# π Modeling of enzyme kinetics: a summary

Contents

Modeling of enzyme kinetics

## Basic principles

For modelling enzyme catalytic cycle

• Structual and stiochiometric data
• Rate equations of elementary reactions
• Reaction graph (cycle) of enzyme stages
• Reduction of rate equations
• Law of mass equation1
• Quasi-equilibrium (slow-fast kinetics)
• King-Altman method23
• Py-substitution4
• Satisfies available experimental data
• Literature of kinetic constants (Km, Ki, Keq, free energy)
• Databases: BRENDA5
• Experiments of both V0 and time series
• Dependence of substrate, product, pH, and temperature

## Examples

Slide Player example

Tilde (~) means specific concentrations ($x / K_x$)

### Random Bi-Bi

\begin{aligned} v &= \frac{V_f\widetilde{A} \widetilde{B} - V_b\widetilde{P} \widetilde{Q}}{1 + \widetilde{A} + \widetilde{B} + \widetilde{A} \widetilde{B} + \widetilde{P} + \widetilde{Q} + \widetilde{P} \widetilde{Q}} \end{aligned}

### Ordered Bi-Bi

E <=> EA <=> EAB <-> EPQ <=> EQ <=> E Cleland \begin{aligned} v &= \frac{V_f\widetilde{A} \widetilde{B} - V_b\widetilde{P} \widetilde{Q}}{1 + \widetilde{A} + \alpha_A\widetilde{B} + \alpha_Q\widetilde{P} +\widetilde{Q} + \alpha_Q\widetilde{A}\widetilde{P} + \alpha_A\widetilde{B}\widetilde{Q} + \widetilde{P} \widetilde{Q} + \widetilde{A} \widetilde{B} \widetilde{P} + \widetilde{B} \widetilde{P} \widetilde{Q} } \cr \widetilde{A} &= \frac{a}{K_i^A} \cr \widetilde{B} &= \frac{b}{K_m^B} \cr \widetilde{P} &= \frac{p}{K_i^P} \cr \widetilde{Q} &= \frac{q}{K_m^Q} \cr \alpha_A &= \frac{K_m^A}{K_i^A} \cr \alpha_Q &= \frac{K_m^Q}{K_i^Q} \cr \end{aligned}

### Theorell-Chance mechanism

The Theorell-Chance mechanism is a simplified version of an ordered mechanism where the steady-state level of central complexes is very low. Therefore, the rate equatioin is identical with the ordered mechanism, except that the terms in ABP and BPQ are missing from the denominator.6

### Ping Pong Bi-Bi

E <=> EA <=> FP <=> F <=> FB <=> EQ <=> E

\begin{aligned} v &= \frac{V_f\widetilde{A} \widetilde{B} - V_b\widetilde{P} \widetilde{Q}}{\widetilde{A} + \alpha_A\widetilde{B} + \widetilde{P} + \alpha_P\widetilde{Q} + \widetilde{A}\widetilde{B} + \widetilde{A}\widetilde{P} + \alpha_A\widetilde{B}\widetilde{Q} + \widetilde{P} \widetilde{Q} } \cr \widetilde{A} &= \frac{a}{K_i^A} \cr \widetilde{B} &= \frac{b}{K_m^B} \cr \widetilde{P} &= \frac{p}{K_i^P} \cr \widetilde{Q} &= \frac{q}{K_m^Q} \cr \alpha_A &= \frac{K_m^A}{K_i^A} \cr \alpha_P &= \frac{K_m^P}{K_i^P} \cr \end{aligned}

## Allosteric enzymes

### Hill models

\begin{aligned} \frac{v}{V_{max}} &= \frac{\phi}{1 + \phi} \cr \phi &= (\frac{x}{k})^n \end{aligned}

### MWC models

\begin{aligned} v &= \frac{nV_S(1 + \frac{V_T}{V_S}Q)}{1+Q} \cr Q &= L_0 \left( \frac{1 + Effector / K_{ef}^T}{1 + Effector / K_{ef}^R} \frac{E_R}{E_T} \right)^n \end{aligned}

## Reference

1. King, E. L., & Altman, C. (1956). A Schematic Method of Deriving the Rate Laws for Enzyme-Catalyzed Reactions. The Journal of physical chemistry, 60(10), 1375β1378. ↩︎

2. Qi, F., Dash, R. K., Han, Y., & Beard, D. A. (2009). Generating rate equations for complex enzyme systems by a computer-assisted systematic method. BMC Bioinformatics, 10, 238. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2729780 ↩︎

3. Loriaux, P. M., Tesler, G., & Hoffmann, A. (2013). Characterizing the relationship between steady state and response using analytical expressions for the steady states of mass action models. PLoS Computational Biology, 9(2), e1002901. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3585464 ↩︎

4. Leskovac, V., TriviΔ, S., Pericin, D., & Kandrac, J. (2006). Deriving the rate equations for product inhibition patterns in bisubstrate enzyme reactions. Journal of Enzyme Inhibition and Medicinal Chemistry, 21(6), 617β634. https://www.tandfonline.com/doi/full/10.1080/14756360600829381?scroll=top&needAccess=true& ↩︎