R. E. BUCHANAN 51 



Equation (8) is in the form of an equation of a straight line. It follows, therefore, that 

 if the logarithms (to base e) of the numbers of bacteria are plotted against time, a 

 straight line will be developed with slope k (rate of growth per cell) and intercept on 

 the y-a,xis at In B. It follows that 



b = Bck^. (10) 



This is another form of equation (3), the equation of the logarithmic growth curve. 

 It is sometimes convenient to evaluate k (rate of growth per cell) in terms of g 

 (generation time) . Since the generation time is inversely proportional to the rate of 

 growth per cell, • 



g=J- (II) 



The value of the proportionality constant C may be determined from equations (5) 

 and (9). 



g=^ (12) 



and 



C=ln 2 = 2.307 log,o2 = 0.692 



PHASE OF NEGATIVE GROWTH ACCELERATION 



This phase succeeds the logarithmic phase when conditions become progressively 

 more unfavorable to growth, due either to decrease in concentration of nutrients or 

 to the accumulation of toxic products. Mathematical analysis of this phase has been 

 attempted, but the adequacy is questionable. 



MAXIMUM STATIONARY PHASE 



This is reached when the cells cease to increase in numbers. A count at this time 

 gives the maximum crop yield. The rate of growth is o. 



SPECIAL MODIFICATIONS OF GROWTH CURVES 



In some cases growth curves are found to be more complex than the type indicated 

 above. They may, for example, exhibit more than one logarithmic phase. One may 

 find cultures of organisms which both produce CO. and are stimulated by it. A small 

 seeding of such an organism (in the logarithmic growth phase) would for a time show 

 a constant rate of growth per cell, later the CO2 would increase to a point where its 

 stimulating action would be manifest, and the value of k would increase with increase 

 in concentration of CO.. Eventually saturation with CO2 would occur and the rate 

 of growth per cell would again become constant. Similarly, more than one of certain 

 other growth phases may be manifested. 



GROWTH CURVES INTERPRETED AS AUTOCATAKINETIC PHENOMENA 



McKendrick and Pai (loc. cit) and later Robertson {he. cit.) and Lotka {loc. cit.) 

 have suggested that the entire growth curve shows marked resemblance to a curve 

 of autocatalysis. The relationship may be derived as follows: 



It has previously been shown that under constant environmental conditions the 

 rate of increase of bacteria is constantly proportional to the number of bacteria pres- 



