BIOSYNTHESIS OF MACROMOLECULES 627 
been applied to amino acid precursor), the permeases and enzymes approach maxi- 
mum concentrations in a manner similar to the so-called logistic curve of growth. 
Since protoplasmic mass is proportional inter alia to cell protein, this agreement is 
not only reasonable in itself, but is also in concordance with the idea that the logistic 
course of growth depends on synthesis of cell material by systems similar to the one 
we have been considering!?. Equations (5) and (7) can be viewed as approximately 
the early phases of (9), (12), and (13) in the same sense as the simple exponential 
growth equation approximates the logistic. 
From the foregoing, it would appear that one could influence the protein com- 
position of a cell rather readily, by supplying some amino acids in definitely limiting 
amount, others at much higher levels, since each protein tends toward a limit set by 
the total supply of its terminal amino acid. However, let us recall the rigorous 
analysis of competition between proteins, which predicts that each rate of synthesis 
will in general be a function of all the efficiencies for all the amino acids that occur 
terminally in any protein. In view of this, it will not be possible to treat individual 
proteins separately in terms of the supply of their respective terminal amino acids 
alone. Accordingly, while some cases may occur in which a specific juggling of the 
amino acid supply will affect the protein composition of the cell, we may expect in 
most instances to find cells relatively insensitive to such differential procedures. 
The exceptions would arise with respect to protein classes of exceptional composi- 
tion (like the histones, for example), where the buffering effect of competitive 
interaction might fail. 
PROTEOLYSIS 
We now consider how proteolytic enzymes might influence biosynthetic processes. 
As a very crude first approximation, we may consider that the proteolytic enzyme E » 
(which we will take as representative of proteolytic effects in general) does not 
vary with time. We ask about the fate of a permease and another enzyme (say P 
and £) as in the preceding section. The equations for these compounds are: 
(14) dP/dt =HP—mPE,  ; dE/dt = H,P —nEE>, 
approximating the rate of proteolytic activity, as we have done previously, by ex- 
pressions that are linear in the concentration of substrate. The solutions are: 
(15) P = Poel — mE, )t : E = Ce-nE,t + HyPoe\# — mE,)t/(H — mEy + nEp), 
where C is a constant of integration, obtained as usual by setting E = E, andt = 0 
in the equation. 
Evidently, there are just two possible outcomes for constant Fy. If H — mE, > 0, 
both P and E will rise without limit, but at a slower rate than in the absence of Fp. 
If H — mE, < 0, both P and E£ will fall steadily to zero levels. 
Suppose Fy is not constant, but increasing. Then it may reach levels at which 
dP/dt and/or dE /dt may become zero or even negative. Under these circumstances, 
however, the solution (15) is no longer valid, and we must begin from the beginning 
once more. Suppose first of all that Ey is given by: 
(16) dE»|dt = HP. 
References p. 632 
