624 



The Determination of Size 



a percentage of the preceding size per unit 

 of time, and is thus exponential. 



A typical growth curve representing the 

 complete period of growth of an organism 

 is generally concave to start with, changing 

 through a relatively straight and steeply 

 ascending phase into an increasing convex- 

 ity, until it levels off as a straight horizontal 

 line. The so-called point of inflection is 

 where the concavity changes to convexity, 

 and it represents the period at which the 

 absolute growth rate, or increment per unit 

 time, is greatest. At this time the curve for 

 the absokite growth rate normally curves 

 downwards again, rapidly at first but more 

 and more slowly until maximum size and 

 zero rate are reached (cf. Fig. 214b). 



A great deal of attention has been given 

 to the significance of the point of inflection — 

 by Brody ('45) for example — but the S- 

 shaped curve is characteristic of many things, 

 equally of the growth of organisms, organs, 

 and populations. Growth of each has its 

 beginning and draws to its natural end, and 

 to quote D'Arcy Thompson, "the motion of 

 a body in a resistant medium . . . for so 

 beginning and so ending the curve must pass 

 through a point of inflection, and it must be 

 an S-shaped curve." 



The curve of growth, being essentially a 

 generalization, tells nothing about the nature 

 of growth beyond the facts that it has a 

 beginning and an end, attains a certain max- 

 imal velocity and has a certain duration. Its 

 value lies primarily in its ability to indicate 

 the presence and operation of restricting 

 factors, cycles of growth, temperature and 

 nutritional inflviences, etc. It does not express 

 the essence of growth itself nor the nature 

 of the decrement in growth rate. The growth 

 curve indicates that a decrement exists from 

 the beginning and that the duration of 

 growth is determined by the magnitude of 

 the early maximal relative growth rate and 

 the magnitude of the decrement. Accord- 

 ingly, the size finally attained by any grow- 

 ing unit is a resultant of the quantity of 

 material initially present, the maximum 

 relative growth rate (which is at or close to 

 the initiation of growth and not at a time 

 corresponding to the point of inflection), and 

 the decrement. 



Algebraic equations may be made to ex- 

 press the curve of growth, facilitating its 

 analysis and yielding information by math- 

 ematical rule of thumb, though there is no 

 such thing as a universal growth equation 

 (Medawar, '45). D'Arcy Thompson worked 

 with the curve of growth and its derivatives. 



the curve of growth rate and the curve of 

 acceleration, but the curve of specific growth 

 is generally more amenable to analysis (cf. 

 Fig. 214e, after Medawar), where the logar- 

 ithm of size (in place of size itself) is plotted 

 against time. The curve of specific growth 

 indicates a progressive decline in the en- 

 ergies of growth, that is, specific acceleration 

 is always negative under actual conditions 

 of development, in contrast to tissue cultures 

 or yeast populations growing in a constant 

 environment (Richards, '28). This is Minot's 

 law, and it follows that the specific growth 

 rate declines more and more slowly as the 

 organism increases in age, or as Minot put 

 it, organisms age fastest when young. 



An attempt at an experimental measure of 

 the rate at which the specific growth-rate de- 

 clines has been made by Medawar ('40). In 

 a later review ('45) on size, shape, and age 

 he concludes, "only one fundamental gen- 

 eralization can be made abovit the relationship 

 between the size of an organism and its age: 

 that which is represented by the equation 



dt 



where K.f{t) is a positive quantity such 

 that f{t) decreases with t, and df(t)/dt in- 

 creases with t towards a zero bound." 



A more biological analysis of growth has 

 been made by Sinnott ('45 and earlier) on 

 cucurbit fruits, with respect to the relation 

 of cell division and cell enlargement to 

 growth rate. During the first part of the 

 growth period, svibstance (whether measured 

 by wet or dry weight) increases at a constant 

 exponential rate regardless of what the con- 

 stituent cells are doing. When cell division 

 ceases and rapid vacuolation begins there is 

 no change in this rate. He concludes that the 

 organ rather than its constituent cells is the 

 dominant entity, but that the mechanisms 

 controlling size and form are not known. 



In certain slime moulds (Raper, '41; Bon- 

 ner, '44) the developmental cycle is divided 

 into vegetative (or growth), aggregation, 

 migration, and culmination stages. The last 

 of these corresponds to the late differen- 

 tiation phase, for example, of the cvicurbit 

 fruits in which cell multiplication and differ- 

 entiation are also separate in time, although 

 it is more purely a morphogenetic movement 

 resulting in change in shape. Bonner and 

 Eldredge ('45) conclude that the motivating 

 force is an internal and not a surface force, 

 and that smaller masses rise up compara- 

 tively slowly because cohesive forces are 

 greater in proportion to the internal morpho- 



