52 GROWTH CURVES OF BACTERIA 



ent. It may be assumed that the rate of increase is also proportional to the concen- 

 tration of the available nutrients, or to that of some single nutrient which acts as a 

 limiting factor. If concentrations of cells and of nutrients are the only two factors 

 governing the increase, the rate will be jointly proportional to the two. A convenient 

 method of estimating the amount of available nutrient is to determine the total max- 

 imum number of bacteria which may be produced in the culture. The difference be- 

 tween the number of bacteria present at any instant and the maximum number of 

 bacteria which may be developed is proportional to the available remaining nutrients. 

 If /3 = maximum bacterial count, then 



Upon integration, 

 When t = o,b = B, and 



j^=Kb{^-h) (13) 



ln-^^=K t+C (14) 



Equation (14) shows a straight-line relationship between In ~ — r and time, the 



straight line having a slope, K. This relationship may be used to determine whether 

 in any case the growth curve resembles that of autocatalysis, or whether (according 

 to Ostwald) the growth curve is autocatakinetic. 



The equation of an autocatakinetic growth curve may be derived from equation 



(14): 



A convenient evaluation of C may be made by taking /i as the time which has elapsed 



to the instant when b = -, i.e., until the number of bacteria has reached one-half the 

 2 



maximum. 



C=-Ki^ 



and 



^w«ir5=^' (^-^') 



i^r^+FKins- <"' 



A curve of this type is illustrated in Figure 6. It will be found to be symmetrical, and 

 asymptotic to the lines 6 = and 6 = /(3. The point of inflection occurs at - after 

 time ti. The y-axis is cut at b = B. 



