Voltage-dependent block by internal spermine of the murine inwardly rectifying K⁺ channel, Kir2.1, with asymmetrical K⁺ concentrations

Abstract

Effects of internal spermine on outward single-channel currents through a strongly inwardly rectifying K⁺ channel (Kir2.1) were studied at asymmetrical K⁺ concentrations (30 mm external and 150 mm internal K⁺). The current–voltage (I–V) relation for the single channel was almost linear and reversed at −37 ± 3 mV (V_R; n= 19). The channel conductance was 26.3 ± 1.3 pS (n= 24). The open-time and closed-time histograms were fitted with a single exponential function. Internal spermine at a concentration of 1–100 nm reduced the open time of the outward currents in a concentration-dependent manner and produced a blocked state. The steady-state open probability of the outward current decreased with larger depolarizations in both the absence and presence of internal spermine. The steady-state open probability with asymmetrical K⁺ and symmetrical (150 mm external and internal K⁺) concentrations plotted against driving force (V−V_R) coincided with smaller depolarizations in the absence of spermine and larger depolarizations and higher spermine concentrations in the presence of spermine. The blocking rate constants and unblock rates with 30 mm and 150 mm external K⁺ were similar at the same driving force. The dissociation constant–membrane potential relation for 30 mm external K⁺ was shifted in the negative direction from that for 150 mm external K⁺ by 36 mV. These results suggested that the blocking kinetics depends on driving force to produce driving force-dependent inward rectification when the equilibrium potential for K⁺ is altered by changing external K⁺ and that the energy barriers and wells for blocking ions from passing or lodging are not stable but affected by external K⁺ ions.

Introduction

Inward rectification, whereby K⁺ conductance increases under hyperpolarization and decreases under depolarization, has been demonstrated in a variety of cell types (Katz, 1949; Hall et al. 1963; Kandel & Tauc, 1966; Hagiwara & Takahashi, 1974). This phenomenon plays an important role in permitting long depolarizing responses. Inward rectification of cardiac inwardly rectifying K⁺ channels was ascribed to a voltage-dependent block of the channel pore by intracellular Mg²⁺ (Matsuda et al. 1987; Vandenberg, 1987; Matsuda, 1988) and an intrinsic gating mechanism that closes the channels under depolarization. Thereafter, the blockade of cloned inwardly rectifying K⁺ channels (Kir2.1; Kubo et al. 1993) by internal polyamines such as spermine, spermidine and putrescine (Ficker et al. 1994; Lopatin et al. 1994) replaces the intrinsic gating mechanism, as supported by the finding that Kir2.1 has no equivalent to the voltage sensor in voltage-dependent channels. However, in our previous studies, unitary outward currents through Kir2.1 expressed in COS-1 cells were barely recorded at holding potentials more positive than +50 mV with symmetrical 150 mm K⁺, even after prolonged washing of inside-out patches with a Mg²⁺- and polyamine-free solution (Omori et al. 1997; Oishi et al. 1998; Matsuda et al. 2003).

We previously studied voltage-dependent gating and the spermine block in channels expressed by cDNA encoding a wild-type (WT) channel, a mutant in which Asp 172 was replaced with Asn (D172N), and a tandem tetramer, WT-(D172N)2-WT (Matsuda et al. 2003). Gating analysis in the absence of internal spermine revealed that a rate constant from C₂ to C₁ (in the linear sequential model as C₂⇌ C₁⇌ O) increased with the number of D172N subunits. Unblock rates also increased with the number of D172N mutant subunits (0.15 s⁻¹ in WT, 72 s⁻¹ in WT-(D172N)2-WT, and 300 s⁻¹ in D172N at +42 mV), supporting that D172 is a spermine-binding site acting as a key rectification controller (Ficker et al. 1994; Lopatin et al. 1994). These results suggest that D172 is involved in both intrinsic gating and the spermine block.

Inward rectification depends on driving force when the equilibrium potential for K⁺ (E_K) is altered by changing external K⁺ and on voltage when E_K is altered by changing internal K⁺ in the inwardly rectifying K⁺ channel of egg cells (Hagiwara et al. 1976; Hagiwara & Yoshii, 1979) and skeletal muscle (Hestrin, 1981; Leech & Stanfield, 1981). Similarly, the blocking effect of intracellular Mg²⁺ in cardiac inwardly rectifying K⁺ channels depends on driving force when E_K is altered by changing external K⁺ and on voltage when E_K is altered by changing internal K⁺ (Matsuda, 1991).

In the present study, we recorded single-channel currents through Kir2.1 in the absence and presence of intracellular spermine with 30 mm external and 150 mm internal K⁺. Comparison of the present data with data obtained with 150 mm external and internal K⁺ (Matsuda et al. 2003) revealed that dependence on driving force differs between the spermine block and voltage-dependent decrease in the control, suggesting that the voltage-dependent decrease in the control is not due to a residual spermine block. The blocking kinetics depends on driving force and so the spermine block depends on driving force when E_K is altered by changing external K⁺.

Methods

Molecular biology

The Kir2.1 gene (Kubo et al. 1993) was digested with HindIII and StuI. The HindIII–StuI fragment, which contains the 5′ untranslated sequence and part of the 3′ untranslated sequence, was subcloned into a pTZ19R vector (Pharmacia, Uppsala, Sweden). For expression in COS-1 cells (Riken Cell Bank, Tsukuba, Japan), cDNAs were fused to enhanced green fluorescent protein (EGFP) by subcloning in pEGFP-C1. The expression plasmid (0.5–2 μg per 35 mm dish) was transiently transfected into cells using Lipofectamine (Invitrogen, Carlsbad, CA, USA) according to the manufacturer's directions. Currents were recorded from cells exhibiting green fluorescence 24–72 h after transfection.

Electrophysiology

The coverslips (3 mm × 18 mm) on which COS-1 cells were grown were transferred to a recording chamber mounted on an inverted fluorescence microscope (IX-70, Olympus). Single-channel currents were recorded using a heat-polished patch electrode (Hamill et al. 1981) in the inside-out configuration with an EPC-7 patch clamp amplifier (HEKA Elektronik, Lambrecht/Pfalz, Germany). Pipettes were made from capillaries of hard borosilicate glass (Pyrex), and coated near the tips with silicone to reduce electrical capacitance. The electrode's resistance ranged between 8 and 20 MΩ when it was filled with a pipette solution containing (mm): KCl, 30; NaCl, 120; CaCl₂, 1; Hepes, 5 (pH 7.4 with KOH). The chamber was first perfused with Tyrode solution containing (mm): NaCl, 140; NaH₂PO₄, 0.33; KCl, 5.4; CaCl₂, 1.8; MgCl₂, 0.5; Hepes, 5; glucose, 5.5; pH adjusted to 7.4 with NaOH. After Kir2.1 channel activity was identified in a patch, the solution perfusing the chamber was switched to a high-K⁺ solution and perfusion of the cell with this solution using a Y-tube method (Murase et al. 1990) was started. Then, the inside-out configuration was constructed. The high-K⁺ solution contained (mm): potassium aspartate, 60 (65 without K₂ATP); KCl, 65; KH₂PO₄, 1; K₂EDTA, 5; K₂ATP, 3; Hepes, 5; pH adjusted to 7.4 with KOH. Since the binding constant of spermine for ATP is 5.95 × 10²m⁻¹ (Watanabe et al. 1991), approximately two-thirds of spermine exists as a complex with ATP and the rest as free ions in the presence of 3 mm ATP. Spermine was diluted with an ATP-free, high-K⁺ solution on the day of the experiments. Experiments were carried out at 23–26°C.

Data were collected on digital audiotape using a PCM data recorder (RD-120TE, TEAC) and stored for subsequent computer analysis (PC-98XL, NEC). The unitary currents were filtered using a four-pole low-pass Bessel filter (FV-665, NF) with a −3 dB corner frequency of 0.2 kHz and sampled at 1 kHz for steady-state kinetics and amplitude histograms. The open probability was calculated from the amplitude histogram or traces reconstructed with a threshold set at around half of the open level. Open and zero-current times were measured in patches where only one channel was active, using a cursor set mid-way between open and zero-current levels. Open and zero-current time histograms were formed and then fitted with exponential(s) using a least-squares algorithm. Membrane potentials were corrected for liquid junction potential at the tip of the patch pipette in the Tyrode solution and for that at the tip of the indifferent reference electrode filled with Tyrode solution and placed in the bathing solution.

Averaged results are given throughout this paper as the mean ±s.d. Student's unpaired t test was performed, and P values of less than 0.05 were regarded as being statistically significant.

Results

Single-channel currents and kinetic analysis of outward currents in the absence of internal blockers with a reduced external K⁺ concentration

We previously studied the voltage-dependent block of Kir2.1 by internal spermine with 150 mm internal and external K⁺ concentrations (Matsuda et al. 2003). In the present study, we replaced 120 mm K⁺ with an equimolar concentration of Na⁺ to reduce the external K⁺ concentration from 150 mm to 30 mm. The K⁺ concentrations after titration were approximately 150–155 mm in the high-K⁺ internal solution and 32 mm in the pipette solution. E_K, predicted from transmembrane K⁺ concentrations, is −40 mV. Figure 1A illustrates single-channel currents through Kir2.1 recorded under steady-state conditions in the inside-out configuration in the absence of internal blockers (polyamines and Mg²⁺). Currents of both directions show long-lasting openings characteristic of an inwardly rectifying K⁺ channel. The I–V relation for the single channel was almost linear and reversed at −37 ± 3 mV (V_R; n= 19; Figure 1B). The channel conductance was 26.3 ± 1.3 pS (n= 24). Since the conductance with 150 mm internal and external K⁺ concentrations was 36.9 ± 1.5 pS (n= 29), the exponent of the K⁺ dependence of the unitary conductance was 0.21. This value is similar to that of the cardiac inwardly rectifying K⁺ channel, 0.22 (Matsuda, 1991).

Figure 1. Single-channel currents and I–V relation in the absence of internal blockers with 30 mm external and 150 mm internal K⁺ concentrations.

A, currents were filtered at 0.2 kHz and sampled at 1 kHz. The dotted line indicates the zero-current level. The numbers to the left of each trace refer to the holding potential. B, I–V relationship obtained from the same patch as in A. The slope conductance was 26.6 pS.

Open-time and zero-current-time histograms were constructed for outward currents recorded at −15 and −10 mV. Previously (Matsuda et al. 2003), we filtered unitary currents with a −3 dB corner frequency of 1 kHz and sampled at 5 kHz for the analysis of fast kinetics with 150 mm external and internal K⁺. The small amplitude with 30 mm external K⁺ forced us to filter the currents with a −3 dB corner frequency of 0.2 kHz and sample at 1 kHz in the present study. Figure 2 shows open-time and closed-time histograms at −10 mV in the absence of spermine. The lifetimes of the openings were distributed according to a single exponential function. The average of the mean open time was 241 ± 26 ms (n= 3) at −10 mV and 311 ± 29 ms (n= 3) at −15 mV. The faster component of the closed time detected with a higher sampling frequency was missing and the closed-time histogram was fitted with a single exponential function. The average of the mean closed time was 4.39 ± 0.20 ms (n= 3) at −10 mV and 4.53 ± 0.10 ms (n= 3) at −15 mV.

Figure 2. Kinetic analysis in the control.

Open-time and closed-time histograms constructed for currents at −10 mV. Both are fitted with a single exponential function with the time constant indicated. The last bin of the closed time histogram includes events longer than 100 ms.

Single-channel currents in the presence of internal spermine with a reduced external K⁺ concentration

Intracellular spermine at a concentration of 1–100 nm blocked the outward currents in a concentration- and voltage-dependent manner (Fig. 3). Figure 3A shows steady-state currents at −15 mV in the absence and presence of internal spermine. The open time decreased as the spermine concentration increased, while a blocked state 250–600 ms in duration occurred in the presence of spermine. Figure 3B shows steady-state currents with 100 nm spermine at different voltages. Mean blocked time increased with greater depolarization. Spermine at 100 nm did not affect inward currents. Results of the kinetic analysis are given below.

Figure 3. Effects of internal spermine on outward currents with a reduced external K⁺ concentration.

A, outward currents recorded at −15 mV in the absence and presence of spermine. Spermine decreased the open time dependent on the concentration and induced a blocked state of 250–600 ms duration. B, currents in the presence of 100 nm spermine. The numbers to the left of each trace refer to the holding potential. Inward currents were not affected.

The steady-state open probability in the absence and presence of internal spermine

The steady-state open probability decreased as holding potential increased even in the absence of spermine (control) and outward currents were hardly recordable at potentials more positive than +30 mV. The open probability was plotted against membrane potential (Fig. 4, upper panel). The data were fitted to Boltzmann distributions. The slope factor and voltage of half-activation in the control were 9.9 mV and +4.7 mV, respectively. The open probability–membrane potential relation shifted in the negative direction with increasing concentrations of spermine: the half-activation voltage was −13.3 mV with 1 nm, −21.3 mV with 10 nm and −29.2 mV with 100 nm.

Figure 4. Steady-state open probability–membrane potential relations.

Upper panel, smooth lines are fits of a Boltzmann function to average data (n= 3–9). The voltages of half-activation and slope factors were respectively +4.7 mV and 9.9 mV in the control (▪), −13.3 mV and 4.1 mV with 1 nm spermine (•), −21.3 mV and 3.0 mV with 10 nm spermine (▴), and −29.2 mV and 3.2 mV with 100 nm spermine (▾). Middle and lower panels, the steady-state open probability was plotted against driving force. For simplicity, s.d. was omitted. •, 30 mm external K⁺; ▪, 150 mm external K⁺. The data with 150 mm external K⁺ are the same as in Fig. 2B of Matsuda et al. (2003).

To study the dependence of the open probability on driving force, we plotted data obtained with a reduced external K⁺ concentration and with 150 mm external K⁺ (Matsuda et al. 2003) against driving force (V−V_R; Fig. 4, middle and lower panels). The open probability in the control was identical during small depolarizations but that with 30 mm external K⁺ was higher than that with 150 mm K⁺ above a driving force of +22 mV. As a result, the relation shifted in the positive direction with reduced K⁺: the half-activation voltage was +41.7 mV with 30 mm K⁺ and +35.9 mV with 150 mm K⁺. A significant difference was observed between the average of the half-activation voltage obtained in each experiment: +42.8 ± 1.9 mV (n= 4) with 30 mm K⁺ and +36.1 ± 1.1 mV (n= 4) with 150 mm K⁺. The relation shifted in the positive direction with reduced K⁺ also in the presence of spermine: the half-activation voltage was +23.7 mV with 30 mm K⁺ and +21.4 mV with 150 mm K⁺ (1 nm), +15.7 mV with 30 mm K⁺ and +12.4 mV with 150 mm K⁺ (10 nm), and +7.8 mV with 30 mm K⁺ and −2.1 mV with 150 mm K⁺ (100 nm). However, curves were superposable at more positive potentials, suggesting that the inward rectification produced by intracellular spermine depends on driving force with larger depolarizations and higher spermine concentrations when E_K is altered by changing external K⁺.

Analysis of blocking kinetics

To study the mechanism of the driving force-dependent spermine block, we constructed open-time and zero-current-time histograms for outward currents recorded at −15 and −10 mV. Figure 5 shows open-time and zero-current-time histograms in the presence of 10 and 100 nm spermine. The open-time histogram was fitted with a single exponential function. The average of the mean open time was 25.3 ± 2.5 ms (10 nm; n= 3) and 3.36 ± 0.61 ms (100 nm; n= 3) at −15 mV, and 23.2 ± 1.1 ms (10 nm; n= 3) and 2.54 ± 0.13 ms (100 nm; n= 4) at −10 mV. The difference between −15 mV and −10 mV with 30 mm external K⁺ was not significant, though the mean open time differed significantly between +22 mV and +42 mV with 150 mm external K⁺ (Matsuda et al. 2003). As shown in Fig. 2, the closed-time histogram was fitted with a single exponential function with a time constant of 4–5 ms. Another zero-current state occurred in the presence of spermine. Most of the zero-current times with 10 and 100 nm spermine were distributed according to a single exponential function with a time constant of 250–600 ms at −15 mV and 1 s at −10 mV, and are attributable to the block by internal spermine (Matsuda et al. 2003). The average of the mean blocked time was 374 ± 155 ms (10 nm; n= 3) and 438 ± 134 ms (100 nm; n= 3) at −15 mV, and 1039 ± 141 ms (10 nm; n= 3) and 1072 ± 112 ms (100 nm; n= 4) at −10 mV. The averaged mean open time and blocked time at −15 mV (V−V_R=+22 mV) are similar to those at +22 mV with 150 mm external K⁺: the open time (n= 4) was 22.6 ± 2.0 ms with 10 nm and 2.43 ± 0.50 ms with 100 nm and the blocked time (n= 4) was 394 ± 36 ms with 10 nm and 419 ± 39 ms with 100 nm (Matsuda et al. 2003). This result indicates that the blocking kinetics depends on driving force when E_K is altered by changing external K⁺.

Figure 5. Analysis of blocking kinetics.

Histograms of open and zero-current times constructed for outward currents in the presence of 10 nm spermine at −15 mV (left) and 100 nm spermine at −15 mV (middle) and −10 mV (right). The open-time histogram was fitted with a single exponential function. The mean open time decreased in a concentration-dependent manner. The currents were filtered with 0.2 kHz and sampled at 1 kHz. Spermine induced a component of zero-current times with the time constant indicated, which represents blocked times. Depolarization at 5 mV prolonged the mean blocked time threefold.

Dissociation constant–membrane potential relation shifts as the V_R shifts by changing external K⁺

To study the voltage dependence of the spermine block, we calculated the dissociation constant. The open probability at −20 and −10 mV was normalized to that in the absence of spermine and plotted against the spermine concentration (Fig. 6A). The relations fitted well with the concentration–effect curves predicted by assuming one-to-one binding of spermine to a site. The dissociation constant obtained thus was 7.1 nm at −20 mV and 0.53 nm at −10 mV. The dissociation constant at −15 mV calculated from the blocking rate constant and unblocking rate was 0.76 nm (see Discussion). The data were plotted against membrane potential in a semilogarithmic scale with the data obtained at symmetrical K⁺ concentrations (Matsuda et al. 2003). Data points at each K⁺ concentration can be fitted by a straight regression line, indicating that the dissociation constant decreases exponentially as the membrane potential is increased (Fig. 6B). The line fitted to the data at 30 mm external K⁺ almost parallels that at 150 mm external K⁺ 36 mV apart.

Figure 6. Dissociation constants in the spermine block.

A, the open probability was normalized to that in the absence of spermine and plotted against the spermine concentration on a logarithmic scale. The curve was fitted by saturation kinetics with a Hill coefficient of 1 and a dissociation constant of 7.1 nm at −20 mV (•) and 0.53 nm at −10 mV (▪). B, dependence of the dissociation constant on membrane potential. In a semilogarithmic plot, data at 30 mm external K⁺ (•) were fitted by a straight line almost in parallel with that fitted to data at 150 mm external K⁺ (▪) 36 mV apart.

The dissociation constant at a membrane potential of V, K_D(V), is described as:

where z′ is the so-called effective valency of the blocking ion, equal to actual valency multiplied by the fractional electrical distance between the internal mouth of the aqueous pore and the blocking ion binding site (Woodhull, 1973). F, R and T have their usual meaning. The slope of the regression line gave a value for z′ of 6.6 with 30 mm K⁺ and 5.3 with 150 mm K⁺. The value of K_D at 0 mV, K_D(0), is 0.0299 nm for 30 mm K⁺ and 117 nm for 150 mm K⁺.

Discussion

We studied effects of internal spermine on outward single-channel currents through Kir2.1 with 30 mm external and 150 mm internal K⁺. The open-time and closed-time histograms were fitted with a single exponential function. Spermine reduced the open time dependent on concentration and produced a blocked state. The mean blocked time depended not on concentration but on voltage. Plotting of the steady-state open probability obtained with 30 mm external K⁺ and with 150 mm external K⁺ (Matsuda et al. 2003) against driving force revealed that the block by internal spermine depends on driving force at larger depolarizations and higher concentrations when E_K is altered by changing external K⁺.

The steady-state open probability of the outward current decreased with larger depolarizations in the control. As discussed previously (Matsuda et al. 2003), this was not ascribed to usage of Hepes (Guo & Lu, 2000b): the steady-state open probabilities (0.27 at +42 mV and 0.021 at +52 mV) with an internal solution where 5 mm Hepes was replaced by 1 mm KH₂PO₄ and 4 mm K₂HPO₄ (pH 7.4) were not different from those obtained with Hepes. It is also unlikely that the voltage-dependent block by residual polyamines caused the decrease in the steady-state open probability. First, spermine was washed out in the control with the high-K⁺ solution containing 3 mm ATP, which bound two-thirds of all the spermine. Second, the mean open time in the control was much shorter than that expected from a residual spermine block in our previous study (Matsuda et al. 2003). Third, dependence on driving force differs between the spermine block and voltage-dependent decrease in the control (Fig. 4). The open probability in the control was identical during small depolarizations but deviated during large depolarizations: that with 30 mm external K⁺ was higher than that with 150 mm K⁺ above a driving force of +22 mV. This is in contrast with the plot in the presence of spermine where the open probability deviated during small depolarizations but not during large depolarizations. Thus it is suggested that voltage-dependent gating works in Kir2.1 in the outward current independent of spermine, though intracellular polyamines and Mg²⁺ at a physiological concentration block Kir2.1 almost completely, making its significance in inward rectification small.

Reduction of the external K⁺ concentration from 150 mm to 30 mm decreased the averaged single-channel conductance from 36.9 pS to 26.3 pS. The current amplitude with 30 mm external K⁺ was around 0.6 pA at −15 mV where the driving force was +22 mV. Because of the small amplitude, the currents were filtered with a −3 dB corner frequency of 0.2 kHz and sampled at 1 kHz. In this condition, most of the closed times were distributed according to a single exponential function with a time constant of 4–5 ms. Spermine induced another component of zero-current times with a much longer time constant, which is attributed to the block by internal spermine.

Therefore, we consider the model for the spermine block to be:

where C is the closed state, O is the open state, B is the blocked state, α and β are open and closing rate constants, and μ and λ are blocking and unblocking rates. The mean open time at −15 mV in the control, 311 ms, gives a value of 3.2 s⁻¹ as β. The mean open time at −10 mV in the control, 241 ms, gives a value of 4.1 s⁻¹ as β. From the mean open time in the presence of spermine and β, the blocking rates were calculated: 36.3 s⁻¹ (10 nm) and 294 s⁻¹ (100 nm) at −15 mV and 39.0 s⁻¹ (10 nm) and 390 s⁻¹ (100 nm) at −10 mV. The blocking rate is linearly proportional to the spermine concentration. The blocking rate constant is approximately 3.3 × 10⁹ s⁻¹m⁻¹ at −15 mV and 3.9 × 10⁹ s⁻¹m⁻¹ at −10 mV. The unblocking rate estimated from the mean blocked time was 2.5 s⁻¹ at −15 mV and 0.95 s⁻¹ at −10 mV. The unblock rate depends on voltage more strongly than the blocking rate (constant). The blocking rate constant and the unblock rate at +22 mV with 150 mm external and internal K⁺ were 4.0 × 10⁹ s⁻¹m⁻¹ and 2.5 s⁻¹, respectively (Matsuda et al. 2003). Thus the kinetics, and therefore spermine block, depends on driving force when E_K is altered by changing external K⁺.

Our study is the first to show that the blocking rate constant and unblock rate for spermine depend on driving force as the V_R or E_K shifts by changing external K⁺. In agreement with this, the dissociation constant–membrane potential relation shifts as the V_R shifts with a change in external K⁺ (Fig. 6B). The effective valency of spermine was 5.3 with 150 mm external K⁺. This is similar to that obtained for macroscopic currents with 85–100 mm external K⁺ (5.2–5.7; Guo & Lu, 2000a,b; Xie et al. 2003). The effective valency with 30 mm external K⁺ was 6.6. The limited voltage range where the dissociation constants were measured with 30 mm external K⁺ may make the estimation inaccurate. External K⁺ did not change the effective valency in the Mg²⁺ block (Matsuda, 1991). An effective valency higher than the charge number of spermine has been ascribed to the displacement of K⁺ to the outside of the membrane (Pearson & Nichols, 1998; Xie et al. 2003).

The result that the spermine block depends on driving force when E_K is shifted by changing external K⁺ may be expressed in a different way: the blocking effect of spermine is decreased by increasing the external K⁺ concentration at a given voltage. It has been described for different K⁺ channels that external cations relieve the block by internally applied ions or drugs (Armstrong, 1971; Yellen, 1984). Neyton & Miller (1988) showed in a Ca²⁺-activated K⁺ channel that an external K⁺ concentration higher than 100 mm raises the rate at which Ba²⁺ dissociates and ascribed this to the repelling of Ba²⁺ back to the cytoplasmic side by the K⁺ ion nearby in a multi-ion channel.

In a four-barrier three-site model with a monovalent internal blocking ion, Hille & Schwartz (1978) successfully simulated inward rectification that depended on driving force at different external K⁺ concentrations. They pointed out that when the blocking ion is divalent the block shifts much less than the change in E_K caused by changing external K⁺. Indeed we could not reproduce the driving force-dependent block by tetravalent spermine in a four-barrier three-site model. The results may suggest that the energy barriers and wells for blocking ions from passing or lodging are not stable but affected by external K⁺ ions (Matsuda, 1991). In accordance with this, Robertson et al. (2008) calculated the electrostatic free energy of an ion along a long extended pore of mammalian Kir channels by solving the Poisson–Boltzmann equation and showed that the electrostatic interaction free energies of spermine and K⁺ in the Kir2.1 channel cavity (where D172 is located) at the centre of the membrane decrease when K⁺ ions are absent in the selectivity filter. Recently, the crystal structure of Kir2.2 has been reported (Tao et al. 2009). Structural analyses of Kir2.1 channels at different K⁺ concentrations may determine the precise molecular mechanism underlying the driving force-dependent blocking kinetics.

Author contributions

Conception and design of the experiments: H.M.; collection, analysis and interpretation of data: H.M., M.H. and M.O.; drafting the article or revising it critically for important intellectual content: H.M. All authors approved the final version to be published. The experiments were done at Department of Physiology, Kansai Medical University.