/* * Modelling The Effect Of Intracellular Ca2+ On Sinoatrial Node * Action Potentials, 2001 * * Model Status * * This model is valid CellML. However the model will not run in * either OpenCell or COR and the model requires further curation. * * Model Structure * * ABSTRACT: The possible effects of intracellular Ca2+ on the * pacemaker of the heart, the sinoatrial node, are reviewed. In * mammalian sinoatrial node, reduction or abolition of the intracellular * Ca2+ transient by ryanodine, sarcoplasmic reticulum Ca2+ pump * block or 1,2-bis(2-aminophenoxy)ethane-N, N, N', N'-tetraacetic * acid (BAPTA) reduces the spontaneous rate by 21-32%, whereas * in amphibian sinus venosus it abolishes spontaneous activity. * In rabbit sinoatrial node, ryanodine/BAPTA reduces the T-type * Ca2+ current (iCa,T), perhaps slows inactivation of the L-type * Ca2+ current (iCa,L), reduces the inward Na+-Ca2+ exchange current * (iNaCa), and reduces the rapid and slow delayed rectifier K+ * currents (iK,r and iK,s, respectively). Other evidence shows * that a reduction of intracellular Ca2+ inhibits the hyperpolarization-activated * current (if). These putative intracellular Ca2+-dependent changes * in ionic currents have been incorporated into different models * of rabbit sinoatrial node action potentials. In the models, * block of the Ca2+ transient reduced the spontaneous rate by * 24 and 26% in the central and peripheral models of Zhang and * others, 13% in the Oxsoft model (Noble et al.), 9% in the model * of Wilders and others, and 41% in the model of Demir and others. * In all models, the reduction in rate was not primarily the result * of the decrease in iNaCa, but instead the combination of all * changes in ionic currents. * * The original paper reference is cited below: * * Control of the pacemaker activity of the sinoatrial node by * intracellular Ca2+. Experiments and modelling, M.R. Boyett, * H. Zhang, A. Garny and A.V. Holden, 2001, Phil. Trans. R. Soc. * Lond. A., 359, 1091-1110. (note there is no PubMed ID for this * publication). * * cell diagram of the Boyett et al SAN model showing ionic currents, * pumps and exchangers within the sarcolemma and the sarcoplasmic * reticulum * * [[Image file: boyett_2001.png]] * * A schematic diagram describing the current flows across the * cell membrane that are captured in the Boyett et al 2001 model * of the action potentials in the peripheral cells of the SA node. */ import nsrunit; unit conversion on; 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 millimolar predefined unit millimolar4=1 meter^(-12)*mole^4; 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 microlitre_per_second=1E-9 meter^3*second^(-1); unit millimole_per_second=.001 second^(-1)*mole^1; unit per_second_millimolar=1 meter^3*second^(-1)*mole^(-1); math main { realDomain time second; time.min=0; extern time.max; extern time.delta; real dCell dimensionless; dCell=0; real FCell dimensionless; real V(time) millivolt; when(time=time.min) V=-39.013558536; real R millijoule_per_mole_kelvin; R=8314; real T kelvin; T=310; real F coulomb_per_mole; F=96845; real Cm microF; real CmCentre microF; CmCentre=2e-5; real CmPeriphery microF; CmPeriphery=6.5e-5; real i_Na(time) nanoA; real i_Ca_L(time) nanoA; real i_Ca_T(time) nanoA; real i_to(time) nanoA; real i_sus(time) nanoA; real i_K_r(time) nanoA; real i_K_s(time) nanoA; real i_f_Na(time) nanoA; real i_f_K(time) nanoA; real i_b_Na(time) nanoA; real i_b_Ca(time) nanoA; real i_b_K(time) nanoA; real i_NaCa(time) nanoA; real i_p(time) nanoA; real i_Ca_P(time) nanoA; real i_st(time) nanoA; real g_Na microlitre_per_second; real g_Na_Centre microlitre_per_second; g_Na_Centre=0; real g_Na_Periphery microlitre_per_second; g_Na_Periphery=1.2e-6; real E_Na millivolt; real Na_o millimolar; Na_o=140; real m(time) dimensionless; when(time=time.min) m=0.092361701692; real h(time) dimensionless; real m_infinity(time) dimensionless; real tau_m(time) second; real F_Na(time) dimensionless; real h1(time) dimensionless; when(time=time.min) h1=0.015905380261; real h2(time) dimensionless; when(time=time.min) h2=0.01445216109; real h1_infinity(time) dimensionless; real h2_infinity(time) dimensionless; real tau_h1(time) second; real tau_h2(time) second; real g_Ca_L_Centre microS; g_Ca_L_Centre=0.0058; real g_Ca_L_Periphery microS; g_Ca_L_Periphery=0.0659; real g_Ca_L microS; real E_Ca_L millivolt; E_Ca_L=46.4; real d_L(time) dimensionless; when(time=time.min) d_L=0.04804900895; real f_L(time) dimensionless; when(time=time.min) f_L=0.48779845203; real alpha_d_L(time) per_second; real beta_d_L(time) per_second; real d_L_infinity(time) dimensionless; real tau_d_L(time) second; real alpha_f_L(time) per_second; real beta_f_L(time) per_second; real f_L_infinity(time) dimensionless; real tau_f_L(time) second; real g_Ca_T_Centre microS; g_Ca_T_Centre=0.0043; real g_Ca_T_Periphery microS; g_Ca_T_Periphery=0.0139; real g_Ca_T microS; real E_Ca_T millivolt; E_Ca_T=45; real d_T(time) dimensionless; when(time=time.min) d_T=0.42074047435; real f_T(time) dimensionless; when(time=time.min) f_T=0.038968420558; real alpha_d_T(time) per_second; real beta_d_T(time) per_second; real d_T_infinity(time) dimensionless; real tau_d_T(time) second; real alpha_f_T(time) per_second; real beta_f_T(time) per_second; real f_T_infinity(time) dimensionless; real tau_f_T(time) second; real g_to_Centre microS; g_to_Centre=0.00491; real g_to_Periphery microS; g_to_Periphery=0.03649; real g_to microS; real g_sus_Centre microS; g_sus_Centre=6.65e-5; real g_sus_Periphery microS; g_sus_Periphery=0.0114; real g_sus microS; real E_K millivolt; real q(time) dimensionless; when(time=time.min) q=0.29760539675; real r(time) dimensionless; when(time=time.min) r=0.064402950262; real q_infinity(time) dimensionless; real tau_q(time) second; real r_infinity(time) dimensionless; real tau_r(time) second; real g_K_r_Centre microS; g_K_r_Centre=0.000797; real g_K_r_Periphery microS; g_K_r_Periphery=0.016; real g_K_r microS; real P_a(time) dimensionless; real P_af(time) dimensionless; when(time=time.min) P_af=0.13034201158; real P_as(time) dimensionless; when(time=time.min) P_as=0.46960956028; real P_i(time) dimensionless; when(time=time.min) P_i=0.87993375273; real P_af_infinity(time) dimensionless; real tau_P_af(time) second; real P_as_infinity(time) dimensionless; real tau_P_as(time) second; real P_i_infinity(time) dimensionless; real tau_P_i second; tau_P_i=0.002; real g_K_s_Centre microS; g_K_s_Centre=0.000518; real g_K_s_Periphery microS; g_K_s_Periphery=0.0104; real g_K_s microS; real E_K_s millivolt; real xs(time) dimensionless; when(time=time.min) xs=0.082293827208; real alpha_xs(time) per_second; real beta_xs(time) per_second; real g_f_Na_Centre microS; g_f_Na_Centre=0.000548; real g_f_Na_Periphery microS; g_f_Na_Periphery=0.0069; real g_f_Na microS; real g_f_K_Centre microS; g_f_K_Centre=0.000548; real g_f_K_Periphery microS; g_f_K_Periphery=0.0069; real g_f_K microS; real y(time) dimensionless; when(time=time.min) y=0.03889291759; real alpha_y(time) per_second; real beta_y(time) per_second; real g_b_Na_Centre microS; g_b_Na_Centre=5.8e-5; real g_b_Na_Periphery microS; g_b_Na_Periphery=0.000189; real g_b_Na microS; real g_b_K_Centre microS; g_b_K_Centre=2.52e-5; real g_b_K_Periphery microS; g_b_K_Periphery=8.19e-5; real g_b_K microS; real g_b_Ca_Centre microS; g_b_Ca_Centre=1.32e-5; real g_b_Ca_Periphery microS; g_b_Ca_Periphery=4.3e-5; real g_b_Ca microS; real E_Ca millivolt; real k_NaCa_Centre nanoA; k_NaCa_Centre=2.7e-6; real k_NaCa_Periphery nanoA; k_NaCa_Periphery=8.8e-6; real k_NaCa nanoA; real d_NaCa dimensionless; d_NaCa=0.0001; real gamma_NaCa dimensionless; gamma_NaCa=0.5; real Na_i millimolar; Na_i=8; real Ca_i millimolar; Ca_i=0.0001; real Ca_o millimolar; Ca_o=2; real K_m_Na millimolar; K_m_Na=5.64; real K_m_K millimolar; K_m_K=0.621; real i_p_max_Centre nanoA; i_p_max_Centre=0.0478; real i_p_max_Periphery nanoA; i_p_max_Periphery=0.16; real i_p_max nanoA; real K_o millimolar; K_o=5.4; real K_i millimolar; K_i=140; real g_st microS; g_st=0.00214; real d_s(time) dimensionless; when(time=time.min) d_s=0.230; real f_s(time) dimensionless; when(time=time.min) f_s=0.283; real alpha_d_s(time) per_second; real beta_d_s(time) per_second; real d_s_infinity(time) dimensionless; real tau_d_s(time) second; real alpha_f_s(time) per_second; real beta_f_s(time) per_second; real f_s_infinity(time) dimensionless; real tau_f_s(time) second; real U_d(time) dimensionless; real U_s(time) dimensionless; real i_Ca_P_max nanoA; i_Ca_P_max=0.24; real J_Ca_ds(time) millimole_per_second; real J_Ca_r(time) millimole_per_second; real J_Ca_P(time) millimole_per_second; real J_Ca_P_max millimole_per_second; J_Ca_P_max=0.00000000096; real J_Ca_u(time) millimole_per_second; real J_Ca_u_max millimole_per_second; J_Ca_u_max=0.00000000096; real J_Ca_ur(time) millimole_per_second; real J_Ca_1(time) millimole_per_second; real Ca_d(time) millimolar; when(time=time.min) Ca_d=0.001544; real Ca_s(time) millimolar; when(time=time.min) Ca_s=0.05487; real Ca_u(time) millimolar; when(time=time.min) Ca_u=6.983; real Ca_r(time) millimolar; when(time=time.min) Ca_r=4.909; real B_d millimolar; B_d=0.4; real B_s millimolar; B_s=0.2; real Vol_u litre; real Vol_r litre; real Vol_d litre; real Vol_s litre; real Vol_c litre; Vol_c=8.0e-12; real K_m_b millimolar; K_m_b=0.002; real K_m_r millimolar; K_m_r=0.001; real K_m_u millimolar; K_m_u=0.005; real f_d dimensionless; f_d=0.1; real f_r dimensionless; f_r=0.3; real f_u dimensionless; f_u=0.06; real alpha_ds per_second; alpha_ds=3300.0; real alpha_fR per_second_millimolar; alpha_fR=6670.0; real alpha_1 per_second; alpha_1=0.172; real alpha_r per_second; alpha_r=333.5; real alpha_ur per_second; alpha_ur=533.6; real beta_fR per_second; beta_fR=2.0; real f_R(time) dimensionless; when(time=time.min) f_R=0.188; real i_Ca(time) nanoA; // // FCell=(1.07*(3*dCell-.1)/(3*(1+.7745*exp((-1)*(3*dCell-2.05)/.295)))); Cm=(CmCentre+FCell*(CmPeriphery-CmCentre)); V:time=((-1)*1/Cm*(i_Na+i_Ca_L+i_Ca_T+i_to+i_sus+i_K_r+i_K_s+i_f_Na+i_f_K+i_b_Na+i_b_Ca+i_b_K+i_NaCa+i_p+i_Ca_P)); // g_Na=(g_Na_Centre+FCell*(g_Na_Periphery-g_Na_Centre)); i_Na=(g_Na*m^3*h*Na_o*F^2/(R*T)*(exp((V-E_Na)*F/(R*T))-1)/(exp(V*F/(R*T))-1)*V); // m:time=((m_infinity-m)/tau_m); m_infinity=((1/(1+exp((-1)*V/(5.46 millivolt))))^(1/3)); tau_m=((6.247E-4 second)/(.832*exp((-1)*(.335 per_millivolt)*(V+(56.7 millivolt)))+.627*exp((.082 per_millivolt)*(V+(65.01 millivolt))))+(3.9999999999999996E-5 second)); // F_Na=(.0952*exp((-1)*(.063 per_millivolt)*(V+(34.4 millivolt)))/(1+1.66*exp((-1)*(.225 per_millivolt)*(V+(63.7 millivolt))))+.0869); h=((1-F_Na)*h1+F_Na*h2); h1:time=((h1_infinity-h1)/tau_h1); h2:time=((h2_infinity-h2)/tau_h2); h1_infinity=(1/(1+exp((V+(66.1 millivolt))/(6.4 millivolt)))); h2_infinity=h1_infinity; tau_h1=((3.7169999999999997E-6 second)*exp((-1)*(.2815 per_millivolt)*(V+(17.11 millivolt)))/(1+.003732*exp((-1)*(.3426 per_millivolt)*(V+(37.76 millivolt))))+(5.977E-4 second)); tau_h2=((3.186E-8 second)*exp((-1)*(.6219 per_millivolt)*(V+(18.8 millivolt)))/(1+7.188999999999999E-5*exp((-1)*(.6683 per_millivolt)*(V+(34.07 millivolt))))+(.003556 second)); // g_Ca_L=(g_Ca_L_Centre+FCell*(g_Ca_L_Periphery-g_Ca_L_Centre)); i_Ca_L=(g_Ca_L*(f_L*d_L+.006/(1+exp((-1)*(V+(14.1 millivolt))/(6 millivolt))))*(V-E_Ca_L)); // d_L:time=((d_L_infinity-d_L)/tau_d_L); alpha_d_L=((-1)*(28.38 per_millivolt_second)*(V+(35 millivolt))/(exp((-1)*(V+(35 millivolt))/(2.5 millivolt))-1)-(84.9 per_millivolt_second)*V/(exp((-1)*(.208 per_millivolt)*V)-1)); beta_d_L=((11.42 per_millivolt_second)*(V-(5 millivolt))/(exp((.4 per_millivolt)*(V-(5 millivolt)))-1)); tau_d_L=(2/(alpha_d_L+beta_d_L)); d_L_infinity=(1/(1+exp((-1)*(V+(23.1 millivolt))/(6 millivolt)))); // f_L:time=((f_L_infinity-f_L)/tau_f_L); alpha_f_L=((3.12 per_millivolt_second)*(V+(28 millivolt))/(exp((V+(28 millivolt))/(4 millivolt))-1)); beta_f_L=((25 per_second)/(1+exp((-1)*(V+(28 millivolt))/(4 millivolt)))); tau_f_L=(1/(alpha_f_L+beta_f_L)); f_L_infinity=(1/(1+exp((V+(45 millivolt))/(5 millivolt)))); // g_Ca_T=(g_Ca_T_Centre+FCell*(g_Ca_T_Periphery-g_Ca_T_Centre)); i_Ca_T=(g_Ca_T*d_T*f_T*(V-E_Ca_T)); // d_T:time=((d_T_infinity-d_T)/tau_d_T); alpha_d_T=((1068 per_second)*exp((V+(26.3 millivolt))/(30 millivolt))); beta_d_T=((1068 per_second)*exp((-1)*(V+(26.3 millivolt))/(30 millivolt))); tau_d_T=(1/(alpha_d_T+beta_d_T)); d_T_infinity=(1/(1+exp((-1)*(V+(37 millivolt))/(6.8 millivolt)))); // f_T:time=((f_T_infinity-f_T)/tau_f_T); alpha_f_T=((15.3 per_second)*exp((-1)*(V+(71.7 millivolt))/(83.3 millivolt))); beta_f_T=((15 per_second)*exp((V+(71.7 millivolt))/(15.38 millivolt))); tau_f_T=(1/(alpha_f_T+beta_f_T)); f_T_infinity=(1/(1+exp((V+(71 millivolt))/(9 millivolt)))); // g_to=(g_to_Centre+FCell*(g_to_Periphery-g_to_Centre)); g_sus=(g_sus_Centre+FCell*(g_sus_Periphery-g_sus_Centre)); i_to=(g_to*q*r*(V-E_K)); i_sus=(g_sus*r*(V-E_K)); // q:time=((q_infinity-q)/tau_q); q_infinity=(1/(1+exp((V+(59.37 millivolt))/(13.1 millivolt)))); tau_q=((.0101 second)+(.06517 second)/(.57*exp((-1)*(.08 per_millivolt)*(V+(49 millivolt))))+(2.3999999999999997E-5 second)*exp((.1 per_millivolt)*(V+(50.93 millivolt)))); // r:time=((r_infinity-r)/tau_r); r_infinity=(1/(1+exp((-1)*(V-(10.93 millivolt))/(19.7 millivolt)))); tau_r=(.001*((2.98 second)+(15.59 second)/(1.037*exp((.09 per_millivolt)*(V+(30.61 millivolt)))+.369*exp((-1)*(.12 per_millivolt)*(V+(23.84 millivolt)))))); // g_K_r=(g_K_r_Centre+FCell*(g_K_r_Periphery-g_K_r_Centre)); i_K_r=(g_K_r*P_a*P_i*(V-E_K)); P_a=(.6*P_af+.4*P_as); // P_af:time=((P_af_infinity-P_af)/tau_P_af); P_af_infinity=(1/(1+exp((-1)*(V+(14.2 millivolt))/(10.6 millivolt)))); tau_P_af=((1 second)/(37.2*exp((V-(9 millivolt))/(15.9 millivolt))+.96*exp((-1)*(V-(9 millivolt))/(22.5 millivolt)))); // P_as:time=((P_as_infinity-P_as)/tau_P_as); P_as_infinity=P_af_infinity; tau_P_as=((1 second)/(4.2*exp((V-(9 millivolt))/(17 millivolt))+.15*exp((-1)*(V-(9 millivolt))/(21.6 millivolt)))); // P_i:time=((P_i_infinity-P_i)/tau_P_i); P_i_infinity=(1/(1+exp((V+(18.6 millivolt))/(10.1 millivolt)))); // g_K_s=(g_K_s_Centre+FCell*(g_K_s_Periphery-g_K_s_Centre)); i_K_s=(g_K_s*xs^2*(V-E_K_s)); // xs:time=(alpha_xs*(1-xs)-beta_xs*xs); alpha_xs=((14 per_second)/(1+exp((-1)*(V-(40 millivolt))/(9 millivolt)))); beta_xs=((1 per_second)*exp((-1)*V/(45 millivolt))); // g_f_Na=(g_f_Na_Centre+FCell*(g_f_Na_Periphery-g_f_Na_Centre)); i_f_Na=(g_f_Na*y*(V-E_Na)); g_f_K=(g_f_K_Centre+FCell*(g_f_K_Periphery-g_f_K_Centre)); i_f_K=(g_f_K*y*(V-E_K)); // y:time=(alpha_y*(1-y)-beta_y*y); alpha_y=((1 per_second)*exp((-1)*(V+(78.91 millivolt))/(26.62 millivolt))); beta_y=((1 per_second)*exp((V+(75.13 millivolt))/(21.25 millivolt))); // g_b_Na=(g_b_Na_Centre+FCell*(g_b_Na_Periphery-g_b_Na_Centre)); i_b_Na=(g_b_Na*(V-E_Na)); // g_b_K=(g_b_K_Centre+FCell*(g_b_K_Periphery-g_b_K_Centre)); i_b_K=(g_b_K*(V-E_K)); // g_b_Ca=(g_b_Ca_Centre+FCell*(g_b_Ca_Periphery-g_b_Ca_Centre)); i_b_Ca=(g_b_Ca*(V-E_Ca)); // k_NaCa=(k_NaCa_Centre+FCell*(k_NaCa_Periphery-k_NaCa_Centre)); i_NaCa=(k_NaCa*(Na_i^3*Ca_o*exp((.03743 per_millivolt)*V*gamma_NaCa)-Na_o^3*Ca_i*exp((.0374 per_millivolt)*V*(gamma_NaCa-1)))/((1 millimolar4)+d_NaCa*(Ca_i*Na_o^3+Ca_o*Na_i^3))); // i_p_max=(i_p_max_Centre+FCell*(i_p_max_Periphery-i_p_max_Centre)); i_p=(i_p_max*(Na_i/(K_m_Na+Na_i))^3*(K_o/(K_m_K+K_o))^2*1.6/(1.5+exp((-1)*(V+(60 millivolt))/(40 millivolt)))); // // E_Na=(R*T/F*ln(Na_o/Na_i)); E_K=(R*T/F*ln(K_o/K_i)); E_Ca=(R*T/(2*F)*ln(Ca_o/Ca_i)); E_K_s=(R*T/F*ln((K_o+.12*Na_o)/(K_i+.12*Na_i))); // i_st=(g_st*d_s*f_s*(V-(18 millivolt))); // d_s:time=((d_s_infinity-d_s)/tau_d_s); alpha_d_s=((1E3 per_second)/(.15*exp((-1)*V/(11 millivolt))+.2*exp((-1)*V/(700 millivolt)))); beta_d_s=((1E3 per_second)/(16*exp(V/(8 millivolt))+.2*exp(V/(50 millivolt)))); tau_d_s=(1/(alpha_d_s+beta_d_s)); d_s_infinity=(alpha_d_s/(alpha_d_s+beta_d_s)); // f_s:time=((f_s_infinity-f_s)/tau_f_s); alpha_f_s=((1E3 per_second)/(3100*exp((-1)*V/(13 millivolt))+700*exp((-1)*V/(70 millivolt)))); beta_f_s=((1E3 per_second)/(16*exp(V/(8 millivolt))+.2*exp(V/(50 millivolt)))); tau_f_s=(1/(alpha_f_s+beta_f_s)); f_s_infinity=(alpha_f_s/(alpha_f_s+beta_f_s)); // U_d=(1-B_d/(K_m_b+Ca_d+B_d)); J_Ca_ds=(alpha_ds*Vol_d*(Ca_d-Ca_s)); i_Ca_P=(i_Ca_P_max*(Ca_s/(Ca_s+(4E-4 millimolar)))); J_Ca_r=(alpha_r*f_R*(Ca_d^2/(K_m_r^2+Ca_d^2))*Vol_r*Ca_r); f_R:time=((-1)*alpha_fR*Ca_d*f_R+beta_fR*(1-f_R)); U_s=(1-B_s/(K_m_b+Ca_s+B_s)); J_Ca_P=(J_Ca_P_max*(Ca_s/(Ca_s+(4E-4 millimolar)))); J_Ca_u=(J_Ca_u_max*(Ca_s^2/(K_m_u^2+Ca_s^2))); J_Ca_ur=(alpha_ur*Vol_u*(Ca_u-Ca_r)); J_Ca_1=(alpha_1*Vol_u*Ca_u); i_Ca=(i_Ca_L+i_Ca_T); Ca_d:time=(U_d/Vol_d*(J_Ca_ds-.95*i_Ca/(2*F))); Ca_s:time=(U_s/Vol_s*(J_Ca_ds-((.05*i_Ca-2*i_NaCa+i_b_Ca)/(2*F)+J_Ca_u)+J_Ca_r+J_Ca_1)); Ca_u:time=((J_Ca_u-(J_Ca_1+J_Ca_ur))/Vol_u); Ca_r:time=((J_Ca_ur-J_Ca_r)/Vol_r); Vol_u=(f_u*Vol_c); Vol_r=(f_r*Vol_c); Vol_d=(f_d*Vol_c); Vol_s=(Vol_c-(Vol_u+Vol_d)); }