## Abstract

Stable coexpression of human (h)KCNQ1 and hKCNE1 in human embryonic kidney (HEK)-293 cells reconstitutes a nativelike slowly activating delayed rectifier K^{+} current (HEK-*I*_{Ks}), allowing β-adrenergic modulation of the current by stimulation of endogenous receptors in the host cell line. HEK-*I*_{Ks} was enhanced two- to fourfold by isoproterenol (EC_{50} = 13 nM), forskolin (10 μM), or 8-(4-chlorophenylthio)adenosine 3′,5′-cyclic monophosphate (50 μM), indicating an intact cAMP-dependent ion channel-regulating pathway analogous to the PKA-dependent regulation observed in native cardiac myocytes. Activation kinetics of HEK-*I*_{Ks} were accurately fit with a novel modified second-order Hodgkin-Huxley (H-H) gating model incorporating a fast and a slow gate, each independent of each other in scale and adrenergic response, or a “heterodimer” model. Macroscopically, β-adrenergic enhancement shifted the current activation threshold to more negative potentials and accelerated activation kinetics while leaving deactivation kinetics relatively unaffected. Modeling of the current response using the H-H model indicated that observed changes in gating could be explained by modulation of the opening rate of the fast gate. Under control conditions at nearly physiological temperatures (35°C), rate-dependent accumulation of HEK-*I*_{Ks} was observed only at pulse frequencies exceeding 3 Hz. Rate-dependent accumulation of *I*_{Ks} at high pulsing rate had two phases, an initial staircaselike effect followed by a slower, incremental accumulation phase. These phases are readily interpreted in the context of a heterodimeric H-H model with two independent gates with differing closing rates. In the presence of isoproterenol after normalizing for its tonic effects, rate-dependent accumulation of HEK-*I*_{Ks} appeared at lower pulse frequencies and was slightly enhanced (∼25%) over control.

- slowly activating cardiac delayed rectifer potassium current
- human embryonic kidney-293 cells
- β-adrenergic modulation
- KCNQ1
- KCNE1

the slowly activating cardiac delayed rectifier K^{+} current (*I*_{Ks}) modulates the late repolarization phase of the cardiac action potential in cardiac myocytes (8, 37, 41). The critical contribution of *I*_{Ks} to normal cardiac electrophysiology is underscored by the finding that congenital long QT syndrome, a condition that predisposes affected individuals to ventricular arrhythmias, is linked to mutations in the genes encoding human (h)KCNQ1 and hKCNE1 (45, 53). Whereas KCNQ1, which belongs to the Kv family of voltage-gated K^{+} channels, reconstitutes a rapidly activating voltage-dependent K^{+} conductance, cotransfection of the KCNE1 regulatory subunit confers the characteristic delayed slowly activating gating kinetics as well as the cAMP/protein kinase A (PKA) pathway-dependent modulation of the native current, *I*_{Ks} (1, 25, 29, 40).

The role that *I*_{Ks} plays in cardiac repolarization is particularly important for its ability to counterbalance the depolarizing effect of enhanced L-type Ca^{2+} current during sympathetic stimulation of the heart (4, 5, 12, 23). This is highlighted by the finding that LQT1 mutations are usually symptomatically silent in many carriers until sudden exertion or emotional upset triggers cardiac events (7, 54). *I*_{Ks} is distinguished from the rapidly activating delayed rectifier K^{+} current (*I*_{Kr}) by its gating kinetics, pharmacological sensitivity, and other properties, notably its considerable enhancement by β-adrenergic stimulation (20, 42). The properties of *I*_{Ks} constitute an important cardiac repolarization reserve invoked especially during elevated sympathetic tone (33).

Modeling of the effect of direct β-adrenergic modulation of *I*_{Ks} (e.g., by isoproterenol) has traditionally been limited to native cardiac myocytes. Electrophysiological study of native *I*_{Ks} is complicated by the technical difficulty of isolating *I*_{Ks} due to the presence of other cardiac K^{+} currents, such as *I*_{Kr}, whose kinetics overlap those of *I*_{Ks}, as well as the relatively small size of native *I*_{Ks}, especially at lower temperatures. The difficulty of obtaining healthy human cardiac myocytes for native *I*_{Ks} studies has limited studies of β-adrenergic modulation of human *I*_{Ks} (23).

Until recently, modulation of recombinant *I*_{Ks} in heterologous expression systems has relied on addition of second messengers such as cAMP and phosphatase inhibitors to the intracellular recording solution (47) rather than β-adrenoceptor-mediated activation of endogenous G protein-coupled signaling pathways that are present in the host background. In this context, we stably expressed hKCNQ1 and hKCNE1 in human embryonic kidney (HEK)-293 cells. Expression of hKCNQ1 and hKCNE1 in HEK-293 cells reconstitutes not only the voltage- and temperature-dependent gating characteristics and pharmacological responses but also the β-adrenergic modulation of native *I*_{Ks} (11). We have taken advantage of this isolated *I*_{Ks} expression system to derive an accurate mathematical model of *I*_{Ks} gating that allows the effects of β-adrenergic stimulation to be easily quantified. The accuracy and simplicity of this model of *I*_{Ks} lends itself to incorporation into mathematical models of the cardiac action potential as well as to differentiation of the effects of pharmacological modulation or genetic alteration of *I*_{Ks}. These studies validate the stable expression of hKCNQ1/hKCNE1 in the HEK-293 background as a highly useful model of native *I*_{Ks}.

## MATERIALS AND METHODS

#### Establishment and maintenance of the HEK-I_{Ks} cell line.

hKCNQ1 (hKvLQT1) and hKCNE1 (hminK) (licensed and obtained from the University of Utah) were respectively cloned into pZeoSV2+ and pcDNA1neo vectors (Invitrogen) and stably transfected into HEK-293 cells (ATCC). Cells were grown in Dulbecco's modified Eagle's medium supplemented with 10% fetal bovine serum, 2 mM l-glutamine, 10 U/ml penicillin, 50 μg/ml streptomycin, 0.5 mg/ml Zeocin, and 0.4 mg/ml G418 antibiotics (Invitrogen Life Technologies). Cells were incubated at 37°C in 5% CO_{2} and grown in T75 tissue culture flasks (Corning Costar) to near confluence, at which time they were passaged and plated in six-well plates. Before electrophysiological recording, cells were gently trypsinized in 0.05% trypsin-EDTA solution (Invitrogen Life Technologies), dislodged from the growth substrate, and then resuspended and maintained in the culture medium until electrophysiological recording.

#### Electrophysiological recordings.

A small aliquot of HEK-293 cells stably expressing hKCNQ1/hKCNE1 was placed into a glass-bottom perfusion chamber containing a HEPES-buffered solution (HBS). Single HEK-*I*_{Ks} cells were studied using the whole cell configuration of the patch-clamp technique with glass microelectrodes (Warner Instruments G150F-4) with tip resistances of 1.5–3 MΩ containing (in mM) 119 K-gluconate, 15 KCl, 3.2 MgCl_{2}, 5 HEPES, 5 EGTA, and 5 K_{2}ATP, adjusted to pH 7.35 with KOH. Cells were continuously superfused with HBS containing (in mM) 132 NaCl, 4 KCl, 1.8 CaCl_{2}, 1.2 MgCl_{2}, 10 HEPES, and 11.1 glucose, adjusted to pH 7.35 with NaOH. HEK-*I*_{Ks} was recorded at monitored temperatures ranging from 19 to 36°C and was quantified as the peak tail current amplitude during repolarization to the holding potential (*V*_{h}) following a step depolarization (*V*_{step}). The temperature of the bath perfusate was measured with a calibrated thermistor positioned near the cell and was maintained at desired settings with a temperature controller (Warner Instruments TC-344B). For cardiac ventricular myocyte recordings, the bath solution was nominally Ca^{2+}-free HBS with 0.4 μM nisoldipine to block L-type Ca^{2+} channels. The pipette solution contained (in mM) 500 K-gluconate, 25 KCl, and 5 K_{2}ATP, adjusted to pH 7.35 with KOH. Guinea pig ventricular myocytes were isolated by enzymatic digestion of the heart using the Langendorff perfusion method (38). Guinea pig myocyte isolations were conducted according to study protocols reviewed and approved by the Institutional Animal Care and Use Committee. In myocytes, cardiac *I*_{Ks} was quantified as the amplitude of the time-dependent activating outward current at the end of a 1-s *V*_{step} to +50 mV from a *V*_{h} of −50 mV.

The benzodiazepine-selective *I*_{Ks} inhibitors L-768,673 and L-735,821 were prepared as 2 mM stock solutions in DMSO. Forskolin (Sigma F-6886) was prepared as a 100 mM stock solution in DMSO. The cAMP analog 8-(4-chlorophenylthio)cAMP sodium (CPT-cAMP; Sigma C-3912) was prepared as a 50 mM stock solution in distilled water. Isoproterenol (Sigma I-6504) was prepared as a 10 mM stock solution in 0.1% ascorbic acid in distilled water. Working dilutions were made fresh before each recording by diluting stock solutions directly into bath solution. Final DMSO concentrations did not exceed 0.1%.

Whole cell currents were elicited with a voltage-clamp amplifier (Axopatch 200B; Axon Instruments) and recorded through a Digidata 1322a interface (Axon Instruments) to an IBM-compatible computer (Compaq). Voltage-clamp protocols and data acquisition and analyses were performed using Clampex v8.2 software (Axon Instruments). The voltage drop across the access resistance was dynamically compensated 80%, while the whole cell capacitance current was subtracted from the clamp current before acquisition using the existing Axopatch 200B circuitry. The resulting whole cell currents were amplified, low-pass filtered with a cutoff at 1 kHz, digitally sampled at 2–5 kHz, and stored to disk. The bath temperature was synchronously recorded for off-line analysis and to monitor temperature-induced changes in current amplitude.

#### Fitting of whole cell currents.

Activating outward currents elicited with depolarizations to activating potentials, *V*_{step}, were fitted with Hodgkin-Huxley (H-H) model-derived equations in the form of products of single exponentials, where the order of the H-H model, *n*, designates the number of independent, first-order gates that must transition from a closed to an open state before ions can flux through the channel (21), and τ_{Ai} is the time constant of the reequilibration of each independently activating gate with index *i* = 1, 2,… *n*. In contrast to standard H-H models, where τ_{Ai} values are identical, we allowed each τ_{Ai} value to vary independently. Least-squares fits to the data, with *n* ranging from 1 to 4, were evaluated.

Deactivating HEK-*I*_{Ks} tail currents on repolarization to the tail potential, *V*_{tail}, were fit with either a single exponential or the sum of two decaying exponentials of the form where *Ai* is the amplitude of each exponential component with deactivating time constant τ_{Di} and *B* is the steady-state current, if any. If *V*_{tail} was below the activation threshold of the current, *B* became negligible and the deactivating tail currents were accurately fit with a single decaying exponential.

#### Calculation of Q_{10}: temperature dependence.

Time constants (τ) derived from fits of activation or deactivation kinetics were plotted as a function of temperature and fit with the Arrhenius equation: 1/τ = *k* = *A*·exp(−*Ea*/*RT*), where *k* is a rate constant at the absolute temperature *T*, *Ea* is the activation energy, *R* is the gas constant, and *A* is a constant factor. Rearranging this equation as ln τ = (*Ea/R)/T* − ln *A* indicates that the logarithm of τ varies linearly as a function of 1/*T* with the ratio *Ea*/*R* defining the slope of this linear relationship. Thus Q_{10} values were calculated from the slope of the Arrhenius plot.

For evaluating the voltage dependence of activation and deactivation, τ values were derived from fitting of the activation and deactivation time courses. It was assumed that voltage-sensitive gating transition rates followed the relation 1/τ ∼ exp[(*zF*/*RT*)(*V*)], where *z* is the gating charge, *F* is Faraday's constant, *R* and *T* are as defined above for the Arrhenius equation, and *V* is the transmembrane potential. According to this relation, at a fixed temperature, the logarithm of τ varies as a linear function of the membrane potential, and the slope of this relation yields the span of voltage over which an *e*-fold change in τ can be expected. To fully define the relation between τ and voltage, the value of τ at a given membrane potential was averaged for all cells and reported with the averaged slope for all cells. For determination of *V*_{mid}, the potential of half-maximal activation of HEK-*I*_{Ks}, normalized *I*_{Ks} tail amplitudes were fit with the Boltzmann relation, {1/(1 + exp[(*z*_{d}*F*/*RT*)(*V* − *V*_{mid})])}, where constants are as defined above and (*z*_{d}*F*/*RT*) is the slope factor *k*.

#### Statistics.

The reported significance of observed changes in selected parameters was calculated using the two-tailed paired Student's *t*-test. The *P* values obtained are indicated.

## RESULTS

#### Pharmacological isolation and characterization of HEK-I_{Ks}.

The pharmacological sensitivity of the outward currents resulting from the stable coexpression of hKCNQ1 and hKCNE1 in HEK-293 cells (HEK-*I*_{Ks}) was probed using potent and selective small molecule inhibitors of native *I*_{Ks}. L-768,673 and L-735,821 are benzodiazepines that have been shown to selectively inhibit native *I*_{Ks} of guinea pig ventricular myocytes at nanomolar concentrations (39). Figure 1*A* illustrates the concentration dependence of inhibition by L-768,673 and L-735,821 of HEK-*I*_{Ks} outward time-dependent activating and deactivating tail currents recorded at 35°C. An initial variable instantaneous HEK-293 background current (58) was insensitive to these *I*_{Ks} inhibitors at concentrations (≥100 nM) that completely inhibited HEK-*I*_{Ks}. Thus the time-independent kinetics of a background current, *I*_{const}, was effectively revealed. Figure 1*B* illustrates a comparable effect of L-768,673 on native *I*_{Ks} in guinea pig ventricular myocytes at 35°C. The recording of native myocyte current illustrates the difficulty of isolating the *I*_{Ks} current in myocytes given that other K^{+} currents, notably the inward rectifier current *I*_{K1} as well as *I*_{Kr}, can introduce considerable instantaneous and holding current offsets as well as contribute to the time-dependent activating and deactivating tail currents. Figure 1*C* illustrates the concentration dependence and reversibility of the effect of L-768,673 on the amplitude of the HEK-*I*_{Ks} tail current. L-768,673 was applied at increasing concentrations of 3, 10, and 100 nM, separated by an intervening washout period (3 to 10 nM) that revealed the reversibility of the HEK-*I*_{Ks} inhibition. The average fraction of control tail current amplitude remaining after equilibration at various inhibitor concentrations is plotted in Fig. 1*D*, and fitting of the Hill equation to these data yielded IC_{50} values of 2.4 and 0.8 nM for inhibition of HEK-*I*_{Ks} by L-768,673 and L-735,821, respectively. The IC_{50} value for L-768,673 inhibition of HEK-*I*_{Ks} (2.4 nM) was comparable to that (within 3-fold) for native *I*_{Ks} (7.3 nM) of guinea pig ventricular myocytes, whereas L-735,821 was 10 times more potent on HEK-*I*_{Ks} (0.8 nM) compared with native *I*_{Ks} (8 nM).

#### Mathematical modeling of activating and deactivating whole cell HEK-I_{Ks}.

A unique characteristic of the native cardiac *I*_{Ks} is a high temperature sensitivity of gating kinetics relative to other voltage-gated K^{+} currents (52). The temperature sensitivity of a rate-dependent gating process may be expressed as the degree of change over a span of 10°C, or Q_{10}. Typically, the kinetics of voltage-gated channels exhibit Q_{10} values in the range of 1.5–3. The native cardiac *I*_{Ks} current has Q_{10} values near 4 (52). To examine the temperature sensitivity of activating HEK-*I*_{Ks}, we applied 1-s *V*_{step} to +50 mV from a *V*_{h} of −50 mV to activate HEK-*I*_{Ks} while raising the temperature from 20 to 35°C (Fig. 2*A*).

To model the activating currents at different temperatures, we fit H-H models with the number of independent gates *n* ranging from 1 to 4 to the time-dependent activating current during *V*_{step}. A second-order (*n* = 2) H-H model most accurately fitted the activation time course, yielding correlation coefficients routinely exceeding 0.99. In these second-order fits, the activating time constant τ_{Ai} for each of the two gates had to vary independently to achieve a high correlation across the temperature range (Fig. 2*B*, ≥29°C). The modified second-order H-H model of activation of the form (1) was fitted to the outward current during the voltage step, *I*_{step}, where *I*_{Ks amp} is the peak amplitude of the time-dependent slowly activating current and *I*_{const} is the time-independent instantaneous constant background current (Fig. 1). The fitted model currents are superimposed over the actual currents obtained at the indicated temperatures (Fig. 2*A*). The activating time constants for the currents, τ_{A1} and τ_{A2}, at each temperature are plotted in Fig. 2*B*.

The tail current deactivation kinetics at −50 mV were analyzed and fit with the following monoexponential decaying function: (2) where *I*_{Ks peaktail} is the peak amplitude of the tail current on repolarization to −50 mV, which varies with temperature, and τ_{D} is the deactivation time constant at a given temperature. In Fig. 2*A*, *inset*, the tail currents at −50 mV are plotted on a logarithmic ordinate axis to demonstrate the linear time course expected for a monoexponential decay of the tail current. Figure 2*B* plots the deactivation time constants τ_{D} for the currents in Fig. 2*A* as a function of temperature.

At 20°C, there is only a minimal amount of time-dependent activating current in response to the 1-s pulse to +50 mV, and the activation and deactivation is relatively very slow with time constants τ_{A1}, τ_{A2}, and τ_{D} in the range of 1–1.5 s. Raising the temperature to 25°C substantially speeded activation (τ_{A1} and τ_{A2} = 0.39 s), increased the magnitude of the time-dependent activating and deactivating tail currents, and speeded deactivation (τ_{D} = 0.74 s). At a nearly physiological temperature of 35°C, the HEK-*I*_{Ks} apparently activated fully within the 1-s step with rapid time constants of activation (τ_{A1} = 0.13 s, τ_{A2} = 0.04 s), and the tail current deactivated rapidly (τ_{D} = 0.2 s). On average, the peak tail current at −50 mV following a 1-s step to +50 mV was 0.17 ± 0.01 nA at room temperature (19–20°C) and increased ∼6.5-fold to 1.1 ± 0.4 nA at 35–36°C (*n* = 6). At potentials where the current could be fitted along a trajectory to a near steady-state level by the end of the 1-s pulse, a faster and slower component of activation with different temperature sensitivities emerged (Fig. 2*B*, ≥29°C). The temperature dependence of the two activation time constants was determined for eight cells, and the Q_{10} for each time constant was averaged and is shown in Fig. 2*C*. Q_{10} values of 3.2 ± 0.3 and 9.5 ± 1.2 were determined for the slow and fast activation time constants τ_{A1} and τ_{A2}, respectively, as shown in Fig. 2*C*.

A Q_{10} of 4.3 ± 0.9 was calculated for the temperature sensitivity of deactivation. The deactivation Q_{10} values obtained with HEK-*I*_{Ks} compare favorably with those of *I*_{Ks} studies in cardiac myocytes. For example, Walsh et al. (52) demonstrated a Q_{10} of 3.8 for *I*_{Ks} recorded in guinea pig ventricular myocytes based on the change in the time constant for deactivation of *I*_{Ks} tail current at −30 mV when the temperature was increased from 22 to 32°C.

#### Effect of temperature on voltage-dependent activation of HEK-I_{Ks}.

Experiments to gauge the effect of temperature on the gating kinetics above used a limited-duration 1-s depolarizing step to a single activating potential of +50 mV. Although this step duration was adequate to resolve activation time courses at higher temperatures, longer step durations were required for more complete saturation of the activating current at lower temperatures and examination of the effect of changing the step potential on the activation time constants of the model (*Eq. 1*). Figure 3 illustrates the voltage and temperature dependence of the activation time course of HEK-*I*_{Ks} at 25 and 35°C. Voltage-dependent activation of the current was measured at increasing *V*_{step} with a longer duration of 3 s. The peak amplitude of the ensuing tail current following repolarization to −50 mV was used to quantify the extent of isochronal activation of the HEK-*I*_{Ks} current during the 3-s steps (Fig. 3*C*, circles). At nearly physiological temperature (35°C), the tail current activated at *V*_{step} greater than −40 mV and saturated as *V*_{step} exceeded +40 mV. HEK-*I*_{Ks} activation was faster at more positive potentials, and eventually the currents saturated during the 3-s *V*_{step}. As shown by fits of *Eq. 1*, the time constants of the two first-order activation processes, τ_{A1} and τ_{A2}, varied in an apparently log-linear fashion with *V*_{step} (Fig. 3*C*). Figure 3*D* shows the averaged time constants from six cells. The straight line fits of the time constants on the log-linear plot yielded *e*-fold changes of the time constants every −55 ± 8 mV for the faster activation process, τ_{A2}, and every −63 ± 4 mV for the slower activation process, τ_{A1} (Table 1). The averaged time constants of activation at 35°C with a *V*_{step} of +20 mV were 0.12 ± 0.02 s for τ_{A2} and 0.38 ± 0.04 s for τ_{A1}. The midpoint of activation, *V*_{mid}, was +3.1 ± 4.6 mV from fits of the Boltzmann equation to the normalized tail currents with an average slope factor, *k*, of 12.2 ± 1.4 mV, which translates to an equivalent of 2.2 elementary gating charges per channel. By comparison with native *I*_{Ks}, Li et al. (26) determined a *V*_{mid} of +9.4 mV and a *k* of 11.8 mV for the *I*_{Ks} current in human ventricular myocytes at 35°C.

Reducing the temperature to 25°C shifted *V*_{mid} of HEK-*I*_{Ks} activation to more positive potentials and reduced the slope factor *k* (Fig. 3*C*). The tail currents at 25°C did not saturate completely during the 3-s step even during *V*_{step} to +100 mV or higher. The *V*_{mid} of isochronal (3 s) activation at 25°C was 14.2 ± 1.7 mV. The average time constants of activation from fits of *Eq. 1* at 25°C at a *V*_{step} of +20 mV were 0.59 ± 0.07 s for τ_{A2} and 1.83 ± 0.19 s for τ_{A1} (Fig. 3*D*). These values represent about a fivefold increase in both activation time constants when the temperature was lowered from 35 to 25°C (Table 1). Comparison of the activation time constants at 25 and 35°C across the voltage range in Fig. 3*D* shows that temperature shifts the time constants of activation at any given *V*_{step} without changing the sensitivity of the gates to voltage; i.e., there was no temperature-induced change in the slopes of the straight line fits of the log-linear time constant plots (expressed as mV/*e*-fold change) (Fig. 3*D* and Table 1).

#### Effect of external K^{+} concentration on reversal potential and deactivation kinetics of HEK-I_{Ks}.

Two separate studies in human ventricular myocytes (26, 50) have measured the reversal potential (*E*_{rev}) of native *I*_{Ks} (Ref. 26, at −70 mV, or Ref. 50, at −82 mV). To examine the K^{+} selectivity of HEK-*I*_{Ks} and to reveal possible changes in gating kinetics with altering the K^{+} driving force, we lowered the external K^{+} concentration ([K^{+}]_{e}) from 4 to 1 mM (Fig. 4). The *E*_{rev} of the HEK-*I*_{Ks} tail current with a [K^{+}]_{e} of 4 mM was −70 ± 1.6 mV (*n* = 4), considerably less negative than the expected Nernst potential for K^{+} if *I*_{Ks} were exclusively K^{+} selective. This finding is consistent with various studies of native *I*_{Ks}, which have established *E*_{rev} for *I*_{Ks} in the range of −60 to −80 mV. The *E*_{rev} of the HEK-*I*_{Ks} current shifted from −70 ± 1.6 to −102 ± 1.2 mV when [K^{+}]_{e} was lowered from 4 to 1 mM (Fig. 4*C*), demonstrating the primarily K^{+}-selective property of HEK-*I*_{Ks} and consistent with findings in native guinea pig *I*_{Ks} (43). Peak activating and deactivating tail current amplitudes scaled with membrane potentials *V*_{m}, according to the expected *E*_{rev}-dependent change in current driving force, *V*_{m} − *E*_{rev}, indicating that changes in tail current amplitude result simply from the shift in *E*_{rev} (Δ*E*_{rev}). Monoexponential fits (*Eq. 2*) of the decaying tail currents at potentials ≤30 mV revealed little or no change in the deactivation time constants with lowered [K^{+}]_{e} (Fig. 4*D*). The *V*_{mid} of activation of HEK-*I*_{Ks} was insensitive to lowering of [K^{+}]_{e} (not shown). Together, these results indicate that altering [K^{+}]_{e} affects HEK-*I*_{Ks} only insofar as it shifts *E*_{rev} of the channel but has little or no effect on gating kinetics.

#### Effect of agents activating the β-adrenergic modulator pathway on HEK-I_{Ks}.

Native *I*_{Ks} is modulated through cAMP-dependent phosphorylation by PKA during sympathetic stimulation of the heart, and this modulation plays a critical role in enhancing *I*_{Ks} and providing a greater repolarization “reserve” during β-adrenergic stimulation of the heart. HEK-293 cells express β-adrenergic G protein-coupled receptors that activate endogenous forskolin-sensitive adenylate cyclase, leading to robust activation of cAMP-dependent PKA (9). The ability to modulate reconstituted HEK-*I*_{Ks} through the endogenous β-adrenergic signaling pathway present in HEK-293 cells would be advantageous, because it would allow investigation of the effect of β-adrenergic stimulation on a variety of aspects of regulation of *I*_{Ks} and perhaps other currents in isolation, which is not feasible in the native myocyte setting. We investigated and verified that HEK-*I*_{Ks} could be modulated by the β-adrenergic agonist isoproterenol and other downstream β-adrenergic modulators as reported recently by Dong et al. (11). Figure 5*A* demonstrates the time course of enhancement of HEK-*I*_{Ks} tail current during application of 10 and 100 nM isoproterenol. The corresponding currents are shown in Fig. 5*B*. Figure 5*C* shows the concentration-response curve for isoproterenol, yielding an EC_{50} of 12.9 nM (24°C) and an average 2.6 ± 0.45-fold enhancement of HEK-*I*_{Ks} with 1 μM isoproterenol. Figure 5*C* also compares the modulation of HEK-*I*_{Ks} by β-adrenergic modulators acting along downstream points within the β-adrenergic PKA-dependent signaling cascade.

The membrane-permeant cAMP analog CPT-cAMP, which acts as a direct activator of PKA, was applied at a bath concentration of 50 μM and resulted in an average 1.81 ± 0.26-fold increase in tail current amplitude at 30°C (*n* = 5; Fig. 5*C*). The highest average increase in the current could be obtained by application of 10 μM forskolin, resulting in a 3.5 ± 1.0-fold increase (*n* = 4, 24°C; Fig. 5*C*). Forskolin directly stimulates adenylate cyclase to elevate cAMP levels, but increases are localized and compartmentalized, allowing locally high concentrations of cAMP to accumulate near the PKA target (10). This may explain why forskolin enhanced HEK-*I*_{Ks} more than the membrane-permeant cAMP analog. We examined the effect of the modulators along the β-adrenergic pathway on the kinetics of gating as interpreted through the success of the heterodimeric H-H gating model to accurately fit the temperature-dependent changes in gating of HEK-*I*_{Ks}. Activating currents at a *V*_{step} of +50 mV were fitted with *Eq. 1*, and deactivating tail currents at −50 mV were fitted with a monoexponential decaying function (*Eq. 2*). The resulting effects on the time constants of activation (τ_{A1}, τ_{A2}) and deactivation (τ_{D}), expressed as percent change from control values, were compared among β-adrenergic modulators (Fig. 5, *D–F*). The primary effect on gating kinetics of the β-adrenergic modulators was quantified as an acceleration of the activating transitions of the H-H gates during channel activation (Fig. 5, *D–F*). Fits with *Eq. 1* show that at the *V*_{step} of +50 mV, the β-adrenergic modulators overall increased both the slow (τ_{A1}) and the fast (τ_{A2}) time constants of the activation gates. The speeding effect was more pronounced on τ_{A2} and was comparable among the different β-adrenergic activators: 24.0 ± 4.2% with isoproterenol, 25.5 ± 6.2% with CPT-cAMP, and 26.6 ± 2.4% with forskolin. Figure 5*D* shows that, on average, all the β-adrenergic activators slowed slightly but not significantly the τ_{D} at −50 mV: 7.6 ± 8.0% with isoproterenol, 4.2 ± 4.9% with CPT-cAMP, and 7.9 ± 6.5% with forskolin.

Given that the effects on the kinetics of activation by the different agonists were comparable and reproducible, we next examined the effects of isoproterenol on the voltage dependence of activation. Figure 6 shows the effect on voltage-sensitive activation kinetics of isoproterenol from a single cell at a temperature of 24°C together with the fits of the activation time course at each potential. Isoproterenol shifted the *V*_{mid} of activation to more negative potentials. Application of 10 nM and 1 μM isoproterenol at 24°C resulted in average shifts in the *V*_{mid} of activation of −5.3 ± 1.0 and −13.4 ± 2.9 mV (*n* = 3), respectively (not shown). Moreover, isoproterenol speeded the activation kinetics as revealed by fits of the second-order modified H-H model (*Eq. 1*). Figure 6*D* shows the effects of isoproterenol on the step current amplitude as well as the activation time constants τ_{A2} and τ_{A1} from a single cell at 24°C. At activating *V*_{step} below +50 mV, the fitting algorithm yielded identical time constants for τ_{A1} and τ_{A2}, presumably due to incomplete definition of a fast and slow component as a result of incomplete saturation of HEK-*I*_{Ks} during the 2-s *V*_{step}, analogous to the low temperature fits of Fig. 2. However, at potentials of +50 mV or higher, the fitting algorithm was able to discriminate a faster and slower time constant. Again, the fits were highly correlated with the data, regularly exceeding correlation coefficients of ≥0.99. Together, these results indicate that the increase in HEK-*I*_{Ks} in response to adrenergic stimulation results from lowering of the activation voltage and speeding of activation kinetics, as well as increasing the overall open probability of the channel.

#### Forskolin activation of adenylate cyclase and more evidence for the heterodimeric H-H model of I_{Ks} gating.

From Fig. 5 it can be seen that, on average, forskolin enhanced HEK-*I*_{Ks} most of all the modulators, consistent with the understanding that adenylate cyclase activity is the final step in the cascade of stimulator G protein (G_{s})-coupled cAMP elevation. We used this strong effect of forskolin on HEK-*I*_{Ks} to explore further the quantitative effect of adrenergic stimulation on voltage-dependent activation and deactivation kinetics within the framework of the heterodimeric H-H model of HEK-*I*_{Ks}. Figure 7, *A* and *B*, shows the effects of forskolin on HEK-*I*_{Ks} with the use of 2-s activating *V*_{step} to potentials ranging from −50 to +20 mV at 30°C. Figure 7*C* plots the averaged peak tail current amplitudes normalized to the maximum amplitude before (control) and after application of 10 μM forskolin following activating *V*_{step} ranging from −50 to +50 mV. Fits with the Boltzmann equation yielded *V*_{mid} of 6.4 ± 1.7 mV for control and −3.6 ± 2.5 mV for forskolin. The average shift of the *V*_{mid} of activation at 30°C by 10 μM forskolin was −10.0 ± 0.94 mV (*n* = 5). Figure 7*D* shows the effects of forskolin on the second-order H-H activation model time constants (*Eq. 1*). At 30°C, the activation time course at steps ≥10 mV is clearly resolved into a fast (τ_{A2}) and a slow (τ_{A1}) process, and the τ_{A} values follow a near-logarithmic relationship with respect to voltage (filled symbols). Forskolin decreased both τ_{A2} and τ_{A1}. The main effect of forskolin on the activation kinetics at 30°C was to reduce τ_{A2} approximately twofold across the voltage range tested. The voltage sensitivity of τ_{A2} and τ_{A1} was unchanged by forskolin; there was no significant change in the slope of the log-linear fits of the time constants (Table 2).

#### Effect of forskolin on the reversal potential and deactivation kinetics of HEK-I_{Ks}.

We next examined the effects of adrenergic stimulation on the *E*_{rev} and the deactivation kinetics of *I*_{Ks} current with 10 μM forskolin. Deactivating tail currents were measured at 30°C with repolarizing step potentials ranging from −10 to −100 mV (Fig. 8, *A* and *B*) in decrements of −10 mV following activation for 2 s with a *V*_{step} of +50 mV (not shown). In the heterodimeric H-H formalism, two kinetically different first-order gates act independently to produce the characteristic sigmoidal activation time course of the activating HEK-*I*_{Ks} current at a given *V*_{step}. On deactivation to subthreshold *V*_{tail}, where the opening rates of the gates are negligible, the deactivating current must follow a monoexponential time course whose time constant reflects the sum of the closing rates of the two postulated first-order gates.

Figure 8*C* summarizes the effect of forskolin on slope conductance and *E*_{rev} of HEK-*I*_{Ks} at a [K^{+}]_{e} of 4 mM. The average peak tail current amplitudes are plotted at various *V*_{tail} values. The instantaneous current voltage relation followed a nearly ohmic (linear) relation around *E*_{rev}, consistent with *I*_{Ks} from native preparations (43, 50). The *E*_{rev} of the tail current at 30°C was −72 ± 2 mV (*n* = 5). After application of forskolin, the slope conductance of HEK-*I*_{Ks} increased, on average, two- to threefold, and *E*_{rev} was shifted slightly more positive, with a Δ*E*_{rev} of +4 ± 0.4 mV (*n* = 5).

Figure 8*D* quantifies the effects of 10 μM forskolin on deactivation kinetics. To examine effects of forskolin at potentials below the activation threshold of HEK-*I*_{Ks}, we averaged and compared the time constants τ_{D} from monoexponential fits of the decaying tail currents obtained at potentials less than or equal to −30 mV (Fig. 8*D*, 30°C). Although forskolin increased tail current amplitudes two- to threefold at all potentials (Fig. 8*C*), it had little or no effect on deactivation kinetics. This result confirmed those obtained with isoproterenol, forskolin, and cAMP at a single potential (Fig. 5). The voltage-dependent changes in deactivation were also examined at 20°C, where the τ constants are significantly slower (see Fig. 2), and minimal changes in τ_{D} were observed (Fig. 8*D*). Log-linear fits of the averaged τ_{D} time constants revealed that τ_{D} increased *e*-fold over a change in potential of 31 mV at 30°C and over a change in potential of 48 mV at 20°C.

#### Rate-dependent accumulation of HEK-I_{Ks} at 35°C and the effect of isoproterenol.

An important physiological function of *I*_{Ks} is its potential enhanced role during elevated sympathetic tone and, consequently, at faster heart rates. In addition to the scaling and more rapid activation of *I*_{Ks} observed above, an unresolved question is whether or not *I*_{Ks} “accumulates” at faster heart rates as a consequence of incomplete deactivation. Given the speeding of deactivation kinetics observed with increases in temperature, we examined whether HEK-*I*_{Ks} could accumulate appreciably at near-physiological temperatures (35°C), where the τ_{D} constants are <50 ms at a ventricular resting potential of −90 mV (Figs. 2 and 4). If the deactivation of *I*_{Ks} were the primary determinant for accumulation, then it would be expected to occur at pulse rates and interpulse intervals where deactivation is incomplete between successive action potentials. To test this hypothesis, we applied trains of simulated action potential voltage-clamp pulses consisting of a 200-ms *V*_{step} to +40 mV followed by a 17-ms repolarizing ramp to the *V*_{h} of −90 mV at pulse rates of 0.5, 2, 3, and 4 Hz (Fig. 9*A*). Figure 9*B* illustrates the relatively small amount of outward time-dependent *I*_{Ks} that is activated over such a short voltage pulse. Large rate-dependent accumulation of *I*_{Ks} over a pulse train of 10-s duration was evident only at the highest pulse rate of 4 Hz as shown in Fig. 9, *B* and *C*. At 4 Hz, a pulse repeat interval of 250 ms leaves only a short 33-ms rest interval at −90 mV for deactivation of the current before reactivation by the subsequent pulse. This allows for a rapid staircaselike accumulation at 4 Hz of the *I*_{Ks} from pulse to pulse, leading to a fourfold (4.0 ± 0.5, *n* = 7) rate-dependent enhancement of the current by the end of the 10-s pulse train. In contrast, at the pulse rate of 3 Hz, where the interpulse interval increases from 33 to 116 ms, much less rate-dependent accumulation of current was observed (1.4 ± 0.2-fold increase, *n* = 7).

As shown in Figs. 5–8, β-adrenergic modulation of HEK-*I*_{Ks} increases current magnitude, shifts activation gating to more hyperpolarized potentials, and speeds activation but has little or no effect on the rate of deactivation at potentials below the activation threshold. If use-dependent accumulation of *I*_{Ks} were strictly dependent on deactivation kinetics, then we would not expect changes in rate-dependent accumulation during β-adrenergic stimulation beyond the speeding and scaling of the time-dependent current. Figure 9*B* compares the HEK-*I*_{Ks} currents elicited by the first and last pulses of 10-s pulse trains applied at 2, 3, and 4 Hz before (control) and after application of 1 μM isoproterenol. After addition of 1 μM isoproterenol at 35°C, the HEK-*I*_{Ks} time-dependent outward current magnitude nearly doubled (fold increase of 1st pulse currents: 1.84 ± 0.16, *n* = 7). As under control conditions, pulsing-dependent accumulation of current over 10 s was most pronounced at 4 Hz, although the relative maximal fold increase over the initial (1st pulse) current was significantly less (3.0 ± 0.6, *n* = 7) than that observed under control conditions (*P* = 0.01). In contrast, even at 2- and 3-Hz pulsing rates, a slight enhancement of rate-dependent accumulation of current over control was observed after application of 1 μM isoproterenol (Fig. 9*C*), although these effects were not significant (*P* ≥ 0.1). The rate-dependent enhancing effect beyond the tonic effect of isoproterenol is further illustrated (Fig. 9*D*) by plotting the average enhancement of current at the end of the 10-s pulse trains normalized to control as a quasiestimation of a “steady-state” at the given pulse rates. At 4 Hz, where deactivation kinetics are the primary determinant of rate-dependent accumulation of current, no enhancement but rather a depression of the steady-state current was observed relative to control, after the “tonic” isoproterenol effect was factored out. Only at the intermediate pulsing rates of 2 and 3 Hz could a small increment in the fold increase over control be observed: from 1.7 ± 0.14 at a pulse rate of 0.5 Hz to 2.1 ± 0.30 at a pulse rate of 3 Hz, a net gain of 25%, although this gain was not highly significant (*P* = 0.12).

## DISCUSSION

#### Study implications.

This study presents a systematic characterization and quantification of the gating kinetics of the human KCNQ1/KCNE1 K^{+} channels stably expressed in HEK-293 cells in response to β-adrenergic, temperature-dependent, and pharmacological modulation. These findings validate a recent report that the expression of hKCNQ1/hKCNE1 in the HEK-293 background (HEK-*I*_{Ks}) provides a unique model of native human *I*_{Ks} because β-adrenergic modulation with isoproterenol is achievable without recourse to cotransfection of other accessory proteins (11). This model of the human *I*_{Ks} provides an alternate means to study the role of *I*_{Ks} in ventricular repolarization and its modulation in response to β-adrenergic receptor and adenylate cyclase activating pharmacological agents, which has not been demonstrated in other expression backgrounds.

#### Modeling of the hKCNQ1/hKCNE1 current activation and deactivation kinetics.

A modified second-order H-H model postulating two independent but nonidentical gates, each of which follows first-order kinetics, was able to describe with high fidelity (*R* ≥ 0.99) the biophysical characteristics of the HEK-*I*_{Ks} at varying potentials and temperatures and during modulation with agents in the β-adrenergic stimulatory pathway. Conventional second-order H-H models with two identical gates or processes have been used to model slow delayed rectifier current activation in both amphibian (18) and mammalian cardiac cells (30). Our results demonstrate that under conditions where the slow gate is not at steady state by the end of the activating voltage pulse, i.e., under conditions of lower temperature, lower stepping potentials, or shorter step durations, the activating current kinetics are well fit with identical time constants for the two gates (see Figs. 2 and 6). Our results further demonstrate that a second-order H-H model with nonidentical gates best defines activation when the current reaches a steady state during the activating step and the time course is fully resolved, i.e., using longer pulses, higher temperatures, or more depolarized potentials.

A second-order H-H model using two nonidentical gates (termed a heterodimeric H-H model to distinguish it from a homodimeric H-H model) was first proposed for native *I*_{Ks} gating in action potential simulation studies by Viswanathan et al. (51) based on activation kinetics obtained in guinea pig ventricular myocytes at the physiological temperature of 34–36° (48). This model proposed two gates with first-order kinetics that are strictly scaled versions of one another. In our fits of the activation kinetics of HEK-*I*_{Ks}, τ_{A2} and τ_{A1} differ by a relatively constant ratio of ∼3–5 across the span of activating voltages (Figs. 3, 6, and 7). Although our observed activation kinetics at first glance are in accord with the model of Viswanathan et al., where a factor of four was adopted, the congruence between our model and the Viswanathan group's model does not hold across the entire voltage range, because the closing rates at negative potentials are not scalar multiples of one another.

Figure 10 shows the overall fit of the second-order H-H model (see appendix for the mathematical formulation of the heterodimeric model) with two nonidentical independent gates for the forskolin data presented in Figs. 7 and 8 obtained at 30°C. This model of *I*_{Ks} gating allows direct interpretation of the time constants obtained in our fits of activation and deactivation current kinetics under limiting membrane potential conditions (see appendix). Table 3 presents the parameter values obtained by the fit of the model to the HEK-*I*_{Ks} current at a temperature of 30°C. Comparison of the individual parameters determined for each gate indicates that the kinetics of the two gates are not simply scaled versions of one another but, rather, independently deterministic. Figure 10 shows the model fit superimposed over the constraining data under control and forskolin-stimulated conditions.

#### Effects of β-adrenergic stimulation on I_{Ks} gating kinetics.

The effects of isoproterenol and forskolin on HEK-*I*_{Ks} activation gating kinetics in this study included an average of a two- to threefold gain of the current, an approximately −10-mV shift in *V*_{mid} of activation and a speeding of the activation kinetics. The increases in HEK-*I*_{Ks} amplitude with β-adrenergic activators were comparable to those for native *I*_{Ks} in mammalian (2, 52, 56) and amphibian (13, 17) cardiac cells, ranging from two- to fivefold. A negative shift in the voltage dependence of activation with β-adrenergic stimulation also has been observed for native *I*_{Ks} in amphibian (13, 17) and mammalian preparations (49, 56), although less reliably (3, 24). Yazawa and Kameyama (56) compared the actions of isoproterenol, forskolin, and cAMP on *I*_{Ks} in guinea pig ventricular myocytes and observed equivalent effects on current amplitude and voltage dependence of activation that are comparable to those observed on HEK-*I*_{Ks} in this study. The effects of β-adrenergic stimulation on activation kinetics of native *I*_{Ks} have been quantified in amphibian (13, 17) and mammalian preparations (3), with speeding of activation observed in both preparations, although more pronounced in amphibians.

In contrast to the speeding of the activation kinetics of HEK-*I*_{Ks}, we found little or no effect of β-adrenergic modulation on the deactivation kinetics of HEK-*I*_{Ks} with repolarization to potentials below −40 mV (Fig. 8). We limited our analysis of deactivation to potentials less than or equal to −30 mV (approximate activation threshold), such that deactivation reflected primarily the transitioning of the channel to the fully closed state [i.e., where the opening rates are small, α_{i}(*V*) → 0] rather than reequilibration among closed and open states.

Interpreted through the heterodimeric H-H gating model in Fig. 10, the data presented in Figs. 7 and 8 show that β-adrenergic modulation primarily affects the opening rates of the HEK-*I*_{Ks} channel, leaving the closing rates unchanged. Fitting of the data with forskolin required only two adjustments to the opening rate of the fast gate: *1*) a change in the constant λ_{α2} from 78 to 65 and *2*) a 18 mV negative shift in the voltage offset from 21 to 3 mV (Table 3). Such a shift could arise presumably as a result of the phosphorylation of an intracellular residue, such as occurs with Ser^{27} of KCNQ1 during cAMP-dependent modulation of the channel (29). Because deactivation kinetics are relatively unaffected, no changes were required to the closing rates to model β-adrenergic modulation of HEK-*I*_{Ks}. Studies of native *I*_{Ks} deactivation in response to β-adrenergic modulation are equivocal in mammalian myocytes, reporting no changes (56), a slight increase (52), or a reduction (2) of the time constant of deactivation. A slowing of deactivation has been reported in amphibian preparations (13, 17). Bearing in mind the difficulties in specifically isolating *I*_{Ks} from *I*_{Kr} tail currents in native preparations, the disparities reported in the relatively minor effects of β-adrenergic modulation on deactivation kinetics could potentially arise from experimental variation. For example, such variation could arise from small temperature fluctuations during experimentation if temperature were not actively monitored and controlled. Alternatively, an apparent speeding of deactivation could occur if the deactivation potential were insufficiently negative to rule out the effects of enhanced opening rates on overall rates of deactivation.

#### β-Adrenergic modulation of rate-dependent facilitation of I_{Ks}.

We examined rate-dependent facilitation of HEK-*I*_{Ks} at 35°C using fixed-duration (217 ms) pulses, roughly mimicking action potentials, from a physiological holding potential of −90 mV (Fig. 9). A large rate-dependent facilitation of *I*_{Ks} under control conditions was evident only at the highest pulse rate of 4 Hz, where the pulses repeated every 250 ms, leaving only a short interpulse (or diastolic) interval of 33 ms, during which the current can deactivate. At −90 mV, given a deactivation time constant, τ_{D}, of 84 ms at 30°C (Fig. 8) with a Q_{10} for tail deactivation of ∼4 (Fig. 2), the calculated τ_{D} of HEK-*I*_{Ks} at 35°C would be ∼40 ms. Thus, the strong initial rate-dependent facilitation of *I*_{Ks} arises from beat-to-beat accumulation of *I*_{Ks} as a result of incomplete deactivation during the 33-ms “diastolic” interval between action potential pulses at 4 Hz (Fig. 9*C*). Human diastolic intervals, measured as the time between the electrocardiographic T wave and subsequent QRS complex, scale nonlinearly with cardiac cycle length (or RR interval). Consequently, even at the heart rate of 180 beats/min (3 Hz, RR interval = 333 ms) in humans, the diastolic interval would normally exceed 100 ms (28). This suggests that the observed initial staircase accumulation of HEK-*I*_{Ks} observed under control conditions used in this study at 4 Hz would likely not occur under physiological conditions (46).

In the presence of isoproterenol, a small enhancement of rate-dependent facilitation of *I*_{Ks} over that which occurred under control conditions was observed (Fig. 9*D*). This rate-dependent enhancement of the isoproterenol effect peaked at ∼25% at the pulse frequency of 3 Hz, where the diastolic interval is 116 ms (Fig. 9*C*). This type of rate-dependent enhancement of *I*_{Ks} during β-adrenergic stimulation with isoproterenol has been demonstrated in studies of native *I*_{Ks} from guinea pig (36) and dog (46) ventricular myocytes. Studies of native *I*_{Ks} have shown that deactivation time constants in human ventricular myocytes stimulated by 1 μM forskolin at physiological temperature are comparable to those observed in this study (∼100 ms at −50 mV at 37°C; Ref. 50) and that rate-dependent enhancement of *I*_{Ks} in the absence of β-adrenergic stimulation is dependent on species as well as experimental conditions (27, 35). For example, rate-dependent accumulation of native *I*_{Ks} is increased with repolarization to more positive potential (46) or at a lower temperature, both of which slow deactivation (see Figs. 2 and 8).

During β-adrenergic stimulation, two apparently paradoxical findings in this study are the slight enhancement of rate-dependent facilitation of *I*_{Ks} at higher pulsing rates (Fig. 9*C*) and the lack of significant slowing of current deactivation (Fig. 5*D*). Besides tonic current amplification, the main effects of β-adrenergic modulation of HEK-*I*_{Ks} are a negative shift in the voltage dependence of activation and acceleration of the activation kinetics (Figs. 6, 7, and 10). These two effects lead to enhanced rate-dependent facilitation due to the net effect of cycling rapidly between the holding potential of −90 mV and the test pulse plateau of +40 mV. This is illustrated by comparing the time course of activation for control vs. isoproterenol-enhanced *I*_{Ks} during the last pulse in a train (Fig. 9*B*, 3 Hz). By speeding the channel opening kinetics, β-adrenergic enhancement reduces the activating current delay that is evident upon depolarization under control conditions.

Further insight into the ability of the heterodimeric gating model of *I*_{Ks} to correctly predict the observed characteristics of rate-dependent accumulation of *I*_{Ks} current arises from the nature of the H-H gating model: channels are conducting only when all gates are open but become nonconducting when any of the gates are closed. If two gates are nonidentical, as in the heterodimeric model presently proposed for *I*_{Ks}, current activation kinetics is dominated by the slowest opening gate, whereas current deactivation kinetics is dominated by the fastest closing gate. Understanding this key concept allows us to explain the observation in Fig. 9*B* at 4 Hz, where, after an initial staircase accumulation of current during the first few seconds of pulsing, a slower accumulation of peak current amplitude becomes evident that spans the entire 10-s pulsing duration, even in the absence of a significant effect of β-adrenergic stimulation on current deactivation (Fig. 5*D*) as shown in Table 3, where neither closing rate is affected. However, the basal closing rate of the fast gate closing rate (parameter β_{2}) is about nine times faster than the slower gate closing rate (parameter β_{1}). The rapid cycling of the channel at the higher pulsing rates allows accumulation of open probability in the slow gate due to its slower closing rate. This accumulation remains “hidden” during deactivation, since deactivation is dominated by the fast gate. The hidden accumulation of the open probability of the slow gate persists beyond observable current deactivation and is evident as the growing near-instantaneous component of subsequent test pulses, as illustrated in the last pulse of the train of pulses (Fig. 9*B*, 4 Hz). The hidden rate-dependent reserve of *I*_{Ks} that we observe is dependent on the slow closing rate of the slow H-H gate of our model and is unmasked under the physiological conditions of limited duration action potentials only when pulsing rates are rapid.

Because the heterodimeric H-H model fits both single-pulse activation gating as well as β-adrenergic modulation on temperature-controlled recordings within a human-derived background cell line, we did not attempt kinetic analyses using more complex Markov models of voltage-gated K^{+} channels (57). Silva and Rudy (44) utilized such Markov modeling of *I*_{Ks} gating to fit experimental data obtained by a variety of investigators using both native cardiac and expressed *I*_{Ks} currents (29). Their model is a significantly more complex one that builds on gating at the microscopic level, starting with observations of activation gating of KCNQ1 expressed in isolation and then expanding to incorporate coexpression of KCNE1 (or native *I*_{Ks} results). Our model attempts to primarily explain macroscopically observed whole cell currents on a multisecond time scale. Compared with the model of Silva and Rudy, our model is computationally more tractable and would allow incorporation into more large-scale simulation of the action potential in response to β-adrenergic modulation. Interestingly, Silva and Rudy observed that interaction between KCNQ1 and KCNE1 confers kinetic properties on *I*_{Ks} that make it suitable for adaptation to rate changes; in particular, the channel develops an available reserve of closed states near the open state that can open rapidly on demand. This mechanism would be provided by the slow closing rate of the slow gate of the heterodimeric model.

Our model also differs from another recently proposed model of *I*_{Ks} adrenergic modulation based on expression of KCNQ1/KCNE1 together with the PKA-anchoring protein (AKAP) yotiao in Chinese hamster ovary (CHO) cells and simulation of PKA-dependent stimulation of the current by addition of intracellular cAMP and okadaic acid (47). Under such experimental conditions, a slowing of deactivation was observed. A gating model incorporating these effects proposed that β-adrenergic stimulation causes both speeding of activation, as in our model, as well as slowing of deactivation. A slowing of *I*_{Ks} deactivation would likely result in a shortening of action potential duration (APD) and could increase postrepolarization refractoriness. However, this potential latter effect should be outweighed by the overriding shortening of APD by *I*_{Ks} enhancement during β-adrenergic stimulation. Overall, any changes in gating of *I*_{Ks} need to support the dynamic rapid cycling of the membrane potential underlying the increased heart rate, decreased APD, and decreased refractory period that are associated with ventricular electrophysiology during β-adrenergic stimulation (15). It is not known whether the combined action of cAMP and okadaic acid, which blocks dephosphorylation of the channel, is the equivalent of receptor-mediated β-adrenergic stimulation with isoproterenol or forskolin, as used in our studies. We did not investigate how cotransfection of yotiao, an AKAP that has been shown to be necessary for the adrenergic modulation of *I*_{Ks} in the CHO cell expression system, may potentially affect the deactivation kinetics of the KCNE1/KCNQ1 current in the HEK expression system. The identity of the endogenous AKAP that presumably facilitates the adrenergic modulation of HEK-*I*_{Ks} is not known. Thus the model of human *I*_{Ks} presented in this study should help extend our current understanding of *I*_{Ks} gating and its contribution to cardiac repolarization in response to β-adrenergic stimulation of the heart.

The HEK-293 cell line (19) has served extensively as a host for both transient and stable expression of receptors and ion channels for pharmacological, biophysical, and modulator characterization. The endogenous G protein-coupled pathways in HEK-293 cells have been used to examine receptor and ion channel coupling to second messenger cascades. HEK-293 cells have a documented endogenous second messenger pathway that incorporates the hallmarks of the prototypical β adrenergic receptor cascade involving β-adrenergic receptors, which couple via G_{s} to adenylate cyclase and cAMP-dependent PKA (9). Daaka et al. (9) have shown, for example, that treatment of HEK-293 cells with isoproterenol or forskolin increases intracellular cAMP levels 17- and 45-fold, respectively. The endogenous adenylate cyclase activating pathway in HEK-293 cells has been harnessed previously to test whether expressed L-type Ca^{2+} (Ca_{V}1.2) channels can be modulated by the cAMP-dependent β-adrenergic pathway (14), but nativelike enhancement of Ca_{V}1.2 current has been found only on cotransfection of AKAPs (16, 22). Recent studies have shown that multiple AKAPs are endogenously present in HEK-293 cells (55), but it is unknown whether the AKAP yotiao (29), shown to be involved in PKA-dependent *I*_{Ks} modulation, is endogenously present.

#### Possible structural interpretations of a heterodimeric H-H gating model.

It is generally accepted that the native *I*_{Ks} channel consists of a core of four KCNQ1 subunits. Although it is not certain whether two or four KNCE1 subunits are contained within an *I*_{Ks} channel, recent evidence favors the belief that two KCNE1 subunits are expressed per channel (6, 32). Depending on whether two or four KNCE1 subunits are expressed per channel, the resulting channel would possess a physical two- or fourfold symmetry. The simplicity of the proposed heterodimeric H-H formalism and its straightforward biophysical interpretation suggest structural changes that may underlie β-adrenergic modulation of *I*_{Ks} gating. As a H-H model, the heterodimeric model places strong constraints on how the channel may function.

In a H-H model with two gating processes with two time scales, current activation is dominated by the slower opening gate, whereas current deactivation is dominated by the faster closing gate. With the assumption that like subunits behave in a concerted, limited two-state gating scheme, it is attractive to speculate that the faster gate in the heterodimeric model is constituted by the cooperative movement of the KCNQ1 subunits. This hypothesis is supported by the faster activation kinetics of KCNQ1 expressed in isolation and the observation that current deactivation of KCNQ1 expressed in isolation shows deactivation time scales similar to those of *I*_{Ks}.

The collapsing of the gating transitions of the KCNQ1 subunit into two states, closed and open, is a simplification given what is known about KCNQ1 gating in isolation (34), but it is one that can be made based on the goodness of fit at time scales on which our fits are based. A cooperative single functional transition of the KCNQ1 tetramer as a consequence of association with KCNE1 can be envisioned based on the allosteric models of function proposed by Monod et al. (31) for symmetric heterooligomeric enzyme structures. The assignment of the fast gating process to KCNQ1 also correlates the finding that PKA-dependent *I*_{Ks} channel phosphorylation targets the KCNQ1 subunit (29). The primary effect of adrenergic modulation in our model of *I*_{Ks} gating is the increase of the opening rate of the fast gate via a shift in its voltage sensitivity and voltage offset (Table 3).

The concerted movements of the KCNE1 subunits, whether two or four, would in this interpretation constitute the functional slow activation gate and thus would dominate the time scale of current activation once KCNE1 coassembles with the KCNQ1 channel. Bearing in mind that the model parameter fits of Table 3 were obtained using a set of limited test conditions, additional aspects of the slow gate need to be further refined. Specifically, incorporation of the kinetics of the second phase of pulsing-dependent current accumulation (Fig. 9, 4-Hz pulsing rates) would yield a more accurate estimate of the closing rate of the slow gate. Despite the limited conditions of model testing and validation used presently, however, the lack of a strong effect of adrenergic stimulation on rate-dependent accumulation of HEK-*I*_{Ks} (Fig. 9, *C* and *D*) suggests limited involvement of the slow gate in the adrenergic response.

An alternate possibility arising from a potential twofold symmetry of the channel complex is to interpret *I*_{Ks} gating as that of a structural dimer, with each protomer consisting of one KCNE1 and two KCNQ1 subunits. In our model the two time scales could no longer be reconciled with independent movements of each half of a channel dimer, because each half would be structurally identical and thus would be expected to gate identically in a H-H model. To reconcile structural movements of a dimer with our findings, the H-H model requirement of the independent action of the two gates would have to be abandoned.

#### Conclusion.

The demonstration of the enhancement of the hKCNQ1/hKCNE1 current, HEK-*I*_{Ks}, in HEK-293 cells in the presence of isoproterenol validate the recent finding that the molecular constituents necessary for β-adrenergic modulation of *I*_{Ks} are present in this host line (11). The ability to isolate and modulate the *I*_{Ks} current in the HEK-293 background has allowed us to derive a simple yet precise model of human *I*_{Ks} gating on the time scales of seconds, termed the heterodimeric Hodgkin-Huxley model of *I*_{Ks} gating. The effect of β-adrenergic modulation on *I*_{Ks} is incorporated in this model, which can explain two observed rates of rate-dependent accumulation of *I*_{Ks} at high pulsing rates. The quantitative model presented in this study may further refine the mathematical model of human *I*_{Ks} in the cardiac myocyte. These results indicate that the HEK-293 line would be very attractive for molecular biology studies of the genetic mutations of *I*_{Ks} with respect to their effect on β-adrenergic modulation, since it would require only the transfection of the mutated channel subunits themselves.

## APPENDIX: THE HODGKIN-HUXLEY FORMALISM APPLIED TO *I*_{Ks} GATING

The heterodimeric gating model uses the mathematical formalism introduced by Hodgkin and Huxley (21), who proposed that the potassium conductance they studied in the squid giant axon could be modeled by a set of four identical, independently operating voltage-sensitive gates. Each gate is denoted by a time- and voltage-dependent state variable, *n*(*t*, *V*), denoting the open probability of each gate. The overall open probability of a channel in the original H-H model is given by the product of the open probabilities of all of the four identical gates, *n*^{4}(*t*, *V*). Because voltage-gated potassium channels of the type that Hodgkin and Huxley studied are based on the structurally homotetrameric Kv family potassium channel, the original H-H model could be called a homotetrameric fourth-order H-H gating model, because there are four identical gates to each channel. By analogy, a heterodimeric second-order H-H gating model proposes two nonidentical gates, *n*_{1} and *n*_{2}, with the overall open probability of the channel given by the product of the open probabilities of each of the two nonidentical gates at any given time, *t*, and membrane potential, *V*. All other aspects of the heterodimeric model are the same as in the original H-H formalism.

Each *I*_{Ks} gate in the heterodimeric model is characterized by an opening rate, α_{i}, and a closing rate, β_{i}, which are voltage sensitive, with *i* = 1, 2. *Equations 3a* and *3b* formulate the voltage dependence of the opening and closing rates of each gate as a function of membrane potential, *V*: where *k*_{αi}, *k*_{βi}, λ_{αi}, and λ_{βi} are constants (see *Eqs. 12* and *13* of Ref. 21). We attempted to use an expression of the form of *Eq. 3b* for fitting the activation time constant data, but this expression was poor at fitting the 1/τ_{A} values across the depolarized voltage range.

The steady-state open probability of each gate at a given membrane potential, *n*_{i}(∞, *V*), is given by the equation (4)

The product of the open probabilities of the two gates at steady state determines the steady-state voltage-dependent open probability of the channel, for which the isochronal activation curve of Fig. 7*C* is an approximation (since true steady state would require the step duration to be of infinite duration).

The time-dependent relaxation of the open probability of each gate from an initial condition at potential *V*_{0} to a new steady state at the potential *V* is given by The open probability of the channel at any time *t* is thus the product of *n*_{1}(*t*, *V*) and *n*_{2}(*t*, *V*).

In our studies of the HEK-*I*_{Ks} current, two special cases of *Eq. 5a* are used to constrain the values of α_{i}(*V*) and β_{i}(*V*) for a given voltage. In the first case, current activation from resting potential, the variable *V*_{0} is negative enough that the initial condition, *n*_{i,∞}(*V*_{0}) → 0 such that *Eq. 5a* is simplified to *n*_{i,∞}(*V*)[1 − exp(−*t*/τ_{i})]. *Equation 1* is the implementation of this special case, and the time constants derived from fits of *Eq. 1* are used to place constraints on α_{i}(*V*) and β_{i}(*V*) during activating voltage steps. A second special case of *Eq. 5a* is encountered when repolarizing the HEK-*I*_{Ks} current from an activated state to resting potentials where α_{i}(*V*) → 0, such that *n*_{i,∞}(*V*) → 0. In this case, the time-dependent open probability of each gate is described by *n*_{i,∞}(*V*_{0})exp(−*t*/τ_{i}), and the overall open probability of the channel is thus given by *n*_{1,∞}(*V*_{0})*n*_{2,∞}(*V*_{0})exp[−*t*(β_{1} + β_{2})], which is the same form as *Eq. 2*.

Summarizing from the previous paragraphs, the following constraints were placed on the opening and closing rates of each gate to obtain the fitting constants in *Eqs. 3a* and *3b*: (6) where τ_{Ai} is as defined in *Eq. 1* and *i* = 1, 2; (7) where τ_{D} is as defined in *Eq. 2*; and (8) the normalized tail current-voltage relation.

By limiting the fits of the model to the limiting cases of membrane potentials, the constraints imposed by *Eqs. 6*–*8* yielded the parameter constants shown in Table 3.

## Acknowledgments

We thank Armando Lagrutta for critical reading of the manuscript and David T. Yue for insightful discussion.

Present address of W. D. Irving: Beth Israel Medical Center, New York, NY 10003.

## Footnotes

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- Copyright © 2008 by the American Physiological Society