/* * A Mathematical Model of the Generation of Action Potentials * in Corticotrophs * * Model Status * * This model is not currently functional. * * Model Structure * * Corticotropin-releaseing hormone (CRH) is an important regulator * of adrenocorticotropin (ACTH) secretion from the pituitary corticotroph * cells. CRH induces the secretion of ACTH through the actication * of the cAMP second messenger system, which results in the activation * of protein kinase A (PKA). Secretion of ACTH alo requires an * influx of Ca2+, which occurs mainly through voltage-sensitive * Ca2+ channels. Corticotrophs generate both spontaneous and CRG-induced * action potentials. L-type voltage-sensitive Ca2+ channels are * the main channel type that underlie Ca2+-induced action potential * generation. It is likely that following activation by CRH, PKA * phosphorylates the L-type channel and promotes Ca2+ action potential * generation with subsequent Ca2+ influx. The rise in the intracellular * concentration of Ca2+ ([Ca2+]i), then leads to the activation * of exocytotic pathways, resulting in the secretion of ACTH. * * Although the PKA-induced action potential activity is known * to play an important role in this secretory pathway, the mechanism * by which PKA activates the L-type Ca2+ channel is currently * unknown. In the publication described here, LeBeau et al. investigate * PKA regulation of the L-type Ca2+ channel. They develop a Hodgkin-Huxley-type * mathematical model of action potential generation in corticotrophs * (see below). The model includes descriptions of four plasma * membrane ionic channels, which allows the analysis of the roles * of each channel type in corticotroph electrical responses. * * The complete original paper reference is cited below: * * Generation of Action Potentials in a Mathematical Model of Corticotrophs, * Andrew P. LeBeau, A. Bruce Robson, Alan E. McKinnon, Richard * A. Donald, and James Sneyd, 1997,Biophysical Journal , 73, 1263-1275. * PubMed ID: 9284294 * * cell diagram * * [[Image file: lebeau_1997.png]] * * Schematic diagram of the model of a corticotroph. The arrows * represent ionic currents and fluxes across the plama membrane * and across the membrane of the endoplasmic reticulum. * * Using model simulations, the authors found that an increase * in the L-type Ca2+ current was sufficient to generate action * potentials from a previously resting state. The favoured mechanism * which was thought to underlie this increase in the L-type Ca2+ * current was a shift in the voltage-dependence of the current * towards more negative potentials. The model also showed that * the T-type Ca2+ current plays a role in establishing the excitability * of the plasma membrane, but it doesn't plauy a major role in * action potential generation. * * The model has been described here in CellML (the raw CellML * description of the LeBeau et al. 1997 model can be downloaded * in various formats as described in ). */ import nsrunit; // Warning: unit conversion turned off due to unit errors in 9 equation(s) unit conversion off; // unit micrometre predefined unit per_micrometre=1E6 meter^(-1); unit micrometre2=1E-12 meter^2; unit picoL=1E-15 meter^3; unit picoF=1E-12 kilogram^(-1)*meter^(-2)*second^4*ampere^2; unit picoA=1E-12 ampere^1; // unit micromolar predefined // unit millimolar predefined unit micromolar_micrometre_per_s_pA=1E3 meter^(-2)*second^(-1)*ampere^(-1)*mole^1; unit nanoS_per_millimolar=1E-9 kilogram^(-1)*meter^1*second^3*ampere^2*mole^(-1); unit nanoS=1E-9 kilogram^(-1)*meter^(-2)*second^3*ampere^2; // unit millivolt predefined // unit millisecond predefined unit flux=1E3 meter^(-3)*second^(-1)*mole^1; unit micromolar_micrometre_per_s=1E-9 meter^(-2)*second^(-1)*mole^1; unit joule_per_mole_kelvin=1 kilogram^1*meter^2*second^(-2)*kelvin^(-1)*mole^(-1); unit coulomb_per_mole=.001 second^(-1)*ampere^(-1)*mole^1; math main { //Warning: the following variables were set 'extern' or given // an initial value of '0' because the model would otherwise be // underdetermined: V, Cai, L_type_calcium_channel_current.m, // T_type_calcium_channel_current.m, h, n realDomain time millisecond; time.min=0; extern time.max; extern time.delta; real V(time) millivolt; //Warning: Assuming zero initial condition; nothing provided in original CellML model. when(time=time.min) V=0; real Cm picoF; Cm=7.0; real i_CaT(time) picoA; real i_CaL(time) picoA; real i_K_DR(time) picoA; real i_K_Ca(time) picoA; real i_Leak(time) picoA; real E_Ca(time) millivolt; real E_K(time) millivolt; real Ki millimolar; Ki=140.0; real Ke millimolar; Ke=5.6; real Cae millimolar; Cae=2.0; real R joule_per_mole_kelvin; R=8.3144; real T kelvin; T=310.15; real F coulomb_per_mole; F=96485; real Cai(time) millimolar; //Warning: Assuming zero initial condition; nothing provided in original CellML model. when(time=time.min) Cai=0; real g_CaL nanoS_per_millimolar; g_CaL=9.0; real L_type_calcium_channel_current.m(time) dimensionless; //Warning: Assuming zero initial condition; nothing provided in original CellML model. when(time=time.min) L_type_calcium_channel_current.m=0; real L_type_calcium_channel_current_m_gate.m_infinity(time) dimensionless; real L_type_calcium_channel_current_m_gate.tau_m(time) millisecond; real L_type_calcium_channel_current_m_gate.tau_m_ millisecond; L_type_calcium_channel_current_m_gate.tau_m_=27.0; real g_CaT nanoS_per_millimolar; g_CaT=10.0; real T_type_calcium_channel_current.m(time) dimensionless; //Warning: Assuming zero initial condition; nothing provided in original CellML model. when(time=time.min) T_type_calcium_channel_current.m=0; real h(time) dimensionless; //Warning: Assuming zero initial condition; nothing provided in original CellML model. when(time=time.min) h=0; real T_type_calcium_channel_current_m_gate.m_infinity(time) dimensionless; real T_type_calcium_channel_current_m_gate.tau_m(time) millisecond; real T_type_calcium_channel_current_m_gate.tau_m_ millisecond; T_type_calcium_channel_current_m_gate.tau_m_=10.0; real h_infinity(time) dimensionless; real tau_h millisecond; tau_h=15.0; real g_K_DR nanoS_per_millimolar; g_K_DR=0.1; real n(time) dimensionless; //Warning: Assuming zero initial condition; nothing provided in original CellML model. when(time=time.min) n=0; real n_infinity(time) dimensionless; real tau_n millisecond; tau_n=20.0; real g_K_Ca nanoS_per_millimolar; g_K_Ca=0.09; real KC micromolar; KC=0.4; real g_L nanoS; g_L=0.3; real E_L millivolt; E_L=-67.0; real f dimensionless; f=0.01; real beta per_micrometre; beta=0.4; real j_exch(time) flux; real j_in(time) flux; real j_eff(time) flux; real tau millisecond; tau=500.0; real alpha micromolar_micrometre_per_s_pA; alpha=7.4; real vp micromolar_micrometre_per_s; vp=40.0; real Kp micromolar; Kp=0.08; real Ca_eq micromolar; Ca_eq=0.1; // // V:time=((-1)*(i_CaL+i_CaT+i_K_DR+i_K_Ca+i_Leak)/Cm); // E_Ca=(V*((Cai-Cae*exp((-1)*(2*F*V/(R*T))))/(1-exp((-1)*(2*F*V/(R*T)))))); E_K=(V*((Ki-Ke*exp((-1)*(F*V/(R*T))))/(1-exp((-1)*(F*V/(R*T)))))); // i_CaL=(g_CaL*L_type_calcium_channel_current.m^2*E_Ca); // L_type_calcium_channel_current_m_gate.m_infinity=(1/(1+exp((-1)*((V-((-12 millivolt)))/(12 millivolt))))); L_type_calcium_channel_current_m_gate.tau_m=(L_type_calcium_channel_current_m_gate.tau_m_/(exp((V-((-60 millivolt)))/(22 millivolt))+2*exp((-1)*(2*((V-((-60 millivolt)))/(22 millivolt)))))); L_type_calcium_channel_current.m:time=((L_type_calcium_channel_current_m_gate.m_infinity-L_type_calcium_channel_current.m)/L_type_calcium_channel_current_m_gate.tau_m); // i_CaT=(g_CaT*T_type_calcium_channel_current.m^2*h*E_Ca); // T_type_calcium_channel_current_m_gate.m_infinity=(1/(1+exp((-1)*((V-((-30 millivolt)))/(10.5 millivolt))))); T_type_calcium_channel_current_m_gate.tau_m=(T_type_calcium_channel_current_m_gate.tau_m_/(exp((V-((-60 millivolt)))/(22 millivolt))+2*exp((-1)*(2*((V-((-60 millivolt)))/(22 millivolt)))))); T_type_calcium_channel_current.m:time=((T_type_calcium_channel_current_m_gate.m_infinity-T_type_calcium_channel_current.m)/T_type_calcium_channel_current_m_gate.tau_m); // h_infinity=(1/(1+exp((V-((-57 millivolt)))/(5 millivolt)))); h:time=((h_infinity-h)/tau_h); // i_K_DR=(g_K_DR*n*E_K); // n_infinity=(1/(1+exp((-1)*((V-((-20 millivolt)))/(4.5 millivolt))))); n:time=((n_infinity-n)/tau_n); // i_K_Ca=(g_K_Ca*(Cai^4/(Cai^4+KC^4))*E_K); // i_Leak=(g_L*(V-E_L)); // Cai:time=(j_exch+f*beta*(j_in-j_eff)); // j_exch=((Ca_eq-Cai)/tau); j_in=((-1)*alpha*(i_CaL+i_CaT)); j_eff=(vp*(Cai^2/(Cai^2+Kp^2))); }