An algorithmic method for determining the kinetic system of receptor-channel complexes

The mathematical study of receptor-channel kinetics involving numerous sites and conformations of the channel calls for specific analytic methods generally based on stochastic formulation in terms of Markov processes. These methods allow the determination of the number of states from the experimental data. When the number of states is known, it is necessary to try numerous kinetic diagrams to find the best one. The construction of the kinetic diagram and the corresponding kinetic system are based on physiological hypotheses. When the number of states is large, the kinetic schema becomes difficult to establish. We present a method that uses an algorithmic scheme to deduce a kinetic system directly from physiological hypotheses. This method takes into account any number of ligands and sites. The set of all the states given by the combination of site occupation and channel conformations is reduced by using two types of hypothesis: (1) molecular constraints that specify the transitions physically possible between states and (2) kinetic considerations related to the assumed physiology of the system, which gives the conditions necessary for a transition between two states. These hypotheses are expressed in terms of rules operating on the initial states of transitions. The expression of rules does not ensure their coherence (i.e., the fact that each kinetic transition is defined by one and only one rule). A mathematical condition has been found that ensures the coherence of rules. When coherence has been established, the corresponding dynamic system can be automatically generated. Because the rules are established in a systematic way and their coherence can be mathematically established, the computer implementation of this method makes it easy to test various kinetic hypotheses for problems where the number of states is large.