VOL. 12 (1953) GROWTH OF Chlorella 35 



culture is iV = [L] + [D]. In steady state of growth, the ratio [D]/[L] is constant, 

 and the relative growth rate {k^ assumes the same value whether it is calculated in 

 terms of increase of cell number (iV) or packed cell volume (F) or dry weight of cells 

 {W) per unit volume of culture, namely 



I dA^ _ I dF _ I dTF (5) 



^ ~ ]V "dT ~ y "d7 ^ Tf "dT 

 The solution of differential equations (3) and (4), with due considerations of above 

 mentioned relations, leads to the following expression for the relative rate of growth in 

 steady state: 



° 2{n-i] 



This expression is too complicated to be used for practical purposes, but if we 

 restrict our considerations to the cases in which k^ is sufficiently small or sufficiently 

 large compared with k^,, then it may be replaced by the simpler form : 



^p^D (6) 



k 



kp ^ kj) 



As is well known, the photosynthetic rate as a function of light intensity (/) may 

 most simply, though rather approximately, be expressed by an equation of a rectangular 

 hyperbola, namely 



akj,I (7) 



kp -\- al 



* Strictly speaking, this substitution postulates some modification in our concept of light cells 

 and of the dark process. As we have shown, the rate of growth of dark cells in process (i) was equal 

 to kp only at the earlier stages of the process, and it decreased gradually owing to the decrease of 

 photosynthetic activity accompanying the enlargement of cells. If we denote the rate constant 

 corresponding to^p by ^£, and the smaller rate constant corresponding to the decreased photosynthetic 

 activity at later .stages by A'l, then the course of events occurring in the whole process may be expressed 

 symbolically b}' 



D -A_^ L' I 



L > L > nu ' 



where L' is a cell which is larger than dark cell but smaller than the light cell according to our de- 

 finition. From the kinetic point of view, the above process may be described as 





(ii) 



where k'-Q is the rate constant of the reaction step which involves a certain photosynthetic process, 

 though consisting mainly of dark processes. In a strict sense, the whole theory developed in the 

 following should be construed as dealing with the kinetic picture represented by formula (ii). 



** If the rate constants of processes (i) and (2) are expressed in terms of the number of cells 

 undergoing the transformations, these constants (denoted by k^ and k^, respectively) are related to ^l 

 and k-Q in the following manner : 



Aj = Al/(w — i) ; k^y = kol{n — i) 



Explanation : Denoting the weight of a single dark cell by w, the rate of increase (in weight) of organic 

 substance (W) in process (i) is given by dWIdt = ki^w['D']. Since the weight of a light cell is 72 -times 

 that of a dark cell, the net increase in weight occurring in the formation of a light cell from a dark cell 

 is (n — l)w. Therefore, we may also write: dlVldt = {n — l}kiw[D]. Comparing this equation with 

 the above one, we have: Al = (^^ — 1)^1- 



On the other hand, the rate of increase in cell number in process (2) is given by dNjdt = A:d[L , 

 where AT = [L] -f [D] . Using A(j we may write the relations: d[D]/di! = nk^fL], -d[L]/d^ = ^dW ; 

 therefore, diV/d; = (n — l))^d[L] .which, compared with the above equation, shows that Ad = (" — -l)^:!- 



References p. 40. 



