📒 Qi 2009

Generating rate equations for complex enzyme systems by a computer-assisted systematic method1


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;

  1. 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


  • 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.


  1. 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↩︎