# Emergence of ion channel modal gating from independent subunit kinetics

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Edited by Richard W. Aldrich, The University of Texas at Austin, Austin, TX, and approved July 13, 2016 (received for review March 14, 2016)

## Significance

Many key features of the behavior of cells are controlled by ion channels—pores in cell membranes that are sometimes open and sometimes closed. It is therefore critically important to understand what controls these opening and closing events. This is challenging because ion channels exhibit stochastic dynamics over several timescales, from the rapid kinetics of a single opening to the slow switching between distinct levels of activity known as modal gating. By mathematically modeling the basic biophysical events that control ion channel opening, we introduce a new principle for understanding the origin of modal gating. Although we focus on the inositol 1,4,5-trisphosphate receptor channel, the framework can be applied more generally to other ion channels.

## Abstract

Many ion channels exhibit a slow stochastic switching between distinct modes of gating activity. This feature of channel behavior has pronounced implications for the dynamics of ionic currents and the signaling pathways that they regulate. A canonical example is the inositol 1,4,5-trisphosphate receptor (IP_{3}R) channel, whose regulation of intracellular Ca^{2+} concentration is essential for numerous cellular processes. However, the underlying biophysical mechanisms that give rise to modal gating in this and most other channels remain unknown. Although ion channels are composed of protein subunits, previous mathematical models of modal gating are coarse grained at the level of whole-channel states, limiting further dialogue between theory and experiment. Here we propose an origin for modal gating, by modeling the kinetics of ligand binding and conformational change in the IP_{3}R at the subunit level. We find good agreement with experimental data over a wide range of ligand concentrations, accounting for equilibrium channel properties, transient responses to changing ligand conditions, and modal gating statistics. We show how this can be understood within a simple analytical framework and confirm our results with stochastic simulations. The model assumes that channel subunits are independent, demonstrating that cooperative binding or concerted conformational changes are not required for modal gating. Moreover, the model embodies a generally applicable principle: If a timescale separation exists in the kinetics of individual subunits, then modal gating can arise as an emergent property of channel behavior.

The regulation of cytosolic Ca^{2+} ion concentration is fundamental to a wide range of cellular processes, including immune responsivity (1), synaptic plasticity (2), axon guidance (3), and apoptosis (4). Several processes contribute to the spatial and temporal dynamics of Ca^{2+}, such as diffusion and buffering, exchange with the extracellular space, and uptake and release from intracellular stores. The inositol 1,4,5-trisphosphate receptor (IP_{3}R) ion channel is a key component in shaping Ca^{2+} signals, as it controls the local efflux from the endoplasmic reticulum (ER), where Ca^{2+} is sequestered at high concentration (5, 6). The IP_{3}R is a ligand-gated channel, subject to regulation by binding of IP_{3} and also Ca^{2+} itself. Recently, it has been revealed that the main method of ligand regulation is to affect a slow switching between distinct levels of channel activity—a phenomenon known as modal gating (7⇓–9). IP_{3}R modal gating has pronounced implications for the dynamics of Ca^{2+} release events (10), and its dysfunction has been implicated in the pathogenesis of familial Alzheimer’s disease (11, 12). A detailed understanding of this important feature of IP_{3}R behavior is therefore crucial for unraveling the complexity of Ca^{2+} signaling and its role in cell function and disease. Modal gating has been observed in the kinetics of many other ion channels, such as K^{+} (13⇓⇓⇓⇓–18), Cl^{−} (19), glutamate receptors (20, 21), plasma membrane Ca^{2+} (22⇓–24), and ryanodine receptor Ca^{2+} (25, 26). However, the underlying biophysical basis of modal gating in the IP_{3}R and most other channels remains unknown.

Structurally, IP_{3}Rs are large, homotetrameric proteins of which there are three main subtypes. Binding of IP_{3} to a cytosolic domain contributes to channel activation via a conformational change in channel subunits. However, the location and action of Ca^{2+} binding sites (believed to be at least two per subunit) are unresolved (27). Here, we focus on type 1 IP_{3}Rs, which are the main neuronal subtype and for which a range of single-channel kinetic properties have been determined from within the same cell type (Sf9) (7, 28, 29). Under fixed ligand conditions, the type 1 IP_{3}R gates in three modes that are characterized by high, intermediate, and low open probability. The within-mode open probabilities are approximately ligand independent, whereas the proportion of time spent in each mode is regulated by the IP_{3} and Ca^{2+} concentration (7, 30), leading to the well-described bell-shaped open probability curve (28, 31, 32). In contrast to the slow switching between modes, the channel exhibits individual opening and closing events of millisecond durations and responds rapidly to changing ligand concentrations (29).

The challenge in developing a complete biophysical understanding of the IP_{3}R is to bridge the gap between the microscopic picture that is evolving from molecular studies and the macroscopically observable statistics of channel gating. Markov models of channel gating are an excellent tool for approaching this problem in quantitative terms. It is clear that any description of the IP_{3}R must encompass stochastic dynamics over several timescales and ideally be relatable to the underlying biology. Stochastic implementations of the classic De Young–Keizer (DYK) model assume four independent subunits and explicitly incorporate ligand binding and conformational change (33, 34). Variants of this scheme have been widely used in studies of the IP_{3}R and Ca^{2+} dynamics (35⇓⇓⇓⇓⇓⇓⇓⇓⇓–45), although none exhibit modal gating. This raises the question of whether this intuitive, bottom–up approach is compatible with current knowledge of IP_{3}R regulation. Whereas two recent top–down IP_{3}R models were able to fit this feature of the data (46, 47), the coarse graining of channel states precludes predictions as to its origin at finer spatial scales.

Here, we show how modal gating can emerge from subunit-based models of ion channel gating. We propose that modal gating in the IP_{3}R is a consequence of a timescale separation in the kinetics of individual subunits. Motivated by the DYK approach, we first introduce two simple motifs for subunit kinetics, where the first one gives rise to channel bursting, and the second one exhibits slow modal gating as an emergent statistical property. We show that the subunit kinetics induce a natural partition of the full channel state space that underlies three distinct modes of activity. We then use these ideas as a basis for constructing a stochastic model of the type 1 IP_{3}R, comprehensively fitted to both equilibrium and transient kinetic data. The result is a bottom–up model that transparently describes all aspects of gating behavior in terms of elementary binding events and conformational changes. In this way we show that minimal coupling is required between the kinetics of IP_{3}R subunits to account for the full complexity of gating patterns. Although we use the IP_{3}R as a model problem, the general principle may be applicable to other ligand-regulated channels.

## Results

We model a channel subunit of the IP_{3}R as a continuous-time Markov chain with a discrete state space corresponding to binding occupancy of ligand molecules and protein conformation. We assume that each of the four subunits that compose the channel are identical and independent and have a single active state that becomes accessible after ligand binding. Previous DYK schemes assume that the channel opens when at least three of four subunits are active (35⇓⇓⇓⇓⇓⇓⇓⇓⇓–45). This yields a channel with multiple open states, consistent with experimental observations (37). However, by itself this rule is inconsistent with the recent discovery that channel opening requires all subunits to be bound by IP_{3} (48). Therefore, we impose the additional constraint that the channel must be fully occupied by activating ligands to open. We assume further that binding of Ca^{2+} to the inhibitory site of a single subunit is sufficient to close the channel. The stoichiometry of Ca^{2+} regulation of the IP_{3}R is unknown; however, we found that these assumptions were the most consistent with observed features of channel gating. We return to address the implications and interpretation of these assumptions in *Discussion*.

We denote the states of a subunit Markov chain by *Q*. Detailed balance is assumed and enforced with Kolmorogov’s criterion (that the products of forward and reverse rates around all cycles are equal). The equilibrium distribution of subunit state is denoted by the vector *w*, given by the solution to

### Bursting Motif.

We first consider a four-state motif for subunit kinetics in the presence of a saturating IP_{3} concentration. This motif is a subset of the 10-state model structure of ref. 37, which we have simplified by removing IP_{3} dependence and assuming sequential binding of activating and inhibitory Ca^{2+}. This simple description captures the essential features of Ca^{2+} regulation and bursting behavior. The diagram for the Markov chain is shown in Fig. 1. State transitions corresponding to a change in binding occupancy are described by mass action kinetics with rate constants *c*, the cytosolic Ca^{2+} concentration. A subunit in state ^{2+} site and an unoccupied inhibitory site, can undergo a ligand-independent conformational change to the active state ^{2+} dependence of the open probability curve arises because the inhibitory site is of much lower affinity than the activating site and therefore suppresses activity only when the concentration is sufficiently high.

Following ref. 37, we calculate the equilibrium open probability (Eq. **2**) by considering the unnormalized probabilities **2** and **4** that have been predominantly used to fit previous subunit-based models of the IP_{3}R channel. We now decompose these further to isolate the contributions from bursts and gaps.

### Timescale Separation.

In their seminal work on the aggregated Markov model approach to channel gating, the authors of ref. 49 demonstrated that rich kinetic behavior could be attributed to a timescale separation that partitions the channel into open states and both short- and long-lived closed states. We consider an analogous principle, although we apply it at the level of a channel subunit. We define the set *c*, although this extends to include *c* becomes large (

In this motif, H and L are simply the regions of the state space associated with channel bursts and burst-terminating gaps, respectively. For *c* not too large, we have*π* of the channel being within either H or L as

The kinetics within bursts are mostly determined by the rates of fast conformational change *c* that **7**, but as **4** (given the parameters in Table S1,

The dynamic switching between H and L is controlled instead by the ligand-dependent transitions. If we consider the states in

This analysis suggests that the essential ingredients for modal gating are present in this simple motif. However, it cannot account for the slow regulation of IP_{3}R modes characterized by ref. 7, where, for example, L-mode dwell times are on the order of seconds even in the high

### Modal Gating Motif.

The ligand-independent conformational change that we have included in the previous motif is supported by experimental observations of channel flickering and reduced sensitivity to Ca^{2+} during bursts (37). We make a simple extension to the bursting motif by assuming that there exists a protein conformation that suppresses this activating step (Fig. 2). Modal gating of the channel emerges if suppression, by any Ca^{2+}-independent mechanism such as phosphorylation or binding of accessory proteins, occurs on a slow timescale.

The modal gating motif illustrates a key principle. There are three timescales present in the subunit kinetics: fast conformational change to an active state (

We scale the rates of activation and inhibition to retain approximately the same total open probability curve as the bursting motif (Fig. 3*A*). Equilibrium properties are calculated as before, in this case with the normalization factor

This motif generates an additional open channel configuration to the two that mediate H-mode openings in the previous model. In this configuration, where a single subunit is sequestered in state *c* is large. Taking the slow sequestration into account, the channel state space is partitioned into three subsets associated with H, intermediate (I), and L modes of activity:

These are occupied with ligand-dependent probabilities**8**. This result gives the decomposition of the total channel open probability**2**. As the open probabilities within each mode are independent of ligand concentration (Fig. 3*B*, with these parameters, *C*).

The mean open time (Eq. **4**) can be decomposed similarly by conditioning on the state in which opening begins (*SI Materials and Methods*). For I-mode openings, we find *D*).

### Simulations.

Gillespie simulations of channel gating exhibit the main qualitative features of experimental current traces (Fig. 4). At subactivating (

The signature of this slow regulation is seen in the autocorrelation function of the binary (open/closed) channel state, which we computed from a ^{2+} concentration (Fig. S2). This result shows that correlations in channel activity persist for seconds. This is because the probability of the channel state is biased by the underlying gating mode, leading to correlated activity on the timescale of mode switches. For comparison, we removed the slowest timescale by scaling up the rates ^{2+} binding event.

This general motif provides a basis by which modal gating can be understood at the level of subunit kinetics. The behavior of individual subunits combines to yield a channel that stochastically switches between three interconnected modes. Channel bursts and gaps are produced by ligand-binding events at the activating and inhibitory sites. Slow suppression of a ligand-independent activation step means that at any point the channel may transition into an intermediate mode or longer-lived periods of quiescence. We now extend this to a full model of the IP_{3}R channel, including IP_{3} binding and fitting parameters to stationary and transient kinetic data.

### Full Model.

Similar to the objectives of ref. 47, we aim to construct a gating model of type 1 IP_{3}R, regulated by both Ca^{2+} and IP_{3}, that can account for the following: equilibrium open probability (_{3} binding (with concentration denoted *I*) in a similar way to that of previous DYK schemes, renumbering rate constants accordingly (Fig. 6). We limit IP_{3} binding to the nonsequestered, ligand-dependent component of the modal gating motif. The corresponding biophysical assumption is that the protein conformation that suppresses activation becomes available only after the change in structure resulting from IP_{3} binding. We continue to assume sequential binding of activating and inhibitory Ca^{2+}, as has been suggested previously as a simplification of the DYK scheme (40). With these simplifying assumptions, we require only three additional states to be added to the modal gating motif and can account for almost all of the experimental data.

The normalization factor for this model is given by**2** with **4**).

### Model Fitting.

We perform a heuristic fit of the model to data taken from refs. 28 and 29, which was recorded from single type 1 IP_{3}Rs in native nuclear membrane of Sf9 insect cells. We use the equilibrium open probability to determine the dissociation constants for the model (Fig. 7*A*) and the mean open and closed times to choose the rates *B* and *C*). We use the remaining parameters to shape the transient dynamics of the system. To keep the number of parameters small we began by assuming the same symmetries in binding rates as in the modal gating motif and as in previous DYK schemes for the additional component. However, consideration of activation latencies led us to increase the rates between _{3} and Ca^{2+} activation. Thus, in contrast to previous schemes, binding of IP_{3} results in an increase in the activating Ca^{2+} binding and dissociation rates (and vice versa).

Ref. 29 characterized the transient behavior of the channel by recording the response of the IP_{3}R to rapid step changes in Ca^{2+} and IP_{3} concentration. We fitted the model by comparing the experimental activation and recovery latency distributions to first passage times of the model. We calculate the cumulative distribution function (cdf) for the time to reach the set of open states from the initial condition given by the equilibrium distribution at the initial concentrations used in the experiments (*Materials and Methods*). We make an exception for the condition with initial concentrations Ca^{2+} _{3}

Even with a simple, heuristic fit, the model structure permits good agreement with the data. The model accounts for the three peaks in the latency histogram when activated from a condition of optimal Ca^{2+} _{3} *A*). This is a very interesting feature of the experimental data as it demonstrates that there are channel activation pathways on timescales spanning three orders of magnitude. In the context of the model, it is this timescale separation that leads to modal gating. The model Ca^{2+} activation latencies in saturating IP_{3} (Fig. 8*B*) exhibit the same peak probability as the experimental data at _{3} and Ca^{2+} (Fig. 8*C*) also captures very well the primary component of the experimental latencies. The smaller component may be recovered by adding additional states to the model (e.g., more components of the original DYK scheme); however, as this has minimal influence on overall channel dynamics, we choose to keep the simpler model structure. Finally, the model captures the very long recovery latencies (mean ^{2+} concentration (Fig. 8*D*). The slow recovery was posited by ref. 29 to account for the similar refractory period observed between Ca^{2+} puffs in vivo.

Ref. 29 also reported deactivation and inhibition latencies, defined as the time taken for the channel to enter a closed state for a duration greater than _{3} deactivation is very good, and although there is a discrepancy in the overall shape of the Ca^{2+} deactivation histograms, the model mean latency of _{3}) replicates the value of ^{2+} and IP_{3} yielded a longer mean deactivation latency for the model of ^{2+} concentration is increased to

### Modal Gating Analysis.

We consider the modal gating properties of the model at saturating and subactivating IP_{3} concentrations of *c* threshold for inclusion of additional states in _{3} is similar, although we omit _{3} binding.

The decomposition of channel open probability (Fig. 9*A*) demonstrates the regulation of the channel by both of its ligands. The within-mode conditional open probabilities are independent of both Ca^{2+} and IP_{3} (Fig. 9*B*). As with the simpler motifs above, the proportion of time spent in each mode (Fig. 9*C*) can be understood very simply as a biasing of individual subunits toward _{3} concentration is low, there is much less probability for subunits to reside in

To assess our theoretical results we apply the algorithm of ref. 7 to segment simulated traces into underlying gating modes. In ref. 7, experimental IP_{3}R traces were burst filtered to ignore any closed durations of less than a threshold

We simulated long traces (^{2+} concentrations. We compared the time series of gating modes from segmentation to the time series of modes given by occupancy of H, I, and L (both resampled to ^{2+} concentrations and robust to the choice of filtering parameter in the range

We calculated the modal gating statistics from the segmented traces (we use notation ^{2+} concentrations of 0.1 μM, 1 μM, and _{3} and ^{2+} at _{3}. At saturating levels of IP_{3}, channel activity can be clearly separated into three distinct modes with approximately ligand-independent open probabilities (Fig. 10*B*). The proportion of time spent in each mode is regulated by the Ca^{2+} concentration; H mode and L mode display an inverse dependence and predominate over the contribution from I mode (Fig. 10*C*). L mode exhibits long dwell times at all concentrations, whereas H-mode dwell times peak in the optimal regime and are otherwise of a similar duration to that of I mode (Fig. 10*D*).

The main quantitative difference between the model and experimental data is the I-mode open probability of

Overall, the results suggest that the modal gating properties observable in traces of the binary channel state can indeed be ascribed to the underlying partitioning of the state space we have described. The Ca^{2+} dependence of modal gating revealed by segmentation at subactivating IP_{3} (Fig. S6) also agrees with our theoretical results and predicts that within-mode open probabilities remain largely independent of both Ca^{2+} and IP_{3} concentration.

## Discussion

We have shown how ion channel modal gating can be understood at the level of subunit kinetics. We first considered two simple motifs for stochastic kinetics, showing how three distinct modes of channel activity can be understood in terms of a natural partition of the channel state space. From this we constructed a detailed model of the type 1 IP_{3}R that accounts for equilibrium channel properties, transient response kinetics, and modal gating. The model demonstrates that coupling of ligand binding and conformational change between subunits is not necessary for persistent time correlations in channel activity. Instead, modal gating is an emergent property that arises from a timescale separation in subunit kinetics.

### Model Assumptions.

As in previous subunit-based IP_{3}R models, we have assumed that at least three of four subunits must enter an active state for channel opening. However, we differ by considering separately the role of binding occupancy as a permissive factor. Whereas the necessity of full occupancy by IP_{3} that we have included in the model has been established experimentally (48), the location of Ca^{2+} binding sites and stoichiometry of regulation are not yet known. In our model, each subunit must have an occupied activating Ca^{2+} site and an unoccupied inhibitory site for the channel to open. Although the steady-state channel properties can be suitably captured by the model if this assumption is relaxed, its enforcement accounts for several additional features of dynamic behavior.

Single-channel recordings at low Ca^{2+} and saturating IP_{3} show abrupt and frequent switching between long channel closures and high activity bursts (7) (as in Fig. 4*A*). If opening were permissible in the model with only three occupied activating sites, then transitions in and out of bursts at low Ca^{2+} would instead exhibit short segments with I-mode kinetics. In the same way, channel closure from a single inhibited subunit in our model yields the isolated, high-activity bursts observable at high Ca^{2+} concentrations. As direct transitions between H and L are possible in the model from a single binding event, all three modes are completely interconnected, as described by ref. 7.

The updated rule for inhibition also addresses an issue that would otherwise likely arise from the fast conformational change our model shares with ref. 37. Although this component permitted ref. 37 an impressive fit to single-channel data, detailed simulations with dynamic Ca^{2+} feedback showed the time to termination of calcium puffs to be unreasonably long (42, 43). This is because the model requires the high Ca^{2+} concentration associated with an open channel pore to become inhibited, but can do so only from a closed state where the concentration quickly collapses to resting levels (34). Our model, however, which accounts accurately for both steady-state and transient data, inhibits directly from an open state. This should allow for appropriate puff termination by self-inhibition, as in the computational studies of refs. 41 and 50. Therefore, we argue that within this general framework the conditions we impose on Ca^{2+} regulation are the most consistent with IP_{3}R behavior and stand as a testable prediction of the model.

We interpret the separation of ligand-dependent and ligand-independent requirements for channel opening by analogy with the proposed gating mechanism for the ATP-sensitive K^{+} channel (K_{ATP}). Conduction by the K_{ATP} is considered to be controlled by two gates: a slow, ligand-dependent gate formed by constriction of the pore and a fast, ligand-independent gate associated with the selectivity filter (51). Ligand regulation determines the duration of bursts and gaps, with minimal effect on intraburst kinetics. Conversely, point mutations near the K_{ATP} selectivity filter alter the kinetics within bursts, but not the burst or gap durations (52). A similar mechanism has been suggested previously for the IP_{3}R to explain the upper bound on ^{2+}. Thus, in our model, all subunits can be understood as necessary for opening a ligand-dependent gate, whereas three are sufficient to allow passage through a fast gate that controls the kinetics within bursts.

Modal gating emerges in the model from the interplay of the “3 out of 4” rule with a slow transition that leads to a set of sequestered states. Several consistent possibilities have been suggested to explain modal gating in other channels, such as phosphorylation (16, 17), binding of accessory proteins (13⇓–15, 17, 20, 22, 24), ligand-independent conformational changes (13, 19), or alterations at the channel pore (18). Our model demonstrates that, in general, these mechanisms can act locally, by influencing even a single subunit, and that this is sufficient to generate the many distinct channel states assumed in whole-channel models.

Although the structure of the IP_{3}R is not known in sufficient detail to confirm or rule out such a mechanism, with the interpretation above, the model is testable at a more macroscopic level. Several mutations near the selectivity filter have been shown to inactivate the type 1 IP_{3}R (55). If the methods of ref. 48, which constructed channels with mutations in the IP_{3} binding domain of a known number of subunits, can be extended to the mutations identified by ref. 55, then the model makes strong, parameter-free predictions of channel behavior. With one defective subunit, the model predicts a channel that gates only in I and L modes, given that the H mode requires four subunits. Similarly, with two defective subunits the model predicts that the channel gates only in L mode. More generally, a role for the selectivity filter suggests that IP_{3}R modal gating is subject to regulation from the luminal side of the ER membrane.

### Role of IP_{3}R Modal Gating in Alzheimer’s Disease.

The enhancement of IP_{3}R activity by mutant presenilins is a key contributor to the pathogenesis of familial Alzheimer’s disease. This is attributed to an increase in the prevalence of H mode, at the expense of L mode (12), leading to a disruption of Ca^{2+} homeostasis. This was recently examined using the whole-channel model of ref. 47. It was concluded that the mutation confers increased sensitivity of the channel to IP_{3}, reflected in a change in occupancy of two particular aggregated states of the model (56). We cannot relate this result directly to our model, as although the model of ref. 47 incorporates putative ligand binding, it allows for a maximum of only three bound Ca^{2+} ions over the whole tetrameric channel. It is therefore unclear how the various aggregated states can be understood in terms of the underlying subunits of our model. Instead, we argue more directly from our model that interaction with mutant presenilins facilitates the IP_{3} binding reaction.

The key evidence for this result is that the increase in channel _{3} concentration (11). Furthermore, experimental traces in ref. 12 show that channels exhibiting high ^{2+} activation or relief of inhibition when IP_{3} is bound is also unlikely. At the Ca^{2+} concentration of

This leaves the possibility of either facilitation of IP_{3} binding or relief of Ca^{2+} inhibition when IP_{3} is unbound. We favor the former as a more direct mechanism and because it gives the more pronounced increase in _{3} binding serves a dual purpose as it biases subunits more strongly toward the active state and allows more subunits to participate in bursts (i.e., _{3} concentration). In terms of the equilibrium modal gating properties (Fig. 9), increasing the rate of IP_{3} binding is equivalent to increasing the IP_{3} concentration. Thus, the curves for the two IP_{3} concentrations plotted in Fig. 9 demonstrate the effect of such a perturbation at subsaturating IP_{3}: a large increase in

Therefore, disruption of modal gating implicated in familial Alzheimer’s disease does not require alteration of channel dynamics at the quaternary level, but rather just the kinetics of an elementary binding event. The model supports a hypothesis of an allosteric interaction between presenilins and IP_{3}R subunits in the membrane of the ER and suggests a search for sites of interaction near the IP_{3} binding domain. The reasoning here depends only weakly on the parameters of the model and therefore encompasses the wide variety of cell types studied by ref. 12, regardless of differences in finer kinetic detail. Our model provides a useful tool to pursue this issue in concert with molecular studies and to explore further questions such as the potential connection to IP_{3} sensitization by PKA (53) and cAMP (57).

### Application to Other Ion Channels.

Although modal gating is a ubiquitous feature of ion channel dynamics, it is unlikely that it has a universal origin (21). However, where there is strong evidence of a subunit-based mechanism, a compatible modeling approach will allow greater synergy between theory and experiment and serve to highlight commonalities. We discuss two such examples: G-protein–coupled inwardly rectifying potassium (GIRK) channels and large conductance calcium-activated potassium (BK) channels.

A subunit model can unify the mode switching behavior in GIRK1/4 channels that has been the characterized at two different timescales. Four gating modes of GIRK1/4 channels in atrial myocytes were posited to arise directly from the independent binding of up to four G-protein G_{βγ} subunits to the tetrameric channel (14, 15). At much longer timescales of tens of seconds, GIRK1/4 channels expressed in *Xenopus* oocytes switch between periods of high and low activity even at saturating G_{βγ} (17). A recent whole-channel model was developed to account for the G_{βγ}-dependent switching, although it did not account for the slow regulation (58). However, these are precisely the channel behaviors predicted by the modal gating motif, implemented without the constraint on ligand occupancy we assumed for the IP_{3}R. The ligand-dependent activation step in the motif accounts for the faster G_{βγ}-dependent switching observed by refs. 14 and 15, whereas the slow sequestration incorporates the G_{βγ}-independent regulation observed by ref. 17. An interesting point of difference is that although the GIRK1/4 channel requires three functional subunits for opening (59), a greater number of modes were identified than for the IP_{3}R. This result raises the intriguing possibility that an additional mode is due to the heterotetrameric nature of the channel. If the two subunit types exhibit different kinetics, then two intermediate open probabilities would be expected, each relating to a particular composition of three contributing subunits. As our approach is easily generalized to heteromeric channels, the role of kinetic differences between subunit types can be explored directly in this and other channels.

BK channels in rat skeletal muscle have been found to exhibit four modes when held at constant voltage and Ca^{2+} concentration (13). Moving from the higher to lower open probability modes is associated with a restriction of a common state space and, in particular, a sequential removal of the longest-lived open states. As our decomposition of the IP_{3}R mean open time demonstrated, this is consistent with a model where the gating mode is determined by the number of contributing subunits. This hypothesis for BK channels can be tested with the detailed 50-state model of ref. 60, which characterizes gating in the highest (and most common) activity mode and explicitly represents the kinetics of four identical channel subunits. The hypothesis predicts that sequential removal of the subsets of the model corresponding to the highest number of contributing subunits will yield the gating activity observed in lower modes. A related question is whether the model structure of ref. 60 can be expressed more compactly in a form similar to our full IP_{3}R model (with voltage playing the role of IP_{3} binding). Such a reduction would drastically reduce the number of free parameters (from 210 to

## Materials and Methods

We investigate channel dynamics and perform simulations using the equivalent aggregated model that arises from the independent subunits and channel opening rule. Aggregated model states *k* subunits with *n* subunit states will have a total of

We compute first passage time distributions used for model fitting, using the absorption method. We consider an initial distribution of aggregated states *O*. We denote by *A*. We form the matrix *A* corresponding to *O* have been set to zero. The cdf of first passage times from *O* is then given by the sum over open states of the solution to the Kolmorogov forward equation,

## SI Materials and Methods

### Conditional Mean Open Time.

In the main text we noted for the modal gating motif that the mean open time in H mode is longer than that of I mode by a factor of 2. Here we provide details of the calculation and the expressions plotted in Fig. 3*D* of the main text.

We calculate mean open times by using an equivalent aggregated representation of the model. Because the subunits are assumed to be independent, the channel states can be grouped into complexes based on the total number of subunits populating each subunit state (*Materials and Methods*). The aggregated representation of the modal gating motif has 210 channel states, of which 3 are open. All open states have three active subunits, but are distinguished by the state of the fourth. We use the shorthand

Using standard methods (ref. 61, section 3.3), we find**S3**. This gives **S**2, accounting for the fact that sojourn to

The two quantities given by Eqs. **S3** and **S4** are plotted in Fig. 3*D* of the main text. Weighting these by the conditional probability of being in

## Acknowledgments

We thank Peter Dayan for valuable discussions. This work was supported by an Australian Postgraduate Award (to B.A.B.).

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: g.goodhill{at}uq.edu.au.

Author contributions: B.A.B. and G.J.G. designed research; B.A.B. performed research; and B.A.B. and G.J.G. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1604090113/-/DCSupplemental.

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