Concise Whole-Cell Modeling of BK_Ca-CaV Activity Controlled by Local Coupling and Stoichiometry

Abstract

Large-conductance Ca²⁺-dependent K⁺ (BK_Ca) channels are important regulators of electrical activity. These channels colocalize and form ion channel complexes with voltage-dependent Ca²⁺ (CaV) channels. Recent stochastic simulations of the BK_Ca-CaV complex with 1:1 stoichiometry have given important insight into the local control of BK_Ca channels by fluctuating nanodomains of Ca²⁺. However, such Monte Carlo simulations are computationally expensive, and are therefore not suitable for large-scale simulations of cellular electrical activity. In this work we extend the stochastic model to more realistic BK_Ca-CaV complexes with 1:n stoichiometry, and analyze the single-complex model with Markov chain theory. From the description of a single BK_Ca-CaV complex, using arguments based on timescale analysis, we derive a concise model of whole-cell BK_Ca currents, which can readily be analyzed and inserted into models of cellular electrical activity. We illustrate the usefulness of our results by inserting our BK_Ca description into previously published whole-cell models, and perform simulations of electrical activity in various cell types, which show that BK_Ca-CaV stoichiometry can affect whole-cell behavior substantially. Our work provides a simple formulation for the whole-cell BK_Ca current that respects local interactions in BK_Ca-CaV complexes, and indicates how local-global coupling of ion channels may affect cell behavior.

Introduction

Mathematical modeling has played an important role in investigations of cellular electrophysiology at least since the works on neuronal action-potential generation by Hodgkin and Huxley (1). In the Hodgkin-Huxley model and most of its descendants, the system of ion channels is coupled globally via the membrane potential or the bulk cytosolic Ca²⁺ concentration. However, some ion channels are colocalized, implying that the activity of one channel may affect the other via local control. Electrical activity is thus a result of the complex interactions of local and global coupling of ion channels. Of note, the standard Hodgkin-Huxley formulation does not take into account local coupling of channels.

Large-conductance Ca²⁺- and voltage-dependent K⁺ (BK_Ca, KCa1.1) channels, ubiquitously found in excitable cells where they shape electrical activity (2), provide an example of such ion channels, whose activity is influenced locally by associated voltage-gated Ca²⁺ channels (CaVs). BK_Ca channels have a single-channel conductance of ∼100 pS in physiological conditions (3), and are activated by Ca²⁺ and transmembrane voltage, which is seen as a Ca²⁺-dependent left-shift of the BK_Ca activation curve (4, 5, 6). In neurons (7, 8, 9, 10) and vascular myocytes (11), BK_Ca channels colocalize with CaVs, which exposes the BK_Ca channels to the Ca²⁺ nanodomains below the mouth of the CaV channels (12, 13, 14, 15), where the local Ca²⁺ concentration reaches the tens of micromolar that are required for activating the BK_Ca channels at physiological voltages (2, 16). There is increasing evidence for a direct coupling between BK_Ca and CaV channels, forming BK_Ca-CaV ion channel complexes with a stoichiometry of 1–4 CaV channels per BK_Ca channel (2, 11), and differences in stoichiometry likely affect channel activity. Intuitively, we expect that more CaVs per complex would augment the BK_Ca open probability, both because of higher local Ca²⁺ concentration when the CaVs open simultaneously, and because of greater probability that at least one of the CaVs is open at any given time.

Recently, Cox (17) presented a Markov chain model for a BK_Ca-CaV complex with 1:1 stoichiometry, and performed Monte Carlo simulations that provided important insight into the open probability of BK_Ca channels during depolarizations and action potentials, and how e.g., inactivation of CaVs directly influence BK_Ca channel activity. Such Monte Carlo simulations are computationally intensive and explicit mathematical relations between assumptions and consequences are not available. Monte Carlo simulations have also been performed for whole-cell simulations of electrical activity to investigate the effects of stochastic ion channel kinetics, for example for Ca²⁺-sensitive SK and BK_Ca channels controlled by local Ca²⁺ dynamics (18, 19). When stochasticity is not of interest, to speed up simulations, many models of whole-cell electrical activity that include BK_Ca channels express this current in a simplified way that neglects local effects due to the BK_Ca-CaV complexes (20, 21) or use heuristic expressions involving the whole-cell Ca²⁺ currents (22, 23), which may not respect the dynamics within BK_Ca-CaV complexes. Alternatively, diffusion of Ca²⁺ around a CaV (or a cluster of synchronized CaVs) has been simulated to investigate, e.g., how BK channels inherit properties of the CaVs, and how distance between channels influence BK_Ca activity (10). Another frequent approach (which, however, neglects local interactions) is to model Ca²⁺ dynamics in one or more shells beneath the cell membrane, which then drives BK_Ca channels (24, 25, 26). The computational intensity is increased in such a model because local Ca²⁺ concentrations resulting from buffering and diffusion must be simulated in addition to ion channel gating.

It would therefore be advantageous to have a simple but mechanistically correct model of the BK_Ca current, which respects the local effects of BK_Ca-CaV coupling, and that can be inserted in Hodgkin-Huxley-type models of whole-cell electrical activity. Such a model would also make explicit how local effects and stochastic ion channel kinetics are reflected in average, whole-cell behavior of BK_Ca channels with the advantage compared to simulations that the dependence on parameters can be read directly from the formulas of the reduced model. Here we achieve both these aims. Our approach is similar to analyses of Ca²⁺-dependent inactivation of Ca²⁺ channels (27), and local control of ryanodine receptors in dyadic subspaces (28, 29). Importantly, in the nanodomains controlling BK_Ca activity, Ca²⁺ is fast enough to avoid the need for, e.g., a probability-density approach for handling local Ca²⁺ dynamics correctly at the whole-cell level (30). We use the mechanistically correct description of single BK_Ca-CaV complexes with 1:1 stoichiometry developed by Cox (17) as the natural starting point for constructing a reduced model for BK_Ca-CaV complexes with 1:n stoichiometry to be inserted into a whole-cell model of electrical activity. Our results give insight into the simulations of single BK_Ca-CaV complexes, and clarify that it is the local effects of ion channel kinetics rather than stochasticity per se that determine whole-cell activity.

Materials and Methods

BK_Ca channel model

We describe the BK_Ca channel with a model of single-channel gating with two states (closed and open). Fig. S1 A shows a schematic representation of the model, where X corresponds to the closed state and Y to the open state. The mathematical description of BK_Ca voltage- and calcium-dependent activation is given by the following:

$$\frac{d{p}_Y}{dt}=-k^{-}{p}_Y+k^{+}\left(1-p_Y\right),$$ (1)

where p_Y represents the open probability for the BK_Ca channel, and k⁻ and k⁺ are the voltage- and calcium-dependent rate constants. As shown in the Supporting Material, from relatively mild assumptions and experimental evidence, we can express these rates as the following:

$$k^{-}=w^{-}\left(V\right)f^{-}\left(Ca\right),$$ (2)

$$k^{+}=w^{+}\left(V\right)f^{+}\left(Ca\right),$$ (3)

where Ca denotes the Ca²⁺ concentration at the BK_Ca channels. At fixed Ca²⁺ levels, BK_Ca activation is well described by Boltzmann functions (6, 8, 16). Hence, we assume that the voltage-dependent rate constants, w⁻, for the transition from the open to closed state, and w⁺, for the transition from the closed to open state, have the following standard forms:

$$w^{-}\left(V\right)=w_0^{-}{e}^{-w_{yx}V},$$ (4)

$$w^{+}\left(V\right)=w_0^{+}{e}^{-w_{xy}V},$$ (5)

where the parameters w⁻₀ and w⁺₀ are voltage-independent.

There is evidence that at fixed V, Ca²⁺ stabilizes the open state (4), i.e., f⁻ should decrease with Ca, and that >1 Ca²⁺ ion is needed for BK_Ca activation, which is a sigmoidal function of the Ca²⁺ concentration (4, 16). The calcium-dependent relations are therefore modeled by the following:

$$f^{-}\left(Ca\right)=1-\frac{C{a}^{n_{yx}}}{K_{yx}^{n_{yx}}+C{a}^{n_{yx}}}=\frac{1}{1+{\left(\frac{Ca}{K_{yx}}\right)}^{n_{yx}}},$$ (6)

$$f^{+}\left(Ca\right)=\frac{C{a}^{n_{xy}}}{K_{xy}^{n_{xy}}+C{a}^{n_{xy}}}=\frac{1}{1+{\left(\frac{K_{xy}}{Ca}\right)}^{n_{xy}}},$$ (7)

where K_yx and K_xy are the calcium affinities when the channel closes and opens, respectively, and n_yx and n_xy are the corresponding Hill coefficients. By using the relationships in (4), (5), (6), (7), we get the following formulas for the equilibrium open fraction of BK_Ca channel activation, $p_{Y_{\infty }}$, and the corresponding time constant ${\tau }_{p_Y}$:

$$p_{Y_{\infty }}=\frac{k^{+}}{k^{-}+k^{+}}=\frac{1}{1-e^{-\frac{V-V_0}{S_0}}},$$ (8)

$${\tau }_{p_Y}=\frac{1}{k^{-}+k^{+}}=\frac{e^{w_{xy}V}}{w_0^{+}}\left(1+{\left(\frac{K_{xy}}{Ca}\right)}^{n_{xy}}\right)\frac{1}{1-e^{-\frac{V-V_0}{S_0}}},$$ (9)

where

$$V_0=\left(\text{log}\frac{w_0^{-}}{w_0^{+}}+\text{log}\left(1+{\left(\frac{K_{xy}}{Ca}\right)}^{n_{xy}}\right)-\text{log}\left(1+{\left(\frac{Ca}{K_{yx}}\right)}^{n_{yx}}\right)\right)S_0,$$ (10)

$$S_0=\frac{1}{w_{yx}-w_{xy}}.$$ (11)

We use global optimization to estimate the model parameters providing the best fit to the experimental data (17), consisting of BK_Ca open probabilities and time constants as functions of voltage, at different Ca²⁺ concentrations. In particular, we formulate an optimization problem to minimize the sum of the squared errors between the simulated responses produced by the model and the corresponding experimental data as follows:

$$\underset{\theta }{\text{min}}J=\sum _j\sum _i{\left(p_{Y_{{\infty }_j}}\left(V_i\right)-{\hat{p}}_{Y_{{\infty }_j}}\left(V_i,\theta \right)\right)}^2+{\left({\tau }_{p_{Yj}}\left(V_i\right)-{\hat{\tau }}_{p_{Yj}}\left(V_i,\theta \right)\right)}^2,$$ (12)

where θ is the set of model parameters, and $p_{Y_{{\infty }_j}}\left(V_i\right)$ and ${\tau }_{p_{Yj}}\left(V_i\right)$ are the experimental BK_Ca steady-state open fraction and corresponding time constants, respectively, at the given voltage V_i for the jth experiment (corresponding to a given Ca²⁺ concentration). ${\hat{p}}_{Y_{{\infty }_j}}\left(V_i,\theta \right)$ and ${\hat{\tau }}_{p_{Yj}}\left(V_i,\theta \right)$ are the simulated equilibrium open fraction of the BK_Ca channel and the corresponding time constants of the model, respectively, at the given V_i for the jth experiment. For the optimization, we use a hybrid genetic algorithm (GA) (31) that combines the most well-known type of evolutionary algorithm with a local gradient-based algorithm (32). We use the function “ga” from the software MATLAB (The MathWorks, Natick, MA) Global Optimization Toolbox and fmincon from the MATLAB Optimization Toolbox as the local algorithm. We repeat the hybrid GA algorithm several times and select the parameter set that gives the best fitting. Table S1 reports the optimal model parameters, and Fig. S1, B–G shows the fits to the data.

CaV channel model

We describe the calcium channel dynamics with the following model (27):

$$\frac{dc}{dt}=\beta o-\alpha c,$$ (13)

$$\frac{do}{dt}=\alpha c+\gamma b-\left(\beta +\delta \right)o,$$ (14)

$$b=1-c-o=1-h,$$ (15)

where c corresponds to the closed state, o to the open state, and b to the inactivated (blocked) state of the calcium channel; h represents the fraction of Ca²⁺channels not inactivated, δ is the rate for channel inactivation, and γ is the reverse reactivation rate; and α and β represent the voltage-dependent Ca²⁺ channel opening rate and closing rate, respectively, and have the following forms:

$$\alpha \left(V\right)={\alpha }_0{e}^{-{\alpha }_{1}V},$$ (16)

$$\beta \left(V\right)=\rho \left({\beta }_0{e}^{-{\beta }_{1}V}+{\alpha }_0{e}^{-{\alpha }_{1}V}\right).$$ (17)

As shown in Sherman et al. (27), the processes of activation and inactivation can be approximately separated in time, because activation is much faster than inactivation. In particular, we achieve the following model for the activation variable, m_CaV,

$$\frac{d{m}_{\text{CaV}}}{dt}=\frac{m_{\text{CaV},\infty }-m_{\text{CaV}}}{{\tau }_{\text{CaV}}},$$ (18)

where

$$m_{\text{CaV},\infty }=\frac{\alpha }{\alpha +\beta },{\tau }_{\text{CaV}}=\frac{1}{\alpha +\beta },$$ (19)

and the following equation for inactivation:

$$\frac{db}{dt}=m_{\text{CaV},\infty }\delta -\left(m_{\text{CaV},\infty }\delta +\gamma \right)b.$$ (20)

As for the BK_Ca channel, we use a global optimization method to optimize the parameters of (16), (17) to fit the experimental data presented by Cox (17), i.e., peak open probabilities and time constants as functions of voltage. For the values of γ = 0.0020 ms⁻¹ and δ = 0.0025 μM⁻¹ ms⁻¹ ×[Ca_CaV], we use those reported by Cox (17). Ca_CaV is the Ca²⁺ concentration at the internal mouth of the channel and defined by Eq. S1 with r = 7 nm, representing the distance of the sensor for Ca²⁺-dependent inactivation from the channel pore. Note that the relation given by Eq. 17 allows scaling of the amount of channel activation at high voltage values according to the experiments (i.e., not all the calcium channels are open even for high voltages). Table S1 reports the optimal parameters for the CaV activation model.

BK_Ca-CaV complex with 1:1 and 1:n stoichiometries

Combining the models for BK_Ca and CaV channels, we obtain the models of the 1:1 (see Results and Supporting Material, Model of the 1:1 BK_Ca-CaV Complex and Timescale Analysis and Model Simplifications) and 1:n BK_Ca-CaV complexes (see Results and Supporting Material, Model for BK_Ca Activation in Complexes with k Noninactivated CaVs and its Approximation). Ca²⁺ levels sensed by the BK_Ca channel were assumed to reach steady state immediately after CaV opening or closure (17), and the steady-state Ca²⁺ concentration Ca_o resulting from influx through a single CaV was calculated by an explicit formula (see Eq. S1), assuming that CaV and BK_Ca channels are r = 13 nm apart (2, 9). At V = 0 mV, Ca_o ≈ 19 μM (see Supporting Material, Model of the 1:1 BK_Ca-CaV Complex and Table S2 for further details). In the case of more than one CaV per complex, the linear buffer approximation (33) was used to summarize Ca²⁺ levels when more than one CaV is open. We note that k⁺_c ≈ 0 (see Supporting Material, Model of the 1:1 BK_Ca-CaV Complex and Table S1) because the background Ca²⁺ concentration Ca_c is much below the levels needed for BK_Ca activation at physiological voltages (2). Thus, a BK_Ca channel opens only when a CaV in the complex is open. This approximation is used widely in our derivations, and is supported by the fact that Ca²⁺ influx via CaVs is needed to open BK_Ca channels (34), and that the submembrane Ca²⁺ concentration of some hundreds of nanomolar that a BK_Ca in a complex without open CaVs would sense, is too low to activate BK_Ca channels at physiological voltages (2, 17).

We refer to the Supporting Material for details on mathematical analysis of the time to first BK_Ca channel opening using phase-type distributions (35) (see Supporting Material, Model of the 1:1 BK_Ca-CaV Complex), and timescale analysis used for model reduction borrowing ideas from enzyme kinetics (36) (see Supporting Material, Timescale Analysis and Model Simplifications and Model for BK_Ca Activation in Complexes with k Noninactivated CaVs and its Approximation), as well as for details on the whole-cell models investigated (see Supporting Material, Whole-Cell Models).

Availability of models and computer code

MATLAB code containing the files for generating the results presented in the main text and Supporting Material is provided as an additional Supporting File S1.

Results

A simple Markov chain model of the BK_Ca-CaV complex

Cox (17) presented a stochastic model of a single CaV2.1 (P/Q-type) controlling a BK_Ca channel (α-subunits only) via local Ca²⁺. The channels were located 13 nm apart, corresponding to physical coupling (2, 9). The CaV was described by a seven-state Markov chain, and when the Ca²⁺ channel opened or closed, the local Ca²⁺ level was assumed to reach equilibrium instantaneously, in accordance with simulations of Ca²⁺ diffusion (12, 13, 17). The calculated local Ca²⁺ concentration was then assumed to drive a 10-state Markov chain model of the BK_Ca channel, and Monte Carlo simulations were performed.

We set out to simplify the description of the 7 × 10-state Markov chain model of the BK_Ca-CaV complex. This was achieved by assuming a three-state model for CaV (27) with states closed (C), open (O), or inactivated (B, for blocked) (see Materials and Methods). Parameters were adjusted to reproduce traces from Cox (17). The BK_Ca channel was represented by a model with only two states, closed (X) or open (Y) (see Materials and Methods). The transitions between states were supposed to depend on voltage and local Ca²⁺, which was assumed to reach equilibrium instantaneously, and depend on voltage via the single-channel Ca²⁺current (17). Parameters describing BK_Ca kinetics were fitted to data from Cox (17). Combining these two models, we obtain a six-state model of the BK_Ca-CaV complex (Fig. 1 A) that shows behavior similar to the 70-state model used by Cox (17) (Fig. 1, B and C). Our simplified BK_Ca model does not describe details of single-channel kinetics, which is not our scope here, but reproduces satisfactorily activation curves and times (Fig. S1), as well as whole-cell currents (Figs. S4 and S5), thus making it appropriate for analysis of whole-cell BK_Ca activity.

Figure 1.

Six-state model of the BK_Ca-CaV complex with 1:1 stoichiometry and its simplification. (A) Shown here is a scheme indicating the six states and voltage-dependent transitions. C, O, and B refer respectively to closed, open, and inactivated states of the CaV, whereas X and Y indicate the closed and open states of the BK_Ca channel. The subscripts o and c on the horizontal transition rates indicate dependence on the Ca²⁺ concentration below an open (respectively, closed) Ca²⁺ channel. At physiological voltages, the transition to a state with an open BK_Ca channel occurs virtually only when the CaV is open (k⁺_c ≈ 0). The green box indicates states with noninactivated CaVs, whereas the blue box highlights states with inactivated CaVs. The transitions between the colored boxes are slow compared to transitions within boxes. (B) Shown here are simulated CaV open probabilities in response to a voltage step from −80 to 0 mV, obtained from the seven-state Markov chain model (gray; (17)), the three-state Markov chain model C, O, B (black; (27)), the ODE model corresponding to the three-state model ((13), (14), (15); blue), and the corresponding model assuming instantaneous activation m_CaV = m_CaV,∞ (Eq. 20; dash-dotted green). (C) Shown here are simulated open probabilities, in response to a voltage step from −80 to 0 mV, for BK_Ca channels controlled by CaVs in complexes with 1:1 stoichiometry, obtained from the original 70-state Markov chain model (gray; (17)), the six-state Markov chain model (A; black), the ODE model corresponding to the six-state model (Eqs. S6–S11; blue), the simplified Hodgkin-Huxley-type model (Eq. 25; dashed red), and the corresponding model assuming instantaneous activation m_CaV = m_CaV,∞ (dash-dotted green; see main text). In (B) and (C), one-thousand realizations were simulated for the Markov chain models, and the average of these Monte Carlo simulations is shown. To see this figure in color, go online.

Time to first opening

Interestingly, Cox (17) found that not all simulated BK_Ca channels open during 20 ms depolarizations or imposed action potentials. We now study the time to the first opening of the BK_Ca channel during a depolarization, which mathematically corresponds to the first time the Markov chain Z corresponding to Fig. 1 A visits one of the states CY, OY, or BY starting from state CX. We denote the time to first opening T_CX,Y, which is a random variable. Simulations show that eventually all BK_Ca channels open, and that the probability of channel opening before a given time t, P(T_CX,Y < t), shows biphasic behavior (Fig. S2). Taking advantage of the fact that transitions from CX to CY, and from BX to BY have virtually zero probability (BK_Ca channels open only if the CaV is open), we obtain explicit formulas for the average time to first opening E(T_CX,Y) and, more generally, for the distribution function P(T_CX,Y < t) using phase-type distribution results for Markov chains (35) (see Supporting Material, Time to First Opening and Phase-Type Distributions), as follows:

$$E\left(T_{CX,Y}\right)=\frac{1}{\alpha }+\frac{1}{k_o^{+}}+\frac{1}{k_o^{+}}\left(\frac{\beta }{\alpha }+\frac{\delta }{\gamma }\right),$$ (21)

$$P\left(T_{CX,Y}<t\right)=1-\sum _{\psi \in \left\{C,O,B\right\}}{\left(\text{exp}\left(t\overline{Q}\right)\right)}_{CX,\psi X},$$ (22)

where $\overline{Q}$ is the subtransition rate matrix of Z corresponding to states {CX, OX, BX}. Thus, the average time to first opening is inversely related to the opening rates of the CaV and BK_Ca, and to the rate of reactivation after inactivation of the CaV. The involvement of these two processes explains the biphasic behavior, because escape from inactivation is much slower than channel opening. Eq. 22 states that P(T_CX,Y < t) is 1 minus the probability of not having left {CX, OX, BX} before t, and makes it explicit that ∼15% of BK_Ca channels do not open during a 20 ms depolarization (17), because P(T_CX,Y < 20 ms) ≈ 85% with our parameters (Fig. S2).

A concise deterministic model of cellular BK_Ca activity derived from multiscale principles

1:1 stoichiometry

For Hodgkin-Huxley-type whole-cell simulations, we do not need to know the state of each single BK_Ca channel, but it suffices to follow the BK_Ca open probability p_Y over time, because in the presence of many channels the whole-cell BK_Ca current is I_BK = g_BK p_Y(V − V_K), where g_BK is the maximal whole-cell BK_Ca conductance and V_K is the K⁺ reversal potential.

The time evolution of the probability distribution of the Markov chain Z corresponding to the six-state model in Fig. 1 A can be described by a system of five ordinary differential equations (ODEs) because the probabilities sum to 1. Denote, for ψ ∈ {C, O, B} and ξ ∈ {X, Y}, the state probabilities p_ψξ(t) = P(Z(t) = ψξ). Then p_Y(t) = p_CY(t) + p_OY(t) + p_BY(t). As shown in Fig. 1 C, the average fraction of open channels calculated from Monte Carlo simulations of the Markov chain is well approximated by p_Y obtained from the ODE system.

Although the reduction to five ODEs for the description of the BK_Ca-CaV complexes is already a substantial reduction compared to Monte Carlo simulations, we wish to obtain an expression for the BK_Ca current of Hodgkin-Huxley form. Such a simplification provides further insight into the regulation of BK_Ca activity by CaVs, and provides the base for concise handling of BK_Ca-CaV complexes with 1:n stoichiometry.

We performed detailed timescale analysis (see Supporting Material, Timescale Analysis and Model Simplifications) based on the fact that re- and inactivation of CaVs are slower than (de-)activation. Thus, on a fast timescale, the average fraction of noninactivated CaVs, h = 1 –(p_BX + p_BY), is assumed to be constant, and the model splits into two submodels with, respectively, four and two states (Fig. 1 A, green and blue).

In the system of ODEs describing the state probabilities of the corresponding reduced four-state Markov chain (Fig. 1 A, green), it turns out that the dynamics of state CY is the fastest because CaV kinetics and BK_Ca-channel closure, when the CaV is closed, are faster reactions than BK_Ca gating in the presence of an open CaV (see Fig. S3). Assuming quasi-steady state for CY, we derive a single ODE describing the gating variable m_BK, which models the fraction of open BK_Ca channels in complexes with noninactivated CaV (see Supporting Material, Model Simplifications), as follows:

$$\frac{d{m}_{BK}}{dt}=\frac{m_{BK,\infty }-m_{BK}}{{\tau }_{BK}},$$ (23)

with steady state and time constant given by the following:

$$\begin{array}{cc}{m}_{BK,\infty }& =\frac{m_{CaV}{k}_o^{+}\left(\alpha +\beta +k_c^{-}\right)}{\left(k_o^{+}+k_o^{-}\right)\left(k_c^{-}+\alpha \right)+\beta k_c^{-}},\\ {\tau }_{BK}& =\frac{\alpha +\beta +k_c^{-}}{\left(k_o^{+}+k_o^{-}\right)\left(k_c^{-}+\alpha \right)+\beta k_c^{-}}.\end{array}$$ (24)

Here, m_CaV is defined by Eq. 18 and denotes the activation variable for the CaV in the complex, which is routinely characterized in patch-clamp experiments and included in models of electrical activity via the time-constant, τ_CaV, and the steady-state activation function, m_CaV,∞ (see Eq. 19). From these quantities, α = m_CaV,∞/τ_CaV and β = 1/τ_CaV − α can be calculated. Note that Eq. 24 makes it explicit how m_BK,∞ inherits properties of the associated Ca²⁺ channel type, as has been found experimentally (10, 37).

Now, because BK_Ca channels close rapidly in complexes with inactivated CaVs (blue in Fig. 1 A), we have p_Y ≈ m_BKh. Thus, the BK_Ca current is approximated by the standard Hodgkin-Huxley expression

$$I_{BK}=g_{BK}{m}_{BK}h\left(V-V_K\right),$$ (25)

where m_BK is given by Eq. 23, and h is the inactivation function of the CaVs (see (15), (20)). As shown in Fig. 1 C, the open-probability expression m_BKh approximates the Monte Carlo simulations very well. From Eq. 25 it is evident that the BK_Ca channels in BK_Ca-CaV complexes exhibit inactivation because of inactivation of the associated CaVs, and with approximately identical dynamics, as found in experiments (8) and Monte Carlo simulations (Fig. 1; (17)).

In many whole-cell models (e.g., (20, 21, 22, 23)), the Ca²⁺ currents are assumed to activate instantaneously, which precludes calculation of α and β. Implicitly, such models assume that CaV gating is infinitely faster than the kinetics of other channels in the model. In our setting, this assumption corresponds to investigating the BK_Ca-CaV model defined by (23), (24), (25) in the limit α, β → ∞. This leads to ${\tau }_{BK}\approx 1/\left[k_c^{-}-m_{\text{CaV},\infty }\left(k_c^{-}-k_o^{+}-k_o^{-}\right)\right]$ and $m_{BK,\infty }=k_o^{+}{m}_{\text{CaV},\infty }{\tau }_{BK}$, which are completely defined from BK_Ca kinetics and m_CaV,∞. In combination with (23), (25), this model approximates the full system decently, except for the initial phase before CaV activation reaches equilibrium (Fig. 1 C, green). For whole-cell models neglecting CaV activation kinetics, this initial-phase error should be of no more concern that the error in the Ca²⁺ current resulting from the steady-state assumption for CaV activation (Fig. 1 B, green).

Complexes with multiple Ca²⁺ channels

As mentioned, a BK_Ca channel can bind up to four CaVs (2, 11). We extend our model to incorporate such cases, assuming that the n CaVs are all located 13 nm from the BK_Ca channel (2, 9, 17). Near the CaVs, the linear buffer approximation (33) holds, and the Ca²⁺ profile from n channels can be calculated by superimposing n nanodomains found for single, isolated CaVs.

One could in principle extend the Markov chain model in Fig. 1 A to a model with 3 × n × 2 states. We take another approach to keep the model tractable. As discussed in the previous section, CaV inactivation is slow compared to other processes. We therefore assume that on a fast timescale, the fraction h of noninactivated CaVs is constant, and note that the BK_Ca channel closes rapidly when all CaVs in the complex are inactivated.

Consider a BK_Ca-CaV complex with k ∈ {1,..., n} noninactivated CaVs. Neglecting inactivated CaVs, because they do not contribute to BK_Ca activation, such a complex can be described on the fast timescale by a Markov chain model with 2 × (k + 1) states (Fig. 2 A). As for the case of 1:1 stoichiometry, we can approximate the dynamics of the BK_Ca open probability by a single ODE (see Supporting Material, Model for BK_Ca Activation in Complexes with k Noninactivated CaVs and its Approximation). Denote this open probability by $m_{BK}^{\left(k\right)}$, and note that $m_{BK}^{\left(1\right)}=m_{BK}$ in Eq. 23. Then, we have the following:

$$\frac{d{m}_{BK}^{\left(k\right)}}{dt}=\frac{m_{BK,\infty }^{\left(k\right)}-m_{BK}^{\left(k\right)}}{{\tau }_{BK}^{\left(k\right)}},$$ (26)

where $m_{BK,\infty }^{\left(k\right)}$ and ${\tau }_{BK}^{\left(k\right)}$ are explicit functions of V, directly or via the local Ca²⁺ concentration (see Eq. S36). The probability that k noninactivated CaVs are present in a complex with n CaVs is $\left(\begin{array}{c}n\\ k\end{array}\right)h^k{\left(1-h\right)}^{n-k}$, and the whole-cell BK_Ca current, is approximated by the following:

$$I_{BK}=g_{BK}\sum _{k=1}^n\left(\begin{array}{c}n\\ k\end{array}\right)h^k{\left(1-h\right)}^{n-k}{m}_{BK}^{\left(k\right)}\left(V-V_K\right),$$ (27)

which involves n ODEs (Eq. 26) for the activations variables $m_{BK}^{\left(k\right)}$, and one ODE for h (h = 1 – b, where b is given by Eq. 20). As shown in Fig. 2 C, this expression provides a good approximation to the results from Monte Carlo simulations of the full Markov Chain. Note that if the CaVs do not inactivate, Eq. 27 reduces to the following:

$$I_{BK}=g_{BK}{m}_{BK}^{\left(n\right)}\left(V-V_K\right).$$ (28)

Figure 2.

Multiple CaVs per BK_Ca-CaV complex. (A) Shown here is the Markov chain model for complexes with k noninactivated CaVs. (B) Shown here are steady-state BK_Ca activation functions (upper) and time constants (lower) for BK_Ca channels in complexes with 1 (cyan), 2 (green), or 4 (red) CaVs given from Eq. 26 (see Eq. S36 for the details; solid) or from the approximation defined by Eq. 29 (dashed). The gray dashed curve shows the CaV activation function m_CaV,∞, for comparison. (C) Shown here are simulated BK_Ca open probabilities in response to a voltage step from −80 to 0 mV, obtained from Monte Carlo simulations of the Markov model of n inactivating independent CaVs controlling a BK_Ca channel (black), from the ODE model of all states in (A) coupled to CaV inactivation (Eq. 27; Eqs. S19–S25; solid blue), from the reduced ODE model considering CaV activation kinetics ((26), (27); dashed red), and from the simplification assuming m_CaV = m_CaV,∞ ((29), (27); dash-dotted green). To see this figure in color, go online.

We can now easily investigate how different stoichiometries of the BK_Ca-CaV complexes influence, e.g., activation of the BK_Ca channels. As expected, we find that the activation curve is shifted upwards as the number of CaVs per complex increase (Fig. 2 B, upper). Interestingly, a left shift of the activation curve is seen when n increases. For example, with n = 4 CaVs per BK_Ca channel, BK_Ca activation is half-maximal at V ≈ −14 mV, compared to V ≈ −5 mV when n = 1, and half-maximal CaV activation at V ≈ −12 mV. This result is due to the fact that the probability of at least one CaV being open is greater with more channels in the complex. For higher voltages, the single channel current decreases and the CaV open probability increases, with the result that, at strongly positive voltages, BK_Ca activation decays more gradually at n = 1 than for higher n. This difference is because the local Ca²⁺ level obtained with a single open CaV is insufficient for complete BK_Ca activation, and therefore the presence of more CaVs per complex becomes advantageous, because the CaVs may open simultaneously, leading to higher local Ca²⁺ levels. This interpretation also underlies the finding that BK_Ca activation is faster with higher n at positive voltages (Fig. 2 B, lower).

As mentioned above, many whole-cell models assume instantaneous activation of CaVs. This assumption implies that vertical transitions in Fig. 2 A are in quasi-equilibrium, and hence that, e.g., $p_{C_i{O}_{k-i}Y}=\left(\begin{array}{c}k\\ i\end{array}\right){\left(1-m_{CaV,\infty }\right)}^{k-i}{m}_{CaV,\infty }^i{p}_Y$, with notation as for the case of 1:1 stoichiometry. Then, $m_{BK}^{\left(k\right)}$ follows Eq. 26 with

$$\begin{array}{c}{\tau }_{BK}^{\left(k\right)}={\left[\sum _{i=1}^k\left(\begin{array}{c}k\\ i\end{array}\right){\left(1-m_{\text{CaV},\infty }\right)}^{k-i}{m}_{\text{CaV},\infty }^i\left(k_{o_i}^{+}+k_{o_i}^{-}\right)+{\left(1-m_{\text{CaV},\infty }\right)}^k{k}_c^{-}\right]}^{-1},\\ m_{BK,\infty }^{\left(k\right)}=\left[\sum _{i=1}^k\left(\begin{array}{c}k\\ i\end{array}\right){\left(1-m_{\text{CaV},\infty }\right)}^{k-i}{m}_{\text{CaV},\infty }^i{k}_{o_i}^{+}\right]{\tau }_{BK}^{\left(k\right)}.\end{array}$$ (29)

This simplified expression provides decent fits to activation functions (Fig. 2 B, upper) and simulated currents (Fig. 2 C), and—in our experience—yields reliable results in whole-cell simulations for cells with relatively slow action potential dynamics, as shown below, despite a slight underestimation of ${\tau }_{BK}^{\left(k\right)}$ at negative voltages (Fig. 2 B, lower).

Whole-cell simulations of electrical activity shaped by BK_Ca-CaV complexes

We now illustrate the type of whole-cell modeling that can be performed readily with our Hodgkin-Huxley-type model of the BK_Ca current controlled locally by CaVs in BK_Ca-CaV complexes.

BK_Ca-CaV stoichiometry controls fAHP in a neuronal model

It is well established that in many neurons, BK_Ca channels play an important role in action potential (AP) repolarization and fast after-hyperpolarization (fAHP), i.e., the undershoot seen after an AP (2, 38), which is important, e.g., for controlling firing frequency and transmitter release. We here adapt a model of AP generation and fAHP in hypothalamic neurosecretory cells (20) to investigate how BK_Ca-CaV complexes influence fAHP. In the original model, CaVs are assumed not to inactivate, and to activate instantaneously. We modified the model to include CaV activation dynamics with time constant τ_CaV = 1.25 ms (37, 39), and inserted our whole-cell BK_Ca model (Eq. 28) in place of the original representation of BK_Ca currents.

Our results suggest that more than one CaV channel is needed in the BK_Ca-CaV complex to develop fAHP that is reduced by BK_Ca-channel blockers (Fig. 3 A). The difference between 1:1 and 1:n BK_Ca-CaV stoichiometry is not a simple result of more BK_Ca conductance. Increasing the BK_Ca conductance fourfold in the case of 1:1 stoichiometry, much more than the difference between the activation functions $m_{BK,\infty }^{\left(1\right)}$ and $m_{BK,\infty }^{\left(4\right)}$ (Fig. 2 B), leads to less fAHP than for 1:4 stoichiometry (Fig. 3 A, inset). Thus, differences in BK_Ca activation kinetics and the shapes of activation functions (Fig. 2 B) play a nontrivial role in shaping APs.

Figure 3.

Whole-cell simulations. (A) Shown here is a simulated AP in a neuronal model (20) with 1:n stoichiometry BK_Ca-CaV complexes with n = 1 (cyan), n = 2 (green), or n = 4 (red). The whole-cell BK_Ca current is described by Eq. 28 (i.e., BK_Ca coupled with noninactivating CaVs), where the BK_Ca activation, $m_{BK}^{\left(n\right)}$, is modeled by Eq. 26 (see Eq. S36 for the details) and g_BK = 1 mS cm⁻². The blue curve shows the case of BK_Ca block (g_BK = 0 mS cm⁻²), and the trace in black displays the result with n = 1, g_BK = 4 mS cm⁻². The inset shows a zoom-in on the fAHP. (B–D) Shown here are simulated APs in a model of human β-cells (22) with BK_Ca channels located in complexes with n T-type (B), L-type (C), or P/Q-type (D) CaVs, with n = 1, 2, or 4. The whole-cell BK_Ca current is described by Eq. 27 (with inactivating T- and L-type CaVs) or Eq. 28 (with noninactivating P/Q-type CaVs), where $m_{BK}^{\left(n\right)}$ is modeled by Eq. 29. Color coding as in (A). (E–G) Shown here is simulated activity in a model of lactotrophs (21) with 1:n BK_Ca-CaV complexes with n = 1 (E), n = 2 (F), or n = 4 (G). The whole-cell BK_Ca current is described by Eq. 28, where $m_{BK}^{\left(n\right)}$ is modeled by the complete BK_Ca model with 2 × (n + 1) states (Fig. 2A) described using Eqs. S19–S25 (upper traces), by Eq. 26 (middle traces), and by Eq. 29 (lower traces). To see this figure in color, go online.

Different CaV types affect electrical activity differently in a model of human β-cell electrophysiology

In our recent model of electrical activity in human β-cells (22, 23), we modeled the BK_Ca-current heuristically. The BK_Ca open probability was proportional to the whole-cell Ca²⁺ current, and this expression was found to reasonably reproduce published data (40) regarding the BK_Ca activation function and the effects of BK_Ca block on AP firing (22).

We now assume that the BK_Ca channels form complexes with either T-, L-, or P/Q-type CaVs (22, 40), and vary the BK_Ca-CaV stoichiometry. As explained in greater detail in the Supporting Material, the different types of CaV differ with respect to activation and inactivation properties, and whole-cell conductance (22, 23). The resulting BK_Ca model is then fit to experimental I-V data (40) (Fig. S7), and inserted in the whole-cell model. T-type CaVs inactivate rapidly (22, 40), and do not activate much BK_Ca current during the relatively broad action potentials. For this reason, simulated BK_Ca block results in almost no increase in AP height (Fig. 3 B), in contrast to experiments (40).

In human β-cells, L-type Ca²⁺ channels show inactivation on a timescale comparable to the duration of an AP (22, 40). When coupled to BK_Ca channels in the model, good fits to the BK_Ca I-V activation curve are obtained, but for different values of the maximal whole-cell BK_Ca conductance g_BK (Fig. S7). In simulations of electrical activity, BK_Ca currents controlled by L-type CaVs reduce AP height, independently of the number of CaVs per complex (Fig. 3 C).

BK_Ca-CaV complexes with P/Q-type Ca²⁺ channels, which activate at very depolarized potentials and show very slow inactivation in human β-cells (22, 40), lead to BK_Ca currents that activate at slightly more depolarized potentials than in experiments, except for the case of 1:4 BK_Ca-CaV stoichiometry (Fig. S7). Simulated application of a BK_Ca channel antagonist increases AP height ∼15 mV, in good correspondence with experiments. Assuming fewer CaVs per complex, leads to poorer fit of the I-V curve and to less difference between APs obtained with operating and blocked BK_Ca channels (Fig. 3 D).

We conclude that AP firing is affected differently by BK_Ca currents depending on the CaV type controlling BK_Ca activity, due to differences in activation and inactivation properties. Because BK_Ca block stimulates insulin secretion in human (40) and mouse (41) β-cells, a better understanding of the interaction between different types of CaVs and BK_Ca channels may provide novel insight into insulin release in health and disease.

Bursting behavior depends on BK_Ca-CaV stoichiometry in a model of pituitary cells

In pituitary cells, BK_Ca channels have been found to be intimately involved in the genesis of so-called plateau bursting, which consists of a few small oscillations riding on a depolarized plateau, and is important for secretion (42, 43). We now investigate how BK_Ca-CaV properties affect such bursting activity in a model of electrical activity in pituitary lactotrophs (21). In this model a single Ca²⁺-channel type is present, which is assumed to activate instantaneously and not to inactivate. The BK_Ca current was modeled as a purely voltage-dependent current, neglecting Ca²⁺ dependency (21). In place of this simplified representation, we substitute our concise BK_Ca model controlled by CaVs in complexes.

With 1:1 stoichiometry, spiking electrical activity is observed, because insufficient BK_Ca current is generated (Fig. 3 E). In contrast, with more than one CaV per complex, plateau bursting appears with the number of small oscillations per burst depending on the number of CaVs per BK_Ca-CaV complex (Fig. 3, F and G). Although the quantitative behavior is independent of the approximation for $m_{BK}^{\left(n\right)}$, minor qualitative differences are present. The approximation given by Eq. 26 reproduces very well the behavior obtained from the complete model for the BK_Ca-CaV complex (Fig. 3, F and G, upper and middle panels), whereas the further simplification given by Eq. 29 produces smaller and more spikes per burst. Nonetheless, considering parameter uncertainties and experimental variations, even Eq. 29 produces reliable results.

Discussion

Models of cellular electrical activity typically do not consider local control in ion channel complexes. This fact is probably to a large extent because of the large computational costs of detailed simulations of Markov chain models (17) or reaction-diffusion models (10) that consider single complexes. In contrast, in the field of Ca²⁺ modeling, global procedures that respect local mechanisms have been presented (28, 29, 30).

We here applied similar methods to the BK_Ca-CaV complex to obtain Hodgkin-Huxley representations of the BK_Ca current that correctly take local control into account. Importantly, in our approach the effects of ion channel colocalization are handled via a deterministic model representation by averaging the stochastic dynamics in single ion channel complexes appropriately. Our timescale analysis allowed us to handle scenarios with more than one CaV per BK_Ca-CaV complex, thus providing important insight into the role of channel stoichiometry. Treating such cases via direct stochastic simulations of the BK_Ca and CaV simulations would be computationally cumbersome, and would not provide the same kind of analytical understanding. For example, we found explicit expressions for the time to first opening of a BK_Ca channel, thus providing theoretical insight into simulation results (17). Our findings also highlighted that $n>1$ CaV per complex left shifts the BK_Ca activation curve, because the presence of more CaVs increase the probability that at least one CaV is open and activates the associated BK_Ca channel.

We illustrated the usefulness of our theoretical results by applying the concise representations of BK_Ca currents to previously published whole-cell models of electrical activity. We chose a model of neuronal APs that has previously been used to investigate how BK_Ca channels contribute to fAHP (20). The simulations based on our BK_Ca-CaV model suggest that the kinetics of BK_Ca activation, which depends on the number of associated CaVs (Fig. 2), influence fAHP generation. It would be interesting to investigate experimentally whether defective BK_Ca-CaV coupling underlies disturbances in fAHP generation, as predicted by the model. In Xenopus motor nerve terminals, BK_Ca-CaV coupling differs between the release face and the nonsynaptic surface of varicosities (44), which, in the light of our simulations, may indicate spatial heterogeneity with respect to, e.g., fAHP.

We went on to investigate how the activation and inactivation properties of specific types of Ca²⁺ channels assumed to be present in BK_Ca-CaV complexes influence whole-cell electrical activity in a model of human β-cells (22). Because both the coupling of BK_Ca channels to L- and P/Q-type CaVs and the different stoichiometries of the complexes allow for simulations comparable to experiments, our findings do not allow us to conclude on the structure of BK_Ca-CaV complexes in human β-cells. Further insight into the control by CaVs of BK_Ca channels, which are involved in regulation of insulin release (40, 41), may lead to a better understanding of β-cell function and how it becomes disturbed in diabetes.

Finally, a model of pituitary cells (21) was used to study the role of BK_Ca channels in the generation of plateau bursting, which is important for secretion of pituitary hormones (42). We found that a reduced number of CaVs per complex, for example because of disturbed BK_Ca-CaV coupling, may abolish bursting activity. Our simulations showed that even the simplification given by Eq. 29 provided reliable results (Fig. 3, E–G). Similar conclusions hold for the β-cell model (see Fig. S7). Interestingly, this was not the case in the neuronal model (20) (Fig. S6), likely because of the shorter neuronal AP being more sensitive to the kinetics of BK_Ca activation.

A general strategy to distinguish between different configurations of the BK_Ca-CaV complex could be to first estimate the maximal whole-cell BK_Ca conductance, for example by depolarizations to highly positive voltages to activate BK_Ca channels independently of CaV activity (16), and then to fit I-V curves obtained from voltage-clamp depolarizations (37, 40) using the expressions presented here.

In summary, we have presented a concise Hodgkin-Huxley-type model of BK_Ca currents that take into account local control in BK_Ca-CaV complexes with different stoichiometries. Our model should be useful for whole-cell simulations of electrical activity in neurons and other excitable cells. The approach should be relatively straightforward to apply to other ion channel complexes, e.g., the Cav3-Kv4 complex (45).

Author Contributions

All authors performed research, prepared Supporting Material, revised the article, and approved the final version. F.M., A.T., and M.G.P. prepared figures. F.M. developed methods. M.G.P. conceived research and wrote the article.

Editor: Arthur Sherman.