For decades, the contribution of I_K,ACh to the modest chronotropic effects of ‘physiological’ or low-dose ACh has been debated [10–12]. Based on the M2R voltage-dependent properties, we predict that subsaturating ACh concentrations exert a larger effect during diastolic (hyperpolarized) membrane voltages, compared to the voltages during the cardiac AP (depolarized) [3, 5]; thus preferentially slowing the pacemaker rate with minimal effect on APD. To test this hypothesis, we simulated the effects of varying low, sub-saturating concentrations (e.g., 20–100 nM) of ACh on sinus node AP properties (Fig 1). The most striking effect of increasing ACh concentration was reduced slope of phase 4 depolarization and the corresponding increase in basic cycle length (Fig 1A, Table 1, V_shift = 0), with minimal shortening of APD₉₀ (Table 2, V_shift = 0). Thus, the basic cycle length (BCL) increased from 827 ms in the absence of ACh, to 1585 ms in the presence of 0.1 μM ACh. The AP amplitude decreased steadily by up to 4.7 mV at the highest tested concentration of 0.1 μM ACh (inset of Fig 1A). Simulated open probability O and I_K,ACh for different ACh concentrations are shown in Fig 1B and 1C. Taken together, these simulations indicate that subsaturating concentrations of ACh slow spontaneous excitation of the SN cell by inhibiting the rise of phase 4 depolarization, without appreciably shortening APD₉₀ or reducing the amplitude of the AP.

In our previous experimental studies using isolated feline left atrial myocytes, the voltage dependence of M2R was explored by measuring the ACh concentration-I_K,ACh response relationship at hyperpolarized (-100 mV) and depolarized membrane voltages (+50 mV) [3, 5]. These experiments indicated that the affinity of the receptor for ACh was greater at hyperpolarized membrane voltages, compared to depolarized voltages. We reasoned that, similar to voltage-gated ion channels, putative disease-associated mutations in M2R might alter the voltage sensitivity of the M2R, with unique consequences for sinus node AP properties and heart rate responses. We modified rate parameters (Eqs 5 and 6) to shift the receptor occupancy towards a hyperpolarized state (higher affinity) or a depolarized state (lower affinity) (S1A Fig). Thus, we simulated the effects of shifting the M2R voltage sensitivity (Fig 2, S3C Fig). S1B Fig highlights the effects of voltage shifts on the state O. Negative shifts (e.g., V_shift of -30 mV and -150 mV), which caused a more hyperpolarized state of the receptor, increased the occupancy of the U1 and B1 states, as well as the state O. Negative voltage shifts resulted in a slight leftward shift in the concentration-response curve when the cell was held at +50 mV (Fig 3E). Likewise, positive shifts in M2R voltage sensitive parameters caused a slight rightward shift in the concentration-response curve for a holding potential of V_h = -100 mV (Fig 3A). To avoid local minima during the parameter fitting, the model was forced to favor the open state (state O equal to 1), at the maximum concentration of 10 μM ACh and to favor the closed state (O equal to 0), with no ACh present. Further, U1 was forced to be as high as possible at a holding potential of -100mV and U2 at +50 mV, in the absence of ACh. Thus, negative and positive voltage shifts did not move the steady-state concentration-response relationships outside of these ranges. Notwithstanding, the voltages experienced by the single cell model vary between -60 and +30 mV (Fig 1A), within the minimum and maximum ranges defined by the parameter fitting.

We previously described the kinetics of “relaxation” gating of I_K,ACh in terms of activation and deactivation of K_ACh channels in the setting of subsaturating ACh concentrations [5]. Activation kinetics were measured by first stepping to a depolarized voltage (+60 mV) to close a large portion of K_ACh channels at a physiological voltage, followed by stepping through a range of voltages to measure activation of I_K,ACh. Deactivation kinetics were assessed by a pre-pulse to a hyperpolarized voltage (-100 mV) to open channels, followed by variable test voltage steps to measure the rate of K_ACh channel closure. Accordingly, simulated I_K,ACh evoked by 0.1 μM ACh using the activation and deactivation voltage protocols are presented in S1 Fig and the kinetic parameters are described in Table 3. The simulations recapitulate the experimental features of I_K,ACh relaxation gating [5], as shown in S1 Fig. The effects of voltage shifts in M2R voltage sensitive parameters on I_K,ACh relaxation gating parameters are shown in Fig 3.

Next, we simulated the effects of negative and positive voltage shifts in M2R voltage-sensitive parameters on sinus node APs, together with the corresponding effects on state O and I_K,ACh (Fig 4). Negative shifts (e.g., V_shift of -10 mV and -30 mV), which would be predicted to increase ACh affinity, reduced the slope of phase 4 depolarization in a concentration-dependent manner, with minimal effects on AP amplitude or APD₉₀ (Fig 4A versus 4D, Tables 1 and 2). The reduction in the slope of phase 4 depolarization induced by a negative V_shift was due to an increase in the open probability of K_ACh channels relative to the control condition, thereby increasing the magnitude of I_K,ACh. These results indicate that shifts in the M2R voltage sensitivity impact the rate of spontaneous depolarization of sinus node APs.

To further characterize the physiological consequences of positive and negative voltage shifts, we studied the effects of variable M2R voltage sensitivity on the ACh concentration-heart rate response relationship (Fig 5A). Our model recapitulates experimental data [12] indicating that ACh concentrations ranging from 0.01 to 0.1 μM induce slowing of the spontaneous activity by 10% and 45%, respectively. Hyperpolarizing shifts in the M2R voltage dependent parameters shifted the relationship toward progressive spontaneous activity slowing. By contrast, depolarizing shifts in these parameters antagonized the spontaneous activity slowing induced by ACh. Taken together, these results suggest that shifts in M2R voltage sensitive parameters exert significant physiological effects on SN firing rate and AP parameters.

Next, we quantified the effects of the individual ACh-influenced cardiac currents on heart rate slowing. Fig 5B illustrates the relative contributions of ACh-sensitive currents, including our new model of I_K,ACh, together with formulations of the L-type Ca2+ current (I_Ca,L) and the hyperpolarized-activated ‘funny current’ (I_f) from Fabbri et al. [6]. Inspection of the individual contributions of the ACh-sensitive currents reveals that our model of I_K,ACh (incorporating the voltage-sensitivity of M2R) accounts for roughly 50% of the slowing at ~30nM and more for higher concentrations. These results highlight the important contribution of M2R voltage sensitivity to heart rate slowing induced by ACh.

There are several limitations inherent in the application of the model to describe the kinetics and behavior of I_K,ACh. First, the receptor-channel model was fit and optimized to recreate the kinetics of I_K,ACh occurring at a concentration of 0.1 μM ACh. While this is within the range of measured concentrations of ACh, measurements at lower concentrations could enable a more accurate reconstruction of the behavior of I_K,ACh and a direct comparison to the effect of I_f at such concentrations. Furthermore, we acknowledge that the description of the process of dissociation of G_βɣ from the M2R to the activation of the K_ACh channel is highly simplified. We used a simple channel opening description with a 2-state Markovian model neglecting different binding properties of different Kir subunits or cooperativity mechanisms in binding of G_βɣ. This simplification was necessary as parameters for more complex models are not identifiable with the existent experimental data. Also, the model is not able to reproduce the distinct characteristics of the deactivation protocol time constants. The experimental data indicate that the deactivation time constant is nearly voltage-independent (Fig 3H), the model predicts an increase with membrane depolarization. Nonetheless, because the other simulated features have a very high similarity to the measured values (Fig 3B, 3C, 3F and 3G) and the time constants are in the range of less than half the length of an action potential, we believe that any error introduced does not significantly influence our findings. Additionally, dissociation or binding of G_βɣ in our model does not account for changes in the process due to other influences or any binding to sites other than the K_ACh channel.

The SN cell model recapitulates the experimental findings that ACh inhibits I_f and I_Ca,L, which also contribute to slowing of spontaneous pacing rate [23]. We argue that unlike I_K,ACh, the M2R voltage dependent effects do not influence I_f and I_Ca,L on the time scale of the AP. I_f and I_Ca,L inhibition are mediated by inhibition of the cAMP-dependent protein kinase A cascade that functions on a much slower time scale than the APD. By contrast, our simulations demonstrate that voltage dependent conformational changes in M2R influence I_K,ACh throughout the cardiac AP, modulating both firing rate and APD. Finally, because our model does not fully integrate all the components of the autonomic nervous system, it is possible that the effects of the putative genetic mutations might be mitigated by compensatory changes in the autonomous nervous system’s response to changes in heart rate.

The recent observation that M2Rs are intrinsically voltage sensitive has important implications for understanding the physiology and pathophysiology of parasympathetic regulation of heart rate and APD. By optimizing and integrating a new Markovian model into a human SN model, we show that low ACh concentrations preferentially slow beating rate, without shortening APD, and thereby provide additional support that I_K,ACh participates in the purely chronotropic effects of basal vagal tone. Moreover, we explore the effects of altered M2R voltage sensitivity and provide a proof-of-principle foundation that altered sensitivity could result in clinical manifestations of disease states such as vagally-mediated atrial fibrillation and syndrome of inappropriate sinus tachycardia. Given the importance of parasympathetic regulation of atrial vulnerability, M2Rs represent an important therapeutic target to control or prevent atrial arrhythmias.

We developed a Markovian model to reconstruct the behavior of K_ACh channels at different ACh concentrations and varying transmembrane voltages (Fig 2). The model comprises 3 sub-models: (1) A Markovian model describing the kinetics of the M2R depending on different concentrations of ACh at different voltages (M2R model), (2) a Markovian model describing the activation of the K_ACh channel based on dispersion of G_βɣ protein from the receptor to the channel (K_ACh channel model), and (3) a model of potassium current through K_ACh channels.

The parameters of the model were determined by iterative stochastic optimization as previously described [5]. Model parameterization was implemented in Matlab R2017a (The Mathworks Inc., Natick, MA) and the Matlab Parallel Computing Toolbox. An error function based on the root mean squared differences of the measured versus simulated features of the activation protocol, deactivation protocol, and concentration response curve from [5] was minimized.

Measured features were based on whole-cell voltage-clamp experiments [5]. I_K,ACh for the activation and deactivation clamp protocols was recorded in the presence of 0.1 μM ACh. The 3 best measured currents, see S1 Fig, from [5] of each protocol were then fitted to a mono-exponential equation, averaged and then normalized to C_off at -10 mV: I(t)=C+Aexp(-tτ) (10) with the constants A and C, and the time-constant τ of activation (on) and deactivation (off). We choose this strategy, as extracting mono-exponential features and then averaging them introduces less error in the overall reconstructed behavior than averaging the signals themselves. The same fitting approach was used during the model parameterization.

Concentration response curves were measured as the resulting I_K,ACh at either a holding potential of -100 mV or +50 mV at different ACh concentrations (Fig 5A and 5E). The currents were then normalized to the current measured at maximal ACh concentration for each holding potential.

A total of 8 simulated features f_s,i, consisting of six features for the voltage clamp protocols (i.e. C_on, C_off, A_on, A_off, τ_on, and τ_off) and two for the concentration response curves (i.e. I_{KACh,norm,50mV} and I_{KACh,norm,-100mV}), were compared to their corresponding measured features f_m,i (Fig 5B–5D and 5F–5H): E2=∑i=18(||fm,i−fs,i||2||fs,i||2)2+(1−max(O))+((1−max50mV(U2))2+(1−max−100mV(U1))2)0μM+((1−max50mV(B2))2+(1−max−100mV(B1))2+(1−(B1+B2)2))10μM (11)

Other components of the cost function were used to constrain the behavior of the model at specific ACh concentration and V_m, and to ensure high open probability of the channel. Without bound ACh and V_m of +50 mV, the state U2 was forced to be maximal. Respectively, U1 was forced to be maximal without bound ACh and a transmembrane voltage of -100 mV. Equivalently, with bound ACh, state B2 and B1 were forced to be maximal at -100 mV and 50 mV, respectively. Further, the sum of B1 and B2 was forced to be maximal with bound ACh.

For simulations of single cell electrophysiology, the new receptor-channel model was integrated in the publically available model of human SN cells [6]. The formulations for the effect of ACh on I_CaL and I_f were left unaltered throughout the experiments. The model was first exported to Matlab using OpenCOR (www.opencor.ws) and then modified accordingly. Numerical integration was performed by using the integrated ode15s formulation provided by Matlab. We measured BCL to characterize the rate of spontaneous activation of the simulated SN cell at varying concentration of ACh and V_shift. The maximum conductivity g_K,ACh,max was set 0.0022 μS to reproduce previously published heart rate slowing in the presence of 0−0.1 μM ACh [12]. Furthermore, we measured the APD at 90% repolarization (APD₉₀) to characterize the cardiac AP.

The stochastic parameterization yielded the model parameters (Table 3). In comparison to the parameterization of our previous model [5], the fit error of the features from the activation and deactivation protocols was reduced from 1.5 to 0.68 despite the additional error terms. Respectively the squared fit error for each feature is; C_on: 0.06, C_off: 0.025, A_on: 0.11, Aoff: 0.014, t_on: 0.04, t_off: 0.18, I_{KACh,norm,50mV}: 0.03, I_{KACh,norm,-100mV}: 0.003. The total error of the features is equal to the square root of the sum of the respective squared fit errors. The corresponding modeled and measured current traces of the voltage protocols are shown in S2 Fig.