Nonlinear drift-diffusion model of gating in the fast Cl channel

Vaccaro, S. R.

doi:10.1103/physreve.76.011923

Cited by 7 publications

(9 citation statements)

References 28 publications

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“…͑7͒, ͑9͒, and ͑13͒ are a more general form of the assumptions previously considered in the derivation of a rate equation ͑T m Ϸ m / 2 when U mb is small͒. 24 Relation ͑20͒ is satisfied if U bc is sufficiently large for each membrane depolarization, and from Eqs. ͑14͒ and ͑15͒, the lowest frequency…”

Section: ͑20͒mentioning

confidence: 99%

“…18 An analytical solution of the stochastic diffusion equation also has an exponential relaxation for a large ramp potential, but as the voltage dependent barrier is attenuated, the response to a membrane depolarization is characterized by a power law for intermediate times, and therefore is in accord with the distribution of closed times recorded during a patch clamp of an ion channel. [19][20][21][22][23][24] If it is assumed that the diffusion barrier is sufficiently large at the entrance to the protein region of the gating pore, the activation process may be described by a rate equation for each membrane depolarization. 25 In this paper, an expression for the energy of an S4 helix is derived which is dependent on the transverse displacement and rotation of the sensor within a gating pore.…”

Section: Introductionmentioning

confidence: 99%

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Stochastic diffusion model of multistep activation in a voltage-dependent K channel

Vaccaro

2010

The Journal of Chemical Physics

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The energy barrier to the activated state for the S4 voltage sensor of a K channel is dependent on the electrostatic force between positively charged S4 residues and negatively charged groups on neighboring segments, the potential difference across the membrane, and the dielectric boundary force on the charged residues near the interface between the solvent and the low dielectric region of the membrane gating pore. The variation of the potential function with transverse displacement and rotation of the S4 sensor across the membrane may be derived from a solution of Poisson's equation for the electrostatic potential. By approximating the energy of an S4 sensor along a path between stationary states by a piecewise linear function of the transverse displacement, the dynamics of slow activation, in the millisecond range, may be described by the lowest frequency component of an analytical solution of interacting diffusion equations of Fokker-Planck type for resting and barrier regions. The solution of the Smoluchowski equations for an S4 sensor in an energy landscape with several barriers is in accord with an empirical master equation for multistep activation in a voltage-dependent K channel.

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Section: ͑20͒mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Stochastic diffusion model of multistep activation in a voltage-dependent K channel

Vaccaro

2010

The Journal of Chemical Physics

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“…Of course as the number of states goes up, their exponents become indistinguishable, usually forming a power-law distribution. Some information can still be gained from such a distribution, although not of the detail as obtained from exponentials (9,29,32,37,40,44). Both the exponential and power-law distribution approaches have been applied to properly aggregated states (i.e., a single open or closed conductance).…”

Section: Discussionmentioning

confidence: 98%

Gating of maxi channels observed from pseudo-phase portraits

Parsons

Huizinga

2013

American Journal of Physiology-Cell Physiology

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Phase space has been used to visualize and analyze the dynamic behavior of stochastic and chaotic systems. We applied this concept to maxi channels recorded from excised inside-out patches of in situ interstitial cells of Cajal. Pseudo-phase portraits of channel current were fairly homogeneous from patch to patch. They showed three main peaks, α, β, and γ, in increasing conductance. These represented single or near aggregated states. The α-peak was the closed state. The β-peak was small, consisting of a single conductance state, or in some cases two (a doublet). The β-peak state(s) had a long lifetime and displayed a characteristic behavior of frequent short transitions to γ but not to α. It was always preceded by a short series of α/γ-transitions. The γ-peak was the largest and consisted of a large number of conductance states with fast state transitions, sometimes to the extent of causing a diffusive-type behavior. Phase portraits allowed us to construct a provisional gating scheme for the maxi channel and suggest that further analysis of recordings in higher dimensional phase space and with related techniques may be promising.

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“…The expression for α n in Eq. (2) may be obtained from a solution to the Smoluchowski equation for the probability density of states of the voltage sensor, when the potential function is linear in the transverse coordinate Z [6,7], and a rate equation for activation may be derived if there is a large diffusion or potential barrier between closed and open states [8]. However, in view of the presence of negative residues on the S2 and S3 segments within the voltage sensing domain (VSD), as well as induced charge at the dielectric boundary of the membrane, the potential function for the S4 sensor is a nonlinear function of Z for each potential V [9,10].…”

Section: Introductionmentioning

confidence: 99%

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

Vaccaro

2014

Phys. Rev. E

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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.

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Nonlinear drift-diffusion model of gating in the fast Cl channel

Cited by 7 publications

References 28 publications

Stochastic diffusion model of multistep activation in a voltage-dependent K channel

Stochastic diffusion model of multistep activation in a voltage-dependent K channel

Gating of maxi channels observed from pseudo-phase portraits

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

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