Short-Term Desensitization of Muscarinic K⁺ Current in the Heart

Abstract

Acetylcholine (ACh) rapidly increases cardiac K⁺ currents (I_KACh) by activating muscarinic K⁺ (K_ACh) channels followed by a gradual amplitude decrease within seconds. This phenomenon is called short-term desensitization and its precise mechanism and physiological role are still unclear. We constructed a mathematical model for I_KACh to examine the conditions required to reconstitute short-term desensitization. Two conditions were crucial: two distinct muscarinic receptors (m₂Rs) with different affinities for ACh, which conferred an I_KACh response over a wide range of ACh concentrations, and two distinct K_ACh channels with different affinities for the G-protein βγ subunits, which contributed to reconstitution of the temporal behavior of I_KACh. Under these conditions, the model quantitatively reproduced several unique properties of short-term desensitization observed in myocytes: 1), the peak and quasi-steady states with 0.01–100 μM [ACh]; 2), effects of ACh preperfusion; and 3), recovery from short-term desensitization. In the presence of 10 μM ACh, the I_KACh model conferred recurring spontaneous firing after asystole of 8.9 s and 10.7 s for the Demir and Kurata sinoatrial node models, respectively. Therefore, two different populations of K_ACh channels and m₂Rs may participate in short-term desensitization of I_KACh in native myocytes, and may be responsible for vagal escape at nodal cells.

Introduction

Vagal nerve stimulation causes the release of acetylcholine (ACh) from axonal termini and then decelerates the heartbeat by increasing the amplitude of muscarinic K⁺ current (I_KACh) in pace-making cells (1–4). I_KACh activation is gradually decreased to a quasi-steady-state level despite the continuous presence of ACh (2). This phenomenon is classified into two distinct categories by timescale: short-term desensitization, which occurs within seconds immediately after exposure to ACh, and long-term desensitization, which is observed on a scale of minutes to hours. Although the latter appears to be attributable to sequestration of available receptors by modulations such as phosphorylation and internalization (5–9), the precise mechanism and physiological role of the former are still a matter of debate (2,10–16).

ACh binding to M₂ muscarinic receptor (m₂R) liberates the βγ subunits (G_βγ) from pertussis-toxin-sensitive G proteins (G_i/o) that activate I_KACh. Interestingly, prestimulation of the A₁-adenosine receptor, another G_i/o-coupled receptor, prevents ACh-induced short-term desensitization (2). This cross talk between receptors suggests that m₂R is not involved in short-term desensitization. Constituents downstream of receptor activation, such as G protein and the muscarinic K⁺ (K_ACh) channel, have been proposed to cause this phenomenon (2,10–16). However, how these constituents could quantitatively account for the physiological I_KACh response has not been fully examined.

Short-term desensitization of I_KACh is a cellular response against overstimulation by ACh. At the organ level, excess vagal nerve stimulation eventually causes cardiac asystole, followed by resumption of the heartbeat. This desensitization is called vagal escape and has been explained by compensation from the sympathetic system (17–20). Although ACh causes short-term desensitization of I_KACh and vagal escape, the functional relevance of these two different phenomena has not been previously examined.

A previously proposed mathematical model for I_KACh activation consisted of only a single population of m₂R, K_ACh channel, and G protein with the chemical cycle reactions (21). However, this linear system was insufficient to reproduce the temporal behavior of I_KACh, suggesting the need for another hypothesis for the simulation of I_KACh activation. At the macroscopic level, short-term desensitization is observed at higher ACh concentrations ([ACh]s) (2). The open probability of the K_ACh channel declines until a steady state is reached (22), and is modulated by the concentration of phosphatidylinositol 4,5-bisphosphate (PIP₂) and the phosphorylation state (23,24). These observations allowed us to hypothesize an additional fraction of K_ACh channels that decrease their activity in a time-dependent manner. Because this fraction was expected to develop at higher [ACh]s, we designated them as K_ACh channels with low affinity for G_βγ (K_ACh/low channel). To differentiate between two populations of K_ACh channels, another fraction was presumed to have high affinity to G_βγ (K_ACh/high channel). This assumption contributed to reproduction of the apparent response of the peak and quasi-steady-state components in short-term desensitization, the effects of ACh preperfusion, and the recovery from short-term desensitization experimentally observed in atrial myocytes. We also implemented two populations of m₂Rs with high and low affinities for ACh (25–29) to confer the I_KACh response over a wide range of [ACh]s. Furthermore, integration of the these hypotheses reconstituted vagal escape when incorporated into mathematical models of action potential in the sinoatrial node (30–32). These results suggest that short-term desensitization may contribute to vagal escape originating in the sinus node.

Materials and Methods

K_ACh channel model

In this study, we used Monod’s allosteric model (33,34) to simulate the interaction between G_βγ and the K_ACh channels. Previously, we showed that G_βγ does not affect channel gating at steady state, but does increase the number of functionally active channels to enhance total channel activity (34). The allosteric model with four subunits could describe this regulation of functionally available channels; therefore, we used the allosteric model of a previous I_KACh model study to reproduce the major features of steady-state I_KACh and relaxation (21). To study the temporal behavior of I_KACh, we adopted the experimental observation that high [ACh] modulates the gating kinetics of the K_ACh channel during short-term desensitization (22). To present the kinetically different populations in this phenomenon (22,35), we used two allosteric models with different affinities. Although the allosteric models of the K_ACh/low and K_ACh/high channels are structurally identical, they have different dissociation constants (Fig. 1, top). Similarly to the previous allosteric model (21,34), the K_ACh channel consists of four subunits. The K_ACh channel is a heterotetramer of Kir3.1 and Kir3.4. Because Kir3.1 homotetramers are not functional and the electrophysiological properties of functional Kir3.4 homotetramers are quite different from those of the K_ACh channel (36), we used four subunits with equal affinity and aimed at the reproduction of the experimental activation curve (Fig. S1 A in the Supporting Material). All of the subunits in a given channel are defined as being in the same state, either tense (T) or relaxed (R), and the subunits change state together (concerted transition). One G_βγ protein equilibrates with one subunit in either the tense or relaxed state with distinct dissociation constants defined as K_T for tense and K_R for relaxed. A tetramer without any G_βγ transitions between the tense and relaxed states according to the equilibrium constant L. The channels in the relaxed state are considered available to open with fast gating kinetics, and the channels in the tense state are considered unavailable. G_βγ generated by the G-protein cycle model was used in the allosteric model. By solving a system of equations (21,34), the channel availabilities for populations with high and low affinities (NP_o,high and NP_o,low, respectively) can be expressed by Eqs. 1 and 2:

$$N{P}_{o,high}=\frac{\left(1+L{\left(\frac{{\text{K}}_{\text{R,high}}}{{\text{K}}_{\text{T,high}}}\right)}^4\right){\left(1+\frac{\left[{\text{G}}_{\beta \gamma }\right]}{{\text{K}}_{\text{R,high}}}\right)}^4}{L{\left(1+\left(\frac{{\text{K}}_{\text{R,high}}}{{\text{K}}_{\text{T,high}}}\right)\frac{\left[{\text{G}}_{\beta \gamma }\right]}{{\text{K}}_{\text{R,high}}}\right)}^4+{\left(1+\frac{\left[{\text{G}}_{\beta \gamma }\right]}{{\text{K}}_{\text{R,high}}}\right)}^4}$$ (1)

$$N{P}_{o,low}=\frac{\left(1+L{\left(\frac{{\text{K}}_{\text{R,low}}}{{\text{K}}_{\text{T,low}}}\right)}^4\right){\left(1+\frac{\left[{\text{G}}_{\beta \gamma }\right]}{{\text{K}}_{\text{R,low}}}\right)}^4}{L{\left(1+\left(\frac{{\text{K}}_{\text{R,low}}}{{\text{K}}_{\text{T,low}}}\right)\frac{\left[{\text{G}}_{\beta \gamma }\right]}{{\text{K}}_{\text{R,low}}}\right)}^4+{\left(1+\frac{\left[{\text{G}}_{\beta \gamma }\right]}{{\text{K}}_{\text{R,low}}}\right)}^4}$$ (2)

The same parameters (L, the ratio between K_T,i and K_R,i) validated in the previous study were used so that the experimental densitometry profiles of membranes containing Kir3.4 preincubated with G_βγ (i.e., the binding relationship between K_ACh channel and G_βγ) could still be reproduced (21,37) (Table 1). The values of K_T,high and K_R,high successfully reproduced the experimental activation curve of the K_ACh channel by G_βγ (38) (Fig. S1 A). A simple sensitive analysis was conducted to see the effects of the K_D of the K_ACh/low channel on the quantitative reproduction of the peak current (I_P) and quasi-steady-state current (I_S) (Fig. S2 A). Based on the analysis results the values of K_T,low and K_R,low were set to be increased threefold compared with those of K_T,high and K_R,high to represent low affinity of the K_ACh/low channel.

Figure 1.

Schematic diagrams of the allosteric model for the K_ACh channel and the model for receptor-G-protein interaction. Top: Schematic representation of the allosteric model. In this scheme, each K_ACh channel with either high or low affinities to G_βγ (K_ACh/high or K_ACh/low, respectively) is assumed to be an oligomer composed of four identical subunits. Each subunit is in either the available (R, relaxed) or unavailable (T, tense) state, which is represented by a circle or a square, respectively. Each subunit in the R or T state binds to one dissociated G-protein βγ subunit (solid circles) independently of the other subunits, with the microscopic dissociation constants K_R or K_T, respectively. In this model, all subunits in the same oligomer must change their conformations simultaneously. R₄ and T₄ are in equilibrium with the allosteric constant L. Bottom: Models of receptor-G-protein interaction. The low-affinity state of m₂Rs was incorporated into our previous model (21). A, ACh; R_high and R_low, high-affinity and low-affinity m₂Rs, respectively; G, G protein.

Table 1.

Parameters of the allosteric and G-protein cycle models

Parameter	Value	Units
L	1.41 × 103	–
KR,high	7.50 × 10−10	M
KT,high	1.36 × 10−8	M
KR,low	2.25 × 10−9	M
KT,low	4.09 × 10−8	M
k1H	2.50 × 106	s−1·M−1
k1L	2.50 × 104	s−1·M−1
k-1H	0.25	s−1
k-1L	0.25	s−1
k2	5.40 × 10−1	s−1·M−1
k-2	1.0 × 103	s−1
k3,4	2.86 × 103	s−1 M−1
k-4,-3[GDP]	0.68	s−1·M−1
k5	10	s−1
k6	0.03 + 2.4/(1 + e(−v−60)/−17)	s−1
[Rhigh]total	1.87 × 10−3	M
[Rlow]total	1.87 × 10−3	M
[G]total	5.60 × 10−2	M

The current-voltage relationship of I_KACh with 5.4 mM [K⁺]_o was obtained from a previous study (1) and fitted to the Boltzmann function: (v − E_K) / (1 + exp[{v − E_K − v_1/2} / k]). The fitted equation reproduces the current-voltage relationship of I_KACh (Fig. S1 B), which is used to calculate I_KACh in Eq. 3. To present the kinetically different gating kinetics of the two populations (22,35), we calculated I_KACh using Eq. 3:

$$I_{KACh}=g_{KACh}\left(N{P}_{o,high}+f\cdot N{P}_{o,low}\right)\frac{v-E_K}{1+e^{\frac{v-E_K+265.7}{53.6}}}$$ (3)

where v is the membrane potential, g_KACh is the maximum conductance for I_KACh, f is the desensitization gate variable for the K_ACh/low channel, and E_K is the equivalent potential for K⁺. The fast decrease in channel activity in short-term desensitization was associated with the gating kinetics of the K_ACh/low channel and characterized by exponential convergence to a steady state (22). Therefore, to represent this decrease, the kinetics of f were calculated using Hodgkin-Huxley-type equations:

$$f_{ss}=\frac{2.5}{1+e^{\frac{\left[G_{\beta \gamma }\right]-7.0\cdot {10}^{-9}}{1.0\cdot {10}^{-9}}}}$$ (4)

$${\tau }_f=10+\frac{110}{1.0+e^{\frac{\left[G_{\beta \gamma }\right]-7.0\cdot {10}^{-9}}{1.0\cdot {10}^{-9}}}}$$ (5)

where f_ss is the steady state of f, τ_f is the time constant for f, and [G_βγ] is the concentration of G_βγ in M.

G-protein cycle model

ACh causes a biphasic response in m₂R over a wide range of [ACh]s (25–29). Therefore, we simulated G_βγ generation by the G-protein cycle model (21) using two m₂Rs with either high or low affinity for ACh (Fig. 1, bottom). In the G-protein cycle model presented here, the reaction rates are represented by differential rate equations with rate constants and concentrations, as done in other studies (39–41). We calculated the values of 10 concentrations using the following ordinary differential equations (Eqs. 6–15):

$$\frac{d\left[R_{high}\right]}{dt}=-k_1^{high}\left[A\right]\left[R_{high}\right]+k_{-1}^{high}\left[A{R}_{high}\right]$$ (6)

$$\frac{d\left[R_{low}\right]}{dt}=-k_1^{low}\left[A\right]\left[R_{low}\right]+k_{-1}^{low}\left[A{R}_{low}\right]$$ (7)

$$\frac{d\left[A{R}_{high}\right]}{dt}=k_1^{high}\left[A\right]\left[R_{high}\right]-k_{-1}^{high}\left[A{R}_{high}\right]+k_{-2}\left[A{R}_{high}G-GDP\right]-k_2\left[A{R}_{high}\right]\left[G-GDP\right]+k_5\left[A{R}_{high}G-GTP\right]$$ (8)

$$\frac{d\left[A{R}_{low}\right]}{dt}=k_1^{low}\left[A\right]\left[R_{low}\right]-k_{-1}^{low}\left[A{R}_{low}\right]+k_{-2}\left[A{R}_{low}G-GDP\right]-k_2\left[A{R}_{low}\right]\left[G-GDP\right]+k_5\left[A{R}_{low}G-GTP\right]$$ (9)

$$\frac{d\left[A{R}_{high}G-GDP\right]}{dt}=-k_{-2}\left[A{R}_{high}G-GDP\right]+k_2\left[A{R}_{high}\right]\left[G-GDP\right]-k_{3,4}\left[A{R}_{high}G-GDP\right]\left[GTP\right]+k_{-4,-3}\left[A{R}_{high}G-GTP\right]\left[GDP\right]$$ (10)

$$\frac{d\left[A{R}_{low}G-GDP\right]}{dt}=-k_{-2}\left[A{R}_{low}G-GDP\right]+k_2\left[A{R}_{low}\right]\left[G-GDP\right]-k_{3,4}\left[A{R}_{low}G-GDP\right]\left[GTP\right]+k_{-4,-3}\left[A{R}_{low}G-GTP\right]\left[GDP\right]$$ (11)

$$\frac{d\left[A{R}_{high}G-GTP\right]}{dt}=k_{3,4}\left[A{R}_{high}G-GDP\right]\left[GTP\right]-k_{-4,-3}\left[A{R}_{high}G-GTP\right]\left[GDP\right]-k_5\left[A{R}_{high}G-GTP\right]$$ (12)

$$\frac{d\left[A{R}_{low}G-GTP\right]}{dt}=k_{3,4}\left[A{R}_{low}G-GDP\right]\left[GTP\right]-k_{-4,-3}\left[A{R}_{low}G-GTP\right]\left[GDP\right]-k_5\left[A{R}_{low}G-GTP\right]$$ (13)

$$\frac{d\left[G_{\alpha }-GTP\right]}{dt}=\frac{d\left[G_{\beta \gamma }\right]}{dt}=k_5\left[A{R}_{high}G-GTP\right]+k_5\left[A{R}_{low}G-GTP\right]-k_6\left[G-GTP\right]$$ (14)

$$\frac{d\left[G-GDP\right]}{dt}=k_6\left[G-GTP\right]+k_{-2}\left(\left[A{R}_{high}G-GDP\right]+\left[A{R}_{low}G-GDP\right]\right)-k_2\left(\left[A{R}_{high}\right]+\left[A{R}_{low}\right]\right)\left[G-GDP\right]$$ (15)

where A is ACh; R_high and R_low are m₂Rs with high and low affinities for ACh, respectively; G is the G_i/o protein; and k is the reaction rate constant. The value of each rate constant was fixed, except for k₆, which was defined as a function of membrane voltage (Table 1) to incorporate the regulation of G-protein signaling (RGS). RGS proteins regulate G-protein signaling by accelerating GTP hydrolysis in a voltage-dependent manner and are required to reconstitute short-term desensitization (42–44). On the basis of the voltage- and time-dependent characteristics of I_KACh in atrial myocytes known as relaxation (44–46), k₆ is defined as a function of membrane voltage so that depolarization of the membrane potential can accelerate GTPase activity. This relaxation of I_KACh reflects an increasing suppression of channel open probability during depolarization (i.e., strong accelerated GTP hydrolysis at depolarization) and a gradual recovery during hyperpolarization (i.e., less accelerated GTP hydrolysis at hyperpolarization). The voltage-dependent Ca²⁺ influx and Ca²⁺/CaM modulation of RGS protein activity have been suggested as one of the underlying mechanisms (44). We did not incorporate the possible voltage dependence of m₂R into our model because its electrophysiological function and properties have not yet been fully elucidated. The parameters in the G-protein cycle were mainly taken from the previous model (21) with minor modifications to incorporate the low-affinity interaction of m₂R with ACh. On the basis of experimental results (25–29), we introduced m₂R with low affinity to ACh into our model. The values of $k_1^{low}$ and $k_{-1}^{low}$ were selected so that low affinity and high affinity differed by a factor of 100 (47). By a simple, sensitive analysis, we verified that this value could reproduce the unique characteristics of short-term desensitization in a wide range of [ACh]s, whereas factors of 10 and 1000 could not reproduce the constant increase in I_P in the wide range of [ACh]s (Fig. S2 B). The values of k₂ and k_-2 were increased from those in the previous model so that the calculated G_βγ concentration would be sufficient to activate the allosteric model with a realistic activation curve. The G_βγ concentration generated by the G-protein cycle model was added to the allosteric model, and then channel availability and I_KACh were calculated.

Incorporation of the I_KACh model into the sinoatrial node models

We examined the effect of short-term desensitization on the action potential by incorporating the I_KACh model into two mathematical models of action potential in the rabbit sinoatrial node (30–32). The original I_KACh model in the sinoatrial node model was replaced with the I_KACh model constructed in this study. The membrane potentials were calculated at 0.1 μM and 10 μM ACh. ACh activates G_i proteins that reduce adenylyl cyclase activity and therefore inhibits the hyperpolarization-activated current (I_f) and the L-type Ca²⁺ current (I_Ca,L) in sinoatrial nodes. On the basis of the experimental dose-dependent effects of ACh for I_f and I_Ca,L in single cells isolated from rabbit sinoatrial nodes (48), the effects of ACh on I_f and I_Ca,L were modeled and incorporated into the sinoatrial models. In the simulation, 0.1 μM ACh inhibits only I_f by a negative shift of its activation curve by 7.0 mV, whereas 10 μM ACh inhibits I_f by a 9.9 mV shift, and inhibits I_Ca,L by reducing its maximum conductance by 12.5%.

Results

Simulation of I_KACh in response to ACh

First, induction of I_KACh by various [ACh]s was simulated using the constructed model (Fig. 2) and quantitatively compared with experimental results (Fig. 3). The simulation conditions were the same as in the previous study (2), i.e., the membrane potential was held at −53 mV and E_K was −87 mV. When various [ACh]s were applied, the outward I_KACh responses exhibited the typical characteristics of short-term desensitization (Fig. 2 A, solid lines). As [ACh] increased, the maximum I_KACh amplitude increased and the response was faster. At [ACh] > 0.1 μM, the I_KACh gradually decreased after reaching a peak. The time constants for the simulated decreases were 12.9 s, 10.5 s, and 10.3 s for [ACh] = 1 μM, 10 μM, and 100 μM, which are comparable to values observed in previous experiments (11.8 s (2) and 7.3–12.7 s (16)). The I_P during application of ACh was used to quantify the dependence of short-term desensitization on [ACh] (Fig. 3, row 1, upper solid line). The I_P increased constantly over a wide range of [ACh]s, in accord with the values recorded experimentally (Fig. 3, row 1, upper circles) (2). In contrast to the I_P, the I_S that is measured at the end of ACh application exhibited saturation of current increase at [ACh]s > 1 μM (Fig. 3, row 1, lower solid line). This response of I_S is consistent with that observed in isolated myocytes (Fig. 3, row 1, lower circles). The two K_ACh channel populations contributed to the responses of I_P and I_S: the I_P is composed of the current response of the K_ACh/low and K_ACh/high channels (Fig. 3, rows 2 and 3), whereas the I_S mainly consists of only the K_ACh/high channel (Fig. 2 A, dotted lines). When m₂R with low affinity for ACh was removed from the model, the increase in the amplitude of I_P was not observed at high [ACh]s (Fig. 2 B, solid line; Fig. 3, row 1, dashed lines). Further, removal of the K_ACh/low channels resulted in complete loss of short-term desensitization (Fig. 2 B, dotted lines). Previous mathematical models that dealt with the G-protein cycle (11) and were equipped with a single population of K_ACh channels (21) did not reproduce the responses of I_P and I_S over a wide range of [ACh]s. Therefore, mechanisms based on two different m₂Rs and K_ACh channels quantitatively account for the short-term desensitization of I_KACh.

Figure 2.

Simulated short-term desensitization. (A) Time course of I_KACh activation by various [ACh]s. The simulated responses of I_KACh at various [ACh]s (0.01–100 μM) are shown as solid lines. The value of g_KACh was set to 0.16 to reproduce the I_KACh amplitude in a quasi-steady state at −53 mV and 5.4 mM K_o (1). I_KACh in the absence of the K_ACh/low channel is shown as dotted lines. The bar above the current traces represents the period of ACh perfusion. The [ACh]s are indicated in μM at each current trace. Arrows indicate the zero current. (B) I_KACh in the absence of m₂R with low affinity for ACh. Simulated I_KACh without low-affinity m₂R to ACh is shown in the same manner as in (A). I_KACh in the absence of the K_ACh/low channel is shown as dotted lines.

Figure 3.

Quantitative analysis of simulated short-term desensitization. Dose-response curves for ACh-induced peak and quasi-steady I_KACh. Top panel: The simulated peak current (I_P) and quasi-steady-state current (I_S) are shown as solid lines. Experimental data (2) for peak and quasi-steady I_KACh are shown as circles with the standard deviation (SD). In the graph, the quasi-steady I_KACh induced by 100 μM ACh is expressed as one. I_KACh in the absence of m₂R with low affinity for ACh is shown as dashed lines. Middle panel: Dose-response curve for I_P through the K_ACh/high channel. The I_P through K_ACh/high channel is shown as a solid line. The currents are normalized as in row 1. Bottom panel: Dose-response curve for I_P through the K_ACh/low channel. The I_P through the K_ACh/low channel is shown in the same manner as in the middle panel.

Characterization of short-term desensitization in the I_KACh model

When atrial myocytes were serially perfused with bath solutions containing different [ACh]s, the amplitude of I_P in the second ACh application was decreased by the high [ACh] of the first application (2). Therefore, the degree of experimental short-term desensitization is influenced by preperfusion with ACh. Using our model, we calculated I_P during the second ACh application (I₁, 10 μM) after applying various [ACh]s (0.01–1 μM) (Fig. 4 A), and then compared the amplitude of the I₁ with that of the I_P elicited by 10 μM ACh applied after a 5-min interval (I₂) (Fig. 4 B). The I₁ amplitude during preapplication of ACh at 0.1 μM did not differ significantly from the I₂ amplitude (Fig. 4 A, top). However, when the [ACh] was increased to 1 μM, I₁ decreased to approximately one-half of I₂ (Fig. 4 A, bottom). The calculated ratio of I₁ to I₂ (Fig. 4 B, solid line) was comparable to that experimentally observed in atrial myocytes (Fig. 4 B, circles) (2). In contrast to the change in I₁ amplitude, the [ACh] in the first application did not influence the fraction of I_KACh through the K_ACh/high channel in I₂ (Fig. 4 A, dotted lines). Therefore, our model reproduces the decreased I_P amplitude after perfusion of high [ACh], suggesting that the desensitization of I_KACh during preperfusion of ACh is attributable to desensitization of the K_ACh/low channel.

Figure 4.

Analysis of simulated ACh preperfusion and recovery from short-term desensitization. (A) Traces of simulated I_KACh with ACh preperfusion. The bars above the current traces represent perfusion with various [ACh]s (indicated in μM). (B) Simulated dose response after ACh preperfusion. I_P induced with 0.01 to 1 μM ACh preperfusion (I₁) was plotted with reference to the control I_P value (I₂). The experimental data (2) are shown as circles. (C) Traces of simulated recovery from short-term desensitization by ACh. ACh (10 μM) was applied during the two periods indicated by the bars below the current traces. (D) Time dependence of simulated recovery from short-term desensitization. The values of the control I_P at 10 μM ACh before a washout period (I₃) were set at one. The ratio of I₄/I₃ in (C) was plotted to determine the time dependence of recovery from short-term desensitization. The experimental data (2) are shown as circles.

As shown in Fig. 4 A, the response of I_KACh to ACh is recovered from the desensitized state by perfusion with an ACh-free bath solution (2). Therefore, we determined the time course of recovery from desensitization. The I_P amplitudes elicited by the first and second applications of 10 μM ACh (I₃ and I₄, respectively) were compared when the administration interval was changed (Fig. 4 C). As the washout time was prolonged, the ratio of I₄ to I₃ gradually increased and returned to one within 5 min (Fig. 4 D, solid line). This temporal response of simulated recovery is nearly identical to that measured experimentally (Fig. 4 D, circles) (2). The recovery rate was dependent on the time constant for G_βγ-dependent desensitization of the K_ACh/low channel. Since the current amplitude through the K_ACh/high channel was nearly constant regardless of the interval duration (Fig. 4 C, dotted lines), this recovery can be attributable to the recovery of the K_ACh/low channel from the desensitized state. Thus, the I_P and I_S in simulated short-term desensitization are comparable to those obtained in experiments. Short-term desensitization in experiments is usually evaluated quantitatively only by using the percentage of desensitization of I_KACh at a certain, high [ACh] (11,14–16,49,50). We further validated the model by quantitatively comparing the simulated results of ACh preperfusion and recovery with the experimental results. Therefore, the major characteristics of short-term desensitization can be well captured by our I_KACh model on the basis of the assumption that the current is due to two populations of K_ACh channels with different affinities for G_βγ.

Effects of nucleotide-bound states of G-protein on short-term desensitization

The transient change in the population of nucleotide-bound G proteins has been proposed to be responsible for the short-term desensitization of I_KACh (11,14). Using our model, we examined the effects of nucleotides on the I_KACh response (Fig. 5). In the presence of 0.1 μM ACh, application of high GTP (200 μM) over a short time period induced short-term desensitization (Fig. 5 A, solid line). The continuous presence of excess GDP (1 mM) in the bath solution (Fig. 5 B) and reduction of the m₂R concentration by 50% (Fig. 5 C) effectively suppressed the GTP-induced short-term desensitization. These results are consistent with previously reported experimental results (11) and can be accounted for by the different degrees of the activation of the G-protein cycle (Fig. S3, B, D, and F). Next, we changed the rate parameters for nucleotide exchange on the G protein to reveal how its nucleotide-bound state affects short-term desensitization. When the rates for GDP/GTP exchange increased fivefold, ACh-induced short-term desensitization was strongly enhanced (Fig. 5 D, solid line). In contrast, when the rates were lowered to one-half, ACh-induced short-term desensitization was suppressed (Fig. 5 E). This is because fast GDP/GTP exchange enhances the formation of GTP-bound G protein (Fig. S3, H and J). Therefore, the nucleotide-bound state of G protein appears to be important in the development of short-term desensitization. However, when the K_ACh/low channel was omitted from the model, short-term desensitization was not produced under any condition (Fig. 5, dotted lines). Therefore, our theoretical analysis suggests that the K_ACh/low channel plays an essential role in reconstitution of short-term desensitization, and an additional role in modulating the nucleotide-bound state of G proteins.

Figure 5.

Effect of GDP/GTP exchange on short-term desensitization. (A) Traces of simulated I_KACh elicited by 200 μM GTP. The bar above the current traces represents the period of GTP perfusion. To represent basal I_KACh activity in the corresponding experimental condition (11), 0.1 μM ACh was applied during the simulation period. I_KACh through the K_ACh/low channel only is shown as a dotted line. The arrowheads indicate the zero current level of each trace. The original values of initial [R_high] and [R_low] were used to represent high receptor expression in the experiments (11). (B) Effect of 1 mM GDP on I_KACh induced by 200 μM GTP. I_KACh was elicited in the same way as in A except that 1 mM GDP was applied during the entire simulation period. (C) Traces of simulated I_KACh elicited by 200 μM GTP with low receptor expression. I_KACh was elicited in the same way as in A except that [R_high] and [R_low] were lowered to 50% to represent low receptor expression in the experiments (11). (D) Effect of high GDP/GTP exchange activity on ACh-induced short-term desensitization. GDP/GTP exchange was enhanced by increasing the rate constants for GDP/GTP exchange (k_3,4 and k_-4,-3) by a factor of 5. The period during which ACh was perfused is shown as a bar above the current trace. (E) The effect of low GDP/GTP exchange activity on ACh-induced short-term desensitization. GDP/GTP exchange was lowered by reducing the rate constants to one-half.

Simulation of the effect of membrane potentials on short-term desensitization

M₂R-dependent G protein signaling is enhanced at hyperpolarization (44,46). Because our I_KACh model retains the voltage dependence incorporated into the previous model (21), we tested how membrane potential affects short-term desensitization of I_KACh (Fig. 6). At 10 μM ACh, short-term desensitization occurred even at depolarized potential (0 mV), but it became more distinct at hyperpolarized potential (Fig. 6 A). Short-term desensitization was calculated in the presence of various [ACh]s at −80 mV, −50 mV, and 0 mV (Fig. 6 B). At hyperpolarization (−80 mV), I_KACh started to exhibit short-term desensitization even at 0.1 μM ACh. In contrast, at depolarization (0 mV), this transient current response developed at [ACh] > 1 μM. These simulations demonstrate that hyperpolarization shifts the [ACh] threshold for short-term desensitization to lower concentrations.

Figure 6.

Voltage dependence in short-term desensitization. (A) The effect of 10 μM ACh on short-term desensitization at different membrane voltages. I_KACh was activated by 10 μM ACh at potentials of 0 mV, −50 mV, and −80 mV. Arrows indicate the zero current levels of each trace. (B) Voltage-dependent dose-response curves for peak and quasi-steady I_KACh. The peak current (I_P) and quasi-steady-state current (I_S) at clamped membrane potentials of −80 mV, −50 mV, and 0 mV are shown as blue, red, and yellow lines, respectively. I_S induced by 100 μM ACh was set to one.

The role of short-term desensitization in action potential generation in the sinoatrial node

The two incorporated mechanisms quantitatively reconstituted short-term desensitization in the I_KACh model. We next investigated how their characteristics were involved in the regulation of cardiac function. To address this question, we replaced the I_KACh portion in the action potential models in the rabbit sinoatrial node with our I_KACh model. Two models of cardiac pacemaker activity, the Demir model (30,31) and the Kurata model (32), were subjected to the simulation. The former is a classical Hodgkin-Huxley-type model, and the latter is a model with updated ion channels and intracellular Ca²⁺ models. Application of low [ACh] (0.1 μM) increased outward I_KACh (Fig. 7 A, rows 2 and 4) and reduced the action potential frequency by 14.5% in the Demir model (Fig. 7 A, row 1) and by 12.4% in the Kurata model (Fig. 7 E, row 1). These values are in the range of the experimental responses to the applied [ACh]: 0.1 μM ACh decreased the action potential frequency by 0% (51) or 30% (30,52). The cycle length was prolonged by 46.0 ms and 43.0 ms, the maximum diastolic potential was reduced by 1.0 mV and 9.7 mV, and action potential amplitude was increased by 1.1 mV and 9.0 mV in the Demir model and Kurata model, respectively. In contrast, application of 10 μM ACh suspended the spontaneous generation of action potential in the two models (Fig. 7, B and F, row 1). This suspension was caused by the strong activation of I_KACh (Fig. 7, B and F, row 2). However, action potential firing resumed in 8.9 s in the Demir model and in 10.7 s in the Kurata model. These time periods are comparable to those reported for the recovery from vagal escape observed in myocytes isolated from rabbit sinoatrial nodes (52). This phenomenon was observed even when a wide range of different values of g_KACh were used (Fig. S4) or when ACh modulation on I_f and I_Ca,L was not incorporated. Resumption was mainly due to the gradual decrease in the outward current. Moreover, the hyperpolarization may facilitate short-term desensitization (see Fig. 6).

Figure 7.

Role of short-term desensitization in spontaneous firing of the sinoatrial node. (A and E) Effects of 0.1 μM ACh on spontaneous firing in the Demir and Kurata sinoatrial node models. The current I_KACh model was incorporated into the Demir (A) and Kurata (E) sinoatrial node models (30–32) and 0.1 μM ACh was applied during the period indicated by the bar. The dashed lines indicate the zero level of each trace. The value of g_KACh was set at 1.1 (pS/pF) for the Demir model and at 6 (pS/pF) for the Kurata model. (B and F) Effects of 10 μM [ACh] on spontaneous firing. Parameters are the same as in A and E except that 10 μM ACh was applied. (C and G) Effects of 10 μM ACh on the K_ACh/high channel only. Parameters are the same as in B and F except that the K_ACh/low channel was replaced with the K_ACh/high channel. (D and H) Effects of 10 μM ACh on the K_ACh/low channel only. Parameters are the same as in B and F except that the K_ACh/high channel was replaced with the K_ACh/low channel.

To examine the effects of the two populations of K_ACh channels on the resumption, we replaced our I_KACh model with an I_KACh model that contained only K_ACh/high channels or only K_ACh/low channels. In both action potential models of the sinoatrial node, incorporation of the I_KACh model with the K_ACh/high channel alone resulted in cardiac asystole with 10 μM ACh, and failed to resume during ACh application (Fig. 7, C and G). In contrast, the I_KACh model with the K_ACh/low channel alone resulted in brief suspension of spontaneous firing (Fig. 7, D and H). These results suggest that the K_ACh/low channel is critical for the linkage between short-term desensitization and vagal escape.

Discussion

Short-term desensitization of I_KACh is an adaptation mechanism for excessive vagal nerve stimulation in cardiac myocytes. In this study, we addressed the mechanisms underlying short-term desensitization of I_KACh theoretically. On the basis of experimental evidence, we hypothesized the existence of two populations of m₂R and K_ACh channels with different affinities to ACh and G_βγ, respectively. The introduction of m₂Rs enabled the model to respond to a wide range of [ACh]s. The introduction of K_ACh channels resulted in quantitative reproduction of the temporal behavior of the I_KACh current in short-term desensitization. Furthermore, the model conferred vagal escape on the mathematical action potential models of sinus node cells. These results allow us to propose that two functionally distinct populations of m₂Rs and K_ACh channels underlie the physiological I_KACh response of short-term desensitization.

To date, both G protein and the K_ACh channel have been proposed to be responsible for short-term desensitization. Experimental manipulations that lead to a rapid increase in free G_βγ induce short-term desensitization (11,14,16,42,43,53). However, neither previous models (11,21) nor our model without the K_ACh/low channel quantitatively reconstituted this phenomenon. This is simply because a linear model of the G-protein cycle cannot represent the transient response of I_KACh. We hypothesized the K_ACh/low channel as the missing component. This fraction is likely a K_ACh channel with a different gating kinetic (35) or modulatory system, such as PIP₂ (23,54) and phosphorylation (24), as discussed below. The K⁺ ion passing through the K⁺ channels has been proposed to account for short-term desensitization (15). This theory is based on the shift of E_K by K⁺ flux in the microspace in close proximity to the plasma membrane. This explanation, however, cannot account for the constant I_S after various amplitudes of I_P under a wide range of [ACh]s. Although our mathematical model reproduced the effects of not only m₂R stimulation (Figs. 2–4) but also G-protein-cycle modification (Fig. 5), there may be some discrepancy between the physiological phenomena and the model design and parameters. However, the theory we propose here is robust to various perturbations (Figs. 2–5; Fig. S4). Therefore, possible deviations in the model structure and parameters would not weaken our conclusion.

Although two different m₂R populations with high and low affinities for ACh are well known, this has been ignored in previous models (25–29). Experimentally, cardiac m₂R shows a biphasic response over a wide range of [ACh]s (to a factor of 10⁴) that cannot be detected by a single receptor population with a single affinity. One cellular mechanism that could generate two distinct populations of K_ACh channel is different levels of PIP₂. PIP₂ is known as a prerequisite for K_ACh channel activation (23) and as a modulator for its gating kinetics (54). It has become apparent that PIP₂ is enriched in membrane microdomains such as lipid rafts (55). Furthermore, the constituents of G-protein signaling and PIP₂ metabolism are distributed differently within the plasma membrane, and after various stimuli, these constituents change their position with respect to these microdomains (49,56). In contrast, whereas phosphorylation of the K_ACh channel subunits (Kir3.1 and Kir3.4) by protein kinase A reportedly increases their open probability (24), phosphorylation of Kir3.1 by protein kinase Cδ has been shown to reduce PIP₂ sensitivity (57). These reports suggest that phosphorylation state also has the potential to generate quantitatively different K_ACh channel populations. Therefore, although the precise mechanisms that contribute to the generation of two distinct K_ACh channel populations are not known, various cellular signals may account for this phenomenon.

This study also demonstrates the role of short-term desensitization in vagal escape. Vagal escape usually refers to the two responses of the heart to continuous vagal stimulation: the gradual return of the heart rate toward control level (58,59) and the development of pacemaker activity in multimodal origins (60). Compensation from the sympathetic system has been proposed to cause vagal escape (17–20). However, since this phenomenon could be observed even in isolated sinoatrial nodes in the presence of a β blocker (52), the cellular machinery associated with the postjunctional mechanism was expected to participate in vagal escape. The two assumptions that we proposed in this study are in agreement with this criterion. Therefore, the function of these two quantitatively different populations of m₂Rs and K_ACh channels in short-term desensitization might be the missing cellular mechanisms in vagal escape from excessive parasympathetic nerve stimulation at the organ level. In general, the high-affinity m₂R is most often measured and assumed to be physiologically relevant. However, it has been suggested that activation of the low-affinity m₂R, rather than the high-affinity m₂R, is responsible for atrial bradycardia in mice (27). The effects of RSG4 are another interesting link between short-term desensitization and vagal escape. In one study, the spontaneous generation of action potential firing in RGS4-null sinoatrial node myocytes was suspended by application of high concentrations of carbachol, whereas that of the wild-type was resumed (53). RGS proteins are also required to reconstitute short-term desensitization without affecting the amplitude of the I_KACh (43). Because acceleration of the G-protein cycle enhances short-term desensitization (Fig. 5), RGS proteins are also important components to identify links between short-term desensitization and vagal escape in vivo.

In this study, we simulated short-term desensitization with a linear model of the G-protein cycle, which was improved on the basis of previous models (21). Although our model traced the I_KACh recorded from electrophysiological experiments performed with isolated atrial myocytes under physiological conditions well (2) (Figs. 2–4), qualitative comparisons of the effects of nucleotides on the experiment and simulation remain to be done. Our model structure is different from that proposed by Chuang et al. (11), in which the nucleotide-free state of G proteins is responsible for short-term desensitization. In their qualitative reproduction of short-term desensitization, a large amount of G protein was required to populate the nucleotide-free state. The lack of consideration for such a drastic case might appear as a limitation of this study. However, when a model structure is linear, intracellular signaling cascades within the G-protein cycle do not appear to reproduce short-term desensitization quantitatively (Fig. 5). In other words, if a signal component is identified to transiently influence the receptor-dependent K_ACh channel activation, as we hypothesized in this study for the K_ACh/low channel, it could become another candidate for the mechanism of short-term desensitization. On the other hand, as shown in Fig. 6, our model possesses voltage dependence as a modulator of RGS protein activity (44–46). Ligand binding in m₂R has been reported to show voltage dependence (61–63). How these voltage-dependent components should be considered with regard to the regulation of I_KACh may need to be clarified in future studies. By incorporating new knowledge, the model presented here can be used to gain more insights into short-term desensitization as well as other phenomena associated with I_KACh.