Fluctuations and noise in kinetic systems. Application to K+ channels in the squid axon

We consider the equilibrium or steady-state noise power density spectrum in the quantity N = Sigma(x) (i=0)a(i)N(i) for an ensemble of independent and equivalent systems each of which can exist in the discrete set of states i = 0, 1, ..., x. N(i) is the number of systems of the ensemble in state i and the a(i)'s are constants. There is a transition rate constant alpha(ij) for an arbitrary transition i --> j; the kinetic equations are linear. There are possible applications to enzyme and biochemical kinetics generally, to membrane transport, muscle contraction, binding on macromolecules, etc. In each case, noise measurements would provide information about the kinetic scheme. The particular application considered here is to K(+) channels or gates (one channel = one system) in the squid axon membrane: a(i)g(K) is the K(+) conductance of a channel in state i and the kinetic scheme is of the Hodgkin-Huxley type (HH). Here we allow an arbitrary set of a(i)'s. This is a generalization of our treatment of K(+) channel noise in an earlier paper. The theory is discussed and some calculations made using Fishman's recent experimental results on K(+) channel noise as a guide. Preliminary indications are that the HH choice of a(i)'s may be oversimplified and that a(0) congruent with 0, a(1) not equal a(0), a(x) not equal a(x-1). Quite possibly the a(i)'s increase from a(0) to a(x), though the early a(i)'s must be relatively small to give the observed induction behavior in g(K)(t). An increase in equal steps is unsatisfactory because this is essentially HH with x = 1 (no induction). More refined experiments may modify these tentative conclusions. In any case, it appears from Fishman's work that noise measurements will probably be very useful in distinguishing between rival models of K(+) channels.