PHASIC ACTIVATION AND STATE-DEPENDENT INHIBITION - AN EXPLICIT SOLUTION FOR A 3-STATE ION-CHANNEL SYSTEM

Ion channels can exist in three broad classes of states: closed (C), o pen (O), and desensitized or inactivated (I). Many ion channel modulat ors interact preferentially with one of these states giving rise to us e or state dependent effects and often complex interactions with phasi c stimulation. Although mathematical descriptions of three-state syste ms 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 repor ted. In physiological systems, ion channels typically experience phasi c stimulation and an explicit mathematical description of the interact ion between phasic activation and use-dependent modulation within the framework of a three-state system should be useful. Here we present de rivations of generalized, recurrent and explicit formulae describing t his interaction that allow prediction of the degree of use dependent m odulation 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 resti ng phases, respectively. For a train of repeated stimuli we defined ve ctor (Z) over right arrow(2n) that has coordinates z(2n)(0) and z(2n)( 1), representing the values for O and I states at the end of the n-th resting phase. We then defined a recurrent relationship, (Z) over righ t arrow(2n) = F (Z) over right arrow(2n-2) + (Z) over right arrow Ther efore, for the steady state: Z = (E - F)(-1)(G) over right arrow, wher e [GRAPHICS] E is the identity matrix. Matrix and vector elements, c(i j), are defined in terms of duration of the repeated stimulation and r esting 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 t o know at least two occupancy levels-either of the same state but rela ted to different phases of the stimulus protocol or of different state s 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(2 n-2)(I)) = ...; (2) the solution (Z) over right arrow(2n) can be appro ximated by a matrix-exponential function, with the precision of the ap proximation depending on the interval between stimuli; (3) for all ste ady-state solutions, the matrix F is such that [GRAPHICS] is a zero-ma trix. Application of this approach is illustrated using experimentally derived parameters describing desensitization of GABA(a) receptors an d modulation of that process by the anesthetic propofol. (C) 1996 Acad emic Press Limited