Ion channel gating: a first-passage time analysis of the Kramers type

Abstract

The opening rate of voltage-gated potassium ion channels exhibits a characteristic knee-like turnover where the common exponential voltage dependence changes suddenly into a linear one. An explanation of this puzzling crossover is put forward in terms of a stochastic first passage time analysis. The theory predicts that the exponential voltage dependence correlates with the exponential distribution of closed residence times. This feature occurs at large negative voltages when the channel is predominantly closed. In contrast, the linear part of voltage dependence emerges together with a nonexponential distribution of closed dwelling times with increasing voltage, yielding a large opening rate. Depending on the parameter set, the closed-time distribution displays a power law behavior that extends over several decades.

Figure 1

Dependence of opening (k_o) and closing (k_c) rates on voltage for a Shaker IR K⁺ channel from ref. at T = 18°C. The opening rate is described by Eq. 1 with the following parameters (6): a_c = 0.03 msec/mV, b_c = 0.8 mV⁻¹, and V_c = −46 mV. The closing rate is given by k_c = 0.015 exp(−0.038V) msec⁻¹ (V in mV) (6, 7). Inset shows the same dependencies on a semilogarithmic scale.

Figure 2

Gating dynamics as an activated diffusion on a complex free energy landscape. Two global minima correspond to open and closed macroconformations. One assumes a large number of quasidegenerate (within k_BT) and voltage-independent closed substates separated from the open conformation by a voltage-dependent potential barrier. This idea is sketched by a simplified model of the Fokker–Planck–Kramers type, and by a discrete model with open (O), closed (C), and inactivated (I) states.

Figure 3

Studied model and its diffusion counterpart.

Figure 4

Closed residence time distribution for a diffusion-limited case. The numerical precise result (solid line) is compared with the analytical approximation in Eqs. 15–17 (broken line). The latter one coincides with the exact solution of the diffusion model by Millhauser et al. (12) in the scaling limit.