The Determination of Size 



627 



tween the power equation and the assump- 

 tion of the sigmoid nature of the growth 

 curves of both x and y, in connection with 

 the autocatalytic and the Gompertz function. 

 He concludes that when the growth of the 

 parts is determinate, the concept of a con- 

 stant growth-partition coefficient is not valid 

 over the entire growth period. Richards and 

 Kavanagh ('45), on the other hand, believe 

 they may both be sigmoid without invalida- 

 ting the formula. 



There is general agreement that the for- 

 mula is purely empirical, and various efforts 

 have been made to find some significance for 

 it in simple postulates about growth or in 

 the time laws of growth, but without success. 

 It does make possible a quantitative study 

 of growth gradients by comparing the suc- 

 cessive growth ratios of a series of organs, 

 but does not lead to any conclusion concern- 

 ing the physiological basis of the gradients 

 themselves. 



Inasmuch as the primary purpose of the 

 mathematical analysis of biological data is 

 to suggest or develop the significance of the 

 data, the persisting enigma of the relative 

 growth formula indicates possibly an over- 

 expenditure of unrewarded effort, and at- 

 tempts to understand growth phenomena in 

 other ways may be more successfid. Differ- 

 ent approaches, more closely bound to the 

 nature of the events, are employed by Meda- 

 war ('45), who develops the method of 

 transformations of D'Arcy Thompson by 

 introducing the concepts of time and gra- 

 dients, utilizing tissue cultures as experi- 

 mental material, and by Weisz ('46, '47), 

 who has analyzed the growing and changing 

 form of the brine shrimp Artemia in a 

 mathematical manner remote from the stand- 

 ard allometry technique. More recently 

 Bertalanffy ('51) has developed a concept 

 of growth as a counteraction of anabolism 

 and catabolism of building materials, accord- 

 ing to the following basic expression: 



dr , 



The change of body weight r is given by the 

 difference between the processes of building 

 up and breaking down: -q and k are constants 

 of anabolism and catabolism, respectively, 

 while the exponents m and n indicate that 

 the latter are proportional to some powers 

 of body weight y. 



REGENERATION LIMITS 



If an organ or part capable of regeneration 

 is removed, the regenerating organ grows 



more rapidly than normal, eventually ap- 

 proaching normal size. The allometry equa- 

 tion therefore does not express the growth 

 rate of an organ but only the limit of its 

 relative size. The regenerating part itself 

 grows in the same manner as a whole or- 

 ganism, exhibiting the typical growth curve. 

 In most cases the part regenerated attains 

 its normal size relative to the size of the 

 whole, after which further growth is in 

 unison with the whole. 



In some cases a blastema is formed that 

 appears to develop as a unit without evidence 

 of growth gradients, in flatworms and in 

 ascidian buds such as those of Botryllus, etc. 

 (Berrill, '41), and determination of final 

 size is the same as before, a product of initial 

 size, maximum growth rate, and the growth 

 decrement. In others, the growth of the 

 blastema is polarized, at once more complex 

 and yet possibly more susceptible to inves- 

 tigation. In limb regeneration in Amphibia, 

 Litwiller ('39) found a peak of mitotic ac- 

 tivity in mesenchyme and epithelium near 

 the base of the young regenerate, but the 

 peak shifts distally in older regenerates as 

 basal regions progressively differentiate. 



Similar but more striking phenomena are 

 exhibited by annelids. In the case of both 

 anterior and posterior regeneration a zone 

 of growth maintains a maximum growth 

 rate at its posterior border, while tissvie pro- 

 duced by it progressively differentiates an- 

 teriorly. Thus in anterior regeneration the 

 oldest tissue is the most anterior and the 

 youngest is in contact with the original an- 

 terior cut surface, while in posterior regen- 

 eration the oldest tissue is also anterior but 

 is accordingly in contact with the original 

 posterior cut svirface. In these cases the quan- 

 tity of tissue is most readily estimated by 

 number of segments. In many forms the 

 number of segments replaced is approxi- 

 mately, even exactly, the number removed — 

 both anteriorly and posteriorly in syllidean 

 polychaetes (Allen, '21) and posteriorly in 

 earthworms (Moment, '46) — while in others 

 it may be less but is never significantly 

 more. Moment states that the number of 

 segments regenerated posteriorly is a linear 

 function of the distance (as measured by 

 segments and not millimeters) of the cut 

 from the anterior end of the worm. For- 

 mation of new segments stops when approxi- 

 mately the species number of segments has 

 been formed, regardless of the size of the 

 segments. He suggests that both in normal 

 growth and in regeneration new cells are 

 added in series tmtil a critical inhibitory 



