have been led, not so much perhaps to erroneous conclusions, as to blind alleys, to barren fields. True progress can be expected only by retracing one’s steps from such tentative excursions and striking out in a new direction; forsaking the way of quasi-dynamics, and breaking a trail toward a system of true dynamics, both of the individual (micro-dynamics) and of the system as a whole (macro-dynamics).

Of intra-species evolution, as expressed in changes in the parameters Q that define the character of the several species, little will be said here. The reasons and justification for this step-fatherly treatment of so important an aspect of our topic have been set forth in the preceding text. A discussion of at least one phase of intra-species evolution, falling under the head of dynamics, will, however, be presented when dealing with that phase of the subject; and we shall briefly note, in due course, some aspects of intraspecies evolution, as discussed more particularly by J. B. 8. Haldane.

These, in broad outline, are some of the principal land-marks in the territory ultimately to be covered by Physical Biology, and to be given a preliminary survey here. In concentrated form the lay of the land, as set forth above, is sketched in the diagram or scheme table 1, which should be found helpful both in presenting to the eye the salient features of the field of investigation, and also in furnishing a logical basis for the systematic arrangement of the subject in the ensuing chapters.

It remains, in this chapter, to enumerate the methods by which Physical Biology may be expected to develop. For the gathering of data two methods are available: observation in natural condi- (a]dioutg 8,1otjeyeyO eT) (seq8}8 Apveqg) untiqimby jo puauroe[dstq Biiqimby Zuracw Biiquinby tions, and observation under experimental (laboratory) conditions. Examples of both these methods will be noted. For the elaboration of data, the establishment of regularities (laws), there is available in this field, as everywhere in science, the method of induction, aided, if need be, by statistical technique. In this volume, however, emphasis will be laid upon deductive methods of mathematical analysis, as applied either to data furnished by observation, or to “unknown” quantities, blanks, as it were, in our formulae, ready for numerical substitution whenever concrete data become available.

The principal subdivisions of our topic, and the relations between them, as outlined above, are summarised graphically in table 1. L’emploi des signes mathématiques est chose naturelle toutes les fois qu’il s’agit de discuter des relations entre des grandeurs.—A. Cournot. General Case. We now proceed to the systematic study of the subject in accordance with the general plan laid down in the preceding survey of the Program of Physical Biology. According to this schedule (table 1)we approach first of all the section of Macromechanics (that is to say, the mechanics of evolving systems regarded as built up of component species in the gross), without carrying the analysis down to the finer details of individual organisms. And, of the field of Macromechanics, we shall here take up the section of Stoichiometry, that is to say, we shall for the present confine our attention to the relations between the masses involved, leaving aside for a later section the associated energy changes. Of the general field of Stoichiometry the first division to be taken up, according to our schedule, is the Kinetics of Evolution, and we shall here begin with a brief consideration of the Fundamental Equations in their general form

One phase of the general problem before us must evidently be the study (by observation, experiment, or any other method available) of the character or form of the functions F which tell us how the growth of each component is dependent upon the other components and the parameters P and Q. It might be supposed, indeed, that until this phase of the problem had received consideration, the system of equations (1), would be at best a barren expression of facts. But this is a misconception. Without knowing anything regarding the precise form of these functions, a good deal of information of considerable interest can be derived from these equations; and before proceeding to the consideration of concrete

cases, in which something is known regarding the particular form of the functions /, it is proper, at this point, to extract from the fundamental equations (1) all the information that we can. This requires a little mathematical manipulation of comparatively simple character. The variables X, the masses of the several components of the system, are not, in general, wholly independent. They are subject to certain constraining relations. For example, for any self-contained system, we shall have an equation of the form

expressing the constancy of the total mass of the system; and a similar equation holds separately for every chemical element.! These equations prohibit certain changes of mass in the system. They are analogous to equations introduced in mechanical problems by the geometric constraints that limit possible displacements (as in the case of a ball rolling down an inclined surface, and prevented, by the resistance of the surface, from falling vertically). They are commonly spoken of as equations of constraint. Equations of constraint such as

will in general enable us to eliminate certain of the variables X, expressing them in terms of the other X’s and of certain constants A. If, by the aid of m such equations of constraint, m variables have been eliminated, the system of equations (1) can be reduced to a simpler one, of identical form, but containing only (n — m) variables apd the same number of equations. In all that follows we shall assume that this has been done, that we now read equations (1) in this sense: the n variables X1 X» .

| X, are those left over after eliminating as many of the X’s as the equations of constraint permit. The functions F will in general, after this elimination, contain constants A introduced by the equations of constraint. We shall now, except where otherwise stated, restrict our considerations to the case in which the parameters P and Q and A are either constant, or change so slowly that we may disregard * Except in those rare cases in which radioactive or other atomic disintegrations occur.

their variation. This means that we restrict ourselves to the consideration of simple inter-group evolution under constant conditions. With this understanding, the first point we may observe about the system of equations (1) is that they define certain conditions of equilibrium, or, to be more precise, of steady states. For such a LX Nal gs, ; state ensues whenever all the velocities op vanish, that is to say, according to (1) whenever Fi=F.=...=Fa=0O (3)

We thus have n equations between the n variables Xj, Xo, X,, which, in general, determine certain values such that when masses of the several components have these values, they persist in these values; the system is at rest, as regards changes in the distribution of matter among its components S1, So, . . . Sy. In general there will be a number of such possible equilibria, some of which will be stable, some unstable. The determination of their number and character is a technical point in the theory of equations, for the general treatment of which the reader must be referred to the pertinent literature (see, for example, Picard, Traité d’Analyse, vol. 1, 1891, pp. 83, 123; vol. 2, 1893, pp. 183, 193, 196, footnote).