/* * Calcium-activated K+ Channels 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 * * Pancreatic beta cells of the islets of Langerhans play an important * role in the control of blood glucose levels by secreting insulin * in response to an increase in extracellular glucose concentration. * Increased glucose concentration also induces rhythmical electrical * activity in beta cells. This activity consists of the periodic * plateaus and spikes of Ca2+-dependent action potentials, which * enhance Ca2+ influx and insulin secretion. Two KCa channels * are present in beta cells: a voltage dependent, "fast", large * conductance KCa, and a slowly activating and deactivating KCa * current, Kslow. Kslow is thought to be responsible for islet * pacemaking. * * Experimental data suggest that both cytosolic calcium and calcium * from the endoplasmic reticulum (ER) play an essential role in * the activation of Kslow channels. In their 2002 paper, P.B. * Goforth, R. Bertram, F.A. Khan, M. Zhang, A. Sherman, and L.S. * Satin develop two mathematical models to test this theory (see * below). Their first minimal model lacks a separate subspace, * and they found that this two component model (cytosol and ER) * was insufficient to explain Ca2+-induced activation of Kslow * channels. However, their second model incorporated a localised * pool of Ca2+ in a subspace located between the plasma membrane * and the ER. In this model, subspace [Ca2+] is positively correlated * with changes in cytosolic [Ca2+], but with a gradient that reflects * Ca2+ efflux from the ER. The simulations from this model were * consistent with experimental data. * * The complete original paper reference is cited below: * * Calcium-activated K+ Channels of Mouse Beta-cells are Controlled * by Both Store and Cytoplasmic Ca2+: Experimental and Theoretical * Studies, P.B. Goforth, R. Bertram, F.A. Khan, M. Zhang, A. Sherman, * and L.S. Satin, 2002, Journal of General Physiology, 120, 307-322. * (Full text and PDF versions of the article are available to * subscribers on the Journal of General Physiology website.) PubMed * ID: 12198088 * * The raw CellML descriptions of the model can be downloaded in * various formats as described in . * * cell_diagram * * [[Image file: goforth_2002.png]] * * Schematic diagram of the model of Kslow activation by Ca2+. * Ca2+ entering the beta-cell through voltage gated Ca2+ channels * (ICa) is either pumped out of the cell by the Ca2+ ATPase (JPMCA), * or by Na+/Ca2+ exchange, or it is pumped into the ER by SERCA * (JSERCA). Ca2+ leaves the ER through specialised channels close * to the Kslow channels (JRELEASE and JX). This produces a local * calcium concentration gradient in the subspace. Subspace calcium * is passively exchanged with cytosolic calcium. */ import nsrunit; // Warning: unit conversion turned off due to unit errors in 1 equation(s) unit conversion off; // unit millisecond predefined unit per_millisecond=1E3 second^(-1); unit femtoA=1E-15 ampere^1; unit femtoF=1E-15 kilogram^(-1)*meter^(-2)*second^4*ampere^2; // unit millivolt predefined // unit micromolar predefined unit flux=1 meter^(-3)*second^(-1)*mole^1; unit picoS=1E-12 kilogram^(-1)*meter^(-2)*second^3*ampere^2; unit micromolar_per_fA_ms=1E15 meter^(-3)*second^(-1)*ampere^(-1)*mole^1; unit coulomb_per_mole=1 second^(-1)*ampere^(-1)*mole^1; // unit microlitre predefined math main { //Warning: the following variables had initial values which were // suppressed because the model would otherwise be overdetermined: // alpha realDomain time millisecond; time.min=0; extern time.max; extern time.delta; real V(time) millivolt; when(time=time.min) V=-65.0; real Cm femtoF; Cm=5300.0; real i_Ca(time) femtoA; real i_K(time) femtoA; real i_KATP(time) femtoA; real i_KCa(time) femtoA; real i_Leak(time) femtoA; real V_Ca millivolt; V_Ca=30.0; real g_Ca picoS; g_Ca=1450.0; real m_infinity(time) dimensionless; real Vm millivolt; Vm=-13.0; real Sm millivolt; Sm=8.0; real V_K millivolt; V_K=-70.0; real g_K picoS; g_K=2500.0; real n(time) dimensionless; when(time=time.min) n=0.0001; real n_infinity(time) dimensionless; real tau_n millisecond; tau_n=10.8; real Vn millivolt; Vn=-15.0; real Sn millivolt; Sn=5.6; real g_KCa picoS; g_KCa=1200.0; real c_SS(time) micromolar; when(time=time.min) c_SS=0.29; real w(time) dimensionless; real Kd micromolar; Kd=0.7; // Var below replaced by constant in model eqns to satisfy unit correction // real q dimensionless; // q=8.0; real g_KATP picoS; g_KATP=63.0; real g_Leak picoS; g_Leak=14.0; real V_Leak millivolt; V_Leak=-30.0; real c(time) micromolar; when(time=time.min) c=0.027; real c_ER(time) micromolar; when(time=time.min) c_ER=111.17; real f_CYT dimensionless; f_CYT=0.01; real f_ER dimensionless; f_ER=0.005; real f_SS dimensionless; f_SS=0.04; real alpha micromolar_per_fA_ms; //Warning: CellML initial value suppressed to prevent overdetermining model. Original initial value: alpha=4.5E-6; real F coulomb_per_mole; F=96845.0; real V_CYT microlitre; real V_ER microlitre; real V_SS microlitre; real J_PMCA(time) flux; real J_SERCA(time) flux; real J_RELEASE(time) flux; real J_X(time) flux; real k_PMCA per_millisecond; k_PMCA=0.18; real k_SERCA per_millisecond; k_SERCA=0.1; real p_ER per_millisecond; p_ER=0.0015; real p_X per_millisecond; p_X=0.025; // // V:time=((-1)*(i_Ca+i_K+i_KATP+i_KCa+i_Leak)/Cm); // i_Ca=(g_Ca*m_infinity*(V-V_Ca)); // m_infinity=(1/(1+exp((Vm-V)/Sm))); // i_K=(g_K*n*(V-V_K)); // n:time=((n_infinity-n)/tau_n); n_infinity=(1/(1+exp((Vn-V)/Sn))); // i_KCa=(g_KCa*w*(V-V_K)); // w=(c_SS^8/(c_SS^8+Kd^8)); // i_KATP=(g_KATP*(V-V_K)); // i_Leak=(g_Leak*(V-V_Leak)); // c:time=(f_CYT*((-1)*(alpha*i_Ca+J_PMCA+J_SERCA)+J_X)); alpha=(1/(2*F*V_CYT)); V_CYT=(25*V_ER); V_SS=(.4*V_CYT); V_ER=(.1*V_SS); c_ER:time=(f_ER*(V_CYT/V_ER*J_SERCA-J_RELEASE)); c_SS:time=(f_SS*(V_ER/V_SS*J_RELEASE-V_CYT/V_SS*J_X)); // J_PMCA=(k_PMCA*c); J_SERCA=(k_SERCA*c); J_RELEASE=(p_ER*(c_ER-c_SS)); J_X=(p_X*(c_SS-c)); }