Figure 3

Various isotope exchange fluxes created in the TK-catalyzed reaction: xu5p + r5p <-> g3p + s7p. Designations are the same as in Figure 1. A. Isotope exchange between xu5p and g3p in the presence of labeled g3p results in labeling of xu5p. This exchange flux could be calculated as follows. Forward flux of the last n-2 atoms of ketose-substrate to the pool of aldose-product (e.g., x5p → g3p) implies delivering the x5p atoms through the three steps (x5p + E → E*x5p → EG*g3p → EG + g3p). The intermediates E*x5p and EG*g3p contain aldose fragments originated from two sources, either x5p or g3p, and the respective fractions of isotopomers are specified by the relative values of the elementary rates. Specifically, the rate of delivery of x5p atoms into g3p is a part of the rate v₃ (v_i is a unitary rate corresponding to the rate constant k_i); it is proportional to the content of carbon atoms originated from x5p in EG*g3p. The proportionality constant or fraction of x5p atoms in EG*g3p (P^x1_EGg, where the superscript x1 denotes the last carbons originated from x5p, and the subscript EGg denotes the form EG*g3p) depends on the fraction of former x5p atoms in E*x5p, that partly consists also of former g3p atoms that enter via the reactions whose rates are v_-2 and v_-3; thus it is expressed as a ratio of the input of the donor atoms from E*x5p (whose fraction is P^x1_Ex) to the total input to EGa₁:

P^x1_EGg = (v₂P^x1 _Ex)/(v₂ + v_-3).     (f1)

The proportion of atoms in E*x5p that originated from x5p (p^x1_Ex) in turn is given by the ratio of influx of this kind of atom to the total influx into the compound Ec₁ at steady state:

P^x1 _Ex = (v₁ + v_-2P^x1_EGg)/(v₁ + v_-2).     (f2)

Solving Eqs fl and f2 yields the expression:

P^x1_EGg = (v₁ v₂)/(v_-2 v_-3 + v_-3 v₁ + v₁ v₂).     (f3)

The flux of former x5p atoms into g3p, v_xg, where the subscript xg denotes the x5p->g3p direction, is given by

v_xg = v₃ P^x1_EGg = (v₃ v₁ v₂)/(v_-2 v_-3 + v_-3 v₁ + v₁ v₂).     (f4)

Equation f4 gives the rate of forward delivery of the last n-2 atoms in x5p to g3p, expressed by using the unitary rates. Since these atoms can originate only from either x5p or g3p, the fraction of atoms originating from g3p is expressed as P^g_Ex = 1 - p^x1_Ex. and the reverse flux of the aldose (g3p) to the ketose pool (x5p) can be described similarly to Eq. f4 as

v_gx = v_-1 P^g_Ex,     (f5)

B. Isotope exchange between s7p and r5p in the presence of labeled s7p results in labeling the r5p. The exchange of atoms between s7p and r5p can be described in the same way as it is done in A.

C. Isotope exchange between s7p and x5p in the presence of labeled s7p results in labeling of x5p. Forward flux (v_xs) of the first two atoms of x5p to a second ketose/donor, s7p, implies delivery of the atoms through six steps (x5p → E*x5p → EG*g3p → EG → EG*r5p → E*s7p → s7p). This is a part of the rate of s7p production (v₆) and it is proportional to the fraction of former x5p carbon atoms in E*s7p, namely P^xf_Es, where the superscript xf denotes that the first part of the molecule originates from x5p. This proportion is determined similarly to that described above, i.e. by solving the five equations for the fractions of atoms that originated from x5p in all the species (similar to the Eqns fl and f2). The reverse flux (v_sx) of the first two atoms of s7p to x5p could be described in the same way.

Thus the following fluxes of the carbon skeleton parts are expressed through the same elementary steps of the catalytic mechanism:

v_xg: x5p -> g3p

v_gx: g3p -> x5p

v_xs: x5p -> s7p

v_sx: s7p -> x5p

v_sr: s7p -> r5p

v_rs: r5p -> s7p

The difference between forward and reverse fluxes of isotope exchange between all pairs of pools is the same and corresponds to the net flux:

v_xg - v_gx = v_xs - v_sx = v_rs - v_sr = v_net     (f6)

It follows from (f6)

v_xg - v_xs = v_gx - v_sx, and v_sr - v_sx = v_rs - v_xs     (f7)

The whole reaction related to exchange between x5p and s7p expressed by the fuxes v_xs and v_sx is accompanied by the exchange inside half-reactions, i.e. between x5p and g3p, and also between s7p and r5p. These exchanges in fact constitute a part of the fluxes v_xg and v_sr deduced above and the differences (f7) describe the "pure" exchange between ketose and the product of its splitting, which is the same in both directions. Taking into account equality of the "pure" exchanges expressed by equations (f7), the four fluxes define all of the isotope exchanges associated with the considered TK reaction:

- forward flux x5p->s7p (v_xs)

- reverse flux s7p->x5p (v_sx)

- pure exchange x5p<->g3p (v_xg - v_xs)

- pure exchange s7p<->r5p (v_sr - v_sx)

The above fluxes could be expressed through the elementary rates, as exemplified by Equation f4. The elementary rates, in turn, could be expressed through the elementary rate constants and substrate and product concentrations using, for instance, King and Altman algorithm (as described e.g. in [48]). Thus, all TK fluxes are considered not as independent but as interrelated through the elementary rate constants, which could be determined in independent experiments as described elsewhere [33].