Computational analysis of the human sinus node action potential: model development and effects of mutations

Abstract

Key points

• We constructed a comprehensive mathematical model of the spontaneous electrical activity of a human sinoatrial node (SAN) pacemaker cell, starting from the recent Severi–DiFrancesco model of rabbit SAN cells.

• Our model is based on electrophysiological data from isolated human SAN pacemaker cells and closely matches the action potentials and calcium transient that were recorded experimentally.

• Simulated ion channelopathies explain the clinically observed changes in heart rate in corresponding mutation carriers, providing an independent qualitative validation of the model.

• The model shows that the modulatory role of the ‘funny current’ (I _f) in the pacing rate of human SAN pacemaker cells is highly similar to that of rabbit SAN cells, despite its considerably lower amplitude.

• The model may prove useful in the design of experiments and the development of heart‐rate modulating drugs.

Abstract

The sinoatrial node (SAN) is the normal pacemaker of the mammalian heart.  Over several decades, a large amount of data on the ionic mechanisms underlying the spontaneous electrical activity of SAN pacemaker cells has been obtained, mostly in experiments on single cells isolated from rabbit SAN. This wealth of data has allowed the development of mathematical models of the electrical activity of rabbit SAN pacemaker cells. The present study aimed to construct a comprehensive model of the electrical activity of a human SAN pacemaker cell using recently obtained electrophysiological data from human SAN pacemaker cells.  We based our model on the recent Severi–DiFrancesco model of a rabbit SAN pacemaker cell. The action potential and calcium transient of the resulting model are close to the experimentally recorded values. The model has a much smaller ‘funny current’ (I _f) than do rabbit cells, although its modulatory role is highly similar. Changes in pacing rate upon the implementation of mutations associated with sinus node dysfunction agree with the clinical observations. This agreement holds for both loss‐of‐function and gain‐of‐function mutations in the HCN4, SCN5A and KCNQ1 genes, underlying ion channelopathies in I _f, fast sodium current and slow delayed rectifier potassium current, respectively. We conclude that our human SAN cell model can be a useful tool in the design of experiments and the development of drugs that aim to modulate heart rate.

Key points

• We constructed a comprehensive mathematical model of the spontaneous electrical activity of a human sinoatrial node (SAN) pacemaker cell, starting from the recent Severi–DiFrancesco model of rabbit SAN cells.

• Our model is based on electrophysiological data from isolated human SAN pacemaker cells and closely matches the action potentials and calcium transient that were recorded experimentally.

• Simulated ion channelopathies explain the clinically observed changes in heart rate in corresponding mutation carriers, providing an independent qualitative validation of the model.

• The model shows that the modulatory role of the ‘funny current’ (I _f) in the pacing rate of human SAN pacemaker cells is highly similar to that of rabbit SAN cells, despite its considerably lower amplitude.

• The model may prove useful in the design of experiments and the development of heart‐rate modulating drugs.

Abbreviations

AP

action potential

APA

action potential amplitude

APD₂₀, APD₅₀ and APD₉₀

action potential duration at 20%, 50% and 90% of repolarization

Ca_i range

diastolic–systolic range of intracellular free calcium concentration

CaT

calcium transient

CL

cycle length

DD

diastolic depolarization

DDR₁₀₀

diastolic depolarization rate over the first 100 ms of diastolic depolarization

(dV/dt)_max

maximum rate of rise of membrane potential

HCN4

hyperpolarization‐activated cyclic nucleotide‐gated cation channel 4

Iso

isoprenaline

KCNQ1

potassium voltage‐gated channel subfamily Q member 1

MDP

maximum diastolic potential

OS

overshoot

RyR

ryanodine receptor

SAN

sinoatrial node

SCN5A

sodium voltage‐gated channel α subunit 5

SERCA

sarco‐endoplasmic reticulum Ca²⁺‐ATPase

TA

intracellular CaT amplitude

TD₂₀, TD₅₀ and TD₉₀

intracellular CaT duration at 20%, 50% and 90% of calcium decay

Introduction

There is no need to explain the important role of the sinoatrial node (SAN) in cardiac function. Many studies have focused on this small but crucial piece of cardiac tissue and provided detailed knowledge of the physiological processes governing its function. Yet we still have an incomplete understanding of the cellular basis of the pacemaker activity of the SAN and, specifically, the degree of contribution of the different mechanisms involved is still debated (Lakatta & DiFrancesco, 2009; DiFrancesco, 2010; Lakatta, 2010; Maltsev & Lakatta, 2010; Noble et al. 2010; Verkerk & Wilders, 2010; Himeno et al. 2011; DiFrancesco & Noble, 2012; Lakatta & Maltsev, 2012; Rosen et al. 2012; Monfredi et al. 2013; Yaniv et al. 2013, 2015).

To date, almost all experiments on SAN electrophysiology have been carried out on animals, particularly rabbits. These experiments have shed light on several aspects, such as characteristics of membrane currents, effects of ion channel blockers, calcium handling and beating rate modulation. This considerable amount of data has allowed the development of increasingly comprehensive and detailed action potential (AP) models (Wilders, 2007) subsequent to the first mathematical models reproducing pacemaker activity being created (McAllister et al. 1975; Yanagihara et al. 1980; Noble & Noble, 1984; DiFrancesco & Noble, 1985). Recently, novel SAN AP models have been proposed, incorporating detailed calcium‐handling dynamics and providing in‐depth descriptions of the underlying events at the cellular level in guinea‐pig, mouse and rabbit (Himeno et al. 2008; Maltsev & Lakatta, 2009; Kharche et al. 2011; Severi et al. 2012).

However, the translation of animal data/models to humans is not straightforward (probably even less so for SAN pacemaker cells than working myocardial cells), given the big difference in their main ‘output’ (i.e. pacing rate) between human and laboratory animals.

Very few measurements are available for human SAN cells because failing explanted hearts, as obtained during heart transplantation, usually do not contain the SAN region, which remains inside the receiver's chest. Drouin (1997) was the first to record human adult APs from SAN tissue, obtained from four subjects affected by left ventricle infarction. Ten years later, Verkerk et al. (2007 b) investigated the electrophysiological properties of three isolated SAN cells from a woman who underwent SAN excision because of a paroxysmal non‐treatable tachycardia. They recorded APs and characterized the funny current using whole‐cell patch clamp in current clamp and voltage clamp mode. In a subsequent study (Verkerk et al. 2013), the same group reported further data from the same patient, regarding the calcium transient (CaT) measured by indo‐1 fluorescent dye.

Some proof‐of‐concept attempts have been made to model spontaneous activity in human cardiac pacemaker cells. Seemann et al. (2006) presented the first human SAN AP model as part of a wider 3D model of human atria. Their model started from the model of Courtemanche et al. (1998) of the human atrial AP and achieved automaticity by including rabbit SAN currents (e.g. I _f, I _CaT, etc.), formulated in accordance with Zhang et al. (2000). Unfortunately, no reliable experimental comparisons could be made with the model of Seemann et al. (2006). Chandler et al. (2009) described the ‘molecular architecture’ of adult human SAN tissue obtained from healthy hearts, comparing it to the non‐specialized tissue of the right atrium. They developed a SAN model, again starting from the model of Courtemanche et al. (1998); by scaling the maximal conductances based on mRNA data, they provided a proof of concept that automaticity can be the result of the specific gene expression pattern of human SAN tissue. However, some mechanisms were not included, and no quantitative AP features were computed or compared with the human data available at that time. Accordingly, as Chandler et al. (2009) themselves affirmed, ‘the resulting model cannot be considered definitive and is a guide only’.

Recently, Verkerk and Wilders (2015) computationally investigated mutations affecting hyperpolarization‐activated cyclic nucleotide‐gated cation channel 4 (HCN4) channels, but they ‘… preferred to study I _f in simulated action potential clamp experiments, thus ensuring that the action potential followed the course of that of a human SAN pacemaker cell’ because a human SAN AP model was not formulated yet. Thus, they highlighted the need for such a model.

Much more recent is the human SAN AP model formulated by Pohl et al. (2016). In their model, to investigate the cardiac neuromodulation (as exerted by vagal stimulation) that aims to reduce heart rate, they focused their attention on the effects that ACh has on the rate. Starting from the rabbit SAN model reported by Dokos et al. (1996), they updated current formulations, integrating data from human SAN cells (Verkerk et al. 2007 b) and human ion channels expressed in heterologous systems (HEK or tsA‐201 cells, Xenopus oocytes, etc.). The simulated cycle length (CL) was in accordance with the experimental recordings of human SAN cell APs by Verkerk et al. (2007 b) but, unfortunately, the AP morphology, both during the diastolic depolarization (DD) phase and AP, was very unlike that obtained experimentally.

The main aim of the present study is the formulation of a human SAN AP model strictly based on and constrained by the available electrophysiological data. We started from the recent rabbit SAN model by Severi et al. (2012), which integrates the two principal mechanisms that determine the beating rate: the ‘membrane clock’ and ‘calcium clock’ (Maltsev & Lakatta, 2009; Lakatta et al. 2010). Several current formulations were updated based on available measurements. A set of parameters, for which no specific data were available, were tuned to reproduce the measured AP and calcium transient data. We then used the model to assess the effects of several mutations affecting heart rate and investigated the rate modulation in some relevant conditions.

Methods

Model development

The starting point of our work was the rabbit SAN cell model by Severi et al. (2012). Several currents, pumps and exchangers were reviewed, based on electrophysiological data from human SAN cells (Verkerk et al. 2007 b), self‐beating embryonic human cardiomyocytes (Danielsson et al. 2013), as selected for their automaticity, and data on gene expression patterns in SAN vs. atrial human cells (Chandler et al. 2009). Specifically, the model was constrained by: AP parameters obtained from three isolated human SAN pacemaker cells by Verkerk et al. (2007 b); the voltage clamp data on I _f in the same three cells (Verkerk et al. 2007 b); the effect of 2 mm Cs⁺, as an I_f blocker, on the AP of a single isolated human SAN pacemaker cell (Verkerk et al. 2007 b); and the Ca²⁺ transient data of a single isolated human SAN pacemaker cell (Verkerk et al. 2013). For a small set of parameters for which no specific experimental data were available, values were obtained via an automatic optimization procedure (see below).

Figure 1 shows a schematic diagram of the human SAN AP model. The compartmentalization, essential for the calcium handling description, is inherited from the parent model, as were the sarcolemmal ionic currents, pumps and exchangers. The ultrarapid delayed rectifier K⁺ current (I _Kur) was developed independently. Table 1 reports the changes with respect to the rabbit SAN model and the rationale for each. All model equations and parameter values are provided in Appendix 1.

Figure 1. Schematic diagram of the human SAN cell model.

The cell is divided into three compartments: subsarcolemma, cytosol and sarcoplasmic reticulum (SR), which is subdivided into junctional and network SR. Ca²⁺ handling is described by: two diffusion fluxes (J _diff from the subsarcolemmal space to the cytosol and J _tr from the network SR to the junctional SR), the Ca²⁺ uptake from the cytosol into the network SR by the SERCA pump (J _up) and the Ca²⁺ release (J _rel) from the junctional SR into the subsarcolemmal space by the RyRs. Sodium, calcium and potassium ions pass the sarcolemmal membrane through 11 different ionic channels, pumps and exchangers, as indicated.

Table 1.

Changes in human SAN cell model with respect to the parent model

	Update	Rationale
Cell capacitance and dimensions	C m = 57 pF, L cell = 67 μm, R cell = 3.9 μm	Human SAN data (Verkerk et al. 2007 b)
I f	New formulation	Human SAN data (Verkerk et al. 2007 b)
I Kr	New steady‐state activation, g Kr = +10%	Embryonic human data (Danielsson et al. 2013)
I Ks	New steady‐state activation, g Ks = −78%	Embryonic human data (Danielsson et al. 2013)
I Kur	Added, g Kur = 6% of atrial formulation from Maleckar et al. (2009) atrial cell model	Gene expression data (Chandler et al. 2009) and automatic optimization (APD)
I to	g to = −1.5%	Automatic optimization (OS)
I K,ACh	g K,ACh = −77.5%	Fitting of ACh effects in rabbit
I NaK	I NaK,max = −28%	Automatic optimization (CL)
I CaL	V ½,dL = −16.45 mV, k dL = 4.337 mV (dL gate) P CaL = +28.5%	Automatic optimization (CL)
I CaT	P CaT = +15%	Automatic optimization (early DD)
I NaCa	K NaCa = −53%	Automatic optimization (diastolic [Ca2+]i)
SR uptake (J up)	Sigmoidal formulation	Fitting of diastolic [Ca2+]i
SR release (J rel)	k s = 1.48 × 108  s−1 k om = 660 s−1	Automatic optimization (Ca i range)
Calmodulin	kf CM = 1.64 × 106 (mm s)−1	Automatic optimization (Ca i range)
Calsequestrin	kf CQ = 175.4 (mm s)−1	Automatic optimization (Ca i range)
Calcium diffusion	τdifCa = 5.469 × 10−5 s	Automatic optimization (Ca i range)
Intracellular Na	Fixed at 5 mm	As used in human SAN experiments (Verkerk et al. 2007 b)

C _m, membrane capacitance; L _cell, R _cell, length and radius of the cell; g _i, maximal conductance for the i type channel; I _NaK,max = maximal activity of Na⁺/K⁺ pump; P _CaT, P _CaL, permeability for the T‐ and L‐type calcium currents; V _½,dL, k _dL, half‐activation potential and slope factor for voltage‐dependent dL gate; K _NaCa, maximal current of NCX; k _s, Ca²⁺ diffusion rate for RyR channels; k _om, RyR deactivation rate; kf _CM, kf _CQ, association constants for calmodulin and calsequestrin. The reported features in parentheses (APD, CL, Ca _i range, DD, OS) are those that were most impacted during the automatic optimization procedure.

Automatic optimization procedure

An automatic optimization was performed to tune the parameters for which human experimental data were not available to fit the recorded AP and cytosolic calcium transient traces (Verkerk et al. 2007 b, 2013).

The cost function of the optimization procedure was based on quantitative data on AP features [action potential amplitude (APA), maximum diastolic potential (MDP), CL, maximum rate of rise of membrane potential [(dV/dt)_max], action potential duration (APD₂₀, APD₅₀, APD₉₀), diastolic depolarization rate (DDR₁₀₀)], intracellular calcium transients [diastolic [Ca²⁺]_i, systolic [Ca²⁺]_i, intracellular CaT duration (TD₂₀, TD₅₀, TD₉₀)] and the effect of the administration of 2 mm Cs⁺, a funny current blocker, on CL (Verkerk et al. 2007 b, 2013). The most critical features (CL, MDP and CL prolongation in response to Cs⁺) were weighted more heavily.

After each simulation, the set of quantitative descriptors was extracted and compared with the experimental data. If a particular observed feature fell out of the experimental mean ± SEM range, the contribution of this feature to the overall cost was increased in a linear way; otherwise, no handicap was added. The search for the optimal solution was conducted using the Nelder–Mead simplex method (Lagarias et al. 1998). The obtained parameters are shown in Table 1. Further details on the optimization procedure are reported in Appendix 2.

Cell capacitance and dimensions

We assumed a membrane capacitance (C _m) of 57 pF, and a cylindrical cell shape with a length of 67 μm and a diameter of 7.8 μm, in accordance with the experimental data of Verkerk et al. (2007 b), who reported values of 56.6 ± 8.7 pF, 66.7 ± 6.3 μm and 7.8 ± 0.4 μm (mean ± SEM, n = 4), respectively. The dimensions of the intracellular compartments, expressed as a percentage of cell volume, were adopted from the parent model.

Membrane currents

Here, we specify each of the sarcolemmal currents flowing through the ionic channels, pumps and exchangers shown in Fig. 1, except for I _K,ACh, which is described in the section ‘Autonomic modulation’. All membrane currents were scaled up by the new value of the cell capacitance, aiming to maintain the conductance densities adopted from the parent model. Changes to the conductance densities, as a result of parameter tuning, are reported in the corresponding current subsections.

Funny current (I _f)

From their experiments on human SAN cells, Verkerk et al. (2007 b) reported a maximal I _f conductance of 75.2 ± 3.8 pS/pF (mean ± SEM, n = 3). Consequently, we assumed a maximal I _f conductance (g _f) of 4.3 nS, given our C _m of 57 pF. The funny current was implemented by splitting it into Na⁺ and K⁺ components, with a $\frac{g_{\mathrm{fNa}}}{g_{\mathrm{fK}}}$ conductance ratio of 0.5927, thus arriving at an I _f reversal potential (E _f) of –22 mV, in accordance with the experimentally determined value of –22.1 ± 2.4 mV (mean ± SEM, n = 3) (Verkerk et al. 2007 b). A first‐order Hodgkin and Huxley‐type kinetic scheme was assumed for I _f activation, as described by the formulations presented by Verkerk et al. (2007 a) and Verkerk and Wilders (2010). The activation time constant τ_y was formulated in accordance with Verkerk et al. (2007 a), who used a Dokos et al. (1996)‐type equation to fit the experimental data obtained from three human adult SAN cells by Verkerk et al. (2007 b).

Rapid delayed rectifier K⁺ current (I _Kr)

The steady‐state activation curve of I _Kr (pa gate) was fitted to data from embryonic human cardiomyocytes by Danielsson et al. (2013). The measured tail current density following activation pulses from −70 to +50 mV was normalized with respect to the maximal measured value and then fitted with a Boltzmann equation. The conductance g _Kr was set to 4.2 nS (+10% compared to parent model) to hyperpolarize the maximum diastolic potential (MDP) and obtain the value experimentally observed in human SAN (Drouin, 1997; Verkerk et al. 2007 b).

Slow delayed rectifier K⁺ current (I _Ks)

The steady‐state activation curve for I _Ks (n gate) was updated in accordance with the experimental data of Danielsson et al. (2013). The reported current density vs. voltage data were normalized to the maximal measured value and subjected to square‐root, in line with the second‐order Hodgkin and Huxley‐type kinetic scheme. Data were then fitted with a Boltzmann equation. The conductance g _Ks was set to 0.65 nS (11.4 pS/pF), −78% with respect to the parent model, as a result of the automatic optimization procedure. On one hand, Chandler et al. (2009) reported an mRNA expression level in the human SAN equal to 69% of that in non‐specialized atrium cells. On the other hand, very discordant values have been reported for g _Ks in human atrial cells up to now, ranging from 3.5 pS/pF (Grandi et al. 2011) to 20 pS/pF (Nygren et al. 1998), whereas g _Ks was adjusted to 129 pS/pF in the model of Courtemanche et al. (1998) simply to match AP duration.

Ultrarapid delayed rectifier K⁺ current (I _Kur)

Chandler et al. (2009) reported the expression of K_V1.5 channels, responsible for I _Kur, in human SAN tissue. Because I _Kur was not present in the parent model, we added this current, formulating it as in the human atrial cell model of Maleckar et al. (2009). The g _Kur conductance was set to 0.1539 nS, 6% of the corresponding atrial value, based on the automatic optimization procedure.

Transient outward K⁺ current (I _to)

We maintained the parent model formulation, which, in turn, was adopted from the model of Maltsev & Lakatta (2009). The conductance g _to was set to 3.5 nS, slightly reduced (−1.5%) compared to the rabbit SAN model.

Sodium/potassium pump current (I _NaK)

The formulation by Severi et al. (2012), which was in turn derived from that of Kurata et al. (2002), was adopted. The maximal activity of the Na⁺/K⁺ pump was reduced by 28% (I _NaK,max = 0.08105 nA) through automatic optimization.

Sodium current (I _Na)

The presence of fast Na⁺ current (I _Na) in human SAN cells has been reported by Verkerk et al. (2009 b). The steady‐state activation and inactivation curves (gates m and h) of the parent model have been simply rewritten in a sigmoidal formulation (see Appendix 1) to facilitate the implementation of mutations related to Na_v1.5 channels.

T‐type Ca²⁺ current (I _CaT)

The mathematical formulation of I _CaT was inherited from the parent model and thus based on the constant field equation by Sarai et al. (2003). The Ca²⁺ permeability P _CaT was set to 0.04132 nA mm ⁻¹ (+15%), as obtained by automatic optimization.

L‐type Ca²⁺ current (I _CaL)

Changes in I _CaL kinetics were limited to the voltage‐dependent steady‐state activation curve dL _∞. The half‐maximal activation voltage (V _½,dL) was slightly shifted towards less negative potentials (from −20.3 to −16.45 mV) and the slope factor k _dL was slightly increased (from 4.2 to 4.337 mV). The Ca²⁺ permeability was increased by 28% (P _CaL = 0.4578 nA mm ⁻¹). All of these parameters were updated using automatic optimization.

Sodium/calcium exchange current (I _NaCa)

The set of equations describing the Na⁺/Ca²⁺ exchanger activity was adopted from the parent model and thus originally derived by Kurata et al. (2002). The maximal current provided by the exchanger was set to 3.343 nA, reduced by 53% as a result of the automatic optimization procedure.

Calcium handling

As in the parent model, the mathematical formulation of Ca²⁺ handling was based on Maltsev and Lakatta (2009), who provided an advanced description of SR behaviour. The parameter updates, which play an important role in Ca²⁺ handling, were achieved by automatic optimization.

SR Ca²⁺ uptake (J _up)

The Ca²⁺ uptake flux was formulated by a sigmoidal curve, instead of the Michaelis–Menten equation of the parent model. The sigmoidal formulation permitted a higher control of Ca²⁺ uptake, in particular during the diastolic phase.

SR Ca²⁺ release (J _rel)

The Ca²⁺ diffusion rate k _s, and the ryanodine receptors (RyRs) Ca‐dependent activation rate, k _om, were set to 148 × 10⁶ s⁻¹ and 660 s⁻¹, respectively, through automatic optimization.

Ca²⁺ diffusion and Ca²⁺ buffers

The time constant for Ca²⁺ diffusion from the subsarcolemma to the cytosol (τ_difCa) was set to 5.469 × 10⁻⁵ s. The Ca²⁺ association constants for calmodulin (kf _CM) and calsequestrin (kf _CQ) were set to 1.642 × 10⁶ and 175.4 (mm s)⁻¹, respectively. Calmodulin is involved in Ca²⁺ buffering in the cytosolic compartment, whereas calsequestrin binds Ca²⁺ in the junctional SR (jSR).

Ion concentrations

Ca²⁺ dynamics for the four compartments were described by the mass balance equations. Intracellular Na⁺ was fixed at 5 mm, the Na⁺ concentration in the pipette solution used in the whole cell configuration by Verkerk et al. (2007 b). In such a configuration, intracellular Na⁺ is expected to equilibrate with the pipette solution.

Sensitivity analysis

The sensitivity analysis was performed according to the approach proposed by Sobie (2009). The randomization procedure involved the parameters that underwent automatic optimization and the remaining maximal conductances, for a total of 18 parameters. The conductances were randomized through scaling factors chosen from a log‐normal distribution with a median value of one and an SD σ = 0.1873; thus, an increase of 20% represents 1 SD away from the control value.

Shifts of the steady‐state gating variables and the sarco‐endoplasmic reticulum Ca²⁺‐ATPase (SERCA) pump calcium dependence were extracted from a normal distribution centered on zero, with SDs of σ = 2 mV and σ = 50 nm, respectively. The randomization was run for a population of 500 models. All of them were simulated and the corresponding AP and CaT features were computed.

Parameters and their corresponding features were collected in the X (n × p) and Y (p × m) matrices, respectively, where n corresponds to the number of simulations showing an auto‐oscillating behaviour (< 500), p is the number of parameters and m is the number of computed features. Next, the matrix B (p × m) containing the sensitivity coefficients was computed using the formula:

$$\mathit{B}={({\mathit{X}}^{\mathbf{T}}\times \mathit{X})}^{-1}\times {\mathit{X}}^{\mathbf{T}}\times \mathit{Y}$$

Functional effects of mutations

Mutations in genes encoding (subunits of) ion channels may lead to changes in electrophysiological properties. We incorporated such functional effects into the model by modifying the parameters of the affected ion channel according to values reported in the literature (Tables 2, 3, 4). Only studies reporting changes in heart rate as (one of) the clinical effects of mutations were included.

Table 2.

Changes introduced in the I _f model parameters to reproduce the electrophysiological characteristics of mutant HCN4 channels, and associated clinically reported heart rate

			Activation
Mutation	Type of expression	Expression system	V 1/2 shift (mV)	Slope (%)	Tau shift (mV)	Current density (%)	Heart rate (beats min–1)	Reference
G480R	Heterozygous	Oocytes, HEK	−15	–	−15	−50%	NC: 73 ± 11 C: 48 ± 12	Nof et al. (2007)
Y481H	Heterozygous	CHO	−43.9	NS	−43.9	NS	IP1: < 30 IP2: 40	Milano et al. (2014)
G482R a	Heterozygous	CHO	−38.7	NS	−38.7	NS	NC: 63 C: 48 ± 10.7	Milano et al. (2014)
G482R b	Heterozygous	HEK	NS	NS	NS	−65%	NC: 63 C: 41 ± 5	Schweizer et al. (2014)
A485V	Heterozygous	Oocytes, HEK	−30	–	−30	−66.4%	NC: 77 ± 12 C: 58 ± 6	Laish‐Farkash et al. (2010)
R524Q	Heterozygous	HEK	+4.2	–	+4.2	NS	IP: 98.5 ± 14.2	Baruscotti et al. (2017)
K530N	Heterozygous	HEK	−14.2	NS	−14.2	–	NC: 74 C: 60.6 ± 6.9	Duhme et al. (2013)
D553N	Heterozygous	COS	NS	NS	–	−63%	IP: 39	Ueda et al. (2004)
S672R	Heterozygous	HEK	−4.9	NS	−4.9	–	NC: 73.2 ± 1.6 C: 52.2 ± 1.4	Milanesi et al. (2006)

Experimental data are changes relative to wild‐type currents. These changes are used to simulate the effect of HCN4 mutations. Oocytes, Xenopus oocytes; HEK, HEK‐293 cells; CHO, CHO cells; COS, COS‐7 cells. V _1/2 shift, shift in I _f steady‐state activation curve; slope, slope factor of steady‐state activation curve; tau shift, shift in voltage dependence of time constant of activation. –, not reported. NS, no significant change. Heart rate is resting heart rate in index patient (IP), mutation carriers (C), or non‐affected family members (non‐carriers, NC).

Table 3.

Changes introduced in the I_Na model parameters to reproduce the electrophysiological characteristics of mutant SCN5A channels, and associated clinically reported heart rate

			Activation			Inactivation
Mutation	β subunit	Expression system	V 1/2 shift (mV)	Slope (%)	Tau shift (mV)	V 1/2 (mV)	Slope (%)	Tau	I Na Late (%)	Current density (%)	Heart rate (beats min–1)	Reference
E161K	Yes	tsA201	+11.9	+17.9	+11.9	NS	NS	NS	–	−60	NC: 51 ± 0.6* C: 39 ± 1*	Smits et al. (2005)
G1406R	Yes	COS	–	–	–	–	–	–	–	−100	NC: 74 ± 2 C: 68.4	Kyndt et al. (2001)
ΔKPQ	No	HEK	+9.0	+26.8	+9.0	NS	NS	NS	0.60	–	NC: 76.4 C: 62.1	Nagatomo et al. (1998) Moss et al. (1995)
E1784K a	Yes	tsA201	+8.8	+78.1	+8.8	−14.4	NS	NS	1.5	–	IP: 42	Deschênes et al. (2000)
E1784K b	Yes	tsA201	+12.5	+32.6	+12.5	−15.0	NS	NS	1.85	−40	SND#	Makita et al. (2008)
D1790G	Yes	HEK	+6	+33.3	+6	−15	−6.8	x0.5	NS	–	NC: 64.9 C: 58.6	Wehrens et al. (2000) Benhorin et al. (2000)
1795insD **	Yes	HEK/Oocytes	+9.1	NS	+9.1	−9.7	NS	NS	1.4	−77.8	NC: 74.5 ± 13.5 C: 67.5 ± 16.4	Bezzina et al. (1999) Veldkamp et al. (2000)

Experimental data are changes relative to wild‐type currents. These changes are used to simulate the effect of SCN5A mutations. Most experimental data are in the presence of the SCN5A channel β‐subunit. HEK, HEK‐293 cells; COS, COS‐7 cells; tsA201, tsA201 cells; Oocytes, Xenopus oocytes. V _1/2 shift, shift in I _Na steady‐state (in)activation curve; slope, slope factor of steady‐state (in)activation curve; tau shift, shift in voltage dependence of time constant of activation; tau, time constant of inactivation; I _Na Late, persistent sodium current as percent of peak current under voltage clamp conditions. –, not reported. NS, no significant change. Heart rate is resting heart rate in index patient (IP), mutation carriers (C) or non‐affected family members (non‐carriers, NC). ^*Absolute minimum heart rate. ^#Sinus node dysfunction (SND) observed in 16/41 mutation carriers.

^**Parameters employed for simulations were selected from two different experimental studies (Bezzina et al. 1999; Veldkamp et al. 2000). In particular, the set of parameters reported by Veldkamp et al. (2000) was extended with a depolarizing shift of the activation and a remarkable current density reduction as observed by Bezzina et al. (1999).

Table 4.

Changes introduced in the I _Ks model parameters to reproduce the electrophysiological characteristics of mutant KCNQ1 channels, and associated clinically reported heart rate

			Activation
Mutation	Type of expression	Expression system	V 1/2 shift (mV)	Slope (%)	Tau (%)	Current density (%)	Heart rate (beats min–1)	Reference
R231C	Homozygous	Oocytes	–	–	–	Voltage‐dependent*	C:58.5 ± 7.7	Henrion et al. (2012)
V241F	Homozygous	HEK	−43.2	+52.9	–	–	IP1:30IP2: 36	Ki et al. (2013)
ΔF339	Heterozygous	Oocytes	+25.0	–	–	−69.5	NC: 63 ± 3.5C: 62.0 ± 6.6	Thomas et al. (2005)

Experimental data are changes relative to wild‐type currents. These changes are used to simulate the effect of KCNQ1 mutations. All experimental data are in the presence of the KCNE1 β‐subunit. Oocytes, Xenopus oocytes; HEK, HEK‐293 cells. –, not reported. Heart rate is resting heart rate in index patient (IP), mutation carriers (C) or non‐affected family members (non‐carriers, NC).

^*The ratio between the current density of I _Ks in wild‐type condition and with R231C mutation (I _KsR231C/I_KsWT) used to simulate the voltage‐dependent gain of function was estimated from Henrion et al. (2012).

The altered functionality was implemented as a shift of the steady‐state (de)activation or inactivation curve, a change in its slope factor or a reduction in maximal conductance. Mutations may also cause a change in the voltage‐dependent time constant (τ) curve. When changes in the time constant had been experimentally reported, they were implemented by a constant multiplication factor. Otherwise, τ underwent the same voltage shift applied to steady‐state (in)activation curve.

When I _Na channels are incompletely inactivated, an additional non‐inactivating term is added to the control formulation of I _Na, to reproduce a persistent, ‘late’ inward sodium current (I _Na,L), which is not normally present in the model. Its maximal conductance was derived from the ratio I _Na,L:I _Na peak, as observed experimentally.

Autonomic modulation

The effects of 10 nm ACh on I _f activation (–7.5 mV shift in voltage dependence of activation), I _CaL (3.1% reduction of maximal conductance) and SR Ca²⁺ uptake (7% decrease of maximum activity) were all adopted from the parent model (Severi et al. 2012). The administration of ACh also activates I _K,ACh, which is zero in the default model. The I _K,ACh formulation was derived from the parent model. The maximal conductance g_K,ACh was set to 3.45 nS (reduced by 77.5%) to achieve an overall reduction of the spontaneous rate by 20.8% upon administration of 10 nm ACh, as observed by Bucchi et al. (2007) in rabbit SAN cells.

The targets of isoprenaline (Iso) are I _f, I _CaL, I _NaK, maximal Ca²⁺ uptake and I _Ks. Changes in currents were adopted from the parent model, except for the modulation of I _CaL, where the effect of Iso induced a slightly smaller decrease of the slope factor k _dL (−27% with respect to control conditions, instead of the −31% assumed by the parent model). The I _CaL current was modulated to fit the experimental data reported by Bucchi et al. (2007) on rabbit SAN cells (26.3 ± 5.4%; mean ± SEM, n = 7) for the same Iso concentration.

Hardware and software

The human SAN model was built in Simulink (The Mathworks, Inc., Natick, MA, USA). Simulations ran on an OS X Mavericks (version 10.9.5) Apple computer (Apple, Cupertino, CA, USA) equipped with an Intel i7 dual core processor (Intel, Santa Clara, CA, USA). Numerical integration was performed by ode15s, a variable order solver based on numerical differentiation formulas, provided by MatLab (The Mathworks, Inc.). Simulations were run until steady‐state was reached, which occurred after 50 s, based on the observation of calcium concentrations in each compartment. The automatic optimization and feature extraction were performed by custom code in MatLab 2013a. Model code is available at: http://www.mcbeng.it/en/downloads/software/hap-san.html and will also be published in the CellML Model Repository (http://models.cellml.org/).

Results

Human SAN model behaviour in basal conditions

Simulated AP and calcium transient

The simulated AP waveform reproduces the available experimental traces well (Fig. 2 A). Indeed, most of the quantitative parameters that describe AP morphology (i.e. CL, MDP, APD₉₀ and DDR₁₀₀) are within the mean ± SD range of the experimental ones (Verkerk et al. 2007 b) (Table 5). In particular, the AP generated by the model is characterized by a CL of 814 ms, corresponding to a beating rate of 74 beats min^–1. However, the model presents a higher (dV/dt)_max and overshoot (OS) and a longer APD₂₀ (predicted features beyond the experimental mean ± SD range).

Figure 2. Action potential and intracellular Ca²⁺ transient of a single human SAN cell.

A, simulated action potential of a single human SAN cell (black thick line) and experimentally recorded action potentials of three different isolated human SAN cells (grey traces). Experimental data from Verkerk et al. (2009 a) and Verkerk and Wilders (2010). B, simulated (black thick line) and experimentally recorded (grey line) Ca²⁺ transient. Experimental data from Verkerk et al. (2013).

Table 5.

Action potential features

AP feature	Unit	Experimental value	Present model
MDP	mV	−61.7 ± 6.1	−58.9
CL	ms	828 ± 21	814
(dV/dt)max	V s−1	4.6 ± 1.7	7.4
APD20	ms	64.9 ± 23.9	98.5
APD50	ms	101.5 ± 38.2	136.0
APD90	ms	143.5 ± 49.3	161.5
OS	mV	16.4 ± 0.9	26.4
DDR100	mV s−1	48.9 ± 25.5	48.1

Comparison between experimental (mean ± SD; n = 3) (Verkerk et al. 2007 b) and simulation data.

The simulated Ca²⁺ transient qualitatively reproduces the single experimental trace acquired by Verkerk et al. (2013), showing a smaller transient amplitude and longer duration compared to rabbit data. Even if the model predicts slightly lower values for both diastolic and systolic [Ca²⁺]_i, the CaT amplitude (intracellular CaT amplitude; TA) is close to the experimental data (Fig. 2 B and Table 6).

Table 6.

Calcium transient features

Calcium transient	Unit	Experimental value	Present model
Ca i range	nm	105–220	85–190
TA	nm	115	105
TD20	ms	138.9	136.7
TD50	ms	217.4	206.3
TD90	ms	394.0	552.3

Comparison between experimental (Verkerk et al. 2013) and simulation data.

The time courses of the underlying currents, Ca²⁺ fluxes and Ca²⁺ concentrations in the four intracellular compartments are shown in Fig. 3. The main inward (depolarizing) and outward (repolarizing) membrane currents are I _CaL and I _Kr, respectively (Fig. 3 C). However, the peak amplitude of I _tot (Fig. 3 B), and therefore the maximum AP upstroke velocity, is determined not only by I _CaL, but also by I _NaCa (Fig. 3 E). Other currents are much smaller in amplitude than I _CaL and I _Kr. Yet, several are important determinants of the net ionic current during DD, and thus of CL (see below). Of note, as set out below, sinus node dysfunction can result from mutations in I _f, I _Na and I _Ks, which are among the smallest currents (Fig. 3 D). The calcium‐induced calcium release from the jSR upon calcium entry through the I _CaL channels is reflected by the rapid increase in J _rel (Fig. 3 G) and the accompanying rapid drop in Ca _jSR (Fig. 3 K). The resulting increase in Ca _sub (Fig. 3 I) and Ca _i (Fig. 3 J) generates an increase in Ca²⁺ uptake (J _up) (Fig. 3 H).

Figure 3. Time courses of simulated AP and underlying ionic currents.

Simulated AP (A) and associated currents (B–F), fluxes (G and H), and calcium concentrations (I–K) in control steady‐state conditions. Not shown is I _K,ACh, which is zero under control conditions. Note differences in ordinate scale.

Ionic currents during the DD phase

Figure 4 shows the time course of individual currents that play a relevant role during DD. The main inward currents during DD are I _CaT, I _f, I _NaCa and I _CaL. I _CaT activates before the membrane potential reaches MDP and reaches its maximal current density in the first 100 ms of DD (early DD), thus contributing to DDR₁₀₀, and then slowly decreases. I _f is considerably smaller than I _NaCa but its amplitude during DD is comparable with that of the net inward current (Fig. 4 B, dash‐dotted line). Similar to I _CaT, I _f starts activating when the membrane is still repolarizing and it provides its maximal contribution in the first half of diastole. I _NaCa is a high‐density inward current; it slowly diminishes during diastole, whereas it rapidly increases at the end of DD, providing an important contribution to the AP upstroke. A small amount of fast I _Na window current is active during DD. It is smaller than I _f, yet it is not negligible; it is able to modulate beating rate, as demonstrated by our simulations of mutations in the sodium voltage‐gated channel α subunit 5 (SCN5A) gene (see below). I _CaL follows a progressive increase during DD and becomes the major contributor to the net inward current at the end of DD; it has a relevant role during both DD and AP.

Figure 4. Membrane currents underlying diastolic depolarization.

Membrane potential (A) and associated net membrane current (B) (I _tot, dash‐dotted trace) during diastolic depolarization, as well as contributing inward and outward currents. t _MDP and t _TOP indicate the time at which V _m reaches MDP and take‐off potential, respectively.

I _NaK, I _Kr and I _Ks are the main outward currents. I _NaK slowly increases during DD and reaches its maximal current density during AP, contributing to repolarization. I _Kr is the major driver in the repolarization process. It contributes to DD through its progressive decrease during this phase. The contribution of I _Ks to DD is almost negligible under control conditions, although gain‐of function mutations can lead to a remarkable slowdown of pacemaking (see below).

Sensitivity analysis

Among the randomly generated population of 500 models, more than 300 showed an auto‐oscillating behaviour. The results of the sensitivity analysis are reported in Fig. 5 A, which shows the coefficients of the sensitivity matrix B, coded in a colour map. Each value in the matrix shows how a change in the parameter P displayed at the top is capable of affecting the feature F displayed on the left, and the coefficients <−0.2 and >0.2 are reported in the corresponding pixel. The obtained map highlights that nine out of the starting 18 parameters show coefficients lower or higher than the selected thresholds of −0.2 and 0.2, respectively.

Figure 5. Sensitivity analysis.

A, colour coded map of sensitivity matrix B. Columns show how a specific parameter P affects AP and CaT features; rows show how each feature is affected by different parameters. Red, blue and white pixels represent positive, negative and no substantial correlation, respectively, between parameters and features. Coefficients < −0.2 or > 0.2 are considered substantial. B and C, graphs indicating how changes in each of the parameters affect CL and DDR₁₀₀. The two panels display the information reported in rows 3 and 8 of (A) in different ways.

As illustrated in Fig. 5 A, changes in the permeability of I _CaL (P _CaL) and its activation kinetics (slope factor k _dL and shift in half‐activation voltage shift_dL) have a large impact on upstroke velocity (dV/dt), AP duration (APD_20,50,90) and calcium transient. Furthermore, the kinetics of the I _CaL activation gate dL strongly affect CL and DDR₁₀₀. Similarly, a change of the maximal conductance of I _Kr (g _Kr) has a high impact on APA, MDP and APD, whereas the maximal activity of NCX (K _NaCa) strongly influences APA, dV/dt and calcium transient. Finally, a shift in the working point of the SERCA pump (shift_up) clearly affects the diastolic calcium concentration (Ca _i,min).

Figure 5 B shows that CL is largely determined by the I _CaL activation kinetics (through its parameters k _dL and shift_dL). Figure 5 C shows that DDR₁₀₀ is also largely determined by the activation kinetics of I _CaL but in the opposite way. In addition, it reveals that the permeability of I _CaT (P _CaT) is also an important determinant of DDR₁₀₀.

The parameter randomization, the first step of the linear regression approach, allowed us to explore a neighbourhood, in the parameter space, of the parameter set obtained as a result of the automatic optimization procedure. Only a few parameter sets (out of 500 tested) led to comparable values of the cost function. In particular, only four led to values of MDP, CL, APD₉₀ and DDR₁₀₀ within the target range (i.e. mean ± SEM of experimental values) and only one parameter set led to a slightly lower value of the cost function than the one obtained from the ‘optimized’ set of parameters. To compare these two models, the effects on the pacemaking rate of mutations affecting I _f and I _Na were compared for the two parameter sets (as detailed in the section ‘Model validation through the simulation of ion channel mutations’ below). The ‘alternative’ set of parameters produced effects in close agreement with the ‘optimized’ set: the difference in mutation‐induced changes in the pacemaking rate was always lower than 2.7%.

The presence of multiple parameter sets compatible with experimental ranges allowed us to compute estimation intervals for the parameters that underwent the optimization procedure (Sarkar & Sobie 2010). Nominal values and ranges for the optimized parameters are reported in Table A1 in Appendix 2.

To confirm that parameter values outside the estimated confidence interval lead to non‐physiological (or at least non‐basal) conditions, we tested the effect of changing P _CaT by 50% more or 50% less with respect to the parent rabbit model (instead of +15% as in the ‘optimized’ model). In accordance with the above observation that P _CaT is an important determinant of DDR₁₀₀, a large increase in P _CaT led to a notably higher beating rate (+17%; from 74 to 86 beats min^–1), whereas a large decrease in permeability resulted in a slower beating rate (−18%; from 74 to 60 beats min^–1). These results indicate that I _CaT could play a substantial role in pacemaking and underscore that the proposed value for P _CaT is quite well constrained (see also Table A1 in Appendix 2).

Model validation through the simulation of ion channel mutations

The implementation of the functional changes in ion channels as a result of genetic mutations allowed validation of the present model of the human SAN AP. A prerequisite of the validation process was that mutations included in the analysis had to be characterized both clinically and electrophysiologically. Specifically, the criteria that a mutation had to satisfy were: (i) the availability of clinical data on adult patients affected by sinus node dysfunction related to the ion channel mutation of interest and (ii) the availability of an electrophysiological characterization of the mutation, usually carried out in heterologous cell systems (COS‐7, CHO, tsA201 and HEK‐293 cells; Xenopus oocytes), into which the mutation was transfected. Furthermore, to represent the clinical conditions as faithfully as possible, the electrophysiological data were ideally obtained from systems in which the expression of the mutation was heterozygous. However, in the absence of data on heterozygously expressed gain‐of‐function mutations in potassium voltage‐gated channel subfamily Q member 1 (KCNQ1), data from homozygously expressed mutations were used.

We used the data from electrophysiological studies to simulate the effects of mutations in the HCN4, SCN5A and KCNQ1 genes, encoding the HCN4, Na_V1.5 and K_V7.1 pore‐forming α‐subunits of the I _f, I _Na and I _Ks channels, respectively. Next, we compared the results of our simulations with the clinical data. When clinical data relative to non‐carrier family members were available, the model was tuned through autonomic modulation to reproduce the average heart rate reported for non‐carriers as the control beating rate. Furthermore, if the clinical control rate was lower than the intrinsic beating rate of the model (74 beats min^–1), a not‐null basal level of ACh was simulated, whereas a fixed percentage of the effects of isoprenaline on all its targets was simulated to obtain higher control rates. When clinical data relative to non‐carriers were not available, the effects of mutations were evaluated starting from the intrinsic beating rate of the model.

Effects of I _f mutations

All the initially observed mutations in HCN4 resulted in a loss of function of I _f (Verkerk & Wilders, 2015), although, recently, a gain‐of‐function mutation in HCN4 (R524Q) has also been reported by Baruscotti et al. (2017). A complete list of the mutations taken into consideration in the present study, together with a short description of their electrophysiological characterization and the parameter values used in simulations, is reported in Table 2.

A loss‐of‐function‐induced decrease in the intensity of I _f could be the result of (i) a hyperpolarizing shift of the steady‐state activation curve y _∞ and time constant τ_y [e.g. for the Y481H and G482R (Milano et al. 2014) and the K530N (Duhme et al. 2013) mutations]; (ii) a reduction of the maximal conductance [e.g. for the G482R mutation (Schweizer et al. 2014) and the D553N mutation (Ueda et al. 2004)]; and (iii) both of the aforementioned changes together (e.g. for the G480R, A485V and S672R mutations; Verkerk & Wilders, 2015). Furthermore, there can be mutation‐induced changes in the activation and/or deactivation rate of the I _f channel that contribute to the decrease of I _f intensity (e.g. the slowed activation in case of the G480R mutation; Nof et al. 2007).

In the simulations, all three types of loss‐of‐function mutations led to an increase in CL and thus a reduction of pacemaking rate (Fig. 6 A). A more detailed look at the AP features reveals that DDR₁₀₀ showed a substantial decrease for all loss‐of‐function mutations, varying from 8.5% for the relatively mild S672R mutation (pacemaking rate of 69 beats min^–1) to 40.7% for the A485V mutation (pacemaking rate of 61 beats min^–1), whereas APD₉₀ was almost unchanged in our simulations. Similarly, changes in MDP and APA were almost negligible: the maximum hyperpolarization of the MDP amounted to 0.5 mV and the maximum increase in APA was 0.9 mV (both for the A485V mutation).

Figure 6. Clinically observed vs. simulated effects of I _f, I _Na and I _Ks mutations.

A, comparison between clinical data (solid and hatched light grey bars) and simulation data (solid and hatched dark grey bars) on the effects of mutations affecting HCN4 channels (I _f). For the Y481H, R524Q and D553N mutations, no clinical data from non‐carrier family members were available. SDs of the clinical data, where available, are indicated. B, comparison between clinical data (solid and hatched light grey bars) and simulation data (solid and hatched dark grey bars) on the effects of mutations affecting SCN5A channels (I _Na, group on the left) and KCNQ1 (I _Ks, group on the right). SDs of the clinical data, where available, are indicated. For the E1784K (a) mutation, no clinical data from non‐carrier family members were available. Clinical data on the E1784K (b) mutation were only reported in a qualitative way (‘sinus node dysfunction’) (Table 3). For the R231C and V241F mutations, no clinical data from non‐carrier family members were available. SDs of the clinical data, where available, are indicated.

The only gain‐of‐function mutation, R524Q, led to an increase in I _f by shifting the steady‐state activation curve and the time constant curve towards less negative potentials (Baruscotti et al. 2017). The simulation of the mutation showed a faster pacemaking rate (79 beats min^–1). DDR₁₀₀ had an increase of 8.2%, whereas MDP, APD₉₀ and APA were unchanged.

The comparison between the pacemaking rates in our simulation data and the clinically observed heart rates (Fig. 6 A) highlighted the ability of the model to reproduce the effects of the mutations on HCN4 channels, at least qualitatively. Quantitatively, the effects in the simulation data are consistently smaller than those observed clinically, which might be expected (see Discussion).

Effects of I _Na mutations

A large number of mutations identified in SCN5A are related to sinus dysfunction, such as sinus bradycardia and sinus pauses (often associated with the LQT3 phenotype) and sick sinus syndrome (Veldkamp et al. 2003; Lei et al. 2007).

A loss of function of the I_Na channel can have several causes: (i) a rapid inactivation of the ion channel; (ii) a depolarizing shift of activation; (iii) a hyperpolarizing shift of inactivation; and (iv) a reduction of the current density, which, in some cases, can lead to the non‐function phenotype (i.e. a full loss of function). In several cases, multiple causes can be present at the same time, enhancing the reduction of I _Na (Table 3).

The simulations of the effects of loss‐of‐function I _Na mutations showed a decrease in pacemaking rate, up to 11.8% for the E161K mutation. DDR₁₀₀ showed a decrease for all mutations (with a maximum of 9.0% for E161K). APD₉₀, MDP and APA were almost unchanged for all mutations.

In simulations of ΔKPQ (Nagatomo et al. 1998) and E1784K [parameters from Deschênes et al. (2000) or Makita et al. (2008)] and 1795insD mutations [parameters from Bezzina et al. (1999) and Veldkamp et al. (2000)], an incomplete inactivation (gain of function) was combined with a loss of function; the absence of a complete inactivation induced a sustained (late) Na⁺ current. The DDR₁₀₀ resulting from this combined gain and loss of function showed a decrease (−5.8% for ΔKPQ, −7.9% and −7.7% for E1784K, and −8.0% for 1795insD mutation). By contrast, APD₉₀ showed an increase for all three mutations (+1.3% for ΔKPQ, +2.2% and +1.5% for E1784K, and +0.6% for 1795insD mutation). The overall effect was a reduction of the pacemaking rate for all simulated mutations (by −4.8% for ΔKPQ, −2.9% and −3.7% for E1784K, and −6.2% for 1795insD mutation). To assess the contribution of an incomplete inactivation of I _Na channels in these ‘mixed mutations’, we investigated the effect of I _NaL alone by introducing this current into a cell in control conditions. The presence of I _NaL caused two opposing effects: enhanced inward current during late DD resulted in a steeper DDR, whereas inward I _Na current during AP prolonged APD. In our model for g _NaL set to 1% of g _Na in the control condition, the higher inward current during late DD prevailed. The overall effect of I _NaL was a minor increase of the pacemaking rate (+3.2%). However, in the four mixed mutations analysed, the loss‐of‐function effect prevailed and led to an overall slowdown of the pacemaking rate.

A comparison of pacemaking rates predicted by the model and clinical data is shown in Fig. 6 B. The simulation data for the G1406R, D1790G and 1795insD mutations were in close agreement with the clinical data. By contrast, for the other three mutations, the simulations showed a decrease of the beating rate compatible with the loss‐of‐function behaviour of the mutations, although it was less close to the clinically observed effect.

Effects of I _Ks mutations

Both loss‐of‐function and gain‐of‐function mutations in KCNQ1, encoding the K_V7.1 pore‐forming α‐subunit of the I _Ks channel, have been reported in relation to sinus bradycardia.

A gain of function of K_V7.1 channels could be the result of (i) a higher current density, as observed for the R231C mutation (Henrion et al. 2012) or (ii) a hyperpolarizing shift of the steady‐state activation curve, as observed for the V241F mutation (Ki et al. 2013). With either mutation, DDR₁₀₀ was markedly increased. However, the overall DDR was reduced, leading to a longer DD phase duration. Both mutations showed a slight depolarization of MDP (to −57.6 and −57.1 mV for R231C and V241F, respectively), whereas APD₉₀ was considerably reduced (by 10.8% and 19.8% for R231C and V241F, respectively). Thus, opposing effects were present. However, the prolongation of the DD phase far outweighed the effects of a higher DDR₁₀₀ and a shorter APD₉₀, leading to slower pacemaking rates (57 and 40 beats min^–1 for R231C and V241F, respectively), which were in good agreement with clinical data (Fig. 6 B).

On the other hand, a loss of function of K_V7.1 channels can be attributed to (i) a lower channel expression or (ii) a depolarizing shift of the steady‐state activation curve. The ΔF339 mutation shows both effects (Table 4) (Thomas et al. 2005). The model predicted a negligible change in pacemaking rate (from 63 to 63.5 beats min^–1), which is consistent with clinical data (Fig. 6 B); DDR₁₀₀ was slightly reduced (by 2.4%) and APD₉₀ was increased (by 1.3%), whereas MDP was almost unchanged.

Model‐based analysis: effects of I _f and Ca²⁺ handling modulation on pacemaking

Contribution of I _f to pacemaking

To quantitatively assess the sensitivity of CL, and thus of pacing rate, to the funny current, we carried out two simulation experiments: (i) a progressive block of funny current, by reducing the maximal conductance g _f and (ii) a shift in voltage dependence of the steady‐state activation curve, with a range from −15 to +15 mV.

Four levels of block were assessed: 30%, 70%, 90% and 100% (i.e. full block) (Fig. 7 A). The total block of funny current led to an increase in CL of 28.1% to 1043 ms, and thus a decrease in pacemaking rate of 22% to 58 beats min^–1, in good accordance with the 26% increase in CL that was experimentally observed by Verkerk et al. (2007 b), who administered 2 mm Cs⁺ as a blocker of I _f to a single isolated SAN cell (Fig. 7 B). At this concentration, Cs⁺ almost completely blocks I _f in the pacemaker range of potentials, whereas the delayed rectifier K⁺ current, which is also sensitive to Cs⁺, is only affected slightly (Denyer & Brown, 1990; Zaza et al. 1997; Liu et al. 1998).

Figure 7. Functional effect of I _f block.

A, simulated AP under control conditions (CTRL) and upon 30%, 70%, 90% and full block of I _f. B, effects of administration of Cs⁺ (2 mm) on the AP of an isolated human SAN myocyte. Experimental trace adapted from Verkerk et al. (2007 b).

The increase in CL was mainly the result of a lower DDR and longer DD phase (with DDR₁₀₀ decreasing by 39.7% to 29 mV s⁻¹ upon full block), whereas APD₉₀ and MDP remained almost unchanged (Fig. 7 A). Of note, the increase in CL was also largely the result of a lower DDR and longer DD phase in the experiment of Verkerk et al. (2007 b) (Fig. 7 B).

Negative shifts of the activation led to an increase in CL, up to 957 ms (+17.5% with respect to control) with a −15 mV shift, whereas positive shifts shortened CL, up to 577 ms (−29.2%) with a +15 mV shift (Fig. 8 A). As with the I _f block, DDR₁₀₀ was the main descriptive parameter that changed (to 37.7 mV s⁻¹ (−25.8%) with a −15 mV shift and to 75.4 mV s⁻¹ (+56.8%) with a +15 mV shift; Fig. 8 B), whereas APD₉₀ and MDP remained almost unaffected, as for the I _f block (Fig. 8 C and D).

Figure 8. Functional effect of changes in voltage dependence of I _f activation.

Simulations of the effect of –15 to +15 mV shifts (with steps of 5 mV) in voltage dependence of the y _∞ steady‐state activation curve of the funny current on (A) cycle length, (B) DDR₁₀₀, (C) MDP and (D) APD₉₀.

Contribution of I _NaCa to pacemaking and spontaneous calcium oscillations

The Na⁺/Ca²⁺ exchanger is an important actor in the generation of the AP and also contributes to the diastolic depolarization phase. To assess the impact that I _NaCa has on CL, we performed a progressive reduction of the maximal activity of Na⁺/Ca²⁺ (K_NaCa) (Fig. 9).

Figure 9. Functional effect of I _NaCa block.

Simulated AP (A) and associated [Ca²⁺]_i time course (C) under control conditions (CTRL) and upon 50%, 75% and 90% block of I _NaCa. Simulated AP (B) and associated I _NaCa time course (D) relative to the dashed box of (A).

Reductions in K_NaCa of 50% and 75% unexpectedly led to faster pacemaking, with a rate of 83 beats min^–1 (+12.2%) and 93 beats min^–1 (+25.7%), respectively. DDR₁₀₀, APD₉₀ and MDP contributed synergistically towards a shorter CL: DDR₁₀₀ increased remarkably, APD₉₀ shortened and MDP depolarized (Fig. 9 A). DDR₁₀₀ was steeper as a result of a more intense I _NaCa density during DD (Fig. 9 B and D) because of a higher concentration of [Ca²⁺]_i in the cell (Fig. 9 C). The reduced maximal activity of the Na⁺/Ca²⁺ exchanger also resulted in a lower contribution of I _NaCa to the AP: the overshoot potential was lower and repolarization was faster, resulting in a shorter APD.

Higher levels of block of the exchanger (reduction of maximal activity by 90%) stopped the automaticity of the cell. The inward current provided by the Na⁺/Ca²⁺ exchanger was no longer sufficient to make the cell reach its threshold potential for an AP. Membrane potential stabilized at a level approaching –40 mV (Fig. 9 A and B), whereas [Ca²⁺]_i stabilized at a constant value close to 200 nm (Fig. 9 C).

The presence in the model of the ‘isolated Ca²⁺ oscillator’ was also tested, by reproducing the simulations in which all membrane currents are set to zero. Spontaneous oscillations of intracellular and subsarcolemmal calcium were indeed achieved upon a large increase in the SERCA pump activity. Notably, the frequency of these oscillations was definitely higher than the physiological human heart rate (> 5 Hz). In particular, calcium oscillations were not sustained for P _up values up to more than 90 nm s⁻¹; sustained oscillations, with a frequency of ≈5 Hz, arose upon increasing P _up to a value approaching 100 nm s⁻¹. This frequency stayed almost constant as P _up was increased to 120 mm s⁻¹, suggesting that it might be not easy to tune the oscillation rate in a robust way, at least in this lumped‐parameter model of calcium handling.

Autonomic modulation of pacemaking

The simulated administration of 10 nm ACh resulted in a reduction of spontaneous rate by 20.8%. DDR₁₀₀ was decreased by 9.8% and APD₉₀ was shortened from 161.5 to 154 ms (−4.6%); MDP was not altered. Even though ACh had opposing effects on DDR₁₀₀ and APD₉₀, the increase in CL showed that the DDR₁₀₀ reduction prevailed (Fig. 10 A).

Figure 10. Functional effect of ACh and isoprenaline.

Time course of (A) membrane potential, (B) net current, target currents (C–G) and SERCA‐pump uptake (H) in control conditions (CTRL, black lines), upon administration of 10 nm ACh (dark grey lines) and 1 μm Iso (light grey lines). Targets for ACh are I _f, I _CaL, I _K,ACh and J _up; targets for Iso are I _f, I _CaL, I _NaK, I _Ks and J _up. Note the differences in ordinate scales.

We also assessed the contribution of each individual ion current by applying the ACh effect to each current separately. Changes in I _K,ACh and I _f were dominant, inducing a reduction in the pacemaking rate of 11.6% and 6.9%, respectively. Exposure of only I _CaL to ACh resulted in a minor effect (1.1% reduction of pacemaking rate), whereas the modification of J _up led to negligible changes. Simulating the ACh effects on all targets but one led to similar results: I _K,ACh and I _f played a primary role, as demonstrated by the model‐predicted reduction of the pacemaking rate by 8% and 13% (when I _K,ACh and I _f were the only unchanged currents), respectively.

The overall effect induced by 1 μm Iso resulted in an increase in the pacemaking rate of 27.9% to 94 beats min^–1 (Fig. 10 A). DDR₁₀₀ increased from 48.1 mV s⁻¹ in control conditions to 53.4 mV s⁻¹ with Iso, whereas APD₉₀ and MDP were almost unchanged.

The assessment of the effect of Iso through each of the Iso‐sensitive currents alone showed that the modified I _f and I _CaL both led to a substantial increase in pacemaking rate (14.2% and 16.4%, respectively). The increased activity of I _NaK, on the other hand, slowed the pacemaking rate (−13.1%), whereas Iso‐induced changes in I _Ks and J _up only had a small effect (+1.5% and −0.4%, respectively).

Considering the effects of all currents but one confirmed the above results; when all the currents but one, either I _CaL, I _f, I _NaK, I _Ks or J _up, were affected by 1 μm Iso, an increase in beating rate equal to 2.1%, 9.6%, 40.6%, 21.3% and 28.7%, respectively, was observed. Of note, these values also show that I _Ks becomes more prominent at higher rates, when there is less time for its deactivation. The effect of I _Ks alone is small, with a 1.5% increase in beating rate, although the ‘all but one’ data (21.3% increase) reveal a more prominent role of I _Ks at higher rates; indeed, the net contribution of I _Ks (overall effect – all but I _Ks effect) shows an increase of beating rate equal to 6.6% (27.9–21.3%).

The model was capable of reproducing the full clinical range of human heart rate, assumed to be from 40 to 180 beats min^–1. In particular, the model predicted that a concentration of ACh equal to 25 nm was able to slow down the pacemaking rate to 40 beats min^–1 (−45.9%). This concentration was responsible for a negative shift of the (de)activation steady‐state curve y_∞ and time constant τ_y of −6.3 mV, a reduction of P _CaL equal to −6.7% and a maximal activity of SERCA pump P _up reduced to 85% with respect to control conditions. The I _K,ACh was also affected, showing a peak value of 0.33 pA/pF. The model also showed stable oscillations at a pacing frequency equal to 182 beats min^–1. Changes in the Iso target parameters responsible for this pacemaking rate were quantified (with respect to control conditions) as: y _∞ and τ_y were shifted by +12 mV; I _NaK,max and g _Ks were increased by 92%; and n _∞ and τ_n were shifted by −22 mV. I _CaL was affected by a 96.8% increase of P _CaL, a shift of dL _∞ and τ_dL by –12.8 mV, and a slope factor k _dL reduced by −29.7%. Thus, we directly tuned the parameters affected by Iso, increasing the aforementioned effects of 1 μm Iso to arrive at a pacemaking rate approaching 180 beats min^–1.

Discussion

In the present study, we formulated a comprehensive human SAN AP model, starting from a state‐of‐the‐art model of rabbit cardiac pacemaker cells, and converting it into a species‐specific description using human experimental data as far as possible. A novel aspect is the adoption of an automatic optimization procedure, tightly bounded by the AP features, to identify the parameters for which experimental data are presently unavailable.

Previously, Chandler et al. (2009) showed that scaling ion current densities, according to the gene expression pattern they found in human SAN vs. atrial tissue, gives rise to automaticity in the human atrial cell model of Courtemanche et al. (1998). However, the values of the descriptive parameters of AP, in particular MDP, APA and APD, were far from the experimentally observed ones.

Our model is able to reproduce the main experimentally observed electrophysiological features (AP waveform, calcium transient) and the simulated changes in pacemaking rate as a result of mutations affecting ionic channels are in line with clinical data. Furthermore, it allows the investigation of relevant conditions such as I _f block, NCX block and autonomic stimulation, as we have demonstrated above.

Construction of the model

AP models of cardiac cells are commonly based on voltage clamp data for ionic current identification and current clamp data for model validation. We tried to maximally exploit the scarce data available on human SAN cells, aiming at a detailed reproduction of experimental AP morphology. Updating the description of the funny current, based on the experimental data obtained by Verkerk et al. (2007 b), was also crucial. In our human cell model, I _f current density during DD is considerably lower than in rabbit SAN cells, although it remains comparable to the net inward current as reported by Verkerk and Wilders (2010) in their virtual AP clamp experiments.

In control conditions, the model reproduces the experimental AP features well. The simulated AP is characterized by CL = 814 ms, which is close to the experimentally observed values (Verkerk et al. 2007 b) and critical parameters such as MDP, APD₉₀ and DDR₁₀₀ are all within the range of the experimental data (Table 5). As a consequence, the model shows a representative behaviour during both DD (subthreshold) and AP (Fig. 3).

To achieve a calcium transient close to the experimental trace, several actors of Ca²⁺ handling underwent changes; in particular, maximal NCX activity was reduced by 53%, and a sigmoidal equation was introduced to describe SR uptake. These updates allowed a higher diastolic [Ca²⁺]_i and finer control during DD.

The Cs⁺‐induced CL prolongation that was experimentally observed was also taken into account to tune parameters. Simulating the administration of 2 mm Cs⁺ led to an increase in CL of 28.1%, close to the experimentally observed 26% (Fig. 7). This Cs⁺ concentration was reported to selectively and almost fully block the funny current (Verkerk et al. 2007 b) and so the reported prolongation established the maximal extent of the contribution of I _f to the pacemaking rate decrease in our model. Thus, the experimentally observed levels of I _f block by Cs⁺ were highly relevant. Accordingly, we should not forget that a voltage‐dependent partial block of funny channels by 5 mm Cs⁺ has been reported elsewhere (DiFrancesco et al. 1986). If the parameters in our model were tuned to reproduce a similar partial Cs⁺‐induced block, a larger effect of complete I _f block would be observed in simulations.

Sensitivity analysis

The sensitivity analysis highlighted the strong impact of I _CaL, in particular its activation kinetics (gate dL), on CL and on the calcium transient. A slight increase/decrease of the slope factor k _dL and a shift of the working point of the steady‐state curve of the activation gate are able to substantially change these two features; indeed, the availability of open I _CaL channels during the highly sensitive DD phase is strongly affected by slight changes of the dL kinetics.

The same AP can be produced by multiple sets of parameters. However, the observation that only a few parameter sets, out of a total of 500, produced simulation data within the experimental range suggests that the parameters should fall within a narrow range. Ranges in Table A1 in Appendix 2 are a first attempt to determine such physiological intervals for human SAN cells exploiting the limited experimental data. Future measurements will help to refine the information extracted by our investigation.

The illustrative investigation on the effects of changes in P _CaT suggests that this parameter is well constrained, and also that I _CaT plays a role in pacemaking, which is consistent with the report by Mangoni et al. (2006) who studied mice with a disrupted gene coding for Ca_V3.1 channels, as responsible for T‐type calcium current.

Ion channel mutations

Having obtained satisfying behaviour in control conditions, we further validated the model through the implementation of mutations affecting ion channels. Because of the clinical importance of this research, a large number of studies have previously been carried out and a considerable amount of both clinical and electrophysiological data are available. HCN4 is the most abundant isoform of the HCN family. Most HCN4 mutations result in a loss of function of I _f, although, recently, Baruscotti et al. (2017) found a mutation that results in a gain of function. In our simulations, I _f exerted its contribution during DD, influencing the amount of net inward current, and thus the slope of this phase. In this way, a loss of function of I _f leads to a slower rate, whereas a gain of function leads to a faster one. A remarkable aspect of this finding is that I _f is able to modulate DDR at the same time as leaving other important parameters (such as APD and MDP) unchanged.

The simulations of loss‐of‐function mutations in HCN4 show a considerable slowing but not a complete cessation of pacemaker activity. The slowing is consistently smaller than that observed in the associated mutation carriers (Fig. 6). The difference between clinical and simulated effects of mutations, however, is to be expected because the hyperpolarizing effect of the surrounding atrium will result in a more prominent role of I _f in the SA node of the intact heart. Of note, our current model appears to be more promising for assessing the effects of HCN4 mutations in humans than the comprehensive rabbit SAN cell models described by Maltsev and Lakatta (2009) and Severi et al. (2012). As recently reported by Wilders and Verkerk (2016), the mutation effects appear to be highly underestimated in the Maltsev–Lakatta model, in which a slowing of only 5.2% was observed for the most severe of the 11 mutations tested (A485V; 21.4% in the present model) and highly overestimated in the Severi–DiFrancesco model, where pacemaker activity ceased for five of the 11 mutations, including A485V.

In the validation stage, a large number of mutations in the SCN5A‐encoded Na_v1.5 channels were assessed. We observed two kinds of mutations: the first kind was characterized by changes in electrophysiological properties that led to a loss of function of channels (E161K, G1406R and D1790G); the second kind showed the concurrent presence of loss of function and gain of function (ΔKPQ, E1784K and 1795insD). The latter effect was the result of an incomplete inactivation of I _Na channels, which was incorporated into the model by means of the I _Na,L current.

Loss‐of‐function mutations reduced the inward current carried by I _Na and therefore decreased DDR. Thus, I _Na regulated heart rate in a way analogous to that of the funny current; however, the lower weight of I _Na in the overall inward current resulted in quantitatively fewer extensive effects with respect to I _f.

Similar to the mutations in HCN4, the slowing in pacemaking rate associated with mutations in SCN5A is consistently smaller in simulations than in the intact heart (Fig. 6). Again, this is to be expected because the hyperpolarizing effect of the surrounding atrium will also result in a more prominent role of I _Na in the SA node of the intact heart. In our current model, we observed a decrease in the pacemaking rate of up to 14.9% for the most severe SCN5A mutation. We could not test the SCN5A mutation effects in the Maltsev–Lakatta rabbit SAN cell model because this model lacks I _Na. In the Severi–DiFrancesco rabbit SAN cell model, the decrease in pacemaking rate was almost negligible for each of the SCN5A mutations tested, with a maximum near 1%.

The simulated effect of the ΔF339 loss‐of‐function mutation in KCNQ1 that affects K_V7.1 channels is almost negligible. The gain of function of I _Ks was more critical. The increase in outward current resulted in a DDR reduction that far outweighed the accompanying APD reduction. Thus, the gain‐of‐function mutations R231C and V241F resulted in a rate slowdown compatible with the clinically observed sinus bradycardia. However, it should be noted that the electrophysiological characterizations of both R231C and V241F were obtained in homozygous expression (Table 4). Although no experimental data are available concerning the R231C and V241F mutations in heterozygous expression, we decided to include these simulations anyway to provide a lower limit of the beating rate that the model can achieve for these mutations.

The comparison between clinical findings and simulated effects of mutations on beating rate showed good overall behaviour of the model. Except for the I _Ks ΔF339 mutation, simulated effects on beating rate were in accordance with the reported clinical data (Fig. 6). It should be noted that there is always a gap between clinical and simulated mutation effects, given the difference in the systems analysed: the heart rate is a macroscopic phenomenon, resulting from the overall SAN tissue behaviour and its interaction with the autonomic nervous system, whereas the beating rate predicted by the model considers only the microscopic single cell level. At this level, there is (for example) no hyperpolarizing effect of the surrounding atrium, which may increase I _f and I _Na and thus the functional effects of mutations in the associated genes. It should also be noted that carriers of the same mutation may show widely different phenotypes, even within the same family. This holds in particular for the SCN5A mutations; it is possible that only a minority of the affected family members show sinus node dysfunction. Finally, other factors, such as physical training level, can affect the difference in basal heart rate found between healthy subjects and mutation carriers.

As emphasized by Verkerk and Wilders (2015) for HCN4 mutations, although it can be readily extended to other ion channels, there are inconsistencies between clinical and experimental data. Many factors can underlie these inconsistencies: (i) data on current density can be misleading because heterologous systems (in vitro) can express different levels of the gene affected by the mutation with respect to the subject (in vivo) under investigation; (ii) experimental data can be incomplete and may strongly depend on the expression system and the experimental protocol employed; (iii) clinical data are often limited to a small number of patients, or even a single patient; (iv) the affected isoform is not the only one expressed in vivo; and (v) remodelling processes can occur in mutant carriers and affect their clinical phenotypes.

Physiological insights from simulations

Simulating I _f and NCX blocks and autonomic stimulation provided insights into the model behaviour and shed light on underlying phenomena.

Lower levels of I _f block and shifts of the steady‐state activation curve were able to modulate pacemaking rate, mainly by varying DDR₁₀₀. By contrast, MDP and APD₉₀ showed no changes (Figs. 7 and 8). These results suggest that I _f exerts its capability to modulate beating rate during the DD phase. As previously shown for the rabbit SAN AP, controlling the steepness of early DD is an efficient way of determining beating rate: small changes in currents are able to substantially modify the slope of DD, and thus the overall duration of the AP. This is even more true in human SAN because the pacing rate is even more sensitive to changes in DD slope. As a result of the non‐linear relationship between dV/dt and time (Zaza, 2016), the same change in current is bound to produce larger changes in CL when basal CL is longer. Unlike the parent model, pacemaker activity does not cease when I _f is fully blocked. Yet, the modulatory role of I _f is as important as in the parent model.

NCX provided an important contribution to automaticity, both during DD and during the AP upstroke. The considerable reduction of its maximal activity relative to the parent model, consistent with the gene expression pattern reported by Chandler et al. (2009), allowed [Ca²⁺]_i to reach a value closer to experimental values. The NCX block showed two different effects: for low‐to‐moderate levels of block, the beating rate became faster than under control conditions, whereas for high‐to‐almost‐full block, the automaticity stopped (Fig. 9). The latter result is consistent with experiments showing an inhibitory effect of Na⁺ replacement by Li⁺, thus abolishing NCX, on spontaneous beating in rabbit SAN cells (Bogdanov et al. 2001). The unexpected effect of rate acceleration upon low‐to‐moderate block can be the result of a counterintuitive, although physiological, process: a lower maximal NCX activity results in a higher [Ca²⁺]_i during DD and, consequently, a more intense I _NaCa. Nevertheless, the rate increase induced by I _NaCa reduction is actually a prediction of our model, and experimental data (not available yet) are needed to confirm this prediction in human SAN cells.

Unlike I _f modulation, NCX block affected several main features: DDR₁₀₀, MDP, APD₉₀ and even APA underwent considerable changes (Fig. 9). Of note, both the absolute and relative amplitude of I _f and I _NaCa during DD were highly similar to those predicted by Verkerk et al. (2013) through numerical reconstruction.

It is worth noting that the ‘calcium clock’ was integrated in the parent model and it is also present in this human model but, as expected, the NCX current cannot effectively modulate the pacemaking rate. Based on the rabbit model simulations, the SERCA pump activity could be expected to modulate pacemaking rate. This was not the case in our model, in which changes in SERCA maximal activity had a negligible effect on rate. This leaves open the question about the actual relevance of the ‘calcium clock’ in human SAN.

The adopted calcium handling considered the presence of a subsarcolemmal space, which is also present in the rabbit SAN cell models of Kurata et al. (2002), Maltsev and Lakatta (2009) and Severi et al. (2012). Because T‐tubules are poorly defined in SAN cells, this subsarcolemmal space is proposed as a ‘fuzzy space’ (Lederer et al. 1990). NCX1 and RyRs are co‐located there, as reported by Lyashkov et al. (2007), who used immunolabelling techniques and confocal imaging. Through further simulations, we tested the effect of a different cell arrangement of RyRs in the cell space. We progressively switched the SR release from the subspace to the cytosol until a 100% Ca²⁺ release, targeted to the cytosol and sensitive to Ca _i instead of Ca _sub, was reached. The model did not show dramatic changes; in an SR release configuration fully targeted to cytosol, we only observed a minor change of pacemaking rate (+5.2%; from 74 to 78 beats min^–1), as well as a minor increase in Ca _i (Ca _i,min from 84 to 99 nm, Ca _i,max from 189 to 199 nm).

The only study to report experimental data on the administration of ACh and Iso in human SAN tissue was carried out by Drouin (1997). However, the observed intrinsic pacemaking frequency was not coherent with clinical observations (30 beats min^–1 in vitro vs. 70–112 beats min^–1 in situ); it is therefore most probable that the effects of ACh and Iso were exacerbated. Furthermore, ACh and Iso targets are not electrophysiologically characterized in humans, and so we used experimental data from rabbit SAN. Thus, our simulations of ACh and Iso effects provide theoretical insights about the mechanisms underlying autonomic modulation that may require updates with respect to future data that become available from human tissue.

The administration of 10 nm ACh led to a reduction of the pacemaking rate as a result of the activation of I _K,ACh together with changes to I _f, SR uptake and I _CaL. The ACh‐induced reduction of I _f and activation of I _K,ACh appeared to be the major determinants of rate slowdown.

The overall pacing rate acceleration as a result of the administration of 1 μm Iso was the result of a balance between opposing contributors. An assessment of the role of each of the five targets (I _f, I _NaK, I _CaL, I _Ks and SR uptake) showed that both I _f and I _CaL changes led to a faster beating rate. However, the underlying mechanisms were different: I _f worked on the early phase of DD, modifying DDR₁₀₀ (also shown in I _f modulation caused by shifts in voltage dependence), whereas I _CaL exerted its effect during late DD. I _NaK showed an enhanced activity with Iso that decreased DDR and therefore counteracted the overall pacing rate acceleration.

An important test of the validity of the model was its ability to reproduce the clinical range of human heart rate. Our model was able to provide stable automaticity at ∼40 beats min^–1 upon administration of 25 nm of ACh, whereas the maximum rate (180 beats min^–1) was obtained by enhancing the effects of Iso on its target parameters.

The overall message from model‐based simulations of autonomic modulation of pacemaking is that I _f plays an important role in rate regulation, as in rabbit. However, at least one other current (I _K,ACh for ACh and I _CaL for Iso administration) needs to be modulated to achieve pacemaking rate regulation in the full physiological range.

Limitations and future developments

One evident limitation of the present study is that there are insufficient experimental data available on human SAN electrophysiology to fully constrain the parameters of the model. It is therefore possible that some non‐human‐specific aspects of the parent (rabbit) model are still present. This could be the case for some aspects of the calcium handling, which could underlie the very high frequency of spontaneous calcium oscillations. The present model thus only represents one step forward in the quantitative description of human SAN electrophysiology. Further experiments carried out on human adult SAN cells, or perhaps even on human induced pluripotent stem cell‐derived cardiomyocytes, will challenge the current model, providing confirmation and/or clarifying any changes that need to be made.

Specific limitations of the model include: the lack of intracellular sodium dynamics, the stop of pacemaking beating when [Ca²⁺]_o is set to a physiological in vivo value (1.15 mm; Severi et al. 2009) and the lack of a detailed description of local calcium releases.

Conclusions

In the present study, we present a human sinus node AP model based on electrophysiological data. The model is able to reproduce the experimentally observed AP morphology, calcium transient and CL prolongation upon I _f block by 2 mm Cs⁺, as well as the effects of mutations responsible for cardiac ion channelopathies associated with sinus node dysfunction.

In addition to contributing to a better understanding of the events that occur on the cellular scale, the model could be a useful tool in the design of future experiments and the study (at a preliminary stage) of the effects of drugs that aim to modulate heart rate.

Additional information

Competing interests

The authors declare that they have no competing interests.

Author contributions

SS was responsible for the study conception. AF and SS were responsible for the study design. AF was responsible for simulations AF, MF, RW and SS were responsible for data analysis. AF, MF and SS were responsible for data interpretation. AF, RW and SS were responsible for writing the manuscript. AF, MF, RW and SS were responsible for the final approval of the manuscript submitted for publication. All authors agree to be accountable for all aspects of the work. All persons designated as authors qualify for authorship, and all those who qualify for authorship are listed.

Funding

No funding was received for the present study.

Appendix 1.

Model parameters and equations

Cell compartments

 

$C=57pF$:	cell capacitance
$L_{\mathrm{cell}}=67\mu \mathrm{m}$:	cell length
$L_{\mathrm{sub}}=0.02\mu \mathrm{m}$:	distance between jSR and surface membrane (submembrane space)
$R_{\mathrm{cell}}=3.9\mu \mathrm{m}$:	cell radius
$V_{{\mathrm{i}}_{\mathrm{part}}}=0.46$:	part of cell volume occupied with myoplasm
$V_{\mathrm{js}{\mathrm{r}}_{\mathrm{part}}}=0.0012$:	part of cell volume occupied by junctional SR
$V_{\mathrm{ns}{\mathrm{r}}_{\mathrm{part}}}=0.0116$:	part of cell volume occupied by network SR
$V_{\mathrm{cell}}=\pi ·{R_{\mathrm{cell}}}^2·L_{\mathrm{cell}}$:	cell volume
$V_{\mathrm{sub}}=2·\pi ·L_{\mathrm{sub}}·(R_{\mathrm{cell}}-\frac{L_{\mathrm{sub}}}{2})·L_{\mathrm{cell}}$:	submembrane space volume
$V_{\mathrm{i}}=V_{{\mathrm{i}}_{\mathrm{part}}}·V_{\mathrm{cell}}-V_{\mathrm{sub}}$:	myoplasmic volume
$V_{\mathrm{jsr}}=V_{\mathrm{js}{\mathrm{r}}_{\mathrm{part}}}·V_{\mathrm{cell}}$:	volume of junctional SR (Ca2+ release store)
$V_{\mathrm{nsr}}=V_{\mathrm{ns}{\mathrm{r}}_{\mathrm{part}}}·V_{\mathrm{cell}}$:	volume of network SR (Ca2+ release store)

Fixed ion concentrations, mm

 

$Cao=1.8$:	extracellular Ca2+ concentration
$Ki=140$:	intracellular K+ concentration

$Ko=5.4$:	extracellular K+ concentration
$Nao=140$:	extracellular Na+ concentration
$N{a}_{\mathrm{i}}=5.0$:	intracellular Na+ concentration
$Mgi=2.5$:	intracellular Mg2+ concentration

Variable ion concentrations, mm

 

$Cai$:	intracellular Ca2+ concentration
$C{a}_{\mathrm{jsr}}$:	Ca2+ concentration in the junctional SR
$C{a}_{\mathrm{nsr}}$:	Ca2+ concentration in the network SR
$C{a}_{\mathrm{sub}}$:	subsarcolemmal Ca2+ concentration

Ionic values

 

$F=96485\frac{C}{mol}$:	Faraday constant
$R=8314.472\frac{J}{(kmolK)}$:	universal gas constant
$T=310K$:	absolute temperature for 37°C
$RTonF=\frac{R·T}{F}=26.72655\mathrm{mV}$
$E_{\mathrm{Na}}=RTonF·\ln \frac{Nao}{Nai}$	reversal potential for Na+
$E_{\mathrm{mh}}=RTonF·\ln \frac{Nao+0.12·Ko}{Nai+0.12·Ki}$	reversal potential for fast Na+ channel
$E_{\mathrm{K}}=RTonF·\ln \frac{Ko}{Ki}$	reversal potential for K+
$E_{\mathrm{Ks}}=RTonF·\ln \frac{Ko+0.12·Nao}{Ki+0.12·Nai}$	reversal potential for slow rectifier K+ channel
$E_{\mathrm{Ca}}=0.5·RTonF·\ln \frac{Cao}{C{a}_{\mathrm{sub}}}$	reversal potential for Ca2+

Sarcolemmal ion currents and their conductances

 

I f:	hyperpolarization‐activated current ($g_{\mathrm{fNa}}=0.00268\mu \mathrm{S},g_{\mathrm{fK}}=0.00159\mu \mathrm{S}$)
I CaL:	L‐type Ca2+ current $(P_{\mathrm{CaL}}=0.4578\frac{\mathrm{nA}}{\mathrm{mM}})$
I CaT:	T‐type Ca2+ current $(P_{\mathrm{CaT}}=0.04132\frac{\mathrm{nA}}{\mathrm{mM}})$
I Kr:	delayed rectifier K+ current, rapid component ($g_{\mathrm{Kr}}=0.00424\mu \mathrm{S}$)
I Ks:	delayed rectifier K+ current, slow component ($g_{\mathrm{Ks}}=0.00065\mu \mathrm{S}$)
I K, ACh:	ACh‐activated K+ current ($g_{\mathrm{K},\mathrm{ACh}}=0.00345\mu \mathrm{S}$)
I to:	transient outward K+ current $(g_{\mathrm{to}}=3.5·{10}^{-3}\mu \mathrm{S}$)
I Na:	fast Na+ current ($g_{\mathrm{Na}}=0.0223\mu \mathrm{S}$)
I NaK:	Na+/K+ pump current $(I_{\mathrm{NaK},\max }=0.08105\mathrm{nA})$
I NaCa:	Na+/Ca2+ exchanger current ($K_{\mathrm{NaCa}}=3.343\mathrm{nA}$)
I Kur:	delayed rectifier K+ current, ultrarapid component ($g_{\mathrm{Kur}}=1.539·{10}^{-4}\mu \mathrm{S}$)

Modulation of sarcolemmal ion currents by ions

 

$K{m}_{\mathrm{fCa}}=0.000338\mathrm{mM}$:	dissociation constant of Ca2+‐dependent I CaL inactivation
$K{m}_{\mathrm{Kp}}=1.4\mathrm{mM}$:	half‐maximal K o for I NaK
$K{m}_{\mathrm{Nap}}=14\mathrm{mM}$:	half‐maximal Na i for I NaK
${\alpha }_{\mathrm{fCa}}=0.0075{\mathrm{s}}^{-1}$:	Ca2+ dissociation rate constant for I CaL

Na⁺/Ca²⁺ exchanger (NaCa) function

 

$K1ni=395.3\mathrm{mM}$:	intracellular Na+ binding to first site on NaCa
$K1no=1628\mathrm{mM}$:	extracellular Na+ binding to first site on NaCa
$K2ni=2.289\mathrm{mM}$:	intracellular Na+ binding to second site on NaCa
$K2no=561.4\mathrm{mM}$:	extracellular Na+ binding to second site on NaCa
$K3ni=26.44\mathrm{mM}$:	intracellular Na+ binding to third site on NaCa
$K3no=4.663\mathrm{mM}$:	extracellular Na+ binding to third site on NaCa
$Kci=0.0207\mathrm{mM}$:	intracellular Ca2+ binding to NaCa transporter
$Kcni=26.44\mathrm{mM}$:	intracellular Na+ and Ca2+ simultaneous binding to NaCa
$Kco=3.663\mathrm{mM}$:	extracellular Ca2+ binding to NaCa transporter
$Qci=0.1369$:	intracellular Ca2+ occlusion reaction of NaCa
$Qco=0$:	extracellular Ca2+ occlusion reaction of NaCa
$Qn=0.4315$:	Na+ occlusion reaction of NaCa

Ca²⁺ diffusion

 

${\tau }_{\mathrm{di}{\mathrm{f}}_{\mathrm{Ca}}}=5.469·{10}^{-5}\mathrm{s}$:	time constant of Ca2+ diffusion from the subsarcolemmal space to the myoplasm
${\tau }_{\mathrm{tr}}=0.04\mathrm{s}$:	time constant of Ca2+ transfer from the network to junctional SR

SERCA pump

 

$K_{\mathrm{up}}=286\mathrm{nM}$:	half‐maximal Ca i for Ca2+ uptake into the network SR
$P_{\mathrm{up}}=5\frac{\mathrm{mM}}{\mathrm{s}}$:	rate constant for Ca2+ uptake by SERCA pump into the network SR
$slop{e}_{\mathrm{up}}=50\mathrm{nM}$:	slope factor for Ca2+ uptake by SERCA pump into the network SR

RyR function

 

$kiCa=500\frac{1}{\mathrm{mM}\mathrm{s}}$:	RyR Ca‐dependent inactivation rate
$kim=5{\mathrm{s}}^{-1}$:	RyR repriming rate
$koCa=10000\frac{1}{\mathrm{m}{\mathrm{M}}^2\mathrm{s}}$:	RyR Ca‐activation rate
$kom=660{\mathrm{s}}^{-1}$:	RyR deactivation rate
$ks=1.48·{10}^8{\mathrm{s}}^{-1}$:	Ca2+ diffusion rate
$EC{50}_{\mathrm{SR}}=0.45\mathrm{mM}$:	EC50 for Ca jsr‐dependent activation of SR Ca release
$HSR=2.5$:	Hill coefficient for Ca jsr‐dependent activation of SR calcium release
$MaxSR=15$:	parameter for maximum SR calcium release
$MinSR=1$:	parameter for minimum SR calcium release

Ca²⁺ and Mg²⁺ buffering

 

$C{M}_{\mathrm{tot}}=0.045\mathrm{mM}$:	total calmodulin concentration
$C{Q}_{\mathrm{tot}}=10\mathrm{mM}$:	total calsequestrin concentration
$T{C}_{\mathrm{tot}}=0.031\mathrm{mM}$:	total concentration of the troponin‐Ca2+ site
$TM{C}_{\mathrm{tot}}=0.062\mathrm{mM}$:	total concentration of the troponin‐Mg2+ site
$k{b}_{\mathrm{CM}}=542{\mathrm{s}}^{-1}$:	Ca2+ dissociation constant for calmodulin
$k{b}_{\mathrm{CQ}}=445{\mathrm{s}}^{-1}$:	Ca2+ dissociation constant for calsequestrin
$k{b}_{\mathrm{TC}}=446{\mathrm{s}}^{-1}$:	Ca2+ dissociation constant for the troponin‐Ca2+ site
$k{b}_{\mathrm{TMC}}=7.51{\mathrm{s}}^{-1}$:	Ca2+ dissociation constant for the troponin‐Mg2+ site
$k{b}_{\mathrm{TMM}}=751{\mathrm{s}}^{-1}$:	Mg2+ dissociation constant for the troponin‐Mg2+ site
$k{f}_{\mathrm{CM}}=1.642·{10}^6\frac{1}{\mathrm{mM}\mathrm{s}}$:	Ca2+ association constant for calmodulin
$k{f}_{\mathrm{CQ}}=175.4\frac{1}{\mathrm{mM}\mathrm{s}}$:	Ca2+ association constant for calsequestrin
$k{f}_{\mathrm{TC}}=88800\frac{1}{\mathrm{mM}\mathrm{s}}$:	Ca2+ association constant for the troponin‐Ca2+ site
$k{f}_{\mathrm{TMC}}=227700\frac{1}{\mathrm{mM}\mathrm{s}}$:	Ca2+ association constant for the troponin‐Mg2+ site
$k{f}_{\mathrm{TMM}}=2277\frac{1}{\mathrm{mM}\mathrm{s}}$:	Mg2+ association constant for the troponin‐Mg2+ site

Model equations

Membrane potential

$$\begin{array}{ccc}\hfill \frac{\mathrm{d}V}{\mathrm{d}time}& =& \frac{-I_{\mathrm{tot}}}{C}\hfill \\ \hfill I_{\mathrm{tot}}& =& I_{\mathrm{f}}+I_{\mathrm{CaL}}+I_{\mathrm{CaT}}+I_{\mathrm{Kr}}+I_{\mathrm{Ks}}+I_{\mathrm{K},\mathrm{ACh}}+I_{\mathrm{to}}\hfill \\ & & +I_{\mathrm{Na}}+I_{\mathrm{NaK}}+I_{\mathrm{NaCa}}+I_{\mathrm{Kur}}\hfill \end{array}$$

Ion currents

 

$x_{\infty }$:	steady‐state for gating variable x
τx:	time constant for gating variable x [s]
αx and βx:	opening and closing rates for channel gate x [s−1]

Hyperpolarization‐activated, ‘funny current’ (I_f)

$$I_{\mathrm{f}}=I_{\mathrm{fNa}}+I_{\mathrm{fK}}$$

$$I_{\mathrm{fNa}}=y·g_{{\mathrm{f}}_{\mathrm{Na}}}·(V-E_{\mathrm{Na}})$$

$$I_{\mathrm{fK}}=y·g_{{\mathrm{f}}_{\mathrm{K}}}·(V-E_{\mathrm{K}})$$

$$y_{\infty }=\left\{\begin{array}{c}0.01329+\frac{0.99921}{1+e^{\frac{V+97.134}{8.1752}}},\mathrm{if}V<-80\\ 0.0002501·e^{\frac{-\mathrm{V}}{12.861}},\mathrm{otherwise}\end{array}\right.$$

$${\tau }_{\mathrm{y}}=\frac{1}{\frac{0.36·\left(V+148.8\right)}{e^{0.066·\left(V+148.8\right)}-1}+\frac{0.1·\left(V+87.3\right)}{1-e^{-0.2·\left(V+87.3\right)}}}-0.054$$

$$\frac{\mathrm{d}y}{\mathrm{d}time}=\frac{y_{\infty }-y}{{\tau }_{\mathrm{y}}}$$

L‐type Ca²⁺ current (I _CaL)

$$\begin{array}{ccc}\hfill I_{\mathrm{CaL}}& =& I_{\mathrm{siCa}}+I_{\mathrm{siK}}+I_{\mathrm{siNa}}\hfill \\ \hfill I_{\mathrm{siCa}}& =& \frac{2·P_{\mathrm{CaL}}·\mathrm{V}}{RTonF·\left(1-e^{\frac{-2\mathrm{V}}{RTonF}}\right)}·\left(C{a}_{\mathrm{sub}}-Cao·e^{\frac{-2\mathrm{V}}{RTonF}}\right)·\hfill \\ & & dL·fL·fCa\hfill \end{array}$$

$$\begin{array}{ccc}\hfill I_{\mathrm{siK}}& =& \frac{0.000365·P_{\mathrm{CaL}}·\mathrm{V}}{RTonF·\left(1-e^{\frac{-\mathrm{V}}{RTonF}}\right)}·\left(Ki-Ko·e^{\frac{-\mathrm{V}}{RTonF}}\right)·\hfill \\ & & dL·fL·fCa\hfill \end{array}$$

$$\begin{array}{ccc}\hfill I_{\mathrm{siNa}}& =& \frac{0.0000185·P_{\mathrm{CaL}}·\mathrm{V}}{RTonF·\left(1-e^{\frac{-\mathrm{V}}{RTonF}}\right)}·\left(Nai-Nao·e^{\frac{-\mathrm{V}}{RTonF}}\right)·\hfill \\ & & dL·fL·fCa\hfill \end{array}$$

$$d{L}_{\infty }=\frac{1}{1+e^{\frac{-(V+16.45)}{4.337}}}$$

$${\alpha }_{\mathrm{dL}}=\frac{-0.02839·(V+41.8)}{e^{\frac{-(V+41.8)}{2.5}}-1}-\frac{0.0849·(V+6.8)}{e^{\frac{-(V+6.8)}{4.8}}-1}$$

$${\beta }_{\mathrm{dL}}=\frac{0.01143·(V+1.8)}{e^{\frac{V+1.8}{2.5}}-1}$$

$${\tau }_{\mathrm{dL}}=\frac{0.001}{{\alpha }_{\mathrm{dL}}+{\beta }_{\mathrm{dL}}}$$

$$\frac{\mathrm{d}dL}{\mathrm{d}time}=\frac{d{L}_{\infty }-dL}{{\tau }_{\mathrm{dL}}}$$

$$fC{a}_{\infty }=\frac{K{m}_{\mathrm{fCa}}}{K{m}_{\mathrm{fCa}}+C{a}_{\mathrm{sub}}}$$

$${\tau }_{\mathrm{fCa}}=\frac{0.001·fC{a}_{\infty }}{{\alpha }_{\mathrm{fCa}}}$$

$$\frac{\mathrm{d}fCa}{\mathrm{d}time}=\frac{fC{a}_{\infty }-fCa}{{\tau }_{\mathrm{fCa}}}$$

$$f{L}_{\infty }=\frac{1}{1+e^{\frac{V+37.4}{5.3}}}$$

$${\tau }_{\mathrm{fL}}=0.001·\left(44.3+230·e^{-{\left(\frac{V+36}{10}\right)}^2}\right)$$

$$\frac{\mathrm{d}fL}{\mathrm{d}time}=\frac{f{L}_{\infty }-fL}{{\tau }_{\mathrm{fL}}}$$

T‐type Ca²⁺ current (I _CaT)

$$\begin{array}{ccc}\hfill I_{\mathrm{CaT}}& =& \frac{2·P_{\mathrm{CaT}}·V}{RTonF·\left(1-e^{\frac{-2·V}{RTonF}}\right)}·\left(C{a}_{\mathrm{sub}}-Cao·e^{\frac{-2·V}{RTonF}}\right)\hfill \\ & & ·dT·fT\hfill \end{array}$$

$$d{T}_{\infty }=\frac{1}{1+e^{\frac{-(V+38.3)}{5.5}}}$$

$${\tau }_{\mathrm{dT}}=\frac{0.001}{1.068·e^{\frac{V+38.3}{30}}+1.068·e^{\frac{-(V+38.3)}{30}}}$$

$$\frac{\mathrm{d}dT}{\mathrm{d}time}=\frac{d{T}_{\infty }-dT}{{\tau }_{\mathrm{dT}}}$$

$$f{T}_{\infty }=\frac{1}{1+e^{\frac{V+58.7}{3.8}}}$$

$${\tau }_{\mathrm{fT}}=\frac{1}{16.67·e^{\frac{-(V+75)}{83.3}}+16.67·e^{\frac{V+75}{15.38}}}$$

$$\frac{\mathrm{d}fT}{\mathrm{d}time}=\frac{f{T}_{\infty }-fT}{{\tau }_{\mathrm{fT}}}$$

Rapidly activating delayed rectifier K⁺ current (I _Kr)

$$I_{\mathrm{Kr}}=g_{\mathrm{Kr}}·(V-E_{\mathrm{K}})·(0.9·paF+0.1·paS)·piy$$

$$pa{F}_{\infty }=pa{S}_{\infty }=p{a}_{\infty }=\frac{1}{1+e^{\frac{-(V+10.0144)}{7.6607}}}$$

$${\tau }_{\mathrm{paS}}=\frac{0.84655354}{4.2·e^{\frac{V}{17}}+0.15·e^{\frac{-V}{21.6}}}$$

$${\tau }_{\mathrm{paF}}=\frac{1}{30·e^{\frac{V}{10}}+e^{\frac{-V}{12}}}$$

$$\frac{\mathrm{d}paS}{\mathrm{d}time}=\frac{p{a}_{\infty }-paS}{{\tau }_{\mathrm{paS}}}$$

$$\frac{\mathrm{d}paF}{\mathrm{d}time}=\frac{p{a}_{\infty }-paF}{{\tau }_{\mathrm{paF}}}$$

$${\tau }_{\mathrm{piy}}=\frac{1}{100·e^{\frac{-V}{54.645}}+656·e^{\frac{V}{106.157}}}$$

$$pi{y}_{\infty }=\frac{1}{1+e^{\frac{V+28.6}{17.1}}}$$

$$\frac{\mathrm{d}piy}{\mathrm{d}time}=\frac{pi{y}_{\infty }-piy}{{\tau }_{\mathrm{piy}}}$$

Slowly activating delayed rectifier K⁺ current (I _Ks)

$$I_{\mathrm{Ks}}=g_{\mathrm{Ks}}·(V-E_{\mathrm{Ks}})·n^2$$

$$n_{\infty }=\sqrt{\frac{1}{1+e^{\frac{-(V+0.6383)}{10.7071}}}}$$

$${\tau }_{\mathrm{n}}=\frac{1}{{\alpha }_{\mathrm{n}}+{\beta }_{\mathrm{n}}}$$

$${\alpha }_{\mathrm{n}}=\frac{28}{1+e^{\frac{-(V-40)}{3}}}$$

$${\beta }_{\mathrm{n}}=1·e^{\frac{-(V-5)}{25}}$$

$$\frac{\mathrm{d}n}{\mathrm{d}time}=\frac{n_{\infty }-n}{{\tau }_{\mathrm{n}}}$$

ACh‐activated K⁺ current (I _K,ACh)

$$\begin{array}{ccc}& & I_{\mathrm{K},\mathrm{ACh}}\hfill \\ & & =\left\{\begin{array}{cc}{g}_{\mathrm{K},\mathrm{ACh}}·(V-E_{\mathrm{K}})·\left(1+e^{\frac{V+20}{20}}\right)·a,& \mathrm{if}ACh>0\hfill \\ 0,& \mathrm{otherwise}\hfill \end{array}\right.\hfill \end{array}$$

$${\alpha }_{\mathrm{a}}=\frac{3.5988-0.025641}{1+\frac{0.0000012155}{{(ACh)}^{1.6951}}}+0.025641$$

$${\beta }_{\mathrm{a}}=10·e^{0.0133·(V+40)}$$

$$a_{\infty }=\frac{{\alpha }_{\mathrm{a}}}{{\alpha }_{\mathrm{a}}+{\beta }_{\mathrm{a}}}$$

$${\tau }_{\mathrm{a}}=\frac{1}{{\alpha }_{\mathrm{a}}+{\beta }_{\mathrm{a}}}$$

$$\frac{\mathrm{d}a}{\mathrm{d}time}=\frac{a_{\infty }-a}{{\tau }_{\mathrm{a}}}$$

Transient outward K⁺ current (I _to)

$$I_{\mathrm{to}}=g_{\mathrm{to}}·(V-E_{\mathrm{K}})·q·r$$

$$q_{\infty }=\frac{1}{1+e^{\frac{V+49}{13}}}$$

$$\begin{array}{ccc}& & {\tau }_{\mathrm{q}}=0.001·0.6\hfill \\ & & ·\left(\frac{65.17}{0.57·e^{-0.08·(V+44)}+0.065·e^{0.1·(V+45.93)}}+10.1\right)\hfill \end{array}$$

$$\frac{\mathrm{d}q}{\mathrm{d}time}=\frac{q_{\infty }-q}{{\tau }_{\mathrm{q}}}$$

$$r_{\infty }=\frac{1}{1+e^{\frac{-(V-19.3)}{15}}}$$

$$\begin{array}{ccc}& & {\tau }_{\mathrm{r}}=0.001·0.66·1.4\hfill \\ & & ·\left(\frac{15.59}{1.037·e^{0.09·(V+30.61)}+0.369·e^{-0.12·(V+23.84)}}+2.98\right)\hfill \end{array}$$

$$\frac{\mathrm{d}r}{\mathrm{d}time}=\frac{r_{\infty }-r}{{\tau }_{\mathrm{r}}}$$

Na⁺ current (I _Na)

$$I_{\mathrm{Na}}=g_{\mathrm{Na}}·m^3·h·(V-E_{\mathrm{mh}})$$

$$m_{\infty }=\frac{1}{1+e^{\frac{-(V+42.0504)}{8.3106}}}$$

$$E{0}_{\mathrm{m}}=V+41$$

$${\alpha }_{\mathrm{m}}=\frac{200·E{0}_{\mathrm{m}}}{1-e^{-0.1·E{0}_{\mathrm{m}}}}$$

$${\beta }_{\mathrm{m}}=8000·e^{-0.056·(V+66)}$$

$${\tau }_{\mathrm{m}}=\frac{1}{{\alpha }_{\mathrm{m}}+{\beta }_{\mathrm{m}}}$$

$$\frac{\mathrm{d}m}{\mathrm{d}time}=\frac{m_{\infty }-m}{{\tau }_{\mathrm{m}}}$$

$$h_{\infty }=\frac{1}{1+e^{\frac{V+69.804}{4.4565}}}$$

$${\alpha }_{\mathrm{h}}=20·e^{-0.125·(V+75)}$$

$${\beta }_{\mathrm{h}}=\frac{2000}{320e^{-0.1(\mathrm{V}+75)}+1}$$

$${\tau }_{\mathrm{h}}=\frac{1}{{\alpha }_{\mathrm{m}}+{\beta }_{\mathrm{m}}}$$

$$\frac{\mathrm{d}h}{\mathrm{d}time}=\frac{h_{\infty }-h}{{\tau }_{\mathrm{h}}}$$

Na⁺/K⁺ pump current (I _NaK)

$$\begin{array}{ccc}\hfill I_{\mathrm{NaK}}& =& I_{\mathrm{NaK},\max }·{\left(1+{\left(\frac{K{m}_{\mathrm{Kp}}}{Ko}\right)}^{1.2}\right)}^{-1}\hfill \\ & & ·{\left(1+{\left(\frac{K{m}_{\mathrm{Nap}}}{Nai}\right)}^{1.3}\right)}^{-1}·{\left(1+e^{\frac{-(V-E_{Na}+110)}{20}}\right)}^{-1}\hfill \end{array}$$

Na⁺/Ca²⁺ exchanger current (I _NaCa)

$$I_{\mathrm{NaCa}}=\frac{K_{\mathrm{NaCa}}·(x2·k21-x1·k12)}{x1+x2+x3+x4}$$

$$x1=k41·k34·(k23+k21)+k21·k32·(k43+k41)$$

$$x2=k32·k43·(k14+k12)+k41·k12·(k34+k32)$$

$$x3=k14·k43·(k23+k21)+k12·k23·(k43+k41)$$

$$x4=k23·k34·(k14+k12)+k14·k21·(k34+k32)$$

$$k43=\frac{Nai}{K3ni+Nai}$$

$$k12=\frac{\frac{C{a}_{\mathrm{sub}}}{Kci}·e^{\frac{-Qci·V}{RTonF}}}{di}$$

$$k14=\frac{\frac{\frac{Nai}{K1ni}·Nai}{K2ni}·\left(1+\frac{Nai}{K3ni}\right)·e^{\frac{Qn·V}{2·RTonF}}}{di}$$

$$k41=e^{\frac{-Qn·V}{2·RTonF}}$$

$$\begin{array}{ccc}\hfill di& =& 1+\frac{C{a}_{\mathrm{sub}}}{Kci}·\left(1+e^{\frac{-Qci·V}{RTonF}}+\frac{Nai}{Kcni}\right)\hfill \\ & & +\frac{Nai}{K1ni}·\left(1+\frac{Nai}{K2ni}·\left(1+\frac{Nai}{K3ni}\right)\right)\hfill \end{array}$$

$$k34=\frac{Nao}{K3no+Nao}$$

$$k21=\frac{\frac{Cao}{Kco}·e^{\frac{Qco·V}{RTonF}}}{do}$$

$$k23=\frac{\frac{\frac{Nao}{K1no}·Nao}{K2no}·\left(1+\frac{Nao}{K3no}\right)·e^{\frac{-Qn·V}{2·RTonF}}}{do}$$

$$k32=e^{\frac{Qn·V}{2·RTonF}}$$

$$\begin{array}{ccc}\hfill do& =& 1+\frac{Cao}{Kco}·\left(1+e^{\frac{Qco·V}{RTonF}}\right)+\frac{Nao}{K1no}\hfill \\ & & ·\left(1+\frac{Nao}{K2no}·\left(1+\frac{Nao}{K3no}\right)\right)\hfill \end{array}$$

Ultra‐rapid activating delayed rectifier K⁺ current (I _Kur)

$$I_{\mathrm{Kur}}=g_{\mathrm{Kur}}·r_{\mathrm{Kur}}·s_{\mathrm{Kur}}·(V-E_K)$$

$$\frac{\mathrm{d}{r}_{\mathrm{Kur}}}{\mathrm{d}time}=\frac{r_{\mathrm{Ku}{\mathrm{r}}_{\infty }}-r_{\mathrm{Kur}}}{{\tau }_{{\mathrm{r}}_{\mathrm{Kur}}}}$$

$$r_{\mathrm{Ku}{\mathrm{r}}_{\infty }}=\frac{1}{1+e^{\frac{V+6}{-8.6}}}$$

$${\tau }_{{\mathrm{r}}_{\mathrm{Kur}}}=\frac{0.009}{1+e^{\frac{V+5}{12}}}+0.0005$$

$$\frac{\mathrm{d}{s}_{\mathrm{Kur}}}{\mathrm{d}time}=\frac{s_{\mathrm{Ku}{\mathrm{r}}_{\infty }}-s_{\mathrm{Kur}}}{{\tau }_{{\mathrm{s}}_{\mathrm{Kur}}}}$$

$$s_{\mathrm{Ku}{\mathrm{r}}_{\infty }}=\frac{1}{1+e^{\frac{V+7.5}{10}}}$$

$${\tau }_{{\mathrm{s}}_{\mathrm{Kur}}}=\frac{0.59}{1+e^{\frac{V+60}{10}}}+3.05$$

Ca²⁺ release flux (J _rel) from SR via RyRs

$$J_{\mathrm{SRCarel}}=ks·O·(C{a}_{\mathrm{jsr}}-C{a}_{\mathrm{sub}})$$

$$kCaSR=MaxSR-\frac{MaxSR-MinSR}{1+{\left(\frac{EC{50}_{\mathrm{SR}}}{C{a}_{\mathrm{jsr}}}\right)}^{\mathrm{HSR}}}$$

$$koSRCa=\frac{koCa}{kCaSR}$$

$$kiSRCa=kiCa·kCaSR$$

$$\begin{array}{ccc}\hfill \frac{\mathrm{d}R}{\mathrm{d}time}& =& kim·RI-kiSRCa·C{a}_{\mathrm{sub}}·R\hfill \\ & & -\left(koSRCa·C{a}_{\mathrm{sub}}^2·R-kom·O\right)\hfill \\ \hfill \frac{\mathrm{d}O}{\mathrm{d}time}& =& koSRCa·C{a_{\mathrm{sub}}}^2·R-kom·O\hfill \\ & & -(kiSRCa·C{a}_{\mathrm{sub}}·O-kim·I)\hfill \\ \hfill \frac{\mathrm{d}I}{\mathrm{d}time}& =& kiSRCa·C{a}_{\mathrm{sub}}·O-kim·I\hfill \\ & & -\left(kom·I-koSRCa·C{a}_{\mathrm{sub}}^2·RI\right)\hfill \\ \hfill \frac{\mathrm{d}RI}{\mathrm{d}time}& =& kom·I-koSRCa·C{a_{\mathrm{sub}}}^2·RI\hfill \\ & & -(kim·RI-kiSRCa·C{a}_{\mathrm{sub}}·R)\hfill \end{array}$$

Intracellular Ca²⁺ fluxes

 

$J_{\mathrm{C}{\mathrm{a}}_{\mathrm{dif}}}$:	Ca2+ diffusion flux from submembrane space to myoplasm
J up:	Ca2+ uptake by the SR
J tr:	Ca2+ diffusion flux from the network to junctional SR

$$J_{\mathrm{C}{\mathrm{a}}_{\mathrm{dif}}}=\frac{C{a}_{\mathrm{sub}}-Cai}{{\tau }_{\mathrm{di}{\mathrm{f}}_{\mathrm{Ca}}}}$$

$$J_{\mathrm{up}}=\frac{P_{\mathrm{up}}}{1+e^{\frac{-(\mathrm{Cai}-{\mathrm{K}}_{\mathrm{up}})}{slop{e}_{\mathrm{up}}}}}$$

$$J_{\mathrm{tr}}=\frac{C{a}_{\mathrm{nsr}}-C{a}_{\mathrm{jsr}}}{{\tau }_{\mathrm{tr}}}$$

Ca²⁺ buffering

 

f CMi:	fractional occupancy of calmodulin by Ca2+ in myoplasm
f CMs:	fractional occupancy of calmodulin by Ca2+ in subspace
f CQ:	fractional occupancy of calsequestrin by Ca2+
f TC:	fractional occupancy of the troponin‐Ca2+ site by Ca2+
f TMC:	fractional occupancy of the troponin‐Mg2+ site by Ca2+
f TMM:	fractional occupancy of the troponin‐Mg2+ site by Mg2+

$$\frac{\mathrm{d}fTC}{\mathrm{d}time}={\delta }_{\mathrm{fTC}}$$

$${\delta }_{\mathrm{fTC}}=k{f}_{\mathrm{TC}}·Cai·(1-fTC)-k{b}_{\mathrm{TC}}·fTC$$

$$\frac{\mathrm{d}fTMC}{\mathrm{d}time}={\delta }_{\mathrm{fTMC}}$$

$$\begin{array}{ccc}\hfill {\delta }_{\mathrm{fTMC}}& =& k{f}_{\mathrm{TMC}}·Cai·(1-(fTMC+fTMM))\hfill \\ & & -k{b}_{\mathrm{TMC}}·fTMC\hfill \end{array}$$

$$\frac{\mathrm{d}fTMM}{\mathrm{d}time}={\delta }_{\mathrm{fTMM}}$$

$$\begin{array}{ccc}\hfill {\delta }_{\mathrm{fTMM}}& =& k{f}_{\mathrm{TMM}}·Mgi·(1-(fTMC+fTMM))\hfill \\ & & -k{b}_{\mathrm{TMM}}·fTMM\hfill \end{array}$$

$$\frac{\mathrm{d}fCMi}{\mathrm{d}time}={\delta }_{\mathrm{fCMi}}$$

$${\delta }_{\mathrm{fCMi}}=k{f}_{\mathrm{CM}}·Cai·(1-fCMi)-k{b}_{\mathrm{CM}}·fCMi$$

$$\frac{\mathrm{d}fCMs}{\mathrm{d}time}={\delta }_{\mathrm{fCMs}}$$

$${\delta }_{\mathrm{fCMs}}=k{f}_{\mathrm{CM}}·C{a}_{\mathrm{sub}}·(1-fCMs)-k{b}_{\mathrm{CM}}·fCMs$$

$$\frac{\mathrm{d}fCQ}{\mathrm{d}time}={\delta }_{\mathrm{fCQ}}$$

$${\delta }_{\mathrm{fCQ}}=k{f}_{\mathrm{CQ}}·C{a}_{\mathrm{jsr}}·(1-fCQ)-k{b}_{\mathrm{CQ}}·fCQ$$

Dynamics of Ca²⁺ concentrations in cell compartments

$$\begin{array}{ccc}\hfill \frac{\mathrm{d}Cai}{\mathrm{d}time}& =& \frac{1·(J_{\mathrm{C}{\mathrm{a}}_{\mathrm{dif}}}·V_{\mathrm{sub}}-J_{\mathrm{up}}·V_{\mathrm{nsr}})}{V_{\mathrm{i}}}\hfill \\ & & -(C{M}_{\mathrm{tot}}·{\delta }_{\mathrm{fCMi}}+T{C}_{tot}·{\delta }_{\mathrm{fTC}}+TM{C}_{\mathrm{tot}}·{\delta }_{\mathrm{fTMC}})\hfill \end{array}$$

$$\begin{array}{c}\hfill \\ \hfill \frac{\mathrm{d}C{a}_{\mathrm{sub}}}{\mathrm{d}time}& =& \frac{J_{\mathrm{SRCarel}}·V_{\mathrm{jsr}}}{V_{\mathrm{sub}}}\hfill \\ & & -\left(\frac{i_{\mathrm{siCa}}+i_{\mathrm{CaT}}-2·i_{\mathrm{NaCa}}}{2·F·V_{\mathrm{sub}}}\right.\hfill \\ & & \left.\phantom{\frac{1}{2}}+j_{\mathrm{C}{\mathrm{a}}_{\mathrm{dif}}}+C{M}_{\mathrm{tot}}·{\delta }_{\mathrm{fCMs}}\right)\hfill \end{array}$$

$$\frac{\mathrm{d}C{a}_{\mathrm{nsr}}}{\mathrm{d}time}=J_{\mathrm{up}}-\frac{J_{\mathrm{tr}}·V_{\mathrm{jsr}}}{V_{\mathrm{nsr}}}$$

$$\frac{\mathrm{d}C{a}_{\mathrm{jsr}}}{\mathrm{d}time}=J_{\mathrm{tr}}-(J_{\mathrm{SRCarel}}+C{Q}_{\mathrm{tot}}·{\delta }_{\mathrm{fCQ}})$$

Rate modulation experiments

ACh 10 nm

 

I f:	shift of y ∞ and τy by −5 mV
I CaL:	reduction of the maximal conductance by 3%
SERCA pump:	decrease of P up by 7%
I K,ACh:	activation

Isoprenaline 1 μm

 

I f:	shift of y ∞ and τy by 7.5 mV
I NaK:	increase of I NaK,max by 20%
I CaL:	increase of the maximal conductance by 23%; shift of dL ∞ and τdL by −8 mV; reduction of the slope factor k dL by 27%
I Ks:	increase of g Ks by 20%; shift of n ∞ and τn by −14 mV
SERCA pump:	increase of P up by 25%

Appendix 2.

Parameters selected for automatic optimization

 

g Kur =	maximal conductance of I Kur
K NaCa =	maximal current of NCX
K up =	Ca2+ concentration for half‐maximal activity of SERCA pump
P CaT =	permeability of T‐type Ca2+ current
P CaL =	permeability of L‐type Ca2+ current
k dL =	slope factor of L‐type voltage‐dependent activation gate dL
V dL =	half‐maximal activation voltage of voltage‐dependent activation gate dL
τdifCa =	time constant of Ca2+ diffusion from the submembrane to myoplasm
k s =	maximal rate of calcium release from RyR channels
kf CM =	Ca2+ association constant for calmodulin

kf CQ =	Ca2+ association constant for calsequestrin
I NaK,max =	maximal Na+/K+ pump current

Action potential features used to constrain model parameters

Experimental data on APA, MDP, CL, V _max, APD₂₀, APD₅₀, APD₉₀, DDR₁₀₀ and CL prolongation induced by Cs⁺ of single isolated human SAN cells as reported by Verkerk et al. (2007 b) were used to constrain model parameters.

Data on CL prolongation induced by Cs⁺ are from only one cell. We arbitrarily adopted an SEM of 10% of the experimentally observed CL prolongation.

Calcium transient features used to constrain model parameters

Experimental data on diastolic [Ca²⁺], systolic [Ca²⁺], TD₂₀, TD₅₀ and TD₉₀ of a single isolated human SAN cell as reported by Verkerk et al. (2013) were used to constrain model parameters.

Because the experimental data on the calcium transient are from only one cell, we arbitrarily set the SEM of each of the calcium transient features to 40% in our optimization procedure.

Cost function

In our optimization procedure, we minimized the overall cost function FCost, which was defined as:

$$FCost=\sum _{\mathrm{i}}Cos{t}_{\mathrm{i}}$$

in which Cost _i denotes the individual contribution of feature i to the overall cost.

Cost _i is described as (Fig. A1):

$$\left\{\begin{array}{cc}{\mathit{Cost}}_{\mathrm{i}}=\frac{\left|Featur{e}_{\mathrm{i},\mathrm{Exp}}-Featur{e}_{\mathrm{i},\mathrm{Sim}}\right|-{\mathit{SEM}}_{\mathrm{i}}}{{\mathit{SEM}}_{\mathrm{i}}}{\mathit{weight}}_{\mathrm{i}}& if\left|{\mathit{Feature}}_{\mathrm{i},\mathrm{Exp}}-{\mathit{Feature}}_{\mathrm{i},\mathrm{Sim}}\right|>\mathit{SEM}\\ 0& \mathit{otherwise}\end{array}\right.$$

Figure A1. Cost function in optimization procedure.

Contribution Cost _i of feature i to the overall cost function (FCost).

We set weight _i = 2 for MDP, CL and CL prolongation induced by Cs⁺, whereas weight _i was set to 1 for all other features.

Search method and stop criterion

The initial values of the selected parameters were obtained from the parent (rabbit) model and then manually tuned to achieve beating rate and action potential morphology close to the experimental human data. Such initial values were then employed to perform the above‐mentioned finer automatic optimization.

The minimization of the overall cost function FCost was based on the Nelder–Mead simplex method (Lagarias et al. 1998). The automatic optimization procedure stopped when FCost converged on a local minimum.

Parameter constraining

To determine whether the parameters are well constrained we followed the approach by Sarkar and Sobie (2010). Accordingly, the extended Bayes theorem was exploited to define a range for parameters, for which a population of models is able to reproduce behaviour in agreement with experimental data. The parameter constraining is performed by increasing the number of features to satisfy: the higher the number of selected features, the lower the number of compatible parameter sets. In a similar way, we selected MDP, CL, APD₉₀ and DDR₁₀₀ from the available 13 extracted features, within the target range (i.e. mean ± SEM of experimental values) and obtained the ranges in Table A1.

Table A1.

Nominal values and ranges for the optimized parameters

Parameters	Units	Nominal	Range (minimum – maximum)
g Kur	μS	1.539 × 10−4	0.9895–1.633 × 10−4
P CaT	nA mm −1	0.04132	0.04132–0.05191
P CaL	nA mm −1	0.4578	0.3475–0.5262
V dL	mV	−16.45	−20.97 to −13.73
k dL	mV	4.337	3.892–4.401
I NaK,max	nA	0.08105	0.0623–0.1063
K NaCa	nA	3.343	2.733–4.964
K up	nm	286.1	204.6–356.2
k s	s−1	1.480 × 108	1.460–1.847 × 108
τdifCa	s	5.469 × 10−5	5.469–7.951 × 10−5
kf CM	mm s−1	1.642 × 106	1.164–1.839 × 106
kf CQ	mm s−1	175.4	155.2–194.6

The collected values show an adequately constrained range for the optimized parameters. To achieve this result, we exploited a priori knowledge during the optimization procedure; accordingly, we avoided non‐physiological values (negative conductances or values too far from data reported in literature), assigning high scores to their cost function.

The achievement of more than one parameter set that falls into experimental ranges could attest the robustness of the model, which is able to adequately compensate for parameter changes without adopting parameter values beyond the physiological range.