Phasic activation and state-dependent inhibition: An explicit solution for a three-state ion channel system

Ion channels can exist in three broad classes of states: closed (C), open (O), and desensitized or inactivated (I). Many ion channel modulators interact preferentially with one of these states giving rise to use or state dependent effects and often complex interactions with phasic stimulation. Although mathematical descriptions of three-state systems at steady-state or following a single perturbation are well known, a solution to the boundary problem of how such a system interacts with regular phasic perturbations or stimuli has not previously been reported. In physiological systems, ion channels typically experience phasic stimulation and an explicit mathematical description of the interaction between phasic activation and use-dependent modulation within the framework of a three-state system should be useful. Here we present derivations of generalized, recurrent and explicit formulae describing this interaction that allow prediction of the degree of use dependent modulation at any point during a train of repeated stimuli. Each state is defined by two functions of time (y or z) that define the fraction of channels in that state during the alternating stimulation and resting phases, respectively. For a train of repeated stimuli we defined vector Z(2n) that has coordinates Z(2n)(O) and Z(2n)(I) representing the values for O and I states at the end of the n-th resting phase. We then defined a recurrent relationship, Z(2n) = FZ(2n-2) + G. Therefore, for the steady state: Z = (E-F)^-1G, where F = (C₁₁C₁₂C₂₁C₂₂), G = (C₁₃C₂₃), E is the identity matrix. Matrix and vector elements, c(ij), are defined in terms of duration of the repeated stimulation and resting phases and the two sets of six rate constants that describe the three-state model during those two phases. Several conclusions can be deduced from the formulation: (1) in order to determine an occupancy of any state under the cyclic stimulus-rest protocol it is necessary to know at least two occupancy levels - either of the same state but related to different phases of the stimulus protocol or of different states at the same point in the stimulus protocol, for instance: Z(2n)(I) = f(Z(2n-2)(O), Z(2n-2)(I)) = h(Z(2n)(O), Z(2n-2)(O)) = g(Z(2n)(O), Z(2n-2)(I)) = ...; (2) the solution Z(2n) can be approximated by a matrix-exponential function, with the precision of the approximation depending on the interval between stimuli; (3) for all steady-state solutions, the matrix F is such that lim n→∞nΠF = ∞ΠF = 0 is a zero-matrix. Application of this approach is illustrated using experimentally derived parameters describing desensitization of GABA(a) receptors and modulation of that process by the anesthetic propofol.