# ðŸ“’ Qi 2009

Contents

Generating rate equations for complex enzyme systems by a computer-assisted systematic method

^{1}

## King and Altman method

- Applying graph theory to enzyme kinetics (state transitions)
- Enzyme internals are in rapid equilibrium
- State transitions are in the pseudo-first order

### Example of fumarase

A = fumarate; B = proton; C = hydroxyl; P = malate;

- listing all of the valid KA patterns for the enzyme mechanism.

2. And determine all of the directional diagrams associated with each state in the enzyme mechanism.

- The arrows are directed toward state i with no diverging edges.
- Each directional diagram is associated with a product of pseudo-first order rate constants for the arrows in the directional diagram.
- Numerator are the terms pointing to one state, and the denominator is the sum of all terms associated with the directional diagrams for all states in the system.
- the method used to generate trees from linear graphs can be applied to complex enzymatic reaction mechanisms

## Motivation and the software

- Manually deriving the steady-state rate equations for non-trivial enzyme mechanisms can be cumbersome and error-prone.
- MATLAB GUI, KAPattern, for generating rate equations in complex enzyme systems, using the KA method as well as the topological theory of linear graphs, called Wang Algebra.
- No limitation on the size of the system
- output as .m or Math ML file
- provides visualization of all the valid KA patterns.
- may obtain insights on catalytic mechanism
- it can deal with the irreversible reaction step

## Algorithm

- Method described by Lam and Priest, with Wang alebra, the addition or multiplication operation on two or more identical elements leads to zero.
- Link matrix(similar to adjecency matrix) and kinetic matrix (with rates)
- randomly selecting n - 1 nodes from the linear graph, and determining the links connected to the n -1 nodes by deleting a row (column) and then listing separately all the nonzero entries from the remaining n - 1 rows (columns).
- the links listing obtained in previous step are alphanumerically multiplied
- 1 * 2 = > 12

- the links (edges) in each KA pattern are assigned directions so that the reaction steps, individually or in sequence, lead to a given enzyme state Ei.

## Reference

Qi F, Dash RK, Han Y, Beard DA. Generating rate equations for complex enzyme systems by a computer-assisted systematic method. BMC Bioinformatics. 2009;10:238. Published 2009 Aug 4. doi:10.1186/1471-2105-10-238 PMC2729780. ↩︎