Applications of graph theory to enzyme kinetics and protein folding kinetics. Steady and non-steady-state systems

Graphic methods have proved to be very useful in enzyme kinetics, as reflected in both raising the efficiency of performing calculations and aiding in the analysis of catalytic mechanisms. The kinetic relations among protein folding states are very similar to those between enzyme-catalyzed species. Therefore, it should be equally useful to provide a visually intuitive relation between kinetic calculations and folding mechanisms for protein folding kinetics, as manifested by the graphic rules in enzyme kinetics. It can actually be anticipated that, due to increasing interest in protein folding, the graphic method will become an important tool in folding kinetics as well. Based on the recent progress made in graphic methods of enzyme kinetics, in this review four graphic rules are summarized, which can be used to deal with protein folding systems as well as enzyme-catalyzed systems. Rules 1-3 are established for deriving the kinetic equations for steady-state processes and Rule 4 for those in the case of non-steady-state processes. In comparison with conventional graphic methods, which can only be applied to a steady-state system, the current rules have the following advantages: (1) Complicated and tedious calculations can be greatly simplified. (2) A lot of wasted labor can be turned away. (3) Final results can be double-checked by a formula provided in each of the graphic rules. (4) Transient kinetic systems can also be treated. The mathematical proof of Rules 1-4 is given in appendices A-D, respectively.