V GROWTH IN TIME OF THE TOTAL ORGANISM 179 



down; the organism reaches a steady state if and when both processes are equal. 

 The overall result of both kinds of processes may, according to physiological 

 evidence, be expected to be expressed by some power function of the size of the 

 organism. This is expressed in the basic equation: 



dwidt ^ r^w"' — yew" (5- 1 6) 



In words : The change of body weight w is given by the difference between the 

 processes of building up and breaking down; t) and x are constants of anabolism 

 and catabolism respectively, and the exponents m and n indicate that the latter 

 are proportional to some power of the body weight w. 



The degradative processes, expressed by the constant x, are represented by the 

 "wearing out quota" (Rubner, 1 902) , that is, the continuous loss of building material 

 as it takes place in every living organism. Cytologically, this means the continuous 

 loss of cells and cell parts {e.g. replacement of epidermis, squamous tissue, etc., 

 gland secretion, perishing of cells in inner organs, of blood cells), and the corre- 

 sponding cell renewal found in many tissues and organs, often at an unexpectedly 

 high rate (p. 164^; Leblond and Walker, 1956). Biochemically, it means the 

 continuous degradation of building materials, measured by isotope techniques, 

 nitrogen excretion, or protein loss during starvation. In the adult steady state 

 this degradation is compensated by regeneration so that the catabolic rate equals 

 turnover rate. In a first approximation, the catabolic constant (x) of the growth 

 equation (or Rubner's wearing-out quota) can be equated to the turnover rate 

 of total protein (r, p. i9gf.). Naturally, however, turnover is not limited to proteins. 

 The constant of the equation rather means the resultant of all growth-limiting 

 factors including, beside manifest protein loss, factors such as progressive differen- 

 tiation (products of which more or less withdraw from turnover), changes in 

 water content, factors of ageing, etc. However, in a number of cases (Table 1 1 , p. 200) 

 protein loss appears to be the limiting factor determining the course and eventual 

 stopping of growth. 



In a physiologically plausible approximation (Bertalanffy, 1951a) and with par- 

 ticular consideration of the fact that the loss of weight in starvation is proportional 

 to body weight : 



dw ^, , s 



— -^^cw;w^WQt-" (5.17) 



dt 



and N or protein content remains nearly constant in starving animals, the rate 

 of catabolism can be assumed to be directly proportional to body weight. On the 

 other hand, mathematical analysis of equation (5.16; cf. p. i99ff.) shows that it is 

 rather insensitive to smaller deviations of the exponent n from unity. Hence, n can 

 be equated to i without considerable loss of generaHty, thus reducing the number 

 of parameters and facilitating mathematical treatment: 



dwjdt = ■fiw'" — yiw (5-18) 



The solution of this general growth equation is : 



Literature p. 253 



