The Functional Role of Hyperpolarization Activated Current (If) on Cardiac Pacemaking in Human vs. in the Rabbit Sinoatrial Node: A Simulation and Theoretical Study

The cardiac hyperpolarization-activated “funny” current (I_f), which contributes to sinoatrial node (SAN) pacemaking, has a more negative half-maximal activation voltage and smaller fully-activated macroscopic conductance in human than in rabbit SAN cells. The consequences of these differences for the relative roles of I_f in the two species, and for their responses to the specific bradycardic agent ivabradine at clinical doses have not been systematically explored. This study aims to address these issues, through incorporating rabbit and human I_f formulations developed by Fabbri et al. into the Severi et al. model of rabbit SAN cells. A theory was developed to correlate the effect of I_f reduction with the total inward depolarising current (I_total) during diastolic depolarization. Replacing the rabbit I_f formulation with the human one increased the pacemaking cycle length (CL) from 355 to 1,139 ms. With up to 20% I_f reduction (a level close to the inhibition of I_f by ivabradine at clinical concentrations), a modest increase (~5%) in the pacemaking CL was observed with the rabbit I_f formulation; however, the effect was doubled (~12.4%) for the human I_f formulation, even though the latter has smaller I_f density. When the action of acetylcholine (ACh, 0.1 nM) was considered, a 20% I_f reduction markedly increased the pacemaking CL by 37.5% (~27.3% reduction in the pacing rate), which is similar to the ivabradine effect at clinical concentrations. Theoretical analysis showed that the resultant increase of the pacemaking CL is inversely proportional to the magnitude of I_total during diastolic depolarization phase: a smaller I_f in the model resulted in a smaller I_total amplitude, resulting in a slower pacemaking rate; and the same reduction in I_f resulted in a more significant change of CL in the cell model with a smaller I_total. This explained the mechanism by which a low dose of ivabradine slows pacemaking rate more in humans than in the rabbit. Similar results were seen in the Fabbri et al. model of human SAN cells, suggesting our observations are model-independent. Collectively, the results of study explain why low dose ivabradine at clinically relevant concentrations acts as an effective bradycardic agent in modulating human SAN pacemaking.

Introduction

The pacemaker activity of sinoatrial node (SAN) cells in the mammalian heart arises from the integrated action of multiple sarcolemmal ionic channel currents and the interaction between the intracellular calcium handling and sarcolemmal electrogenic processes (Irisawa et al., 1993; Mangoni and Nargeot, 2008; Lakatta et al., 2010). The hyperpolarization-activated “funny” current, I_f, present in the SAN and other regions of the cardiac conduction system (Boyett, 2009; Difrancesco, 2010), is produced by the hyperpolarization-activated cyclic nucleotide gated (HCN) channel isoforms (of which there are four: HCN1-4), each of which is comprised of six transmembrane domains with four subunits combining to produce functional tetrameric channels, as occurs for voltage-gated potassium channels (Bois et al., 2007; Difrancesco, 2010). Previous studies of the rabbit SAN have shown that although HCN isoforms 1, 2, and 4 are all expressed in the heart, HCN4 is the most abundant in the SAN and the I_f density within SAN sub-regions correlates strongly with HCN4 expression levels (Thollon et al., 2007; Brioschi et al., 2009). The unique features of HCN channels lie in the fact that they are activated not on depolarization but on hyperpolarization of cell membrane potential (to voltages negative to ~−40 to ~−50 mV) (Hagiwara and Irisawa, 1989; Accili et al., 1997; Baruscotti et al., 2005) and are permeable to both Na⁺ and K⁺ ions, with an approximate reversal potential of −30 mV (Van Ginneken and Giles, 1991; Verkerk et al., 2009a). Upon hyperpolarization, HCN channels generate an inward current over the pacemaking potential range which, together with the current generated by other electrogenic processes of the intracellular calcium handling (i.e., the Ca²⁺ clock), contributes to the genesis of intrinsic pacemaker activity of the SAN (Lakatta and Difrancesco, 2009). HCN channels are also modulated by adrenergic agonists via cAMP (Bucchi et al., 2003; Craven and Zagotta, 2006).

I_f channels are also present in the human SAN. It has been shown that HCN4 is strongly expressed in the human SAN, with the measured mRNA level of other isoforms accounting for <16% (with HCN3 being negligible, accounting for only 0.5%) of the total mRNA measurement (Chandler et al., 2009). Though the HCN expression in the human SAN is similar to that in the rabbit, properties and kinetics of I_f are clearly different. It has been shown the fully-activated I_f conductance in the human is about 3–4 times smaller than that in the rabbit (Verkerk et al., 2007a). Additionally, I_f in human SAN cells has a more negative half-maximal activation voltage, and a greater time constant of deactivation/activation process which is also negatively shifted (Verkerk et al., 2007a). With such a marked species differences in I_f conductance and kinetics between the rabbit and human, a question arises as to whether a smaller I_f in the human SAN cells plays the same important role in regulating cardiac pacemaking activity as in the rabbit SAN?

Although I_f in human SAN cells is much smaller than those in other mammals, it may play a comparable role to that in the rabbit in modulating cardiac pacemaking. In their study, (Verkerk et al., 2007b) observed about a 26% increase in pacemaking cycle length in human SAN cells on complete block of I_f by using 2 mM Cs⁺, which is close to that seen in the rabbit (Verkerk and Wilders, 2010; Fabbri et al., 2017). Pharmacological targeting of I_f by ivabradine has also shown the clinical value of I_f in controlling the heart rate in patients who need heart rate control in conditions of coronary artery disease (CAD) (Tardif et al., 2005; Camici et al., 2016; Niccoli et al., 2017) and heart failure (HF) (Bohm et al., 2015; Yancy et al., 2016). In both conditions, slowing down the heart rate by ivabradine increases the diastolic interval, thereby reducing the metabolic load on the working myocardium; this reduces the risks of ischemia of the heart leading to reduced risk of sudden death (Niccoli et al., 2017).

Although inhibition of I_f by ivabradine provides an efficient pharmacological control of heart rate in the clinic, it is still unclear how the clinical concentration range of ivabradine [about 20–140 nM (Choi et al., 2013; Jiang et al., 2013); blocking I_f < 20% (Bois et al., 1996; Bucchi et al., 2002)] can produce a significant effect in reducing human heart rate. In pre-clinical animal model studies, ivabradine has been reported to inhibit I_f in SAN cells with a half-maximal inhibitory concentration of 1.5–2.8 μM (Bois et al., 1996; Bucchi et al., 2002), and recombinant HCN4 channels with an IC₅₀ values between 0.5 and 2.0 μM (Bucchi et al., 2002, 2006, 2013). In rabbit SAN cells, 1 μM ivabradine has been observed to reduce the pacemaking rate by 12.3% (Thollon et al., 1994), whilst about 16.2 and 23.8% of heart rate reduction at 3 μM have been seen (Thollon et al., 1994; Bucchi et al., 2007).

When ivabradine was administered intravenously (0.2 mg kg⁻¹) to patients with normal baseline electrophysiology, mean heart rate reductions of 12.9 and 14.1 beats min⁻¹ (at 0.5 and 1 h respectively following administration) were observed (Camm and Lau, 2003). The drug is usually administered orally, however, and after repeated oral dosing at 5 mg, mean maximal plasma (C_max) levels of 11–16 ng ml⁻¹ (23.5–34.1 nM) have been measured, whilst for repeated dosing at 10 mg, mean C_max levels of 29–42 ng ml⁻¹ (61.8–89.6 nM) have been seen (Choi et al., 2013; Jiang et al., 2013). Repeated dosing with a high ivabradine concentration of 20 mg has been associated with a C_max of 137 nM (Jiang et al., 2013). On the basis of the pre-clinically observed concentration-dependence for I_f inhibition, comparatively low levels of I_f block might be expected at such plasma levels (Thollon et al., 1994; Bucchi et al., 2007). However, clinical concentrations of ivabradine produce about 18–20% reduction in human heart rate (Camm and Lau, 2003; Doesch et al., 2007).

Previous animal model studies (Difrancesco, 1991, 2010) have found that inhibition of I_f by ivabradine slowed down the spontaneous firing rate of the rabbit SAN cell in a use-dependent manner (Bois et al., 1996), and about 15% reduction of the pacemaking rate was observed at a concentration of 3 μM, which produced about 60% I_f reduction at membrane potential of −92 mV, and about 41% I_f reduction at physiologically relevant membrane potentials (~60 mV) (Yaniv et al., 2012). Numerically, in a recent in silico exploration of the role of I_f in SAN pacemaking using a rabbit SAN model, the effect of ivabradine was simulated through implementing a 66% reduction of I_f (mimicking an experimentally reported effect of 3 μM ivabradine), leading to a 22% reduction in spontaneous rate (Severi et al., 2012). However, due to the non-linear concentration-dependent action of ivabradine on I_f, it is hard (if not impossible) to derive the effect of a low concentration of ivabradine from that of a high concentration on modulating cardiac pacemaking rate. To date, therefore, the effect of blocking I_f at the level of I_f reduction at clinical plasma levels of ivabradine (<20%) on pacemaking rate of the rabbit SAN has not been elucidated, as is how such concentration can produce a marked effect on the human SAN, in which I_f is much smaller than that in the rabbit SAN. Most importantly, it is unclear either how clinical concentrations of ivabradine affect cardiac pacemaking activity in vivo as compared to those predicted by single cell experiments in vitro, in which vagal tone modulation of cardiac pacemaking is missing. The aim of the present study was therefore to investigate through simulations and theoretical analysis the effect of I_f reduction over a wide range of values on cardiac pacemaking activity in the presence and absence of vagal tone modulation of cardiac pacemaking activity by acetylcholine.

Methods

SAN Cell Model and I_f Formulations

In this study, the contemporary model of rabbit SAN cells developed by Severi et al. (2012) was used as a basal model. The model was chosen as it represents the most updated progress in the model development of rabbit SAN cells, in particular it incorporates an updated I_f formulation based on recently available experimental data (Altomare et al., 2003; Barbuti et al., 2007). The basal model code was downloadable from cellML at the following URL: https://models.physiomeproject.org/e/139; and the source codes used for this study are available on request to: aGVuZ2d1aS56aGFuZyYjeDAwMDQwO21hbmNoZXN0ZXIuYWMudWs=. In brief, the dynamics of the membrane action potential the SAN cell were modeled as:

$\begin{array}{ll}\frac{dV}{dt}=-\frac{i_{total}}{C_m}& (1)\end{array}$

$\begin{array}{r}{i}_{total}=I_{CaL}+I_{CaT}+I_{Kr}+I_{Ks}+I_{sus}+\\ I_{to}+I_{NaK}+I_{NaCa}+I_{Na}+I_f\end{array}$

Where V is the membrane potential, C_m the membrane capacitance, t the time, I_total the total membrane current generated by potassium (I_Kr, I_Ks, I_sus, I_to), calcium (I_CaL, I_CaT), sodium (I_Na), Na⁺-K⁺ pump (I_NaK), Na⁺-Ca²⁺ exchanger (I_NaCa), and funny (I_f) channels. More details of the basal model are documented in the study of Severi et al. (2012).

Previous experimental studies have shown some distinctive differences in the maximal macroscopic conductance, the steady-state activation curve and the time constant of the channel activation of I_f between the rabbit and the human SAN cells (Difrancesco et al., 1989; Altomare et al., 2003; Barbuti et al., 2007; Verkerk et al., 2007b). In order to take into consideration the reported species difference in I_f properties, in our simulations we implemented two different sets of I_f formulations, one is the original model formulations developed by Severi et al. (2012) (rabbit-like formulation), and the other is Fabbri et al. formulation (Fabbri et al., 2017) based on the human I_f data (Verkerk et al., 2007b) (human-like formulation), which takes the form:

$\begin{array}{l}{I}_{\text{f}}=I_{\text{f,Na}}+I_{\text{f,K}}\end{array}$

$\begin{array}{l}{I}_{\text{f,Na}}=g_{\text{f,N}a}·y·(V-E_{\text{Na}})\end{array}$

$\begin{array}{l}{I}_{\text{f,K}}=g_{\text{f,K}}·y·(V-E_{\text{K}})\end{array}$

$\begin{array}{l}{\tau }_y=\frac{1}{(0.36·(V+148.8)/(e^{0.066·(V+148.8)}-1)+0.1·(V+87.3)/(1-e^{-0.21·(V+87.3)})}-0.054\end{array}$

$\begin{array}{r}{y}_{\infty }=0.01329+0.99921/(1+e^{(V+97.134)/8.1752}),\\ \text{if V}<-80\text{mV}\end{array}$

$\begin{array}{l}{y}_{\infty }=0.0002501·e^{(-V/12.861)},\text{if V}\ge -80\text{mV}\end{array}$

$\begin{array}{l}\frac{dy}{dt}=\frac{y_{\infty }-y}{{\tau }_y}\end{array}$

where I_f,Na and I_f,K are Na⁺ and K⁺ components of I_f, g_f,Na (0.00268 μS) and g_f,K (0.00159 μS) conductance of I_f,Na and I_f,K. E_Na and E_K the equilibrium potential for Na⁺ and K⁺. y_∞ is the steady state activation variable, τ_y the time constant of activation variable (y).

To determine and validate the parameters in the equations of human-like and rabbit-like I_f formulations, the equations for the steady-state activation curves (Figure 1A), and the equation for the activation time constant (Figure 1B) were fitted to experimental data obtained from human and rabbit SAN cells respectively (Difrancesco et al., 1989; Altomare et al., 2003; Barbuti et al., 2007; Verkerk et al., 2007b). The developed I_f formulations were validated by their ability to reproduce experimental I-V relationship data (Figure 1C) obtained by running a series of voltage-clamp commands (Figure 1D) for both the human-like and rabbit-like formulations. Figure 1 shows clearly that I_f in the human SAN has a more negative half-maximal activation voltage (Figure 1A), a greater activation time constant (i.e., slower activation process (Figure 1B) and a smaller current density (Figure 1C) than that in the rabbit SAN.

FIGURE 1

Figure 1. Kinetics of I_f from experimental and mathematical modeling data. (A) Steady-state activation curves in rabbit and human and SAN cells, which were used to fit I_f formulations for the Severi et al. (2012) rabbit (solid lines) and the modified Fabbri et al. (2017) human (dashed lines) models. Experimental data from Altomare et al. (2003) (▴), Barbuti et al. (2007) (▾), and Verkerk et al. (2007b) (♦) are also shown. (B) Corresponding activation time constant of I_f for human and rabbit SAN cells. Experimental data from Difrancesco et al. (1989) (□) and Verkerk et al. (2007b) (■) are also shown. (C) Computed I-V relationship for the human- (dashed line) and rabbit-like (solid line) models, which are compared with experimental data from the study of Verkerk et al. (2007b) (□) for the human SAN cells, Zaza et al. (1996) (■) and Goethals et al. (1993) (▴) for the rabbit SAN cells. (D,E) Voltage-clamp protocol and simulated time traces of I_f during voltage-clamp for the rabbit and human I_f model. (F) Dose-dependent effect of ivabradine on I_f in rabbit SAN cells. Experimental data of Bois et al. (1996) (•) and Bucchi et al. (2002) (▴) were based to develop the concentration-response curves as shown by the solid line and dashed line respectively.

Simulating the Effects of I_f Blockade

It has been shown that ivabradine blocks I_f without affecting channel kinetics, with block leading to a constant level of I_f reduction after a period of transition (Bucchi et al., 2002, 2013). Therefore, in this study, we implemented a pore-block theory (Yuan et al., 2015) to simulate the steady-state effect of I_f blocking by ivabradine over a wide concentration range by reducing its conductance by a factor k (k∈ (0,1), mimicking 0–100% (Bucchi et al., 2013) of I_f reduction. With varying levels of I_f inhibition, by the pore-block theory the I_f conductance became:

$\begin{array}{l}\{\begin{array}{l}{g}_{\text{fNa}}^{\prime }=(1-k)·g_{\text{fNa}}\\ g_{\text{fK}}^{\prime }=(1-k)·g_{\text{fK}}\end{array}\end{array}$

Simulating the Effect of Acetylcholine (ACh)

In vivo, ACh released from vagal activity slows down the spontaneous pacing rate of the SAN mainly by inhibiting I_f and I_CaL (Boyett et al., 1995), and activating acetylcholine-dependent K⁺ current (I_KACh) (Voigt et al., 2014). Vagal activity may also play a significant role in slowing down the pacing rate when ivabradine blocks I_f. To test the combined action of ivabradine and ACh, we simulated the ACh effect on SAN spontaneous APs, based on the formulations of Severi et al. (2012) for I_f and I_CaL inhibition, as well as I_KACh activation. In simulations, the values of g_K,ACh used were the same as those used in the Severi et al. (2012) and Fabbri et al. models (Fabbri et al., 2017) for the human-like model (Severi model with human-like I_f) ((Fabbri et al., 2017) and the rabbit-like model (i.e., the Fabbri et al. model with rabbit-like I_f; see details in the Supplementary Material) (Severi et al., 2012). Effects of ACh on pacemaking APs were qualitatively analyzed and compared with the implementation of rabbit-like and human-like I_f formulation, as well as I_f blocking. Details of the I_KACh formulation were listed in the Supplementary Material.

Study of Model-Dependence

In order to test the model-dependence of results, simulations were also conducted in the Fabbri et al. model of the human SAN cell (Fabbri et al., 2017), the source code of which was downloadable from cellML at the following URL: https://models.physiomeproject.org/e/568?searchterm=human++si. In this case, the Fabbri et al. model with the human-like I_f formulation was taken as the basal model, which was then modified by replacing the I_f formulation by the rabbit-like one.

Although simulated action potentials from the original Fabbri et al. (2017) model closely match experiment data of AP properties and calcium transient of human SAN cells, some modification was necessary as most of the ionic currents in the model (except I_f, I_Kr, and I_Ks) were based on rabbit SAN cell model, densities of which were modified by automatic optimization to match simulated action potential characteristics to experimental data. Such automatic optimization of model parameters may deviate from physiological relevance, resulting in some limitations. For example, a full block of I_CaT in the model abolished the pacemaking action potential. Though there are no direct experimental data from human SAN cells to validate the simulation result, data from rabbit sinoatrial node (Hagiwara et al., 1988; Takeda et al., 2004) and human patients (Madle et al., 2001) suggested a more modest change in the pacemaking cycle length when I_CaT was blocked. Therefore, we updated the model to address some of the limitations. Details about model updating and validations are presented in Supplementary Figure 1. Using the updated model, simulations in the Serveri et al. model were repeated in the Fabbri et al. model, and results are included in the Supplementary Material to support the major conclusion of the study.

Numerical Scheme

A fourth-order Runge-Kutta-Merson numerical integration method was used to solve the ordinary differential equations of the model. The time step was 5 × 10⁻⁶ s, which gives a stable solution of the equations and maintains the accuracy of the computation of membrane current and potential. In simulations, action potentials after the 20th one were recorded for analysis. For solving the Severi et al. model with the human I_f formulation, a set of initial values were used, which were taken from the recorded state variables when the model reached its steady state (see details about the initial values for solving the model in the Supplementary Material). This allows the secondary effect of different I_f formulations to other channel variables of the models to be considered in the simulations.

Theoretical Analysis

Theoretical analysis of the effect of I_f block on altered cardiac pacemaking cycle length (CL) was conducted following a similar approach as implemented in previous studies (Rocchetti et al., 2000; Zaza and Lombardi, 2001; Monfredi et al., 2014; Winter and Shattock, 2016; Zaza, 2016).

Figure 2 shows a schematic illustration of a cycle of action potentials of SAN cells. During the time course of the action potential, the voltage difference between the MDP and the voltage at the beginning of AP upstroke (V_up) (ΔV_m) can be discretized as many small steps by ΔV_i, each taking a time period DI_i to complete. Here dV_i/dt represents the local diastolic depolarization rate (DDR). During the diastolic depolarization phase, with small time interval (dt), |dV_i/dt| can be approximately considered as a constant and denoted by |dV/dt|. And the total diastolic interval (DI) can be expressed as:

$\begin{array}{ll}\text{DI}={\displaystyle \sum }\text{D}{\text{I}}_i={\displaystyle \sum }\frac{\Delta {\text{V}}_i}{|\text{d}{\text{V}}_i/\text{d}t|}=\frac{{\displaystyle \sum }\Delta {\text{V}}_i}{|\text{dV}/\text{d}t|}=\frac{\Delta {\text{V}}_m}{|\text{dV}/\text{d}t|}& (2)\end{array}$

Considering Equation (2), the pacemaking cycle length (CL) can be denoted as:

$\begin{array}{l}\text{CL}=\text{APD}+\text{DI}=\text{APD}+\frac{\Delta {\text{V}}_m}{|\text{dV}/\text{d}t|}=\text{APD}+\frac{{\text{C}}_m\Delta {\text{V}}_m}{\left|I_{total}\right|}\end{array}$

Where I_total denotes total membrane currents during the diastolic depolarization phase.

FIGURE 2

Figure 2. Schematic illustration of a cycle of SAN cell action potentials. During one cycle, the time interval between two consecutive action potentials measures the intrinsic pacemaking CL, which can be considered as a sum of the AP duration (APD) and the diastolic interval (DI) which starts from the maximum diastolic potential (MDP) to the beginning of AP upstroke [V_up (Fabbri et al., 2017)]. ΔV_m denotes the voltage difference between the MPD and the V_up. During the DI, multiple time intervals (dt) were needed for the membrane potential to transit from the MDP to the V_up, during each of which a ΔV_i is generated.

In response to I_f block, a new total ion channel current during the diastolic depolarization phase I' _total is generated, which takes the form

$\begin{array}{l}{I}_{total}^{\prime }=I_{total}-\Delta I\end{array}$

ΔI is the change of I_total caused by I_f reduction. This produces a new pacing cycle length (CL′), which can be represented as:

$\begin{array}{l}\text{C}{\text{L}}^{\prime }={\text{APD}}^{\prime }+\frac{{\text{C}}_m\Delta {\text{V}}_m}{\left|I_{total}^{\prime }\right|}=\text{APD}+\frac{C_m\Delta {\text{V}}_m}{|I_{total}-\Delta I|}\end{array}$

where APD' is the new action potential duration in response to I_f blockade. As a small I_f block in response to a low dose of ivabradine mainly affects the diastolic depolarization phase and has little or no impact on the AP duration and the MDP, APD′ is approximately equal to APD. Therefore, the increased cycle length (ΔCL) can be represented as

$\begin{array}{l}\Delta \text{CL}=\text{C}{\text{L}}^{\prime }-\text{CL}={\text{C}}_m\Delta {\text{V}}_m(\frac{1}{|I_{total}-\Delta I|}-\frac{1}{\left|I_{total}\right|})\\ ={\text{C}}_m\Delta {\text{V}}_m\frac{|\Delta I|}{\left|I_{total}\right|·|I_{total}-\Delta I|}\end{array}$

Then the relative change of the pacing cycle length is:

$\begin{array}{l}\frac{\Delta \text{CL}}{\text{CL}}=\frac{{\text{C}}_m\Delta {\text{V}}_m}{(\text{APD}+\frac{{\text{C}}_m\Delta {\text{V}}_m}{\left|I_{total}\right|})}·\frac{1}{\left|I_{total}\right|}·\frac{|\Delta I|}{|I_{total}-\Delta I|}\end{array}$

By denoting p as I_f current block potency (p = |ΔI/I_f|), and x as the proportion of I_f to I_total during the diastolic phase (x = |I_f/I_total|), then we have:

$\begin{array}{l}\frac{\Delta \text{CL}}{\text{CL}}=\frac{{\text{C}}_m\Delta {\text{V}}_m}{(\text{APD}+\frac{{\text{C}}_m\Delta {\text{V}}_m}{\left|I_{total}\right|})}·\frac{1}{\left|I_{total}\right|}·\frac{|\Delta I/I_f|}{|I_{total}/I_f-\Delta I/I_f|}\\ =\frac{p}{(\frac{\text{APD}}{{\text{C}}_m\Delta {\text{V}}_m}\left|I_{total}\right|+1)}·\frac{1}{|\frac{1}{x}-p|}\end{array}$

By setting

$\begin{array}{l}{\text{C}}_1=\frac{\text{APD}}{{\text{C}}_m\Delta {\text{V}}_m}\end{array}$

where C₁ (C₁ > 0) can be considered as a constant during the diastolic depolarization phase with a fixed level of I_f reduction (i.e., p is fixed), which has no significant effect on the difference between MDP and the V_up (i.e., Δ V_m), then we have:

$\begin{array}{ll}\frac{\Delta \text{CL}}{\text{CL}}=\frac{p}{({\text{C}}_1\left|I_{total}\right|+1)}·\frac{1}{|\frac{1}{x}-p|}\propto {\text{C}}_2·\frac{1}{\left|I_{total}\right|}·\frac{1}{|\frac{1}{x}-p|}& (3)\end{array}$

Where C₂ also can be seen as a constant related to C₁ and p, also with a fixed level of I_f reduction.

With I_f block, the resultant relative change of the CL predicted by Equation (3) is inversely proportional to the amplitude of I_total during the diastolic depolarization phase, which determines the intrinsic CL of the pacemaking action potential. It is also related to the level of I_f reduction and the ratio between I_f and I_total. With a small level of I_f block, the resultant relative change of CL is greater for a smaller I_total (i.e., greater when the CL is larger or the heart rate is lower).

Results

Simulation Results

Figure 3 shows the simulated action potentials from the Severi et al. model with rabbit (Figures 3Ai–Fi) and human (Figures 3Aii–Fii) I_f formulations. In the figure, action potentials (Figures 3Ai,Aii) are shown together with membrane currents during the genesis of action potentials, including I_CaL (Figures 3Bi,Bii), I_f (Figures 3Ci,Cii), I_Na (Figures 3Di,Dii), I_CaT (Figures 3Ei,Eii), I_NaCa (Figures 3Fi,Fii), I_Kr (Figures 3Gi,Gii), I_Ks (Figures 3Hi,Hii), I_to (Figures 3Ii,Iii), and I_NaK (Figures 3Ji,Jii). By replacing the rabbit I_f formulation with the human I_f formulation, the pacemaking activity was slowed down, with a pacemaking CL that increased from 355 ms to 1,139 ms, which was associated with a slight increase in I_Na and I_CaT at the late period of the diastolic depolarization phase (DDP) (with no noticeable difference at the initial period of the DDP), and a slight decrease in I_Kr, I_Ks, I_to, and I_NaK during the DDP. Such a small increase in the above-mentioned inward currents and a decrease in the outward currents, though contributory factors, are not the major determinants of the prolonged diastolic depolarization phase. The slowing down in the pacemaking activity of the human-like formulation model is mainly attributable to the smaller I_f, I_CaL, and I_NaCa during the diastolic depolarization phase.

FIGURE 3

Figure 3. Simulated spontaneous action potentials and underlying currents by the Severi et al. model with rabbit I_f (Ai–Fi) and human I_f (Aii–Fii) formulations. Effects of blocking I_f by 20% on action potentials and underlying ionic currents (dotted lines) are also shown and superimposed with those in control condition (solid lines). (Ai,Aii): action potentials; (Bi,Bii): I_CaL; (Ci,Cii): I_f; (Di,Dii): I_Na; (Ei,Eii): I_CaT; (Fi,Fii): I_NaCa; (Gi,Gii): I_Kr; (Hi,Hii): I_Ks; (Ii,Iii): I_to; (K) increase of pacing CL.

In Figure 3, effects of reducing I_f by 20% on the pacemaking activity of the two models are also shown.

Reduction of I_f by 20% produced an increase of the CL in the rabbit-like I_f formulation model by about 4.6% (Figure 3K). This is in agreement with experimental data from isolated rabbit SAN cells, which showed that a low level of I_f block by ivabradine [<0.5 μM, a concentration not affecting SAN I_CaL which only slightly decreased by 18.12 ± 0.66% at 10 μM (Bois et al., 1996)] produced only a slight slowing down of the pacemaking rate, while >50% I_f blockade by 3 μM ivabradine (see Figure 1F) only reduced the spontaneous pacing rate by 11–17.7% (Bucchi et al., 2007; Yaniv et al., 2012).

However, in the human-like I_f formulation model, I_f reduction by 20% produced a more than 2-fold increase in the pacemaking CL by 12.4% (i.e., equivalent to about 11.1% reduction in the heart rate) as compared to the rabbit-like model (Figure 3K). The pacing rate reduction though was slightly less than the effect of intravenous administration of ivabradine by 0.2 mg·kg⁻¹ (~23.5–34.1 nM of mean maximal ivabradine plasma levels) produced a reduction of heart rate by 18–20% (i.e., mean heart rate reductions of 12.9 and 14.1 beats min⁻¹; Camm and Lau, 2003; Jiang et al., 2013), but close to experimental data of the pacing rate reduction when I_f was blocked by 3 μM ivabradine in rabbit SAN (Bucchi et al., 2007; Yaniv et al., 2012). This illustrates that the small human-like I_f has a greater effect on slowing down the pacing rate than the rabbit-like one when I_f is inhibited by ivabradine.

The results above suggested that the increased CL induced by 20% I_f reduction is proportional to the intrinsic cycle length of the model, i.e., the greater the intrinsic CL (e.g., the model with human-like I_f formulation) the greater the increase of the CL. This observation was model-independent as shown in Supplementary Figure 3, in which Fabbri et al. model of the human SAN was implemented by using rabbit-like and human-like I_f formulations. In the basal condition (Fabbri et al. model with human-like I_f formulation), the CL was 810 ms, which was increased by 44 ms with 20% I_f reduction (i.e., 5.5%). When the rabbit-like I_f formulation was used, the pacemaking rate was increased due to a larger I_f, resulting a CL of 355 ms. With 20% I_f reduction, the CL was increased by 18 ms (i.e., 4.7%), which was smaller than that when the human-like I_f formulation was used.

As shown in Figures 3Ci,Cii, a 20% reduction in the channel conductance did not necessarily produce 20% reduction in I_f amplitude during the time course of action potential, due to the dependence of I_f on membrane voltage. To further investigate this, we computed the average I_f during the diastolic phase before and after 20% reduction in its channel conductance. Results are shown in Supplementary Figure 2. It was found that 20% reduction in the channel conductance produced a similar decrease in the avarage I_f in the rabbit-like (by 0.34 pA) and the human-like model (by 0.22 pA), but the relative change was greater in the latter model because of its smaller I_f in the control condition. Such difference in the relative change of I_f may also be one of the important reasons for the more pronounced prolongation of the diastolic phase in model with human I_f formulation. Note that in both models (rabbit-like and human-like models), the relative change of I_f was smaller than 20% though the channel conductance was reduced by 20%, due to the voltage-dependence of the channel's activation.

Further simulations were carried out to analyze possible effects of the cross talk between the membrane clock (I_f) and Ca²⁺ clock on modulating pacemaking action potentials in response to I_f reduction. Figure 4 shows results for the rabbit-like (Figures 4Ai–Ei) and human-like (Figures 4Aii–Eii) I_f formulation models. Blocking I_f by 20% decreased intracellular (cytoplasmic) Ca²⁺ transients ([Ca²⁺]_i, Figures 4Bi,Bii), the intracellular subspace Ca²⁺ concentration ([Ca²⁺]_sub, Figures 4Ci,Cii) and the Ca²⁺ content in the network SR ([Ca²⁺]_nsr, Figures 4Ei,Eii). The diastolic level of [Ca²⁺]_sub was reduced by 2.9 and 5% in the rabbit-like and human-like I_f formulation model respectively, the decreased [Ca]_sub during the diastolic depolarization phase of the action potential (Figures 4Ci,Cii) leads to a decreased I_NaCa (Figures 3Fi,Fii) and I_CaL (Figures 3Bi,Bii), especially during the later phase of diastolic depolarization, which prolonged the later phase of the diastolic depolarization in the human-like I_f formulation model (Figure 4Aii). There was a negligible change in I_Na (Figures 3Di,Dii) and I_CaT (Figures 3Ei,Eii) during the diastolic phase. Taken together with the observation of changes in I_NaCa (Figure 3F), our simulation results imply that the slowing down of the spontaneous action potentials in response to a low level of I_f block (<20%) may be mainly attributable to I_f reduction (i.e., membrane clock), with some contribution from a crosstalk between the membrane clock and Ca²⁺ clock in the later diastolic depolarization phase.

FIGURE 4

Figure 4. Effect of I_f block on the Ca²⁺ transient in rabbit-like (Ai–Ei) and human-like (Aii–Eii) SAN cell models. (Ai,Aii): the action potentials; (Bi,Bii): Ca²⁺ transient in the myoplasmic space; (Ci,Cii): Ca²⁺ transient in the myoplasmic sub-space; (Di,Dii): Ca²⁺ transients in the junctional SR; (Ei,Eii): Ca²⁺ transient in the network SR space; (Fi,Fii): intracellular Na⁺ concentration. Simulation results in control (solid line) are superimposed on those of I_f blocking (dotted lines).

Effects were also investigated of how a systematic change in I_f density affects spontaneous pacemaker activity. Results are shown in Figure 5, in which the computed CL (Figure 5A) and its increase (Figure 5B) with I_f being blocked from 0 to 100% with 1% increment for the rabbit-like (closed circles) and human-like I_f formulations (open circles). It was shown that over the range 0–80%, I_f block produced a greater CL increase with the human-like formulation than in the rabbit-like model. With a low level of I_f reduction, a linear relationship between the CL increase and I_f block was seen (Figure 5B). However, the relationship changed to an exponential one (Figure 5B) when a high level of I_f block was implemented. With about 66% I_f reduction, the CL was increased by about 26% (reduced pacemaking rate by 21%) in the rabbit-like model, which is consistent with experimental data (Thollon et al., 1994; Bucchi et al., 2007), but by about 42% in the human I_f formulation. A low level of I_f reduction (<20%) resulted in a negligible change in the MDP (<-0.3 mV in both models) in both rabbit-like and human-like I_f formulation models, and 100% I_f reduction hyperpolarised MDP by only 3.2 mV in the latter model (Figure 5C). Corresponding changes in APD₉₀ (Figure 5D) and the voltage at the beginning of AP upstroke (Figure 5E) were also shown (also see Supplementary Table 1). While a high level of I_f reduction hyperpolarized the MDP, it had modest secondary effect on APD₉₀ (mainly prolonging the late repolarization phase which may be due to the decreased I_Kr and I_NaK as shown in Supplementary Figures 4Gi,Ji, and there was no significant change in the fundamental morphology of the action potential as shown in Supplementary Figure 4Ai) and AP upstroke potential in the rabbit-like model.

FIGURE 5

Figure 5. Simulated effect of I_f block on the CL increase. (A,B) Pacing cycle length and the relative percentage changes of CL when I_f blocked from 0 to 100% with 1% increment in the human-like (open circle) and rabbit-like (closed circle) SAN cell models; (C,D) the maximum diastolic potential (MDP) and action potential duration at 90% of repolarization (APD₉₀); (E) change of the beginning of action potential upstroke voltage, which was calculated as the diastolic depolarization rate reached about 50% of its maximum value. I_f block produced greater effect in the human-like (gray circle) and rabbit-like (black square) SAN cell models.

Changes of the membrane currents and ion concentrations with a systematic change in I_f density in the two models were also investigated (Supplementary Figures 3, 4). During the diastolic depolarization phase of the action potentials, [Ca]_sub decreased as I_f density gradually decreased (Supplementary Figures 3Ci,Cii), further leading to a slower activation of the I_CaL (Supplementary Figures 4Bi,Bii) and a decrease in I_NaCa (Supplementary Figures 4Fi,Fii) during this phase. Reduction of I_NaCa decreased [Na]_i (Supplementary Figures 3Fi,Fii), leading to a reduction in I_NaK (Supplementary Figures 4Ji,Jii) which further exacerbated the reduction of [Na]_i. The outward currents (I_Kr, I_Ks, I_to, I_NaK) also showed a gradual decrease with a decreased I_f density during the diastolic depolarization phase (Supplementary Figures 4Gi–Ji,Gii–Jii).

The simulated action potentials from the two models showed some differences when I_f was fully blocked (Figure 5A), with the rabbit-like model failing to generate spontaneous action potentials. Such differences may be attributable to the use of different sets of initial values recorded from their steady state variables as there was no other changes in model equations or parameters, except for the use of rabbit-like I_f or human-like I_f formulations. In order to determine potential factors contributing to such differences or contributing to the pacemaking action potentials in the two models, further analyses on membrane currents and the intracellular Ca²⁺ transients between control and 99% reduction of I_f were conducted. Results are shown in Supplementary Figures 3, 4. It was shown that in both models, in addition to I_f, I_CaT, I_NaCa, and I_Na contributed to the diastolic depolarization. With a high level of I_f reduction (99%), there was a significant decrease in [Ca]_sub during the diastolic depolarization phase (Supplementary Figure 3Ci), causing a significant reduction in I_NaCa (Supplementary Figure 4Fi). Consequently, the spontaneous membrane depolarization was not able to reach the I_CaL activation potential, terminating the action potentials in the rabbit-like model.

In the human-like model, a reduction of [Ca]_sub was also observed with a high level of I_f reduction, resulting in a decreased I_NaCa. However I_NaCa was sufficient to maintain the spontaneous depolarization to generate a full action potential.

The focus of this study was on the action of a modest extent (i.e., <20%) of I_f bock, mimicking the clinical use of ivabradine, rather than on the action of a large percentage of I_f block. With <50% I_f block, the increase in CL was about <30%, which is reasonably close to the experimental data observed in rabbit SAN cells when I_f is blocked by use of Cs⁺ (Nikmaram et al., 1997), validating the physiological relevance of the results obtained.

Further simulations were conducted to investigate the combined action of I_f reduction by ivabradine and actions of ACh (Boyett et al., 1995), mimicking the autonomic regulation of cardiac pacemaking in vivo. In simulations, acetylcholine-dependent inhibition of I_f and I_CaL, and activation of K⁺ current (I_K,ACh) (Voigt et al., 2014) were considered. Results are shown in Figure 6, in which computed time courses of APs in control (with both rabbit-like and human-like I_f formulations), I_f reduction alone (by 20%) and I_f reduction together with actions of 0.1 nM ACh were compared (Figures 6A,B). It was shown that ACh augmented the effect of I_f reduction on the increase of CL in the model with both the rabbit-like and human-like I_f formulations. With the action of 0.1 nM ACh, 20% of I_f block increased the CL by 37.5% (about 27.3% reduction the pacing rate) (Figure 6C) in the model with human-like I_f formulation, which is remarkably greater than that of 10.1% (about 9.1% reduction in the pacing rat) in the model with the rabbit-like I_f formulation. This observation held true when different ACh concentrations were considered (Figure 6D). These results illustrate that the combined action of I_KACh and I_f reduction further slowed down the pacemaking AP due to a reduced total depolarization current during diastolic depolarization phase, resulting in a greater CL prolongation. It suggested that the clinical observed effect of low ivabradine on reducing pacing rate may partially result from the action of ACh due to active vagal tone in vivo. Similar observations were also seen in the Fabbri et al. model as shown in Supplementary Figure 4, in which ACh augmented more the effect of I_f reduction on pacemaking rate in the model with human-like I_f formulation than that with rabbit-like I_f formulation. Results from the Fabbri et al. model were similar, as shown in Supplementary Figure 5.

FIGURE 6

Figure 6. Effects of combined action of ACh and 20% I_f reduction on sponteneous action potentials in the Severi et al. model with the rabbit-like and human-like I_f formulations. (A,B) Time courses of action potentials in control, I_f reduction and combined effect of I_f reduction and ACh (0.1 nM). (C) The increase of pacemaking CL in the conditions of 20% I_f block alone and combined action with ACh (0.1 nM). (D) Combined effect of variant concentrations of ACh (from 0.1 to 10 nM) and 20% I_f block on pacemaking CL in the rabbit-like and human-like I_f formulations models.

Theoretical Analysis Validation

The simulation results presented above showed that I_f block produced a greater impact on slowing down the pacemaking rate with the human-like I_f formulation than that in the rabbit-like cell model in both the Severi et al. and Fabbri et al. model (see Supplementary Material). This is paradoxical as the I_f density over the pacemaker range with the human-like formulation is much smaller than that in the rabbit-like model, and one would expect a smaller I_f contribution to the spontaneous action potentials (therefore a smaller CL increase with I_f block). However, such a paradoxical effect of I_f reduction on the increase of CL as observed in the two models matched the theoretical prediction shown in the Method section (Equation 2), which showed an inverse relationship between an increased CL and the amplitude of total ionic membrane currents during the diastolic depolarization phase. With the theoretical prediction, a greater CL increase in the human-like I_f formulation SAN cell model can be attributed to a smaller I_total during the diastolic depolarization phase.

To test the theoretical prediction, further analyses were conducted to compute the averaged I_total amplitude during the DDP. Results from the Severi et al. model are shown in Figure 7 for control and 20% I_f reduction for action potentials (Figures 7Ai,Aii), the time course of I_total (Figures 7Bi,Bii) and the averaged amplitude of I_total (Figure 7C) during the DDP. It was found that during the time course of diastolic depolarization, the averaged I_total amplitude in the cell model with the human I_f formulations was much smaller (<30%) than that in the rabbit-like model, which produced a slower pacemaking rate (i.e., longer CL) and greater CL increase in response to 20% I_f reduction, matching the theoretical prediction. The changes in currents and Ca²⁺ ion concentration associated with the change in I_total (Figures 7Bi,Bii) are shown in Figures 3, 4. A similar observation was also seen in the Fabbri et al. model as shown in Supplementary Figure 6C.

FIGURE 7

Figure 7. Simulation of the inverse relationship of I_total and the increasing of CL in control (solid lines) and 20% I_f reduction (dotted lines) to validate theoretical analysis result (Equation 2) in method section (p = 0.2, modeling 20% I_f reduction). (Ai,Aii) Time course of action potentials for rabbit-like and human-like SAN cell models respectively (gray lines represents diastolic depolarization voltage changes from the MDP to the voltage at the beginning of AP upstroke in control condition); (Bi,Bii) time courses of I_total (gray lines represents diastolic depolarization voltage changes from the MDP to the voltage at the beginning of AP upstroke in control condition). (C) Averaged value of I_total amplitude during the diastolic depolarization phase for the rabbit-like (black) and human-like (gray) SAN cell models. In each case, the integral interval was set to the time interval between the MDP and beginning of the upstroke membrane potential (V_up).

In theoretical analysis it was shown that the relative increase of CL was also influenced by a factor of $\frac{1}{|1/x-p|}$, which was related to the ratio of I_f to I_total (x) and I_f block potency (p). In simulations, we further computed the values of $\frac{1}{|1/x-p|}$ and $\frac{1}{\left|I_{total}\right|}·\frac{1}{|1/x-p|}$ for control and I_f reduction cases. Results from the Severi et al. model are shown in Figure 8, in which the time courses of $\frac{1}{|1/x-0.2|}$ (Figures 8Ai,Aii) and $\frac{1}{\left|I_{total}\right|}·\frac{1}{|1/x-0.2|}$ (Figures 8Bi,Bii) were plotted for the control (black) and 20% I_f reduction (p = 0.2) for the rabbit-like (left panels) and human-like (the right panels) models. The computed $\frac{1}{|1/x-0.2|}$ (acting as a piecewise function) was set to $\frac{1}{\mathrm{\text{1}}\times \mathrm{\text{1}}{\mathrm{\text{0}}}^{\mathrm{\text{-3}}}}$ when |1/x − 0.2| was smaller than 1 × 10^-3 to avoid the value close to positive infinity. It was shown that in both models with a small I_f block the difference in the computed value of $\frac{1}{|1/x-p|}$between control (solid line) and 20% I_f reduction (dotted line) was very small during diastolic depolarization phase, and also small when the value was normalized against I_total ($\frac{1}{\left|I_{total}\right|}·\frac{1}{|1/x-0.2|}$). This provides support for the notion that the relative increase of CL was mainly determined by the amplitude of I_total during the diastolic depolarisation phase in response to I_f block.

FIGURE 8

Figure 8. Computed values of $\frac{1}{|1/x-p|}$ and $\frac{1}{\left|I_{total}\right|}·\frac{1}{|1/x-p|}$ to validate theoretical analysis result (Equation 2) in method section (p = 0.2, modeling 20% I_f reduction). The highlighted lines shown in this figure are during the diastolic depolarization phase of the action potentials of control (solid lines) and 20% I_f reduction (dotted lines) cases for the rabbit-like (Ai,Bi) and human-like (Aii,Bii) SAN cell models.

Note that during the last period of diastolic depolarisation phase (i.e., during time period of 310–387 ms as shown in the figure for the rabbit-like model and 890–1,331 ms for the human-like model), the difference in the computed values of $\frac{1}{|1/x-0.2|}$ and $\frac{1}{\left|I_{total}\right|}·\frac{1}{|1/x-0.2|}$ became more noticeable. This may be attributable to the different timings by which the upstroke of pacemaking actions potentials occurred between control and I_f reduction conditions. Note that the value of $\frac{1}{|1/x-0.2|}$was also greater in the human-like model than that in the rabbit, which amplified the contribution of $\frac{1}{\left|I_{total}\right|}$toward a relative increase of CL. Results from the Fabbri et al. model were similar, as shown in Supplementary Figure 7.

A marked difference in the V_1/2 of the steady-state activation relationship (y_∞) of I_f between the rabbit and the human SAN cells has been observed (Difrancesco et al., 1989; Altomare et al., 2003; Barbuti et al., 2007; Verkerk et al., 2007b). In this study, we used V_1/2 of −52.5 and −97.1 mV for the rabbit-like and the human-like I_f formulation respectively. In order to systematically determine a possible role of varying V_1/2 in modulating I_f amplitude, and thus the I_total and the effect of I_f reduction on the increase of CL, we changed V_1/2 of y_∞ in the human-like I_f formulation in a border range from −50 to −70 mV. Results from the Serveri et al. model are shown in Figure 9 for the pacemaking CL (Figure 9A), the averaged I_total during diastolic depolarization phase (Figure 8B), and increase of CL (Figure 9C) with 20% I_f block. Shifting the V_1/2 from −50 mV (about rabbit V_1/2) to −70 mV (about human V_1/2), the I_total was decreased (Figure 9B), which were correlated with an increased CL (Figure 9A) as well as an increased effect of I_f reduction on CL (Figure 9C). These results supported our theoretical analysis on that a smaller I_f in the model resulted in a smaller I_total amplitude, resulting in a slower pacemaking rate; and as such the same reduction in I_f resulted in a more significant change of CL in the cell model with a smaller I_total. Similar results were also observed using the Fabbri et al. model as shown in Supplementary Figure 8.

FIGURE 9

Figure 9. (A–C) Effect of varied half-activation voltage (V_1/2) of the steady state activation variable (y_∞) on pacemaking CL in the Serveri et al. model with human-like I_f formulation. V_1/2 changed from −50 to −70 mV, covering the experimental data range for rabbit (−50 mV) and human (−97.1 mV) SAN cells.

Discussion

This study was conducted to determine the mechanism by which a low level of I_f block by clinical drug (ivabradine) concentration is able to reduce the heart rate of patients by about 18–20%. In a previous study, Verkerk and Wilders found that though I_f has a small magnitude in the human SA node cells, it has an equally important role as in the rabbit (Verkerk and Wilders, 2010). In another study, Maltsev and Lakatta argued that I_f provides a relatively modest contribution to spontaneous beating rate regulation of human and rabbit sinoatrial node cells, and its major role in human SAN cells is to prevent excessive hyperpolarization during AP repolarization, thus representing an anti-bradycardic mechanism, rather than a primary rate regulatory mechanism (Maltsev and Lakatta, 2010). Though these previous studies addressed how a smaller I_f can produce the same effect in humans with respect to rabbit (Verkerk and Wilders, 2010; Fabbri et al., 2017) on complete I_f block, however, the question on how <20% I_f block produces a marked heart rate reduction in humans in vivo had remained unclear. This is due to the nonlinear relationship between the extent of I_f block and heart rate reduction (see Figure 5); effects of partial I_f block cannot be predicted with certainty from the complete block data already in the literature.

The principal contributions of the work are insights into how a small extent of I_f block (<20%; as may occur during clinical use of ivabradine) produces a marked heart rate reduction in human SAN, greater than that predicted by single cell experiments in the rabbit though the latter has a greater I_f density. Our principal findings are: (i) blocking I_f by 20% resulted in only about a 4.6 % increase in the CL in the rabbit-like SAN cell model, but about 12.4% in the cell model with human I_f formulation. This finding suggests that I_f block has a greater effect with the human-like SAN cell I_f formulation than with rabbit-like I_f formulation, despite the fact that the former has a smaller I_f current density over diastolic potentials, based on which one might expect a smaller contribution of I_f to pacemaking; (ii) there is a cross-talk between the membrane clock and Ca²⁺ clock with 20% I_f block in the later phase of the diastolic depolarization of the action potential; (iii) a theoretical analysis matching the simulation data has been produced, providing a numerical formalism explaining the relationship between I_f block effects and its contribution to total current during diastolic phase. In this study, both numerical simulations and theoretical analysis here have attributed the paradoxical effect of I_f reduction (i.e., a greater effect in SAN cells with smaller I_f current density and therefore a slower heart rate) to an inverse relationship between the relative increase of CL and the amplitude of the total current during the diastolic depolarization phase in response to I_f block; and (iv) significantly, vagal tone activity via ACh augments the effects of ivabradine on heart rate reduction, providing a possible mechanism(s) by which the clinical concentrations can have larger effects in vivo than those predicted by single cell experiments in vitro. It has been shown that combined action of ACh (0.1 nM) and 20% I_f reduction markedly increase the pacemaking CL by 37.5%, close to the clinical effect of ivabradine when human-like I_f formulation was used, which is significantly greater than that of 10.1% when the rabbit-like I_f formulation was used. These findings were also observed in the Fabbri et al. model.

The results of the present study demonstrate and explain why a smaller human I_f has a greater effect on prolonging the diastolic depolarization phase when it is partially blocked, using both mathematical theoretical analysis and computer simulation, which is clearly distinct from the previous studies (Verkerk and Wilders, 2010; Fabbri et al., 2017). Collectively, they add mechanistic insight into the understanding of how a low dose of clinical use of ivabradine (<137 nM) can effectively slow down the human heart rate by about 18–20% (Camm and Lau, 2003; Joannides et al., 2006; Doesch et al., 2007; Jiang et al., 2013), which contrasts with a negligible effect predicted by experimental studies in the rabbit (<4% at 10 min after administration of ivabradine; Thollon et al., 2007).

Role of I_f in Generating Cardiac Pacemaking Activity

I_f channels encoded by HCN genes are richly expressed in cardiac conduction systems including the primary pacemaker, the SAN (Altomare et al., 2003; Ravagli et al., 2016). Previous studies from animal models have suggested an important role of I_f in the SAN (Difrancesco and Noble, 2012; Baruscotti et al., 2016; Kozasa et al., 2018). It has been shown that a complete block of I_f by Cs⁺ (2 mM) slowed down the pacemaking rate by 30% in the rabbit SAN cells (Denyer and Brown, 1990), a 17.6% reduction of pacemaking rate also seen when blocking of I_f with 0.5 mmol/L Cs⁺ (Liu et al., 2008). In transgenic mice, knocking down HCN4 produced bradycardiac effects as well as atrioventricular node conduction block (Herrmann et al., 2007; Hoesl et al., 2008; Baruscotti et al., 2011). All of this evidence demonstrates that I_f, together with the more recently identified Ca²⁺ clock [arising from the coupling between the intracellular Ca²⁺ cycling and electrogenesis of membrane currents (e.g., I_NaCa) (Maltsev and Lakatta, 2008; Lakatta and Difrancesco, 2009)], provide a major driving force for generating the spontaneous depolarization during the diastolic phase that leads to intrinsic pacemaker activity.

I_f is also present in human SAN cells (Verkerk et al., 2009b; Li et al., 2015) and contributes to pacemaking. It has been shown that loss-of-function of HCN channel mutations is associated with sick sinus syndrome, which manifests with symptoms of bradycardia and conduction block (Schweizer et al., 2014; Verkerk and Wilders, 2014, 2015). However, the functional role of I_f in generating pacemaking action potentials of human SAN is less well-characterized as compared to that from small mammals. Limited data have shown that I_f in the human SAN cells has a smaller current density, more negative depolarised membrane potential of half maximal activation and slower activation rate as compared to the rabbit (Verkerk et al., 2007a,b, 2009b). All of these suggested a smaller I_f current during the diastolic depolarization phase, which may result in a slower heart rate. Indeed this is the case as shown in the present simulation study. In the Severi et al. model (Severi et al., 2012) with the rabbit-like I_f formulation, the measured pacemaking cycle length was 354.8 ms. However, when the rabbit-like I_f formulation was replaced by the human one, the pacemaking rate was slowed down and the pacemaking cycle length increased to 1,139.4 ms, greater than the intrinsic pacemaking cycle length of native human SAN cells [about 828 ± 21 ms (Verkerk et al., 2007b)]. Note that in the model, such a significant increase of the pacemaking CL from the one of rabbit SAN cells to the one close to human SAN cells was mainly attributable to a small I_f, as no change or negligible secondary changes of other ion channels were implemented. This suggests that I_f exerts a strong influence on pacemaker rhythm. Note that there is a marked difference in the pacemaking CL between the human-like model (1,139.4 ms) and native human SAN cells [about 828 ± 21 ms (Verkerk et al., 2007b)], which may be due to possible species differences in the properties of other membrane currents (Fabbri et al., 2017) and Ca²⁺ clock (Tsutsui et al., 2018). With the increase of I_f blocking, the pacemaking CL increased non-linearly in the model with either rabbit-like or human-like I_f formulations.

In simulations, the action of 100% I_f block abolished the pacemaking in the model with rabbit-like I_f formulations. This may over emphasize the role of I_f in the rabbit SAN pacemaking and is non-physiological. However, the focus of this study is on the action of small I_f block (i.e., <20%), mimicking the use of effect of ivabradine in practice, rather than on the action of large percentage of I_f block. With <50% I_f block, the increase in CL is about <30%, which is reasonably close to the experimental data observed in rabbit SAN cells when I_f is blocked by use of Cs⁺ (Nikmaram et al., 1997).

Mechanism for the Action of Low Dose Ivabradine on Human SAN

Our theoretical and numerical simulation results have shown that a low level of I_f block, mimicking the clinical concentrations of ivabradine, produced a more marked effect in reducing the heart rate of the human-like I_f formulation SAN cell model than the rabbit-like I_f formulation SAN cell model. When the action of ACh (0.1 nM) was considered, there was a further reduction of sponteneous pacing rate (reduced upto by 27.3%), which is close to the effect of ivabradine at clinical concentration. Results from the Fabbri et al. model were similar, showing that these observations are model-independent. All these results suggest that a combined action of I_f reduction by ivabradine at clinical concentrations and ACh are attributable to the heart rate reduction as seen clinically (Camm and Lau, 2003; Doesch et al., 2007).

It is possible that ivabradine regulates the heart rate by a cross-talk between the membrane clock and Ca²⁺ clock via the electrogenic Na⁺-Ca²⁺ exchangers (Yaniv et al., 2013). While the direct action of ivabradine on the intracellular Ca²⁺ handling is unclear, our simulation results showed that a 20% I_f reduction produced secondary modulations of other ionic currents (e.g., I_NaCa) and the intracellular Ca²⁺ handling, suggesting there is a cross-talk between the membrane clock and Ca²⁺ clock in the late phase of the diastolic depolarization of the action potential (Figures 4Bii–Eii). All these illustrated that ivabradine affected the heart rate at clinical concentrations mainly through regulating membrane clock and Ca²⁺ clock, and combining with the action of ACh.

Limitations of the Study

Possible limitations of the Severi et al. model of the rabbit SAN cells have been well-discussed and documented (Severi et al., 2012). For example, as highlighted by Verkerk and Wilders (2014), the reversal potential of I_f in the Severi et al. model (−4.39 mV) was more positive than that experimentally reported (about −30 mV) (Difrancesco et al., 1986; Van Ginneken and Giles, 1991; Verkerk et al., 2009a), which is a limitation of the Severi et al. model, and therefore also of studies employing this model. However, we did not modify the I_f equation of the Severi et al. model for the following reasons: (1) the simulated I-V relationship data shown in (Figure 1C) lie within the range of experimental data, and I_f is very small when the voltage is more positive than −40 mV; (2) when the voltage is more positive than −30 mV, the activation of I_CaL is dominant, which contributes mainly to the upstroke phase (non-diastolic depolarization phase) of the action potential. Therefore, even if the reversal potential of I_f is changed from −30 mV to −4.39 mV, it has negligible effect on the diastolic depolarization of spontaneous action potentials. Moreover, in simulations in which the rabbit-like I_f was replaced by the human-like formulations in the Severi et al. model, the reversal potential of I_f used was −27.5 mV, close to the experimentally determined value of −22.1 ± 2.4 mV (Verkerk et al., 2007b).

Another potential limitation of the present study relates to modification of the Severi et al. model to incorporate the human I_f formulation developed by Fabbri et al. based on experimental data from human SAN cells. Due to lack of experimental data from human SAN cells, equations and parameters for other ion channels and transporters in the Severi et al. model were not updated. Note that experimental data on the Ca²⁺ clock of the human SAN became available recently (Tsutsui et al., 2018), which are not incorporated in the model yet. This may explain why the computed CL of the human-like SAN cell is greater than that of natural human SAN cells, and consequently the smaller effect of I_f blocking on the reduction of the heart rate the clinical data. In addition, the action of low dose ivabradine was simulated by considering its action on blocking I_f only, and did not incorporate its possible actions on I_Kr as seen in some experimental studies on ventricular cells/tissue at low dose (Melgari et al., 2015) or high doses (Lees-Miller et al., 2015). However, the present study deliberately focused on the difference in direct effects of ivabradine on I_f in the SAN cells between species (I_f in the rabbit SAN and in the human-like SAN). Whilst it is necessary to point out these potential limitations, nevertheless, the simulation data strongly supported the mechanism demonstrated by the theoretical analysis in showing the inverse correlation between I_total during the diastolic depolarization phase and the relative increase of the CL. Therefore, these limitations do not alter our major conclusion on the role of I_f block in modulating cardiac pacemaking activities in the human SAN by low concentrations of ivabradine.

Conclusion

An inverse correlation between the relative increase of CL and the amplitude of the total ion channel current during the diastolic depolarization phase has been observed. Both theoretical analysis and simulations have shown that a low level of I_f block (<20%) can produce a more marked reduction of in the pacemaking rate of the human-like SAN cell model than the rabbit-like one due to its smaller I_total during the diastolic depolarization phase. This was particularly the case when ACh actions were considered, which amplified the pacemaking cycle length prolongation. This study thus provides a mechanistic explanation into how a low level of I_f block by the clinical concentrations of ivabradine can effectively reduce the heart rate in humans but produce a small or negligible effect in the rabbit.

Data Availability Statement

The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding author/s.

Code Availability Statement

The code for the models employed is available on request to Henggui Zhang at aGVuZ2d1aS56aGFuZyYjeDAwMDQwO21hbmNoZXN0ZXIuYWMudWs=.

Author Contributions

HZ and JH conceived the study. XB conducted simulation and data analysis. HZ and MB contributed to theoretical analysis. XB, KW, MB, JH, and HZ wrote the paper. All authors contributed to the article and approved the submitted version.

Funding

This work was supported by grants from the British Heart Foundation FS/14/5/30533, EPSRC (UK) (EP/J00958X/1; EP/I029826/1), Shenzhen Science and Technology Innovation Committee (JCYJ20151029173639477; JSGG2016022912504 9615), and Scientific research plan projects of Shaanxi Education Department (20JK0917). JH received a University of Bristol Research Fellowship.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher's Note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Acknowledgments

We are grateful to Dr. Severi's group for providing us the source code of the Severi et al. model.

Supplementary Material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fphys.2021.582037/full#supplementary-material

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