Potassium-selective inward rectifier (Kir) channels are a class of membrane proteins necessary for maintaining stable resting membrane potentials, controlling excitability, and shaping the final repolarization of action potentials in excitable cells. In addition to the strong inward rectification of the ionic current caused by intracellular blockers, Kir2.1 channels possess “weak” inward rectification observed in inside-out patches after prolonged washout of intracellular blockers. The mechanisms underlying strong inward rectification have been attributed to voltage-dependent block by intracellular Mg^{2+} and polyamines; however, the mechanism responsible for weak rectification remains elusive. Hypotheses include weak voltage-dependent block and intrinsic voltage-dependent gating. Here, we performed a conductance Hill analysis of currents recorded with a double-ramp protocol to evaluate different mechanisms proposed for weak inward rectification of Kir2.1 channels. Linkage analysis in the form of a Hill plot revealed that the ramp currents could be best explained by allosteric coupling between a mildly voltage-dependent pore gate (gating charge ∼0.18 e_{o}) and a voltage sensor (gating charge ∼1.7 e_{o}). The proposed voltage sensor stabilized the closing of the pore gate (coupling factor ∼31). We anticipate that the use of linkage analysis will broaden understanding of functional coupling in ion channels and proteins in general.

## Introduction

Members of the inward rectifier (Kir) subfamily of potassium channels mediate K^{+} currents, which are important in maintaining stable resting membrane potentials, controlling excitability, and shaping the final repolarization of action potentials in excitable cells such as neurons and cardiac myocytes (Noble, 1979; Constanti and Galvan, 1983; Hume and Uehara, 1985; Nichols and Lopatin, 1997; Day et al., 2005). The current–voltage relationship of Kir2.x channels displays a characteristic hump near the K^{+} equilibrium potential (Matsuda et al., 1987; Vandenberg, 1987), which is attributed to the presence of a strong inward rectification mechanism, allowing inward currents to pass through the channels more easily than outward currents.

The pore of a Kir2.x channel consists of the cytoplasmic segment, central cavity, and selectivity filter. The mechanism underlying the strong inward rectification of Kir2.1 channels has been ascribed to a voltage-dependent block of outward currents by internal Mg^{2+} and polyamines (Matsuda et al., 1987; Vandenberg, 1987; Ficker et al., 1994; Lopatin et al., 1994; Stanfield et al., 1994). Outward currents through Kir2.1 channels (Kubo et al., 1993) can be inhibited by these molecules with high affinity at the central cavity or with low affinity in the cytoplasmic pore (Yang et al., 1995; Kubo and Murata, 2001).

In addition to the strong inward rectification caused by intracellular blockers, Kir2.1 channels possess “weak” inward rectification observed in inside-out patches after prolonged washout of intracellular blockers. All currently identified prokaryotic Kir channels possess only weak inward rectification, and some eukaryotic Kir channels exhibiting weak but not strong inward rectification are involved in important physiological functions in brain, heart, kidney, pancreas, etc. (Hibino et al., 2010). A systematic comparison of prokaryotic and eukaryotic Kir channels favors the notion that strong inwardly rectifying Kir channels evolved from weakly rectifying channels (Cheng, 2012). Whether weak inward rectification observed in strongly rectifying channels subject to washout reflects the action of an evolutionarily old gating mechanism intrinsic to the channel (Lopatin et al., 1994; Yang et al., 1995; Shieh et al., 1996; Lee et al., 1999; Matsuda et al., 2003, 2010) or is an artifact of residual block (Guo and Lu, 2000; Guo et al., 2003) remains a matter of controversy.

Ion channels are activated (gated) by stimuli such as voltage and ligand binding and transduce this information into conformational changes that open/close the channel pore. One of the key questions in ion channel gating is how changes in environment lead to pore opening/closing. A working hypothesis is that specialized sensors respond to external stimuli through changes in conformation that can be transmitted to the pore gate through energetic coupling between the relevant sensors and the gate. Linkage analysis in the form of a conductance Hill plot described by *W _{H}*

_{[}

_{g}_{]}=

*kT*ln[(

*G*−

*G*)/(

_{min}*G*−

_{max}*G*)] versus voltage (

*V*) can be used to describe the equilibrium relationship between pore opening (tracked by conductance

*G*) and

*V*-dependent modifiers of

*G*(Chowdhury and Chanda, 2010; Sigg, 2013). The Hill plot, which has units of energy, contains one or more linear segments reflecting voltage intervals in which pore opening can be characterized by a simple two-state process. The traditional Hill approach (Hill, 1910) focuses on the region of steepest slope (Hill coefficient), which is a lower estimate for total gating charge (Yifrach, 2004). Here, we are more interested in the asymptotic behavior of the Hill plot for very large positive and negative voltages (outer segments). The vertical separation of positive-voltage and negative-voltage asymptotes in the conductance Hill plot of an allosterically regulated channel is equal to the energy of interaction between the pore and its voltage sensors. The asymptotic behavior of the Hill plot can be used to discriminate between different models of pore gating in a manner to which conventional

*G*(

*V*) activity curves may be insensitive. In this study, we applied linkage analysis to examine various mechanisms proposed for the weak inward rectification observed in Kir2.1 channels perfused with nominal blocker-free solutions.

## Materials and methods

*Xenopus* oocyte preparations

All surgical and anesthesia procedures have been reviewed and approved by the Academia Sinica Institutional Animal Care and Utilization Committee (Protocol 11-12-278). *Xenopus* oocytes were isolated by partial ovariectomy from a frog anesthetized with 0.1% tricaine (3-aminobenzoic acid ethyl ester). The incision was sutured, and the animal was monitored during the recovery period before it was returned to its tank. Oocytes were maintained at 18°C in Barth’s solution containing (in mM): NaCl 88, KCl 1, NaHCO_{3} 2.4, Ca(NO_{3})_{2} 0.3, CaCl_{2} 0.41, MgSO_{4} 0.82, and HEPES 15, pH 7.6, with gentamicin (20 µg/ml) and were used 1–3 d after injection of cRNA obtained from the Kir2.1 cDNA clone (Kubo et al., 1993) by in vitro transcription (mMessage mMachine, Ambion).

### Electrophysiological recordings

Currents were recorded using patch-clamp techniques (Hamill et al., 1981; Hilgemann, 1995) with an Axopatch 200B amplifier (Axon Instruments). Electrodes with pipette diameters ranging from 1 to 5 µm were used. The command voltage pulses, ramp protocols, and data acquisition functions were controlled using a DigiData1440 board and pClamp10 software (Molecular Devices). Sampling and filtering frequencies were 25 and 5 kHz, respectively. The currents were recorded using the inside-out patch-clamp configuration under a symmetrical bath solution composed of 14.15 mM KCl, 1 mM EDTA, 3.6 mM KOH, 1 mM K_{2}HPO_{4}, and 0.25 mM KH_{2}PO_{4}, pH 7.4. The access (series) resistances of the recording ranged between 5 and 10 MΩ and were compensated by 60–70%. All experiments were performed at room temperature (*kT* ∼25 meV). Patches were held at 0 mV, the same as the K^{+} reversal potential (*V _{rev}*). Any offset current was manually zeroed.

### Use of ramp protocol

Because Hill plots are noisy at voltages where *G* approaches *G _{min}* or

*G*, it was essential that ionic currents (

_{max}*I*) be recorded for as many voltages as possible to maximize the number of data points in the

*G*(

*V*) curve. For this purpose, we recorded currents using a slow ramp protocol. Fig. 1 A shows the ramp protocol (

*V*) and current (

*I*) obtained from an inside-out patch expressing Kir2.1 channels exposed to symmetrical 20 mM [K

^{+}]. The ionic current demonstrated inward rectification in a nominal blocker-free solution. This inward rectification is referred to as intrinsic or weak inward rectification, to be differentiated from the “strong” inward rectification induced by polyamine or Mg

^{2+}block. The subphysiological value of [K

^{+}] was chosen to reduce the size of the current for improved voltage control of the membrane patch. Symmetric [K

^{+}] values smaller than 20 mM were avoided because a low concentration of permeant ions reduced pore conductance at extreme negative potentials by an unknown mechanism.

### Constructing the Hill plot

To construct a conductance Hill plot from the quasi-equilibrium *I*(*V*) curve, we needed an expression for the single-channel open probability *P _{O}*(

*V*). Provided the time-averaged conductance of a channel-containing membrane has well-defined maximum and minimum values, then one can define the open probability as

The ionic current from *N* channels was assumed to have the general form *I*(*V*, *t*) = *g _{l}*(

*V*−

*V*) +

_{l}*NP*(

_{O}*V*,

*t*)

*g*(

*V*)(

*V*−

*V*), where the subscript

_{rev}*l*indicates an unavoidable “leak” current. Under conditions of symmetrical [K

^{+}] and after carefully zeroing the offset potential of the voltage clamp, we obtain

*V*=

_{rev}*V*= 0, and we set

_{l}*G*(

*V*) equal to the chord conductance

*I*(

*V*)/

*V*=

*g*+

_{l}*NP*(

_{O}*V*)

*g*(

*V*), assuming quasi-equilibrium conditions. Previous studies have shown the open-pore conductance

*g*(

*V*) in Kir channels to be voltage independent (ohmic) in the range of −100 to 100 mV (Matsuda et al., 1987; Vandenberg, 1987). With these considerations, one obtains a sigmoidal

*G*(

*V*) curve that is related to

*P*(

_{O}*V*) through Eq. 1, after setting

*G*=

_{min}*g*and

_{L}*G*=

_{max}*g*+

_{L}*Ng*.

To construct Hill plots from experimental data, we plotted *G*(*V*) and determined *G _{min}* and

*G*so that the Hill plot was asymptotically linear for positive and negative voltages (Fig. 1, B and C). The conductance Hill energy was defined as

_{max}In practice, conductance limits are better represented by *G _{min}* – ε

*and*

_{b}*G*+ ε

_{max}*, where*

_{a}*ε*are positive-value errors in determining

*G*and

_{min}*G*. As a result of these errors, which are caused by noise distortion as

_{max}*G*(

*V*) approaches either

*G*or

_{min}*G*, we obtain

_{max}where *b* = ε* _{b}*/Δ

*G*,

*a*= ε

*/Δ*

_{a}*G*, and Δ

*G*=

*G*–

_{max}*G*. An important component of Hill analysis is to minimize the proportional errors

_{min}*a*and

*b*, because these cause distortions in the Hill plot, clipping it in the range

*kT*ln

*b*to –

*kT*ln

*a*. Only

*b*was relevant in the present case, because

*G*−

_{max}*G*was greater than noise for all voltages.

The best estimates for *G _{min}* and

*G*were obtained by requiring the Hill plot to be linear in the negative (–200 to –100 mV) and positive (+150 to +200 mV) voltage ranges. Each trace of

_{max}*G*(

*V*) was scaled to unity at

*V*= –150 mV (Fig. 1 B). Assigning

*G*= 1.5 linearized the negative-voltage asymptote of the Hill plot in each of seven experiments (R1–R7, Fig. 2). In experiments R1 and R2, assigning

_{max}*G*= 0 rendered the positive-voltage asymptote linear and parallel to the negative-voltage asymptote. Achieving linear positive-voltage asymptotes in the remaining experiments R3–R7 required

_{min}*G*to acquire a small value in the range of 0.0065 to 0.02, presumably because of lower seal resistance or a slight current offset. A small voltage offset in the clamp apparatus (

_{min}*V*) was corrected manually in most traces by minimizing the conductance spike at

_{os}*V*= 0. Values for the scale factor,

*G*, and

_{min}*V*in each experiment are found in Table 1.

_{os}Because of time correlations in the original data, achieving statistical independence between neighboring points required reducing the original number of 4,000 (experiments R1 and R2) or 16,000 (experiments R3–R7) data points to 400, of which roughly half were in the outer segments (∼100 points in each outer segment). Data reduction was performed in two ways: (1) standard decimation and (2) random sampling of the dataset with replacement (bootstrap method).

### Statistical analysis

We performed least-squares fitting of various equilibrium gating schemes to the Hill plot using the solver routine in Excel (Microsoft). Only the outer asymptotic regions were included in the fit. Because the positive-voltage and negative-voltage asymptotic regions exhibited substantially different noise levels, a weighting factor equal to the inverse variance of linear regression residuals in the original (nondecimated) data was incorporated into the sum of residual squares, yielding the χ^{2} statistic. Confidence intervals for fitted parameters were determined by varying the parameter of interest and determining the boundaries where χ^{2} fits of the remaining parameters exceeded a critical χ^{2} value corresponding to the 95% quantile of the *F* distribution (Kemmer and Keller, 2010). The χ^{2} value was also used in model discrimination using the Akaike information criterion (AIC) score: AIC = *n*ln(χ^{2}/*n*) + 2*k*, where *n* ≈ 200 is the number of independent and normally distributed data points and *k* is the number of fitted variables plus 1 (Horn, 1987; Burnham and Anderson, 2004; Banks and Joyner, 2017). The relative likelihood of two models describing the same dataset is given by exp[(AIC_{1} − AIC_{2})/2], where the smaller AIC value indicates the better model. Distributions of AIC values and fitted parameters were obtained from bootstrap-sampled data, yielding a second source of model discrimination and confidence intervals.

### Equilibrium gating models

Cartoon representations of candidate gating models are shown in Fig. 3. The equilibrium properties of each model are described by their partition function *Z*, a Boltzmann-weighted sum of states that can be expressed as a polynomial function of equilibrium constants and allosteric factors (Sigg, 2013). The open probability *P _{O}* was obtained by multiplying each term in

*Z*by 0 (closed) or 1 (open) or by a fractional conductance φ, and then dividing the weighted sum by

*Z*.

The simplest scheme is the two-state gating particle (Fig. 3 A), where voltage *V* displaces a gating charge Δ*q* and closes the pore. We assigned to the closing transition an equilibrium constant of the form *L* = exp(–η* _{L}*/

*kT*). The equilibrium constant is a Boltzmann function of the transition free energy or “particle potential” given by η

*= –Δ*

_{L}*q*(

_{L}*V*–

*V*), where

_{L}*V*is the half-activation potential. The partition function of the pore gating particle is 1 +

_{L}*L*, yielding

*P*= 1/(1 +

_{O}*L*). The corresponding Hill equation is simply

*W*

_{H}_{[}

_{g}_{]}=

*η*, which is linear with slope –Δ

_{L}*q*.

_{L}### Residual block (COSB) model

Our previous work (Liu et al., 2012) has argued that the Kir channel contains two blocking sites, each binding one polyamine molecule, rather than progressing from partial to complete block in a singly occupied site. The resulting COSB model (Fig. 3 B) follows the scheme

The open state is O, from which the channel may close through intrinsic pore gating (C) or undergo sequential blocking by a polyamine species, first as a partial (subconductance) block (S) by the high-affinity site, followed by complete block (B) by the low-affinity site. The arrows indicate both the direction of positive gating charge displacement and the orientation of equilibrium constants. The equilibrium constants have the following form: *L* = exp(–η* _{L}*/

*kT*) and

*K*= exp(–η

_{i}*/*

_{i}*kT*). The corresponding “particle potentials” are given by η

*= –Δ*

_{L}*q*(

_{L}*V*–

*V*) and η

_{L}*= –Δ*

_{i}*q*(

_{i}*V*–

*V*) – μ, where μ =

_{i}*kT*ln([

*B*]/[

*B*]) is the chemical potential of the blocker, Δ

_{o}*q*

_{1}and Δ

*q*

_{2}are the charge displacements of partial and complete block, respectively, and

*V*

_{1}and

*V*

_{2}are the corresponding half-activation potentials when [

*B*] equals the standard blocker concentration [

*B*], which was arbitrarily assigned the value 1 nM.

_{o}The open probability for the COSB model is

where φ is the fractional conductance of the S state, and the partition function is *Z* = 1 + *L* + *K*_{1} + *K*_{1}*K*_{2}.

The asymptotic behavior of *W _{H}*

_{[}

_{g}_{]}at extreme voltages (

*V*→ ±∞) is

*W*

_{H}_{[}

_{g}_{](–)}= η

*(provided Δ*

_{L}*q*< Δ

_{L}*q*

_{1}) and

*W*

_{H}_{[}

_{g}_{](+)}= η

_{2}+

*kT*lnφ (requiring φ ≠ 0, otherwise η

_{1}+ η

_{2}). The upper limit of the slope in the steepest portion of the plot (the traditional Hill coefficient) is given by –(Δ

*q*

_{1}+ Δ

*q*

_{2}).

### Allosteric model

The partition function for a radial arrangement of *n* voltage sensors J around a single pore L is: *Z* = (1 + *J*)* ^{n}* +

*L*(1 +

*JD*)

*, where*

^{n}*L*and

*J*are voltage-dependent equilibrium constants:

*L*= exp(–η

*/*

_{L}*kT)*and

*J*= exp(–η

*/*

_{J}*kT*), whose particle potentials are given by η

*= –Δ*

_{X}*q*(

_{X}*V*–

*V*). The state diagram is shown in Fig. 3 C. The coupling factor is

_{X}*D*= exp(–η

*/*

_{D}*kT*), where η

*=*

_{D}*W*– Δ

_{D}*q*, and

_{D}V*W*is the interaction energy. The open probability is

_{D}*P*= (1+

_{O}*J*)

*/*

^{n}*Z*. The conductance Hill energy is

*W*

_{H}_{[}

_{g}_{]}=

*η*ln[(1 +

_{L}+ nkT*J*)/(1 +

*JD*)], with asymptotes

*W*

_{H}_{[}

_{g}_{](–)}= η

*and*

_{L}*W*

_{H}_{[}

_{g}_{](+)}= η

*+*

_{L}*n*η

*. Thus*

_{D}*nW*is the vertical separation between positive and negative asymptotes at

_{D}*V*= 0. The upper limit of the slope at intermediate voltages is –(Δ

*q*+

_{L}*n*Δ

*q*+

_{J}*n*Δ

*q*).

_{D}### Two-populations model

The partition function for the sum of independent pore particles is *Z* = (1 + *L*_{1})^{N}^{1}(1 + *L*_{2})^{N}^{2}, where the two populations *N*_{1} and *N*_{2} satisfy *N*_{1} + *N*_{2} = *N*, and the pore equilibrium constants *L*_{1} and *L*_{2} are voltage dependent. The model scheme (Fig. 3 D) invokes two populations of channels with weakly voltage-rectifying pores (gating charges Δ*q*_{1} and Δ*q*_{2}) and possibly distinct conductance values *g*_{1} and *g*_{2}. Setting population fractions to *f*_{1} = *N*_{1}*g*_{1}/(*N*_{1}*g*_{1}+ *N*_{2}*g*_{2}) and *f*_{2} = 1 − *f*_{1}, we can write the open probability as

As usual, *W _{H}*

_{[}

_{g}_{]}=

*kT*ln[

*P*/(1 –

_{O}*P*)]. The channel populations are pure if

_{O}*f*

_{1}equals 0 or 1. In such cases, we obtain straight lines for the Hill plots corresponding to either η

_{1}or η

_{2}. The asymptotic behavior for intermediate

*f*values depends on which species carries the lesser charge:

The “interaction energy” is equal to 2*kT*ln*f*_{1} or 2*kT*ln*f*_{2}, but this is a fictitious quantity, as there is no energy linkage between the two channel species. When the charges of the two populations are of similar value, the result is essentially a straight-line Hill plot with slope Δ*q*_{1} ≈ Δ*q*_{2}.

### Kinetic analysis

Berkeley Madonna (Macey and Oster) was used to solve the kinetics of the equivalent circuit model using a fourth-order Runge–Kutta algorithm. It was also used for least-squares fitting of kinetic parameters in the equivalent circuit model.

## Results

### Comparing proposed mechanisms of weak inward rectification

Previous studies have suggested that blocker-free inward rectification is caused by a weakly charged gating transition (Matsuda et al., 2003, 2010). In a two-state C-O scheme, the theoretical Hill plot is a straight line with slope equal to Δ*q _{L}*, the gating charge of the pore gate. However, the experimental Hill plots are consistently sigmoidal (Fig. 2), suggesting that in nominal blocker-free solutions the pore gate is coupled to a second voltage-dependent process.

We considered whether the second process could be a blocking event. First, we postulated that incompletely washed polyamines are responsible for inward rectification. Our previous study (Liu et al., 2012) suggested that spermidine block can be described by the four-state COSB model, where S is a subconducting state induced by high-affinity polyamine block and B is the completely blocked state attributed to low-affinity binding. To estimate the amount of residual spermine necessary to account for the observed rectification, it was necessary to perform ramp experiments with varying spermine concentrations plus a nominally blocker-free control (Fig. 4) and fit the COSB model to the corresponding Hill-transformed conductance curves. The control plot was compatible with a residual spermine concentration of 1.0 nM, which was applied to subsequent fits of the COSB model to nominally blocker-free experiments. The negative-voltage asymptote corresponding to the closing transition (Δ*q _{L}* = 0.18 e

_{o}) was well represented by the model, but the positive-voltage asymptote is predicted to be steeper, with slope equal to the charge displacement of the low-affinity blocker (Fig. 4, black lines, Δ

*q*

_{2}= 1.5 e

_{o}; Fig. 5 A, green dashed line, Δ

*q*

_{2}= 1.1 e

_{o}). By considering the possibility of a

*G*baseline error, we forced the positive asymptote to level off at the value

_{min}*kT*ln

*b*. The same behavior occurs if the B state demonstrates partial block. However, the clipped curve produced a downward sloping residual in the positive-voltage linear asymptote of the Hill plot (Fig. 5 D), resulting in a poor fit. Thus incomplete polyamine washout is not compatible with weak inward rectification. Parameter values for the COSB model obtained from seven experiments are listed in Table 2.

After evaluating blocking mechanisms, we considered allosteric interactions between a weakly voltage-dependent pore region and a second voltage sensor. The allosteric model fit the Hill plot well in the critical outer asymptotic regions (Fig. 5 B). Like the COSB model, the allosteric model predicted a small gating charge of pore opening (Δ*q _{L}* = 0.18 e

_{o}; Table 3). The value of Δ

*q*(= 0.002 e

_{D}_{o}) was not statistically different from zero (P = 0.80, Student’s

*t*test), implying no difference in the slopes of the positive and negative voltage asymptotes. The proposed voltage sensor J demonstrated a voltage sensitivity (Δ

*q*= 1.65 e

_{J}_{o}) larger than the pore by an order of magnitude. The interaction energy between the two processes was computed to be

*W*= −86 meV (−2.0 kcal/mol), equating to a coupling factor of

_{D}*D*= 31. Because Kir channels contain four subunits, we considered the interaction of the pore with

*n*= 4 identical voltage sensors. The resulting five-particle model had no effect on the outer asymptotic regions, nor did it significantly improve the fit in the intermediate region around

*V*= 0, where residuals from the COSB model are found to be smaller (Fig. 5, D and E). The intermediate region of the Hill plot was more accurately described by the allosteric model with the addition of a second transition to the J particle, resulting in a six-state scheme with three additional parameters. However, since the experiments were designed to measure the outer asymptotic segments and hysteresis and voltage-offset artifacts can produce distortions in the steep intermediate portion of the curve, we decided that pursuing a better overall fit did not warrant the added complexity of the six-state model.

Finally, we explored whether the sigmoidicity of the Hill plot can be attributed to independent populations of simple two-state channels (Ishihara and Ehara, 2004). This scenario corresponds to the traditional method of fitting equilibrium curves to a sum of *s* Boltzmann curves, where in this case *s* = 2. In principle, the lesser charge movement (Δ*q*_{1}) should equal the slopes of the two outer segment asymptotes, as is the case for the allosteric model. The vertical displacement between the asymptotes is a function of the fraction of channels with the smaller charge (*f*_{1}). Equating the asymptotic slopes of the allosteric and two-populations models (Δ*q*_{1} = Δ*q _{L}*), one obtains a value of

*f*

_{1}= 0.16, comparable to estimates between 0.1 and 0.25 from previous work performed in the presence of polyamines (Ishihara and Ehara, 2004). However, this created a sizeable hump as the curve attempted to reach the negative-voltage asymptote after overreaching (Fig. 6). A better match of the theoretical curve to the experimental Hill plot required a larger asymptotic slope (Δ

*q*

_{1}= 0.34 e

_{o}), which led to a more balanced distribution of channel populations (

*f*

_{1}= 0.51; Fig. 5 C and Table 4). However, these parameter adjustments came at a cost, because the slopes of the outer asymptotic segments did not match the experimental Hill plots as well as the allosteric model (Fig. 5 F). Regression analysis of residuals in all seven experiments for the two-populations model demonstrated nonzero slopes in the positive-voltage asymptote (P values from 5 × 10

^{−7}to 0.24,

*F*test), though less compellingly so in the negative-voltage asymptote (P values from 0.02 to 0.38). The same analysis applied to the allosteric model convincingly failed to demonstrate a nonzero residual slope in either the positive-voltage (P values from 0.79 to 0.99) or negative-voltage (P values from 0.92 to 0.99) asymptotes.

### Ranking models with AIC scores

To supplement the above considerations, the three models were ranked according to their AIC score, which awards parsimony by yielding lower scores for smaller numbers of adjustable variables. Because only the outer asymptotic regions are reliably measured by the slow ramp experiment, the χ^{2} value used for AIC scoring included these regions only (Fig. 5, G–I), and only the variables that affect the outer asymptotes of the Hill plot were used in the fitting process; specifically Δ*q _{L}*,

*V*,

_{L}*W*, and Δ

*q*for the allosteric model (

_{D}*k*= 5), Δ

*q*,

_{L}*V*, and

_{L}*b*for the COSB model (

*k*= 4); and Δ

*q*

_{1},

*V*

_{1},

*f*

_{1}, and Δ

*G*for the two-populations model (

_{min}*k*= 5). AIC scores were consistently lower for the allosteric model across all experiments, whereas the COSB and two-populations models received similar scores, with the exception of experiment R4, where the two-populations model ranked higher (Fig. 7).

### Comparing variation of fitted parameters across experiments

To assess the reliability of fitted parameters across experiments, we obtained confidence intervals for the adjustable parameters fitted to the outer asymptotes (Fig. 8), using the critical χ^{2} method in the case of the decimated data, and measuring variability in parameter estimates from the bootstrap-sampled data. In order for the two methods to be comparable, the bootstrap estimates used standard deviations, and so were $n $-fold larger than confidence intervals of the mean, which is based on standard errors. The means and confidence intervals of adjustable parameters across all gating schemes and experiments are plotted in Fig. 9. In all variables analyzed, there was greater variability in their fitted values across experiments than within individual experiments (P < 0.001, ANOVA). However, parameter estimates tended to be more closely correlated when performed on oocytes from the same frog. This is was the case for grouped experiments [R1, R2] and [R6, R7], each of which were performed in one day, though not for the group [R3–R5], which was performed in the span of a week.

### Kinetic models

The Hill plot is an equilibrium curve, and therefore there is the possibility that the dynamic ramps used in our experiments violate the quasi-equilibrium condition. There are two sources of error. First, series resistance in the pipette (5–10 MΩ) combined with large inward currents (>1 nA) can decrease the magnitude of the effective membrane potential in a time- and voltage-dependent manner, potentially causing nonlinearity in the negative voltage segment of the Hill plot. Second, the ramp speed, although fairly slow (1 mV/ms), produces a time delay in the dynamic G-V curves compared with the true equilibrium curves. The reason for the double ramp protocol was to evaluate the time delay, which appears as a small hysteresis between the positive (Pos) and negative (Neg) curves at the steepest portion of the Hill plot (Fig. 1 C), where gating processes are characteristically at their slowest.

To test for these errors, we used kinetic modeling with an equivalent circuit (Fig. 10 A) that was able to account for the effects of membrane capacitance, series resistance, and membrane leak on the ramped current. We used a dynamic formulation of the *n* = 1 allosteric model (Fig. 10 B), which is shown here to fit the Hill plot well in its equilibrium form. The equivalent circuit model adequately accounted for the passive membrane components before and after capacity and series resistance compensation (Fig. 10, C and D). The distortions of the ionic current and membrane potential were minor even though the maximum current approached 4 nA (Fig. 10 E). The gating parameters for the kinetic model that aimed to reproduce the hysteresis in the Hill curves were very similar to those obtained assuming true equilibrium (Fig. 10 F). These results show that the ramp protocol is suitable for obtaining quasi-equilibrium G-V relationships, particularly at extreme voltages where the Hill plot is expected to be linear and the system equilibrates rapidly.

## Discussion

Weak inward rectification of Kir channels plays an important physiological function in many organs including brain, heart, kidney, and pancreas (Hibino et al., 2010). However, the underlying mechanism remains elusive. The purpose of this study was to systematically investigate candidate hypotheses through the use of Hill analysis, which can resolve cooperative models of gating by emphasizing their asymptotic behavior at extreme voltages. Detailed analysis suggested that the intrinsic inward rectification is caused by the allosteric coupling between a very weakly voltage-dependent pore gate (Δ*q _{L}* ∼0.18 e

_{o}) and a voltage sensor with roughly 10-fold greater gating charge (Δ

*q*∼1.7 e

_{J}_{o}). A positive coupling factor (

*D*∼31) between the two gating particles suggests that voltage gating stabilizes the closing of the channel. Hill analysis determined that neither a simple pore gate nor residual polyamine block could account for the weak inward rectification. A simple pore gate would generate a straight-line Hill plot, whereas the lower asymptote for any blocker model such as the COSB model would be steep, reflecting the charge carried by the blocker molecule. On the other hand, the two-populations model, like the allosteric model, is compatible with parallel outer segment asymptotes, but a pronounced hump in the region of

*V*< 0 predicted by the two-populations model was incompatible with the experimental data (Fig. 6). A previous study (Ishihara and Ehara, 2004) examining the two-populations model in the presence of polyamines would not have resolved the hump, because they measured ln(

*G*/

*G*) rather than ln[

_{max}*G*/(

*G*–

_{max}*G*)]. The former is sensitive only to the positive-voltage asymptote. To prevent the hump, the lesser of two gating charges needed to be increased from ∼0.18 to ∼0.34 e

_{o}, which required an adjustment to

*G*and an increase of the population fraction

_{min}*f*

_{1}from ∼0.16 to ∼0.51. Even with these parameter changes, the fit failed to match the slope of the outer asymptotic segments of the Hill plot as well as the allosteric model (Fig. 5, E and F). In short, the demonstration of a sigmoidal transition between equally shallow linear asymptotes separated by a coupling energy is most naturally explained by a two-particle allosteric scheme, a conclusion supported by AIC ranking of the models (Fig. 7).

Most Kir channels possess weak inward rectification in nominal blocker-free solutions. The weak inward rectification in Kir2.1 has been explained by voltage block of residual polyamines or contaminates in solutions (Guo and Lu, 2000; Guo et al., 2003). However, previous studies have shown that extensive washout of inside-out patches with Mg^{2+}- and polyamine-free solutions could not remove the weak inward rectification (Vandenberg, 1987; Kubo et al., 1993; Shieh et al., 1996; Omori et al., 1997; Oishi et al., 1998; Matsuda et al., 2003, 2010; Ishihara and Ehara, 2004). A blocker model lacking the voltage sensor would require consistent trace levels of “residual” blocking ion concentration, which would be difficult to control and would generate Hill plots of varying depth. In this study, fits of the COSB model to conductance Hill plots with known spermine concentrations were compatible with a residual spermine concentration of 1.0 nM for nominally blocker-free preparations (Fig. 4), a larger value than previous upper limits for residual spermine concentrations (0.1 to 0.5 nM; Matsuda et al., 2003; Liu et al., 2011). It is unlikely that washout after pipette excision from the oocyte would yield consistent levels of residual polyamine levels at this concentration. One can generalize this result to any gating process relying on internal polyamine blockers. For example, allosteric interaction between one or more polyamine binding sites and the pore would successfully model the outer asymptotes, but would require a residual polyamine presence. Such an allosteric scheme would also require 1 mM residual spermine, because for a given Δ*q*, the horizontal spacing between curves at intermediate voltages is uniform for constant ratios between spermine concentrations, and the analysis in Fig. 4 applied to the COSB model would also apply in this case.

The nature of the putative voltage sensor J coupled to the pore is unclear, as the Kir channel lacks a canonical voltage sensor. One possibility is that one or more permeant ions bind to sites within the pore (Kurata et al., 2010; Schewe et al., 2016). Given that Δ*q _{J}* = 1.7 e

_{o}, at least two monovalent or one ion divalent species would need to travel nearly the entire distance of the electric field to behave as the voltage sensor, although the distance requirement might be divided by four if voltage-sensing duties were split among the four subunits. We saw no improvement in fitting with a single J particle versus four J particles; however, our experiments were not designed to accurately define the steep intermediate portion of the Hill curve. With permeation-type gating, one expects the half-activation voltage

*V*to depend on the concentration of the contributing ion species. The modulation of Kir gating by permeant K

_{J}^{+}has long been recognized (Hagiwara et al., 1976), and it is conceivable that J “particle” activation represents the voltage-dependent unbinding of K

^{+}from one or more sites within the protein to the extracellular solution. The particle potential describing such an event is −Δ

*q*(

_{J}*V*−

*V*), where the half-action potential

_{J}*V*([Δ

_{J}*W*+ μ

_{J}_{o}]/Δ

*q*) increases with increasing chemical potential μ

_{J}_{o}=

*kT*ln[K

^{+}]

_{o}of the external ion, consistent with an observed rightward shift in outward current with increasing [K

^{+}]

_{o}(Chang et al., 2010). Although a permeation-type gating mechanism is plausible given the importance of blockers and permeant ions in Kir gating, one cannot rule out that charged protein residues located within one of the transmembrane domains could be collectively acting as a voltage sensor. Potential candidates include K182 in the C-terminal portion of the M2 domain, R148 located just two residues beyond the highly conserved GYG sequence in the selectivity filter, D172 in the midportion of the M2 domain (known to be a polyamine binding site), and E138 in the midpore domain. Charge-neutralization experiments (Aggarwal and MacKinnon, 1996; Seoh et al., 1996) would be necessary to determine whether fixed charges contribute to voltage sensing in the blocker-free channel.

Constructing the conductance Hill plot required accurate measurements of channel open probability (Eq. 1). Previous studies (Horrigan et al., 1999; Islas and Sigworth, 1999; González-Pérez et al., 2010) have determined the limiting conductance slope from a series of constant-voltage conductance histograms (*NPo* method), but the technique is not suitable for very large currents when *Po* ≈ 1, so we used the ramp approach, which had the advantage of sampling all voltages in a single trace. A limitation of this study is the potential errors in determining *G _{min}* and

*G*. It was not possible to unambiguously measure the precise limits of conductance because they are approached asymptotically and noise in the ionic current is greatly amplified in the extreme segments of the Hill plot. Our approach was predicated on the assumption that the Hill plots achieved a limiting slope within the range of voltage measured. Even if this were true based on a channel’s gating mechanism, nonlinearities can arise from the effects of incomplete series resistance and linear capacity compensation, although these were accounted for by simulating an equivalent circuit and the effects were found to be small (Fig. 10).

_{max}In conclusion, this study shows that the weak inward rectification in Kir2.1 channels is best explained by an unknown voltage sensor communicating allosterically with the pore gate. The double ramp protocol used in these experiments is a useful tool for uncoupling gating components to reveal both their individual characteristics and the energies of interaction (or coupling constants) between them, but conductance Hill analysis depends on being able to accurately measure asymptotic rates of approach at extreme voltages, and so is primarily useful for weak coupling. With these limitations in mind, Hill analysis has the potential for improving our understanding of allosteric coupling in ion channels and proteins in general.

## Acknowledgments

We thank Dr. Lily Jan (University of California, San Francisco, San Francisco, CA) for providing the Kir2.1 clone. We thank Dr. Shi-Bing Yang at Institute of Biomedical Science and Dr. Grace S. Shieh at Institute of Statistical Science, Academia Sinica, Taiwan, for their comments and discussion.

This work was supported by Academia Sinica andthe Ministry of Science and Technology, Taiwan (103-2320-B-001-007-MY3 and 107-2923-B-001-001-MY3).

The authors declare no competing financial interests.

Author contributions: D. Sigg performed computer modeling; H.-K. Chang and R.-C. Shieh conducted the experiments; D. Sigg and R.-C. Shieh designed research and wrote the paper. All authors analyzed data and reviewed the manuscript.

Richard W. Aldrich served as editor.

## References

## Author notes

Dr. Shieh died on March 14, 2017.