article: 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.
  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 [1,2]. Specialization could result from tuning to different functional environments, or nearly identical genes might play very different functional roles [3]. 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 [4]. Since Hodgkin and Huxley [5], mathematical models have suggested that sodium channel properties need to be tightly constrained for an action potential to propagate. In mammals, there are nine different genes encoding voltage-gated sodium channels [6], many of which are more than ≈90% identical by sequence [7]. On one hand, the sequence similarity of the channels presumably corresponds to similarity of their functional properties; this is consistent with the idea that these properties must be tightly constrained. On the other hand, the multiplicity of genes encoding sodium channels raises the question: why are so many different mechanisms for generating an action potential necessary? Sodium channels are predominantly found in specific anatomical regions, suggesting that they might be tuned for specific functions. For example, the channels Nav1. 1, Nav1. 2, Nav1. 3, Nav1. 6, and Nav1. 7 are predominantly localized in the central and peripheral nervous systems; Nav1. 8 and Nav1. 9 primarily in the dorsal root ganglion; Nav1. 4 primarily at skeletal muscular junctions; Nav1. 5 primarily in cardiac tissue (Table 1) [8]. 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.) [9–48]. The mutant channels tend to be associated with a disease state and hence presumably differ in a physiologically significant way from the wild-type. 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 [7,49–53]. What are the functional differences between these groups? Two groups correspond to channels expressed in nerve and muscle tissue, respectively. Another group (consisting of the single channel Nav1. 8) has the potential for a voltage threshold substantially higher than the other channels. The final group (consisting of the single channel Nav1. 9) can only produce action potentials in a narrow conductance range and even then has a maximum voltage threshold which is less than thermal fluctuations. The separation of the channels into functionally distinct groups suggests that they have evolved to perform specialized tasks.
  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 [5]. The membrane potential V changes due to both sodium and potassium currents, so that Here GNa is the conductance of sodium due to voltage-dependent sodium channels, PNa (V, t) is the time-dependent probability that a sodium channel is open, and is the conductance of non-voltage-dependent sodium channels. The potassium channels are similarly characterized by GK, PK (V, t), and. The reversal potentials for sodium and potassium are fixed by the sodium and potassium concentrations on both sides of the membrane. The temporal dynamics of PNa (V, t) and PK (V, t) are determined by the kinetics of the sodium and potassium channels. 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 (V1/2, k) are reported explicitly, and also includes data that we digitized directly from the literature. Since each activation and inactivation curve is represented by two parameters (V1/2, k), we can represent the corresponding property space in two dimensions. Figure 4 shows this for humans (A, B) and rats (C, D), respectively. The left half of Figure 4A and 4C represents the inactivation curves, while the right half of Figure 4B and 4D shows the data for the activation curves. 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). For each channel type we include measurements for both wild-type and mutant, as well as a range of conditions: these include with or without subunits; with or without external effectors such as calmodulin, etc. Dataset S1 summarizes all of the collected data, including specific references, mutations, and conditions. Figure 4A–4D represents the excitability threshold as a solid line. To the right of the solid line, excitability is possible, and to the left it is impossible. Consistent with our analysis of the excitability constraints, all measured activation curves are on the right side of the excitability threshold. On the other hand, the inactivation curves do not all obey the constraint given by the uniqueness of the resting potential. For the assumed VK = −90 mV, we find that the channels expressed in muscle (Nav1. 4, Nav1. 5) obey the constraint whereas the neuronal channels do not. The implication of this is that for the neuronal channels, there exists a range of conductances (GK, GNa) for which there are multiple fixed points. We cannot say whether there is physiological significance to this parameter regime but note that an unstable resting potential might be advantageous for spontaneous firing. Figure 5 shows this same data for humans (A, B) and rats (C, D), respectively. Now we use different colors for each channel type, 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.
  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 [51], who argue on the basis of sequence similarity that it evolved independently from the other channels. According to our analysis, it is essentially impossible for Nav1. 9 to trigger action potentials. 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 (Dataset S1) includes channels with and without different types of subunits, and with chemicals (lidocaine, tetrodotoxin, etc.). The fact that the constraints and the correlations uncovered here hold so strongly indicates how robust they are. From this point of view, it is not surprising that our analysis has not completely provided a rationalization for all the channel diversity. In particular, we cannot distinguish the differences within either the group of muscular channels, Nav1. 4,1. 5, or the group of neuronal channels, Nav1. 1,1. 2,1. 3,1. 6,1. 7. 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 Figure 9, compiled from a remarkably complete analysis [55] on the effects of calmodulin and other effectors on Nav1. 4,1. 5. The triangle symbols of Figure 9 show while the circle symbols show. The blue symbols correspond to Nav1. 4 and the red to Nav1. 5. This data shows a clear split between the properties of the two channels, which was not apparent from the data in Figures 5 and 7. 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 for invertebrate channels to include them in the present study.
  Our theoretical analysis depended on a model for the membrane potential, given by Equation 1. We used the same model introduced by Hodgkin and Huxley [5], considering only contributions from sodium and potassium ions. Simplifying assumptions to this model are discussed in the text. Here we calculate the excitability threshold. We are interested in characterizing the fixed points of the following equation: Setting dV/dt = 0 and rearranging gives that the fixed points obey where we have made use of the measured functional form of the activation curve, P (V), given by Equation 6. The bifurcation points Θ = Θmin, max are determined by looking for when the right and left hand sides of Equation 14 are tangent to each other, Equations 14 and 15 give two equations for the two unknowns (Θ, V), whose solutions give Θmin and Θmax, and the corresponding voltages at the bifurcation point. These equations can be solved by dividing Equation 14 by Equation 15. We obtain: Combining the solution to Equation 16 for V with Equation 14 gives the bifurcation points Θmin and Θmax. We are interested in computing the excitability threshold, namely the boundary in (V1/2, k) property space beyond which multiple solutions of Equation 14 do not exist for any values of Θ. This is guaranteed if there are no solutions to Equation 16, so that there is no range of Θ for which multiple intersections to Equation 14 exist. To identify the excitability threshold, we therefore find the (V1/2, k) where only a single solution to Equation 16 exists. This corresponds to requiring that the right-hand side of Equation 16 is tangent to the left-hand side. Differentiating the two sides gives Combining Equations 16 and 17 implies that V = VK + 2k. Inserting this into Equation 17 then implies that the excitability threshold is given by Now, since VNa − VK ≫ k, we can simplify Equation 18 to 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 Dataset S1. The dataset includes papers where (V1/2, k) are reported explicitly, and also includes data that we digitized directly from the literature using the program GraphClick (Arizona Software, version 2. 8. 2). The values of (V1/2, k) from the digitized data were obtained by curve-fitting with Matlab.
summary: 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.
keywords: physics
  computational biology
  vertebrates
  biophysics
  neuroscience
title: Excitability Constraints on Voltage-Gated Sodium Channels