/* * Espinosa Model - hypertrophic version * * Model Status * * This CellML model runs in both OpenCell and COR to recreate * the published results. The model is valid CellML, and the units * have been checked and they are consistent. This version of the * model is for a hypertrophic heart. * * Model Structure * * Chronic exposure of adult rats to a simulated altitude of 4500 * meters is accompanied by right ventricular hypertrophy which * is, at least, partially attributable to an enlargement of right * ventricular myocytes. This work contributes to the understanding * of cardiovascular system modifications induced by changes in * the atmospheric environment. * * Hypertrophied myocytes show an important increase in action * potential duration, particularly at the delayed phase of the * repolarisation. Although it is well known that Ito and ICaL * are decreased in hypertrophied rats, we hypothesized that the * Na+/Ca2+ exchange current could be implied in the prolongation * of action potential duration in hypertrophied cells. The electrophysiological * study of myocytes was achieved using the whole cell configuration * of the patch clamp technique. This configuration was obtained * in two ways: the conventional broken patch technique, which * allows to control the internal medium composition and to buffer * the intracellular calcium, and the perforated patch technique * which allows to respect the internal composition of each cell. * * In order to test for the implication of Na+/Ca2+ exchange current, * we compared the action potentials obtained in control and in * hypertrophied cells, in ionic conditions where the Na+/Ca2+ * exchanger was either present or blocked. * * The Na+/Ca2+ exchanger current measured in voltage clamp conditions * was also compared in control and hypertrophied cells. * * Our results clearly show and implication of Na+/Ca2+ exchanger * current in the prolongation of action potentials recorded in * hypertrophied cells. Concerning the underlying mechanisms, the * voltage clamp experiments show that the increased exchanger * current is induced by a modification of the driving force for * the exchanger rather than by a modification of 'conductance' * or density of Na+/Ca2+ exchanger proteins. The modification * of the driving force could be induced by a change in the regulation * of calcium concentration by the sarcoplasmic reticulum or by * the action potential duration itself. * * The experimental data acquired in this study, were used in a * mathematical simulation of the action potential (Oxsoft Heart * model). One of the most interesting result obtained by this * method is the high implication of a persistent sodium current * in the hypertrophied cell action potentials. * * model diagram * * [[Image file: espinosa_1998.png]] * * A schematic diagram of the cell described by the model. * * This model was taken from the PhD thesis of Leon Espinosa: * * L'Echange Na+ /Ca2+ dans l'Hypertrophie Ventriculaire D'Altitude * chez le Rat: Etude Electrophysiologique et Utilisation du Modele * "Oxsoft Heart", Leon Espinosa, 1998. */ import nsrunit; unit conversion on; // unit micrometre predefined unit per_second=1 second^(-1); // unit millivolt predefined unit per_millivolt=1E3 kilogram^(-1)*meter^(-2)*second^3*ampere^1; unit per_millivolt_second=1E3 kilogram^(-1)*meter^(-2)*second^2*ampere^1; unit microS=1E-6 kilogram^(-1)*meter^(-2)*second^3*ampere^2; unit microF=1E-6 kilogram^(-1)*meter^(-2)*second^4*ampere^2; unit nanoA=1E-9 ampere^1; unit mA_nA=1E6 dimensionless; unit nanoA_per_millimolar=1E-9 meter^3*ampere^1*mole^(-1); // unit millimolar predefined unit millimolar4=1 meter^(-12)*mole^4; unit millimolar_per_second=1 meter^(-3)*second^(-1)*mole^1; unit per_millimolar_second=1 meter^3*second^(-1)*mole^(-1); unit millijoule_per_mole_kelvin=.001 kilogram^1*meter^2*second^(-2)*kelvin^(-1)*mole^(-1); unit coulomb_per_mole=1 second^1*ampere^1*mole^(-1); unit micrometre3=1E-18 meter^3; unit litre_micrometre3=1E15 dimensionless; math main { realDomain time second; time.min=0; extern time.max; extern time.delta; real V(time) millivolt; when(time=time.min) V=-85.35765; real R millijoule_per_mole_kelvin; R=8314.472; real T kelvin; T=310; real F coulomb_per_mole; F=96485.3415; real Cm microF; Cm=9.5e-5; real i_K1(time) nanoA; real i_to(time) nanoA; real i_K(time) nanoA; real i_Ca_L_K_cyt(time) nanoA; real i_Ca_L_K_ds(time) nanoA; real i_NaK(time) nanoA; real i_Na(time) nanoA; real i_b_Na(time) nanoA; real i_Ca_L_Na_cyt(time) nanoA; real i_Ca_L_Na_ds(time) nanoA; real i_NaCa_cyt(time) nanoA; real i_NaCa_ds(time) nanoA; real i_Ca_L_Ca_cyt(time) nanoA; real i_Ca_L_Ca_ds(time) nanoA; real i_b_Ca(time) nanoA; real i_b_K(time) nanoA; real i_Stim(time) nanoA; real stim_start second; stim_start=0.02; real stim_end second; stim_end=9; real stim_period second; stim_period=0.5; real stim_duration second; stim_duration=0.002; real stim_amplitude nanoA; stim_amplitude=-1.8; real E_Na(time) millivolt; real E_K(time) millivolt; real E_Ca(time) millivolt; real E_mh(time) millivolt; real K_o(time) millimolar; when(time=time.min) K_o=5.3367; real Na_o millimolar; Na_o=148.5; real K_i(time) millimolar; when(time=time.min) K_i=140.096; real Na_i(time) millimolar; when(time=time.min) Na_i=6.9366; real Ca_o millimolar; Ca_o=2.5; real Ca_i(time) millimolar; when(time=time.min) Ca_i=3.792e-5; real K_mk1 millimolar; K_mk1=10; real g_K1 microS; g_K1=0.12; real I_K(time) nanoA; real i_K_max nanoA; i_K_max=1.5; real x(time) dimensionless; when(time=time.min) x=3.5095e-6; real alpha_x(time) per_second; real beta_x(time) per_second; real delta_x millivolt; delta_x=0.0001; real E0_x(time) millivolt; real g_Na microS; g_Na=0.6; real m(time) dimensionless; when(time=time.min) m=0.00448; real h(time) dimensionless; when(time=time.min) h=0.9704; real alpha_m(time) per_second; real beta_m(time) per_second; real delta_m millivolt; delta_m=1e-5; real E0_m(time) millivolt; real alpha_h(time) per_second; real beta_h(time) per_second; real shift_h millivolt; shift_h=0; real g_bna microS; g_bna=0.0001; real g_bk microS; g_bk=0.004; real i_Ca_L(time) nanoA; real P_Ca_L nanoA_per_millimolar; P_Ca_L=0.045; real P_CaK dimensionless; P_CaK=0.003; real P_CaNa dimensionless; P_CaNa=0.01; real Ca_ds(time) millimolar; when(time=time.min) Ca_ds=0.00077; real d(time) dimensionless; when(time=time.min) d=4.171e-12; real f(time) dimensionless; when(time=time.min) f=0.999997; real f2(time) dimensionless; when(time=time.min) f2=0.99279; real f2ds(time) dimensionless; when(time=time.min) f2ds=0.459; real Km_f2 millimolar; Km_f2=100000; real Km_f2ds millimolar; Km_f2ds=0.001; real R_decay per_second; R_decay=20; real FrICa dimensionless; FrICa=1; real alpha_d(time) per_second; real beta_d(time) per_second; real E0_d(time) millivolt; real speed_d dimensionless; speed_d=10; real alpha_f(time) per_second; real beta_f(time) per_second; real speed_f dimensionless; speed_f=2; real delta_f millivolt; delta_f=0.0001; real E0_f(time) millivolt; real g_bca microS; g_bca=0.00025; real g_to microS; g_to=0.048; real g_tos dimensionless; g_tos=0.15; real s(time) dimensionless; when(time=time.min) s=0.9379; real r(time) dimensionless; when(time=time.min) r=2.6578e-5; real alpha_s(time) per_second; real beta_s(time) per_second; real i_NaK_max nanoA; i_NaK_max=0.7; real K_mK millimolar; K_mK=1; real K_mNa millimolar; K_mNa=40; real i_NaCa(time) nanoA; real k_NaCa nanoA; k_NaCa=0.0002; // Var below replaced by constant in model eqns to satisfy unit correction // real n_NaCa dimensionless; // n_NaCa=3; real d_NaCa dimensionless; d_NaCa=0.001; real gamma dimensionless; gamma=0.5; real FRiNaCa dimensionless; FRiNaCa=0.001; real i_up(time) millimolar_per_second; real K_1 dimensionless; real K_2(time) millimolar; real K_cyca millimolar; K_cyca=0.0003; real K_xcs dimensionless; K_xcs=0.4; real K_srca millimolar; K_srca=0.5; real alpha_up millimolar_per_second; alpha_up=0.4; real beta_up millimolar_per_second; beta_up=0.03; real Ca_up(time) millimolar; when(time=time.min) Ca_up=0.3342; real i_trans(time) millimolar_per_second; real Ca_rel(time) millimolar; when(time=time.min) Ca_rel=0.31007; real i_rel(time) millimolar_per_second; real VoltDep(time) dimensionless; real RegBindSite(time) dimensionless; real CaiReg(time) dimensionless; real CadsReg(time) dimensionless; real ActRate(time) per_second; real InactRate(time) per_second; real K_leak_rate per_second; K_leak_rate=0.05; real K_m_rel per_second; K_m_rel=250; real K_m_Ca_cyt millimolar; K_m_Ca_cyt=0.0005; real K_m_Ca_ds millimolar; K_m_Ca_ds=0.01; real PrecFrac(time) dimensionless; real ActFrac(time) dimensionless; when(time=time.min) ActFrac=0.0112; real ProdFrac(time) dimensionless; when(time=time.min) ProdFrac=0.9059; real SpeedRel(time) dimensionless; real V_i micrometre3; real K_b millimolar; K_b=5.4; real pf per_second; pf=0.7; real V_e micrometre3; real Ca_Calmod(time) millimolar; when(time=time.min) Ca_Calmod=0.001419; real Ca_Trop(time) millimolar; when(time=time.min) Ca_Trop=0.000932; real Calmod millimolar; Calmod=0.02; real Trop millimolar; Trop=0.05; real alpha_Calmod per_millimolar_second; alpha_Calmod=100000; real beta_Calmod per_second; beta_Calmod=50; real alpha_Trop per_millimolar_second; alpha_Trop=100000; real beta_Trop per_second; beta_Trop=200; real radius micrometre; radius=12; real length micrometre; length=74; real V_Cell micrometre3; real V_i_ratio dimensionless; real V_ds_ratio dimensionless; V_ds_ratio=0.1; real V_rel_ratio dimensionless; V_rel_ratio=0.1; real V_e_ratio dimensionless; V_e_ratio=0.4; real V_up_ratio dimensionless; V_up_ratio=0.01; real Kdecay per_second; Kdecay=10; // // i_Stim=(if (((time>=stim_start) and (time<=stim_end)) and ((time-stim_start-floor((time-stim_start)/stim_period)*stim_period)<=stim_duration)) stim_amplitude else (0 nanoA)); V:time=((-1)*1/Cm*(i_Stim+i_K1+i_to+i_K+i_NaK+i_Na+i_b_Na+i_Ca_L_Na_cyt+i_Ca_L_Na_ds+i_NaCa_cyt+i_NaCa_ds+i_Ca_L_Ca_cyt+i_Ca_L_Ca_ds+i_Ca_L_K_cyt+i_Ca_L_K_ds+i_b_Ca+i_b_K)); // E_Na=(R*T/F*ln(Na_o/Na_i)); E_K=(R*T/F*ln(K_o/K_i)); E_Ca=(.5*R*T/F*ln(Ca_o/Ca_i)); E_mh=(R*T/F*ln((Na_o+.12*K_o)/(Na_i+.12*K_i))); // i_K1=(g_K1*K_o/(K_o+K_mk1)*(V-E_K)/(1+exp((V-E_K+(10 millivolt))*F*1.67/(R*T)))); // I_K=(i_K_max*(K_i-K_o*exp((-1)*V*F/(R*T)))/(140 millimolar)); i_K=(x*I_K); // E0_x=(V+(22 millivolt)-(40 millivolt)); alpha_x=(if (abs(E0_x) i_Na=(g_Na*m^3*h*(V-E_mh)); // E0_m=(V+(41 millivolt)); alpha_m=(if (abs(E0_m) alpha_h=((20 per_second)*exp((-1)*(.125 per_millivolt)*(V+(75 millivolt)-shift_h))); beta_h=((2E3 per_second)/(1+320*exp((-1)*(.1 per_millivolt)*(V+(75 millivolt)-shift_h)))); h:time=(alpha_h*(1-h)-beta_h*h); // i_b_Na=(g_bna*(V-E_Na)); // i_b_K=(g_bk*(V-E_K)); // i_Ca_L_Ca_cyt=((1-FrICa)*4*P_Ca_L*d*f*f2*(V-(50 millivolt))*F/(R*T)/(1-exp((-1)*(V-(50 millivolt))*F*2/(R*T)))*(Ca_i*exp((100 millivolt)*F/(R*T))-Ca_o*exp((-1)*(V-(50 millivolt))*F*2/(R*T)))); i_Ca_L_K_cyt=((1-FrICa)*P_CaK*P_Ca_L*d*f*f2*(V-(50 millivolt))*F/(R*T)/(1-exp((-1)*(V-(50 millivolt))*F/(R*T)))*(K_i*exp((50 millivolt)*F/(R*T))-K_o*exp((-1)*(V-(50 millivolt))*F/(R*T)))); i_Ca_L_Na_cyt=((1-FrICa)*P_CaNa*P_Ca_L*d*f*f2*(V-(50 millivolt))*F/(R*T)/(1-exp((-1)*(V-(50 millivolt))*F/(R*T)))*(Na_i*exp((50 millivolt)*F/(R*T))-Na_o*exp((-1)*(V-(50 millivolt))*F/(R*T)))); i_Ca_L_Ca_ds=(FrICa*4*P_Ca_L*d*f*f2ds*(V-(50 millivolt))*F/(R*T)/(1-exp((-1)*(V-(50 millivolt))*F*2/(R*T)))*(Ca_i*exp((100 millivolt)*F/(R*T))-Ca_o*exp((-1)*(V-(50 millivolt))*F*2/(R*T)))); i_Ca_L_K_ds=(FrICa*P_CaK*P_Ca_L*d*f*f2ds*(V-(50 millivolt))*F/(R*T)/(1-exp((-1)*(V-(50 millivolt))*F/(R*T)))*(K_i*exp((50 millivolt)*F/(R*T))-K_o*exp((-1)*(V-(50 millivolt))*F/(R*T)))); i_Ca_L_Na_ds=(FrICa*P_CaNa*P_Ca_L*d*f*f2ds*(V-(50 millivolt))*F/(R*T)/(1-exp((-1)*(V-(50 millivolt))*F/(R*T)))*(Na_i*exp((50 millivolt)*F/(R*T))-Na_o*exp((-1)*(V-(50 millivolt))*F/(R*T)))); i_Ca_L=(i_Ca_L_Ca_cyt+i_Ca_L_K_cyt+i_Ca_L_Na_cyt+i_Ca_L_Ca_ds+i_Ca_L_K_ds+i_Ca_L_Na_ds); // E0_d=(V+(24 millivolt)-(20 millivolt)); alpha_d=(if (abs(E0_d)<(1E-4 millivolt)) (120 per_second) else (30 per_millivolt_second)*E0_d/(1-exp((-1)*E0_d/(3 millivolt)))); beta_d=(if (abs(E0_d)<(1E-4 millivolt)) (120 per_second) else (12 per_millivolt_second)*E0_d/(exp(E0_d/(7.5 millivolt))-1)); d:time=(speed_d*(alpha_d*(1-d)-beta_d*d)); // E0_f=(V+(34 millivolt)-(10 millivolt)); alpha_f=(if (abs(E0_f) f2:time=((1 per_second)-(1 per_second)*(Ca_i/(Km_f2+Ca_i)+f2)); // f2ds:time=(R_decay*(1-(Ca_ds/(Km_f2ds+Ca_ds)+f2ds))); // i_b_Ca=(g_bca*(V-E_Ca)); // i_to=(g_to*(g_tos+s*(1-g_tos))*r*(V-E_K)); // alpha_s=(.26*(.033 per_second)*exp((-1)*V/(14.875 millivolt))); beta_s=(.26*(33 per_second)/(1+exp((-1)*(V+(10 millivolt))/(7 millivolt)))); s:time=(alpha_s*(1-s)-beta_s*s); // r:time=((333 per_second)*(1/(1+exp((-1)*(V+(4 millivolt)-(24 millivolt))/(10 millivolt)))-r)); // i_NaK=(i_NaK_max*K_o/(K_mK+K_o)*Na_i/(K_mNa+Na_i)); // i_NaCa_cyt=((1-FRiNaCa)*k_NaCa*(exp(gamma*(3-2)*V*F/(R*T))*Na_i^3*Ca_o-exp((gamma-1)*(3-2)*V*F/(R*T))*Na_o^3*Ca_i)/(((1 millimolar4)+d_NaCa*(Ca_i*Na_o^3+Ca_o*Na_i^3))*(1+Ca_i/(.0069 millimolar)))); i_NaCa_ds=(FRiNaCa*k_NaCa*(exp(gamma*(3-2)*V*F/(R*T))*Na_i^3*Ca_o-exp((gamma-1)*(3-2)*V*F/(R*T))*Na_o^3*Ca_ds)/(((1 millimolar4)+d_NaCa*(Ca_ds*Na_o^3+Ca_o*Na_i^3))*(1+Ca_ds/(.0069 millimolar)))); i_NaCa=(i_NaCa_cyt+i_NaCa_ds); // K_1=(K_cyca*K_xcs/K_srca); K_2=(Ca_i+Ca_up*K_1+K_cyca*K_xcs+K_cyca); i_up=(Ca_i/K_2*alpha_up-Ca_up*K_1/K_2*beta_up); // i_trans=((50 per_second)*(Ca_up-Ca_rel)); // VoltDep=exp((.08 per_millivolt)*(V-(40 millivolt))); CaiReg=(Ca_i/(Ca_i+K_m_Ca_cyt)); CadsReg=(Ca_ds/(Ca_ds+K_m_Ca_ds)); RegBindSite=(CaiReg+(1-CaiReg)*CadsReg); ActRate=((0 per_second)*VoltDep+(500 per_second)*RegBindSite^2); InactRate=((60 per_second)+(500 per_second)*RegBindSite^2); SpeedRel=(if (V<((-1)*(50 millivolt))) 5 else 1); PrecFrac=(1-ActFrac-ProdFrac); ActFrac:time=(PrecFrac*SpeedRel*ActRate-ActFrac*SpeedRel*InactRate); ProdFrac:time=(ActFrac*SpeedRel*InactRate-SpeedRel*(1 per_second)*ProdFrac); i_rel=(((ActFrac/(ActFrac+.25))^2*K_m_rel+K_leak_rate)*Ca_rel); // // Na_i:time=((-1)*(1 mA_nA)/((1 litre_micrometre3)*V_i*F)*(i_Na+i_b_Na+3*i_NaK+3*i_NaCa_cyt+i_Ca_L_Na_cyt+i_Ca_L_Na_ds)); // // K_o:time=((1 mA_nA)*(i_K1+i_to+i_Ca_L_K_cyt+i_Ca_L_K_ds+(-1)*2*i_NaK+i_K+i_b_K)/((1 litre_micrometre3)*V_e*F)-pf*(K_o-K_b)); // K_i:time=((-1)*(1 mA_nA)/((1 litre_micrometre3)*V_i*F)*(i_K1+i_K+i_Ca_L_K_cyt+i_Ca_L_K_ds+i_to+i_b_K-2*i_NaK)); // V_Cell=(3.141592654*(radius/1E3)^2*length/1E3); V_i_ratio=(1-V_e_ratio-V_up_ratio-V_rel_ratio); V_i=(V_Cell*V_i_ratio); V_e=(V_Cell*V_e_ratio); Ca_i:time=((-1)*(1 mA_nA)/(2*(1 litre_micrometre3)*V_i*F)*(i_Ca_L_Ca_cyt+i_b_Ca-2*i_NaCa_cyt)+Ca_ds*V_ds_ratio*Kdecay+i_rel*V_rel_ratio/V_i_ratio-Ca_Calmod:time-Ca_Trop:time-i_up); Ca_ds:time=((-1)*(1 mA_nA)*i_Ca_L_Ca_ds/(2*V_ds_ratio*(1 litre_micrometre3)*V_i*F)-Ca_ds*Kdecay); Ca_up:time=(V_i_ratio/V_up_ratio*i_up-i_trans); Ca_rel:time=(V_up_ratio/V_rel_ratio*i_trans-i_rel); Ca_Calmod:time=(alpha_Calmod*Ca_i*(Calmod-Ca_Calmod)-beta_Calmod*Ca_Calmod); Ca_Trop:time=(alpha_Trop*Ca_i*(Trop-Ca_Trop)-beta_Trop*Ca_Trop); }