Excitability constraints on voltage-gated sodium channels

Abstract

We study how functional constraints bound and shape evolution through an analysis of mammalian voltage-gated sodium channels. The primary function of sodium channels is to allow the propagation of action potentials. Since Hodgkin and Huxley, mathematical models have suggested that sodium channel properties need to be tightly constrained for an action potential to propagate. There are nine mammalian genes encoding voltage-gated sodium channels, many of which are more than approximately 90% identical by sequence. This sequence similarity presumably corresponds to similarity of function, consistent with the idea that these properties must be tightly constrained. However, the multiplicity of genes encoding sodium channels raises the question: why are there so many? We demonstrate that the simplest theoretical constraints bounding sodium channel diversity--the requirements of membrane excitability and the uniqueness of the resting potential--act directly on constraining sodium channel properties. We compare the predicted constraints with functional data on mammalian sodium channel properties collected from the literature, including 172 different sets of measurements from 40 publications, wild-type and mutant, under a variety of conditions. The data from all channel types, including mutants, obeys the excitability constraint; on the other hand, channels expressed in muscle tend to obey the constraint of a unique resting potential, while channels expressed in neuronal tissue do not. The excitability properties alone distinguish the nine sodium channels into four different groups that are consistent with phylogenetic analysis. Our calculations suggest interpretations for the functional differences between these groups.

Figure 1. Experimental Characterization of Sodium Channels Used in This Paper

A step change in the membrane potential from a very negative value (for t < 0) where all the channels are closed, to V (at t = 0) results in a measurable current. The left panel shows the current for V = 0 mV. The activation curve (upper right) is constructed by plotting the maximum current as a function of V, normalized by the maximum current as V → ∞. The inactivation curve (lower right) is the steady state current as a function of V, also normalized by the steady state value as V → ∞. This figure was produced using Kuo and Bean's model for sodium channels [56].

Figure 2. Current J(V/ ¯_K Given by Equation 3 for Different Values of

(A) The sodium channel is characterized by ()=(−50 mV, 6 mV), and the dot-dashed, dotted, dashed, and solid lines correspond to Θ = 0.5, 1, 5, 20. For Θ = 1, 5, there are multiple fixed points (i.e., zero crossings), and hence this channel is excitable. (B) The sodium channel is characterized by ()=(−80 mV, 6 mV), and the dot-dashed, dotted, dashed, and solid lines correspond to Θ = 0.1, 0.2, 0.4, 0.8. There is a single fixed point for each, and hence for this channel excitability is impossible.

Figure 3. Schematic Showing Projection of Activation and Inactivation Curve Parameters onto the Same Two-Dimensional Space (V _1/2, k)

The activation and inactivation curve parameters are predicted to lie on opposite sides of the excitability threshold given by the line V _1/2 = V _K + 2k.

Figure 4. Summary of Activation and Inactivation Data for Human and Rat Voltage-Gated Sodium Channels

The left plots (A) and (C) show inactivation data for human and rat, while the right plots (B) and (D) show activation data for human and rat. The black symbols represent the neuronal channels (Na_v1.1, 1.2, 1.3, 1.6, 1.7, 1.8, 1.9) and the red symbols represent the muscular channels (Na_v1.4, 1.5). The solid line is the excitability threshold. The activation data is predicted to lie in the shaded region.

Figure 5. Summary of Activation and Inactivation Data for Human and Rat Voltage-Gated Sodium Channels

The left plots (A) and (C) show inactivation data for human and rat. The right plots (B) and (D) show activation data for human and rat. The different colors represent different channel types, with blue, green, red, cyan, magenta, yellow, black, orange, grey representing Na_v1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, respectively. The solid line is the excitability threshold. The activation data is predicted to lie in the shaded region.

Figure 6. Activation Data for Human (Squares) and Rat (Circles), for Different Channel Types and Conditions

As above, the different colors represent different channel types, with blue, green, red, cyan, magenta, yellow, black, orange, grey representing Na_v1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, respectively. The thick black line is the excitability threshold; the thin dark blue, light blue, orange, and red lines represent voltage thresholds of k_BT/e, 2k_BT/e, 3k_BT/e, and 4k_BT/e, respectively.

Figure 7. Correlation between and for Channels Colored by Four Different Groups

The four groups are: (i) the non-muscular channels (Na_v1.1, 1.2, 1.3, 1.6, 1.7) (black), (ii) muscular channels (Na_v1.4, 1.5) (red), (iii) the channel Na_v1.8 (blue), and (iv) the channel Na_v1.9 (green). The solid line shows . This plot contains all data for which we have measurements of both inactivation and activation properties, including wild-type, mutant, and different conditions for human, rat, and mouse.

Figure 8. Correlation between k ^act and k ^inact

As above, the different colors represent different channel types, with blue, green, red, cyan, magenta, yellow, black, orange, grey representing Na_v1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, respectively. This plot contains all data for which we have measurements of both inactivation and activation properties, including wild-type, mutant, and different conditions for human, rat, and mouse.

Figure 9. Activation (Circles) and Inactivation (Triangles) for Human Voltage-Gated Sodium Channels Na_v1.4 (Blue) and Na_v1.5 (Red) from [55]

The solid line is the excitability threshold.