Evaluating sequential and allosteric activation models in I_Ks channels with mutated voltage sensors

The modeling of the processes underlying the activation and subsequent opening of voltage-dependent K⁺ (Kv) channels has always been central to the description of, and differentiation between, potential mechanisms underlying membrane excitation. The original formalism of squid axon conductance and gating by Hodgkin and Huxley contains only a few dynamic elements, which mirrors the simplicity of purpose in a physiological system designed primarily to conduct nerve impulses at high speed and frequency (Hodgkin and Huxley, 1952). Despite this, some fundamental properties of the original Hodgkin and Huxley model remain key features of almost all voltage-dependent channel models, including the delay observed in channel opening requiring multiple activation steps and independent movement of gating particles or voltage sensor (VS) domains, while separation between VS activation and pore domain (PD) dynamics is not considered in the squid axon model.

Activation models of Kv channels have undergone several important reinventions since these foundational studies, increasing in complexity and flexibility as subsequent investigators sought to describe ion channel systems in much greater quantitative detail (Zagotta et al., 1994; Sigworth, 1994) and understand those that did not conform, or only partially conformed, to adapted Hodgkin–Huxley models. Those studies more interested in reproducing action potential waveforms and properties have tended to rely on more classical ion channel model formulations (Hafner et al., 1981; O’Hara et al., 2011; Paci et al., 2013), but the expression of individual ion channels, such as Shaker, in heterologous systems that permit precise time- and voltage-control of the experimental substrate has allowed much more detailed analysis of channel kinetics in the absence of other overlapping ion channel conductances (Zagotta et al., 1994). The aims of these models have been to either refine the earlier models to achieve a previously unattainable quantitatively accurate description of the particular ion channel system under investigation (Schoppa and Sigworth, 1998a, 1998b; Zheng et al., 2001; Rothberg and Magleby, 2000) or to build new models to account for large qualitative differences in activation mechanisms (Horrigan et al., 1999; Altomare et al., 2001; Chowdhury and Chanda, 2012a, 2012b). The original Hodgkin–Huxley formulation for Loligo expects ion conduction when all VSs become activated, but fly and mammalian ion channel schemes allow for a subsequent final or multiple concerted transition(s) (Schoppa and Sigworth, 1998b; Ledwell and Aldrich, 1999) from VS activated-closed states to VS activated-open states. These transitions are included after the VS activation delay to reproduce channel opening that is slower than predicted from a strict Hodgkin–Huxley model, and such concerted events make the overall activation scheme sequential (Zagotta et al., 1994; Sigworth, 1994; Horrigan et al., 1999).

The success of structural modeling approaches to understanding ion channel activation gating and block mechanisms (Ramasubramanian and Rudy, 2018; Maly et al., 2022; Willegems et al., 2022; Chan et al., 2023), along with the use of molecular dynamics simulations to directly predict the underlying molecular motions and atomistic details of permeation, gating, and block (Carnevale et al., 2021; Flood et al., 2019; Abrahamyan et al., 2023), have, in recent years, lessened the popularity of classic kinetic modeling. Still, there remains value in examining how the demonstrated biophysical properties of currents through ion channels conform to the particular stringencies of different types of Markovian kinetic models. Kinetic models with rate constants provide a precise means to describe and/or predict behavior (kinetics) of ion channels by collating the information that must be accounted for by any viable structural models. Also, the structural basis for all of the observed kinetic states summarized by the model needs to be determined to understand gating. Indeed, such models are still widely used to study mutational effects on the kinetics of ion channels (Moreno et al., 2013; Carbonell-Pascual et al., 2016), predict action potential properties and shapes in novel iPSC models (Paci et al., 2013; Kernik et al., 2019), for stratification of genetic variants (Kernik et al., 2020), and the potential pro-arrhythmic response to drugs in development (Bottino et al., 2006; Moreno et al., 2016; Passini et al., 2017; Li et al., 2019).

Models of KCNQ1 and I_Ks, KCNE1:KCNQ1 in a stoichiometry up to 4:4 (Murray et al., 2016), are of particular interest to investigators, and a particular challenge to model as the kinetic interconversion between the two current phenotypes is so extreme upon the coexpression of KCNE1. Activation of KCNQ1 shows time constants in the order of 16–20 ms at +10 to +50 mV (Ruscic et al., 2013), while being closer to 3 s for I_Ks at +60 mV (Westhoff et al., 2019), and is followed by inactivation, which is apparently absent in I_Ks (Hou et al., 2017). The activation delay observed in both channels suggests that the VSs of KCNQ1 and I_Ks, much like those of Shaker, must also undergo multiple transitions during activation, prior to PD opening. At the present time, two classes of Markov-type models are proposed to account for the activation properties of KCNQ1 and/or I_Ks. Early models from Pusch et al. and Silva and Rudy are sequential (see below), and incorporate VSs which transition independently through one or two conformational changes prior to becoming fully activated (Pusch et al., 1998, 2001; Silva and Rudy, 2005). The fully activated VS-channel complex then undergoes a single concerted step to allow the PD to open to one or more open states (Pusch et al., 1998; Silva and Rudy, 2005; Nekouzadeh et al., 2008). Slow activation in I_Ks versus KCNQ1 is reproduced by slowing the VS transitions from resting to intermediate states.

A related model for single-channel recordings was modified to allow channels to open to subconducting levels when individual VS domains were fully activated, and an additional fast closing pathway was connected to channel open states to account for the rapid closing of channels during opening bursts seen during long sweep records of single-channel openings (Werry et al., 2013).

Allosteric models of gating (see above) in the manner described for the voltage gating behavior of Shaker channels (McCormack et al., 1994), HCN channels (Altomare et al., 2001), and large conductance Ca²⁺-activated potassium (BK) channels (Horrigan et al., 1999; Rothberg and Magleby, 2000; Horrigan and Aldrich, 2002; Horrigan, 2012) have also been suggested to best represent KCNQ1 (Ma et al., 2011; Meisel et al., 2012; Osteen et al., 2012) and I_Ks gating (Meisel et al., 2012). In these models, independent activation of VSs increases the channel open probability and this positive allosteric activation is represented by a factor L (Osteen et al., 2012), which increases the equilibrium constants for the C–O transitions L-fold for each VS that is activated (Horrigan et al., 1999). The concordance on the voltage axis of voltage sensor fluorescence (VCF) measurements with the voltage dependence of the conductance–voltage (G-V) relationship in KCNQ1 (Osteen et al., 2010), but not I_Ks, led these authors to propose an allosteric gating model for KCNQ1 alone but to incorporate a concerted opening step in channel gating into their I_Ks models, with the implication that the VSs in all four subunits must be activated before some local conformational rearrangement can lead to PD opening. Subsequently, the idea of a concerted opening step has proven controversial, with some studies in support (Barro-Soria et al., 2014) and others favoring an allosteric model of activation for I_Ks as well as KCNQ1 (Meisel et al., 2012; Zaydman et al., 2014). Recent experiments that demonstrate the opening of I_Ks channels with only a single, two, or three active VS domains have lent further experimental support to the allosteric activation of the VSs and pore conductance (Westhoff et al., 2019), but this idea remains controversial. Structural modeling has identified specific domains of KCNE subunits that may regulate gating via an allosteric network of residues on the S5–S6 domain (Kuenze et al., 2020), and molecular simulations support flexible coupling between VS and pore, thus permitting pore opening at intermediate VS positions (Ramasubramanian and Rudy, 2018).

In this study, we will compare the predictions of different activation models for I_Ks with our recent and new experimental data to determine which class of models best reproduces I_Ks activation kinetics. Our goal in modeling is to use the simplest applicable models with the fewest transitions that are currently understood to make sense in the context of our current appreciation of I_Ks channel activation. We will explore the effect of reducing the number of independently moving VSs (e.g., wild-type [wt] versus E160R-containing VS subunits), as we have already shown how this affects the electrophysiological properties of I_Ks (Westhoff et al., 2019). We started with the most basic of Hodgkin–Huxley type models and increased model complexity and type, from sequential to allosteric, ending with a 30-state activation gating model, all the time trying to define transition rates using experimental electrophysiological and VCF data. Our analysis suggests that allosteric gating models provide the best fit to a broad set of experimentally defined biophysical parameters. Channel opening results from the movement of independent VSs, so that in I_Ks channels a final concerted transition of all four VSs is not obligatory for PD opening. The allosteric factor is relatively small at 1.36, which suggests that activation of less than four VSs may result in meaningful current conduction through open substates of the I_Ks channel at intermediate voltages.

Alexa Fluor 488 C₅-maleimide was obtained from Thermo Fisher Scientific. HMR1556 was from Bio-Techne Canada. All other chemicals were obtained from Sigma-Aldrich. All biohazard and animal care activities were carried out after ethical review and approval by The University of British Columbia Biosafety and Animal Care and Use Committees under protocol numbers B21-0006 and A22-0049-R001, respectively, and their antecedents.

The EQ (KCNE1:KCNQ1), EQQ (KCNE1:KCNQ1:KCNQ1) and EQQQQ (KCNE1:KCNQ1:KCNQ1:KCNQ1:KCNQ1) constructs were generated as described (Murray et al., 2016). The E160R mutation was incorporated into Q1 subunits of the fusion constructs via a gBlocks Gene Fragment (Integrated DNA Technologies) with compatible restriction sites. Throughout this paper, the E160R mutation in Q1 is denoted by an asterisk (Q*). The EQ*QQ*Q* and tandem fluorescence constructs were made by Applied Biological Materials Inc. from precursor constructs that we provided to them. All mutations were confirmed by sequencing. Constructs for transfection into mammalian cells were subcloned into pcDNA3.1 (RRID:Addgene_117272).

tsA201-transformed human embryonic kidney 293 cells (whole cell) or mouse ltk- fibroblast cells (cell attached) were obtained from ATCC or Sigma-Aldrich, certified authentic, validated, and mycoplasma free. Cell lines were cultured and plated as described (Murray et al., 2016; Westhoff et al., 2017). Transfections with Lipofectamine 2000 (Thermo Fisher Scientific) followed the manufacturer’s protocol, and cotransfection with KCNE1-GFP (hereafter referred to as E1) was at a 3:1 ratio of E1:KCNQ1 (E1+Q1). For experiments described in Figs. 1 and 2, to measure the I_Ks activation kinetics, we usually used the concatenated construct EQ, instead of co-transfected E1+Q1, to ensure a 4:4 stoichiometry of expressed E1:Q1 channels. However, we have previously shown that a 3:1 transfection ratio of E1:Q1 construct DNA produces the same activation and deactivation kinetics whether EQ, Q1+E1, QQQQ+E1, QQ+E1, or EQQQQ+E1 are transfected into tsA201 cells (Westhoff et al., 2019). In the extended τ_act–V relationship in Fig. 3 B, we did include time constant data from two wt QQ+E1 cells at +110 and three wt QQQQ+E1 cells at +110 to +130 mV. All experiments were performed 24–48 h after transfection at room temperature, 22°C. Successfully transfected cells were identified by GFP-fluorescence. For recordings in the presence of E1, currents <500 pA or not expressing I_Ks were discarded.

Mature female Xenopus laevis frogs (Xenopus 1) between 3 and 4 yr of age and between 0.2 and 0.5 kg in weight were anesthetized in a solution containing 2 g/l tricaine methanesulfonate and 2 g/l HEPES (pH 7.4 with NaOH). Animals were euthanized in accordance with the University of British Columbia animal care protocol number A22-0049-R001. The ovarian lobes were extracted, divided into smaller sections, and digested for 2–4 h in a solution containing 3 mg/ml collagenase type 4 (Worthington Biochemical Corporation), 82.5 mM NaCl, 2.5 mM KCl, 1 mM MgCl₂, and 5 mM HEPES (pH 7.6 with NaOH). The oocytes were washed and stored in media containing 500 ml Leibovitz’s L-15 medium (Thermo Fisher Scientific), 15 mM HEPES, 1 mM glutamine, and 500 μM gentamicin, brought up to 1 liter with distilled water (pH 7.6 with NaOH). Stage IV and V oocytes were selected and stored at 18°C.

Oocytes were injected with cRNA synthesized using the Ambion mMessage mMachine T7 transcription kit (Applied Biosystems). Q1 constructs differed from mammalian constructs as they were modified to remove extracellular C214 and C331 cysteine residues and included a G219C for fluorescence purposes. 10 ng of C214A/G219C/C331A Q1 pcDNA3.1+ (psQ; a gift from Dr. Jianmin Cui, Washington University in St. Louis, St. Louis, MO) cRNA was injected, while 50 ng of the tandem constructs in pGEMHE was used. These were psQQ (C214A/G219C/C331A Q1-C214A/C331A Q1) and psQ-E160R Q* (C214A/G219C/C331A Q1-E160R/C214A/C331A Q1*), also named psQQ*. The E160R mutation prevents activation of VSs in two of the four domains. The pGEMHE (RRID:Addgene_114674) vector was a gift from Dr. Yoshihiro Kubo (National Institute for Physiological Sciences, Okazaki, Japan). All constructs were coinjected with 5 ng E1 in pBSTA. Experiments were performed 3–4 days after injection at room temperature. For wt hERG, cRNA was synthesized as above and oocytes were injected with 10–100 ng cRNA. Injected oocytes were stored in ND96 (96 mM NaCl, 3 mM KCl, 2 mM CaCl₂, 1 mM MgCl₂, and 5 mM HEPES at pH 7.4) and incubated at 18°C prior to use.

For whole-cell recordings, the bath solution contained (in mM) 135 NaCl, 5 KCl, 1 MgCl₂, 2.8 NaAcetate, and 10 HEPES (pH 7.4 with NaOH). The pipette solution contained (in mM) 130 KCl, 5 EGTA, 1 MgCl₂, 4 Na₂-ATP, 0.1 GTP, and 10 HEPES (pH 7.2 with KOH). For two-electrode fluorometry (VCF), the bath solution contained ND96 plus 0.1 mM LaCl₃. The pipette solution contained 3 M KCl. For cell-attached recordings (Fig. S1), the bath solution contained (in mM) 135 KCl, 1 MgCl₂, 0.1 CaCl₂, 10 HEPES, and 10 dextrose (pH 7.4 with KOH). The pipette solution contained (in mM) 6 NaCl, 129 MES, 1 MgCl₂, 5 KCl, 1 CaCl₂, and 10 HEPES (pH 7.4 with NaOH; Eldstrom et al., 2021).

Whole-cell, macropatch, and single-channel currents were acquired using an Axopatch 200B amplifier, Digidata 1440A, and pClamp 10 software (Molecular Devices). For whole-cell recordings, a linear multistage electrode puller (Sutter Instrument) was used to pull electrode pipettes from thin-walled borosilicate glass (World Precision Instruments; Murray et al., 2016; Westhoff et al., 2017). Pipettes were fire-polished prior to use. Electrode resistances for whole-cell recordings were between 1 and 3 MΩ, with series resistances <5 MΩ. Series resistance compensation of ∼80% was applied to all whole-cell recordings, with a calculated voltage error of ∼1 mV/nA current. Whole-cell currents were sampled at 10 kHz and filtered at 2–5 kHz (Murray et al., 2016; Westhoff et al., 2017). We had noticed in preliminary experiments that the endogenous delayed rectifier K⁺ current in tsA201 cells contaminated the I_Ks current records at a level of ∼200–400 pA at +80 mV, and this is an issue when measuring the delay time to exponential activation (Δt) for constructs expressed at a lower level, such as EQ*QQ*Q*, as is typical for many concatemers. Thus, wherever possible and for all Δt measurements, whole-cell current data used in this study were obtained from experiments after 0.3–1.0 μΜ HMR1556 subtraction. Examples of tsA201 cells transfected with EQ*Q+E1 and exposed to HMR1556 are shown in Fig. S1. The endogenous current is revealed by exposure to 0.3 μΜ HMR1556, which blocks I_Ks but leaves the endogenous current unaffected. Doses of 1 μΜ HMR1556 were required to affect endogenous currents. The only important contamination of the I_Ks current is the step current that occurs at the start of the pulse and the subsequent development of the I_Ks time course and tail currents are unaffected, as is the G–V relationship (Fig. S1 B). The contamination at the start of the depolarization is removed by subtraction of records obtained after, from those obtained before exposure to 0.3 μΜ HMR1556 (Fig. S1 E).

Oocytes were labeled with 10 μM Alexa Fluor 488 C₅-maleimide in a depolarizing high potassium ion solution containing (in mM) 98 KCl, 1.8 CaCl₂, and 5 HEPES (pH 7.6 with KOH) for 30 min on ice. The oocytes were washed with the bath solution and left on ice prior to recording. Fluorescence and ionic current recordings were obtained simultaneously as described (Es-Salah-Lamoureux et al., 2010) with an Omega XF100-2 filter set (Omega Optical Inc). Two-electrode voltage-clamp experiments were performed using an Oocyte Clamp OC-725C (Warner Instruments) and digitized via a Digidata 1440A (Molecular Devices) running pClamp 10 software (Molecular Devices). Fluorescence recordings from the same oocyte were averaged when necessary to improve the signal-to-noise ratio and photobleaching of the dye between recordings was minimized by using an electronic shutter to limit oocyte UV exposure during the voltage step epochs of the protocol. To correct the fluorescence baseline for dye bleaching, a fluorescence signal was recorded at the −140 mV holding potential in the absence of a voltage step and was subtracted from signals recorded during steps to voltages between −180 and +80 mV. Bleaching did not appear to affect the components or kinetics of the fluorescence–voltage (F-V) relations, but diminished signal amplitude over time.

Data were cumulated and analyzed in MS Excel. G-V plots were obtained from tail current amplitudes. For electrophysiological experiments, the number of replicates is stated in the figure legends and Table 1. Although we often collected many replicates, we used a minimum sample size of three to allow calculation of sample means and standard errors (SEM) using standard functions in MS Excel. Data were discarded if cells were excessively leaky or degraded during experiments. No sample size calculation was performed as a standard procedure in cellular electrophysiology experiments. All results are reported as mean ± SEM, as denoted by error bars on data points unless otherwise stated. For comparison of t_1/2 differences in Fig. S4 B, a statistical comparison was performed using a one-way ANOVA with a Bonferroni post hoc test. P values <0.05 were considered to be statistically significant.

MATLAB R2021, R2022, and R2023a (RRID:SCR_001622; Mathworks) were used to construct and simulate Markov models. Final models were also checked using IonChannelLab software (Santiago-Castillo et al., 2010), and model outputs were plotted using custom-written MATLAB routines. Within MATLAB, the function “ode23s” was used to solve differential equations using a Jacobian matrix, and the function “fminsearchbnd” with boundaries (D’Errico, 2021) was used to optimize rate constants and model fits to steady-state G-V and F-V relationships, τ_act and Δt, and time-dependent current and fluorescence recordings. We followed the methods described by Moreno et al. to use a bounded Nelder Mead optimization algorithm (Nelder and Mead, 1965; Moreno et al., 2016), and we optimized transition rates by minimizing a scalar function that simultaneously incorporated the squared errors (SumSq) between model and experimental G-Vs, F-Vs, τ_act versus Δt (e.g., Figs. 6 B, 12, and 13), and/or the error between experimental current recordings and the model simulations at +60 mV (Figs. 2 C and 12 A). It has been suggested that more predictive models can be obtained from data based on short experimental voltage-clamp protocols rather than those derived from more traditional current–voltage or time–voltage curves (Clerx et al., 2019). Here, we used a combination of different inputs as part of the optimization process. Empirically, we found that the isochronal G-V and F-V relationships were the quickest to optimize together, and gave excellent fits using the optimization algorithm, while including time-dependent current or fluorescence data slowed the process considerably. Usually, simulations were run until the variable SumSq was no longer decreasing, which usually took between 500 and thousands of optimization cycles (e.g., Fig. S7). This process was then repeated a number of times to detect and exclude local minima and attempt to improve the overall fit to all the different data sets, G-V, F-V, τ_act-V, Δt-V, and current recordings. In the Scheme 1 models, the current time course was used as an important optimization parameter, but in Schemes 2, 3, 4, and 5 models, the Δt versus τ_act relationship was extensively used as an experimental input to ensure time-dependent delay and time constant data were considered together with the isochronal G-V and F-V relationships during the optimization process. The experimentally derived Δt versus τ_act graph defines the kinetics of transitions through closed states as well as the kinetics of the opening transition(s) as long as the two processes are temporally distinct. The object was not to build entirely new conceptual models of I_Ks activation gating but to explore the effect of reducing the numbers of functional VSs on already-existing sequential or allosteric models, with the specific aim of observing which ones could best simulate the experimental data we obtained from I_Ks constructs with one to three E160R-containing VSs (Westhoff et al., 2019). In this sense, model transitions were preconstrained and the absolute values of the rates themselves were of secondary importance to how the models themselves dealt with the experimentally observed effects of zero compared with two or three E160R-containing VSs.

In Markov model schemes, subscripts R or A indicate the number of resting or activated subunits. Closed and open channel states are indicated as “C” and “O,” respectively. Forward and backward rate constants in VS state changes are indicated as “α, β” and “κ, λ”, for the first VS and second VS transitions, respectively. The allosteric factor in the model Schemes 4 and 5 is indicated as “D.” Voltage-dependent transition rates related to VS-pore coupling in the opening step of Schemes 2, 3, 4, and 5 are denoted by “δ” and “γ” for opening and closing, respectively.

Fig. S1 details the HMR1556 subtraction methodology used for tsA201 cell data in the study; Figs. S2 and S4 show the rationale for the use of prepulse potentials in tsA201 cells and oocytes; Figs. S3, S5, S6, and S7 show additional results and model outputs.

In wt I_Ks channels, the VS domains respond to depolarizing changes in transmembrane potential as physical charge translation within the electric field in a manner that leads to inner gate opening and ion conduction through the pore (Wang et al., 2020). The conversion of the negatively charged glutamic acid at E160 in the S2 of Q1 into a positively charged arginine prevents individual VS translation as evidenced by the lack of methanethiosulfonate modification of the ion currents or visible changes in the Alexa Fluor 488 fluorescence signal from labeled G229C or G219C residues, respectively, in the E160R subunits (Westhoff et al., 2019). The inclusion of an E160R mutation in all four VS domains of the Q1 subunit results in a non-activatable channel, from which ion currents cannot be observed (Restier et al., 2008; Zaydman et al., 2014; Westhoff et al., 2019), but the experimental consequence of including E160R mutations in one, two, or three tethered-subunit VSs is still to allow channel complexes to activate and pass current (Fig. 1 A, Westhoff et al., 2019). It can be seen that while the expression of E1-GFP (referred to as E1) along with the Q1 subunits produces dramatic changes in the current activation and deactivation recordings from KCNQ1 alone to EQ with a 4:4 KCNE1:KCNQ1 stoichiometry, the inclusion of the E160R mutation, indicated by a *, in two (EQ*Q+E1) or three (EQ*QQ*Q*+E1) Q1 subunits does not, at first glance, alter the channel activation kinetics a great deal, although it does reduce the overall current magnitude from wt (Westhoff et al., 2019). Upon careful inspection, though, activation is slower in channels containing more E160R subunits at equivalent potentials, while EQ*Q+E1 and EQ*QQ*Q*+E1 channels deactivate more quickly than wt EQ at −40 mV.

The aims of the present experiments are to understand in greater detail the activation process of the I_Ks channel complex and to assess the suitability of the various kinetic models proposed for their activation. What is new here is the approach of utilizing changes in I_Ks current kinetics and fluorescence in the presence of the E160R mutation in two or three Q1 VS domains, as a way to reduce the number of activatable subunits and simplify the number of possible model transitions that are required to simulate the data. As well, rate constants derived from the simplest model with the fewest variables can be incorporated into the models with more activatable subunits to compare models to experimental data and check the validity of the overall schemes.

The starting point for the modeling of I_Ks activation kinetics is the application of a Hodgkin–Huxley type system to the behavior of the different E160R constructs. Scheme 1, A–C, are Markov diagrams for the state transitions of four-VS, two-VS, and single-activatable VS channels, based on the Hodgkin–Huxley model of independent gating particle movement, α and β rate constants, and the idea that concomitant activation of the available gating particles is sufficient for channel pore opening (Hodgkin and Huxley, 1952). Subscripts to closed (C_a) and open (O_a) channel states indicate the individual subunit states, resting (R) or activated (A). The advantage of restricting individual subunit activation with the E160R mutation is that models are simplified, and ultimately, with three E160R subunits in a channel complex, activation/opening comprises only a single transition for which rate constants can be analytically derived from experimental data.

In the simplest case in which there are three E160R-containing VS subunits in the complex and the remaining activatable subunit only undergoes a single transition (Scheme 1 C), the Po-V relationship, the τ_act-V, and the τ_deact-V are uniquely defined by the forward and backward rate constants between the resting and activated channel states. Current records (Fig. 1 A) were fit with an exponential starting at 0.5 s, which omits the early current at the start of the records, and activation time constants (τ_act) extracted and plotted against the membrane potential (Fig. 2 A). Tail currents were obtained using a two-pulse protocol and fit with a single exponential decay function to provide data points at negative potentials. Using 10 s isochronal G-V data for EQ*QQ*Q*+E1 (Fig. 1 B), the exponential voltage-dependence of the rate constants, α_v and β_v (Fig. 1 C), and the apparent valence of the forward and backward transitions were obtained by calculation and fitting (Eq. 1 and Table 2). Constants were optimized together using custom-written MATLAB programs and library solvers (see Materials and methods) to minimize the error between model and experimental fits to G-V, τ_act-V data, and between model and current tracings at +30 mV (rates in Table 2). The analytical fits to the experimental data are shown as dashed lines and the optimized model fits as solid lines (Fig. 2).

The forward (α) and backward (β) rate constants versus voltage functions obtained from the fitting described above were incorporated into MATLAB models and used to simulate isochronal G-V relationships and currents for the EQ*QQ*Q*+E1 (Scheme 1 C), EQ*Q+E1 (Scheme 1 B), and wt (Scheme 1 A) constructs (Fig. 1 D and Table 1), and the predicted steady-state G-V relationship for EQ*QQ*Q*+E1 after 50 s pulses (dashed line, Fig. 1 D). The curves generated overlaid the experimental data from the single activatable subunit construct and the dashed curves predicted from Eq. 2 for the other constructs, and they also provided a good fit to current records for the EQ*QQ*Q*+E1 construct in minimizing the difference currents between the experiment and model (Fig. 2 C). Scheme 1 models are highly constrained in that the last VS to move also comprises the gate, and the rate constant for the closed-to-opening transition has to be identical to that for moving a VS. Structural studies show that these events are not identical and thus that the rate constants are unlikely to be identical. Despite the constraints, these models reproduce much of the simple kinetic data obtained from an I_Ks channel with only one activatable VS domain and predict the slope changes in the isochronal G-V when the channel complex contains two or four activatable VSs. However, the models do not predict the progressive hyperpolarization of the G-V seen with two or more activatable subunits (Fig. 1 B) or the slower deactivation of tail currents (Fig. 1 A), and the model currents for EQ*QQ*Q*+E1 (Fig. 2 C) do not show the delay in activation before the exponential rise of current seen experimentally (Fig. 1 A). These differences suggest that Hodgkin–Huxley models, while supporting the idea that the E160R mutation can immobilize VS domains, fail to reproduce both central properties (G-V shifts) and more subtle behaviors of the I_Ks constructs. Thus, experiments are needed that provide a more complete description of activation kinetics along with further development of more representative activation models.

Using HMR1556-subtracted currents, it was relatively straightforward to fit the current time course at positive potentials with an exponential function and obtain Δt for wt EQ channel currents (Fig. 3 A). It is clear that for this construct, Δt ≠ 0 as exponential fits to current activation intersect the time axis at positive time values after the initiation of the voltage step. It was difficult to fit exponentials to activating currents at potentials negative to −20 mV, even during 20-s depolarizations due to extremely slow activation of I_Ks, so tail current deactivation rates were used to obtain time constants between −30 and −60 mV. The averaged bell-shaped activation time constant–voltage relationships (τ_act-V) for the three constructs are shifted in a hyperpolarizing direction with more active VSs (Fig. 3 B). We extended the τ_act-V relations to +180 mV to compare the relative time constants of activation for the three constructs at voltages where the forward rate constant would define the speed of the opening transition. The experimental data diverge from the red, green, and blue Scheme 1 curves which converge as the opening time constant reduces to 1/α in each case. Instead, the τ_act-V ratios relative to wt remain relatively constant at potentials positive to +50 mV, with τ_act ratios of ∼0.5–0.6 for wt/EQ*Q+E1, and ∼0.4 for wt/EQ*QQ*Q*+E1 (Fig. 3 B, inset).

The Δt-V relationships have a shallower dependence on pulse potential than the time constants and are displaced upwards to longer delay values as more activatable VSs are present in the construct (symbols in Fig. 3 C), although this is only significant between wt EQ and the other constructs. Peak values are at ∼0 mV for wt and +20 to +30 mV for EQ*Q+E1 and EQ*QQ*Q*+E1, which are close to the isochronal V_1/2s for each construct (Fig. 1 B), as expected if Δt bears a fixed relationship to the activation time constants. When the Δt values are plotted against τ_act (symbols in Fig. 3 D), the fast time constants of current activation at more positive potentials than +50 mV correlate with shorter Δt values, but overall the relationships are non-linear, τ_act increases (slower activation) while Δt values appear to plateau closer to the V_1/2 and negative potentials. At the higher potentials between +60 and +150 mV for wt EQ data, the maximum slope of Δt / τ_act was 0.336 (R² = 0.96).

The optimized rate constants from the Scheme 1 C model described in Fig. 1 (Table 2, row 2) were used as seed values in optimizing Scheme 1 B and Scheme 1 A models by simultaneous least squares fitting to the experimental G-V and τ_act-V relationships for wt and EQ*Q+E1, and the final rate constants for these two models are shown in Table 2, rows 3 and 4. The activation time courses of simulated currents produced by the final Scheme 1 model were fit with exponential relationships in the same way as the experimental current records were fit to obtain model τ_act and Δt values. Optimized Scheme 1 τ_act-V, Δt-V, Δt-τ_act, and G-V model curves are shown as solid lines overlaying wt, EQ*Q+E1, and EQ*QQ*Q*+E1 experimental data points (Fig. 3, B–D; and Fig. S3 A). Clearly, while the model τ_act–V and G–V curves are good fits, none of the model Δt-V curves are good fits to the experimental data (Fig. 3 C), and for the Scheme 1 C model, Eq. 3 predicts Δt values of zero across the voltage range (red line in Fig. 3 C). Scheme 1 B and Scheme 1 A models output Δt-V curves of a similar shape to those found experimentally but displaced upwards to longer delay times, and these model differences are carried over to the Δt-τ_act relationships (Fig. 3 D).

The activation delay times found experimentally are about half those expected from the Scheme 1 models and one-quarter of expected values from Eq. 3 for the measured exponential relaxation time constants, and may be compared with a value of 0.12 for BK channels previously reported (Horrigan et al., 1999). The classical explanation would be that a much slower process or processes slow current activation beyond the time taken for equilibration of the closed state transitions. More representative models of this type would then require additional steps in the activation process, such as extra or concerted transitions. Alternatively, only one subunit may be required to activate for the channel to open and current to begin its exponential trajectory, irrespective of the number of subunits available for this process, so that the activation delay, Δt, is less than expected from the Scheme 1 models. Notably, this latter idea cannot explain the activation delays found in the EQ*QQ*Q*+E1 channel (red points in Fig. 3 C) with only one active VS, which undergoes a single activating transition.

The model simulations bring up another interesting quirk in the delay Δt versus τ_act relationships, which, according to Eq. 3, should be linear with slopes ln(a). Neither the experimental nor the model Δt-τ_act relationships are straight lines, and we hypothesized this was related to fitting relatively slow time constants to current data obtained from 10-s test pulses. We examined this possibility using the Scheme 1 B and Scheme 1 A models (Fig. S3 B). Simulated currents during 4–100 s pulses were fit with exponentials to extract τ_act and Δt. It appears that longer test pulses improve the linearity of Δt-τ_act relationships which reach steady state with pulses of a 50-s duration or longer. The time constant fits from 10 s pulses deviate from this steady-state relationship with τ_act values >2 s. Regression lines fit to the initial steep regions of the model 50 s Δt – τ_act relations gave slopes of 1.35 and 0.62 for Scheme 1 A and Scheme 1 B, respectively (R² = 0.99 in each case), which compare favorably with the theoretical values from Eq. 3 of 1.39 and 0.69 (Fig. S3 B). The slight underestimations are hypothesized to reflect errors in the automatic fitting of model currents and estimations of Δt. It appears that the non-linearity of the experimental relationships is partly due to the inability to accurately measure Δt-τ_act from fits to currents during 10–20 s depolarizations, but this does not explain the deviation of the experimental Δt- τ_act curves from the Scheme 1 model curves, especially at the shorter values of τ_act.

The observed activation delay after the initiation of the step voltage pulse (Fig. 3 A), suggests the existence of multiple activation steps even for channels with a single activatable VS and fits well with prior literature that divides the movement of each VS into at least two independent steps with their own resolvable fluorescence components, F1 and F2 (Osteen et al., 2010; Zaydman et al., 2014; Barro-Soria et al., 2014; Westhoff et al., 2019). In Scheme 2 models, F1 forward and reverse kinetics are associated with voltage-dependent activating transitions with rates α and β, and F2 kinetics are represented by a concerted transition associated with channel opening, and opening and closing rates δ and ɣ, respectively.

To measure fluorescence and separate the fluorescence components in oocytes, we used a single C214A/G219C/C331A-Q1 pseudo-wildtype construct (psQ) that contains a labelable cysteine, 219C, at the top of the S4 (Fig. 4), or two Q1 subunits in a tandem linked psQQ with a 219C at the top of the S4 of one of the pairs of Q1s. Using Alexa Fluor 488 C₅-maleimide allowed tracking of VS environment changes during activation of the I_Ks channel. To examine the effect of reducing the number of activatable subunits on the F1 and F2 components of VS activation, a dimeric construct was developed that has two E160R-containing VS subunits when it assembles as a tetramer (psQ-E160R Q*, psQQ*+E1 in Fig. 4 A). Previously, we had compared activation data from psQ*Q and psQQ* and found no difference (Westhoff et al., 2019), so here we only use psQQ*+E1. For psQQ*, the presence of the labelable G219C in subunits not containing E160R ensured that the two activatable subunits in the complex could be labeled with the fluorophore Alexa Fluor 488. Unfortunately, it was not possible to record fluorescence from constructs with E160R mutations in three VSs due to difficulties with the synthesis and expression of protein from very large cRNA molecules, so we were limited to the study of psQ, psQQ, and psQQ* constructs. We also tested how prepulse voltage modulated the residence of psQQ+E1 and psQQ*+E1 channels in resting closed states in oocytes. Either the activation half-time (Fig. S4, A and B) for psQQ+E1 and psQQ*+E1 channels or the activation delay, Δt, for psQQ+E1 channels (Fig. S4, C and D) was measured after 5-s prepulses to between −160 and −40 mV. It was noted that the activation half-times of psQQ*+E1 channels were less dependent on the prepulse voltage than the half-times or Δt values for psQQ+E1. This is likely due to the restriction on activation of two of the four VSs in the psQQ*+E1 channels. Nevertheless, in channel constructs using the two methods, a prepulse voltage of −140 mV was sufficient to place channel complexes in resting closed states, and so this prepulse potential was used in experiments to determine the F1 and F2 components of VS fluorescence during activation gating.

Current and fluorescence records are shown for voltage steps from −140 mV to between −180 and +80 mV (Fig. 4 A). After subtraction of the fluorescence baseline (the −140 mV step), which corrected for photobleaching during the pulse, fluorescence records during activation were fit with either one or two exponentials, depending on the step voltage, which allowed separation of data into F1 and F2 components. The graph in Fig. 4 B summarizes the peak fluorescence values as a function of potential along with G-V relations for the psQ+E1 and psQQ*+E1 constructs (G-V only shown for psQQ+E1). The experimental F-V and G-V relations are shown as symbols, the G-V and F-V fits as lines, with the component F1-V and F2-V relationships as broken lines. The F1-V relationships were not clearly separable for the psQ+E1, psQQ+E1, and psQQ*+E1 constructs, with V_1/2s of −96.9, −88.2 (not shown), and −86.5 mV, respectively (Table 1). However, the F2-Vs tracked closely with the respective G-V relationships for the specific constructs. The V_1/2s of the G-V and F2-V for psQ+E1 were 4.2 and 3.0 mV, and for psQQ*+E1 were 23.5 and 21.0 mV, respectively. This tracking of F2 with the G-V has been described previously for pseudo-wt KCNQ1+KCNE1 channels, so was not unexpected in the present experiments (Osteen et al., 2010; Zaydman et al., 2014; Barro-Soria et al., 2014).

The V_1/2 separation between the G-V relationships for psQ+E1 and psQQ+E1 channels compared with that for psQQ*+E1 channels with only two activatable subunits is preserved in a similar manner to that observed in the mammalian cell expression system (Fig. 1 and Table 1) and is accompanied by a hyperpolarizing shift in the F2 component of fluorescence emission in wt, but minimal change in the voltage-dependence of the F1 component. This suggests that the major F1 component of fluorescence and consequently VS activation is independent and unaffected by the presence or otherwise of fixed and/or mobile adjacent VS domains. In contrast, the displacement of the F2-V to more depolarized potentials in the presence of E160R subunits suggests that this secondary VS movement is constrained by the presence of non-activatable subunits within the same channel complex or is tied in some more complex manner to the G-V, opening of the channel gate and development of conductance. Interestingly, it was also noted that the proportion of the total fluorescence signal attributable to F1 decreased from 80% in wt to 57% in psQQ*+E1. This can be clearly inferred from the time course of fluorescence tail decay at −40 mV (Fig. 4 A, lower). The psQ+E1 tails are quite well maintained at −40 mV, as expected if the majority of the fluorescence signal is accounted for by the F1 component, whereas the total psQQ*+E1 signal decays much more. This relative increase of the F2 component compared with F1, in the presence of E160R-containing subunits, is also suggestive that this component of fluorescence may not arise entirely from VS movements alone.

We analyzed the kinetics of the separate fluorescence components in terms of two transitions for independent VSs, according to Scheme 2. The time constants for relaxation of F1 and F2 were obtained from mono-exponential or biexponential fits to the fluorescence traces at different potentials (Fig. 5, A and B; and Materials and methods), and with the steady-state, F1-V and F2-V were used to calculate α, β (for F1, Fig. 5, C and D) and δ, ɣ (for F2, Fig. 5, E and F) for psQ+E1 and psQQ*+E1 as a function of potential. The kinetic parameters that define the curves may be found in Table 2, and the psQ+E1 data are consistent with values reported previously (in brackets) with α₀ = 4.8e-3 ms⁻¹ (4e-3 ms⁻¹), β₀ = 1.6e-4 ms⁻¹ (9.1e-5 ms¹), δ₀ = 3.4e-4 ms⁻¹ (2.6e-4 ms⁻¹), and ɣ₀ = 3.9e-4 ms⁻¹ (1.18e-4 ms⁻¹), considering these authors used slightly different methods to obtain the time constants for F1 and F2 (Barro-Soria et al., 2014).

The experimentally derived kinetic parameters from the psQ+E1 and psQQ*+E1 fluorescence F1-V and the F2-V (Fig. 5), summarized in Table 2, were incorporated into Scheme 2, A and B, models (below), assuming independent transitions for each VS subunit (reflected in F1 and the α, β rate constants) followed by a second, shared, coordinated transition that leads to channel opening (reflected in F2 and the δ, γ rate constants). The fluorescence emission for an α transition versus a δ transition was either set as f1:1 or f1:2 (one-to-one or twice the fluorescence emission for an F2 versus an F1 transition). Isochronal model F-V and G-V curves simulated using these rates are shown overlying the experimental data (symbols, Fig. 6 A). For psQ+E1, Scheme 2 A with f1:1, the model fits overlay the F-V data extremely well and also simulates the voltage-dependence of the G-V exactly. The modeled F-V determined using f1:2 is shown as the long-dash blue line and is displaced below the experimental data as expected if the fluorescence contribution from F2 is being overestimated. In contrast to this result, the simulations using Scheme 2 B show a much better fit to experimental data with the F-V modeled using f1:2. This correlates with the observation made experimentally (Fig. 4) that the F2 fluorescence component comprised a larger proportion of the total fluorescence in the psQQ*+E1 mutant than in psQ+E1. With the F-V modeled using f1:2, the F1, F2 components, and the G-V also show an excellent correlation with the experimental data.

The delay (Δt) and exponential current relaxation kinetics (τ_act) of the oocyte currents (open circles in Fig. 6 B) were very similar to the mammalian cell data previously described (filled circles in Fig. 6 B, from Fig. 3 D), but the transient kinetics were not well fit by model simulations using the experimental rates from the VCF experiments or the F1 and F2 rate constants from Barro-Soria et al. (2014), shown by dashed lines (Fig. 6 B), which model Δt versus τ_act from −20 to +100 mV. To reproduce the transient kinetics, rate constants for both VS transitions needed to be optimized in Scheme 2, A and B, models (Table 2). This required slower rate constants for F1 and F2 transitions, which gave the Δt and τ_act graphs (continuous lines in Fig. 6 B), and model currents and fluorescence (Fig. 6 C), without altering the steady-state kinetics, which were already well fit by the experimentally derived rate constants. The model current tracings show appropriate activation delay and activation time constants in psQQ*+E1 (Scheme 2 B) compared with psQ+E1 (Scheme 2 A), reflecting the model relationships between Δt and τ_act for the two schemes (Fig. 6 B). It is not clear why the kinetic parameters from the fluorescence experiments require scaling and do not directly reproduce the delay and exponential activation of currents. It is likely that the 5-s duration fluorescence test pulses are too short to measure F2 time constants accurately from small signals and reproduce current relaxation time constants in the 10 s range, but this does not explain why modeled Δt measurements, which predominantly reflect the F1 transition, do not reproduce experimental activation delays well.

Although these models produce excellent fits to the data, neither set of rates used in both models reproduce the G-V shift seen experimentally, nor the persistently slower activation time constants at positive potentials in EQ*Q+E1 versus wt constructs (Fig. 3 B). The Scheme 2 results described above were obtained using either a four-state model simulating psQQ*+E1 or a six-state model simulating psQ+E1. In these models, the independently moving VSs are assumed to undergo a first transition from rest, followed by a second coordinated transition of all four activated subunits that induce channel opening. However, published experimental data describing two VS movements in I_Ks do not explicitly demonstrate direct coupling of the second VS movement to the channel opening step (Zaydman et al., 2014; Barro-Soria et al., 2014; Westhoff et al., 2019; Hou et al., 2020), and this raises the possibility that separation of these two events might better predict the current kinetics and the hyperpolarization of the G-V seen in psQQ+E1 compared with the two active VS channel psQQ*+E1.

The incorporation of two independent VS transitions, separated from the coordinated opening transition, in our models with four (Scheme 3 A) or two (Scheme 3 B) activatable VSs, uses fluorescence rate constants from psQ+E1 and psQQ*+E1 experiments in a similar manner to the Scheme 2 models (Fig. 7). Plots of Δt versus τ_act show similarly fast time constants of exponential activation as the Scheme 2 models as the rate-limiting opening step remains the same (Fig. 7 A). However, the predicted Δt values are much closer to experimental values due to the multiple extra closed steps that have to be traversed in Scheme 3 models before channels can open. Activation time constants from the simulated data (not shown) again converge at very positive potentials as did the Scheme 1 model curves, which is not what was observed experimentally (Fig. 3 B). Scheme 3 models did not produce good simulations of the 5-s experimental F-V and G-V data despite the use of the δ, γ rates measured from the fluorescence emissions. The normalized F-V relation was displaced below the experimental points (blue curves below triangles in Fig. 7 B) and although the F2-V overlays the experimental G-V, the model G-V data points are displaced to more positive voltages. The simulations of 5 s F-V data for the Scheme 3 B model using rate constants from the psQQ*+E1 data are quite good if we assume equivalent fluorescence for the F1 and F2 transitions (f1:1, solid green line over the triangles in Fig. 7 B). Still, the model G-V data points are significantly depolarized compared with the experimental G-V data. In this case, the modeled F2-V also overlays the experimental G-V relationship.

The inclusion of a second transition in the activation path of each VS (via the δ, γ rates) in Scheme 3 models, separate from the final concerted opening transition involving all activated VSs, did slow current activation kinetics and improve the activation delay properties, but separated the F2-V from the G-V in both models, which was a significant failure. Unsurprisingly, the use of the same experimental rate constants in both Scheme 3, A and B, models gave identical F-V relationships and the expansion of the 6-state two VS model to the 15-state four VS model failed to predict the hyperpolarization of the psQ+E1 G-V compared with that from psQQ*+E1. There is no simple way, in sequential activation models that maintain independent subunit movement, to account for the G-V hyperpolarization seen experimentally as the number of activatable VSs is increased from one or two to four in both mammalian and oocyte expression systems (Fig. 1 B and Fig. 4 B).

Ion channels that open with a concerted transition, after activation of independently moving VSs, such as Shaker and hERG, reach a limiting slope of e-fold per RT/F increase in Po at very negative voltages from which the equivalent gating charge can be estimated (Schoppa et al., 1992). The equivalent gating charge of ion channels which gate in an allosteric manner with parallel VS gating pathways through closed and open states cannot be estimated using the limiting slope method (Horrigan et al., 1999). At very negative voltages and low Po where the equivalent charge is usually estimated, the Po is higher due to channels opening from multiple closed states along the activation pathway. In allosteric gating models, a weakly voltage-dependent channel opening pathway is introduced between resting closed states and open states as well as between closed and additional open states that are accessible in a voltage-dependent manner to produce the voltage-dependence of activation seen at more positive voltages. Given these different limiting slope behaviors of ion channels that gate in a sequential or allosteric manner, the Po-V plot of I_Ks may reveal clues about its mechanism of gating.

For comparative purposes, this experiment was performed on hERG channels, psQ+E1, psQQ+E1, and psQQ*+E1, expressed in oocytes in separate experiments. To enable measurement of very low Po values from small hERG tail currents, records were made in 20 mM [K⁺]_o, which increases single channel conductance (Materials and methods and Fig. 8) following protocols adapted from a prior study (Zhang et al., 2004). Experimental Po values were obtained from tail currents (shown enlarged) after 2 mV steps to between −72 and −40 mV, and from a second set of data obtained after 20 mV steps from −40 to +40 mV (right panel). Data from the two protocols were combined and normalized to draw an overall Po-V curve. Consistent with previous findings, at very negative voltages, a limiting Po slope was reached for hERG currents, with no deviation at the lowest Po values recorded (Fig. 8 C), which gave an equivalent number of gating charges, z_a of 6.08 ± 0.34 e₀, in keeping with prior measurements from wt and mutant hERG channels. The limiting slope behavior of psQQ+E1 at negative voltages is very different from that of hERG, with the slope weakly voltage-dependent in both individual data sets and also mean data (Fig. 8, B and C). The maximum slope is achieved down to Po values of ∼0.01 after which the slope decreases again at more negative voltages. The mean z_a value for psQQ+E1 measured during the steepest part of the Po-V relationship was 2.07 ± 0.06 e₀, for psQQ*+E1 it was 2.46 ± 0.12 e₀, and for psQ+E1 was 2.41 ± 0.13 e₀ (n = 10, not plotted in Fig. 8 C). Neither the sequential activation Scheme 2 A (model used in Fig. 6) nor Scheme 3 A reproduce this slope decrease at more negative voltages for psQQ+E1 where the experimental Po remains higher than predicted from a sequential activation pathway (Fig. 8 D). The fit is actually reasonable down to Po ∼0.01, and for psQQ*+E1, there is less deviation from a sequential model. The data and fits suggest that I_Ks has access to open states at low Po and supports the idea of independent VS activation regulating I_Ks pore conductance in an allosteric manner. Moreover, adopting an allosteric model of gating might give insight into the hyperpolarizing shift of the four versus two active VS G-Vs (Figs. 1 and 4), which was not successfully reproduced by sequential gating schemes (Figs. 1, 6, and 7).

An allosteric model in which independent VSs undergo a single transition from resting to activated and channel opening occurs when zero to four subunits are activated was examined first and is shown in Scheme 4. For both the 10-state model representing psQ+E1 (Scheme 4 A), the 6-state model representing psQQ*+E1 (Scheme 4 B) and the 4-state model representing EQ*QQ*Q*+E1 expressed in tsA201 cells (Scheme 4 C), an allosteric factor, D, was introduced for the closed to open transitions and between open transitions to maintain microscopic reversibility. The allosteric factor is further increased D-fold for each subunit that is activated, predisposing the channel to pass through several closed states before opening, consistent with the delays seen in I_Ks activation, while the C_RRRR-O_RRRR, C_RRRA-O_RRRA, and C_RRAA-O_RRAA transitions remain accessible, consistent with the higher-than-expected Po at very negative voltages (Fig. 8).

In Scheme 4 models, as α >> δ (Fig. 5), if D is sufficiently large, channel openings will be biased to occur after all VSs are activated (C_AAAA-O_AAAA), which makes the C_RRRR-O_RRRR and other C-O_RRRA, -O_RRAA, and -O_RAAA open state transitions negligible and transforms the parallel allosteric model into a quasi-sequential or an obligatory-coupled one (Chowdhury and Chanda, 2012b). A limiting slope would then be reached at negative voltages and the activation path would be equivalent to that of Scheme 2 models (Fig. 6). As the slope of the experimental psQQ+E1 G-V relationship becomes weakly voltage-dependent for Po less-than ∼0.01 (Fig. 8 C), significant channel opening must occur from early closed states, and indeed, simulations of the Po-V relationship support the idea that D is small (Fig. 10 D).

The feature that sets allosteric models apart from sequential models with different numbers of active subunits is that in those models the rate of the opening transition reduces to δ_V at very positive voltages, whereas in the allosteric models, the rate is determined by δ_VD^a, where a is the number of activated VS subunits. We suggest that this explains the lack of convergence of the τ_act-V relationship (Fig. 3 B) where EQ*QQ*Q*+E1 activation time constants remain persistently slower than wt Q1+E1 or EQ*Q+E1 channels with a rather stable ratio (Fig. 3 B, inset), although the theoretical Scheme 1 curves converge. These data give us a unique opportunity to calculate D from the ratios of the activation time constants for the different constructs. If the time constant ratio for wt/EQ*QQ*Q*+E1 is 0.4, which is δ_VD/δ_VD⁴, then D = 1.357, and if the ratio for wt/EQ*Q+E1 is 0.6 (Fig. 3 B, inset), which is δ_VD/δ_VD², then D = 1.291.

A value for D of 1.357 was selected as the initial condition based on the experimental activation rates and this was combined with the experimentally derived kinetic parameters from psQQ+E1 (Table 2) and incorporated into a 10-state allosteric model for optimization (Scheme 4 A), with the conductance of early open states set to the same value as that for O_AAAA. The relatively small allosteric factor of 1.357 suggests that the C_RRRR-O_RRRR transition and other early closed-open transitions will be significant at voltages around the resting potential, consistent with the idea of multiple open states accessible when zero to four VSs are fully activated (Westhoff et al., 2019). The same rate constants and allosteric factors were incorporated into the psQQ*+E1 6-state allosteric model (Scheme 4 B) and the EQ*QQ*Q*+E1 (Scheme 4 C) four-state model to examine the currents, fluorescence tracings, and model G–V relationships for comparison with experimental data (Fig. 9, A and B). The models predict the depolarizing shift of the G-V with fewer active VSs, and with D set to 1.357 are in quantitative agreement with the experimental V_1/2 values (Fig. 9, B and C; and Table 1). The slopes of the model G-Vs also decrease as the number of active VSs are reduced from 21.4 to 23.3 mV, as did the experimental G-V relationships (Fig. 1).

These model parameters also fit the equilibrium experimental oocyte current and fluorescence data (Fig. 4). Using the single set of rate constants from Table 2 (row 9), the 10-state Scheme 4 A and 6-state Scheme 4 B models fit both the F-V and G-V relationships from the psQ+E1 and psQQ*+E1 constructs very well (Fig. 9 D), predicting the shape of the F-V relationships and the voltage dependence of the F1-V and F2-V components as well as the G-Vs, plus the G-V shift to more positive potentials in psQQ*+E1 (Table 1). Only the foot of the model G-V relationship for psQQ*+E1 between −40 and 0 mV is not so well fit to the experimental data. The model appears to better simulate the mammalian data with only a minor divergence of the G-V data between −50 and −10 mV and a reduction in slope for the EQ*Q+E1 model G-V (Fig. 1 B), rather than the depolarizing shift of the relationship seen in oocytes.

The Scheme 4 τ_act-V relations reproduce the experimental current relaxation time constants well (Fig. 10 A), and as predicted earlier, the model relationships fail to converge at up to +300 mV, the most positive voltages examined, which supports the idea of allosteric activation when more VSs contribute to the activation process. The curves also crossed over negative to −25 mV so that wt (Scheme 4 A) models support slower deactivation of currents than for constructs containing fewer active VSs, as was observed experimentally (Fig. 1 A and Fig. 3 B; Westhoff et al., 2019). The one property not well reproduced by these simulations was the activation delay, Δt. Even though the model exponential activation time constants correlated well with experimental data, the Δt values were too short when all open-state conductance levels were set to 1.0 (Fig. 10, B and C). The other extreme was therefore tested with all conductance levels set to zero except the fully activated open state (O_AAAA). In this situation, not surprisingly, the activation delays were increased significantly and matched the experimental data at potentials close to the activation V_1/2 (Fig. 10 C).

The most significant improvement that the implementation of an allosteric model with one transition per VS makes when fitting the experimental steady-state data is that the depolarization of the EQ*QQ*Q*+E1, and psQQ*+E1 G-V relationships, and the displacement of the F2 component of the F-V relative to the wt EQ and G219C psQ+E1 data are now faithfully reproduced in the models using a single set of rate constants (Fig. 9 and Table 2). The correct voltage dependence and amplitude of current relaxation time constants, and speeding of deactivation in EQ*Q+E1 and EQ*QQ*Q*+E1 compared with wt is also seen (Fig. 10). At more negative potentials, the use of the allosteric Scheme 4 model could reduce the slope of the Po-V relationship at Po values <0.01, and adjustments to the allosteric factor changed the shape of the model Po-V between −50 and −100 mV. Overall, the experimental D value of 1.357 most accurately predicted the limiting slope behavior of psQQ+E1 at very low Po values, as shown by the overlay with the averaged experimental data (Fig. 10 D).

Having established that allosteric models are able to make significant improvements to the modeling of I_Ks activation, we proceeded to include a second VS step (Figs. 11, 12, and 13) to be consistent with our appreciation of experimental data, demonstrating that VSs undergo at least two transitions during the activation process along with the channel opening step (Barro-Soria et al., 2014; Zaydman et al., 2014). In these Scheme 5 models, though, unlike in prior published work (Zaydman et al., 2014; Westhoff et al., 2019), we coupled the δ, γ rates derived from F2 measurements to the opening transitions and generated a new set of rate constants, “κ” and “λ,” to simulate a slow VS transition from intermediate-activated to fully activated states (Fig. 11). The reasons for this are first, that experimental studies have closely linked the F2 measurement with channel opening, even in the presence of mutations and drugs that change its voltage dependence (Osteen et al., 2010; Zaydman et al., 2014; Barro-Soria et al., 2014; Westhoff et al., 2019). Second, no gating currents have been shown to be associated with this second VS movement (Ruscic et al., 2013), and third, that recent cryo-electron microscopy data support a reorientation of the top of S4 which could certainly affect the environment of a fluorophore attached at G219C when PIP₂ binds to the channel and it opens (Mandala and MacKinnon, 2023).

Scheme 5 models (Fig. 11) depict tiered allosteric schemes wherein the VSs undergo two transitions during the activation process before being able to open. Two sets of voltage-dependent rates, α, β and κ, λ, determine the VS transitions from resting to intermediate (R→I) and intermediate to activated (I→A) states. Rate constants δ and γ regulate opening and closing along with the allosteric factor, D, to determine the channel transitions from closed to open states, and similar to Scheme 4, A–C, opening allostery increases D-fold for each activated available VS. (D set to 1.357 for these simulations). Experimentally derived G-V and F-V kinetic parameters from psQ+E1 (Fig. 4), and Δt versus τ_act data from wt EQ (Fig. 3) were incorporated into Scheme 5 A for optimization in MATLAB (see Materials and methods) to obtain a single set of rates used in Scheme 5, A–C, models. The optimized simulations for current and fluorescence tracings and equilibrium relationships are shown in Fig. 12, with summarized model kinetics in Table 1 and optimized transition rates in Table 2 (row 10). The time course and exponential activation of experimental currents are accurately simulated by Scheme 5 A using the optimized transition rates (Fig. 12 A). Simulations of Scheme 5 A currents and the underlying occupancy of channel open states at 0 and +80 mV show that the activation delay and slow exponential relaxation of currents is accounted for by the slow movement of channels through connected open states, especially at 0 mV, by the large numbers of channels retained in closed states in Scheme 5 models (Fig. S5). As was the case for Scheme 4 models, the oocyte G-V and F-V curves are also reproduced by Scheme 5, A and B, models. The F2-Vs are not so closely aligned at the foot with the G-Vs as in Scheme 4 models, but the depolarization of the F2-V and G-V (Fig. 12, B and C) curves in the psQQ*+E1 model versus the psQ+E1 model is reproduced, as is the overall shape of both F-V relations. The change in the G-V V_1/2 values from 7.5 to 34 mV going from wt to the EQ*QQ*Q*+E1 Scheme 5 C model are in quantitative agreement with tsA201 and oocyte data in Table 1, and are just as impressive as for Scheme 4 models (Fig. 9 C). Importantly, the current relaxation time constants (τ_act, Fig.12 D) and activation delays (Δt, Fig. 12 E) are also very well simulated by Scheme 5 A models using a single set of rate constants (Table 2). At very positive voltages, the τ_act-V relationships maintain their separation between the three Scheme 5 models (Fig. 12 D). This is the first model system that we have investigated that has proven adept at simulations of both isochronal and transient I_Ks kinetics across all three constructs using a single set of kinetic constants.

Although the experimental G-V and F-V equilibrium kinetics and the separation of the τ_act-V relationships from the different E160R-containing constructs are reproduced across all Scheme 5 models using a single set of kinetic constants (Fig. 12), the measured activation delay, Δt, and τ_act-V was not well fit for the Scheme 5 C model (Fig. S6). Reasons for this are considered in the Discussion, but in order to simulate the transient kinetics of the tsA201 EQ*Q+E1 and EQ*QQ*Q*+E1 current activation data better, we optimized the Scheme 5, B and C, models individually, minimizing the difference between experimental and computed relationships for the τ_act-V and Δt-V. The results are shown in Fig. 13, and the kinetics for these models are in Table 2. The simulation of the Δt versus τ_act relationship (Fig. 13 A) and its component Δt-V (Fig. 13 C) and τ_act-V relations (Fig. 13 D) for Scheme 5, B and C, models is excellent with the new rates, which are characterized by a 3–5× acceleration of the α, β, δ, γ, rate constants and minor slowing of the κ and λ rate constants (Table 2), compared with the single set of kinetic constants used for all model plots in Fig. 12 (and Fig. S6) and the Scheme 5 A simulations of EQ experimental data in Fig. 13. There is only a slight degradation of the Scheme 5, B and C, model G-V fits to the experimental data using these separate rates (Fig. 13 B), compared with the unified set of rates (Fig. 12 C), and this is almost certainly the result of not prioritizing the experimental G-V data in the optimization process to obtain them.

The aim of the present study was to examine gating models that have been proposed for the I_Ks channel and evaluate their ability to reproduce the gating and activation properties of the channel under a number of different conditions. The new information that we were able to add in this study is built upon prior work from our laboratory, which demonstrated the functionality of channels with less than four active subunits. The mutation E160R in individual subunits allowed novel insights to be obtained from the comparison of channel currents, activation properties, and fluorescence in situations where one to three VS subunits were restrained and unable to contribute to the gating process (Westhoff et al., 2019). The ability of the E160R mutation to restrain the activation of VS was suggested by the lack of current when E160R was present in all four VS domains despite cell-surface protein expression (Zaydman et al., 2014; Westhoff et al., 2019), the lack of methanethiosulfonate reagent effects on ion current activation when labeled subunits also contained the E160R mutation, the reduction in conductance with each additional E160R, and the absence of fluorescence during channel activation from E160R-containing subunits (Westhoff et al., 2019). Further support for the ability of the E160R mutation to negate the contribution of individual VS to channel activation is provided in the present study by the non-convergence of τ_act data at voltages positive to +100 mV (Fig. 3), reduced activation delay in constructs with fewer active VSs, and allosteric activation models, which predict the G-V shifts and reduced slope, and F-V depolarization with fewer active VSs (Figs. 9, 10, 11, 12, and 13).

At first glance, the currents from the E160R-containing constructs, EQ*Q+E1 and EQ*QQ*Q*+E1, look not dissimilar to wt I_Ks, even when three VSs contain the E160R mutation, leaving only one active VS (Fig. 1 A). However, the amplitudes are less (Westhoff et al., 2019), current activation appears slower, and tail currents are faster with fewer activatable VSs (Figs. 1, 2, and 3). Upon closer examination and with further experimentation, some subtle, and not so subtle differences in the behavior of the different channel constructs become apparent. There is a decrease in the slope of the G-V relationship as the number of VSs is reduced, and a progressive shift of the V_1/2 of activation to more positive potentials (Fig. 1 B), which is opposite to the direction expected for Hodgkin–Huxley models of activation. The time constants for exponential relaxation of current (τ_act) after the activation delay accelerate at very positive potentials but do not converge in the different constructs, and the τ_act ratio between wt and EQ*QQ*Q*+E1 stabilizes at ∼0.4 (Fig. 3 B). The activation delay, Δt, is reduced, but ≠ 0 with three restrained VSs in EQ*QQ*Q*+E1 (Fig. 3 D), which indicates that even the single active VS must undergo multiple steps during activation gating. This idea is supported by fluorescence data which divide the fluorescence emission from G219C into two exponential components (Fig. 4). The presence of fewer closed states in oocyte constructs with fewer active VSs (psQQ*+E1) is suggested by Cole–Moore type experiments (Cole and Moore, 1960), while the limiting slope experiments show a prominent deviation from linearity at Po ∼ 0.005 and −80 mV, in psQ+E1 and psQQ+E1 constructs, which supports the presence of multiple open states, some traversed early in the activation pathway.

There were some experimental limitations that were imposed by the need to use a wide range of voltages during experimental protocols and constructs with VSs containing the E160R mutations. Ideally, we preferred to use channels expressed in mammalian cells due to the higher quality of voltage clamp attainable to measure Δt and τ_act. But, cells were generally intolerant of being held at potentials negative to −120 mV (Fig. S2), of voltage pulse steps to greater than +100 mV, and also of voltages pulses longer than 10 or 20 s, which would be required to obtain steady-state G-V relationships. Fluorescence measurements of VS movement could not be made in tsA201 cells due to their small size, while limiting slope measurements at low Po required expression levels at negative voltages that were also unattainable in mammalian cells. These limitations necessitated the use of a mixed set of data from mammalian cells and oocytes, which were not identical. Oocyte constructs modified for fluorescence measurements (see Materials and methods) all showed a negatively shifted G-V relationship ∼10 mV (from +13 to +4 mV, Table 1), but psQQ*+E1 still showed a positive shift of the G-V V_1/2 (from +4 to +24 mV) compared with psQ+E1 and psQQ+E1 (Fig. 4 B), similar to that seen in mammalian cells. Unfortunately, the oocytes only poorly expressed RNA constructs with multiple concatenated subunits, so fluorescence data from psQQ+E1 and psQQ*+E1 were much harder to obtain, signals were smaller than from psQ+E1, and could not be obtained from constructs with three E160R VS subunits. Despite these limitations, new data were obtained that informed the makeup of activation models, as discussed below.

While models based on the Hodgkin and Huxley differential equations of channel activation (Hodgkin and Huxley, 1952) may seem anachronistic in a modern analysis of activation gating systems, the reality is that most models of ion channel gating will adhere to their systematization of activation, perhaps because experimental data from steady-state G-V relationships and activation time constants are still widely used to formulate gating models. Therefore, we first examined sequential models with a single open state based on Scheme 1 to determine if they could account for I_Ks gating. Such models are consistent with a reduced version of Hodgkin and Huxley activation of a K⁺ channel (Hille, 2001) with different numbers of VSs. In these models, the movement of the last voltage sensor directly couples with channel opening and is depicted as a single event. We used the EQ*QQ*Q*+E1 construct, which has only one activatable VS, to record time constant and isochronal G-V data and solve exactly the rate kinetics for the single transition (Scheme 1 C), and optimize them to fit a model to the data. This model was then extended to Scheme 1, B and A, to fit data from EQ*Q+E1 and wt, respectively (Figs. 1, 2, and 3; Figs. S1, S2, and S3; and Tables 1 and 2). The Scheme 1 C model optimized fits reproduce the isochronal G-V and τ_act kinetics of EQ*QQ*Q*+E1 well, including the overall current recordings at different potentials (Fig. 1 and Fig. 2 C), but cannot simulate the delay in activation (Δt) and the rate constants do not transfer well to the schemes with more active subunits.

First, the Δt-V relationship should be zero for the single subunit, but it is not, and in the other two models, the Δt values are depressed below those expected from models based on the current relaxation time constants (τ_act, Fig. 3, C and D), where subunit activation determines both the delay in opening and subsequent activation of I_Ks. Although values negative to 0 mV were difficult to obtain, it appears that the Δt-V relationships have a bell-shaped voltage dependence, which suggests a process governed by a single transition with voltage-dependent forward and backward rate constants (Horrigan et al., 1999), and is consistent with the mainly exponential time course of I_Ks activation (Fig. 3, A and B; and Figs. S2 and S4). The existence of a delay in the EQ*QQ*Q*+E1 single activatable subunit Δt-V data indicates that multiple transition steps must occur for each subunit VS, and wt and EQ*Q+E1 Δt-V data show that models based on the measured activation time constants produce Δt values that are about twice as long as experimental values. This suggests that slower steps in activation gating are not accounted for in the models, which limits the speed of exponential current relaxation during opening. In I_Ks channels, it is the interaction of the KCNE1 subunit, associated with KCNQ1, that causes this very slow exponential relaxation of currents, since it is not present in KCNQ1 channels expressed alone (Fig. 1 A). Further, Scheme 1 models do not take into account that the delay duration may be influenced by the presence of multiple open states if they exist, and therefore also transitions between closed and open states and between open states themselves.

Secondly, the Scheme 1 models predict that τ_act kinetics will converge at potentials between +140 and +180 mV, but no such convergence was seen experimentally as relatively stable time constant ratios between wt, EQ*Q+E1 and EQ*QQ*Q*+E1 of 0.6 and 0.4, respectively, were maintained out to +180 mV (Fig. 3 B), the limit of our ability to record currents and maintain seals in tsA201 cells. Eq. 3 predicts a linear relationship between Δt and τ_act, but this was observed neither experimentally nor in the models (Fig. 3 D). To reproduce the experimental method, simulated Δt values were obtained by fitting the output model currents to obtain Δt and τ_act. Longer simulation test pulses improved the linearity of Δt versus τ_act curves (Fig. S3), and we conclude that to approach the theoretical relationship described by Eq. 3, experimental pulse durations would need to be longer than 20 s. Most of our experimental data were obtained from 10-s pulses, or 20 s at voltages around 0 mV, and the curvature of the experimental relationships closely matches the model curves for 10–20 s pulse durations. It would not have been practical in mammalian cells to use 50- or 100-s duration pulses to get better fits for τ_act at voltages around and below 0 mV. The remaining curvature of the model Δt versus τ_act curves obtained from pulses longer than 50 s can be attributed to other terms in the binomial expansion that defines I_Ks activation with multiple subunits (Horrigan et al., 1999).

Thirdly, Scheme 1 models predict that the optimized isochronal G-V relationships should shift to more positive potentials and become steeper as more subunits are added in Scheme 1, B and A (Fig. 1). Experimentally, the G-V relations do become steeper by about the right amount, but they are hyperpolarized and displaced in the opposite direction, as we had previously observed using data from cells that were not HMR1556-subtracted (Westhoff et al., 2019).

Scheme 2 models incorporate a concerted gating step, with an additional set of rate constants, that represents a slower and concerted conformational change of the VS and channel gate, which is required to regulate the time course of exponential current relaxation, as pointed out during the Scheme 1 analysis (Figs. 1, 2, and 3). Rate constants for such models can be obtained from the activation delay and exponential current activation kinetics as long as VS activation is much faster than the kinetics of the closed to open transition (Horrigan et al., 1999). In our experiments, we avoided this issue by making direct measurements of VS movement during activation using fluorescence spectroscopy.

Apart from being able to measure fluorescence from labeled cysteine residues in oocytes, a further advantage of using oocytes is that they can be held at −140 mV for extended periods to place channels in their ground closed states (Fig. S4), and pulsed to potentials as negative as −180 mV for a number of seconds. Double exponential fits of the resulting fluorescence records allowed separation of F1 and F2 components and calculation of rate constants for the two VS steps (Fig. 5 and Table 2) that were then incorporated into Scheme 2 models. These models simulated the isochronal data very well, but again did not simulate the transient kinetics well. This is perhaps not surprising; both in our experiments (Fig. 4) and those described previously, VS fluorescence pulse durations are limited to 2–5 s to minimize photobleaching of the Alexa Fluor 488 dye (Osteen et al., 2010; Barro-Soria et al., 2014; Zaydman et al., 2014; Westhoff et al., 2019), while current activation can take over 10 s at the V_1/2 in both mammalian cells and oocytes (Fig. 3 B and Fig. 6 B). Optimization slowed rates in the models for both VS transitions (Table 2), and allowed Scheme 2 psQ+E1 and psQQ*+E1 model outputs to fit transient as well as isochronal experimental data (lines in Fig. 6 B). Importantly, however, models still did not predict the hyperpolarization of the G-V as more active subunits are present and the shift is only apparent in the isochronal model fits (Fig. 6 A) because of the different experimental rates pertinent to psQ+E1 and psQQ*+E1 (Fig. 5 and Table 2) that were used in the separate models. The Boltzmann parameters from fits to the F1-V component of fluorescence from psQ+E1 and psQQ*+E1 (Table 1), and the F1 rate constants that were obtained from fits to the F1 fluorescence time course (Table 2) were almost identical, despite the fact that two of the four VSs in the psQQ*+E1 construct were being prevented from outward displacement during activation. This F1 matching, plus the reduction of the total fluorescence attributable to F1 versus F2 in psQQ*+E1 data compared with psQ+E1 (Fig. 4), strongly supports the idea that the component of VS movement reflected in F1 is independent of the position, fixed or otherwise, of the other VSs, and is expected for independent movement of VSs, which is an axiom of all the models examined in this paper.

Experimental data have suggested that the second VS step is very closely associated with channel gate opening, and mutations in some subunits that shifted the voltage dependence of channel opening to more positive potentials, or drugs that prevented channel gate opening also shifted the voltage-dependence or prevent the appearance of the second phase of fluorescence (Barro-Soria et al., 2014). Still, it did seem possible that the second VS step could be divorced from channel gate opening, and so we constructed Scheme 3 models to represent the possible closed channel conformations in situations where the two VS steps occur between closed channel states, but occupancy of a single open state remains contingent on a transition that occurs after the concerted movement of all available subunits. We found two significant mismatches between experimental observations and model simulations using Scheme 3 models which meant that we did not pursue these models further. The first was that the voltage dependence of the F2-V fluorescence component was no longer tied to channel gate opening and therefore to the G-V, so that in most simulation results, the F2-V no longer overlayed with the voltage dependence of the G-V (Fig. 7). This was a significant problem, as in all published data, F2 in pseudo-wt channels or in the presence of mutations like F351A, F232A, or E160R, tracks closely with the G-V (Osteen et al., 2010; Barro-Soria et al., 2014, 2017; Zaydman et al., 2014; Westhoff et al., 2019; Wu et al., 2021). The second was that the hyperpolarization of the isochronal G-V with four versus two VSs was still not observed which meant that one of our key experimental findings could not be simulated by any of the Scheme 2 or 3 models.

At very positive potentials in the Schemes 1, 2, and 3 models which possess a single open state, the current relaxation time constants during the exponential phase approximate to 1/α or 1/δ. In contrast, in the Schemes 4 and 5 models, the channels undergo rate-limiting conformational changes from multiple closed to open states, which are allosterically regulated. The result is that if horizontal transitions equilibrate rapidly, the maximum current relaxation rate will remain faster even at very positive potentials in channels with more activatable VSs, in the ratio of D⁴:D² for a four or two subunit channel, respectively. This was observed experimentally (Fig. 3 B), and the ratios of the τ_act-V between wt, EQ*Q+E1, and EQ*QQ*Q*+E1 allowed us to calculate a value of 1.357 for the allosteric factor (D). The limiting slope experiments (Fig. 8) provide additional support for models with multiple open states, as the deviation of the Po-V curve from linearity at negative voltages is consistent with access of channels to open states from ground closed states and those only part way along the horizontal activation pathways of Schemes 4 and 5 (Fig. 11). Using 1.357 in the model, with equal conductance of early and final open states (O_RRRR, O_RRRA, and O_AAAA) to generate the Po-V curve, gave the closest approximation to the Po-V recorded experimentally (Fig. 10 D), and there was no indication that D was going to be a much higher value. Allosteric factors have been calculated from fits to data from Shaker channels (k = 7; McCormack et al., 1994), and BK channels in the absence of Ca²⁺, where D had a value of 17 (Horrigan et al., 1999) - much higher than our value of 1.357. In HCN channels, the allosteric factor was fixed as a fraction (0.2) of the ratio of reverse:forward rates, and did not depend on the number of “willing” VSs (Altomare et al., 2001). In KCNQ1 channels without KCNE1, values of the allosteric factor for different mutants varied between 2.04 and 6.08 (Osteen et al., 2012), values much closer to ours.

Scheme 4 model simulations with D = 1.357 fit almost all of our experimental data well (Figs. 9 and 10), surprisingly so given the simplicity of this model and the fact that both isochronal and transient kinetic data span a wide range of voltages and two experimental models. A relatively low value for the allosteric factor might suggest that conduction through intermediate open states along the activation pathway would be significant. However, this tendency for opening from intermediate closed states is counteracted by the relatively rapid F1 kinetics so that channels may activate preferentially across models from left to right and then vertically, rather than in a mixed manner (Schemes 4 and 5; and Fig. S5).

The voltage-dependence of the G-Vs and F-Vs for both psQ+E1 and psQQ*+E1 and the magnitudes of the G-V and F-V shifts with different numbers of E160R mutant VSs could be fit in Scheme 4 models after optimization starting from the experimentally obtained rate constants from psQ+E1 (Fig. 9 and Table 2, row 9). It was interesting to note that the foot of the model G-V relationships for psQ+E1 and psQQ*+E1 remain relatively superimposed as seen in G-V data from EQ, EQ*Q+E1, and EQ*QQ*Q*+E1 expressed in tsA201 cells (Fig. 1 B), and this is likely due to the fact that the closed to open transitions for the two models are governed by identical rates for potentials to about −50 mV. While the isochronal G-V and F-V were well modeled, the transient kinetics were again too fast. After optimization to experimental values, the current relaxation time constants, τ_act, showed a good agreement with tsA201 experimental data, and like the experiments, the model curves did not converge for the different constructs at potentials up to +300 mV (Fig. 10 A), and they also showed a crossover at negative potentials to explain the speeding of current tails in EQ*Q+E1 and EQ*QQ*Q*+E1 versus wt. The Δt activation delays were still not well reproduced in this model (Fig. 10 B), and remained about half the experimental values, which depressed the Δt versus τ_act relationships (Fig. 10 C, solid lines). Changing the model into one with a single conducting open state reached only when all VSs are activated, analogous to Scheme 2 models, did increase the delay, and the Scheme 4 A curve trended up toward the EQ data.

At least one other VS transition is required to adequately simulate the transient activation kinetics. Scheme 5 models (Fig. 11) were built to improve the transient kinetic simulations while preserving the ability to accurately model the steady-state kinetics of I_Ks channel currents. As for Scheme 4, the Scheme 5 models heavily relied on rates optimized from experimental F1 fluorescence kinetics to model the horizontal resting-closed to intermediate-closed transitions, and F2 fluorescence kinetics to model the activated-closed to activated-open transitions. A second VS movement was interpolated between these two movements to model intermediate-closed to activated-closed transitions. The concept of intermediate- and activated-closed states was introduced along with the idea of two VS transitions during I_Ks activation (Zaydman et al., 2014). The optimized model rates solved the transient kinetics problem and preserved the ability of allosteric models to reproduce the isochronal G-V and F-V kinetics (Fig. 12). We have no experimental evidence to support a slow VS transition. But fluorescence recordings that would reveal such a slow VS movement have never been made, to our knowledge, for the reasons discussed in the Results. Such transients would probably be small and very slow, and any attempt to record these would face significant photobleaching issues. The unified Scheme 5 rates (Table 2, row 10) applied to Scheme 5, B and C, models did not reproduce the degree of downward displacement of the Δt delay kinetics in Scheme 5 C compared with Scheme 5 A (Fig. S6) that was seen experimentally (Fig. 3 D). In the Scheme 5 models, the conductance of all the viable open states (colored purple in Fig. 11) was set to be the same, for simplicity, but this has the effect that in all the models exponential current development can occur as soon as they access open states in the activation pathway. Experimentally, it has been shown that these open states are likely of different conductance since EQ, EQ*Q+E1, and EQ*QQ*Q*+E1 show progressively diminished peak single-channel amplitudes (Westhoff et al., 2019). Indeed, reducing the peak conductance of the upper tier open states earlier in the activation pathway of Scheme 5 models does separate the Δt delay curves of the Scheme 5 models (not shown), but fully developed models of this type are beyond the scope of our study, which attempts to simulate whole cell/oocyte rather than single channel kinetic data. The unified Scheme 5 rates, which were based on optimization of EQ and psQ+E1 data, fitted the wt experimental activation kinetics extremely well and also fitted much of the kinetic data from EQ*Q+E1, psQQ*+E1, and EQ*QQ*Q*+E1 constructs as well (Fig. 12). Our solution to the problem discussed above is to provide optimized rates specifically for the EQ*Q+E1 (Scheme 5 B) and EQ*QQ*Q*+E1 (Scheme 5 C) models (Table 2, rows 11 and 12) that are able to fully simulate all the current kinetics seen experimentally in these constructs (Fig. 13).

The nature of the allosteric switching between the activated-closed and activated-open conducting channel pore in the presence of individual activated VS in KCNQ1:KCNE1 channels is likely to be extremely complex (reviewed in Wang et al., 2020), and understanding VS–PD coupling is generally beyond the experimental and modeling studies presented here. The reader is directed to experimental studies that examine VS–PD coupling in some detail (Zaydman et al., 2013; Zaydman and Cui, 2014) and models of the VS–PD coupling in KCNQ1 channels (Zaydman et al., 2014; Westhoff et al., 2019) for a comprehensive analysis and model usage. Recent structural data suggest that perhaps the VS may not have evolved in KCNQ1 to open the channel but rather to impart voltage-dependent access to PIP₂ (Mandala and MacKinnon 2023). In a manner analogous to many multimeric proteins, acting as a switch between two parallel conformational pathways (Ackers et al., 1992), PIP₂ binding might then act to displace calmodulin (CaM) from the S2–S3 linker and allow the opening of the channel (Sun and MacKinnon, 2020). As PIP₂ is thought to act after VS activation, it is not surprising that depletion of PIP₂ does not affect the VS, with no apparent change in the F-V of I_Ks, although currents are eliminated (Barro-Soria et al., 2017). On the other hand, the G-V has been shown to be affected by increases in PIP₂ concentration, with the V_1/2 shifting to more negative potentials upon patch excision into PIP₂-containing bath solutions (Loussouarn et al., 2003; Li et al., 2011) and a concurrent slowing of deactivation. This G-V shift suggests that not all channels are saturated with PIP₂ under typical recording conditions and that when the channel is saturated with PIP₂, channel opening is easier. This is in contrast to when E160R-containing VSs are unable to activate and the V_1/2 of the G-V shifts to more depolarized voltages with an increasing number of E160R mutations in the channel complex (Fig. 1 B). We know that this shift takes place even in the absence of KCNE1 (Westhoff et al., 2019), so it is unrelated to the E1 subunit and preferentially affects the V_1/2 of the G-V and F2-V relationships and the relative amplitude of the F2 compared with the F1 component of the F-V (Fig. 4 B). The preservation of F2 may reflect VS–pore interactions that change the environment of the fluorophore attached at 219C even in VSs that contain E160R mutations. This would make sense in light of compounds like UCL 2077 that prevent channel opening and abolish F2 (Barro-Soria et al., 2014), but is not supported by the observation that depletion of PIP₂ using the voltage-sensitive phosphatase from Ciona intestinalis prevents pore opening but does not affect either component of the F-V (Barro-Soria et al., 2017).

Morphing the new low-resolution KCNQ1 resting-state structural model (PDB ID 8sin; Mandala and MacKinnon, 2023) into the activated-open-state structure of KCNQ1/KCNE3/PIP2 (PDB ID 6v01; Sun and MacKinnon, 2020) highlights the rearrangements that take place within the cytoplasmic C-terminus of the complex. To open, the S6 transmembrane domain becomes continuous with the helix A portion of the C-terminal domain of the channel, and as it splays open, CaM rotates around the helix A–helix B segment into the space where a CaM on a neighboring subunit would reside in the closed state. In E160R-containing VSs where activation is prevented, the mutation may create a steric hindrance to the opening where a moving CaM would bump into a static CaM and make the opening transition more energetically difficult. This would explain the depolarizing V_1/2 shift in E160R-containing complexes and in those not fully saturated with PIP₂. In contrast, a hyperpolarizing V_1/2 shift occurs when the pore is propped open by an increasing number of L353K mutations in the channel complex (Meisel et al., 2012), where presumably CaM molecules bound to mutated subunits are already out of the way of those remaining subunits that are starting from a closed pore configuration. Not surprisingly, several mutations in CaM linked to long QT-interval syndrome are also known to shift the V_1/2 of I_Ks to more positive potentials (McCormick et al., 2023).

Having examined both sequential and allosteric models for I_Ks gating, we conclude that only the allosteric models, Schemes 4 and 5, are able to faithfully reproduce the experimentally observed gating behavior, including the Po-V upturn at very low Po values and negative voltages, the depolarization of the EQ*Q+E1 and EQ*QQ*Q*+E1 G-V relationships relative to wt, the relative changes in deactivation rates in the different constructs, and the concordance of the G-V and F2-V relationships from psQ+E1 and psQQ*+E1 oocytes. Sequential Schemes 1, 2, and 3 with and without a concerted opening transition are qualitatively unable to reproduce these phenomena. Our Scheme 5 models present an accurate depiction of I_Ks VS activation and current behavior as, using a single set of rate constants, the models are able to reproduce both transient and quasi steady-state experimental kinetics from mammalian cells and oocytes in models with various numbers of active VSs. It appears that I_Ks is yet another channel whose gating is consistent with rows (or tiers) of open and closed states, like the Ca²⁺-activated K⁺ channel (Cox et al., 1997; Rothberg and Magleby, 1998, 2000; Horrigan et al., 1999; Cui and Aldrich, 2000), ACh receptor isomerization (Auerbach, 2010), the HCN channel (Altomare et al., 2001), and indeed KCNQ1 itself (Ma et al., 2011; Osteen et al., 2012; Zaydman et al., 2014).

Data and the code for the final tiered allosteric model used in the manuscript to generate tables and figures have been deposited in a publicly accessible database at https://doi.org/10.5281/zenodo.10421153.

Crina M. Nimigean served as editor.

We thank Dr. Jianmin Cui for the C214A/G219C/C331A Q1 construct and Dr. Yoshihiro Kubo for the pGEMHE vector. We thank Yundi Wang and David Chen for their help with preliminary experiments, Fariba Ataei for her technical assistance, and members of the Fedida lab, Drs. Peter Backx and Khanh Dao-Duc for valuable discussion.

This research was funded by the Natural Sciences and Engineering Research Council of Canada (grant #RGPIN-2022-03021), Canadian Institutes of Health Research (#PJT-175024), and Heart and Stroke Foundation of Canada (#G-21-0031566) grants to D. Fedida.

Author contributions: D. Fedida: all MATLAB modeling, tsA201 cell data collection, data analysis, figure preparation, writing of paper draft, revisions, and final version of manuscript. D. Sastre: tsA201 cell and oocyte data collection, data analysis, figure preparation, and review of the final version of the manuscript. M. Westhoff: tsA201 cell and oocyte data collection, review of the final version of the manuscript. Y. Dou: tsa201 cell and oocyte data collection, review of the final version of the manuscript. J. Eldstrom: tsA201 and mouse ltk- cell data collection, data analysis, writing and review of the draft paper, and review of the final version of the manuscript.