IV qUANTITATIVE ASPECTS 603 



A number of factors may tend to make the pre-inflectional part of the curve more rounded, 

 for mstance if dedifferentiation continues after growth has begun, if the onset is not syn- 

 chronous (Paulain, 1938, p. 206) throughout the regenerate [cf. Medawar, 1950), if there is 

 an autocatalytic component of growth (Robertson, 1923) or any other component which 

 causes a geometric mode of increase. The growth of a population of microorganisms (Hins- 

 helwood, 1946; Knaysi, 195 1), and of an embryo, no doubt meets this last requirement, 

 but regenerates probably often approximate to linear extrusions from the parent body — or 

 to intrusions into it, in the case of whole-body regenerates of lower animals. In a regener- 

 ating nerve-fibre this is particularly true of the visible manifestation of growth ; whatever 

 the nature of the actual process of synthesis in the cyton, material flows into the growing 

 axon or dendron at a linear rate (Young, 1942, 1945). Here, moreover, (Seddon e< a/.. I.e.) 

 the R-phase must be completely finished in the long period before growth begins. In limb- 

 regenerates, also, the gross increase may be virtually linear, because activity sweeps as a 

 linear wave along the structure (Table 4), even if the activity of individual cells is geometric. 

 It is quite probable (p. 641) that the linearly regenerating nerve determines the linear 

 pattern of growth of the whole regenerate. The post-inflectional part of the curve is more 

 rounded in this type of regenerate because some cell-division persists in the wake of the 

 wave, and even more because growth by cell-hypertrophy which follows is not so sharply 

 "quantised". Moreover, for each small cell proliferated in the pre-inflectional, a large cell 

 ceases growth in the post-inflectional phase, so that the scale of the curve becomes magnified. 



The curves relating to time, the total amount of various substances in the regenerating 

 mouse-liver also are typically sigmoid (Tsuboi et al., 1954). The liver regenerates throughout 

 its remainder and therefore probably geometrically, giving a rounded pre-inflectional 

 portion to the curve. 



Robertson (1923) pointed out the similarity between the sigmoid curve of growth and 

 that for a monomolecular autocatalytic reaction, when the amount of the reagent is limited, 

 and found that actual measurements of growth fitted very tolerably the algebraical relation 

 for this chemical reaction. The resemblance might be explained if the curve of growth were 

 determined by a particular "master reaction". It is doubtful if regeneration, or most other 

 types of growth, are limited by exhaustion of materials, but a progressive retardation, due to 

 a "deliberate " inhibitor which determines the limiting size, is consistent with a relation 

 of the same form. There is in fact some evidence (p. 636) of an auto-accelerator, auto-in- 

 hibitor pair of substances, though probably no results yet obtained are sufficiently accurate 

 to ascertain the exact mathematical expression of their action (Gray, 1929). 



Robertson derived his relation from its first differential: dxjdt=kx{A-x), where dx/d^ is 

 the rate of increase of .v, A is the initial amount of reagent and k is a constant. On the 

 modified hypothesis applied to regeneration, A represents the definitive size and {A-x) 

 measures the effect of the progressively increasing inhibitor. An alternative interpretation 

 of the relation above is that the instantaneous rate of regeneration, dxfdt, is related to the 

 amount still to be regenerated, measured by (^-.v). This relation was tested by Przibram 

 (19 1 7). It follows as a corollary that the initial rate, the maximal rate, or any other suitable 

 measure of regeneration-rate for any particular act, should be related to the amount ampu- 

 tated. This was verified qualitatively by Spallanzani and more quantitatively by many 

 subsequent workers (Zeleny, 1905, 1907; Ellis, 1909; Durbin, 1909; Przibram, 191 7; 

 Paulain, 1938; Needham, 1947a; Liischer, 1948). This relation, again, is sigmoid (Thomp- 

 son, 1942, p. 276), the response, like most physiological "mechanisms", being most sensi- 

 tive over the middle of its normal range. Thompson plotted time, reciprocal of rate, against 

 amount amputated, but the rate itself also gives a sigmoid curve. The results of Needham 

 (1947a) on the limbs oi Asellus also indicate a sigmoid relation. In the tadpole's tail (Ellis, 

 1909) further increase in the amount amputatated beyond that giving a maximal rate of 

 regeneration results in a progressive decline in absolute rate which approaches zero when 

 75% of the tail is amputated. Paulain (1938) obtained a complex curve for the antenna of 

 Gammarus which, after amputations distally, regenerates from material in the rest of the 

 antenna, but from material of the body when amputation is near the base. Regeneration- 

 rate declines as the size of the residuum is decreased, up to the critical point at which body- 

 materials begin to contribute, and then increases progressively, because the latter have a 

 progressively shorter distance to travel. 



Literature p. 64^ 



