102 Growth and Form 



4. Structural limitations. Nature is an engineer and an architect. She 

 builds conservatively and according to sound engineering principles. She 

 pays attention to strength of materials and patterns of stress. She believes 

 that form should follow function. The man to realize these ideas most 

 strongly and to express them most convincingly in terms of mathematical 

 rigor was a great biologist named D'Arcy Thompson, who wrote a book on 

 the subject entitled Growth and Form. 



Equations That Describe Growth 



Many attempts have been made to describe mathematically the 

 growth of a population of cells or of multicellular individuals. It is not 

 difficult to arrive at an equation that can generate an S-shaped curve. 

 What is difficult is to invest the constants of this equation with biological 

 meaning, that is, to interpret them as being due to specific physiological 

 mechanisms that initiate, maintain, and terminate growth. Generally, we 

 start with the growth-affecting processes we know about (production of 

 diffusible subunits, production of toxic wastes, exhaustion of the nutrient 

 supply ) , then describe them in mathematical terms, and derive the proper 

 equation relating the number or mass of organisms to the passage of 

 time. The extent to which the equation ( called the logistic equation ) ac- 

 curately describes growth, as measured in specific organisms, tells how 

 much we really know about growth processes. The extent to which it 

 does not, tells us how much is not known but, more important, provides a 

 testing ground for the inclusion of additional physiological processes that 

 we suspect may affect growth. The following paragraphs contain in ab- 

 breviated form the arguments that led to the derivation of the logistic 

 equation. It is not expected that you will understand the subject com- 

 pletely from this exposition, but I hope you will appreciate from it that a 

 rigorous, quantitative approach to biological phenomena is possible and 

 represents a fertile field of investigation for the mathematically-minded 

 student. 



The first part of the S-shaped curve, the period of so-called exponen- 

 tial growth, is simple to derive. It is described by the equation: 



\og,^ = kt 1 



where N is the number of organisms at time t, No is the starting number, k 

 is a constant, and t is time. As mentioned previously, the growth of a 

 population of cells that divide by binary fission follows the series 1, 2, 4, 

 8, 16, . . . and these can be stated as powers of 2, i.e., 2", 2\ 2-, 2^, ... 

 in which the exponents refer to the number of generations of growth that 



