EA and MPB conceived and designed the experiments, analyzed the data, and wrote the paper.
The authors have declared that no competing interests exist.
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 ≈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.
There are few quantitative examples for how functional constraints bound and shape evolution. Sodium channels are a central player in the propagation of action potentials. Action potentials fire above a critical voltage threshold. Below the voltage threshold the membrane potential recovers to a resting value, which is assumed to be unique. Here we ask whether the properties of mammalian voltage-gated sodium channels are determined by the simplest possible constraints. We demonstrate that the requirements, (1) a voltage threshold and (2) a unique resting potential, severely constrain sodium channel properties. These constraints contain no free parameters, depending only on the concentrations of potassium inside and outside the cell. We test these predictions on functional data from the nine mammalian genes encoding voltage-gated sodium channels. All measurements obey the excitability constraint, whereas channels expressed in the nervous system systematically violate the constraint for a unique resting potential. These properties alone distinguish the nine sodium channels into four groups consistent with phylogenetic analysis. Our calculations suggest that different channel types have evolved to perform different tasks.
Despite the relatively small number of genes in the human genome, there are many examples of groups of nearly identical genes that perform similar functions. Such diversity could either reflect redundancy or evolutionary specialization [
Here we explore how functional constraints bound and shape evolution through an analysis of mammalian voltage-gated sodium channels. The primary function of voltage-gated sodium channels is to allow the propagation of action potentials [
Primary Locations of Mammalian Voltage-Gated Sodium Channels Nav1.1–1.9
In this paper, we address the questions of whether and how sodium channel diversity is bounded by the simplest theoretical constraints on action potential propagation: (i) the sodium channel properties must be tuned to allow the membrane to be excitable, i.e., there must exist a voltage threshold above which an action potential can be produced, and (ii) the constraint of a unique resting potential. Through a theoretical analysis of macroscopic sodium currents, we demonstrate that these two requirements depend only on sodium channel properties, directly constraining the activation and inactivation curves of sodium channels, which are routinely directly measured in experiments. We then compare the constraints with measurements of mammalian sodium channels reported in the literature. Our dataset uses 172 different measurements from 40 distinct publications, including both wild-type and mutant Nav1.1–1.9, in human, mouse, and rat, under a range of different conditions including with and without different types of β subunits, and with chemicals (lidocaine, tetrodotoxin, etc.) [
Our analysis demonstrates that excitability properties alone distinguish the nine sodium channels into four different groups. Within each group there is a strong positive correlation between the voltage dependence of activation and inactivation. The members of each of the four groups are close according to phylogenetic analysis [
How are the properties of a voltage-gated sodium channel constrained by its function? The primary role of voltage-gated sodium channels is to make action potentials. Action potential generation corresponds to two fundamental requirements on the sodium channels. First, sodium channel properties must allow for the membrane to be excitable; namely there must exist a voltage threshold above which action potentials can be produced. Second, the sodium channel properties must give rise to a unique stable resting potential, where the sodium and potassium currents are in steady state. In the following we show that these criteria constrain sodium channel properties which are directly measured in routine experiments used to characterize sodium channels.
The basic model for an action potential was introduced by Hodgkin and Huxley [
First we consider the constraints arising from excitability. The existence of a voltage threshold, above which an action potential can occur, can be analyzed by approximating
Given these assumptions, we now ask, under what conditions is the membrane excitable? In particular, for a sodium channel whose open probability is characterized by the the function
Excitability requires multiple solutions to
We now demonstrate that the requirement that there is a unique resting potential also imposes a constraint on sodium channel properties. The resting potential is determined by
The requirement that there is a unique value of the resting potential thus translates into the requirement that there is only a single solution to the equation
We have thus demonstrated that the excitability constraint and uniqueness of the resting potential present essentially the same mathematical problem and depend only on the sodium channel properties represented by the probability of a channel opening
Both of these quantities are measured with a common experimental protocol, shown in
A step change in the membrane potential from a very negative value (for
The Boltzmann functions corresponding to the activation and inactivation curves
Since both the activation and inactivation curves are given by Boltzmann functions as shown in
We first consider the excitability constraint on the activation curve.
(A) The sodium channel is characterized by (
(B) The sodium channel is characterized by (
We have carried out a mathematical analysis that follows from the excitability constraint given by
The activation
The predicted constraints on sodium channels can be directly compared with measured sodium channel properties. We have collected activation and inactivation curve data of the various sodium channels from papers in the recent literature. The dataset includes papers where (
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 (Nav1.1, 1.2, 1.3, 1.6, 1.7, 1.8, 1.9) and the red symbols represent the muscular channels (Nav1.4, 1.5). The solid line is the excitability threshold. The activation data is predicted to lie in the shaded region.
On the other hand, the inactivation curves do not all obey the constraint given by the uniqueness of the resting potential. For the assumed
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 Nav1.1, 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.
We now extend the excitability argument one step further, and demonstrate that if
For each
We can use this fact to explicitly compute the maximum voltage threshold as a function of
As above, the different colors represent different channel types, with blue, green, red, cyan, magenta, yellow, black, orange, grey representing Nav1.1, 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
The cardiac channels Nav1.5 (magenta symbols) have the lowest voltage thresholds at around 2
What about the range of Θ where excitability is possible? As Θ increases from Θ
The data presented so far demonstrates systematic differences in (
The four groups are: (i) the non-muscular channels (Nav1.1, 1.2, 1.3, 1.6, 1.7) (black), (ii) muscular channels (Nav1.4, 1.5) (red), (iii) the channel Nav1.8 (blue), and (iv) the channel Nav1.9 (green).
The solid line shows
In this representation it is clear that the channels break into four different groups: (i) the channels Nav1.1, 1.2, 1.3, 1.6, 1.7, which are primarily expressed in nervous tissue (black); (ii) those expressed primarily in muscle, Nav1.4, 1.5 (red); (iii) the channel Nav1.8 (blue); and (iv) the channel Nav1.9 (green).
This grouping of the channels has been previously observed in phylogenetic analyses. Plummer and Meisler [
Strikingly, within each group there is a strong correlation between
The offset parameter
One might imagine that the clustering observed in
As above, the different colors represent different channel types, with blue, green, red, cyan, magenta, yellow, black, orange, grey representing Nav1.1, 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.
Indeed, we can understand the correlation between
It is worth remarking explicitly on channel Nav1.9, which has
In this paper, we have considered the functional diversity of mammalian voltage-gated sodium channels as bounded by physical constraints on sodium channel function. We discussed two constraints: first, for a sodium channel to be able to generate an action potential it must be excitable, implying multiple equilibria for the current evoked by a step change in membrane potential; second, there is a constraint associated with a unique value of the resting potential in steady state. We showed that these constraints yielded results depending only on the sodium channel properties represented by the activation and inactivation curves, and collected corresponding data for human, mouse, and rat voltage-gated sodium channels, both wild-type and mutant, under a wide range of conditions. The excitability constraint was obeyed by all the data; on the other hand, the resting potential constraint was only obeyed in the channels expressed in muscle.
Furthermore, we demonstrated that there is a strong correlation between the voltage dependence of activation and inactivation in all of the channels. This correlation naturally breaks the channels into four different groups: (i) channels (Nav1.1, 1.2, 1.3, 1.6, 1.7), which are primarily expressed in nervous tissue, (ii) those expressed primarily in muscle (Nav1.4, 1.5), (iii) channel Nav1.8, and (iv) channel Nav1.9. The groups uncovered by analysis of this physiological data follow the major differences between channels shown by phylogeny. Most strikingly, the uniqueness of the physiological properties of Nav1.9 relative to the other channels is consistent with the phylogenetic assertion of Plummer and Meisler [
In making all these conclusions, we have not distinguished between mutant and wild-type channels of a given type, despite the fact that the mutant channels included in our study are physiologically significant (generally leading to sodium channel disease). Additionally, we have not distinguished between the many different conditions in which the channels are expressed: our dataset (
However, we believe there is significant opportunity for separating the channel properties further, even according to the admittedly crude metrics described here, if experiments were performed under more uniform conditions. An example of this is shown in
The solid line is the excitability threshold.
To these conclusions we need to add a strong caveat: we are not in any way suggesting that the physiological data analyzed here correspond to the most important differences between the different voltage-gated sodium channels. Indeed, it is clear that the kinetic properties of sodium channels are of critical importance for determining how they function. For example, repeated firing characteristics sensitively depend on channel properties and channel kinetics. Our analysis has not addressed the kinetic aspects of this problem at all, and we believe it is for this reason that our study has not been able to distinguish between mutant and wild-type channels of the same type.
Our omission of time constant information is not for lack of interest on our part, but instead because our review of the literature indicates that although a significant number of papers report measurements of both activation and inactivation curves, there are unfortunately far fewer consistent measurements of other channel properties that are critical for fully describing sodium channel function. These include the (voltage-dependent) timescales for activation and inactivation. Although many papers measure the inactivation timescales at positive voltages, a variety of different procedures are used for extracting these timescales from the raw data, and the raw data is not generally presented. This complicates comparing measurements with each other, and prevented our using them in this analysis.
We believe there is significant opportunity for extending the approach outlined here; namely, deriving constraints from simple models and comparing these results to kinetic properties of channel currents or even to single channels. There is little doubt that there are strong constraints on time constants for activation and inactivation in order for action potentials to fire properly. Uncovering these constraints (and hence the origin of the most critical differences between the channels) would lead to an understanding of the reasons for the differences between the different mammalian channels, and perhaps shed some light on how the channels contribute to nervous system function.
Ultimately such an approach could be applied to sodium, potassium, and calcium channels. There are no doubt constraints on sodium channels that arise from potassium channels and vice versa; for example, the time constant for inactivation of sodium channels must be tuned to the activation time constant for potassium channels for action potentials to fire properly. Repeated firing properties depend critically on the interaction of the various time constants. A complete and careful analysis of such constraints could be used as a tool to track and understand the evolution of the channels, perhaps relating to the origin of the nervous system itself. However, at present these ideas are at best immature speculation: for such studies to occur, it is necessary to expand efforts at acquiring kinetic properties of channels to non-mammalian species. There is much work to be done: our literature review was not able to uncover enough information about
Our theoretical analysis depended on a model for the membrane potential, given by
Here we calculate the excitability threshold. We are interested in characterizing the fixed points of the following equation:
Combining the solution to
We are interested in computing the excitability threshold, namely the boundary in (
Combining
Now, since
To test our theoretical predictions on sodium channel properties, we collected activation and inactivation curves from papers in recent literature. The data is summarized in
(91 KB PDF)
We express particular gratitude to Bruce Bean for persistent encouragement and advice, without which this work would not have been possible. We also thank Olivia White for early collaborations, and Ron Milo and Andrew Murray for helpful comments on this manuscript. MPB thanks Marc Kirschner and the Department of Systems Biology at Harvard Medical School for their hospitality during the early stages of this work.