l62 GROWTH PRINCIPLES AND THEORY 2 



1934, ig5ia; Fig. 5). Rodlike bacteria grow with almost constant specific growth 

 rate, and hence show simple exponential growth. In contrast, specific growth 

 rate decreases in spherical bacteria and in yeast, the growth curve presenting a 

 decaying exponential for the radius and a sigmoid curve for the volume, 

 respectively. 



If growth rate is conceived as the difference between anabolic processes 

 dependent on import and hence on surface, and catabolic processes depending 

 on mass or volume, then for rodlike bacteria the equations: 



dv/dt = 7] V — ycv^cv (S-i) 



V = v^e" (3.2) 



/ = lo^"" (3-3) 



{v = cell volume; v], x = constants of anabolism and catabolism, respectively; c = 

 growth constant ; / = length; v^, l^ initial volume, length) will apply because rodlike 

 bacteria grow almost exclusively in length and hence surface responsible for 

 import of nutrients increases nearly proportional to volume. Hence, specific growth 

 rate will be nearly constant, and the growth curve a simple exponential. 



In contrast, in a spherical microorganism the surface-volume ratio is shifted 

 in disfavor of surface. Then the growth equation : 



dvjdt = 7] v~'^ — V.V (3.4) 



will apply indicating that specific growth rate decreases. This gives for linear 

 growth (radius = r) the decaying exponential : 



r =r* — (r* — Oe-'-'' (3.5) 



and for growth of the volume the sigmoid curve : 



V = \\yv* - if/v* - ]yv,)eX']' (3.6) 



with r*, V* = final radius, volume, r^, v^ initial radius, volume, and k = x/3. 



These equations explain the characteristic differences between the growth 

 curves of rodlike and spherical microorganisms, and between the curves of linear 

 and volume growth of the latter, and give excellent fit of empirical data (Schmal- 

 hausen and Bordzilowskaja, 1930; Bertalanffy, 1934, 1951a), so showing that the 

 model concept is essentially correct. 



The theory also explains the temperature dependence of growth visually found 

 in microorganisms, with the exception of certain phenomena in the growth of 

 yeast (p. 205). 



Furthermore, the theory gives a certain insight into the basis of the principle of 

 constant cell size. If a growth equation of the kind of equation (3.4) applies, the 

 growing system is equijinal, i.e. it will attain the same final size independent of 

 initial size. Under the simple condition: 



V* = (ri/x)3 = 2v^ (3.7) 



the steady state is reached when the initial volume is doubled. However, the final 

 volume (y*) depends only on the constants of building up and breaking down 

 (t; and x), not on the initial size (v^). That is, the system may divide into parts 



