Voltage dependence of a stochastic model of activation of an alpha helical S4 sensor in a K channel membrane

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The activation of a K^{+} channel sensor in two sequential stages during a voltage clamp may be described as the translocation of a Brownian particle in an energy landscape with two large barriers between states. A solution of the Smoluchowski equation for a square-well approximation to the potential function of the S4 voltage sensor satisfies a master equation and has two frequencies that may be determined from the forward and backward rate functions. When the higher-frequency terms have small amplitude, the solution reduces to the relaxation of a rate equation, where the derived two-state rate functions are dependent on the relative magnitude of the forward rates (α and γ) and the backward rates (β and δ) for each stage. In particular, the voltage dependence of the Hodgkin-Huxley rate functions for a K^{+} channel may be derived by assuming that the rate functions of the first stage are large relative to those of the second stage-α≫γ and β≫δ. For a Shaker IR K^{+} channel, the first forward and backward transitions are rate limiting (α<γ and δ≪β), and for an activation process with either two or three stages, the derived two-state rate functions also have a voltage dependence that is of a similar form to that determined for the squid axon. The potential variation generated by the interaction between a two-stage K^{+} ion channel and a noninactivating Na^{+} ion channel is determined by the master equation for K^{+} channel activation and the ionic current equation when the Na^{+} channel activation time is small, and if β≪δ and α≪γ, the system may exhibit a small amplitude oscillation between spikes, or mixed-mode oscillation, in which the slow closed state modulates the K^{+} ion channel conductance in the membrane.

Section: -3mentioning

confidence: 69%

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Voltage dependence of Hodgkin-Huxley rate functions for a multistageK+channel voltage sensor within a membrane

Vaccaro

2014

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“…As the relaxation within each deep well is rapid, the probability density may be expressed as the product of the stationary distribution and a survival probability that is the solution of a rate equation [13]. By approximating the potential function for the voltage sensor by a square well potential, the low frequency component of the solution of the Smoluchowski equation may be expressed as differential equations for the survival probabilities of the closed and open states [14,15], and is similar to that obtained from a numerical solution [16].…”

Section: Introductionmentioning

confidence: 99%

Reduction of a kinetic model for Na+ channel activation, and fast and slow inactivation within a neural or cardiac membrane

Vaccaro

2019

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A fifteen state kinetic model for Na+ channel gating that describes the coupling between three activation sensors, a twostage fast inactivation process and slow inactivated states, may be reduced to equations for a six state system by application of the method of multiple scales. By expressing the occupation probabilities for closed states and the open state in terms of activation and fast inactivation variables, and assuming that activation has a faster relaxation than inactivation and that the activation sensors are mutually independent, the kinetic equations may be further reduced to rate equations for activation, and coupled fast and slow inactivation that describe spike frequency adaptation, a repetitive bursting oscillation in the neural membrane, and a cardiac action potential with a plateau oscillation. The fast inactivation rate function is, in general, dependent on the activation variable m(t) but may be approximated by a voltage-dependent function, and the rate function for entry into the slow inactivated state is dependent on the fast inactivation variable. 1 INTRODUCTIONDuring prolonged or repetitive depolarization, in addition to the fast inactivation of Na channels that contributes to repolarization of the membrane [1], a slow inactivation process reduces the number of Na+ channels available for activation. The increase in slow inactivation of Na+ channels during depolarization is associated with a delay to the next spike or a reduction in the firing frequency (spike frequency adaptation) [2] and is the result of a structural rearrangement in the selectivity filter region of the ion channel that generally occurs following the inactivation of the pore [3]. Slow inactivation of the transient and persistent components of the Na+ current in a mesencephalic V neuron is associated with the termination of a bursting oscillation, and the increase in the amplitude of the subthreshold oscillation between bursts occurs during the recovery from slow inactivation [4]. In subicular neurons adjacent to the hippocampus, the transition from bursting to single spiking is influenced by the slow inactivation of Na+ channels, and this may provide a mechanism for enhancing the effect of input signals [5].For Na+ channels with slow inactivation, the Na+ current I N a may be described by the expression m 3 hs(V N a − V ) [2] where V N a is the equilibrium potential, and the activation variable m, the fast inactivation variable h, and the slow inactivation variable s satisfy the equations

“…The master equation may be derived from a Smoluchowski equation applied to the resting and barrier regions of an energy landscape for each of the S4 sensors in the domains DI to DIV [14,15]. The translocation of the S4 segment through the gating pore for Na+ (or K+) channels requires sufficient energy to overcome several barriers that are dependent on the Coulomb force between positively charged residues on the S4 sensor and negatively charged residues on neighboring helices, the dielectric boundary force, the electric field between internal and external aqueous crevices, and hydrophobic forces [16].…”

Section: Voltage Clamp Of a Na+ Channel With Two Ac-tivation Sensorsmentioning

confidence: 99%

Derivation of Hodgkin-Huxley equations for aNa+channel from a master equation for coupled activation and inactivation

Vaccaro¹

2016

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The Na+ current in nerve and muscle membranes may be described in terms of the activation variable m(t) and the inactivation variable h(t), which are dependent on the transitions of S4 sensors of each of the Na+ channel domains DI to DIV. The time-dependence of the Na+ current and the rate equations satisfied by m(t) and h(t) may be derived from the solution to a master equation which describes the coupling between two or three activation sensors regulating the Na+ channel conductance and a two stage inactivation process. If the inactivation rate from the closed or open states increases as the S4 sensors activate, a more general form for the Hodgkin-Huxley expression for the open state probability may be derived where m(t) is dependent on both activation and inactivation processes. The voltage dependence of the rate functions for inactivation and recovery from inactivation are consistent with the empirically determined expressions, and exhibit saturation for both depolarized and hyperpolarized clamp potentials. 1 INTRODUCTIONThe opening and subsequent inactivation of Na+ channels and the activation of K+ channels generate the action potential in nerve and muscle membranes [1]. The time-dependence of the Na+ current in the squid axon may be described in terms of the expression m(t) 3 h(t) where the activation variable m(t) and inactivation variable h(t) satisfy the rate equations