622 J. M. REINER 
The subscript 1, let us remember, refers to the first (e.g., the N-terminal) amino 
acid in the polypeptide sequence of the protemm. The efficiencies H, and H,’ are 
functions of x9, and yo,, the extracellular concentrations of the amino acid precursor 
and amino acid itself. The P, and P,’ are the permeases for these compounds. 
Thus (not altogether surprisingly) it appears that, if a protein is synthesized in 
sequential order, the rate of synthesis in the steady state will vary directly with the 
rates of permeation of the first amino acid in the chain and of its precursor. 
Can we achieve somewhat less banal results? The permeases are presumed to be 
proteins; hence equation (3), which applies to any protein, should apply to each of 
them as well. This, however, does not immediately get us out of trouble. For if we 
put, say, P, for protein in (3), then on the right side there will in general appear a 
new set of Ps—namely, the permeases for the first amino acid of P, and its precursor; 
and this amino acid will not in general be the same as the first amino acid of another 
protein, for which P, is the permease. We appear to be going in circles, or in a messy 
system of loops and regresses. 
An interesting game could be played by talking about a conceivable case—that in 
which all the permeases can be arranged in a cyclical order, such that each permease 
works on the terminal amino acid of the preceding permease. This system can be 
solved fairly easily. Nevertheless, the cyclical order assumed seems too implausible 
to justify the mad pursuit of mathematical curiosity. 
Let us try another approach. Suppose permeases are like group-specific rather 
than compound-specific enzymes (7.e., like phosphatases or esterases). Then we 
may assume that there is a single permease for all of the x;, and another for all of 
the y;. The rates of transport of different “substrates” by these permeases will 
differ only in the efficiency factors, which involve the substrate concentrations and 
rate constants. This leaves us, as far as amino acids and their precursors are con- 
cerned, with just two permeases, which we may denote by P and P’. A given one 
of the x’s may be transported with efficiency £,, rate E,P, another of the x’s with 
efficiency E,, and so on. The equations for the two permeases may then be written, 
modelled on (3): 
(4) dP/dt = E,P © FP’ |: aP"|dt 2 EP + F,P’ 
where E, is the efficiency factor for the precursor of the terminal amino acid of P, 
F, the factor for the amino acid itself, and FE, and F, the corresponding factors for P’. 
The pair (4) has the solution: 
(5) P= aye! + a,emet ; P’ = —a,fyemit — apf.emet; 
with 
(6) i to ean peo are ath , A, = —(Po’ + Pof,)/(f2 — fh); 
m, = 3|S Ls — 4G 5 My = LS — (S? — 4G) 4]; 
S=E,+ Fy;G = EF, — E.F,; f, = (Ei, — ™)/Fii fp = (Ex — ms)/Fias 
where P, and P,’ are the initial values of P and P’. From (6) it is easy to show that my, 
> my, that my has the same sign as G, that m, > 0, that f, < o, and that f, > 0. 
Thus, both P and P’ increase exponentially, but one of the two terms in each may 
decrease exponentially to zero in case my < 0. 
References p. 632 
