3.1 Kinetic scenarios for SucP-catalyzed transglycosylation
To account for a transfer to hydrolysis rate ratio (\(\frac{v_{X}}{v_{H}}\)) dependent on the donor substrate used, we considered possible extensions of the base mechanism (M1) for enzymatic trans-glycosylation, as shown in Figure 1 and more fully in Figure 2. Mechanism M1 is a canonical Ping-Pong reaction with irreversible release of GG (\(k_{+4}\)) and glucose (\(k_{+5}\)). Both extensions of mechanism, M2 and M3, assume that not only does hydrolysis occur from E-Glc, but it can also happen from a non-covalent complex of E-Glc, still containing the leaving group expelled from the donor substrate (M2; step \(k_{+7}\)) or having a second donor substrate bound (M3; step \(k_{+7^{\prime}}\)). While able to undergo hydrolysis, these non-covalent complexes of E-Glc are unreactive towards glycerol. Note: to avoid model complexity not strictly necessary in the analysis, mechanism M2 was formulated to consolidate two chemical reaction steps into one single kinetic step \(k_{+7}\). One chemical step is the formation of E-Glc with X still bound to the enzyme and the other is hydrolysis of the complex to give free enzyme, glucose and X. The idea for mechanism M3 arose from experimental evidence of substrate inhibition in the reaction of Lm SucP with G1P (see later). Upon hydrolysis, these additional E-Glc complexes can regenerate the free enzyme under release of glucose and the substrate leaving group (M2; \(k_{+7}\)) or the substrate (M3; \(k_{+7^{\prime}}\)). For each proposed mechanism, we derived mathematical expressions relating the observable kinetic parameters, including \(\frac{v_{X}}{v_{H}}\) (Figure 2), to the microscopic rate constants of the reaction. A detailed summary is given in the Supporting Information (Table S1). Maximal rate (\(V\)) and binding (\(K\)) parameters for formation of GG (\(V_{X},\)\(K_{\text{GlcX}}\), \(K_{\text{GOH}}\)) and hydrolysis (\(V_{H}\), \(K_{\text{GlcXH}}\)) are thus defined (Supporting Information Table S1; Figure 2, Eqs. 5, 7, 11). Substrate inhibition (\(K_{i,GlcX}\)) is included in all mechanisms. Mechanism M3 involves additional parameters for hydrolysis (\(V_{H2}\)) and substrate binding (\(K_{GlcXH2}\)), arising from donor substrate binding to the E-Glc intermediate and hydrolysis via step \(k_{+7^{\prime}}\). With the help of these expressions (Figure 2), important deductions can be made for the experiment, which we then show, enable clear-cut mechanistic discrimination.
In particular as demonstrated in Figure 2, mechanism M2 predicts\(\frac{v_{X}}{v_{H}}\) to exhibit curved dependence on [GOH] (Eq. 7), whereas in mechanism M1 \(\frac{v_{X}}{v_{H}}\) increases linearly with [GOH] (Eq. 5). The dependence of \(\frac{v_{X}}{v_{H}}\) on [GOH] derived from mechanism M3 is also linear (Eq. 11). However, contrary to the other mechanisms, mechanism M3 involves a unique effect of the donor substrate concentration [GlcX] on\(\frac{v_{X}}{v_{H}}\) dependent on [GOH], as can be easily realized from the denominator term in Eq. 11. The\(\frac{v_{X}}{v_{H}}\) increases when the donor concentration decreases, approaching a limiting value described by Eq. 12. Moreover, Eq. 7 (mechanism M2) and Eq. 11 (mechanism M3) involve rate constants from donor substrate-dependent steps, with the important implication that the \(\frac{v_{X}}{v_{H}}\) for these mechanisms can exhibit dependence on the type of donor substrate used. Eq. 5 for mechanism M1, in contrast, involves only rate constants from steps after the E-Glc intermediate, thus requiring the \(\frac{v_{X}}{v_{H}}\) to be independent of the donor used. In mechanism M2, the\(\frac{v_{X}}{v_{H}}\) curve described by Eq. 7 reaches plateau at high [GOH], with a maximum value equal to (\(\frac{k_{+2}}{k_{+7}}\) + 1) (Eq. 8). In the case that release of X (\(k_{+2}\)) is considerably faster than the hydrolysis according to step \(k_{+7}\), Eq. 7 reduces to Eq. 9 and M2 can no longer be distinguished from M1. Besides its unique feature of \(\frac{v_{X}}{v_{H}}\) dependent on the donor substrate concentration, as shown in Figure 2, the mechanism M3 can additionally involve a characteristic deviation from Michaelis-Menten behavior when initial rates dependent on the donor substrate concentration are recorded in the absence of glycerol (compare Eq. 13 (M3) with Eq. 6 (M1) and Eq. 10 (M2)). Various scenarios are possible from Eq. 13: one is inhibition at high [GlcX]; another is that the initial rate does not reach saturation at high [GlcX]. Both have relevance for the enzymatic reactions, as shown later.