BIOSYNTHESIS OF MACROMOLECULES 631 
the second problem is implicit in an earlier study of ours on non-equilibrium ther- 
modynamics"; but the mathematical difficulties seem likely to be formidable. 
The assumption of the steady state for intermediate steps of synthesis has impli- 
cations that deserve to be considered more fully. In relations like H;P; = A;E,j, 
the factors P; and E,; are, we now know, exponential functions of the time. The 
coefficients of these exponentials will have to have certain relations if these steady- 
state equations are indeed to hold independently of time. In the non-competitive 
model, the relations that will arise are obvious, and can always be fulfilled; they will, 
among other things, define the internal pools of amino acids and their precursors in 
terms of the extracellular concentrations and the various constants of the system. 
For the competitive case, however, the greater freedom of the constants with respect 
to sign means that we will have to ask seriously whether the steady state conditions 
can always be fulfilled, or whether instances may arise when this will not be possible. 
Of all our results, the emergence of the competitive system poses the greatest 
challenges to patience and ingenuity, and hence may perhaps be deemed the most 
interesting feature. The (admittedly oversimplified) illustrative cases, implying me- 
chanisms of differentiation and control, may hint at the possible rewards awaiting 
further work on this feature. 
In connection with the competitive system, it may be well to examine more closely 
the apparent paradox of 20 equations for vastly more than 20 unknown rates of 
protein synthesis—a paradox which we side-tracked by an approximate procedure. 
Let us start counting equations and unknowns more carefully. If there are 20 amino 
acids, and thus also 20 more activated amino acids, the first two lines of equation 
(x) exhibit equations of the type that will govern them; the same holds for pre- 
cursors. The solutions for the activated amino acids will, however, involve all the 
T; for all the proteins. Let us ask how many there are. If protein a contains Ng 
residues, its template has Ng + I states 7;(a); there is a conservation equation that 
says their sum equals the total amount of that kind of template, reducing the number 
of independent 7;(a) to Ng. The recursive equations M,(a) = M,(a) =... = aly, 
are Nq in number. Hence there are in our system, as written for the steady state, 
enough equations to determine all the unknowns. However, if we proceed to eliminate 
the y,’, after solving the recursive equations, we get finally a set of non-linear equa- 
tions for the 7;, each equation involving 7;s from all the proteins. It is clear that a 
general solution is out of the question, and that results will have to come from con- 
sidering special cases that are more tractable. 
The general equations suggest one interesting conclusion, however. If we try to 
write them down carefully rather than sketchily, it becomes evident that our nota- 
tion for enumerating amino acids according to their sequential positions in one 
arbitrarily chosen protein is ambiguous and confusing when we try to apply it to 
all the proteins simultaneously. It becomes necessary to have a notation that sepa- 
rates the enumeration of the kinds of amino acids from the specification of their 
positions in any one protein. Such a notation can be devised, and will be discussed 
in a forthcoming publication. The new notation does not in any way reduce the 
mathematical difficulties of a general solution. What it does, however, is to make it 
clear that the complete system is not only defined by the compositions of the pro- 
teins (in addition to the other factors), but also by their sequential orders —a result 
which I find intriguing, and which was certainly not intuitively obvious to me. 
References p. 632 
