394 S. ARONOFF 



whore kih = l',r and /.•„, + L-,i, + /.-,> = l-sr- One may ol)tain A:., and 

 h\u from the calculations lo he presented l)elow. From (future) 

 knowledge of the enzyme kinetics of the above reactions, one will 

 know k,^j), h-,iA and A^r- ('onse((uently, by solution of a pair of simul- 

 taneous equations describing dA/dt and dB/dt, one should be a})le to 

 solve for />;,■„ and /i-,7, = /.',>. In a similar, though more complicated 

 manner, the solution of compartment *S may be ol)tained. 



The formal set of differential equations describing the rate of in- 

 corporation of radioactivity into each of the compartments of Fig. 



2 is: 



dC 



(It 



dS 



— = kgsG — (/t,,r + A"o)*S 



dt 



dt 



where G, R, and S are specific activities of PGA, the pentose sugars, 

 and the residual sugars, respecti^^ely. The subscripts of the various 

 A;'s denote direction of movement of the isotope. If G, R, and S 

 denote the corresponding steady states of G, R, S, then the set of 

 linear, nonhomogeneous ecjuations may be converted to a set of 

 homogeneous eciuations by substitution of the corresponding dif- 

 ferences, a = G - G, = S -S, y = R - R, following the meth- 

 odology of Denbigh (3). The solution of these difference equations is 

 readily shown to be 



„ — mit I „ — m2t I „— mst 



OL = illge + IJ-lgC -\- fX^igC 



P = Hue + fl'lsC -\- MSs^ 



where the ?n's are the roots of the cubic equation: 



w^ - m-[krg + kg, -t- k,r + A:ol + m[krgkg, -f- (A;,^ + kgs)(k,r + />'n)] 



— kokrgkgs = U 



and the constants have the relation : 



