REVIEW ARTICLE published: 11 June 2012 doi: 10.3389/fphar.2012.00112 Pathophysiological role of omega pore current in

Karin Jurkat-Rott 1*, James Groome 2 and Frank Lehmann-Horn1

1 Department of Neurophysiology, Ulm University, Ulm, Germany 2 Department of Biological Sciences, Idaho State University, Pocatello, ID, USA

Edited by: In voltage-gated cation channels, a recurrent pattern for is the neutralization Gildas Loussouarn, University of of positively charged residues in the voltage-sensing S4 transmembrane segments.These Nantes, France mutations cause dominant ion channelopathies affecting many tissues such as brain, , Reviewed by: Stephen Cannon, University of Texas and skeletal muscle. Recent studies suggest that the pathogenesis of associated pheno- Southwestern Medical Center, USA types is not limited to alterations in the gating of the ion-conducting alpha pore. Instead, Domenico Tricarico, University of Bari, aberrant so-called omega currents, facilitated by the movement of mutated S4 segments, Italy also appear to contribute to symptoms. Surprisingly, these omega currents conduct cations *Correspondence: with varying ion selectivity and are activated in either a hyperpolarized or depolarized volt- Karin Jurkat-Rott, Department of Neurophysiology, Ulm University, age range.This review gives an overview of voltage sensor channelopathies in general and Albert-Einstein-Allee 11, 89069 Ulm, focuses on pathogenesis of skeletal muscle S4 disorders for which current knowledge is Germany. most advanced. e-mail: [email protected] Keywords: and , long QT syndrome, familial hemiplegic migraine, myotonia and paramyotonia, hyperkalemic and hypokalemic periodic paralysis, sodium overload, cytotoxic edema, degeneration

INTRODUCTION than histidine result in a hyperpolarization-induced inward cur- The basic motif of the alpha subunit of voltage-gated cation chan- rent carried by alkali metal cations rather than protons (Tombola nels is a tetrameric association of four domains I–IV, each con- et al., 2005) indicating that this creates a short-circuit sisting of six transmembrane helical segments S1–S6, connected connection of intracellular and extracellular compartments in by intracellular and extracellular loops. At the resting potential, the canaliculus. This current varies with the identity of the open probability is low. Activation results from a depolarization- substituted residue: R1S > R1C > R1V ∼ R1A, has the selectiv- induced conformational change leading to the opening of the ity Cs+ > K+ > Li+, and is not affected by alpha pore blockers. alpha pore. When the pore is pharmacologically blocked, charge Shaker mutants R2H and R3H conduct protons at potentials cor- movements in the electrical membrane field are measurable as so- responding to the voltage-dependent movement of S4 charges, as called gating currents. These gating currents are a result of the the current is maximal at potentials close to midpoint of the QON- outward movement of highly conserved voltage-sensitive S4 seg- voltage relation (Starace et al., 1997; Starace and Bezanilla, 2001). ments which display an arginine or lysine residue at every third Further depolarization blocks the omega pore, consistent with amino acid position, almost vertically aligned (denoted as R1, the model that outward S4 movement shifts the short-circuiting R2, R3, etc.). At resting potential, the outermost S4 charge sep- mutant residues out of the canaliculus. arates the extracellular fluid from the cytoplasm. With membrane Studies on Shaker and domain II of rat depolarization, each S4 moves outwardly shifting deeper situated brain Nav1.2 suggest that substitution of a single arginines to the critical position of extra-intracellular separation. S4 arginine may be insufficient to produce a non-proton omega During their outward movement, the S4 segments move in a spi- current, but that two adjacent mutations (i.e., R2/R3 or II-R1/R2) ral path through canaliculi of the channel protein made up of are required (Sokolov et al., 2005; Gamal El-Din et al., 2010). In segments S1–S3 (Figures 1 and 2). Likewise, during recovery from this respect, the presence of alanine at the position equivalent to channel inactivation, the S4 segments are thought to move back R0 in Shaker may therefore enhance the omega current of R1S. to their original position. Because of the narrow constriction sep- The motif of adjacent arginine replacements recurs in some of the arating extracellular from intracellular compartments, mutations potassium channel disorders and may be explained by the degree of the S4 charges may cause either a hyperpolarization-activated of accessibility of arginines to the extracellular or intracellular or a depolarization-activated non-specific cation leak called the compartments. However, this does not hold true for all channels omega current (Figure 2). suggesting that position and orientation of the S4 segment and An omega current was first described in the voltage-gated its environment play an important role as well. Based on current potassium channel of drosophila Shaker. In this channel the muta- knowledge of S4 position, in this review R0 designates charge posi- tion R1H in S4 generates an inward-directed hyperpolarization- tions of S4 outside of the canaliculus and R1 the first charge inside induced proton current that becomes increasingly prominent as of it. the proton reversal potential EH is shifted to more positive values For human diseases, the channelopathies result from muta- (Starace and Bezanilla, 2004). R1 substitutions to residues other tions in voltage-gated ion channels. In several of these disorders,

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FIGURE1|Scheme of a voltage-gated . Bird’s eye view of the pore. The model also shows one of the four voltage sensors S4, which moves channel consisting of four similar repeats (I to IV). The channel has been cut outward when the membrane becomes depolarized and remains in this open between repeats I and IV to show the opening and closing of the central position until repolarization. mutations of specific charges of S4 segments are known, such EPILEPSY AND NEUROMYOTONIA as in epilepsy, hemiplegic migraine, long QT (LQT) syndrome, Epilepsy is characterized by recurring episodes of synchronized , and periodic paralyzes (Table 1). The electrical discharges of caused by their facilitated depo- underlying mutations generally modify voltage sensitivity of gat- larization within the . The symptoms of a ing and maximal amplitude of alpha current. Additionally, some depend on age of the patient, the underlying cause, and the of these mutations have been reported to produce an omega cur- brain region involved. rent. Most of these are inward currents that result in enhanced with febrile plus (GEFS+)is membrane depolarization, contributing to the symptoms of the a childhood-onset syndrome featuring febrile seizures (FS) and . Depending on the position of the S4 mutation afebrile epileptic within the same pedigree. The pen- and the voltage threshold of activation of the channel, the omega etrance is ∼60%. In two-thirds of affected individuals either FS current can appear in very different voltage ranges. persist after the sixth year of life or afebrile generalized tonic-clonic This review briefly outlines the clinical features and pathogene- seizures additionally occur (FS+). GEFS+ type II is caused among sis of the S4 channelopathies with a focus on mutation-dependent others by missense mutations in the alpha subunit of the neuronal changes of alpha and omega currents as well as their contributions sodium channel Nav1.1, encoded by SCN1A. Mutations frequently to the phenotype. It also introduces the concept of membrane bi- destabilize the fast-inactivated state, resulting in a persistent stability in general and in computer simulations incorporating inward sodium current that depolarizes the membrane potential omega current. Also discussed are the consequences of intra- to a value closer to the Nav1.1 threshold (facilitated spike gen- cellular sodium overload and tissue degeneration to explain the eration). Additionally, a few mutations result in loss-of-function non-episodic features of the channelopathies and the ideas behind by trafficking defects (Catterall et al., 2010; Escayg and Goldin, best practice treatment. For further reading, there are reviews 2010). S4 mutations II-R1C, IV-R4H, and IV-R7C (Figure 3) with more emphasis on S4 structure and function (Catterall, result in additional loss-of-function effects such as decreased cur- 2010), S4 mutations in channels of skeletal muscle (Cannon, 2010; rent amplitude and stabilization of the inactivated state (Alekov Jurkat-Rott et al., 2010), or drug action in periodic paralyzes and et al., 2000; Spampanato et al., 2001; Lossin et al., 2003; Barela related channelopathies (Matthews and Hanna, 2010; Tricarico et al., 2006; Vanoye et al., 2006). Apparently, loss-of-function in and Camerino, 2011). inhibitory neurons produces overexcitability (Martin et al., 2010)

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FIGURE 2 | Omega pores and currents dependent on the R position. structure of NaVAb (activated-closed; crystal structure at 0 mV), using (A) Replacement of the outermost arginine (red) by a neutral amino acid (gray) Modeller. Positions of arginine and lysine residues of DIS4 are shown, relative such as glycine (R1G) opens a conductive pathway through the polarized to the putative gating pore constriction (arrow). Modified from Groome and membrane, resulting in an omega current (red). At depolarized potentials at Winston (2012). (C) Comparison of current-voltage (I/V ) traces for wild type

which the S4 segment moves outward, the conductive pathway is closed by a hNav1.4 and R222G, with plots of raw I/V, linear leak, and normalized current deeper arginine and the omega current ceases. In contrast the replacement (linear leak subtracted from IV and normalized to gating current at +40 mV). of a deeper arginine (R3G) only opens the omega pore if the membrane is The mutation R222G causes HypoPP type 2. External solution contained + depolarized. (B) Homology model of domain I in hNav1.4 based on crystal 120 mM K and 1 μM TTX. Modified from Holzherr et al. (2010). similar to gain-of-function effects in excitatory neurons (Kahlig in patients reveals focal and generalized atrophy. Because some et al., 2006, 2010). patients with SMEI have a family history of febrile or afebrile Severe of infancy (SMEI), or Dravet syn- seizures, and in some families GEFS+ and SMEI overlap, SMEI drome, is characterized by clonic or tonic-clonic seizures that may therefore be regarded as the most severe phenotype of the occur in the first year of life, are often prolonged, and are typically GEFS+ spectrum (Mullen and Scheffer, 2009). SMEI is caused associated with . During the course of the disease, patients by mutations in SCN1A encoding Nav1.1. Most SMEI mutations develop additional afebrile generalized and partial seizures. Cog- cause loss-of-function due to nonsense mutations demonstrat- nitive deterioration appears in early childhood. In contrast to ing that haploinsufficiency of SCN1A is pathogenic (Oakley et al., GEFS+, the syndrome is resistant to pharmacotherapy, although 2011). The loss-of-function of Nav1.1 channels results in reduced stiripentol seems to have a positive effect by enhancing GABAergic action potential (AP) firing in hippocampal inhibitory neurons, neurotransmission. Cranial magnetic resonance (MR) imaging thus generating overexcitability. IV-R4C causes gain-of-function

www.frontiersin.org June 2012 | Volume 3 | Article 112 | 3 Jurkat-Rott et al. Omega pores in channelopathies , , Sokolov Sokolov Wu et al. , , , Franqueza , Kahlig et al. Vanoye et al. , Sokolov et al. , , Bendahhou et al. , Spampanato et al. Struyk et al. (2000) Kahlig et al. (2010) Wuttke et al. (2007) , , , Vanoye et al. (2006) , , Francis et al. (2011) , Sokolov et al. (2008) , Struyk et al. (2008) Chouabe et al. (2000) , , Martin et al. (2010) Kuzmenkin et al. (2002) Kahlig et al. (2006) Struyk and Cannon (2007) , , , , – – – – – et al. (2010) (2001) et al. (2007) Kuzmenkin et al.(2007) (2002) (2011) Miceli et al. (2012) Bartos et al. (2011) Jurkat-Rott et al. (2000) (2001) (2006) (2010) Mohammad-Panah et al. (1999) et al. (1999) Franqueza , Davies et al. Fodstad et al. , , Wuttke et al. (2007) Dedek et al. (2001) , Kim et al. (2007) , Hong et al. (2010) Carle et al. (2006)Matthews et al. (2009) Carle et al. (2006) Lee et al. (2009) Bulman et al. (1999) Bulman et al. (1999) Matthews et al. (2009) Lupoglazoff et al. (2004)(2004) Dedek et al. (2001) Millat et al. (2006) Jurkat-Rott et al.(2001) (2000) Barela et al. (2006)Escayg et al. (2000) Barela et al. (2006) Alekov et al. (2000) Rhodes et al. (2004) Rhodes et al. (2004) Itoh et al. (1998) Nakajima et al. (1999) Mohammad-Panah et al. (1999) Lossin et al. (2003) Lossin et al. (2003) et al. (1999) Vicart et al. (2004) Wu et al. (2008) channelopathies. R1132QR1135H R2 R3 R672H/G/S/C R2 R859CR1648H IIS4 IVS4 R1 R4 R534C R3 R243H/C R5 R1657C R7 + 2 , and Ca + ,Na + + + HypoPP-2/NormoPPHypoPP-2 R675G/Q/W R1129Q R3 IIIS4 R1 MyotoniaHypoPP-2 R225W R669H IIS4 R1 R3 GEFS LQT-1BFNS-PNHLQTS-2 R231C R207W/Q S4 K525N-R528P S4 R1 S4 R3 R0–R1 GEFS SMEI R1648C R4 SCN1A KCNQ1 KCNQ2 KCNH2 1. 1 1.4 SCN4A HypoPP-2 R222W IS4 R2 v v 7. 1 7. 2 11.1 v v v ChannelK Disease Mutation Location Reference genetics Reference function K Na K Na Table 1 | Voltage sensor mutations in K

Frontiers in Pharmacology | Pharmacology of Ion Channels and Channelopathies June 2012 | Volume 3 | Article 112 | 4 Jurkat-Rott et al. Omega pores in channelopathies , , , van , Feath- , Jurkat-Rott Morrill and , , Morrill and Cannon Francis et al. (2011) Groome et al. (1999) , , , Hans et al. (1999) Fan et al. (1996) Banderali et al. (2010) Jurkat-Rott et al. (1998) , , , , Kuzmenkin et al. (2007) , Kuzmenkin et al. (2007) , – – – – – – – – Morrill et al.(1999) (1998) Lehmann-Horn et al.Cannon (1995) (1999) Jurkat-Rott et al. (2009) et al. (2009) den Maagdenberg et al. (2004) Lapie et al. (1996) Makita et al. (1998) Lerche et al. (1996) erstone et al. (1998) Mitrovic et al. (1999) , , Matthews et al. , Lerche et al. (1996) Millat et al. (2006) Bezzina et al. (2003) Lehmann-Horn et al. , , , Yamagishi et al. (1998) , (1995) Chabrier et al. (2008) Matthews et al. (2009) Ptacek et al. (1994) Ducros et al. (2001) Battistini et al. (1999)Friend et al. (1999) Ducros et al. (2001) Kraus et al. (2000) Jurkat-Rott et al. (1994) (2009) Ophoff et al. (1996) Kraus et al. (1998) Splawski et al. (2000) Dumaine et al. (1996) Bezzina et al. (2003) Frigo et al. (2007) Makita et al. (1998) Ruan et al. (2007) Ruan et al. (2007) Splawski et al. (2000) Ptacek et al. (1992) Matthews et al. (2008) Alonso et al. (2004) Tonelli et al. (2006) R897SR900SR1239H/G IIIS4 IVS4R195Q R1 R583Q R2 R1346Q R2 R1661HR1664QR1667W IIS4 IIIS4 IVS4 R1 R1 R1 R1 R2 R3 R1644C/H R7 R814QR1623L/QR1626P IIS4 IVS4 R0 R3 R1 HypoPP-1 R528H/G IIS4FHM R1 R192Q IS4 R0 LQT-3 R225Q/W IS4 R3 Paramyotonia R1448H/C/S/L/P IVS4 R0 CACNA1S CACNA1A SCN5A 1. 5 1. 1 2.1 v v v Ca Ca Bold print: proof of omega currents or omega-analogous leaks in native tissue. Na

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by destabilizing fast inactivation and producing persistent currents (Dedek et al., 2001; Wuttke et al., 2007; Miceli et al., 2012). This (Rhodes et al., 2004) that may be inhibited by the sodium channel suppression of inhibitory neurons explains the hyperexcitability blocker ranolazine (Kahlig et al., 2010; Figure 3). that affects central as well as peripheral neurons. The latter locus Benign familial neonatal seizures (BFNS) are dominantly inher- of hyperexcitability leads to additional neuromyotonia also termed ited with a of 85%. The seizures manifest within the peripheral nerve hyperexcitability (PNH). This disorder is charac- first weeks of life and typically disappear spontaneously after weeks terized by short spells of spontaneous skeletal muscle overactivity to months. Seizures may have a partial onset or may appear as resulting in twitching, undulation, or rippling of the muscles and generalized. Accordingly, ictal EEGs show focal and generalized painful cramps, so that the phenotype for these S4 mutation carri- discharges. Interictal EEGs are mostly normal. The risk of seizures ers is BFNS-PNH. Omega currents have been implicated in BFNS recurring in adulthood is ∼15%, but psychomotor development is for the analogous Kv7.4 channel. There, corresponding R3W/Q usually normal. Causative mutations have been identified in Kv7.2 mutations generate non-specific depolarization-induced cation and Kv7.3 potassium channels encoded by KCNQ2 and KCNQ3 outward currents of ∼1% of the pore current amplitude, at the respectively that interact with each other and constitute the so- +40 mV voltage step (Miceli et al., 2012). In Kv7 channels, single called M-current, an important regulator of membrane potential mutations of R1 and R3 fulfill the concept of adjacent arginine near the AP threshold. Co-expression of heteromeric wild type and replacements because there is a glutamine at position R2. mutant Kv7.2/Kv7.3 channels usually reveals a potassium current reduction of ∼20–30% in the setting of haploinsufficiency. This LONG QT SYNDROME causes BFNS by depolarizing the membrane and facilitating AP fir- Long QT syndrome is named for an elongated QT interval in ing (Maljevic et al., 2008). In contrast, S4 mutations of the pattern the electrocardiogram. It is caused by lengthening of the repolar- R3W/Q (Figure 3) produce a dominant negative effect on co- ization phase of the cardiac ventricular AP. Because duration of expressed wild type brought about by a drastic depolarizing shift the cardiac AP is dispersed in different regions of the of the activation curve and slowing of the activation time course depending on regional channel expression, any disproportionate

FIGURE3|S4sequences of channels with an R0 to R6 mutation. neutral amino acid. An outward movement of S4 due to depolarization Positively charged residues in R0 to R6 positions are delineated with a opens the omega pore if R3 or R4 is replaced by a neutral amino acid. shaded background (gray). The constriction of the omega pore lies These gray-background arginines are boxed. The Shaker K+ channel serves between R2 and R3. Therefore an inward movement of S4 due to as reference. Neutral replacements of its arginines R2 and R3 have been hyperpolarization opens the omega pore if R1 or R2 is replaced by a described as proton transporters.

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prolongation of the AP increases the probability for re-entry of the part of the FHM phenotype. Up to 50% of cases are caused depolarization wave, early after-depolarizations and thus, arrhyth- by mutations in Cav2.1 encoded by CACNA1A. Current patho- mia. Associated ECG findings may be ventricular bigemini and genesis models of migraine with suggests cortical spreading torsade de pointes in which the QRS complex twists around the iso- depression which consists of an initial brief spike of increased neu- electric axis in the electrocardiogram. By this, and sudden ronal activity followed by long-lasting suppression of excitability death may result in young and otherwise healthy individuals. LQT spreading across the cortex at 1–3 mm/min. The depression wave syndrome (LQTS) is caused among others by loss-of-function of is associated with long-lasting depolarization associated with ele- potassium channels Kv7.1 encoded by KCNQ1 or Kv11.1 encoded vation of extracellular potassium and intracellular sodium. The by KCNH2 and by gain-of-function of the Nav1.5 sodium channel progress of FHM correlates to the succession of symptoms during encoded by SCN5A. the aura initiating the migraine attacks. FHM1 includes sporadic Kv7.1 currents deactivate very slowly allowing adaptive short- hemiplegic migraine with progressive cerebellar ataxia. The aura ening of the AP during tachycardia by incomplete deactivation. may be prolonged and confusion and loss of consciousness may Therefore, loss-of-function in Kv7.1 leads to espe- occur. In the interval, some families additionally present with cially at elevated heart rate due to physical or emotional stress. epilepsy, retinal degeneration, hypacusis, and persistent cerebellar R1C decreases current amplitude which explains the LQT in the dysfunction with Purkinje cell atrophy. While migraine is mainly electrocardiogram, but also produces a hyperpolarizing shift of caused by gain-of-function, loss-of-function may lead to addi- the voltage dependence of activation and tail currents, thereby tional ataxia. Of the seven known S4 mutations in Cav2.1, there explaining the additional atrial fibrillation in these patients (Bar- is functional data for only two. I-R0Q increases open probabil- tos et al., 2011). R5H/C produce a slowed rate of activation, a ity and channel density in vitro (Kraus et al., 1998; Hans et al., positive voltage shift of activation and/or dominant suppression 1999). In a knock-in mouse this mutation generates increased of co-expressed wild type (Franqueza et al., 1999; Mohammad- current density in cerebellar neurons, enhanced transmission at Panah et al., 1999; Chouabe et al., 2000). R1, R3, and R5 mutants the neuromuscular junction, and reduced threshold and increased all fulfill the concept of adjacent arginine replacements because velocity of cortical spreading depression, effects that are compat- there is a glutamine at R2 and a histidine at R4 (Figure 3). ible with an omega current (van den Maagdenberg et al., 2004; Kv11.1 activates and deactivates relatively slowly in comparison Figure 3). II-R1Q shifts activation to the left but decreases peak to a rapid inactivation. LQT mutations suppress repolarization of current (Kraus et al., 2000). Omega currents produced by either the myocardial AP and lengthen the QT interval by either loss-of- of these mutations have yet to be demonstrated. function or haploinsufficiency. A dominant negative effect may be achieved by current reduction of the tetrameric channel complex. SODIUM CHANNEL MYOTONIA AND PARAMYOTONIA For the LQT double mutation of R0/R1, there is no functional data Myotonia is an involuntary slowed relaxation after a forceful vol- but there two adjacent arginines are changed (Figure 3). The point untary muscle contraction, experienced by the patient as muscle mutation R3C reduces current amplitude and accelerates deacti- stiffness. After making a forceful fist or eyelid closure, the patient vation thereby slowing repolarization of the cardiac AP in its final cannot reopen their hand or eye. Repetition decreases the myoto- phase (Nakajima et al., 1999). nia, a phenomenon called warm-up. Electrical hyperexcitability of Nav1.5 initiates the cardiac AP. LQT mutants frequently con- the muscle fiber membrane is the basis for myotonia, apparent in duct a persistent inward current during membrane depolarization the form of repetitive APs in the EMG. Needle insertions into the and single channel recordings show fluctuation between normal resting muscle elicit myotonic bursts, i.e., runs of APs with ampli- and non-inactivating gating modes. I-R3Q/W shift the steady-state tude and frequency modulation that sound like dive bombers. inactivation curve to the right (Bezzina et al., 2003). IV-R0L/Q Autosomal dominantly inherited myotonia can be caused among delay inactivation (Makita et al., 1998), IV-R1P impairs inactiva- others by mutations in SCN4A, the gene encoding the alpha tion or shifts the steady-state inactivation curve to the left (Ruan subunit of the voltage-gated sodium channel of skeletal muscle, et al., 2007) and decreases inactivation upon stretch modula- Nav1.4 (Heine et al., 1993), essential for generation of the muscle tion (Banderali et al., 2010). Mutations IV-R7C/H show dispersed fiber AP. Most mutations destabilize the fast-inactivated state so re-openings (Dumaine et al., 1996; Figure 3). It has yet to be estab- that the channel inactivates slower and incompletely (Lerche et al., lished that omega currents play a role in the pathogenesis of Kv or 1993). The resulting subthreshold membrane potential facilitates Nav LQTS channelopathies, but the finding of mutations in these AP generation. I-R3W causes typical myotonia with warm-up phe- cardiac disorders in R1–R3 residues of S4 raises that possibility nomenon and transient weakness (Lee et al., 2009). However, no (Table 1; Sokolov et al., 2007). functional data are available. Paramyotonia (PC) is also caused by dominant SCN4A mis- HEMIPLEGIC MIGRAINE sense mutations. The cardinal symptom is cold-induced muscle Familial hemiplegic migraine (FHM) presents with characteristic stiffness that increases with continued activity called paradoxical unilateral migraine headaches accompanied by nausea, phono-, myotonia, or paramyotonia for short. In most families, on inten- and photophobia. Episodes are typically precipitated by an aura sive cooling the stiffness gives way to flaccid weakness or even with symptoms of both hyper- and hypo-excitability such as apha- to paralysis which is caused by severe membrane depolarization sia, dysarthria, vertigo, homonymous hemianopsia, cheiro-oral (Lehmann-Horn et al., 1987; Lerche et al., 1996). Families with paresthesias, and some degree of (primarily) unilateral paresis. IV-R0H/C/S/L/P also have episodes of generalized periodic paral- Additionally, ataxia and cerebellar degeneration are frequently ysis (Lehmann-Horn and Jurkat-Rott, 1999). Such attacks occur

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spontaneously and can be triggered by rest or elevated serum are still active at the usual resting potential, but are closed potassium. They are of much shorter duration than the cold- by depolarizations large enough to activate and thus, move S4 induced weakness that usually lasts for several hours even after (Figure 3). Their amplitude is about 1% of the alpha pore cur- the muscles are re-warmed. During cooling, PC muscle fibers rent. Substitution of these arginines by the bulky histidine residue slowly depolarize to an extent greater than that observed in normal enables proton transport only; other cations do not seem to have muscle fibers. The depolarization is associated with long-lasting enough space to pass (Sokolov et al., 2007; Struyk and Can- bursts of APs which cease when the membrane potential reaches non, 2007; Francis et al., 2011). Substitution of these arginines −55 mV. At this potential sodium channels inactivate so that with uncharged resides such as glycine allow an inwardly recti- muscle becomes inexcitable and paralyzed. Causative mutations fying flow of small monovalent cations whereby potassium and generally slow the entry into, and accelerate recovery from, fast cesium are preferred over sodium or lithium (Sokolov et al., 2007; inactivation (Yang et al., 1994). IV-R0 mutations slow deactiva- Struyk et al., 2008). Figure 2C shows current-voltage relation- tion, in addition to destabilizing the fast-inactivated state (Fan ships for R222G/hNaV1.4 channels from experiments performed et al., 1996; Lerche et al., 1996; Featherstone et al., 1998; Groome in our lab as an example (Holzherr et al., 2010). Omega pores et al., 1999; Mitrovic et al., 1999). No omega current is observed conduct currents that show an above-linear increase in ampli- (Francis et al., 2011), possibly because R0 is located outside of the tude with hyper- or depolarization although the electrical field is canaliculus (Figures 2 and 3). focused to a single amino acid (Ohm resistor) and not constantly increasing within the membrane (constant field theory). The non- PERIODIC PARALYSIS linearity of the omega current reflects the stochastic process of Autosomal dominant hypokalemic periodic paralysis (HypoPP) is a voltage-dependent open probability and follows a Boltzmann characterized by episodes of flaccid generalized muscle weakness distribution. accompanied by low serum potassium levels. Triggers for the weak- Guanidinium, a derivative of arginine, has greater than 10 ness are carbohydrates and insulin with subsequent , times the conductance through the omega pore than monova- rest after exercise, and cooling. Oral potassium administration lent cations, while divalent and trivalent cations block the omega accelerates the recovery of muscle strength. Native patient mus- pore at mM concentrations (Sokolov et al., 2010). In that study, a cle fibers show long-lasting membrane depolarization paradoxi- screen of guanidine derivatives as potential blockers of the path- cally triggered by lowering of extracellular potassium, resulting in ogenic omega current in HypoPP yielded one compound, 1-(2-4 an inactivation of sodium channels necessary for AP generation xylyl) guanine that exhibited similar potency for block compared (Jurkat-Rott et al., 2000). This effect is exacerbated in the pres- to the divalent cations tested. In addition, there is an inward cation ence of insulin (Bond and Gordon, 1993; Ruff, 1999). Permanent leak current at hyperpolarized potentials in native muscle of R1H progressive weakness is additionally found in ∼60% of patients knock-in mice with omega features (Wu et al., 2011). The conclu- whereby muscle degenerates and is increasingly replaced by fatty sion from these alternative approaches to study omega currents is tissue. The causative mutations are located in the skeletal muscle that they contribute to the depolarization leading to inexcitability Cav1.1 encoded by CACNA1S (HypoPP-1) and and weakness. Nevertheless, despite similarities of sodium channel the skeletal muscle sodium channel Nav1.4 encoded by SCN4A. omega currents with those in Shaker produced by R1H mutations, Given the finding that all but one of these mutations neutralize their blockade is quite different, suggesting differences in shape, a positively charged S4 residue, it is not surprising that a role for size, and exact cation pathway along the canaliculi of these chan- omega currents as a causal factor in the pathogenesis of the chan- nels. Perhaps this phenomenon is related to the postulation that nelopathies has been most clearly demonstrated for the periodic two adjacent mutated charges promote omega currents in voltage- paralyzes. gated potassium channels, a requisite that does not hold true for Functional studies of the alpha currents of either of these sodium channels. channels do not satisfactorily explain the phenotype. In Cav1.1, In contrast to the hyperpolarization-induced omega currents II-R1H/G and IV-R2H/G (Figure 3) lead to loss-of-function in from substitutions of II-R1 and II-R2, II-R3G/Q/W mutations the sense of current amplitude reduction, slowing of activation, generate Nav1.4 cation currents that are activated by depolariza- reduced open probability, left shift of steady-state inactivation, tion (Figure 3). This means that the omega currents are conducted reduced amplitude and broadening of APs (Lehmann-Horn et al., when S4 is in the activated or inactivated state,but not in the resting 1995; Lapie et al., 1996; Jurkat-Rott et al., 1998; Morrill et al., 1998; state. This omega current is deactivated at hyperpolarized mem- Morrill and Cannon, 1999; Kuzmenkin et al., 2007). None of these brane potentials (Sokolov et al., 2008). The associated phenotype gating defects are expected to lead to increased depolarization. differs slightly in the ictal serum potassium levels which can be low Similarly, in Nav1.4, II-R1H, II-R2H/G/S/C, and II-R3G/W/Q or normal (normokalemic periodic paralysis, NormoPP). Also, the show reduced current amplitude, left shift of steady-state fast inac- reaction to oral potassium administration may be different than tivation, enhanced fast and slow inactivation, slowed recovery after for HypoPP – anything from amelioration to worsening of the long depolarizations, and reduced AP amplitudes (Bulman et al., weakness. 1999; Jurkat-Rott et al., 2000; Struyk et al., 2000; Bendahhou et al., For II-R1H/Q and IV-R2H/G Cav1.1 channel mutations 2001; Kuzmenkin et al., 2002; Carle et al., 2006; Wu et al., 2008, (Figure 3), no direct omega currents have been shown. However, 2011). These defects would not contribute to depolarization either. in native patient muscle there is an inward cation leak current at In Nav1.4, II-R1H, II-R2H/G/C/S, and III-R2Q mutations gen- hyperpolarized potentials with omega features (Jurkat-Rott et al., erate omega currents that are activated by hyperpolarization, 2009). Also, based on these measurements, the periodic paralysis

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episodes can be successfully modeled. The underlying idea is that Transitions between the P1- and P2-states can occur when any depolarized muscle fibers will be paralyzed, and that polarized trigger drives the system closer to a limit point or shifts a limit fibers will be fully functional. If transitions between these two point. If the limit points of both states (LP1 and LP2) are situated situations are reversible, the membrane potential is bi-stable. at different K+ values, the system shows hysteresis (Figure 4C). If external K+ is considered as a control parameter, it is shown to BI-STABILITY OF MEMBRANE POTENTIAL play an important role in driving the system to its limits, and set P-states are the electrically stable resting membrane potentials of the stage during “physiological” hypokalemia, for the paradoxical a cell in the nomenclature of Jurkat-Rott et al. (2009). Normal depolarization triggered by a slight increase in sodium conduc- cells reveal a bimodal distribution around the stable membrane tance provided by an omega current. A significant role for the potentials P1 and P2. At physiological conditions, most fibers are activated omega pore as a causal factor in the pathogenesis of highly negative polarized and in the excitable P1-state which fol- HypoPP is supported by the computer model whose details are lows the predictions of the Goldman–Hodgkin–Katz equation. given in the Appendix and whose parameters are given in Table 2. + Upon an increased sodium conductance, e.g., by omega pores, the A drop of serum K will decrease K ir conductance. Although fraction of cells in the (depolarized) P2-state is increased. With the Nernst equation predicts hyperpolarization when external K+ slow depolarization, voltage-gated sodium channels might inacti- is reduced, the hypokalemia-induced reduction of K ir conduc- vate via closed-state inactivation and render the cells inexcitable. tance may shift the cells from the P1- to the P2-state (Figure 4C). In contrast, cells that remain in the P1-state are only slightly depo- Therefore, cells show a bimodal distribution of membrane poten- larized and thus still excitable. This bi-stability is the result of tials especially at low K+. This phenomenon is called paradoxical inwardly rectifying potassium channels (K ir), which open with depolarization (Jurkat-Rott and Lehmann-Horn,2007; Struyk and hyperpolarization and close with depolarization (Figure 4A). The Cannon,2008; Jurkat-Rott et al.,2009). Not surprisingly,decreased bi-stability is the reason that only a small increase in sodium con- conductance in mutant K ir channels contributes to the pathogen- ductance, e.g., by an omega pore, is required to shift cells from esis of episodic paralysis associated with thyrotoxicosis (K ir 2.6; the P1- to the P2-state (Figure 4B). Vice versa, a repolarizing con- Ryan et al., 2010; Cheng et al., 2011), periodic paralysis and car- ductance can shift the membrane back to the P1-state (Figure 4). diac arrhythmia in Tawil–Anderson’s syndrome (K ir 2.1; Plaster

+ FIGURE4|Effects of omega currents schematically (A) and in a increasing [K ]o takes the potential along the curve until LP2 is reached. LP2

computer simulation (B). (A) The current-voltage relationship of K ir is the starting point for the repolarization. The curves of LP1 and LP2 meet in potassium channels shows a voltage range characterized by a “negative” the cusp point, CP.The region inside the cusp (bounded by LP1, LP2, and CP) resistance. This negativity leads to membrane bi-stability. Whereas small is bi-stable. In contrast omega pore conductances larger than 18 μS/cm2 instantaneous changes of P1 will be compensated for (P1 is therefore a result in gradual depolarization without bi-stability. The model reveals that an stable membrane potential), larger depolarizing artifacts will cause a jump of omega pore shifts LP1 and LP2 (the cusp) to the right, i.e., less severe P1 to the limit point LP2. P1 can be regained by substantial repolarizing hypokalemia is required to shift cells from the P1- to the P2-state. The influences like the Na/K pump. (B) The curve is downwardly shifted by an membrane potentials yielded by the computer simulation were compared omega Na+ current. Limit point LP1 is a very instable membrane potential with values measured for muscle fibers by use of microelectrodes: open that will be shifted to P2 by even smallest instantaneous changes. In the symbols stand for human controls (−83 ± 5 mV in 95% of fibers at 4 mM K+, P2-state the will be electrically stable. (C) Membrane −99 ± 3 mV for 87% at 1.5 mM K+, and −58 mV for 91% at 1 mM K+); filled + potentials P for various [K ]o values and for various omega pore symbols for HypoPP patients with either Cav1.1-R1239H (circle: −74 ± 5mV 2 + + conductances (in μS/cm ). Reducing [K ]o first leads to hyperpolarization until in 76% of fibers at 4 mM K )orCav1.1-R528H (triangle: −75 ± 5mVin91% the limit point (LP1) is reached at which the membrane potential becomes of fibers at 4 mM K+); at 1.5 mM K+, 95% of the patients’ fibers −56 mV instable and jumps to the depolarized state of about −58 mV. From there, (square; n = 127). Modified after Jurkat-Rott et al. (2010).

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Table2|Parameters of the catastrophe model.

Parameter Value Unit Parameter Value Unit

GATING PARAMETERS OF NaV,KV, AND CLC PHYSICAL AND STRUCTURAL PARAMETERS ˆ Eomega −110 mV Df 70 μm Aomega 34 mV η 2– ˆ 2 ¯ μ 2 gNaV 268 mS/cm cm 1 F/cm ˆ 2 gNaL 2.75 μS/cm zFix −1. 3 – ˆ 2 −1 −1 gKV 21.6 mS/cm R 8.3145 J mol K 2 gˆClC 0.5 mS/cm T 311.15 K 2 gˆir 0.0485 mS/cm F 96485 C/mol 2 K S 0.0475 mM 2 K K 5mM et al., 2001) and underlies barium-induced augmentation of para- δ 0.2 – doxical depolarization of skeletal muscle fibers in low external K+ ˆ −2 −1 Jp 8 Nmol cm s (K ir 2.1; Struyk and Cannon, 2008). Decreased conductance of the KmK 1mM ATP-sensitive K channel in skeletal muscle may also be a causative KmNa 8mM factor in hypokalemic-induced depolarization, at least in response × −10 −2 −1 −1 LH 1.25 10 lcm N s to insulin (Tricarico et al., 1998). It has been proposed that part of EXTRA AND INITIAL INTRACELLULAR CONCENTRATIONS + the therapeutic effect of acetazolamide in HypoPP is its action to ˆ Na+ + ˆ K ce ce 141.5 mM − promote the activation of several different types of skeletal muscle ˆ Cl ce 108 mM − potassium channels, which may offset the decreased conductance cˆ Fix 25.8 mM e of K ir (for review see Matthews and Hanna, 2010). ˆ Neutral ce 10 mM If the omega pore conductance is >18 μS/cm2 (for the condi- ˆ Na+ ci 15.6 mM + tions in Figure 4C), the bi-stability is replaced by a single stable cˆ K 132.3 mM i resting potential. The value depends on the conductance ratio ˆ S ci 8.9 mM − g K/gω with g K carried by additional (e.g., delayed rectifying) Kv ˆ Cl + ci 4.3 mM − channels which are less dependent on extracellular K than are cˆ Fix 124.1 mM i K ir channels. In this monostable system, the membrane potential + − EH 18.5 mV can vary between −74 and −58 mV (Figure 4C). μ V i 3.38 l Usually the majority of normal cells do not become inexcitable c¯ 55555 mM + H2O in response to decreased serum K , because the limit points are PARAMETERS FOR ION AND WATER FLUXES located at very low K+ that do not occur physiologically. However αˆ −1 m 0.288 ms in the presence of an omega current, the limit points are shifted to βˆ −1 m 1.38 ms higher K+, and transitions between the states are possible under ˆ − Em 42 mV physiological conditions. HypoPP seems to be caused by such a K αm 10 mV shift. For instance, computer simulation of the effect of R669H K βm 18 mV on V rest in response to low external K+ predicts a decrease in the αˆ −1 h 0.0081 ms threshold for paradoxical depolarization of muscle fibers, similar βˆ −1 h 4.38 ms to that observed experimentally with barium poisoning (Struyk ˆ − Eh 45 mV and Cannon, 2008). Cation ionophores like amphotericin B and K αh 14.7 mV gramicidin can mimic the omega pore current and serve as a suc- K βh 9mV cessful, pharmacological in vitro model for HypoPP (Jurkat-Rott ˆ − ES 85 mV et al., 2009). Based on the ionophore experiments, the fraction of AS 6mV fibers in the P2-state may correspond to the chronic weakness in αˆ −1 n 0.0131 ms HypoPP patients. Typical triggers such as glucose and insulin caus- βˆ −1 n 0.067 ms ing hypokalemia will further increase this fraction and precipitate ˆ − En 37 mV a paralytic attack. The pharmacological computer model predicts K αn 7mV that smaller omega currents require a more severe hypokalemia K βn 40 mV of the patients to shift cells into the P2-state than larger omega Eˆ −30 mV hK currents. K τhK 25.75 mV A 7. 5 m V hK SODIUM OVERLOAD, CYTOTOXIC EDEMA, AND TISSUE ˆ Ea 70 mV DEGENERATION Aa 150 mV By tackling the question as to whether sodium influx (e.g., through omega pores) increases the cytoplasmic sodium concentration, a (Continued) 23Na MR sequence was developed (Nielles-Vallespin et al., 2007)

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FIGURE 5 | 1H and 23Na measurements in the calf muscles of arrows pointing at highest Na signal intensities) and their improvement HypoPP patients. (A–F) T2-weighted STIR 1H (left) and 23Na-MR after treatment. The central reference contains 0.3% NaCl solution; images (right) from a healthy control (A,B) and the propositus of a occasional side tubes containing 0.3% NaCl in 1% agarose (left) and

HypoPP family, a 37-year-old female harboring the Cav1.1-R1239H 0.6% NaCl in H2O (right) were additional standards. (G–I) Axial mutation (C–F). The images in (C,D) were taken before treatment and T1-weighted MR images from family members patients: the female’s the images in (E,F) were taken after treatment with 250 mg/d 80-year-old grandmother whose limb muscles were almost completely acetazolamide for 4 weeks. Note the very high proton intensities in replaced with fat (G), the female’s 35-year-old sister (I), and the STIR (C) and the elevated Na+ concentration before treatment (D), 55-year-old uncle (H,G). Modified after Jurkat-Rott et al. (2009). that indeed revealed a muscle sodium overload in HypoPP patients 2011). Likewise, ictal edema is associated with neuronal degener- (Figure 5). In addition, 1H-MR images (MRI) with a short-tau ation and sclerosis of the in FSs (Scott et al., 2003, inversion recovery (STIR) sequence displayed an edema in the 2006; Sokol et al., 2003). Therefore, it is possible that intracellular muscles of these HypoPP patients (Figures 5A–F). Therefore the sodium overload may contribute to neuronal as well as muscle question arose as to whether the edema was caused by a cyto- channelopathies. plasmic sodium accumulation and osmotic imbalance, or by an interstitial edema, e.g., by an inflammation. To answer this ques- REPOLARIZATION AND RELIEF OF THE CYTOTOXIC EDEMA tion the MRI technique was further improved to a 23Na MRI Guided by the experience that acetazolamide has favorable effects inversion recovery sequence (Na-IR) which partially suppresses on the episodic weakness in the periodic paralyzes (Resnick the signal raised by free sodium in the extracellular fluid and et al., 1968), this agent was administered to HypoPP patients and thus mainly represents cytoplasmic sodium (Nagel et al., 2011). reduced the chronic weakness of the patients as well as the sodium In the HypoPP patients the nature of the edema was cytoplasmic. overload and edema in the MRI (Figure 5). In the pharmacologi- It seemed to be cytotoxic as all patients who showed the edema cal in vitro model of HypoPP using an ionophore, acetazolamide presented with a chronic weakness and muscle degeneration with shifted many fibers from the P2- to the P1-state. This in vitro age (Figures 5G–I). effect was considered to be responsible for the positive, in vivo In parallel with the concept of degeneration caused by sodium effects (Jurkat-Rott et al., 2009). Thus, drugs that repolarize the overload, ictal cytotoxic edema during the long-lasting depolar- fiber membrane may prevent progression of the cell degeneration ization (spreading depression) has been frequently reported in by reducing the cytotoxic edema. Supporting the idea that FHM FHM (for example Chabriat et al., 2000; Butteriss et al., 2003) and epilepsy share similar pathomechanisms to HypoPP, it is not and has been suggested to sustain the long-lasting aura (Iizuka surprising that acetazolamide is standard treatment option for et al., 2006) and contribute to neurodegeneration (Carreño et al., both epilepsy (Wolf, 2011) and FHM (Russell and Ducros, 2011).

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ACKNOWLEDGMENTS 1R15NS064556-01 to James Groome. Frank Lehmann-Horn is The work was supported by research grants from the Else Kröner- endowed Senior Research Professor of the non-profit Hertie- Fresenius-Stiftung and the BMBF (IonoNeurOnet), both to Karin Foundation. We thank M. Fauler and V. Winston (ISU) for their Jurkat-Rott and Frank Lehmann-Horn (2010_A27), and by NIH contributions to the computer models.

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dysfunction in SCN1A-associated Wu, F., Mi, W., Burns, D. K., Fu, Y.,Gray, Yamagishi, H., Furutani, M., Kamisago, could be construed as a potential epilepsy. J. Gen. Physiol. 127, H. F., Struyk, A. F., and Cannon, M., Morikawa, Y., Kojima, Y., Hino, conflict of interest. 1–14. S. C. (2011). A sodium channel Y., Furutani, Y., Kimura, M., Ima- Vicart, S., Sternberg, D., Fournier, E., knockin mutant (NaV1.4-R669H) mura, S., Takao, A., Momma, K., Received: 22 April 2012; accepted: 23 May Ochsner, F., Laforet, P., Kuntzer, mouse model of hypokalemic peri- and Matsuoka, R. (1998). A de novo 2012; published online: 11 June 2012. T., Eymard, B., Hainque, B., odic paralysis. J. Clin. Invest. 121, missense mutation (R1623Q) of the Citation: Jurkat-Rott K, Groome J and and Fontaine, B. (2004). New 4082–4094. SCN5A gene in a Japanese girl with Lehmann-Horn F (2012) Pathophysio- mutations of SCN4A cause a Wu, L., Wu, W., Yan, G., Wang, X., and sporadic long QT sydrome. Muta- logical role of omega pore current in chan- potassium-sensitive normokalemic Liu, J. (2008). The construction and tions in brief no. 140. Online. Hum. nelopathies. Front. Pharmacol. 3:112. periodic paralysis. Neurology 63, preliminary investigation of the cell Mutat. 11, 481. doi: 10.3389/fphar.2012.00112 2120–2127. model of a novel mutation R675Q Yang, N., Ji, S., Zhou, M., Ptacek, L. This article was submitted to Frontiers Wallinga, W., Meijer, S. L., Alberink, in the SCN4A gene identified in a J., Barchi, R. L., Horn, R., and in Pharmacology of Ion Channels and M. J., Vliek, M., Wienk, E. D., and Chinese family with normokalemic George, A. L. Jr. (1994). Sodium Channelopathies, a specialty of Frontiers Ypey, D. L. (1999). Modelling action periodic paralysis. Zhonghua Yi channel mutations in paramy- in Pharmacology. potentials and membrane cur- Xue Yi Chuan Xue Za Zhi 25, otonia congenita exhibit similar Copyright © 2012 Jurkat-Rott, Groome rents of mammalian skeletal mus- 629–632. biophysical phenotypes in vitro. and Lehmann-Horn. This is an open- cle fibres in coherence with potas- Wuttke, T. V., Jurkat-Rott, K., Paulus, Proc. Natl. Acad. Sci. U.S.A. 91, access article distributed under the terms sium concentration changes in the W., Garncarek, M., Lehmann- 12785–12789. of the Creative Commons Attribution T-tubular system. Eur. Biophys. J. 28, Horn, F., and Lerche, H. (2007). Non Commercial License, which per- 317–329. Peripehral nerve hyperexcitabil- Conflict of Interest Statement: The mits non-commercial use, distribution, Wolf, P. (2011). Acute drug adminis- ity due to dominant-negative authors declare that the research was and reproduction in other forums, pro- tration in epilepsy: a review. CNS KCNQ2 mutations. Neurology 69, conducted in the absence of any com- vided the original authors and source are Neurosci. Ther. 17, 442–448. 2045–2053. mercial or financial relationships that credited.

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APPENDIX PATHOGENESIS MODEL – CATASTROPHE ON THE CUSP Using catastrophe theory, the effects of an omega current were simulated in a one-compartment computer model of skeletal mus- cle fibers. Both changes of ion concentrations and consecutive water shifts were considered. The size of the T-tubular system was accounted for by a specific factor. It was not considered a separate compartment since the low-resistance T-tubular openings justified the assumption of an equilibrium. Parameters of the catastrophe model are given in Table 2; notations and abbreviations at the end of the Appendix. + The membrane was equipped with the following conductances: a background Na leak conductance (g NaL), the omega pore con- + + + ductance (g leak), a voltage-gated Na (g NaV ) and K (g Kv ) conductance, an inward-rectifying K (g IR) and a voltage-dependent − + + Cl (g Cl) conductance, a hydraulic conductivity (LH), and a Na /K pump flux (J p). The model was based on the charge-difference approach (3) in which the membrane potential E is calculated from the amount of charges in the intracellular compartment Qi and the membrane area Am:

Qi Vi E = with Am = 4 · η · c¯m · Am Df

Qi = F · nion · zion ion ∈ {Na, K , Cl, Fix, S} ion

V i was the cell volume, Df the fiber diameter and η a factor correcting the size of the T-tubular membrane. The cell was only permeable for Na, K, Cl, and H2O. Fix are intracellular anions (e.g., proteins, phosphates), which – in this model – cannot pass through the membrane. Their overall charge valence zFix is set to −1.3. This was necessary to avoid very negative membrane potentials at low potassium concentrations. S are the positive charged ions, that block inward-rectifying K+ channels from inside upon depolarization. 2+ Its valence zS is set to +2, so they represent Mg . Multiple positively charged polyamines that increase the strength of rectification were not considered. The amounts of ions and water in the cell n are state variables in this model. They are integrated over time by numerically solving the differential equations that describe their fluxes J:

t ns =ˆns + Js · dt s ∈ {Na, K , Cl,H2O} 0

∈ dns c p c NaV,KV,Kir, ClC, Aqua J = = J + Js s dt s p ∈ Na/K − pump c p

The fluxes of ions through channels J c and water through an aquaporin J Aqua (modeled by a simple hydraulic conductivity) are s H2O computed by the following equations: ∈ + + − c =− −1 · c · ( − ) ion Na ,K ,Cl Jion F gion E Eion c ∈ NaL, Leak, NaV, KV, Kir, ClC J Aqua = L ·ˆc · ΔΠ H2O H H2O

The driving forces for the fluxes were the electrochemical gradients (E − Eion) for ions or the osmotic pressure difference ΔΠ for water: R · T cion E = · ln e ion · ion zion F ci

ΔΠ = · · sol − sol ∈ { } R T ci ce sol Na, K, Cl, Fix, S sol

The Na/K pump was modeled by −2 −3 = ˆ · + K + · + Na + p =− · p = · Jp Jp 1 KmK ce 1 KmNa ci , JNa + 3 Jp and JK + 2 Jp

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ˆ The parameters K mK and K mNa were taken from Wallinga et al. (1999). The maximum pump flux Jp was chosen to get an intracellular + Na+ Na concentration ci of 15 mM at control conditions. Intracellular ion concentrations were calculated from the cell volume V i and the amount of ions: ns = s ∈ { } Vi ¯ ,s Na, K, Cl, Fix, H2O cH2O

n cion = ion ,ion∈ {Na, K, Cl, Fix} i Vi + + − The voltage-gated Na (NaV), K (KV), and Cl (ClC1) conductances were based on the Hodgkin–Huxley type formalism; identifiers had their classical meaning. The equations taken from Wallinga et al. (1999) are given here for completeness:

− Na+ =ˆ · 3 · · K+ =ˆ · 4 · Cl =ˆ · 4 gNaV gNaV m h S, gKV gKV n hK and gClC gClC a

The gating variables m, h, S (NaV slow inactivation), n, and hK (inactivation of KV) were state variables and defined by the following differential equations dy = α · 1 − y − β · yy∈ {m, n, h} dt y y

αˆ · − ˆ m E Em − − ˆ ˆ E Em Kβm αm = and βm = βm · e − E−Eˆ Kα 1 − e m m

ˆ ˆ − E−E Kα β α = αˆ · e h h and β = h h h h ˆ − E−E Kβ 1 + e h h dS S∞ − S 60 1 = τ = ∞ = , S 2 and S dt τ + · ( + ) E−Eˆ A S 0.2 5.65 E 90 100 1 + e S S

αˆ · − ˆ n E En − − ˆ ˆ E En Kβn αn = and βn = βn · e − E−Eˆ Kα 1 − e n n

dh h ∞ − h − E−E K 1 K = K K τ = hK hK = , hK e and hK ∞ dt τ E−Eˆ A hK 1 + e hK hK

The gating variable of ClC1 was assumed to reach the steady-state instantaneously

1 a = E−Eˆ A 1 + e a a For the inward-rectifying potassium channels the same model as in Wallinga et al. (1999) was used, parameters were chosen to get K+ = μ 2 appropriate values for the limit points at control conditions and a resting conductivity at ce 3.5 mM of 41 S/cm . The equations were: + 2 ˆ · K + gIR c + +   R −δ + K = · = K = K · EK F RT gIR gIR y , gIR 2 , cR ce e K+ + K+ KIR cR

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⎡ ⎛ ⎞⎤ 2 −1 K+ ⎢ K S ⎜ cR ⎟⎥ y = 1 − ⎣1 + IR · ⎝1 + ⎠⎦ S · 2(1−δ)EF/RT K+ ci e KIR

The constant background sodium leak conductance was then adapted to reach a K+ to Na+ permeability ratio between 0.01 and 0.02 K+ = ) and a membrane potential of approximately –86 mV (at ce 3.5 mM . For modeling the H+ leak conductance, it was assumed – based on the 31P MR spectroscopy measurements (Jurkat-Rott et al., 2009) – that the H+ equilibrium potential was constant and further, that every H+ was exchanged by one Na+ ion, presumably by a Na+/H+ exchanger. Thus the omega current was treated as a sodium current (indirectly) driven by the electrochemical gradient for protons. Its conductance was modeled as a pore with a voltage-dependent open probability that follows a Boltzmann distribution. The equation

− ˆ 1 E−Eleak Aleak gomega =ˆgomega · 1 + e was fitted to the HypoPP muscle fiber results. The Na+ flux (indirectly) mediated by the omega pore was calculated by pH + − R · T 10 i Na =− 1 · · − + + = · J F gomega E EH with EH ln omega F 10pHe

The model was implemented in MatLab Version 7.3 (The Mathworks, Inc.). Initial-value problems were solved with ODE15s, a solver for sets of stiff ordinary differential equations. Then bifurcation analysis was done by using the continuation routines of the toolbox CL_MATCONT Version 2.4.

NOTATIONS AND ABBREVIATIONS Indices

Aqua water channel c channel like voltage-gated Na+ channel, voltage-gated K+ channel ClC1 voltage-dependent Cl− channel e extracellular f fiber i intracellular ion Ion: Na+,K+,Cl− IR inwardly rectifying K+ channel + KV voltage-gated K channel Omega omega pore conductance + NaL background Na channels + NaV voltage-gated Na channel p Na+/K+-pump s substance: sodium, potassium, chloride, fixed anion, water ∞ steady-state

Identifiers

a gating variable of the Cl− channel

Am membrane area c concentration Cl− chloride ion cm specific membrane capacity

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Df fiber diameter + 2+ δ electrical distance from outside where binding site for K and S is localized in K ir E membrane potential

Eion equilibrium potential (Nernst) F Faraday constant Fix intracellular fixed anions (e.g., proteinates, phosphates) g conductance H+ protons η scaling factor relating the size of T-tubular membrane to sarcolemma J flux K+ potassium ion + + KmK sensitivity of the Na /K -pump for extracellular potassium + + KmNa sensitivity of the Na /K -pump for intracellular sodium + K K dissociation constant of K on K ir channels 2+ K S dissociation constant of S on K ir channels

LH hydraulic conductivity m, h, S gating variables of the voltage-gated Na+ channel (activation, fast, slow inactivation) + n, hK gating variables of the voltage-gated K channel (activation, inactivation) n amount of substance Na+ sodium ion Π osmotic pressure Q charge R gas constant 2+ S intracellular fixed cation (cytoplasmic blocker of K ir channels) T absolute temperature t time

V i cell volume

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