One Enzyme 2 Substrates, A and B, to 2 Products, C and D, Reversible Second-order enzyme kinetic model with capacitance and reversible enzymatic reaction with binding and release kinetics for two substrates givng two products, thermodynamically constrained.
This model represents the enzymatic conversion of two substrate, A and B, to two products, C and D. First is the binding of one substrate to the enzyme, E, to form EA or EB, a 1-substrate-enzyme complex, ES. This binding is followed by binding the other substrate to form EAB. This is random bi-bi, not ordered. Next is a reaction-release event, which yields one product, C or D, and an enzyme still bound to the other product. The second release yields the free enzyme and the second product. Note 1. The equilibrium between EAB and E is independent of the route taken: the alp1 and alp2 factors are to assure this. Because there is no disinction between EAB and ECD in this model alp2 = alp1 in order to keep the same ratio EAB/E = ECD/E. Note 2. The model parameters can be adjusted to fit Michaelis-Menten conditions (Substrate concentrations high compared to enzyme; on/off reaction fast compared to forward reaction rate) and Briggs-Haldane conditions where the On-rate is low. Reducing the [Enzyme] relative to the substrate and product concnetrations reduces the effect of the capacitance of the enzyme-bound forms on the kinetics of the reactions. This results in faster attainment of the steady state. Note 3: Consuming the products to reduce the back flux drains the system; Segel puts all products to zero in order to calculat the M-M equivalent equations. When without a drain, the system --> equilibrium, as in this version. EquilA is the theoretical value. Note 4. Substrate is preserved: Stot: Sum at beginning = Sum at end. Enzyme is conserved: Etest. Note 5. This code converts the model from Random Bi-Bi to Ordered Bi Bi simply by putting an intital binding, k1 or j1 to zero, and likewise putting K_4 or j_4 to zero. Note 6. Segel's approach is instantaneous steady state, i.e. the binding and release rates are exceeding fast, so his reaction velocities, shown by v and by v/Vmax, are reached instantaneously even as concentrations change. Our approach here is to account for the binding and unbinding that actaully take time, and alos to account for the capacitance in the enzymatic reactions by accounting for the substrates an products bound to the enzyme. This is most important with high enyme concentrations or low substrate concnetrations in the range of the binding affinities. Note 7. Observe that EAB achieves a peak concentration at early times when A and B are first sliding through the series of reaction steps starting from A and B with C and D = 0. This peak is always(?) higher than at equilibrium. Simultaneous substrate binding and simultaneous product release bidirectional: Net reaction A + B <----> C + D E = Enzyme concentration k1 k2 k3 k4 A+E <--> EA + B <-->\ /<----> EC + D <---> E + C k_1 k_2 \ / k_3 k_4 EAB=ECD j1 j2 / \ j3 j4 B+E <--> EB + A <-->/ \<----> ED + C <---> E + D j_1 j_2 j_3 j_4
Figure: Progress curves of substrates (A,B), products (C,D), and enzyme (E). Etest is Ebound plus E (unbound). Stot is total amount of substrates and products.
The equations for this model may be viewed by running the JSim model applet and clicking on the Source tab at the bottom left of JSim's Run Time graphical user interface. The equations are written in JSim's Mathematical Modeling Language (MML). See the Introduction to MML and the MML Reference Manual. Additional documentation for MML can be found by using the search option at the Physiome home page.
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These references do not include either the capacitance or the kinetics of binding or release. Segel, I. H. Biochemical Calculations. 2nd Ed.Wiley 1976, p.294. Segel, I. H. Enzyme Kinetics> Behavior and Analysis of RApid Equilibria and Steady-State Enyme Systems Wiley 1993 p 274-319 Bassingthwaighte JB.: Enzymes and Metabolic Reactions, Chapter 10 in "Transport and Reactions in Biological Systems", Pages 7-8
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