16 Growth 



to fall along an essentially straight line, showing that the fruit was growing 

 at a constant exponential rate. The slope of the line is a measure of this 



rate. 



The resemblance between such organic growth and compound-interest 

 increase has long been noted, but it was particularly emphasized by 

 V. H. Blackman (1919). He proposed the term efficiency index for the 

 "interest rate" in such growth. That living things in the early stages of 

 their development should grow in this way is not surprising, for if 

 embryonic material is self-multiplicative, the increase per unit of time 

 should be proportional to the growing mass. 



To explain the rest of the growth curve is more difficult. Evidently 

 growth cannot proceed in any organism at a continually accelerating rate, 

 if for no other reason than that building material would soon be used up. 

 The gradual slowing down and final cessation of growth are far too 

 regular a process, however, to be due to mere exhaustion of materials. A 

 plant or animal provided with a superabundance of nutrients will rarely 

 exceed the size characteristic for its species. Each organ or body has a 

 specific growth cycle through which it passes, and the second part of 

 this cycle, in which growth is falling off in rate, is much like the first part 

 in reverse, so that the entire growth curve thus tends to be symmetrical. 

 In such a case the periodic increments form a curve (Fig. 2-2) which 

 much resembles the so-called normal curve, or curve of probability. This 

 relationship has been observed and discussed by various workers, 

 especially Pearl and his school (1915), but any causal relationship be- 

 tween the two types of curves is not easy to see. 



The similarity between growth and the chemical phenomenon of auto- 

 catalysis, in which the products of a catalytic process accelerate the 

 process itself, has been noted by many observers. Robertson ( 1923 ) and 

 his followers have attempted to analyze the whole growth curve as a sim- 

 ple ( monomolecular ) autocatalytic reaction by which an enzyme breaks 

 down a mass of substrate. During the first part of the process, growth will 

 therefore accelerate, but when the amount of substrate becomes seriously 

 reduced, rate of growth will also be reduced, and when the substrate is 

 exhausted, growth will cease. 



Robertson derives the typical sigmoid curve from the equation for 

 autocatalysis: 



l ^A^x = k{t ~ tl} 



where x is the volume of the organic structure (amount grown) at any 

 time t; A the final size of the structure; U the time at which it attains half 

 its final size; and k is a specific growth constant, or exponential growth 

 rate. 



