BIOSYNTHESIS OF MACROMOLECULES 629 
The interpretation of this result is unremarkable but makes sense. The maximum 
value of E increases with its rate of synthesis (H,) and that of its permease (H). 
It decreases with the rate of synthesis of the protease F,(H,), and with the proteo- 
lytic rate constant (7). It decreases with the proteolysis constant of P (m, via q), 
and increases with the autoproteolysis constant of the protease itself (g). All the 
factors in the system enter into its capacity to make protein in about the way one 
would intuitively expect. But intuition might not have predicted the remarkably 
simple way most of the factors enter the result. And one might have guessed wrong 
about the role of g: as g increases without bound, the ratio g/q¢ = g/(4m + g) ap- 
proaches the upper limit 1, so that the rise of P and E is not unbounded even if the 
protease destroys itself at an infinite rate. 
COMPETITION AS A LIMITING FACTOR 
The competitive system of equations discussed previously has one interesting feature 
which we have not mentioned so far: The coefficients of P and P’ are not necessarily 
intrinsically positive, as in the simple non-interacting system. This has two conse- 
quences: (a) it is no longer necessarily true that at least one of the exponential 
terms of P and P’ has a positive exponent; and (b) the relations among the constants 
may be such as to permit maxima or minima in the time curves of various proteins 
while this could be excluded by elementary calculations for the non-competitive 
system. 
We shall not enter into these points extensively. It will suffice to give a couple of 
elementary examples that will point up the possibilities. 
Consider for simplicity, as we have at times earlier, the case of just one permease, 
say P’. Suppose this is governed by dP’/dt = —aP’. The solution is P’ = P,'e~™. 
Now take a protein governed by dPr/dt = bP’. Using the value just found for P’ 
in this equation, we can solve for Pr, finding Pr = Pr, + (b/a)P,' (1 — e-@). This 
rises from Pr, to Pr, + (b/a)P,’. At the same time, P’ decreases to zero. This might 
serve as a simple-minded model of 7zvreversible maturation: the various proteins in- 
crease, but always to limiting values, while the permease vanishes, so that it cannot 
be regenerated (since its development is autocatalytic). The model does not admit 
of turnover in the mature organism; but such turnover, independent of cell death 
and concomitant multiplication, is questionable in any case. 
Consider a second example. Suppose both permeases are present. Let the values 
of the various constants of the system be such that the equation of a certain protein 
comes out as dPr/dt = AP,e% — BP,'e-%t, where the constants as written are 
assumed positive. We ask under what conditions this expression may equal zero, 
denoting a maximum or minimum of Pr. The condition is: e(@ +) = BP,’/APo. 
Since the left side of the condition is a positive quantity, and greater than one, for 
real and positive values of ¢, it must be true that BP,’ > AP,. (The same holds for 
boy if @4=b >0; if a+ 6 < 0, we demand BP,’ < AP,.) If this condition is 
fulfilled, there will be a minimum of Pr, as is evident from setting ¢ = 0 in dPr/dt, 
and seeing that it will be negative. Eventually, of course, Py will increase expo- 
nentially; but there may be a long deep trough if BP,’ > AP, and a and b are not 
too large. If Pr were a protease, we would have a mechanism of growth triggering 
and control in which the protein-destroying activity at first decreases (while presum- 
References p. 632 
