626 J. M. REINER 
LIMITATIONS OF SUPPLY 
Up to the present, we have treated the factors H and H’ as constant, implying a 
constant extracellular supply of amino acids and of their precursors. It may be of 
some interest to consider what might happen where this does not hold, as in the ordi- 
nary course of events during the growth of a bacterial culture. 
To keep the mathematical developments from becoming unduly complex, let us 
consider the special case where we have only one of the two key permeases, say P’. 
Let the supply of its terminal amino acid be limited so that the equations (1) would 
include: 
(1’) dyeidt = =H’ P” 
There is no possible steady-state solution for this equation, so we must solve it as 
it stands. We take a linear approximation to H’, and write it as h’y,. To eliminate 
P’ from (1’), we note that, by (3), it is governed by the equation dP’ /dt = H'P’, 
or, using (1’), dP’ /dt = —dy,/dt, hence P’ = k — yo, where k = P’o + Yoo, the sum 
of the initial values. Eliminating P’ from (1’) and solving, we have: 
(8) Vo = RCe kh | (x a Ce-kn’t) 
where C = Yo9/P’o. Since P’ = k — yo, we have at once: 
(9) iP Ri] (ne -- Cenk oy 
the familiar logistic relation. 
Now consider one of the other enzymes, say E. If E has the same terminal amino 
acid as P’, then cepdE /dt = dP/dt, or: 
(10) Ei — BiG | Cen (ei io) 
If E has a terminal member distinct from that of P’, then its rate of synthesis, 
using the same linear approximation as we did for H’ above, is given by: 
(11) dE |dt = —dyq|dt = ha’vaF’, 
where we use yq to represent the external concentration of the terminal amino acid 
of E. Solving (11) for yg, with P’ given by (9), we get: 
(12) Va = VYaol(L + C)eFh’t] (x + Cekh’t)]hg//h’, 
Inasmuch as dE /dt = —dyq/dt, we have (with k’ = Ey + Yao): 
(13) E =R’ — ya. 
(Vao is of course the initial value of yq). This will have a course similar to the logistic 
of (g). The latter will arise initially at the rate h’yo9P,’, the former at the rate Ha’VaoP’. 
With sufficient time P’ will approach the value yoo + P,’, while E approaches 
Yao + Eo. That is to say, the proteins P and E rise to values limited, as should be the 
case, by their supplies of terminal amino acid. The only other noteworthy feature 
of the results is that, while P rises to its maximum at steadily decreasing rate of 
increase, E may have a period of positive acceleration; the condition for this to 
occur is that hq’ > h’ and that 2yo9/Po' > hq’ |h’ — I. 
Thus, when the supply of amino acid is limited (the same argument could have 
References p. 632 
