Mechanisms underlying the distinct K⁺ dependencies of periodic paralysis

Primary periodic paralyzes are autosomal dominantly inherited disorders characterized by episodic attacks of weakness. The attacks are associated with reduced (hypokalemic periodic paralysis, hypoKPP), elevated (hyperkalemic periodic paralysis, hyperKPP), or both reduced and elevated serum K⁺ (Andersen–Tawil syndrome) (Sansone et al., 1997; Venance et al., 2006; Jurkat-Rott et al., 2010; Statland and Barohn, 2013; Cannon, 2015; Maggi et al., 2021). Mutations responsible for each type of periodic paralysis have been identified in muscle Na⁺ (Na_v1.4), Ca²⁺ (Ca_v1.1), and K⁺ (Kir2.1) channels (Venance et al., 2006; Jurkat-Rott et al., 2010; Statland and Barohn, 2013; Cannon, 2015; Sansone, 2019) (Table 1).

In all three forms of periodic paralysis, weakness is triggered by depolarization of muscle caused by a mutation-induced alteration in function of muscle ion channels. In hypoKPP, mutations in either the muscle Na⁺ or Ca²⁺ channel cause formation of a new pathway, termed a gating pore, which allows flow of monovalent cations and depolarization of muscle (Sokolov et al., 2007; Wu et al., 2021). In hyperKPP, gain of function mutations in the muscle Na⁺ channel increase a non-fast inactivating, or persistent, Na⁺ current (NaP), which depolarizes muscle (Cannon et al., 1991; Rojas et al., 1991; Cannon and Strittmatter, 1993). In Andersen–Tawil syndrome, mutations lead to loss of function of Kir channels (Plaster et al., 2001; Maggi et al., 2021). Since K⁺ channels hyperpolarize muscle, loss of their function predisposes to depolarization. Thus, the tendency toward pathologic depolarization is well understood in all three forms of periodic paralysis and has led to the current understanding of periodic paralysis:

• (1)

  Mutations responsible either increase a depolarizing current (hypo- and hyperKPP) or decrease a hyperpolarizing current (Andersen–Tawil syndrome).

• (2)

  Mutations in ion channels causing gating pore currents are associated exclusively with hypoKPP.

• (3)

  Mutations increasing NaP are associated exclusively with hyperKPP.

• (4)

  Attacks of paralysis occur when depolarization of the resting membrane potential (Vm) becomes severe enough that Na⁺ channels inactivate and the ability to generate action potentials is lost.

In hypoKPP, the pathologic depolarizing current subsequently leads to closure of inward rectifying K⁺ (Kir) channels when extracellular K⁺ (K_ex) is low. This leads to paradoxical depolarization in the setting of low K_ex as well as inappropriate retention of K⁺ by muscle, which worsens hypokalemia (Struyk and Cannon, 2008; Cheng et al., 2013, 2015; Cannon, 2015). HyperKPP also has a pathologic depolarizing current due to excess NaP activity, but in this case, paralysis due to depolarization only occurs in response to high K_ex. What is the essential difference between these cases that leads to paralysis with opposite perturbations in K_ex, despite both having excess depolarizing current? Least understood is why patients with Andersen–Tawil develop weakness when K⁺ is either elevated or reduced. Building on the idea that muscle serves as a buffer for K_ex (Struyk and Cannon, 2008; Cheng et al., 2013, 2015; Cannon, 2015), we developed a computer simulation of muscle ion fluxes to identify potential mechanisms underlying the unique K⁺ dependence of each disorder, as well as the development of either elevated or decreased serum K⁺.

All animal procedures were performed in accordance with the policies of the Animal Care and Use Committee of Wright State University and with the United States Public Health Service’s Policy on Humane Care and Use of Laboratory Animals. Both male and female mice were used from 2 to 6 mo of age. Mice were sacrificed using CO₂ inhalation followed by cervical dislocation.

Mice heterozygous for the M1592V Na_v1.4 mutation were used to model HyperKPP (FVB.129S4(B6)-Scn4a<tm1.1Ljh>/J, cat# 011033; Jackson Labs). Mice homozygous for the Na_v1.4 R669H mutation were used to model HypoKPP (cat# 036984-MU; MMRRC). WT mice from each colony were used as controls.

Recordings were performed at 35°C within 3 h of sacrifice. Muscle was stained with 10 µM 4-Di-2-ASP: 4-(4-diethylaminostyrl)-N-methylpyridinium iodide (cat# D289 [discontinued]; Molecular Probes) to allow for visualization of muscle fibers. The recording chamber was continuously perfused with Ringer Solution containing (in mM): NaCl, 118; KCl, 3.5; CaCl₂, 1.5; MgSO₄, 0.7; NaHCO₃, 26.2; NaH₂PO₄, 1.7; and glucose, 5.5 (pH 7.3–7.4) and equilibrated with 95% O₂ and 5% CO₂. Fibers were impaled with a sharp microelectrode filled with 3 M KCl solution containing 1 mM sulforhodamine 101 (Catalogue #S7635; Sigma-Aldrich) to allow for visualization of electrodes. Electrode resistance was between 15 and 25 MΩ.

We used 3.5 mM K⁺ as the K⁺ concentration for experiments, as it is the K⁺ concentration we have used for a number of years (Rich et al., 1998; Novak et al., 2015). While it is on the low end of normal, two factors reassure us that the use of 3.5 mM K⁺ did not lead to an incorrect conclusion regarding Vm of hypoKPP muscle in normal K⁺. (1) Unpublished data from our lab reveal that hypoKPP muscle does not become weak in response to single stimuli in 2 mM K⁺. Thus, weakness (and by inference pathologic depolarization), if present in 3.5 mM K, would be expected to be minimal. (2) Computer simulation was performed to compare hypoKPP Vm in 3.5 and 4 mM K⁺. Vm in 3.5 mM K⁺ was −79.3 mV and in 4 mM K⁺ it was −77.4 mV. This simulation suggests use of 3.5 mM K⁺ did not lead to pathologic depolarization.

Nested analysis of variance was performed using R version 4.2.0 with n as the number of animals in each group.

Muscle was modeled as a two-compartment system with the cylindrical intracellular space occupying 2/3 of the total system volume and the extracellular space occupying 1/3 using MATLAB (Brinkman et al., 2024). Membrane channels included voltage-gated channels: Kv, NaP, and gating pore (when present) based on the Hodgkin–Huxley approach (Hodgkin and Huxley, 1952); Kir (Struyk and Cannon, 2008), and ClC-1 (DiFranco et al., 2011). The Na/K ATPase pump (Wallinga et al., 1999) and a Na⁺ leak channel using constant Na⁺ permeability were also present. A NaKCl (NKCC1) cotransporter depended only on ion concentrations (Fraser and Huang, 2004). All channels except the Na/K ATPase and NKCC1 cotransporter used a constant field driving force (Goldman, 1943). The charge-difference approach was used to calculate ion fluxes, ion concentration changes, and volume changes (Fraser and Huang, 2004; Fraser et al., 2011). Osmolarity of both compartments was fixed at 300 mOsm through inclusion of impermeant intracellular and extracellular osmolyte and anions. Water flux to maintain osmolarity was instantaneous. When K_ex was raised or lowered, the process was accomplished over a 10 s interval.

For simulations of periodic paralysis, the following changes were made compared with WT for each disorder. For hypoKPP, a gating pore selectively permeable to Na⁺ was added. The magnitude of the gating pore current was chosen conservatively. The gating pore current has been estimated to increase resting Na⁺ influx by up to 21-fold (Sokolov et al., 2007, supplementary). In presented simulations, the Na⁺ gating pore plus Na⁺ leak current in hypoKPP at the resting potential under baseline K⁺ conditions increased Na⁺ influx by sixfold compared with the Na⁺ leak current in WT. For simulations of hyperKPP, NaP was increased twofold based on unpublished data from our lab in which NaP was increased by 100% in FDB muscles (unpublished data). For simulations of Andersen–Tawil, Kir conductance was conservatively decreased by 65%, based on findings that the mutations are dominant negative and can greatly decrease Kir currents (Kimura et al., 2012).

The model used is available at: https://github.com/bdf555/Muscle-EPhys-K-Hypothesis.git.

Table S1 shows the concentrations of intracellular and extracellular ions of normal and periodic paralysis muscle 1, 10, and 100 min after lowering K_ex. Table S2 shows the concentrations of intracellular and extracellular ions 1, 10, and 100 min after raising K_ex. Table S3 provides the data underlying Fig. 2.

Control of K⁺ was simulated in a 3-liter system with a 2-liter intracellular compartment and a 1-liter extracellular compartment to mimic the ratio of intracellular to extracellular fluid (Kim et al., 2017). Vm was set by the types of ion channels open at rest and the Nernst potentials (the electrical potential resulting from the concentration gradient) of the ions that flow through the open channels. An ion channel type with a high conductance (many open channels) moves Vm close to the Nernst potential of the conducting ion. In skeletal muscle, the two conductances that predominate at rest are K⁺ and Cl⁻; therefore, the pumps and transporters that establish the Nernst potentials for K⁺ and Cl⁻ are critical in setting Vm.

Ion channels included in the simulation of WT muscle were Kv, Kir, ClC, and Na⁺ leak (Fig. 1). A voltage-gated NaP was included in simulations but is not shown in Fig. 1 as it is not active at the normal Vm of muscle. The NKCC1 (Na⁺, K⁺, 2 Cl⁻) cotransporter and the Na/K ATPase pump were included. To simulate hyperKPP, NaP was increased, and for hypoKPP, a gating pore current was added. For simulation of Andersen–Tawil syndrome, Kir conductance was reduced. K_ex was fixed to 3.5 mM in baseline conditions for each of the disease states to mimic our standard recording solution (Rich et al., 1998). Due to differing channel and pump activities intracellular and total K⁺ in the system differed at baseline between WT and the various forms of periodic paralysis. For simulation of high or low K⁺, K⁺ was altered by adding or removing K⁺ from the extracellular compartment. To maintain charge neutrality, Na⁺ was added or removed from the extracellular compartment such that the total concentration of cations was unchanged. The model was then allowed to reach a new steady state.

At steady state, not only is total current across the membrane 0, but the net flux of each ion must also be 0. K⁺ is brought into muscle in two ways (Fig. 1). The first is via the Na/K ATPase pump that transports K⁺ into, and Na⁺ out of, muscle fibers (Clausen, 2013). This results in high K⁺ and low Na⁺ inside the muscle fiber, creating a concentration gradient for K⁺ to diffuse out of the fiber, which leads to a negative E_K. The second pathway for K⁺ influx is the NKCC1 cotransporter (Lindinger et al., 2011). The inward flow of K⁺ via NKCC1 is driven by the concentration gradient favoring influx of Na⁺ and Cl⁻. The influx of K⁺ is matched by K⁺ efflux via the K⁺ channel open at rest: the Kir channel (Struyk and Cannon, 2008; DiFranco et al., 2015).

At steady state, Vm in WT muscle in the simulation was −86.8 mV. E_K was slightly negative relative to Vm at −88.0 mV and E_Cl was −86.6 mV. Vm was closer to E_Cl than E_K because the relative conductance for Cl⁻ is higher than that for K⁺ (Bryant and Morales-Aguilera, 1971; Palade and Barchi, 1977; Geukes Foppen et al., 2002). The NKCC transporter increased intracellular Cl⁻ such that E_Cl was slightly more positive than Vm (Dulhunty, 1978; Aickin et al., 1989; Cannon, 2015) (however, see Renaud et al., 2023), so Cl⁻ left the cell through ClC channels open at rest to balance the influx of Cl⁻ via NKCC1 (Fig. 1) (Geukes Foppen et al., 2002).

To simulate the responses of hypoKPP and hyperKPP muscle to alterations in K_ex, it was necessary to accurately simulate the pathologic current in each disorder. The gating pore current responsible for hypoKPP increases at hyperpolarized voltages and decreases as Vm becomes depolarized (Sokolov et al., 2007; Wu et al., 2021). Na⁺ channel mutations that cause hyperKPP are gain of function mutations that increase an inactivation-resistant NaP (Cannon et al., 1991; Rojas et al., 1991; Cannon and Strittmatter, 1993). The voltage dependence of NaP is opposite of that of the gating pore current; it begins to activate when Vm reaches approximately −75 mV and increases in amplitude with depolarization (Gage et al., 1989; Hawash et al., 2017). The difference in voltage dependence of mutant currents in hypo- and hyperKPP means that in normal K⁺, the conductance of the mutant current in hypoKPP is high, but in hyperKPP, the conductance of the mutant current is near 0. The result is that when K⁺ is normal, Vm of hypoKPP muscle should be depolarized, while Vm of hyperKPP muscle is the same as in WT muscle.

Published data do not yield a consistent conclusion regarding Vm at baseline in hypoKPP. Two early studies of muscle from patients suggested hypoKPP muscle is more depolarized than WT muscle in solution containing normal K_ex (Rüdel et al., 1984; Ruff, 1999), while in one study, Vm measured in hypoKPP muscle was normal (Creutzfeldt et al., 1963). In a study of muscle biopsies from seven patients with hypoKPP, Vm was depolarized in solution containing normal K_ex (Jurkat-Rott et al., 2009). A study using muscle force velocity recovery measurements in vivo also suggests Vm was depolarized at baseline in hypoKPP (Tan et al., 2020). However, a study using the Na_v1.4 R669H mouse model of hypoKPP reported no difference in Vm between WT and mutant muscle (Wu et al., 2011).

There is similar confusion regarding baseline Vm in hyperKPP. An early study performed by impaling muscle fibers in patients found no difference in Vm between a normal and a hyperKPP patient (Brooks, 1969), but another study reported depolarized Vm in a hyperKPP patient (Creutzfeldt et al., 1963). A recent study of hyperKPP patients using muscle force velocity recovery measurements suggests Vm is not different between disease and control (Tan et al., 2020). However, two studies using a mouse model expressing a hyperKPP Na⁺ channel mutant suggest Vm is depolarized in affected muscle when K_ex is normal (Clausen et al., 2011; Ammar et al., 2015).

Given the lack of clarity regarding Vm in normal K⁺ in both hypoPP and hyperKPP, we measured Vm in EDL muscles from the Na_v1.4 R669H mouse model of hypoKPP and the Na_v1.4 M1592V mouse model of hyperKPP in 3.5 mM K_ex at 35°C (see Materials and methods for discussion of the use of 3.5 mM K⁺). Vm of hypoKPP muscle was depolarized relative to WT, whereas there was no difference between Vm of hyperKPP muscle and WT (Fig. 2 A and Table S3). Our data from mouse models of hypo- and hyperKPP agree with recent studies of patients with both disorders (Jurkat-Rott et al., 2009; Tan et al., 2020).

Based on our recordings of Vm in 3.5 mM K⁺, we simulated hypoKPP with a gating pore current active at the normal Vm of muscle and hyperKPP with an increase in NaP that is not active at the Vm of muscle in 3.5 mM K⁺ (Fig. 2 B). As will be discussed below, this resulted in hypoKPP muscle having different intracellular concentrations of Na⁺ and K⁺ than WT muscle at baseline, whereas hyperKPP muscle had the same concentrations as WT.

Net flux of a given ion is determined by both the movement of that ion through ion channels as well as the movement via transporters. Changes in K_ex have two effects on ion fluxes that shape the responses of WT and periodic paralysis muscles. The first is the effect of changing K_ex on K⁺ current through Kir channels (I_K). I_K is determined by the channel permeability, the voltage dependence of channel open probability, and a constant field driving force determined by the difference between Vm and E_K (Fig. 3 A). There is a maximum of outward K⁺ current at membrane potentials positive to E_K (Fig. 3 B). Reduction of current beyond the maximum is due to block of the channel pore by intracellular polyamines and Mg²⁺ (Nichols and Lopatin, 1997; Hibino et al., 2010). Interestingly, the maximum and the reduction in outward Kir current due to polyamine block shift in parallel with E_K. As a consequence, the entire Kir current-voltage (IV) relationship shifts with E_K (Fig. 3 C) (Chang et al., 2010; Cheng et al., 2015). With elevation of K_ex, there is also an increase in peak outward I_K through Kir channels (Fig. 3 C).

The second effect of changes in K_ex concentration is on ion fluxes mediated by the Na/K ATPase. Na/K ATPase activity is modulated by substrate: K_ex, which is transported into the cell, and intracellular Na⁺ (Na_in), which is transported out of the cell. Both higher K_ex and higher Na_in increase activity, while lower K_ex and lower Na_in reduce activity (Clausen, 2003). Finally, activity of the pump is affected by membrane potential: the more depolarized the membrane potential, the higher the activity (Wallinga et al., 1999; Clausen, 2003).

In simulations, reduction of K_ex concentration was performed by removing 3.0 mmol of K⁺ from the 1-liter extracellular compartment. Without compensation by muscle, this would result in a reduction in concentration of K_ex from 3.5 to 0.5 mM. However, in all cases muscle responded to the reduction in K_ex such that the concentration of K_ex changed over time. One reason for this was that lowering of K_ex made E_K more negative, which increased the voltage difference between Vm and E_K since high Cl⁻ conductance initially anchored Vm near E_Cl. In WT muscle, despite reduction in the maximal outward Kir current, Vm moved to a region where there was an increase in outward Kir current, relative to the normal K_ex situation, which resulted in net outward K⁺ current (Fig. 4).

In WT, reduction of K_ex resulted in hyperpolarization of Vm (Fig. 5 A) and mitigation of the reduction in K_ex (Fig. 5 E and Table S1) due to the following series of events: Reduction of K_ex reduced Na/K ATPase pump activity (Wallinga et al., 1999; Clausen, 2003) (Fig. 5 F, 0–25 min), which caused a net influx of Na⁺ via Na⁺ leak and NKCC (Fig. 5 B, 0–20 min), which slightly elevated Na_in (Fig. 5 D). The net Na⁺ influx would depolarize Vm, but was overwhelmed by the increased efflux of K⁺ (Fig. 5 C, 0–20 min) because E_K and the Kir IV relationship shifted in the hyperpolarized direction more than Vm (due to high Cl⁻ conductance).

HyperKPP mutations increase NaP at membrane potentials depolarized relative to the normal Vm (Cannon et al., 1991; Rojas et al., 1991; Cannon and Strittmatter, 1993). Because NaP was inactive at baseline and in low K, hyperKPP muscle had an identical response to lowering of K_ex as WT muscle. We conclude that because NaP activates at membrane potentials depolarized to the normal Vm of muscle, hyperKPP muscle handles reduction of K_ex normally.

HypoKPP mutations create a gating pore current that increases at hyperpolarized voltages (Fig. 2 B) (Sokolov et al., 2007; Wu et al., 2021). The baseline activity of the gating pore current increased influx of Na⁺ with a secondary increase in Na_in at baseline (Fig. 5 D, 0 min). The increase in Na_in increased Na/K ATPase activity such that Na⁺ efflux was equal and opposite to Na⁺ influx (Fig. 5 F, 0 min). The increase in Na/K ATPase activity also increased K⁺ influx. This was offset by increased K⁺ efflux via Kir channels, caused by depolarization of Vm (caused by the gating pore current, Fig. 5 A, 0 min) such that Vm was closer to the peak of the Kir current IV relationship (Fig. 4). Thus, at steady state in normal K_ex, hypoKPP muscle had depolarization of Vm with balanced, but larger, influx and efflux of both Na⁺ and K⁺ than in WT muscle.

The gating pore current and baseline changes in hypoKPP muscle caused depolarization and inability to mitigate lowering of K_ex due to the following series of events: Similar to WT, reduction of K_ex caused both hyperpolarization of E_K and reduction in Na/K ATPase pump activity (Fig. 5 F, 0–40 min). Kir current was already near its peak before the lowering of K_ex such that the leftward shift of the Kir IV relationship decreased outward Kir current (Fig. 4). This effect overwhelmed reduced K⁺ influx caused by reduction in Na/K ATPase activity such that there was net K⁺ influx via NKCC (Fig. 5 C, 0–30 min). K_ex dropped further (Fig. 5 E, 0–30 min), which further hyperpolarized E_K. K_ex could not recover due to near zero Kir current. A consequence of the persistent low K_ex was persistently reduced activity of the Na/K ATPase (Fig. 5 F, 0–40 min) such that inward Na⁺ leak current resulted in net influx of Na⁺ and elevation of Na_in (Fig. 5, B and D, 0–40 min), which resulted in pathologic depolarization. K⁺ efflux only recovered when Vm became so depolarized that Kv channels opened (Fig. 5 C, 30–50 min) (Beam and Donaldson, 1983; DiFranco et al., 2012; Cannon, 2015; Cheng et al., 2015). The elevation of K_ex (Fig. 5 E, 30–50 min) coupled with elevation of Na_in (Fig. 5 B, 0–50 min), finally allowed Na/K ATPase activity to recover to near baseline (Fig. 5 F, 50 min). Vm stabilized at −55.9 mV (Fig. 5 A and Table S1), a membrane potential that does not support excitation-contraction coupling (Wang et al., 2022).

The mechanism underlying pathologic depolarization in Andersen–Tawil syndrome is not activation of a mutated channel that increases depolarizing current. Instead, there is a reduction in functional Kir channels in muscle (Plaster et al., 2001; Cheng et al., 2015; Simkin et al., 2018; Maggi et al., 2021). There are no studies we are aware of in which Vm of muscle from patients with Andersen–Tawil syndrome was measured. Viral transduction of mouse skeletal muscle with an Andersen–Tawil causing mutation of Kir 2.1 caused both reduction in Kir current and weakness, but Vm was not reported (Simkin et al., 2018). In mouse muscle, in which Ba²⁺ was used to partially block Kir channels to mimic Andersen–Tawil syndrome, there was depolarization of Vm (Struyk and Cannon, 2008).

Depolarization of Vm at baseline caused by reduction in Kir conductance in Andersen–Tawil syndrome caused outward Kir current to be near maximal at baseline (Fig. 4). Reduction of K_ex decreased outward Kir current more than reduction of K⁺ influx via the Na/K ATPase such that there was net K⁺ influx (Fig. 5, C and F, 0–100 min). The continued influx of K⁺ into muscle via Na/K ATPase and NKCC resulted in further reduction of K_ex (Fig. 5 E, 0–100 min and Table S1). The reduction of Na/K ATPase activity caused a net inward Na⁺ current, which depolarized the cell (Fig. 5 B, 0–140 min) and was made worse by activation of the normal small NaP caused by depolarization (Fig. 5 A). Na/K ATPase activity returned to near baseline after Na_in increased (Fig. 5, D and F, 130 min). K_ex stabilized at a low value once Vm was depolarized enough to activate Kv channels (Fig. 5, A and E, 130 min).

Removing amounts of K⁺ from the extracellular space other than 3 mmol produced similar findings. In the case of hypoKPP, removing any amount over 2.1 mmol produced pathologic depolarization. For Andersen–Tawil, removing 2.7 mmol or more produced pathologic depolarization. For WT and hyperKPP, removal of almost all the K_ex (3.45 mmol) was also able to cause pathologic depolarization.

An increase in K_ex was simulated by adding 8 mmol of K⁺ to the 1-liter extracellular compartment. Without compensation by muscle this would result in an increase in K_ex from 3.5 to 11.5 mM. In both WT and periodic paralysis muscle, the addition caused E_K to become more positive than Vm, which depolarized less than E_K because the high resting Cl⁻ conductance anchored Vm. The result was a very large initial K⁺ influx via Kir channels (Fig. 6 C, 0–10 min). A second change was that Na/K ATPase pump activity increased due to increased substrate (K_ex) (Wallinga et al., 1999; Clausen, 2003) (Fig. 6 F, 0–20 min). Both inward I_K through Kir channels and increased Na/K ATPase activity caused a large increase in K⁺ influx initially, which lowered K_ex, but depolarized Vm (Fig. 6, A, C, and E, 0–10 min). The increase in Na/K ATPase activity also increased Na⁺ efflux. Na⁺ efflux was opposed by activation of the normal NaP caused by depolarization of Vm.

In WT muscle, the outward Na⁺ flux due to increased Na/K ATPase activity predominated, resulting in net Na⁺ efflux (Fig. 6 B, 0–50 min). The hyperpolarizing Na⁺ efflux largely offset the depolarizing effect of K⁺ influx such that Vm stabilized with moderately elevated K_ex and the ability to support effective excitation-contraction coupling (Wang et al., 2022) (Fig. 6, A and E, 100 min, Table S2).

In hyperKPP, the initial events triggered by elevation of K_ex were the same as in WT muscle: depolarization of E_K relative to Vm and activation of Na/K ATPase activity. These changes caused an initial K⁺ influx similar to WT (Fig. 6 C, 0–10 min). Deviation from behavior of WT muscle was triggered by activation of the larger than normal NaP, which led to net Na⁺ influx (Fig. 6 B, 10–150 min). Net Na⁺ influx triggered additional depolarization of Vm (Fig. 6 A, 10–150 min), which caused two things: (1) Because NaP current is more steeply voltage dependent than Na/K ATPase activity at voltages between −70 and −65 mV, further depolarization initially led to greater net inward Na⁺ flux (Fig. 6 B). Na_in reached steady state when Na⁺ efflux via Na/K ATPase activity increased due to increased Na_in (Fig. 6, D and F, 50–150 min). (2) Depolarization of Vm increased outward Kir current as Vm moved further from E_K. The increase in K⁺ efflux via Kir channels overwhelmed the increase in K⁺ influx due to increased Na/K ATPase activity. The result was net efflux of K⁺, which worsened elevation of K_ex (Fig. 6, C and E, 30–150 min). The rise in Na_in and K_ex finally triggered a large enough increase in Na/K ATPase activity which caused the system to reach steady state at a high K_ex (Fig. 6, D–F; and Table S2).

As described above, at baseline, hypoKPP muscle had higher K⁺ efflux via Kir channels than WT, which was offset by higher Na/K ATPase activity. The gating pore current was active at baseline and decreased with depolarization triggered by elevation of K_ex. The already high Na/K ATPase activity was further increased by elevation of K_ex (Fig. 6 F, 0–50 min). The result was that hypoKPP muscle lowered K_ex more effectively than WT muscle (Fig. 6 E, 0–50 min, Table S2). The increase in Na/K ATPase activity caused a net outward Na⁺ current, which hyperpolarized Vm (Fig. 6, A and B, 0–50 min). HypoKPP muscle ended up with a Vm similar to that of WT muscle following elevation of K_ex.

In Andersen–Tawil syndrome, Vm was depolarized at baseline due to reduction in Kir conductance. The additional depolarization of Vm caused by elevation of K_ex caused activation of NaP beyond that seen in WT muscle. The increase in Na⁺ influx via NaP caused net inward Na⁺ current and elevation of Na_in despite increased Na⁺ efflux via the Na/K ATPase (Fig. 6, B, D, and F, 20–350 min). This resulted in the same spiral of further depolarization and further activation of NaP seen in hyperKPP (Fig. 6 A). The pathologic depolarization moved Vm to near the maximum of outward Kir current (Fig. 4). However, since Kir conductance was low the net K⁺ flux was small and changes to K_ex were small (Fig. 6 E). The resultant rise in Na_in (Fig. 6 D) finally triggered a large enough increase in Na/K ATPase activity (Fig. 6 F) to counteract NaP and stabilize Na⁺ flux and Na_in (Fig. 6, B and D, 300–400 min). The result was significantly greater depolarization than in WT muscle despite a minimal additional elevation of K_ex (Fig. 6, A and E; and Table S2).

Adding amounts of K⁺ to the extracellular space other than 8 mmol produced similar findings. In the case of hyperKPP, adding 6.9 mmol or more produced pathologic depolarization. In the case of Andersen–Tawil, adding 7.1 mmol or more produced pathologic depolarization. For WT and hypoKPP, adding more K⁺ to the extracellular space produced progressively more depolarization, but no discontinuity in the resting Vm versus additional K⁺ was observed as occurs for hyperKPP and Andersen–Tawil. For WT, the resting Vm became more depolarized than −60 mV when >16 mmol of K⁺ was added.

Building on understanding of the mutations causing periodic paralysis and their effects on Vm, we generated a model of ion fluxes in muscle, which was able to account for both changes in serum K⁺ and the distinct K⁺ dependence of weakness in each of the three forms of periodic paralysis. Conclusions regarding possible mechanisms underlying attacks of weakness in hyperKPP and hypoKPP are summarized in Fig. 7.

Our computer simulations support the idea that muscle serves as a buffer for K⁺ (Struyk and Cannon, 2008; Cheng et al., 2013; Cannon, 2015; Cheng et al., 2015). Ion channel mutations responsible for periodic paralysis disrupt the ability of muscle to buffer K⁺ such that serum K⁺ increases or decreases more than is normal. The swings in K⁺ are accompanied by depolarization and attacks of weakness. A key factor in determining whether muscle depolarizes following changes in K_ex is the net K⁺ and net Na⁺ fluxes. For K⁺, net flux is primarily determined by the balance between K⁺ efflux via Kir channels and influx via the Na/K ATPase. For Na⁺, it is primarily determined by influx via NaP or the gating pore current and efflux via the Na/K ATPase. Regulation of Na/K ATPase activity by changes in Na_in or K_ex (substrate) plays a central role in the response to alterations in K⁺ in the system. The net K⁺ or Na⁺ flux is often very small such that changes in concentration and the resultant change in Na/K ATPase activity evolve over hours. This fits with the duration of attacks in patients (Lehmann-Horn et al., 2008; Charles et al., 2013).

A deeper understanding of mechanisms underlying buffering of serum K⁺ by muscle and pathologic depolarization in periodic paralysis may aid in development of novel approaches to therapy for both periodic paralysis as well as other diseases with dysregulation of muscle excitability. The hope is that attacks of weakness may be preventable by very small changes in net K⁺ or Na⁺ current.

Eduardo Ríos served as editor.

We would like to thank Adam Deardorff and Xueyong Wang for their helpful comments. Figs. 1, 2, 3, and 7 were made using BioRender software.

This work was supported by National Institutes of Health grants AR074985 (M.M. Rich) and 1F30AR081675 (C. Dupont).

Author contributions: B.D. Foy: conceptualization, data curation, formal analysis, investigation, methodology, project administration, resources, software, supervision, validation, visualization, and writing—original draft, review, and editing. C. Dupont: conceptualization, data curation, formal analysis, funding acquisition, investigation, methodology, project administration, supervision, validation, visualization, and writing—original draft, review, and editing. P.V. Walker: investigation. K. Denman: data curation and writing—review and editing. K.L. Engisch: conceptualization, visualization, and writing—original draft, review, and editing. M.M. Rich: conceptualization, data curation, formal analysis, funding acquisition, methodology, project administration, resources, supervision, validation, visualization, and writing—original draft, review, and editing.