Elements of Physical Biology

Consider the case of a population containing two, one, or no ‘doses’ of a completely dominant Mendelian factor A. Mating is at random and selection acts in equal degree in both sexes upon the character produced by the factor. Pearson!’ and Hardy!® have shown that in a population mating at random, the square of the number of heterozygotes is four times the product of the numbers of the two homozygous classes. Let up A.: 1a be the proportion of the types of gametes produced in the (n—1)* generation. Then in the nth generation the initial proportion of the two classes of zygotes will be

Fig. 24. GrowrH oF Favorep Tyrp In Mixep PopuLatTIon oF Two PHENOTYPES Constant total population. Abscissae = number of generations; ordinates = percentage of favored type in total population. After J. B. S. Haldane. The proportion of recessives to the whole population is Yn = (1+ un)? (48) Now only (1—k) of the recessives survive to breed, so that the survivors are in the proportion The numbers of the next generation can most easily be calculated from the new gametic ratio un+1. This is immediately obvious in the case of aquatic organisms who shed their gametes into the water. If each zygote produces N gametes which conjugate, the numbers of gametes of type A and a respectively, are, clearly,

Now if we know the original proportion of recessives yo, we start with a population If we start with a population containing } recessives, the second generation will contain 4 the third 7, the nth Thus 999 generations will be (n + 1)* required to reduce the proportion to 1:1,000,000, and we need not wonder that recessive sports still occur in most of our domestic breed of animals. When selection is not very intense, we can proceed as follows: Fia. 25. Errect or SELECTION ON PopuLaTion Comprisina Two PHENOTYPES

Upper curve, dominants favored; lower curve, recessives favored. Abscissae = generations; ordinates = percentage of population with the favored character. After J. B. S. Haldane. If we start from a population containing 25 per cent recessives u. = 1, kn = Un + logeun — 1 (64) In figure 25 is shown the growth curve for the dominants when k = + 0.001 (upper curve) and for the recessives when k = —0.001 (lower curve), i.e., the favored type has an advantage of 1 in 1000, as in figure 24. In each case 16,582 generations are required to increase the proportion of the favored type

from 1 per cent to 99 per cent, but dominants increase more rapidly than recessives when they are few, and more slowly when they are numerous. The change occurs most rapidly when yn, the proportion of recessives to the total population, is 56.25 per cent. Haldane deals also with several other cases, for the details of which the reader must be referred to the original. A continuation of Haldane’s article is anticipated at the time of this writing.

We fat all creatures else to fat us, and we fat ourselves for maggots.— Shakespeare. Adjustment of Birth Rate to Optimum. In the preceding discussion of the equations (25) and (34) of chapter IX it has been supposed that the form not only of p(a), but also of (a), the rate of procreation at different ages, is fixed. It is only on this supposition that the two equations (25) and (34) of Chapter IX together determine both bandd. In point of fact the function 6(a) is undoubtedly, for most species of organsms, very elastic (much more so than p(a), the survival factor), and capable of adapting itself to varying circumstances. This is especially so in the case of man, who exhibits in particularly high degree the rather astonishing phenomenon of a portion of matter whose growth is at least partially under the control of a willin some manner associated withit. But, in the organic world at large also, there is presumably at least some tendency for the adjustment of the procreation factor so to take place as to make the rate of increase r a maximum under the existing conditions. Too high a procreation factor would lead to excessive sacrifices in progeny that could not be raised to maturity, and would increase the death rate more than the birth rate. On the other hand too small a procreation factor would obviously fail to give the maximum attainable rate of increase. Somewhere between the two extremes a certain optimum procreation factor will make r a maximum. From this point of view the two relations that effectively determine the actual values of the two variables b and d are

The view that the rate of procreation thus expands or contracts in sympathy with the expanding or contracting food supply (or economic conditions generally) has been developed in detail (for the human species) by R. Lascaux in his work La Production et la Population (1921). That some such adjustment occurs with many biological species is a very plausible, one might say an inevitable supposition; but the approach to the ideal optimum is probably often only very imperfect, if only because the nature of the case calls for a large ‘factor of safety.”

In any event it is true that the birth rate does not play so unqualifiedly a dominant réle in determining the rate of growth of a species as might appear on cursory reflection. The equation (5) of Chapter IX, expressing the rate of growth of the species in terms of the birth rate and death rate, while it renders correctly the quantitative relations to which it refers, is only a partial or one-sided representation of the facts, and is even open to misinterpretation. Incautiously construed it might be taken to imply that growth of an aggregate of living organisms takes place by births of new individuals into the aggregate. This, of course, is not the case. The new material enters the aggregate in another way, namely in the form of food consumed by the existing organisms. Births and the preliminaries of procreation do not in themselves add anything to the aggregate, but are merely of directing or catalyzing influences, initiating growth, and guiding material into so many avenues of entrance (mouths) of the aggregate, provided that the requisite food supplies are presented, provided that the system is, in a sense, “supersaturated” with regard to the species seeking to grow therein. The final result may not depend very greatly on the number of births, somewhat as the final state of a crystallizing solution is independent of the number of crystal germs initially sown therein.

It will be desirable to develop our analysis in such manner as to bring out the relations thus involved. Aggregates of Constant Units. In aggregates of a simpler kind, as presented to our view in ordinary physico-chemical systems, each individual unit retains its identical substance unchanged throughout its period of existence as such unit. So, for example, a molecule of water consists of two particular atoms of hydrogen united to one specific atom of oxygen; and these same atoms continue to exist together as the building stones of the molecule as long as this continues in existence as a water molecule.

In these circumstances the mass of each unit is obviously constant throughout its period of existence as such; and furthermore, addition to the component, or elimination therefrom, can take place only by the actual entry or departure of a complete unit. If, therefore, the mass of each unit is m;, and if B; new units are added to the aggregate per unit of time, while D; are eliminated, we have, in this case, very simply, Aggregates of Variable Units. For aggregates of living organisms we can also write an equation identical in form with (3), as has already been noted in Chapter IX; B; is in this case the total birth rate, D; the total death rate, and m; the average mass per head of the living population. But this is an inadequate representation of the significant facts. The equation, thus written, glosses over certain important characteristics of living organisms. Unlike molecules in a system in the course of chemical transformation, each unit in an aggregate of living organisms does not retain its substance unchanged in identity or in total mass. In fact, each unit is itself an aggregate within the larger aggregate that constitutes the species or biological group, and for each individual unit (organism) separately we can write an equation analogous to equation (1) of Chapter [IX

where U’; is the total mass taken up (ingested) per unit of time by the unit organism, and V;’ is the total mass eliminated therefrom per unit of time. So, for example, in the course of one year a boy ten years old and weighing 32.5 kgm. may consume about 600 kgm. of food (inclusive of water and oxygen), may eliminate about 599 kgm of wastes, and will grow in actual mass by about 1 kgm., so that we have arr enn et hao = a Oe (5) dt ‘eis The Stream of Substance Through the Form of the Organism. It will be observed that a portion of the intake, but a portion only, is

expended in adding to the total mass of the unit. The remainder Rj is expended without, apparently,! any resulting increase in the total mass of the unit. This constant expenditure of substance, and the equally constant intake required to balance it, is a fundamental characteristic of the units here under discussion. In the adult, whose mass is (on an average) approximately constant, we have simply dm and the entire intake goes to meet the requirements of maintaining the mass of the unit at constant level.?

Turning now from the consideration of the individual unit to that of the aggregate of N such units, evidently, if the average intake per unit of time per individual is U’;, and if the average elimination is V’;, then we shall have for the rate of increase of the total mass X; of the aggregate. where d; is the death rate per head per unit of time and m’; is the average mass of a unit (organism) at death. Two Types of Organisms: Economical and Lavish Birth Rate. The relative importance of the second and the third term in the right hand member of the equation (9) differs greatly in different biological species. At the one extreme we have a type of which perhaps the most characteristic representative is man. With a mean length of life of about fifty years, his body must be replaced about twice in a century to maintain the population equilibrium. If we assume (as a rough but sufficient approximation) that the average weight of man at death is 50 kgm., this means that the third term, Nm’jd;

1 Indirectly a part of the excess R; of the mass intake over the mass increase may contribute to that increase, namely by furnishing some of the energy required for anabolism. But we are here discussing mass relations only. The energy relations are reserved for separate consideration later. 2 Except during gestation, if the mass of the fetus is reckoned in with that of the mother. <= kgm. for the entire population, or just about 1 kgm., or say 2 pounds per head perannum. To put it crudely, of the food consumed by each human individual in a year, 2 pounds go, on an average, to replace the bodies of his fellows departed that year. This, it will be seen, is an insignificant, almost wholly negligible fraction of the 1000 pounds’ or more than he consumes, in all, ina year. Of the total food consumed by the human race, then, about 0.2 per cent? goes to replace the bodies eliminated by death. The remainder is for current maintenance of the living. And the total food consumption? may be of the order of 7 to 10 times the mass of the population per annum.

But, as already stated, man represents an extreme type, the extreme economy of life, with low death rate and correspondingly low birth rate. Of the opposite extreme, lavish, seemingly wasteful extravagance, examples are exceedingly common, though it may not be easy to give full quantitative detail. Among the most wasteful breeders are, no doubt many aquatic species, including fish, since their young are ill protected and become ready victims of other species. So a ling weighing 54 pounds was found to be carrying twenty-eight million eggs.» An oyster may have sixty million eggs. But some familiar land animals are prolific enough, even if they do fall far behind the standards just exemplified. The brown rat may have five or six litters averaging about eight or ten each, in a year.®

Domestic Animals Kept for Produce. Accurate figures can be obtained in case of domestic animals. While these do not represent so extreme an example, a special interest attaches to them owing to their direct relation to human food economics. The most prolific among domestic animals is the pig. In reasonably good farm conditions a sow should average three litters in two years, each of seven farrows, of which five are successfully raised and marketed.

Even with the high mortality artificially induced by man in his domestic stock the item of running expenditure in feed for mere maintenance is far in excess of the replacement cost, that is to say, the feed ‘For quantitative data on growth in man see C. 8. Minot, The Problem of Age, Growth and Death. stored up and finally utilized in the carcass of the slaughtered animal. From a detailed study of the vital economics of beef production made at the University of Missouri’ figure 26 is reproduced here to show these relations. The convex curve shows the average growth per head in a group of steers fed with a ration regulated to secure a maxi-

Fic. 26. Freep ConsuMED, AND INcREASE IN Live WEIGHT OF Srmprs AT SeveRAL AGES Dry matter consumed is represented on one-tenth the scale of the live weight. After Moulton, Trowbridge and Haigh. mum of growth, without storage of surplus fat; the approximately straight line mounting upward shows the steadily increasing integrated amount of feed consumed since birth. Only the dry weight of the feed is plotted, and the scale employed is ten times more

7 University of Missouri, College of Agriculture, Bulletins 43, 54, 55 (Ric, Trowbridge and L. D. Haigh). condensed than that used for the live weight—else the second curve would rise too steeply as to lie for the most part far beyond the limits of the page. Thus is shown the great disproportion between the feed Average Yearly Gains of Steers* Animals of group II were fed to secure maximum growth without storage of surplus fat. Animals of group III were somewhat underfed, to represent animals not properly cared for.

that is recovered in the body of the growing steer, and the far greater amount that is wasted, so far as the interest of the producer is concerned, in mere maintenance, for the private satisfaction and benefit of the animal, so to speak. The numerical data on which figure 26 is based are exhibited in table 9, together with the corresponding figures observed when the animals are somewhat underfed and overfed respectively. The instances that have been cited—man on the one hand, and the highly prolific species, both feral and captive, on the other—are eloquent illustrations of the elasticity of adaptation. Clearly, a species may hold its own, in the struggle for existence, either by the aid of well-developed protective devices resulting in a low death rate, and requiring only a correspondingly low birth rate; or, a less well protected species may balance a high death rate by an equally high birth rate. Which of these two methods would be chosen in the natural course of.events is a question that it might be difficult to answer on any general a priorz principle, so long as attention remained fixed on a single species. Perhaps one would have expected evolution to turn in a favor of the more economical method of meeting a low death rate with a low birth rate. In point of fact both types of organism—the economical type (as judged by its own standard) with low death rate, and the wasteful with high death rate—exist side by side in abundance. Thisisa good example to illustrate the purely relative character of fitness, and to remind us once more that we cannot expect any success in attempts to define the direction of evolution in terms of a single species. It is not the individual species, the individual components of the system, that evolve, but the system as a whole, comprising all the species and their environment. The species of the economical type, with low death rate, are largely dependent for their subsistence on the presence of species of the opposite type; we must think here of a competition, not between individual species, but between groups of species, groups consisting, in the simplest case, of two species each, a food species or prey, and a feeding or predatory species.

Of two such groups, that one will, other things equal, have the advantage in the struggle, in which high productivity of the food species is accompanied by economy of life on the part of the feeding species. From the point of view of the hog, so to speak, the high mortality in the pen is a disastrous inefficiency and maladaptation, a misfortune to be borne, as best it may, with porcine philosophy. From the point of view of the consumer on the other hand, this high mortality is, quite on the contrary, a measure of the efficiency, the eminent fitness of swine as producers of pork; and his only regret

is that so much of the feed placed in the trough goes merely to carry on ‘what may be called the personal activities of the animals themselves.’’® It isto be noted, however, that only a part of the material accountable as waste from the standpoint of the food species is gain for the feeding species. Deaths from disease are a pure loss to both species. Similar reflections, of course, apply, mutatis mutandis, to those cases in which the feeding species derives its nourishment from some current product of the life activity of the food species or host, instead of from its carcass. The most notable example of this in the food economy of man is his exploitation of the milch cow, who is a far more efficient producer than the beef steer. The latter at best consumes over 6 pounds of nutriment for every pound of product.® According to the investigations of the National Research Council about 18 per cent of the energy of grain fed to cattle is recovered for human consumption in milk, but only about 3.5 per cent in beef. Similarly, crops on a given area will yield about four to five times as much protein and energy when fed to dairy cows as when used for beef production. In providing mineral substances and vitamines the milk of cows contrasts even more favorably with the beef animal. The vitamines and calcium salts contained in hay and grain are storedin the muscular tissue only to a slight extent, but are in relative abundanceinmilk.!° From the standpoint of the dairymana thoroughbred prize cow, such as Glista Ernestine (a Holstein), which gave in one year 833 pounds of butter fat, and in one hundred days 10,000 pounds of milk, is a very model of efficiency, producing more than her own weight in milk each month. But from the point of view of the bovine species such record performances are gross inefficiency, approaching in some cases perilously near to total biological unfitness, for some of the record Jersey cows are probably unable, under the conditions of the stable at any rate, to raise their own calves—the over-rich milk would probably kill the young animal.

Network of Chains of Interrelated Species. The relation between man and the domesticated species of animals and plants on which he aa have here borrowed a felicitous phrase from an anonymous writer on the editorial page of the New York Times, February 10, 1921. so largely depends for food, in the present state of civilization, is only a particularly tangible, a particularly accessible example of an intricate network of relationships that connect more or less closely all living species. In this network each species or component is interlaced, like a link in a meshed coat of mail, with other species, which in turn connect with still others, and so forth. In our effort to get some sort of mental grasp of the complicated interlocking of these elements we seize upon some one link, some one species or component, and we note, first of all, that whatever is eliminated from one component of a self-contained system must pass into one or more other components of the system. So, for example, the component Sj may be a herd of cattle. The matter eliminated from this component goes in part as food to build up or sustain a human population; in part it goes as fertilizer on the fields to furnish nutriment for crops; still other parts are worked up into various industrial products, such as leather, glue, etc. We thus have, in schematic representation,

vig y Si \ o ~ Ks * L N x N\ Human Leather Sj | Sy population Fertilizer Sk On the other hand, the substance of the herd itself is recruited from certain other components of the system, grass, clover, corn, etc., so that we may further develop the scheme Clover Se Grass Corn Sy | Sh \ f pt ler ioe’ Si eA | at | i Human | Leather Sj S} population Sk Transformation Factors and Their Economic Significance. In general any one component thus appears as a link in a complicated chain or rather network of chains; the component S;, for example,

receives a certain fraction a;,; of the mass V;X; eliminated per unit of time from the component S;; it passes on to the component S, a certain fraction Bi, of the mass V;X; eliminated from X; itself. The rate of growth ons of X; is the balance of the sum JU; of all contributions it receives over and above the sum JV; of all the contributions which it makes to other components, thus dX; the first summation being extended over all those components S; which contribute to X; and the second over all those components S; to which S; contributes.

But we may also analyse the contributions to and from the component S; in another way. We may say that, of the total contributions per unit of time w;X; to the mass Xj, a certain fraction y;;wiX; is derived from S;. Then Lastly, if the system is not self-contained, we must add a term J i for “imports” per unit of time, and a term —Z; for “exports”’ per unit of time, that is to say, aa reek SS yXi+ i, — Ej (15) When S; is the human species, the coefficients a, 8, y have an obvious economic significance. The restriction of this remark to the human species must not be taken to imply that there is in this feature something wholly peculiar to man, but rather, that underlying our economic manifestations are biological phenomena which we share in common with other species; and that the laying bare and clearly formulating

of the relations thus involved—in other words, the analysis of the biophysical foundations of economics—is one of the problems coming within the program of physical biology. Hints as to the direction in which we may or must look for light on this phase of our problem have now been noted upon several occasions. So it was observed that the components of a life-bearing system can be divided into two classes, relative to the component S;, namely, on the one hand those

; 0 dX; eed. components S; for which sx. ae Ws positive, or, aS we may say, d those useful to the species S;, those having for it a positive value, and, on the other hand, components S; for which a o "<0, components harmful to S;, or having for it a negative value. Elsewhere we have noted the classification of components into replaceable and indispensable components, a classification that at once recalls elementary economic reflections. These hints we note in passing. They may serve to put ourminds in a state of preparedness for the more formal and decisive attack of the problem, to which we shall be led in the last division of our enquiry, dealing with the dynamics of life-bearing systems.

iia oe a ag y a v od. - ! ve. 7 ' y aoe 7 rad LY _ or - a 7 “ a a _ id ; ae Repeatedly, in preceding chapters, occasion has arisen to refer to stationary states or equilibria. Inevitably, in the discussion of the kinetics of evolution, one is led to consider incidentally certain conditions and special cases in which the velocities of the changes in the evolving system are zero; when, that is to say, the system under discussion is in a steady or stationary state, in equilibrium. Viewed from this avenue of approach equilibrium presents itself as a special case of motion or change, namely motion or change with zero velocity. Indeed, something very like equilibrium occurs also with velocities that are merely small, not vanishingly small. In such case the phenomenon of moving equilibrium may present itself, as we shall have occasion to observe in greater detail in due course.

Stationary states—equilibria and near-equilibria—play an important rdle in nature, and it is desirable at this point to give them something more than incidental consideration; to sketch, at least in outline, their systematic study; to stake out, in the rough, that field which, in our survey of the Program of Physical Biology (Chapter V) was designated as the Statics of Evolution, and was systematized according to the schedule Statics <p =e oa Equilibria Moving equilibria Displacement of (steady states) equilibrium