A single Markov-type kinetic model accounting for the macroscopic currents of all human voltage-gated sodium channel isoforms

Abstract

Modelling ionic channels represents a fundamental step towards developing biologically detailed neuron models. Until recently, the voltage-gated ion channels have been mainly modelled according to the formalism introduced by the seminal works of Hodgkin and Huxley (HH). However, following the continuing achievements in the biophysical and molecular comprehension of these pore-forming transmembrane proteins, the HH formalism turned out to carry limitations and inconsistencies in reproducing the ion-channels electrophysiological behaviour. At the same time, Markov-type kinetic models have been increasingly proven to successfully replicate both the electrophysiological and biophysical features of different ion channels. However, in order to model even the finest non-conducting molecular conformational change, they are often equipped with a considerable number of states and related transitions, which make them computationally heavy and less suitable for implementation in conductance-based neurons and large networks of those. In this purely modelling study we develop a Markov-type kinetic model for all human voltage-gated sodium channels (VGSCs). The model framework is detailed, unifying (i.e., it accounts for all ion-channel isoforms) and computationally efficient (i.e. with a minimal set of states and transitions). The electrophysiological data to be modelled are gathered from previously published studies on whole-cell patch-clamp experiments in mammalian cell lines heterologously expressing the human VGSC subtypes (from NaV1.1 to NaV1.9). By adopting a minimum sequence of states, and using the same state diagram for all the distinct isoforms, the model ensures the lightest computational load when used in neuron models and neural networks of increasing complexity. The transitions between the states are described by original ordinary differential equations, which represent the rate of the state transitions as a function of voltage (i.e., membrane potential). The kinetic model, developed in the NEURON simulation environment, appears to be the simplest and most parsimonious way for a detailed phenomenological description of the human VGSCs electrophysiological behaviour.

Fig 1. Electrophysiological protocols commonly used in the considered studies.

A: Series of depolarizations generated from a holding potential, ranging from less to more depolarized values, used for recording current-voltage curves, voltage-peak relationship, conductance-voltage relationship. B: Conditioning (pre-pulse) long depolarizations at different voltages followed by a test depolarization, used for studying the voltage dependence of steady-state inactivation (availability). C: A conditioning long depolarization followed by repolarizations of increasing duration, before the test depolarization, used for studying the recovery from inactivation (repriming).

Fig 2. Comparison between experimental and modelled data in Na_V1.5.

A: Voltage-clamp curves from -90 mV to 60 mV in steps of 5 mV. B: Voltage dependence of the normalized conductance. C: Steady-state availability during fast inactivation. D: Recovery from fast and slow inactivation. E: Development of slow inactivation. F: Availability curves. G: Deactivation curves. Left column: modelled data. Right column: experimental data. Experimental data reproduced from [27], under the terms of the Creative Commons Attribution License.

Fig 3. State diagram and transitions of the proposed six-state kinetic model.

Fig 4. Different formal mathematical descriptions of transition rates.

A: Curves drawn from the original equations by Hodgkin and Huxley [3] for the sodium channel rate constants α_m (red line), β_m (similar also to α_h) (black line), and β_h (blue line). B: Modified sigmoid curve adopted in the present study (see text for details). Inset: Sigmoid curve with minimum and maximum asymptotes.

Fig 5. Features of activation sequence in Na_V1.2 and Na_V1.9.

Modelled Na_V1.2 (A) and Na_V1.9 (B) voltage-clamp curves from -80 mV to 70 mV in 5 mV increments, from a -120 mV holding potential. Na_V1.2 (C) and Na_V1.9 (D) transition rates dependence from voltage of the activation sequence (comprehensive of fast inactivation). Green: C1 to C2; yellow: C2 to C1; red: C2 to O1; purple: O1 to C2; black: O1 to I1.

Fig 6. Recovery from inactivation in Na_V1.2.

A: Superimposed simulated current traces recorded with increasing duration of repolarization interval. B: Time dependent fractional recovery curve from fast inactivation. C: Fraction of channels in the inactivated states I1 (red) or I2 (blue) as brought about by the repriming protocol depicted in the inset.

Fig 7. Other electrophysiological features.

A: Decay from peak of activation during a voltage-clamp of -10 mV in Nav 1.1 modelled channel. The curve between the two vertical blue lines has been fitted to a double exponential (see text for details). B: Simulated deactivation curves in Na_V1.5 following the protocol depicted in the inset. C: Simulated persistent current in Na_V1.6.

Fig 8. Electrophysiological features of a classical HH model of sodium channel.

A: Voltage-clamp curves from -80 mV to 60 mV in step of 10 mV. B: Voltage dependence of the normalized conductance. C: Voltage dependence of normalized current during fast inactivation. D: Recovery from fast inactivation. E: Steady-state availability during slow inactivation. F: Recovery from slow inactivation.

Fig 9. Comparison between an HH model and a ‘hybrid’ one.

A: A single spike recorded at the soma (black trace) and initial segment (red trace) after a 75 nA, 1 ms long electrical stimulus at soma, in the neuron model by Dodge and Cooley [38]. B: A single spike recorded at the soma (black trace) and initial segment (red trace) after a 100 nA, 1 ms long electrical stimulus at soma, in the neuron model equipped with the Na_V1.6 kinetic model. C: A series of spikes recorded at the soma following a repetitive stimulation in the HH model. D: Same recording and stimulating parameters as in C, in a ‘hybrid’ model. See text for details.