The last model I wish to discuss is the model of growth, honorifically called the Bertalanffy equations (von Bertalanffy, 1957b, 1960b); basic ideas go back to the great German physiologist Putter (1920). Here, too, I am not primarily concerned with details or even the merits and shortcomings of the model; I rather wish to use it to make clear some principles in quantitative metabolism research.
We all know, firstly, that the process of growth is of utmost complexity; and secondly, that there is a large number of formulas on the market which claim satisfactorily to represent observed growth data and curves. The general procedure was that some more or less complex and more or less plausible equation was proposed. Then the investigator sat down to calculate a number of growth curves with that formula, and was satisfied if a sufficient approximation of empirical data was obtained.
Here is a first illusion we have to destroy. It is a mathematical rule of thumb that almost every curve can be approximated if three or more free parameters are permitted—i.e., if an equation contains three or more so-called constants that cannot be verified otherwise. This is true quite irrespective of the particular form of the equation chosen; the simplest equation to be applied is a power series (y = a0 + a1x + a2x2 + . ..) developed to, say, the cubic term. Such calculation is a mere mathematical exercise. Closer approximation can always be obtained by permitting further terms.
The consequence is that curve-fitting may be an indoor sport and useful for purposes of interpolation and extrapolation. However, approximation of empirical data is not a verification of particular mathematical expressions used. We can speak of verification and of equations representing a theory only if (1) the parameters occurring can be confirmed by independent experiment; and if (2) predictions of yet unobserved facts can be derived from the theory. It is in this sense that I am going to discuss the so-called Bertalanffy growth equations because, to the best of my knowledge, they are the only ones in the field which try to meet the specifications just mentioned.
The argument is very simple. If an organism is an open system, its increase or growth rate (G.R.) may, quite generally, be expressed by a balance equation of the form:
i.e., growth in weight is represented by the difference between processes of synthesis and degeneration of its building materials, plus any number of indeterminate factors that may influence the process. Without loss of generality, we may further assume that the terms are some undefined functions of the variables concerned:
Now we see immediately that time t should not enter into the equation. For at least some growth processes are equifinal, i.e., the same final values can be reached at different times (FIG. 6.1). Even without strict mathematical proof, we can see intuitively that this would not be possible if growth rate directly depends on time; for if this were the case, different growth rates could not occur at given times as is sometimes the case.
Consequently, the terms envisaged will be functions of body mass present;
if we tentatively limit the consideration to the simplest open- system scheme. The simplest assumption we can make is that the terms are power functions of body mass. And, indeed, we know empirically that quite generally the size dependence of physiological processes can well be approximated by allometric expressions. Then we have:
where η and K are constants of anabolism and catabolism, re- spectively, corresponding to the general structure of allometric equations.
Mathematical considerations show furthermore that smaller deviations of the exponent m from unity do not much influence the shape of the curves obtained. Thus, for further simplification let us put m — 1. This makes things much easier mathematically, and appears to be justified physiologically, since physiological experience—limited it is true—seems to show that catabolism of building materials, especially proteins, is roughly proportional to body mass present.
Now let us make a big leap. Synthesis of building materials needs energy which, in aerobic animals, is provided by processes of cell respiration and ultimately the ATP system. Let us assume there are correlations between energy metabolism of an animal and its anabolic processes. This is plausible insofar as energy metabolism must, in one way or the other, provide the energies that are required for synthesis of body components. We therefore insert for size dependence of anabolism that of metabolic rates (n = a) and arrive at the simple equation:
The solution of this equation is:
with w0 = weight at time t = 0.
Empirically, we find that resting metabolism of many animals is surface-dependent, i.e., that they follow Rubner’s rule. In this case, we set a = 2/3. There are other animals where it is directly dependent on body mass, and then a = 1. Finally, cases are found where metabolic rate is in between surface and mass proportionality, that is 2/3 < a <1. Let us tentatively refer to these differences in size- dependence of metabolic rate as “metabolic types.”
Now if we insert the different values for a into our basic equation, we easily see that they yield very different curves of growth. Let us refer to them as “growth types.” These are summarized in Table 7.4; corresponding graphs, showing the differences in metabolic behavior and concomitant differences of growth curves, are presented in Figure 7.8. Detailed discussions of the theory have been given elsewhere. It has been shown that the above derivations apply in many cases; no less than fourteen different arguments in verification of the theory can be presented (Table 7.5, FIGS. 7.9, 7.10). We shall limit the present discussion to a few remarks on principle.
All parameters of the growth equations are verifiable experimentally. a, the size dependence of metabolic rate, determines the shape of the growth curve. This correlation has been confirmed in a wide range of cases, as seen in Table 7.4 , constant of catabolism, can in first approximation be identified with turnover of total protein (r) as determined by isotope tracers and other techniques. For example, from the growth curves catabolic rates of 0.045/day for the rat, and 1.165 g. protein/kg. body wt./day for man were calculated (von Bertalanffy, 1938). Determinations of protein catabolism then available did not agree with these predictions: protein loss determined by minimum N-excretion was 0.00282/day for the rat after Terroine, and some 0.4-0.6 g. protein/kg. body wt./day for man, according to the conceptions then prevailing in physiology (von Bertalanffy, 1942, pp. 180ff., 186-188). It was therefore a striking confirmation of the theory when later on determinations using the isotope method (Sprinson and Rittenberg, 1949, Table 6.2) yielded turnover rates of total protein (r) of 0.04/day for the rat, and of 1.3 g. protein/kg. body wt./day for man in an amazing agreement between predicted and experimental values. It may be noted in passing that an estimate of the turnover time of the human organism similar to that found in isotope experiments (r ≈ 0.009, t ≈ 110 days) can be obtained in different ways, e.g., also from calorie loss in starvation (t = 100 days: Dost, 1962a). η, constant of anabolism, is dimensionally complex. It can, however, be checked by comparison of growth curves of related organisms: according to theory, the ratio of metabolic rates should correspond to the ratio of η’s of the animals concerned. This also has been confirmed (FIG. 7.10).
The theory, therefore, fulfills the first postulate indicated above, i.e., verification of calculated parameters in independent experiments. As has been shown elsewhere, it also fulfills the second postulate: Predictions from the theory were made which came as “surprises,” i.e., were unknown at the time, but later on confirmed.
Discussion of some typical objections is in place because it may contribute to better understanding of mathematical models in general.
- The main reproach against models and laws for physiological phenomena is that of “oversimplification.” In a process such as animal growth there is, at the level of cells, a microcosm of in- numerable processes of chemical and physical nature: all the reactions in intermediary metabolism as well as factors like cell permeability, diffusion, active transport and innumerable others.
On the level of organs, each tissue behaves differently with respect to cell renewal and growth; besides multiplication of cells, formation of intercellular substances is included. The organism as a whole changes in composition, with alterations of the content in protein, deposition of fat or simple intake of water; the specific weight of organs changes, not to speak of morphogenesis and differentiation which presently elude mathematical formulation. Is not any simple model and formula a sort of rape of nature, pressing reality into a Procrustean bed and recklessly cutting off what doesn’t fit into the mold? The answer is that science in general consists to a large extent of oversimplifications in the models it uses. These are an aspect of the idealization taking place in every law and model of science. Already Galileo’s student, Torricelli, bluntly stated that if balls of stone, of metal, etc., do not follow the law, it is just too bad for them. Bohr’s model of the atom was one of the most arbitrary simplifications ever conceived—but nevertheless became a cornerstone of modern physics. Oversimplifications progressively corrected in subsequent development are the most potent or indeed the only means toward conceptual mastery of nature. In our particular case it is not quite correct to speak of oversimplification. What is involved are rather balance equations over many complex and partly unknown processes. The legitimacy of such balance expressions is established by routine practice. For example, if we speak of B.R.M. -and are, in fact, able to establish quantitative relationships such as the “surface law’’—it is balances we express which revertheless are important both theoretically and practically (e.g., iiagnostic use of B.M.R.). The regularities so observed cannot re refuted by “general considerations” of oversimplification, but rnly empirically and by offering better explanations. It would re easy to make the growth model seemingly more realistic and o improve fitting of data, by introducing a few more parameters.
However, the gain is spurious as long as these parameters cannot be checked experimentally; and for the reasons mentioned, a closer fit of data tells nothing about the merits of a particular formula if the number of “free constants” is increased.
- Another question is the choice of parameters. It has been noted above that metabolic rate under basal and nonbasal conditions changes not only in magnitude but also with respect to allometry expressing its relation to body size. What is the justification of taking “resting metabolism” as standard and to range various species into “metabolic” and “growth types” accordingly? The answer is that among available measures of metabolism— none of them ideal— resting metabolism appears to approach best those natural conditions which prevail during The B.M.R. standard (i.e., thermoneutrality of environment, fasting and muscular rest) makes the values so determined a laboratory artifact, because at least the first condition is unnatural; although it is most useful because B.M.R. values show the least dispersion. In cold-blooded animals, B.M.R. cannot be used as standard because there is no condition of thermoneutrality, and the fasting condition often cannot be exactly established. Activity metabolism, on the other hand, changes with the amount of muscular action (FIG. 7.4), and the growing animal is not under conditions of hard muscular work all the time. Hence resting metabolic rate is comparatively the best approximation to the natural state; and choice of this parameter leads to a useful theory.
- The most important criticism becomes apparent from the above It was said that there appear to be so-called metabolic types and growth types and correlations between both. However, earlier it has been emphasized that the parameters implied, especially the relation of metabolic rate to body size expressed in the exponent a, can be altered and shifted with experimental conditions (FIGS. 7.4-7.7). Similarly, also growth curves are not fixed. Experiments on the rat have shown that the shape of the growth curve, including location and existence of a point of inflection, can be changed by different nutrition (L. Zucker et al., 1941a, 1941b, 1942; T.F. Zucker et al., 1941; Dunn et al., 1947; Mayer, 1948). None of the characteristics is rigid— and, incidentally, within my own biological concepts, I would be the last to presuppose rigidity in the dynamic order of physiological processes. According to my whole biological outlook, I am rather committed to the ancient Heraclitean concept that what is permanent is only the law and order of change.
However, the apparent contradiction can well be resolved when we remain faithful to the spirit of the theory. What is really invariable is the organization of processes expressed by certain relationships. This is what the theory states and experiments show, namely, that there are functional relationships between certain metabolic and growth parameters. This does not imply that the parameters themselves are unchangeable—and the experiments show that they are not. Hence, without loss of generality, we may understand “metabolic” and “growth types” as ideal cases observable under certain conditions, rather than as rigid species characteristics. “Metabolic” and “growth types” appear in the respective groups of animals if certain standard conditions are met. However, it is clearly incorrect that “the reduction of metabolic rates is a fundamental magnitude, not changing in different external conditions” (Lehmann, 1956). Under natural or experimental conditions, the relationships can be shifted, and then a corresponding alteration of growth curves should take place. There are indications that this is actually the case; it is a clear-cut problem for further investigation.
A case to the point are seasonal changes. Berg (1959, 1961), while in general confirming previous data, found that the size- metabolism relation varies seasonally in snails:………………… “Thus the relation, oxygen consumption to body size, is not a fixed, un- changeable quantity characteristic of all species as supposed by Bertalanffy If (Bertalanffy’s theory) were true, then the observed seasonal variation in metabolic type would imply a seasonal variation in the type of growth rate.”
As a matter of fact, precisely this has been found in our laboratory long ago (von Bertalanffy and Miiller, 1943). Seasonal variations of metabolic rate in snails have been described (FIG. 7.11a). But correspondingly, also the growth curve (exponential in this case because these snails belong to “Type II”) shows breaks and cycles (FIG. 7.115). Therefore, this certainly is a problem deserving more detailed investigation; however, the data available are a hint toward confirmation rather than refutation of the theory.
I would have been much surprised, indeed suspicious, if this first crude model would have provided a conclusive theory. Such things just do not happen, as is witnessed by many examples from history of science. Mendel’s laws were the beginnings of genetics but—with linkage, crossing-over, position effect and what not—it is only a minute part of genetic experience that is described by the classical laws. Galileo’s law is the beginning of physics, but only highly idealized cases—such as bodies falling in vacuo—actually follow the simple law. It is a long way from Bohr’s simple model of the hydrogen atom to present atomic physics, and so on. It would be fantastically improbable if this were different with a proposed model of growth. The most we can say about it is that it is backed by a considerable amount of experimental evidence, has proved to have explanatory and predictive capacities, and offers clear-cut problems for further research.
It is obvious that the theory has been developed for a limited number of cases only, owing to the limited number of good data and the time-consuming nature both of observation and calculation of growth. Hemmingsen (1960) has made this clear: “With n varying as much as the examples show, within any group with allegedly (or at least first allegedly) uniform growth type, it seems impossible to accept Bertalanffy’s generalizations unless a statistically significant correlation between n and growth type can be demonstrated on a much larger number of examples than the few ones which Bertalanffy has repeatedly published.” I entirely agree with this criticism; many more data would be desirable, although one should not cavalierly bypass those offered in confirmation of the theory, even if they are some 20 years old. I would amend Hemmingsen’s criticism by suggesting reexamination on a broader basis. This should include at least the following items: analysis of a large number of growth data, now made possible by electronic computers; concurrent determination of size-dependence of resting metabolism (constant a) in these cases; determinations of protein catabolism (constant K ); determination, in related species, of the ratios between allometry exponents of metabolic rates and the theoretically identical ratios of the anabolic constants ( η ). These are all interesting and somewhat neglected research problems; and if the model does no more than bring them to the fore, it has proved its usefulness.
Such investigation may bring additional confirmation of the model; it may lead to its modification and elaboration by taking into account additional factors; or it may lead to abandoning the model altogether and replacing it with a better one. If the latter should happen, I would be in no way disappointed. This is exactly what models are for—to serve as working hypotheses for further research.
What I have tried to show in the models discussed are general ways of analysis of quantitative data. I wanted to make clear both the usefulness and the limitations of such models. Any model should be investigated according to its merit with a view at the explanations and predictions it is able to provide. General criticism does not help, and the decision whether or not a model is suitable, exclusively rests with facts of observation and experiment. On the other hand, no model should be taken as conclusive; at best it is an approximation to be progressively worked out and corrected. In close interaction between experiment and conceptualization, but not in confinement to experimentation or construction of purely speculative models, lies the further development of a field like quantitative biology of metabolism.
Source: Bertalanffy Ludwig Von (1969), General System Theory: Foundations, Development, Applications, George Braziller Inc.; Revised edition.