202 GROWTH PRINCIPLES AND THEORY 2 



Experience shows that in many cases (Growth Type I), there is a characteristic 

 difference between time growth in linear dimensions and time growth in weight: 

 the first shows monotonic leveHng, the second a sigmoid curve. The conditions 

 necessary for this characteristic, that is, the disappearance of inflexion in the curve 

 of Unear growth, can easily be ascertained. 



In the case of proportional growth, 



^' = qi' (5-5) 



Therefore we obtain (omitting the proportionality constant) : 



dt dt dt 



ril^'" — xl^" 



_ = ^ , I (3m — 2) ^ ^ ^ (3n — 2) 



d^ 3 3 



In order that this curve has no inflexion, it is necessary that / vanishes in one 

 of the two terms in the second derivative. This is possible only if 



(3m — 2) = o 



m = 2/3 (5.41) 



This is the necessary and sufficient condition for the difference between the 

 curves of length growth and weight growth as observed. 



It is not possible to set n =2/3 so that the second term disappears in the 

 second derivative. For, since m < n, we would have, in this case, m < 2/3. But 

 then the exponent (3m — 2) in the first term would be negative, that is, the rate 

 of anabolism would be inversely proportional to length and hence to weight, 

 which is absurd. 



\^ 7n = 2/3, it further follows that the inflection of the weight curve is between 

 o and ije, and for n = i, at 8/27. That is, the sigmoid curve of weight growth 

 is asymmetrical, with an inflexion at about 1/3 of final weight. This is the shape 

 of the curve of weight growth found in the organisms concerned. 



While in our previous considerations we have inferred the several growth 

 types from physiological data on the size dependence of metabolism, we now see 

 that, given the differences of curves, these physiological relations follow for 

 mathematical reasons. If we find the characteristic difference mentioned between 

 weight growth and length growth, then the anabolic processes must be surface- 

 proportional. This deduction is verified by physiological evidence showing that 

 the surface rule of metabolism applies precisely in those organisms where the 

 curve of length growth shows no inflexion, and the length and weight curves are 

 characteristically different. 



3. For mathematical convenience, we have used equation (5.16) in the simplified 

 form of equation (5.18), assuming catabolism to be proportional to weight. This 

 is an approximation which is not necessarily correct. There are few data on 

 size dependence of catabolism and in particular of protein turnover. It has to be 

 borne in mind that, apart from the objections against using N-excretion as a 

 measure of growth-inhibiting factors (p. 207), the methods for determining 



