BIOSYNTHESIS OF MACROMOLECULES 623 
If we introduce the result (5) into equation (3), and solve: 
(7) Pr = Pry + (Po’ + Pofe) (Hi — Hy fi) (em? — 1)/m, 
— (Po’ + Poft) (Ai — Hy’f,) (emet — 1) /m,. 
Now m, > my. Hence, as ¢ increases, the first exponential term will more and 
more outweigh the second one; and the time course of the protein concentration will 
tend to be governed by a simple exponential increase : 
(7’) Py > (Bo Poe) dg — sea fee lo 
Cell composition 
One consequence of these results is easily inferred. To make it easier to see, we 
may abbreviate the expression (7) as Pry + Q,(Pr)R,(t) + Qo(Pr)Ro(t), where the 
notation Q,(Pr), for instance, means that the factor depends on constants charac- 
teristic of the specific protein, while R,(¢) is a factor common to all proteins, depen- 
dent on the time ¢ and on constants of the twin permease system. We now consider 
what fraction Py is out of the total cell protein. By definition, this is Pr/2’Pr. This 
is, in general: 
[Pro + Q,(Pr)Ry(t) + Qe(Pr)Ra(t)|/Z[Pro + Q1(Pr) Ri (t) + Q2(Pr) Re (t)]. 
Without further assumptions, this expression cannot be appreciably simplified. It 
means that the given Py is a fraction of total protein that depends on its individual 
characteristics and on the time ¢, so that the cell composition, in general, is variable. 
However, consider times long enough for (7’) to be a good approximation. The 
fraction of Py will be 
Q, (Pr) R,(t)/2Q, (Pr) Ry (2). 
Here the numerator and denominator have a common factor R,(¢). When this is 
removed, the fraction of Py is simply Q,(Pr)/XQ,(P7). This is independent of the 
time ¢. We conclude that, as a system such as we are considering begins the biosyn- 
thetic process, its protein composition will shift with time. However, it will approach 
a constant composition, characteristic for each protein, as time passes. This result 
is reminiscent of the changes that take place in a bacterial culture during the early 
stages of growth, and of their tendency toward a constancy of cell composition 
during the logarithmic phase of growth. 
It should be noted that the characteristic composition depends on the composition 
of the medium, since the H, and H,’ of each protein depends on the external con- 
centration of amino acids and their precursors. Thus, if one amino acid is omitted 
from the medium, the H,’ for all proteins in which it is terminal will vanish, and 
those proteins will have a lower representation in 2'Pr. 
Competition among proteins 
So far, we have been treating a somewhat idealized situation, in which each amino 
acid occurs but once in a protein, and in which we ignore the fact that it is also 
used in the synthesis of other proteins. The key equation is in (1), in the expression 
for dy,’ /dt, which is written as BjE; — M;,E’. Since the 7-th amino acid in the protein 
we are considering, in its activated form y;’, also enters, in general, other places in 
References p. 632 
