/* * Modelling Ca2+ Flux in Pancreatic Beta-cells * * Model Status * * This is the original unchecked version of the model imported * from the previous CellML model repository, 24-Jan-2006. * * Model Structure * * An increase in the concentration of intracellular free calcium * ([Ca2+]i) is an essential signal for the initiation of insulin * secretion in pancreatic beta-cells. This increase is primarily * due to the opening of Ca2+ channels in the plasma membrane in * response to glucose. Glucose metabolism leads to an increase * in the cytosolic ATP:ADP ratio, which in turn causes the ATP-sensitive * potassium channels to close. The beta-cell membrane becomes * depolarised, Ca2+ channels open, and Ca2+ enters the cell. These * events underlie the glucose-induced electrical activity, which * in pancreatic islets, consists of Ca2+-dependent action potentials. * * There is an abundance of literature that describes beta-cell * electrical activity and its relationship to [Ca2+]i. Complex * and cyclic spike-burst activity, and corresponding [Ca2+]i oscillations * in pancreatic islets and beta-cell clusters are induced in response * to a rise in extracellular glucose concentration. Intermediate * glucose concentrations induce both fast and slow oscillations. * The authors of this current study: Fridlyand, Tamarina and Philipson, * have previously studied slow and fast [Ca2+]i oscillations in * islets in response to a variety of conditions. However, the * experimental results were complex, and precise understanding * was limited by the large number of channels and pumps in the * beta-cell plasma membrane that were simultaneously working. * * In order to better understand the molecular mechanisms underlying * this behaviour, in this publication Fridlyand et al. have developed * a mathematical model of the Ca2+ fluxes in pancreatic beta-cells. * Several other mathematical models of glucose-induced insulin * secretion, with corresponding descriptions of glucose transport, * metabolism and ion regulation, have been published. These include: * * Extracellular and Intracellular Calcium Effects on Pancreatic * Beta Cells, Chay, 1997; * * Na+/Ca2+ Exchange in Models for Pancreatic Beta-Cells, Gall * and Susa, 1999; and * * The Phantom Burster Model for Pancreatic Beta-Cells, Bertram * et al., 2000. * * However, most of these models are focused on describing one * specific phenomenon. They only include a very limited set of * channels and pumps, and therefore it is difficult to apply them * to a another situation. In addition, since their publication, * new experimental data has become available, and these new findings * should be included in a theoretical model. For this reason Fridlyand * et al. have developed the new mathematical model described here * (see the raw CellML description of the Fridlyand et al. 2003 * model in ). They have adopted the more complex style of modelling * that has previously been used successfully to describe the electrophysiology * of cardiac myocytes and other cell types (for example in Human * Atrial Cell Model, Nygren et al. 1998 and in Modelling Interval-Force * Relations in Cardiac Muscle, Rice et al., 2000). * * Their new model includes a wider range of channels and pumps, * as well as endoplasmic reticulum (ER) Ca2+ sequestration mechanisms * (see below). Using this model they were able to simulate whole * cell electrical activity and [Ca2+]i, free calcium in the ER * ([Ca2+]ER), intracellular Na+ ([Na+]i), cytosolic ATP ([ATP]i), * and inositol triphosphate ([IP3]i) concentrations. However, * they acknowledge that this model does not consider metabolic * processes or insulin secretion. * * The complete original paper reference is cited below: * * Modeling of Ca2+ flux in pancreatic beta-cells: role of the * plasma membrane and intracellular stores, Leonid E. Fridlyand, * Natalia Tamarina, and Louis H. Philipson, 2003, American Journal * of Physiology, 285, E138-E154. (Full text (HTML) and PDF versions * of the article are available on the American Jounal of Physiology * website.) PubMed ID: 12644446 * * cell diagram * * [[Image file: fridlyand_2003.png]] * * Schematic representation of currents and ion fluxes, through * the plasma membrane and the endoplasmic reticulum membrane, * which have been included in the whole beta-cell mathematical * model. */ import nsrunit; // Warning: unit conversion turned off due to unit errors in 10 equation(s) unit conversion off; // unit millisecond predefined unit first_order_rate_constant=1E3 second^(-1); unit second_order_rate_constant=1E6 meter^3*second^(-1)*mole^(-1); unit third_order_rate_constant=1E9 meter^6*second^(-1)*mole^(-2); unit forth_order_rate_constant=1E12 meter^9*second^(-1)*mole^(-3); unit flux=1 meter^(-3)*second^(-1)*mole^1; // unit millivolt predefined unit per_millivolt_millisecond=1E6 kilogram^(-1)*meter^(-2)*second^2*ampere^1; unit picoS=1E-12 kilogram^(-1)*meter^(-2)*second^3*ampere^2; unit picol=1E-15 meter^3; unit picol_per_ms=1E-12 meter^3*second^(-1); unit femtoA=1E-15 ampere^1; unit femtoF=1E-15 kilogram^(-1)*meter^(-2)*second^4*ampere^2; //Warning: unit millimolar_ renamed from millimolar, as the latter is predefined in JSim with different fundamental units. unit millimolar_=1E-3 meter^(-3)*mole^1; // unit micromolar predefined unit joule_per_kilomole_kelvin=.001 kilogram^1*meter^2*second^(-2)*kelvin^(-1)*mole^(-1); unit coulomb_per_mole=1 second^1*ampere^1*mole^(-1); unit picoS_per_millivolt=1E-9 kilogram^(-2)*meter^(-4)*second^6*ampere^3; math main { //Warning: the following variables were set 'extern' or given // an initial value of '0' because the model would otherwise be // underdetermined: K_supi, k_ADP realDomain time millisecond; time.min=0; extern time.max; extern time.delta; real V(time) millivolt; when(time=time.min) V=-84.624; real Cm femtoF; Cm=6158.0; real i_VCa(time) femtoA; real i_Ca_pump(time) femtoA; real i_NaCa(time) femtoA; real i_CRAN(time) femtoA; real i_Na(time) femtoA; real i_NaK(time) femtoA; real i_KDr(time) femtoA; real i_KCa(time) femtoA; real i_KATP(time) femtoA; real V_Ca(time) millivolt; real gm_VCa picoS; gm_VCa=770.0; real R joule_per_kilomole_kelvin; R=8314.0; real F coulomb_per_mole; F=96500.0; real T kelvin; T=310.0; real Cai(time) micromolar; when(time=time.min) Cai=0.085; real Cao micromolar; Cao=2.6; real p_VCa(time) dimensionless; real V_Cah millivolt; V_Cah=-19.0; real K_Cah millivolt; K_Cah=9.5; real Pm_Cap femtoA; Pm_Cap=2000.0; real K_Cap micromolar; K_Cap=0.1; real V_NaCa(time) millivolt; real g_NaCa picoS; g_NaCa=271.0; real K_NaCa micromolar; K_NaCa=0.75; real Nai(time) micromolar; when(time=time.min) Nai=9858.0; real Nao micromolar; Nao=140.0; real V_CRAN millivolt; V_CRAN=0.0; real gm_CRAN picoS_per_millivolt; gm_CRAN=0.7; real K_Car micromolar; K_Car=200.0; real f_CRAN(time) dimensionless; real P_CRAN(time) millivolt; real V_Na(time) millivolt; real CaER(time) micromolar; when(time=time.min) CaER=22.8; real gm_Na picoS; gm_Na=1200.0; real p_Na(time) dimensionless; real P_NaK femtoA; P_NaK=600.0; real P micromolar; P=4950.0; extern real K_supi micromolar; real F1(time) first_order_rate_constant; real F4 first_order_rate_constant; real F5(time) first_order_rate_constant; real f1 forth_order_rate_constant; f1=2.5E-10; real f2 first_order_rate_constant; f2=10.0; real f3 first_order_rate_constant; f3=0.172; real f4 third_order_rate_constant; f4=1.5E-8; real f5_ second_order_rate_constant; f5_=0.002; real f5(time) second_order_rate_constant; real f6 first_order_rate_constant; f6=11.5; real b1 first_order_rate_constant; b1=100.0; real b2 second_order_rate_constant; b2=0.0001; real b3 forth_order_rate_constant; b3=1.72E-17; real b4 second_order_rate_constant; b4=0.0002; real b5_ first_order_rate_constant; b5_=0.03; real b5(time) first_order_rate_constant; real b6 second_order_rate_constant; b6=6.0E-7; real B2(time) first_order_rate_constant; real B3(time) first_order_rate_constant; real B4 first_order_rate_constant; real B6 first_order_rate_constant; real D(time) dimensionless; real ATPi(time) micromolar; when(time=time.min) ATPi=932.1; real ADPi(time) micromolar; real Ko micromolar; Ko=8.0; real V_K millivolt; real gm_KDr picoS; gm_KDr=3000.0; real Ki millimolar_; Ki=132.4; real n(time) dimensionless; when(time=time.min) n=0.00123; real tau_n(time) dimensionless; real n_infinity(time) dimensionless; real Vn millivolt; Vn=-14.0; real Vtau millivolt; Vtau=-75.0; real Sn millivolt; Sn=7.0; real a millivolt; a=65.0; real b millivolt; b=20.0; real c millisecond; c=20.0; real gm_KCa picoS; gm_KCa=130.0; real fCa(time) dimensionless; real K_KCa micromolar; K_KCa=0.1; real gm_KATP picoS; gm_KATP=24000.0; real O_KATP(time) dimensionless; real Kdd micromolar; Kdd=17.0; real Ktd micromolar; Ktd=26.0; real Ktt micromolar; Ktt=1.0; real Jer_p(time) flux; real P_CaER flux; P_CaER=0.105; real K_Carp micromolar; K_Carp=0.5; real IP3i(time) micromolar; when(time=time.min) IP3i=0.33; real kIP flux; kIP=0.0003; real kdIP first_order_rate_constant; kdIP=0.00004; real K_IPCa micromolar; K_IPCa=0.4; real J_out(time) flux; real P_leak picol_per_ms; P_leak=0.0001; real P_IP3 picol_per_ms; P_IP3=0.0012; real O_infinity(time) dimensionless; real K_RCa micromolar; K_RCa=3.2; real K_IP3 micromolar; K_IP3=0.077; real fi dimensionless; fi=0.01; real fer dimensionless; fer=0.03; real ksg first_order_rate_constant; ksg=0.0001; real Ver picol; Ver=0.280; real Vi picol; Vi=0.764; extern real k_ADP first_order_rate_constant; real k_ATP first_order_rate_constant; k_ATP=0.00005; real k_ATP_Ca second_order_rate_constant; k_ATP_Ca=0.00005; real Ao micromolar; Ao=4000.0; // // V:time=((i_VCa+i_Ca_pump+i_NaCa+i_CRAN+i_Na+i_NaK+i_KDr+i_KCa+i_KATP)/((-1)*Cm)); // i_VCa=(gm_VCa*p_VCa*(V-V_Ca)); V_Ca=(R*T/(2*F)*ln(Cao/Cai)); // p_VCa=(1/(1+exp((V_Cah-V)/K_Cah))); // i_Ca_pump=(Pm_Cap*(Cai^2/(K_Cap^2+Cai^2))); // i_NaCa=(g_NaCa*(Cai^5/(Cai^5+K_NaCa^5))*(V-V_NaCa)); V_NaCa=(R*T/F*(3*ln(Nao/Nai)-ln(Cao/Cai))); // i_CRAN=(f_CRAN*P_CRAN*(V-V_Na)); f_CRAN=((-1)*gm_CRAN/(1+exp((CaER-K_Car)/3))); P_CRAN=(V-V_CRAN); // i_Na=(gm_Na*p_Na*(V-V_Na)); V_Na=(R*T/F*ln(Nao/Nai)); // p_Na=(1/(1+exp(((104 millivolt)+V)/(8 millivolt)))); // i_NaK=(P_NaK*((F1*f2*f3*F4*F5*f6-b1*B2*B3*B4*b5*B6)/D)); D=(f2*f3*F4*F5*f6+b1*f3*F4*F5*f6+b1*B2*F4*F5*f6+b1*B2*B3*F5*f6+b1*B2*B3*B4*f6+b1*B2*B3*B4*b5); F1=(f1*Nai^3); F4=(f4*Ko^2); F5=(f5*ATPi); f5=(f5_*exp(V*F/(2*R*T))); B2=(b2*ATPi); B3=(b3*Nai^3); B4=(b4*P); B6=(b6*K_supi^2); b5=(b5_*exp((-1)*(V*F)/(2*R*T))); // i_KDr=(gm_KDr*n*(V-V_K)); V_K=(R*T/F*ln(Ko/Ki)); // n:time=((n_infinity-n)/tau_n); n_infinity=(1/(1+exp((Vn-V)/Sn))); tau_n=(c/(exp((V-Vtau)/a)+exp((Vtau-V)/b))); // i_KCa=(gm_KCa*fCa*(V-V_K)); // fCa=(Cai^4/(Cai^4+K_KCa^4)); // i_KATP=(gm_KATP*O_KATP*(V-V_K)); O_KATP=((.08*(1+.33*ADPi/Kdd)+.89*(.165+ADPi/Kdd)^2)/((.165+ADPi/Kdd)^2*(1+.135*ADPi/Ktd+.05*ATPi/Ktt))); // Jer_p=(P_CaER*(Cai^2/(Cai^2+K_Carp^2))); // IP3i:time=(kIP*(Cai^2/(Cai^2+K_IPCa^2))-kdIP*IP3i); // J_out=((P_leak+P_IP3*O_infinity)*(CaER-Cai)); O_infinity=(Cai/(Cai+K_RCa)*(IP3i^3/(IP3i^3+K_IP3^3))); // Cai:time=(fi*(((-1)*i_VCa+i_NaCa*2-i_Ca_pump*2)/(2*F*Vi)-Jer_p+J_out/Vi)-ksg*Cai); CaER:time=(fer/Ver*(Jer_p*Vi-J_out)); Nai:time=((-1)*(3*i_NaCa+3*i_NaK+i_Na+i_CRAN)/(Vi*F)); // ATPi:time=(k_ADP*ADPi-((i_NaK+i_CRAN)/(Vi*F)+Jer_p/2+(k_ATP_Ca*Cai+k_ATP)*ATPi)); ADPi=(Ao-ATPi); // }