/* * A Biophysically Based Mathematical Model of Unitary Potential * Activity in Interstitial Cells of Cajal * * Model Status * * This CellML model is known to run in both COR and PCEnv to replicate * the published results. The units have been checked and are consistent. * We'd like to acknowledge Richard Faville for his help in getting * the CellML model to replicate his published model. * * Model Structure * * ABSTRACT: Unitary potential (UP) depolarizations are the basic * intracellular events responsible for pacemaker activity in interstitial * cells of Cajal (ICCs), and are generated at intracellular sites * termed "pacemaker units". In this study, we present a mathematical * model of the transmembrane ion flows and intracellular Ca(2+) * dynamics from a single ICC pacemaker unit acting at near-resting * membrane potential. This model quantitatively formalizes the * framework of a novel ICC pacemaking mechanism that has recently * been proposed. Model simulations produce spontaneously rhythmic * UP depolarizations with an amplitude of approximately 3 mV at * a frequency of 0.05 Hz. The model predicts that the main inward * currents, carried by a Ca(2+)-inhibited nonselective cation * conductance, are activated by depletion of sub-plasma-membrane * [Ca(2+)] caused by sarcoendoplasmic reticulum calcium ATPase * Ca(2+) sequestration. Furthermore, pacemaker activity predicted * by our model persists under simulated voltage clamp and is independent * of [IP(3)] oscillations. The model presented here provides a * basis to quantitatively analyze UP depolarizations and the biophysical * mechanisms underlying their production. * * model diagram * * [[Image file: faville_2008.png]] * * A schematic diagram of the pacemaker unit illustrating all the * compartmental volumes and ionic conductances, together with * their interactions. * * The complete paper reference is cited below: * * A Biophysically Based Mathematical Model of Unitary Potential * Activity in Interstitial Cells of Cajal, R.A. Faville, A.J. * Pullan, K.M. Sanders, and N.P. Smith, 2008, Biophysical Journal. * PubMed ID: 18339738 */ import nsrunit; unit conversion on; // unit millivolt predefined // unit micromolar predefined unit per_micromolar=1E3 meter^3*mole^(-1); unit per_micromolar2=1E6 meter^6*mole^(-2); // unit picofarad predefined // unit picosiemens predefined // unit picoampere predefined // unit femtoampere predefined unit micromolar_coulomb=1E-3 meter^(-3)*second^(-1)*ampere^(-1)*mole^1; unit femtocoulomb_per_zeptomole=1E6 second^1*ampere^1*mole^(-1); unit attojoule_per_zeptomole_kelvin=1E3 kilogram^1*meter^2*second^(-2)*kelvin^(-1)*mole^(-1); unit first_order_rate_constant=1 second^(-1); unit per_second_squared=1 second^(-2); unit flux=1E-3 meter^(-3)*second^(-1)*mole^1; math main { realDomain time second; time.min=0; extern time.max; extern time.delta; real Vm(time) millivolt; when(time=time.min) Vm=-70.1; real I_iCa(time) picoampere; real I_iNa(time) picoampere; real Cm picofarad; Cm=20.0; real I_Ca(time) picoampere; real I_Na(time) picoampere; real I_NSCC_Ca(time) picoampere; real I_PM(time) picoampere; real I_NSCC_Na(time) picoampere; real gCa picosiemens; gCa=0.01; real ECa(time) millivolt; real T kelvin; T=310.16; real R attojoule_per_zeptomole_kelvin; R=8.314E-3; real F femtocoulomb_per_zeptomole; F=0.09649; real CO micromolar; CO=1.8E3; real CS1(time) micromolar; when(time=time.min) CS1=0.120; real gNSCC_Ca_ picosiemens; gNSCC_Ca_=0.12; real gNSCC_Ca(time) picosiemens; // Var below replaced by constant in model eqns to satisfy unit correction // real hNSCC dimensionless; // hNSCC=3.0; real ENSCC millivolt; ENSCC=0.0; real KNSCC micromolar; KNSCC=0.12; real gNSCC_Na_ picosiemens; gNSCC_Na_=220.0; real gNSCC_Na(time) picosiemens; real gPM femtoampere; gPM=420.0; real KPM micromolar; KPM=1.0; real gNa femtoampere; gNa=1.5E4; real KNa micromolar; KNa=1.0E4; // Var below replaced by constant in model eqns to satisfy unit correction // real hNa dimensionless; // hNa=4.0; real NS1(time) micromolar; when(time=time.min) NS1=1.01E4; real was_JSERCA(time) flux; real VSERCA first_order_rate_constant; VSERCA=1.0E5; real A2 dimensionless; A2=6E-4; real A4 per_micromolar; A4=3.57; real A5 per_micromolar; A5=2.7E-5; real A6 per_micromolar2; A6=2.31E-5; real CER(time) micromolar; when(time=time.min) CER=203.0; real JMCU(time) flux; real VMCU flux; VMCU=800.0; real KMCU micromolar; KMCU=10.0; real epsilon_INH(time) dimensionless; real KINH micromolar; KINH=10.0; // Var below replaced by constant in model eqns to satisfy unit correction // real hINH dimensionless; // hINH=4.0; real CS2(time) micromolar; when(time=time.min) CS2=0.023; real CMT(time) micromolar; when(time=time.min) CMT=0.220; real JNCX(time) flux; real VNCX flux; VNCX=0.5; real KNCX micromolar; KNCX=0.3; real was_JS1S2(time) flux; real mu_S1S2 first_order_rate_constant; mu_S1S2=0.04; real JIPR(time) flux; real kIPR first_order_rate_constant; kIPR=2000.0; real k_1 flux; k_1=6.4; real k1 first_order_rate_constant; k1=0.0; real k2 first_order_rate_constant; k2=4.0; real r2 first_order_rate_constant; r2=200.0; real r_2 flux; r_2=0.0; real r4 first_order_rate_constant; r4=750.0; real R1 micromolar; R1=36.0; real R3 micromolar; R3=300.0; real phi1(time) first_order_rate_constant; real phi_1(time) flux; real phi2(time) first_order_rate_constant; real phi3(time) first_order_rate_constant; when(time=time.min) phi3=0.306; real H(time) dimensionless; when(time=time.min) H=0.787; real g_beta first_order_rate_constant; g_beta=300.0; // Var below replaced by constant in model eqns to satisfy unit correction // real h_beta dimensionless; // h_beta=2.0; real g_alpha per_second_squared; g_alpha=0.02; real K_beta micromolar; K_beta=2.0; real alpha_phi3 per_second_squared; real beta_phi3(time) first_order_rate_constant; real P micromolar; P=1.0; real lambda_MT_S1 dimensionless; real lambda_ER_S1 dimensionless; real delta_s micromolar_coulomb; delta_s=26.0; real gamma_S1 dimensionless; gamma_S1=100.0; real gamma_MT dimensionless; gamma_MT=200.0; real gamma_ER dimensionless; gamma_ER=20.0; real lambda_MT_S2 dimensionless; real lambda_ER_S2 dimensionless; real lambda_S1_S2 dimensionless; real gamma_S2 dimensionless; gamma_S2=1.0; real fm(time) dimensionless; real Km micromolar; Km=0.01; real Bm micromolar; Bm=100.0; // // Vm:time=((-1)*((I_iCa+I_iNa)/Cm)); I_iCa=(I_Ca+I_NSCC_Ca+I_PM); I_iNa=(I_NSCC_Na+I_Na); // ECa=(R*T/(2*F)*ln(CO/CS1)); I_Ca=(gCa*(Vm-ECa)); // gNSCC_Ca=(gNSCC_Ca_*(KNSCC^3/(KNSCC^3+CS1^3))); I_NSCC_Ca=(gNSCC_Ca*(Vm-ENSCC)); // gNSCC_Na=(gNSCC_Na_*(KNSCC^3/(KNSCC^3+CS1^3))); I_NSCC_Na=(gNSCC_Na*(Vm-ENSCC)); // I_PM=(gPM*(CS1^2/(KPM^2+CS1^2))); // I_Na=(gNa*(NS1^4/(KNa^4+NS1^4))); // was_JSERCA=(VSERCA*(CS1-A2*CER)/(1+A4*CS1+A5*CER+A6*CS1*CER)); // JMCU=(VMCU*(CS2^2/(KMCU^2+CS2^2))*epsilon_INH); epsilon_INH=(KINH^4/(KINH^4+CMT^4)); // JNCX=(VNCX*(CMT/(CMT+KNCX))); // was_JS1S2=(mu_S1S2*(CS2-CS1)); // H:time=(phi3*(1-H)-P*phi1*phi2/(P*phi1+phi_1)*H); JIPR=(kIPR*(P*phi1*H/(P*phi1+phi_1))^4*(CER-CS2)); phi1=((k1*R1+r2*CS2)/(R1+CS2)); phi_1=((k_1+r_2)*R3/(R3+CS2)); phi2=((k2*R3+r4*CS2)/(R3+CS2)); phi3:time=(alpha_phi3-beta_phi3*phi3); alpha_phi3=g_alpha; beta_phi3=(g_beta*(CS2^2/(K_beta^2+CS2^2))); // CS1:time=(was_JS1S2+lambda_MT_S1*JNCX-(delta_s/2*I_iCa+lambda_ER_S1*was_JSERCA)); lambda_MT_S1=(gamma_MT/gamma_S1); lambda_ER_S1=(gamma_ER/gamma_S1); // CS2:time=(lambda_ER_S2*JIPR-(lambda_S1_S2*was_JS1S2+lambda_MT_S2*JMCU)); lambda_MT_S2=(gamma_MT/gamma_S2); lambda_ER_S2=(gamma_ER/gamma_S2); lambda_S1_S2=(gamma_S1/gamma_S2); // CER:time=(was_JSERCA-JIPR); // CMT:time=(fm*(JMCU-JNCX)); fm=(1/(1+Km*Bm/(Km+CMT)^2)); // NS1:time=((-1)*(delta_s/1)*I_iNa); // }