Closed and open state dependent block of potassium channels cause opposing effects on excitability – a computational approach

Block of voltage-gated potassium (Kv) channels has been demonstrated to affect neuronal activity described as increasing excitability. The effect has been associated with a closed-state dependent block. However, the block of Kv channels in e.g. local anesthetic and antiarrhythmics, is open state-dependent. Since the reduced excitability in this case mainly is due to sodium channel block, the role of the Kv channel block is concealed. The present investigation aims to analyse the specific role of state-dependent Kv channel block for excitability. Using a computational approach, with introduced blocked states in the Kv channel of the Frankenhaeuser-Huxley axon membrane model, we calculated the effects on threshold, firing and presynaptic Ca influx. The Ca influx was obtained from an N-type Cav channel model linked to the Frankenhaeuser-Huxley membrane. The results suggested that a selective block of open Kv channels decreased the rate of repetitive firing and the consequent Ca influx, thus challenging the traditional view. In contrast, presence of a closed-state block, increased the firing rate and the Ca influx. These findings propose that Kv channel block may either increase or decrease cellular excitability, thus highlighting the importance of further investigating the role of state-specific blocking mechanisms.

There is strong experimental evidence supporting that Kv channel specific blockers increase spontaneous activity, thereby inducing increased neuronal excitability, e.g. the small-molecule Kv channel blockers 4-aminopyridine (4-AP) and tetraethylammonium (TEA) salts have been shown to be epileptogenic in mammalian nervous systems [5][6][7][8] . Similarly, several Kv channel specific peptide and protein toxins from so evolutionary diversified organisms as snakes, scorpions, sea anemones, and cone snails have been shown to be epileptogenic 9,10 . Many of these cases have been shown to concern a resting closed state binding mechanism. Thus, 4-AP has been suggested to interact with Kv1 channels by first binding in an open state and then inducing a bound closed state 11 . This would render functional effects corresponding to a closed state dependent block. An alternative model suggests 4-AP binding in both closed and open state 12 , also rendering the resulting block functionally closed-state dependent 11,12 . TEA salts have both an external state independent binding site and an internal open state dependent site in myelinated axons, suggesting binding in resting state 13 . Dendrotoxins block Kv1 channels in a manner suggesting binding in resting state 14 .
We have in previous investigations suggested that blocking Kv channels in open state may modulate the action potential train and reduce the firing frequency 2 . Assuming an open-state binding mechanism in these cases could theoretically allow for a synergism between the local anesthetic action on voltage-gated sodium (Nav) and Kv channels. For local anesthetics this assumption of open-state Kv channel blocking mechanisms has experimental support 15,16 .
To investigate the influence of different Kv channel blocking mechanisms on the excitability, we attempted a computational approach simulating the well-established membrane model of Frankenhaeuser and Huxley 17 . This model is an offspring of the traditional Hodgkin-Huxley model, describing the features of a myelinated axon, using the Goldman-Hodgkin-Katz permeability concept 18 . Through introduction of closed-and open-state blocked Kv1 channel states into the Frankenhaeuser-Huxley model and linking the model to a Cav channel 19 , the implications of differential state-dependent Kv channel effects on the neuronal firing pattern and presynaptic Ca-influx was simulated, allowing for approximation of the cumulative consequence of the firing pattern on the cellular excitability.

Modulation of firing pattern by state-dependent Kv channel block. The Frankenhaeuser-Huxley
model responds to supra-threshold stimulation by repetitive high-frequency firing and is classified as representing Class 2 excitability in Hodgkin's nomenclature 20 or Type 2 excitability in the nomenclature of Robinson et al. 21 . In Fig. 1A, the firing pattern of this model at near threshold stimulation values (5.3 and 5.6 mA/m 2 ) is compared to that of the adapted model, incorporating binding to the closed or open states of Kv channels in the presence of 200 µM blocking agent. Figure 1A shows the effect on the spiking of the adapted model assuming exclusively closed state binding at a K d value of 200 µM and of the adapted model assuming exclusively open state binding at a K d value of 200 µM compared to the control case. The control case induced a near-threshold spiking behavior, characteristic for Class 2 dynamics (close to an Andronov-Hopf bifurcation), comprising a few action potentials. Closed state binding induced sustained spiking, while open state binding eliminated the last action potentials. Stability of the model was assessed by analyzing the resting membrane potential after the action potential train for all combinations of closed-and open-state blocking drugs between 5.1-6.0 mA/m 2 . In all cases, the resting membrane potential returned to −70 mV when stimulation was abolished.
For open channel (use-dependent) block, the rate of channel closing (i.e. α + 2β) vs. rate of transition between open and blocked states (i.e. L o × κ + λ, where L o is the drug concentration) critically influences the channel behavior. In Fig. 1B the rates are plotted as functions of voltage at different concentrations. The data shows that at a drug concentration of 1 to 3 K d equivalents, the open channel state (OK3) predominantly transitions into the closed channel states (CK1 and CK2) when the membrane is repolarizing. At drug concentrations above 3 K d equivalents, the rate of the transitions between open and blocked states is greater than the rate of closing the channel (L o × κ + λ > α n + 2β n ) between −44 to −13 mV, suggesting increased transitions from the open state (OK3) into the open-blocked state (OB) rather than to the closed channel states (CK1 and CK2). This would allow for the channel displaying burst behavior with extended time spent in the closed state.
Further information on the mechanisms of the differential effect of blocking the Kv channel in open and closed state can be obtained from the distribution of channels in open (OK3), closed-blocked (CB) and open-blocked (OB) states at various stimulating currents and drug concentrations. In Fig. 1C, the distributions at 5.3 and 5.6 mA/m 2 stimulation and a concentration of 1 K d equivalent are shown. The triggering of action potentials depends critically on the opposing forces of the K current and the Nav channel inactivation process.
For the closed-state block case the fraction of channels in open state (OK3) is markedly reduced compared to the no drug situation. The opposing effect of an increased Nav channel inactivation (not shown) was not sufficiently strong to overcome the effect of the K current, thus rendering a lowered impulse generating threshold, which induces spiking in accordance with the results in Fig. 1A.
For the open-state block case, on the other hand, the decreased K current was not sufficient to overcome the enhanced Nav channel inactivation, leading to an increased triggering threshold and a block of the impulse generation in accordance with the results in Fig. 1A. Parallel with these processes was an accumulation of channels in open-blocked state (OB), followed by a slow decrease over time.
This differential response to the two cases of state-specific binding, measured as number of action potentials per stimulation step duration (50 ms), was consistent over a range of stimulation levels for different concentrations (Fig. 1D). Whereas increased closed state Kv channel block increased the number of action potentials for all stimulations, a decrease was noticed for the open state Kv channel block. The induced prolongation of the action potential was present over a range of stimulation levels for both the closed state binding and the open state binding cases (Fig. 1E). The absolute increase in peak width for the closed state Kv channel block was pronounced www.nature.com/scientificreports www.nature.com/scientificreports/

Modulation of presynaptic Ca influx by state-dependent Kv channel block. The implications of
the firing pattern modulations on the excitability on downstream synaptic activity were analysed by modelling the Ca influx through a hypothetical synaptic N-type Cav channel connected to Frankenhaeuser-Huxley membrane model. Using the previous stimulation and drug protocol, the repetitive spiking for both 5.3 and 5.6 mA/m 2 was evident for the closed state binding case, whereas the reduced number of spikes was evident for the open state binding case ( Fig. 2A). The Ca influx was normalized over time to provide an integral over a range of stimulation levels for different concentrations of the blocking drug (Fig. 2B). An increase in Ca influx was noticed for all concentrations and all stimulation levels analysed for the closed state binding case, whereas for the open-state binding case a reduced Ca influx was noticed for corresponding concentrations and stimulation levels.

Discussion
The background to the present study was the widely held belief that pharmacological blocking or genetic elimination of Kv channels in neurons generally induce an increased excitability [5][6][7] . This would leave us with a paradoxical situation with regards to local anesthetics that block both Nav and Kv channels, with the action on Kv channels opposing the inhibiting effect of the Nav channel block. We therefore found it of interest to analyse how the state dependency of the binding affected the effect on the excitability.
The effects of Kv channels in which the pore was occluded in the resting or closed channel state were investigated using the experimentally based computational membrane model of Frankenhaeuser and Huxley, extended by the blocked resting and open states, and linked to the Cav channel model of Patil et al. 19 . The results confirmed that, and detailed how, the hypothetical drug binding to closed Kv channels increased the excitability, measured as a decreased threshold for the stimulation current, an increased firing frequency and an increased Ca inflow at the downstream presynaptic membrane. In contrast, hypothetical drug binding solely to open Kv channels decreased the excitability measures, caused an increased current threshold, a decreased firing frequency and a decreased Ca inflow. Furthermore, assuming binding to both resting and open channels, it was demonstrated that the binding affinity in open state had to be approximately 6-12-fold more potent as compared to that in the resting state in order to cause a reduced excitability.
Information on the mechanisms behind the effects on the excitability measures of the state-specific Kv channel block were obtained from the analysis of the distribution of open, closed-blocked and open-blocked Kv channel states during the action potential train, shown in Fig. 1C. The cause of the differential effects of closed and open state binding was found to be the critical relation between of the opposing actions of the K current and the Nav channel inactivation.
For the closed channel block case, the maximum number of channels in conducting state (OK3) was markedly reduced compared to the no drug case along with a reduction of the number of channels in closed-blocked state (CB) during the stimulation. An increased fraction of channels in closed-state block shifts the balance between channels in closed state (CK2) and closed-blocked state (CB) towards the latter, further decreasing the fraction of channels in conducting state (OK3). The increase of the Nav channel inactivation was not sufficient to counteract the reduction of the K current, resulting in repetitive spiking. For the open channel block case, in contrast, the reduction of the K current was not sufficient to counteract the increased Nav channel inactivation, resulting in a block of impulse generation 22 . An accumulation of channels in the open-blocked state (OB) followed by a gradual reduction suggested a foot-in-the-door mechanism.
Antiarrhythmic drugs of class III inhibit outward potassium currents, increasing the refractory period of cardiomyocytes and suppressing atrial arrhythmias 23 . However, based on the prevalent side effect of the ventricular tachyarrhythmia torsades de pointes 24 , the interest for subtype-and state-specific potassium channel blocking agents has increased. Computational investigations suggested that targeting open-and inactivated-states of the cardiac Kv1.5 channels to be beneficial for antiarrhythmic drug action as compared to the closed Kv1.5 channel state 25,26 . In previous investigations of cardiac Kv4.2/4.3 channels, closed-and, more so, open-state channel block prolonged the action potential durations, although with varying potencies, depending on the fraction of repolarization used to assess the action potential durations 27 . Our results using the neuronal firing model suggest prolongation of the action potential peak widths following both closed-and open-state Kv channel block, with approximately a tenfold prolongation by the former. These findings highlight the discrepancies resulting from state-dependent Kv channel block possibly present in neurons.
The present simulations are based on binding constants estimated from electrophysiological studies of the local anesthetic drug bupivacaine on Kv1.1 channels 15 , which limits the direct generalizability of the findings. In addition, there are about 40 different Kv channel genes in the human nervous system 28 , and each neuron comprises several types of heteromeric Kv channels, presumably with different affinities in different states for different drugs and toxins. A similar channel diversity is noted for cardiac ion channels 29,30 , complicating accurate in silico modelling and discovery of channel-specific therapeutic drugs.
However, in conclusion, the present computational investigation is challenging the common notion that Kv channel blockers per se are proexcitatory. Rather, it stresses the importance of analyzing state dependency of binding when developing drugs with significant affinity to Kv channels. It also stresses the importance of analyzing the state dependency when trying to understand the mechanisms underlying the clinical symptoms of natural toxins as many of them specifically target Kv channels, and thus have been suggested to account for over 150 000 deaths annually 31 . Numerous details regarding their pharmacology remain to be revealed.

Nodal membrane model. The Frankenhaeuser-Huxley model is based on voltage-clamp data from sciatic
nerves from the African clawed toad (Xenopus laevis) 17 . To model the action potential train, the time-derivate of the membrane potential V was numerically calculated where g leak denotes the leak conductance and E leak the equilibrium value for the leak current. Parameter values are listed in Table 1.
Nav channel model. The Nav channel model was derived from the Frankenhaeuser-Huxley model 17 . The model can be described as a Markov model (Fig. 4). The corresponding equation system to be solved can be expressed in matrix form as The voltage-dependent rate constants were defined by the following equations: www.nature.com/scientificreports www.nature.com/scientificreports/  (Fig. 5).
The corresponding equation system can be expressed in matrix form as The voltage dependent rate constants were defined by the following equations:  www.nature.com/scientificreports www.nature.com/scientificreports/ The Kv channel binding rates, for the CB and OB states, e.g. κ and λ, were derived from previously described simulated voltage-clamp data of local anesthetic block on Kv1.1 channels 32 , and are presented in Table 1 19 (Fig. 6). The corresponding equation system can be expressed in matrix form as:   www.nature.com/scientificreports www.nature.com/scientificreports/ (1) The transition between CCa4 and OCa5, and between OCa5 and ICa10, as well as the transitions between closed and inactivated states were assumed to be voltage independent, with values given in Table 2 below. The voltage dependent rate constants were given by where k o is constant (in units of s −1 ), z is the valence of the gating charge associated with the transition, e is an elementary charge, V is membrane voltage, δ is the fraction of gating charge that is moved to reach the transition state, V o is offset voltage, and k B is the Boltzmann constant, see Table 2 derived from Patil et al. 19 . The use of the closed-state inactivating Cav channel model of Patil et al. 19 allowed for simulations of high frequency Ca influx, as compared to previous, open-state inactivating, Cav models 33 . Simulation of spiking in the nodal membrane model. Initially, the pulse current model was equilibrated at a holding potential of −70 mV, with no stimulating current for 50 ms. Production runs of 60 ms stimulation, using pulse currents between 5.1 to 6 mA/m 2 , followed by 10 ms with no stimulating current, were calculated at different concentrations of blocking agents. The integral of Ca current over 70 ms was calculated for comparison between different levels of Kv channel block. The timesteps for all calculations were 5 µs. All simulations were conducted using Matlab R2017a software. Graphical objects were created in GraphPad Prism 6.

Data Availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.