# Effects of Na+ Current and Mechanogated Channels in Myofibroblasts on Myocyte Excitability and Repolarization

Authors: Heqing Zhan; Jingtao Zhang; Jialun Lin; Guilai Han
Journal: Computational and Mathematical Methods in Medicine
 (2016)

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## Abstract

Fibrotic remodeling, characterized by fibroblast phenotype switching, is often associated with atrial fibrillation and heart failure. This study aimed to investigate the effects on electrotonic myofibroblast-myocyte (Mfb-M) coupling on cardiac myocytes excitability and repolarization of the voltage-gated sodium channels (VGSCs) and single mechanogated channels (MGCs) in human atrial Mfbs. Mathematical modeling was developed from a combination of (1) models of the human atrial myocyte (including the stretch activated ion channel current,ISAC) and Mfb and (2) our formulation of currents through VGSCs (INa_Mfb) and MGCs (IMGC_Mfb) based upon experimental findings. The effects of changes in the intercellular coupling conductance, the number of coupled Mfbs, and the basic cycle length on the myocyte action potential were simulated. The results demonstrated that the integration of ISAC, INa_Mfb, and IMGC_Mfb reduced the amplitude of the myocyte membrane potential (Vmax) and the action potential duration (APD), increased the depolarization of the resting myocyte membrane potential (Vrest), and made it easy to trigger spontaneous excitement in myocytes. For Mfbs, significant electrotonic depolarizations were exhibited with the addition of INa_Mfb and IMGC_Mfb. Our results indicated that ISAC, INa_Mfb, and IMGC_Mfb significantly influenced myocytes and Mfbs properties and should be considered in future cardiac pathological mathematical modeling.


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## Body

## 1. Introduction

Recent studies have demonstrated a correlation between atrial fibrillation (AF) and fibrotic remodeling, specifically fibrosis [1–3]. However, their relationship has not been fully understood. Changes have taken place during the process of fibrotic remodeling in terms of gap junction remodeling [4], the deposition of excess collagen [5], and fibroblast phenotype switching [2, 6], which determines the degree of AF initiation and maintenance in atrial fibrosis in AF subjects [7].Different currents have been identified in cardiac fibroblasts by recent electrophysiological studies, including the currents through potassium channels [8, 9], the nonselective transient receptor potential cationic channel subfamily M member 7 (TRPM7) [10], voltage-gated sodium channels (VGSCs) [11], chloride channels [12], single mechanogated channels (MGCs) [13], and voltage-dependent proton currents [14]. Therefore, fibroblasts are no longer considered as nonexcitable cells. As one of the main characteristics of fibrotic remodeling, fibroblasts proliferation and differentiation into myofibroblasts (Mfbs) at the cellular level have been shown to play an important role in cardiac pathological status [15, 16]. During fibroblasts differentiation, significantly increased potassium channels and TRPM7 have been reported [10, 17]. Specifically, with the measurement of the neoexpression of rapid Na+ currents through VGSCs during the process, 75% of the Mfbs derived from the culture of human atrial fibroblasts expressed Na+ current between 8 and 12 days. After 12 days, 100% of the Mfbs expressed Na+ current [11]. In addition, there is strong evidence that this Na+ current is generated by Nav 1.5 α-subunit, a typical VGSC to produce Na+ current in cardiomyocytes [11].Recent studies have also suggested that MGCs were modulated by mechanical deformations of fibroblasts, which may contribute to the cardiac mechanoelectrical feedback under both physiological and pathophysiological conditions [18, 19]. As myocytes start contracting, the interposed fibroblasts are mechanically compressed. The membrane potential of fibroblasts is depolarized by ionic currents through MGCs [13, 18].Computational models of atrial fibrosis have been used to investigate the relationship between fibrotic remodeling and AF. For gap junction remodeling, simulation results showed that increased anisotropy led to sustained reentrant activity [20]. For collagen deposition, it has been demonstrated that patchy distributions of collagen were responsible for atrial conduction disturbances in failing hearts [21, 22]. For fibroblast proliferation and phenotype switching, studies indicated that coupling of fibroblasts to atrial myocytes resulted in shorter duration of the action potential (APD), slower conduction, and the development of AF [23, 24]. Taking Mfbs into account, other studies combined all the above-mentioned elements in the left atrial model to examine the underlying mechanisms of AF initiation and indicated that Mfbs exerted electrotonic influences on myocytes in the lesions [1, 25]. However, all these published simulation studies have not considered the following: first, the stretch activated ion channel current (ISAC) in myocytes, which significantly influences the electrophysiological characteristics of cardiac myocytes under stretching [26, 27]; second, the currents through VGSCs (INa_Mfb) and MGCs (IMGC_Mfb) in Mfbs, which could influence Mfb properties and contribute to mechanoelectrical coupling in cardiac pathologies [11, 13].The present study aimed to investigate the combinational role ofISAC, INa_Mfb, and IMGC_Mfb in the excitability and repolarization of cardiac myocytes by Mfb-myocyte (Mfb-M) coupling. Specifically, a new formulation of INa_Mfb and IMGC_Mfb based on experimental data will be combined with models of human atrial myocyte (including ISAC) and Mfb. Simulation results of human atrial myocyte action potential (AP) dynamics from different gap-junctional conductances (Ggap), number of coupled Mfbs, and basic cycle lengths (BCLs) will be compared.


## 2. Materials and Methods

The framework of the coupled Mfb-M model was developed from Maleckar et al.’s model of the human atrial myocyte [28] (including ISAC equations described by Kuijpers et al. [26]), MacCannell et al.’s model of the human cardiac Mfb [24], and our new formulation of INa_Mfb and IMGC_Mfb based on experimental findings from Chatelier et al. [11] and Kamkin et al. [13]. Figure 1 is a schematic of the Mfb-M coupling model. An overview of the simulations is described as follows.Figure 1
Schematic representation of the mathematical model of Mfb-M coupling.


### 2.1. Mfb-M Coupling

According to [24], the differential equations for the membrane potential of cardiac Mfb and myocyte are given by(1)dVMfb,idt=-1Cm,MfbIMfb,iVMfb,i,t+Ggap⁡VMfb,i-VM,dVMdt=-1Cm,MIMVM,t+∑i=1nGgap⁡VM-VMfb,i,where VMfb,i and VM represent the transmembrane potential of the ith coupled Mfb and the human atrial myocyte, respectively; Cm,Mfb and Cm,M represent the membrane capacitance of the Mfb and the myocyte, respectively; IMfb,i and IM represent the transmembrane current of the ith coupled Mfb and the human atrial myocyte, respectively; and Ggap represents the gap-junctional conductance, which varies between 0.5 and 8 nS in individual simulations based on previous measurements [19, 29]. A negative Igap⁡ (i.e., Ggap (VMfb,i-VM)) indicates that the current is flowing from the myocyte into the ith Mfb, and n is the total number of coupled Mfbs.


### 2.2. Mathematical Model of the Human Atrial Myocyte

The mathematical model from Maleckar et al. was implemented in this study [28], which was based on experimental data and has correctly replicated APD restitution of the adult human atrial myocyte. To describe the influence of stretch on myocyte AP, the original model from Maleckar et al. was modified with the total ionic current of myocyte (IM) given by(2)IMVM,t=INa+ICaL+It+IKur+IK1+IK,r+IK,s+IB,Na+IB,Ca+INaK+ICaP+INaCa+ISAC-IStim,where INa is fast inward Na+ current; ICaL is L-type Ca2+ current; It is transient outward K+ current; IKur is sustained outward K+ current; IK1 is inward rectifying K+ current; IK,r is rapid delayed rectifier K+ current; IK,s is slow delayed rectifier K+ current; IB,Na is background Na+ current; IB,Ca is background Ca2+ current; INaK is Na+-K+ pump current; ICaP is sarcolemmal Ca2+ pump current; INaCa is Na+-Ca2+ exchange current; ISAC is stretch activated current; and IStim is stimulated current.On the basis of experimental observations [30], it has been reported by Kuijpers et al. that ISAC in atrial myocytes is permeable to Na+, K+, and Ca2+ [26], which is defined as(3)ISAC=ISAC,Na+ISAC,K+ISAC,Ca,where ISAC,Na, ISAC,K, and ISAC,Ca represent the contributions of Na+, K+, and Ca2+ to ISAC, respectively. These currents were defined by the constant-field Goldman-Hodgkin-Katz current equation [26].To introduce the effect ofISAC on intracellular Na+, K+, and Ca2+ concentrations ([Na+]i, [K+]i, and [Ca2+]i), we replaced equations of [Na+]i, [K+]i, and [Ca2+]i in Maleckar et al.’s model [28] by(4)dNa+idt=-INa+IB,Na+3INaK+3INaCa+ISAC,NaVoliF,dK+idt=-It+IKur+IK1+IK,s+IK,r-2INaK+ISAC,KVoliF,dCa2+idt=--Idi+IB,Ca+ICaP-2INaCa+Iup-Irel⁡+ISAC,Ca2.0VoliF-dOdt,where F is Faraday’s constant; Voli is cytosolic volume; Idi is Ca2+ diffusion current from the diffusion-restricted subsarcolemmal space to the cytosol; Iup is sarcoplasmic reticulum Ca2+ uptake current; Irel⁡ is sarcoplasmic reticulum Ca2+ release current; andO is buffer occupancy.


### 2.3. Electrophysiological Model of the Human Atrial Mfb

The electrophysiological model of the human atrial Mfb from MacCannell et al. [24] was used in the present study. It included time- and voltage-dependent K+ current (IKv_Mfb), inward rectifying K+ current (IK1_Mfb), Na+-K+ pump current (INaK_Mfb), and Na+ background current (IB,Na_Mfb).In addition,INa_Mfb and IMGC_Mfb were added in the Mfb model. According to the published experimental data [9], the Mfb transmembrane potential VMfb and Cm,Mfb of −47.75 mV and 6.3 pF were used.ForINa_Mfb, the steady-state value of activation gating variable for Na+ current (m-Mfb) was obtained by fitting the experimental data from Chatelier et al. [11] with a Boltzmann function using Origin software: (5)y=A1-A21+eVMfb-V0.5/k+A2,where VMfb is the Mfb potential ranging from −120 mV to 40 mV in 10 mV increments and y is the corresponding value of m-Mfb at each VMfb. A1 and A2 are fitting coefficients and V0.5 and k represent the half-maximum voltage of activation and the Boltzmann steepness coefficient, respectively.After the data were fitted with a Boltzmann function,A1 and A2 were −0.0102 and 1.0063, respectively.V0.5 andk were −42.1 mV and 10.53, respectively. The equation of m-Mfb was then expressed as(6)m-Mfb=-0.0102-1.00631.0+eVMfb+42.1/10.53+1.0063.The steady-state value of inactivation gating variable for INa_Mfb (j-Mfb) was also fitted with the Boltzmann function, with A1 and A2 of 1.04 and 0.004 and V0.5 and k of −84.82 mV and 9.4, respectively. The equation of j-Mfb was given as(7)j-Mfb=1.04-0.0041.0+eVMfb+84.82/9.4+0.004.As suggested by Chatelier et al. that INa_Mfb was generated by the same Nav 1.5 which produced Na+ currents in the atria and ventricle of the adult human heart [11], the equations of activation time constants in the model of Courtemanche et al. [31] were used to describe the activation time constant for INa_Mfb (τmMfb). The equations are given by(8)αmMfb=A1VMfb+Vx1-eA2VMfb+Vx,(9)βmMfb=A3e-VMfb/A4,(10)τmMfb=1αmMfb+βmMfb,where VMfb are original values, ranging from −40 mV to 30 mV in 10 mV increments and τmMfb is the corresponding value at each VMfb. αmMfband βmMfbare extrapolated rate coefficients. After the data were fitted with these equations, A1 to A4 were 0.0077, −0.18, 0.004, and 11.98, and Vx was 68.19 mV.Equation (8) requires evaluation of a limit to determine the value at membrane potential for which its denominator is zero. Equations (8) and (9) are therefore expressed as follows: (11)αmMfb=0.0077VMfb+68.191.0-e-0.18VMfb+68.190.0433,ifVMfb=-68.19βmMfb=0.004e-VMfb/11.98.Similarly, the equations of inactivation time constant (τjMfb) were as follows: (12)αjMfb=-14.5e0.17VMfb+1.8e1.56VMfb·VMfb+35.731.0+e3.31VMfb+3.860,ifVMfb≥-40.0,βjMfb=5.05×10-5e-0.0995VMfb1.0+e-0.01VMfb-84.020.11e-4.13×10-4VMfb1.0+e-0.09VMfb+42.44,ifVMfb≥-40.0,τjMfb=1αjMfb+βjMfb.The model of INa_Mfb was modified from the equation described by Luo and Rudy [32], as follows:(13)INa_Mfb=g-Na,MfbmMfbjMfb0.12VMfb-ENa,Mfb,ENa,Mfb=RTFlog⁡Na+c,MfbNa+i,Mfb,where g-Na,Mfb, the maximum conductance of INa_Mfb, was 0.756 nS; ENa,Mfb is the Nernst potential for Na+ ions; [Na+]c,Mfb, the Mfb extracellular Na+ concentration, was 130.011 mmol/L; [Na+]i,Mfb, the Mfb intracellular Na+ concentration, initial value was 8.5547 mmol/L; mMfb and jMfb were the activation and inactivation parameters, respectively. In order to conform to the experiment data [11], j was modified to bej0.12. Equations of time dependence were given by(14)dmMfbdt=m-Mfb-mMfbτmMfb,djMfbdt=j-Mfb-jMfbτjMfb.For IMGC_Mfb, the equation based on experimental results from Kamkin et al. [13] was given by(15)IMGC_Mfb=g-MGC,Mfb·VMfb-EMGC,Mfb,where g-MGC,Mfb, the maximum conductance of IMGC_Mfb, was 0.043 nS and EMGC,Mfb, the reversal potential of MGCs, was close to 0 mV [13].


### 2.4. Simulation Protocol

Simulations were carried out with (1) two Mfbs coupled to a single atrial myocyte in which Ggap varied between 0.5 and 8 nS and (2) one and four identical Mfbs coupled to a single atrial myocyte at constant Ggap of 3 nS. Next, in order to investigate the role of BCL in myocyte AP, the coupled system was paced with BCLs from 100 to 2000 ms for each Ggap   (0.5 or 8 nS) and for each number of coupled Mfbs (1 or 4). The resting myocyte membrane potential (Vrest), the amplitude of the myocyte membrane potential (Vmax), and APD at 90% repolarization (APD90) at different BCLs were obtained.To ensure that the coupled system reached steady state, stimulation was repeated for 20 cycles. The results from the last cycle in each simulation were used. All state variables of the coupled model were updated by means of the forward Euler method. The time step was set to be 10μs to ensure numerical accuracy and stability.


## 2.1. Mfb-M Coupling

According to [24], the differential equations for the membrane potential of cardiac Mfb and myocyte are given by(1)dVMfb,idt=-1Cm,MfbIMfb,iVMfb,i,t+Ggap⁡VMfb,i-VM,dVMdt=-1Cm,MIMVM,t+∑i=1nGgap⁡VM-VMfb,i,where VMfb,i and VM represent the transmembrane potential of the ith coupled Mfb and the human atrial myocyte, respectively; Cm,Mfb and Cm,M represent the membrane capacitance of the Mfb and the myocyte, respectively; IMfb,i and IM represent the transmembrane current of the ith coupled Mfb and the human atrial myocyte, respectively; and Ggap represents the gap-junctional conductance, which varies between 0.5 and 8 nS in individual simulations based on previous measurements [19, 29]. A negative Igap⁡ (i.e., Ggap (VMfb,i-VM)) indicates that the current is flowing from the myocyte into the ith Mfb, and n is the total number of coupled Mfbs.


## 2.2. Mathematical Model of the Human Atrial Myocyte

The mathematical model from Maleckar et al. was implemented in this study [28], which was based on experimental data and has correctly replicated APD restitution of the adult human atrial myocyte. To describe the influence of stretch on myocyte AP, the original model from Maleckar et al. was modified with the total ionic current of myocyte (IM) given by(2)IMVM,t=INa+ICaL+It+IKur+IK1+IK,r+IK,s+IB,Na+IB,Ca+INaK+ICaP+INaCa+ISAC-IStim,where INa is fast inward Na+ current; ICaL is L-type Ca2+ current; It is transient outward K+ current; IKur is sustained outward K+ current; IK1 is inward rectifying K+ current; IK,r is rapid delayed rectifier K+ current; IK,s is slow delayed rectifier K+ current; IB,Na is background Na+ current; IB,Ca is background Ca2+ current; INaK is Na+-K+ pump current; ICaP is sarcolemmal Ca2+ pump current; INaCa is Na+-Ca2+ exchange current; ISAC is stretch activated current; and IStim is stimulated current.On the basis of experimental observations [30], it has been reported by Kuijpers et al. that ISAC in atrial myocytes is permeable to Na+, K+, and Ca2+ [26], which is defined as(3)ISAC=ISAC,Na+ISAC,K+ISAC,Ca,where ISAC,Na, ISAC,K, and ISAC,Ca represent the contributions of Na+, K+, and Ca2+ to ISAC, respectively. These currents were defined by the constant-field Goldman-Hodgkin-Katz current equation [26].To introduce the effect ofISAC on intracellular Na+, K+, and Ca2+ concentrations ([Na+]i, [K+]i, and [Ca2+]i), we replaced equations of [Na+]i, [K+]i, and [Ca2+]i in Maleckar et al.’s model [28] by(4)dNa+idt=-INa+IB,Na+3INaK+3INaCa+ISAC,NaVoliF,dK+idt=-It+IKur+IK1+IK,s+IK,r-2INaK+ISAC,KVoliF,dCa2+idt=--Idi+IB,Ca+ICaP-2INaCa+Iup-Irel⁡+ISAC,Ca2.0VoliF-dOdt,where F is Faraday’s constant; Voli is cytosolic volume; Idi is Ca2+ diffusion current from the diffusion-restricted subsarcolemmal space to the cytosol; Iup is sarcoplasmic reticulum Ca2+ uptake current; Irel⁡ is sarcoplasmic reticulum Ca2+ release current; andO is buffer occupancy.


## 2.3. Electrophysiological Model of the Human Atrial Mfb

The electrophysiological model of the human atrial Mfb from MacCannell et al. [24] was used in the present study. It included time- and voltage-dependent K+ current (IKv_Mfb), inward rectifying K+ current (IK1_Mfb), Na+-K+ pump current (INaK_Mfb), and Na+ background current (IB,Na_Mfb).In addition,INa_Mfb and IMGC_Mfb were added in the Mfb model. According to the published experimental data [9], the Mfb transmembrane potential VMfb and Cm,Mfb of −47.75 mV and 6.3 pF were used.ForINa_Mfb, the steady-state value of activation gating variable for Na+ current (m-Mfb) was obtained by fitting the experimental data from Chatelier et al. [11] with a Boltzmann function using Origin software: (5)y=A1-A21+eVMfb-V0.5/k+A2,where VMfb is the Mfb potential ranging from −120 mV to 40 mV in 10 mV increments and y is the corresponding value of m-Mfb at each VMfb. A1 and A2 are fitting coefficients and V0.5 and k represent the half-maximum voltage of activation and the Boltzmann steepness coefficient, respectively.After the data were fitted with a Boltzmann function,A1 and A2 were −0.0102 and 1.0063, respectively.V0.5 andk were −42.1 mV and 10.53, respectively. The equation of m-Mfb was then expressed as(6)m-Mfb=-0.0102-1.00631.0+eVMfb+42.1/10.53+1.0063.The steady-state value of inactivation gating variable for INa_Mfb (j-Mfb) was also fitted with the Boltzmann function, with A1 and A2 of 1.04 and 0.004 and V0.5 and k of −84.82 mV and 9.4, respectively. The equation of j-Mfb was given as(7)j-Mfb=1.04-0.0041.0+eVMfb+84.82/9.4+0.004.As suggested by Chatelier et al. that INa_Mfb was generated by the same Nav 1.5 which produced Na+ currents in the atria and ventricle of the adult human heart [11], the equations of activation time constants in the model of Courtemanche et al. [31] were used to describe the activation time constant for INa_Mfb (τmMfb). The equations are given by(8)αmMfb=A1VMfb+Vx1-eA2VMfb+Vx,(9)βmMfb=A3e-VMfb/A4,(10)τmMfb=1αmMfb+βmMfb,where VMfb are original values, ranging from −40 mV to 30 mV in 10 mV increments and τmMfb is the corresponding value at each VMfb. αmMfband βmMfbare extrapolated rate coefficients. After the data were fitted with these equations, A1 to A4 were 0.0077, −0.18, 0.004, and 11.98, and Vx was 68.19 mV.Equation (8) requires evaluation of a limit to determine the value at membrane potential for which its denominator is zero. Equations (8) and (9) are therefore expressed as follows: (11)αmMfb=0.0077VMfb+68.191.0-e-0.18VMfb+68.190.0433,ifVMfb=-68.19βmMfb=0.004e-VMfb/11.98.Similarly, the equations of inactivation time constant (τjMfb) were as follows: (12)αjMfb=-14.5e0.17VMfb+1.8e1.56VMfb·VMfb+35.731.0+e3.31VMfb+3.860,ifVMfb≥-40.0,βjMfb=5.05×10-5e-0.0995VMfb1.0+e-0.01VMfb-84.020.11e-4.13×10-4VMfb1.0+e-0.09VMfb+42.44,ifVMfb≥-40.0,τjMfb=1αjMfb+βjMfb.The model of INa_Mfb was modified from the equation described by Luo and Rudy [32], as follows:(13)INa_Mfb=g-Na,MfbmMfbjMfb0.12VMfb-ENa,Mfb,ENa,Mfb=RTFlog⁡Na+c,MfbNa+i,Mfb,where g-Na,Mfb, the maximum conductance of INa_Mfb, was 0.756 nS; ENa,Mfb is the Nernst potential for Na+ ions; [Na+]c,Mfb, the Mfb extracellular Na+ concentration, was 130.011 mmol/L; [Na+]i,Mfb, the Mfb intracellular Na+ concentration, initial value was 8.5547 mmol/L; mMfb and jMfb were the activation and inactivation parameters, respectively. In order to conform to the experiment data [11], j was modified to bej0.12. Equations of time dependence were given by(14)dmMfbdt=m-Mfb-mMfbτmMfb,djMfbdt=j-Mfb-jMfbτjMfb.For IMGC_Mfb, the equation based on experimental results from Kamkin et al. [13] was given by(15)IMGC_Mfb=g-MGC,Mfb·VMfb-EMGC,Mfb,where g-MGC,Mfb, the maximum conductance of IMGC_Mfb, was 0.043 nS and EMGC,Mfb, the reversal potential of MGCs, was close to 0 mV [13].


## 2.4. Simulation Protocol

Simulations were carried out with (1) two Mfbs coupled to a single atrial myocyte in which Ggap varied between 0.5 and 8 nS and (2) one and four identical Mfbs coupled to a single atrial myocyte at constant Ggap of 3 nS. Next, in order to investigate the role of BCL in myocyte AP, the coupled system was paced with BCLs from 100 to 2000 ms for each Ggap   (0.5 or 8 nS) and for each number of coupled Mfbs (1 or 4). The resting myocyte membrane potential (Vrest), the amplitude of the myocyte membrane potential (Vmax), and APD at 90% repolarization (APD90) at different BCLs were obtained.To ensure that the coupled system reached steady state, stimulation was repeated for 20 cycles. The results from the last cycle in each simulation were used. All state variables of the coupled model were updated by means of the forward Euler method. The time step was set to be 10μs to ensure numerical accuracy and stability.


## 3. Results

### 3.1.INa_Mfb and IMGC_Mfb in Mfbs

Figure2 shows the steady-state activation and inactivation curves, time constants, and the current-voltage (I-V) relationship of INa_Mfb. All the curves were consistent with the experimental data [11].Figure 2
Model representation of parameters describingINa_Mfb. (a) Steady-state activation (solid line) and inactivation (dashed line) curves, (b) gating variable fast (solid line) and slow (dashed line) time constants, and (c)I-V relationship (solid line). Experimental data (filled and hollow circles) from Chatelier et al. is included for comparison.
(a) (b) (c)Figure3 shows the linearI-V relationship of IMGC_Mfb. The curve was consistent with the experimental data [13].Figure 3
TheI-V relationship (solid line) of MGCs. Experimental data (filled circles) from Kamkin et al. is included for comparison.


### 3.2. Effects ofISAC, INa_Mfb, and IMGC_Mfb on Atrial Myocyte and Mfbs

Figure4 shows the combinational effects of ISAC, INa_Mfb, and IMGC_Mfb on the membrane potential of myocytes and Mfbs by coupling two Mfbs to a human atrial myocyte with Ggap of 0.5 and 8 nS. For myocytes, including INa_Mfb, ISAC and IMGC_Mfb to the model of Mfb-M coupling resulted in gradually decreased Vmax and APD90 and increased Vrest depolarization (Figures 4(a) and 4(b)). With Ggap of 0.5 nS, Vmax was decreased by 0.8% (with INa_Mfb), 16.7% (with INa_Mfb and ISAC), and 18.6% (with INa_Mfb, ISAC, and IMGC_Mfb) in comparison with the control (without INa_Mfb, ISAC, and IMGC_Mfb). Correspondingly, for the three conditions, APD90 was decreased by 20.0%, 19.1%, and 20.2%, respectively, and Vrest increased by 8.3%, 20.3%, and 22.4%, respectively. With Ggap of 8 nS, Vmax was decreased by 0.2%, 23.5%, and 18.6%, respectively. APD90 was decreased by 1.6%, 4.1%, and 11.1%, respectively. Vrest was increased by 0.8%, 12.5%, and 15.9%, respectively. In addition, INa_Mfb, ISAC, and IMGC_Mfb decelerated the repolarization of the atrial myocyte AP, especially with a large Ggap. For Mfbs, with Ggap of 0.5 nS, significant electrotonic depolarizations were observed with the integration of the three currents (Figure 4(c)). Vmax of the Mfb was −12.4 mV (control), 7.1 mV (with INa_Mfb), 1.5 mV (with INa_Mfb and ISAC), and −1.0 mV (with INa_Mfb, ISAC, and IMGC_Mfb), respectively. With Ggap of 8 ns, the effects of these currents on Mfbs were similar with that on myocytes (Figure 4(d)).Figure 4
Effects ofISAC, INa_Mfb, and IMGC_Mfb on transmembrane potential of myocyte and Mfb. (a, b) VM and (c, d) VMfb, in Mfb-M coupling with INa_Mfb (blue line), with ISAC and INa_Mfb (green line), and with ISAC, INa_Mfb, and IMGC (red line) as compared to control (black line), by coupling two Mfbs to an atrial myocyte with Ggap of 0.5 and 8 nS.
(a) (b) (c) (d)Figure5 illustrates similar effects of ISAC, INa_Mfb, and IMGC_Mfb on the membrane potential of myocytes and Mfbs when Ggap was fixed at 3 nS and the number of coupled Mfbs was one or four. For myocytes, integrating these currents into the coupled model resulted in decreased Vmax and APD90 and greater depolarization in Vrest (Figures 5(a) and 5(b)). Coupling one Mfb to a human atrial myocyte resulted in 0.1% (with INa_Mfb), 38.1% (with INa_Mfb and ISAC), and 37.4% (with INa_Mfb, ISAC, and IMGC_Mfb) decreases in Vmax when compared with the control. Correspondingly, APD90 was decreased by 1.1%, 7.7%, and 7.8%, and Vrest increased by 0.4%, 25.0%, and 28.6%, respectively. Similarly, coupling four Mfbs to a human atrial myocyte resulted in 7.9%, 30.3%, and 25.9% decreases in Vmax when compared with the control. APD90 was decreased by 1.6%, 8.9%, and 18.9%, and Vrest was increased by 2.3%, 17.4%, and 23.6%, respectively. For Mfbs, the effects of these currents were similar with that on myocytes (Figures 5(c) and 5(d)).Figure 5
Effects ofISAC, INa_Mfb, and IMGC_Mfb on transmembrane potential of myocyte and Mfb. (a, b) VM and (c, d) VMfb, in Mfb-M coupling with INa_Mfb (blue line), with ISAC and INa_Mfb (green line), and with ISAC, INa_Mfb and IMGC (red line) as compared to control (black line), by coupling one and four Mfbs to an atrial myocyte with Ggap of 3 nS.
(a) (b) (c) (d)Figure6 illustrates the changes of  Vrest, Vmax, and APD90 in myocyte with the integration of ISAC, INa_Mfb, and IMGC_Mfb as a function of BCL when coupling two Mfbs to a human atrial myocyte. For both low (0.5 nS, Figures 6(a), 6(c), and 6(e)) and high (8 nS, Figures 6(b), 6(d), and 6(f)) Ggap, Vrest declined and Vmax rose in the control and with INa_Mfb as BCL increased (Figures 6(a) to 6(d)). Vrest decreased initially and then increased slightly with ISAC and INa_Mfb and with ISAC, INa_Mfb, and IMGC_Mfb. Contrarily, Vmax increased with BCL initially and then decreased slightly. Gaps of Vrest and Vmax between the four cases widened as the BCL increased with more pronounced variations for the high Ggap. APD90 was prolonged as BCL increased in all the four situations for both Ggap values (Figures 6(e) and 6(f)). With Ggap of 0.5 nS, similar to Vmax, APD90 gradually dropped with INa_Mfb, ISAC, and IMGC_Mfb in order at each BCL. However, there was no obvious difference between the four situations with Ggap of 8 nS.Figure 6
Effects ofISAC, INa_Mfb, and IMGC_Mfb on AP characteristics as a function of BCL in Mfb-M coupling with INa_Mfb (blue line with hollow triangle), with ISAC and INa_Mfb (green line with hollow rectangle), and with ISAC, INa_Mfb, and IMGC (red line with asterisk) as compared to control (black line with hollow circle), by coupling two Mfbs to an atrial myocyte with Ggap of 0.5 and 8 nS. (a, b) Vrest, (c, d) Vmax, and (e, f) APD90.
(a) (b) (c) (d) (e) (f)Figure7 shows the APs of the human atrial myocyte at the 20st cycle with BCL of 100 (Figures 7(a) and 7(b)), 500 (Figures 7(c) and 7(d)), 1000 (Figures 7(e) and 7(f)), and 2000 ms (Figures 7(g) and 7(h)) when Ggap was fixed at 3 nS and the number of coupled Mfbs was one or four. Integrating ISAC, INa_Mfb, and IMGC_Mfb into the model of Mfb-M coupling resulted in reductions in Vmax and APD90 and greater depolarization in Vrest. However, it is noted that spontaneous excitements in the myocyte appeared when the value of BCL was higher than a certain value (1000 ms in this study). The increased Vrest and the longer stimulus interval were the two major factors for the spontaneous excitement. Meanwhile, integrating ISAC, INa_Mfb, and IMGC_Mfb simultaneously into the coupled model triggered the most and earliest spontaneous excitement at BCL of 1000 and 2000 ms, suggesting that the three currents could result in discordant alternans, especially with large BCL (2000 ms in this study).Figure 7
Effects ofISAC, INa_Mfb, and IMGC_Mfb on VM for several BCLs. (a, b) BCL = 100 ms, (c, d) BCL = 500 ms, (e, f) BCL = 1000 ms, and (g, h) BCL = 2000 ms, in Mfb-M coupling with INa_Mfb (blue line), with ISAC and INa_Mfb (green line), and with ISAC, INa_Mfb, and IMGC (red line) as compared to control (black line), by coupling one and four Mfbs to an atrial myocyte with Ggap of 3 nS. Results corresponded to the last cycle in each simulation.
(a) (b) (c) (d) (e) (f) (g) (h)


## 3.1.INa_Mfb and IMGC_Mfb in Mfbs

Figure2 shows the steady-state activation and inactivation curves, time constants, and the current-voltage (I-V) relationship of INa_Mfb. All the curves were consistent with the experimental data [11].Figure 2
Model representation of parameters describingINa_Mfb. (a) Steady-state activation (solid line) and inactivation (dashed line) curves, (b) gating variable fast (solid line) and slow (dashed line) time constants, and (c)I-V relationship (solid line). Experimental data (filled and hollow circles) from Chatelier et al. is included for comparison.
(a) (b) (c)Figure3 shows the linearI-V relationship of IMGC_Mfb. The curve was consistent with the experimental data [13].Figure 3
TheI-V relationship (solid line) of MGCs. Experimental data (filled circles) from Kamkin et al. is included for comparison.


## 3.2. Effects ofISAC, INa_Mfb, and IMGC_Mfb on Atrial Myocyte and Mfbs

Figure4 shows the combinational effects of ISAC, INa_Mfb, and IMGC_Mfb on the membrane potential of myocytes and Mfbs by coupling two Mfbs to a human atrial myocyte with Ggap of 0.5 and 8 nS. For myocytes, including INa_Mfb, ISAC and IMGC_Mfb to the model of Mfb-M coupling resulted in gradually decreased Vmax and APD90 and increased Vrest depolarization (Figures 4(a) and 4(b)). With Ggap of 0.5 nS, Vmax was decreased by 0.8% (with INa_Mfb), 16.7% (with INa_Mfb and ISAC), and 18.6% (with INa_Mfb, ISAC, and IMGC_Mfb) in comparison with the control (without INa_Mfb, ISAC, and IMGC_Mfb). Correspondingly, for the three conditions, APD90 was decreased by 20.0%, 19.1%, and 20.2%, respectively, and Vrest increased by 8.3%, 20.3%, and 22.4%, respectively. With Ggap of 8 nS, Vmax was decreased by 0.2%, 23.5%, and 18.6%, respectively. APD90 was decreased by 1.6%, 4.1%, and 11.1%, respectively. Vrest was increased by 0.8%, 12.5%, and 15.9%, respectively. In addition, INa_Mfb, ISAC, and IMGC_Mfb decelerated the repolarization of the atrial myocyte AP, especially with a large Ggap. For Mfbs, with Ggap of 0.5 nS, significant electrotonic depolarizations were observed with the integration of the three currents (Figure 4(c)). Vmax of the Mfb was −12.4 mV (control), 7.1 mV (with INa_Mfb), 1.5 mV (with INa_Mfb and ISAC), and −1.0 mV (with INa_Mfb, ISAC, and IMGC_Mfb), respectively. With Ggap of 8 ns, the effects of these currents on Mfbs were similar with that on myocytes (Figure 4(d)).Figure 4
Effects ofISAC, INa_Mfb, and IMGC_Mfb on transmembrane potential of myocyte and Mfb. (a, b) VM and (c, d) VMfb, in Mfb-M coupling with INa_Mfb (blue line), with ISAC and INa_Mfb (green line), and with ISAC, INa_Mfb, and IMGC (red line) as compared to control (black line), by coupling two Mfbs to an atrial myocyte with Ggap of 0.5 and 8 nS.
(a) (b) (c) (d)Figure5 illustrates similar effects of ISAC, INa_Mfb, and IMGC_Mfb on the membrane potential of myocytes and Mfbs when Ggap was fixed at 3 nS and the number of coupled Mfbs was one or four. For myocytes, integrating these currents into the coupled model resulted in decreased Vmax and APD90 and greater depolarization in Vrest (Figures 5(a) and 5(b)). Coupling one Mfb to a human atrial myocyte resulted in 0.1% (with INa_Mfb), 38.1% (with INa_Mfb and ISAC), and 37.4% (with INa_Mfb, ISAC, and IMGC_Mfb) decreases in Vmax when compared with the control. Correspondingly, APD90 was decreased by 1.1%, 7.7%, and 7.8%, and Vrest increased by 0.4%, 25.0%, and 28.6%, respectively. Similarly, coupling four Mfbs to a human atrial myocyte resulted in 7.9%, 30.3%, and 25.9% decreases in Vmax when compared with the control. APD90 was decreased by 1.6%, 8.9%, and 18.9%, and Vrest was increased by 2.3%, 17.4%, and 23.6%, respectively. For Mfbs, the effects of these currents were similar with that on myocytes (Figures 5(c) and 5(d)).Figure 5
Effects ofISAC, INa_Mfb, and IMGC_Mfb on transmembrane potential of myocyte and Mfb. (a, b) VM and (c, d) VMfb, in Mfb-M coupling with INa_Mfb (blue line), with ISAC and INa_Mfb (green line), and with ISAC, INa_Mfb and IMGC (red line) as compared to control (black line), by coupling one and four Mfbs to an atrial myocyte with Ggap of 3 nS.
(a) (b) (c) (d)Figure6 illustrates the changes of  Vrest, Vmax, and APD90 in myocyte with the integration of ISAC, INa_Mfb, and IMGC_Mfb as a function of BCL when coupling two Mfbs to a human atrial myocyte. For both low (0.5 nS, Figures 6(a), 6(c), and 6(e)) and high (8 nS, Figures 6(b), 6(d), and 6(f)) Ggap, Vrest declined and Vmax rose in the control and with INa_Mfb as BCL increased (Figures 6(a) to 6(d)). Vrest decreased initially and then increased slightly with ISAC and INa_Mfb and with ISAC, INa_Mfb, and IMGC_Mfb. Contrarily, Vmax increased with BCL initially and then decreased slightly. Gaps of Vrest and Vmax between the four cases widened as the BCL increased with more pronounced variations for the high Ggap. APD90 was prolonged as BCL increased in all the four situations for both Ggap values (Figures 6(e) and 6(f)). With Ggap of 0.5 nS, similar to Vmax, APD90 gradually dropped with INa_Mfb, ISAC, and IMGC_Mfb in order at each BCL. However, there was no obvious difference between the four situations with Ggap of 8 nS.Figure 6
Effects ofISAC, INa_Mfb, and IMGC_Mfb on AP characteristics as a function of BCL in Mfb-M coupling with INa_Mfb (blue line with hollow triangle), with ISAC and INa_Mfb (green line with hollow rectangle), and with ISAC, INa_Mfb, and IMGC (red line with asterisk) as compared to control (black line with hollow circle), by coupling two Mfbs to an atrial myocyte with Ggap of 0.5 and 8 nS. (a, b) Vrest, (c, d) Vmax, and (e, f) APD90.
(a) (b) (c) (d) (e) (f)Figure7 shows the APs of the human atrial myocyte at the 20st cycle with BCL of 100 (Figures 7(a) and 7(b)), 500 (Figures 7(c) and 7(d)), 1000 (Figures 7(e) and 7(f)), and 2000 ms (Figures 7(g) and 7(h)) when Ggap was fixed at 3 nS and the number of coupled Mfbs was one or four. Integrating ISAC, INa_Mfb, and IMGC_Mfb into the model of Mfb-M coupling resulted in reductions in Vmax and APD90 and greater depolarization in Vrest. However, it is noted that spontaneous excitements in the myocyte appeared when the value of BCL was higher than a certain value (1000 ms in this study). The increased Vrest and the longer stimulus interval were the two major factors for the spontaneous excitement. Meanwhile, integrating ISAC, INa_Mfb, and IMGC_Mfb simultaneously into the coupled model triggered the most and earliest spontaneous excitement at BCL of 1000 and 2000 ms, suggesting that the three currents could result in discordant alternans, especially with large BCL (2000 ms in this study).Figure 7
Effects ofISAC, INa_Mfb, and IMGC_Mfb on VM for several BCLs. (a, b) BCL = 100 ms, (c, d) BCL = 500 ms, (e, f) BCL = 1000 ms, and (g, h) BCL = 2000 ms, in Mfb-M coupling with INa_Mfb (blue line), with ISAC and INa_Mfb (green line), and with ISAC, INa_Mfb, and IMGC (red line) as compared to control (black line), by coupling one and four Mfbs to an atrial myocyte with Ggap of 3 nS. Results corresponded to the last cycle in each simulation.
(a) (b) (c) (d) (e) (f) (g) (h)


## 4. Discussion

This study investigated the roles ofISAC, INa_Mfb, and IMGC_Mfb in the excitability and AP. To address these issues, numerical simulations of the coupled Mfb-M system were performed by employing a combination of models of the human atrial myocyte (including ISAC) and Mfb, as well as modified formulation of INa_Mfb and IMGC_Mfb based on experimental data. Specifically, the effects of these currents with changes in (1) Ggap, (2) the number of Mfbs, and (3) BCL on the human atrial myocyte AP dynamics were investigated. The main results with the integration of ISAC, INa_Mfb, and IMGC_Mfb were as follows: (1) for myocytes, the addition of ISAC, INa_Mfb, and IMGC_Mfb resulted in decreased Vmax and APD90 and increased Vrest depolarization; deceleration of AP repolarization; and trigger of spontaneous excitements even discordant alternans at large BCLs and (2) for Mfbs, significant electrotonic depolarizations were obtained with small Ggap. As the value of Ggap and the number of coupled Mfbs increased, the effects for Mfbs were similar to those in myocytes.


### 4.1. Resting Potential of Human Atrial Mfbs

V r e s t of fibroblasts/Mfbs in human atria was close to −15 mV [33, 34]. At this potential, the persistent entry of Na+ in Mfb would be negligible [11]. Indeed, it has been reported that Vrest of these cells was sensitive to mechanical stress under both contraction and relaxation [13, 35, 36]. A stretch of 3 μm led to a negative shift of Vrest, about −35 ± 5 mV [13]. Reoxygenation after hypoxia also produced a hyperpolarization to Vrest [37]. Based on these findings, we set up the mathematical formulation of IMGC_Mfb and included it in the Mfb-M coupling model. In addition, it has also been shown that K+ current in cardiac fibroblasts and Vrest was between −40.0 and −60.0 mV [8, 9, 38]. These potentials corresponded to the peak of the Na+ window current described by Chatelier et al. [11]. A persistent Na+ entry may be induced based on such a cell polarity. Therefore, we also introduced the mathematical formulation of INa_Mfb and included it in Mfb modeling. Vrest was set to be −47.8 mV, within the range of experimental findings [8, 9].


### 4.2. Role ofISAC, INa_Mfb, and IMGC_Mfb on Vrest and Excitability of Human Atrial Myocytes

Both experimental data and computational work have shown that the value ofGgap and the number of coupled fibroblasts/Mfbs were important in determining the depolarization of the coupled human myocyte at rest [24, 28, 29, 39]. Based on the preceding conclusions, our simulations also considered the effects of ISAC, INa_Mfb, and IMGC_Mfb on Vrest and excitability of human atrial myocytes.ForISAC, previous studies explored the electrophysiological effect of sustained stretch and indicated that ISAC influenced cardiac myocytes repolarization and activation [26, 40]. With pathophysiological conditions, such as hypertension and AF, it has been observed that atrial stretch and dilatation influenced atrial flutter cycle length and facilitated the induction and maintenance of AF [41, 42]. However, the SAC blocker Gd3+ reduced the stretch-induced vulnerability to AF [43]. These observational studies confirmed that ISAC plays a significant role in heart pathology. In our simulations, integrating ISAC into the mathematical model of the human atrial myocyte resulted in a significant depolarization of Vrest and prolongation of repolarization (Figures 4 to 7). These phenomena led to alternating spontaneous excitement propagation at a BCL of 2000 ms (Figure 7). Our simulation agreed with Kuijpers et al., where slow conduction, a longer effective refractory period, and unidirectional conduction block with increasing stretch were observed [26], which were related to the inducibility of AF.ForINa_Mfb, many studies have been conducted to investigate how this current could influence Mfb proliferation, since fibrotic remodeling was closely related to AF. It has been reported that Nav 1.5 α-subunit was responsible of INa_Mfb in human atrial Mfbs, which represented electrophysiological characteristics similar to Na+ channels found in cardiac excitable cells [11, 44]. Based on this, a multiple parameter exponential function was used in this study, which was modeled after the equation of time courses by Courtemanche et al. to fit the time courses of current decay elicited at depolarized voltages [31]. Unlike the assumption from Koivumäki et al. that Vrest of Mfbs resulted in significant inactivation of INa_Mfb [45], the current was activated during an atrial Mfb AP in our simulations. Our results showed that INa_Mfb decreased Vmax and APD90 and increased Vrest depolarization in myocytes (Figures 4 to 7). Like ISAC, this depolarization would be expected to change diastolic Ca2+ levels and conduction velocity.ForIMGC_Mfb, experimental data has indicated that cardiac fibroblasts expressed functional MCGs, contributing to the cardiac mechanoelectrical feedback under both physiological and pathophysiological conditions [18, 19]. Since IMGC_Mfb has been found in Mfbs, we assumed that it could affect Mfb electrophysiological characteristics like ISAC on myocyte. In our simulations, theI-V relation for single MGCs was linear, which agreed with the experimental results [13]. The results showed that integrating IMGC_Mfb in the Mfb model changed the membrane potential of Mfb and further influenced the membrane potential of myocyte by Mfb-myocyte coupling (Figures 4 to 7). Interestingly, MGCs were activated by Mfb compression and inactivated by Mfb stretch [13], implying that IMGC_Mfb should be integrated in cell modeling only during cell compression, such as fibroblasts/Mfbs compression caused by stretching and dilatation of surrounding cardiac myocytes.Table1 shows changes in myocyte Vrest, Vmax, and APD90 in three situations as compared to control. The situations were Mfb-M coupling with (1) INa_Mfb, (2) INa_Mfb and ISAC, and (3) INa_Mfb, ISAC, and IMGC_Mfb, respectively. Values were recorded with two Mfbs coupled to a single atrial myocyte in which Ggap was varied between 0.5 and 8 nS, or with one and four identical Mfbs coupled to a single atrial myocyte at a constant Ggap of 3 nS. In general, it can be seen that INa_Mfb, ISAC, and IMGC_Mfb played a greater role in reducing Vmax and APD90 and increasing the depolarization of Vrest when Ggap was relatively small. Increasing the number of coupled Mfbs contributed to a further decline in APD90 in all cases, while having different influences on Vmax and Vrest. The reduction of Vmax and the rise of depolarization of Vrest were increased in case (1) and both declined in case (2) and (3).Table 1
Percentage changes in myocyteVrest, Vmax and APD90 as compared to control.
G g a p (nS) Coupled Mfbs 0.5 8 1 4 V m a x↓ (%) I N a _ M f b 0.8 0.2 0.1 7.9 I N a _ M f b, ISAC 16.7 23.5 38.1 30.3 I N a _ M f b, ISAC, IMGC_Mfb 18.6 18.6 37.4 25.9 APD↓ (%) I N a _ M f b 20.0 1.6 1.1 1.6 I N a _ M f b, ISAC 19.1 4.1 7.7 8.9 I N a _ M f b, ISAC, IMGC_Mfb 20.2 11.1 7.8 18.9 V r e s t↑ (%) I N a _ M f b 8.3 0.8 0.4 2.3 I N a _ M f b, ISAC 20.3 12.5 25.0 17.4 I N a _ M f b, ISAC, IMGC_Mfb 22.4 15.9 28.6 23.6Previous modeling work suggested that Mfb-M coupling contributed to arrhythmia formation [23, 46]. The key factors included Ggap, the number of coupled Mfbs, and Mfb density. Here, we investigated the functional effect of fibrotic remodeling on AF by integrating INa_Mfb, ISAC, and IMGC_Mfb into Mfb-M coupling. To the best of our knowledge, this has not been examined before. As summarized in Figure 7, these currents made the Mfb act as a current source for the cardiac myocyte, leading to depolarization of Vrest and shortening of APD, thereby preserving myocyte excitability to trigger spontaneous excitement and facilitating reentry (such as those of AF).


### 4.3. Limitations

Two limitations of this mathematical modeling should be mentioned. First, the influence of homologous coupling between adjacent Mfbs was not considered. Since Mfbs can form conduction bridges between myocyte bundles and introduce nonlinearity into Mfb-M coupling to alter electrophysiological properties of the coupled myocyte, coupling between Mfbs may be important in tissue modeling [28, 29]. Second, our modeling of IMGC_Mfb was done using a simplified model based on a linear equation. We did not include the mechanosensitivity of MGCs in our simulation; for example, the conductance depends of the mechanical forces to which the cells are submitted.


## 4.1. Resting Potential of Human Atrial Mfbs

V r e s t of fibroblasts/Mfbs in human atria was close to −15 mV [33, 34]. At this potential, the persistent entry of Na+ in Mfb would be negligible [11]. Indeed, it has been reported that Vrest of these cells was sensitive to mechanical stress under both contraction and relaxation [13, 35, 36]. A stretch of 3 μm led to a negative shift of Vrest, about −35 ± 5 mV [13]. Reoxygenation after hypoxia also produced a hyperpolarization to Vrest [37]. Based on these findings, we set up the mathematical formulation of IMGC_Mfb and included it in the Mfb-M coupling model. In addition, it has also been shown that K+ current in cardiac fibroblasts and Vrest was between −40.0 and −60.0 mV [8, 9, 38]. These potentials corresponded to the peak of the Na+ window current described by Chatelier et al. [11]. A persistent Na+ entry may be induced based on such a cell polarity. Therefore, we also introduced the mathematical formulation of INa_Mfb and included it in Mfb modeling. Vrest was set to be −47.8 mV, within the range of experimental findings [8, 9].


## 4.2. Role ofISAC, INa_Mfb, and IMGC_Mfb on Vrest and Excitability of Human Atrial Myocytes

Both experimental data and computational work have shown that the value ofGgap and the number of coupled fibroblasts/Mfbs were important in determining the depolarization of the coupled human myocyte at rest [24, 28, 29, 39]. Based on the preceding conclusions, our simulations also considered the effects of ISAC, INa_Mfb, and IMGC_Mfb on Vrest and excitability of human atrial myocytes.ForISAC, previous studies explored the electrophysiological effect of sustained stretch and indicated that ISAC influenced cardiac myocytes repolarization and activation [26, 40]. With pathophysiological conditions, such as hypertension and AF, it has been observed that atrial stretch and dilatation influenced atrial flutter cycle length and facilitated the induction and maintenance of AF [41, 42]. However, the SAC blocker Gd3+ reduced the stretch-induced vulnerability to AF [43]. These observational studies confirmed that ISAC plays a significant role in heart pathology. In our simulations, integrating ISAC into the mathematical model of the human atrial myocyte resulted in a significant depolarization of Vrest and prolongation of repolarization (Figures 4 to 7). These phenomena led to alternating spontaneous excitement propagation at a BCL of 2000 ms (Figure 7). Our simulation agreed with Kuijpers et al., where slow conduction, a longer effective refractory period, and unidirectional conduction block with increasing stretch were observed [26], which were related to the inducibility of AF.ForINa_Mfb, many studies have been conducted to investigate how this current could influence Mfb proliferation, since fibrotic remodeling was closely related to AF. It has been reported that Nav 1.5 α-subunit was responsible of INa_Mfb in human atrial Mfbs, which represented electrophysiological characteristics similar to Na+ channels found in cardiac excitable cells [11, 44]. Based on this, a multiple parameter exponential function was used in this study, which was modeled after the equation of time courses by Courtemanche et al. to fit the time courses of current decay elicited at depolarized voltages [31]. Unlike the assumption from Koivumäki et al. that Vrest of Mfbs resulted in significant inactivation of INa_Mfb [45], the current was activated during an atrial Mfb AP in our simulations. Our results showed that INa_Mfb decreased Vmax and APD90 and increased Vrest depolarization in myocytes (Figures 4 to 7). Like ISAC, this depolarization would be expected to change diastolic Ca2+ levels and conduction velocity.ForIMGC_Mfb, experimental data has indicated that cardiac fibroblasts expressed functional MCGs, contributing to the cardiac mechanoelectrical feedback under both physiological and pathophysiological conditions [18, 19]. Since IMGC_Mfb has been found in Mfbs, we assumed that it could affect Mfb electrophysiological characteristics like ISAC on myocyte. In our simulations, theI-V relation for single MGCs was linear, which agreed with the experimental results [13]. The results showed that integrating IMGC_Mfb in the Mfb model changed the membrane potential of Mfb and further influenced the membrane potential of myocyte by Mfb-myocyte coupling (Figures 4 to 7). Interestingly, MGCs were activated by Mfb compression and inactivated by Mfb stretch [13], implying that IMGC_Mfb should be integrated in cell modeling only during cell compression, such as fibroblasts/Mfbs compression caused by stretching and dilatation of surrounding cardiac myocytes.Table1 shows changes in myocyte Vrest, Vmax, and APD90 in three situations as compared to control. The situations were Mfb-M coupling with (1) INa_Mfb, (2) INa_Mfb and ISAC, and (3) INa_Mfb, ISAC, and IMGC_Mfb, respectively. Values were recorded with two Mfbs coupled to a single atrial myocyte in which Ggap was varied between 0.5 and 8 nS, or with one and four identical Mfbs coupled to a single atrial myocyte at a constant Ggap of 3 nS. In general, it can be seen that INa_Mfb, ISAC, and IMGC_Mfb played a greater role in reducing Vmax and APD90 and increasing the depolarization of Vrest when Ggap was relatively small. Increasing the number of coupled Mfbs contributed to a further decline in APD90 in all cases, while having different influences on Vmax and Vrest. The reduction of Vmax and the rise of depolarization of Vrest were increased in case (1) and both declined in case (2) and (3).Table 1
Percentage changes in myocyteVrest, Vmax and APD90 as compared to control.
G g a p (nS) Coupled Mfbs 0.5 8 1 4 V m a x↓ (%) I N a _ M f b 0.8 0.2 0.1 7.9 I N a _ M f b, ISAC 16.7 23.5 38.1 30.3 I N a _ M f b, ISAC, IMGC_Mfb 18.6 18.6 37.4 25.9 APD↓ (%) I N a _ M f b 20.0 1.6 1.1 1.6 I N a _ M f b, ISAC 19.1 4.1 7.7 8.9 I N a _ M f b, ISAC, IMGC_Mfb 20.2 11.1 7.8 18.9 V r e s t↑ (%) I N a _ M f b 8.3 0.8 0.4 2.3 I N a _ M f b, ISAC 20.3 12.5 25.0 17.4 I N a _ M f b, ISAC, IMGC_Mfb 22.4 15.9 28.6 23.6Previous modeling work suggested that Mfb-M coupling contributed to arrhythmia formation [23, 46]. The key factors included Ggap, the number of coupled Mfbs, and Mfb density. Here, we investigated the functional effect of fibrotic remodeling on AF by integrating INa_Mfb, ISAC, and IMGC_Mfb into Mfb-M coupling. To the best of our knowledge, this has not been examined before. As summarized in Figure 7, these currents made the Mfb act as a current source for the cardiac myocyte, leading to depolarization of Vrest and shortening of APD, thereby preserving myocyte excitability to trigger spontaneous excitement and facilitating reentry (such as those of AF).


## 4.3. Limitations

Two limitations of this mathematical modeling should be mentioned. First, the influence of homologous coupling between adjacent Mfbs was not considered. Since Mfbs can form conduction bridges between myocyte bundles and introduce nonlinearity into Mfb-M coupling to alter electrophysiological properties of the coupled myocyte, coupling between Mfbs may be important in tissue modeling [28, 29]. Second, our modeling of IMGC_Mfb was done using a simplified model based on a linear equation. We did not include the mechanosensitivity of MGCs in our simulation; for example, the conductance depends of the mechanical forces to which the cells are submitted.


## 5. Conclusions

This study demonstrated the combinational effects ofINa_Mfb and IMGC_Mfb in Mfbs on myocyte excitability and repolarization. Our results showed that the addition of INa_Mfb and IMGC_Mfb reduced Vmax, shortened APD, depolarized Vrest, and was easy to trigger spontaneous excitement in myocytes. The effects proved that Na+ current and mechanogated channels in Mfbs should be considered in future pathological cardiac mathematical modeling, such as atrial fibrillation and cardiac fibrosis.



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*Source: 6189374-2016-11-17.xml*