628 J. M. REINER 
Then, eliminating P between (16) and (14), we can get the relation: 
(17) P = (HE, — }mE,? + A,)/H, 
where Ag = H,Po — (HEop — 4 mE@y). Putting (17) into (16), we have : 
(18) Ey = (Exp—Eope-m™@) | (1 — Be- mat), 
where: 
(19) B = (Eop — E1p)/(Eop — E 2p) , d= Exp — Erp 
Ey, = (H + (A? + 2Aqm)?|/m_ ; Ean — f 2 + 2A om) *]/m. 
In this case Ey will rise steadily to the limiting value E,,, which is, by (19), greater 
than 2H/m if Ao > 0; if Ag < 0, Ey lies between H/m and 2H /m. 
The consequences of this setup are obvious. From (16), it is clear that Ey will 
stop rising when P = 0; and from (14) it is evident that P will reach its maximum 
and begin declining already when FE, reaches the value H/m. From (14) for dE /dt, 
one can see that dE /dt will be negative when P = 0 and E, is at its maximum. 
Thus E£ will eventually decline and reach a zero value, after rising to some maximum 
previously. It would be farfetched to regard this as a model of life and death; and 
as a model of normal biosynthesis and growth it could serve only if we added some 
assumption that would stop Ey early enough to prevent the total disappearance of 
Pande. 
This could be done in a natural way by assuming that Fy itself is subject to pro- 
teolysis. To avoid introducing another enzyme to devour Fy, let us approximate 
this by having Fy act on itself. We can rewrite (16) as: 
(160’) dE,/dt = H,P — gE,y? 
This modification prevents the simple mathematical device that permitted us to 
derive (17). However, we may assume that g is small enough so that the term gk? 
will be negligible while Ey is small. Thus, we assume that (17) still holds approxi- 
mately. Putting (17) into the complete equation (16’), as the next better approxi- 
mation, we solve for EF, and have the same form as (18), with the modification that 
im is replaced by g = 4m-+ g. The maximum value of Ey, namely E,y, is now 
approximately H/q instead of 2H/m*. At this point, when dE ,/dt = 0, instead of P 
being also zero, we have from (16’) P = gH?/H,g?. It is no longer necessary that 
dE |dt be negative; substituting EF,» = H/qin (14),dE/dt = (H/q) (H,gH/qH, — nE). 
If this is to be zero, and F at its maximum, at this time, we must have FE = H,gH /qHgn. 
That this result for &, rather informally obtained, is reasonable can be seen by 
taking the value (17) for P, substituting into (16’) and (14), treating A, as negligible 
to minimize mathematical complications. This permits us to combine the equations 
(14) and (16’) and get an equation for dE /dE py. Solving for E, we get: 
(20) E =C(H — qEp)"™4 + 4mH,(H — gEp)/H.q(n — q) + HygH/qHn, 
where C is a constant of integration. When E, reaches its maximum value H/q, 
the first two terms of (20) vanish leaving E = H,gH/qH,.n as the limiting value of FE. 

* Whether £,, is equal to H/q exactly, or a little less or a little more, depending on the sign and 
magnitude of 44, is immaterial for the argument. The important point, evident from (16’), is that 
dE,/dt = o does not imply P = o as was true in (16). We use the approximate limiting value 
H/q merely for the sake of some illustrative formulas. 
References p. 632 
