/* * Intracellular Ca2+ Spikes in Non-excitable Cell Models * * Model Status * * This model produces an oscillating output but this result has * not been checked against the publication. * * ValidateCellML verifies this model as valid CellML, but detects * some unit inconsistencies. * * Model Structure * * Intracellular Ca2+ ions are essential for the initiation of * cellular events such as the fertilisation of eggs, muscle contraction * and hormone and peptide secretion. In excitable cells such as * neurons, an increase in intracellular Ca2+ is brought about * by the Ca2+ channels in the plasma membrane when these channels * open during depolarisation, permitting extracellular Ca2+ to * enter the cell. By contrast, in non-excitable cells Ca2+ is * supplied mainly by internal stores. These stores release Ca2+ * when neurotransmitters and hormones engage in the phospholinositide * signalling pathway. * * Oscillatory signals in [Ca2+]i in non-excitable cells are frequency * encoded. These patterns are similar to action potential responses * observed in excitable cells upon stimulation. A model exclusively * studied in this paper is based on the receptor-operated model * of Cuthberson and Chay (1991) (see below). The essence of this * model is that the resting period between the Ca2+ spikes are * timed by the build up of GTP-binding proteins combined with * positive feedback effects and co-operation, leading to the sudden * activation of phospholipase C (PLC). Activated PLC converts * phosphatidylinositol 4,5-bisphosphate (PIP2) into two second * messengers, diacylglycerol (DAG) and inositol 1,4,5-triphosphate * (IP3), which releases Ca2+ from intracellular stores (see below). * * The complete original paper reference is cited below: * * Appearance of phase-locked wenckebach-like rhythms, devil's * staircase and universality in intracellular calcium spikes in * non-excitable cell models, Teresa Ree Chay, Young Seek Lee and * Yin Shui Fan, 1995, Journal Of Theoretical Biology , 174, 21-44. * PubMed ID: 7643604 * * The raw CellML description of the intracellular Ca2+ spikes * in non-excitable cells model can be downloaded in various formats * as described in . * * A schematic diagram of the G-protein pathway * * [[Image file: chay_1995a.png]] * * A schematic diagram of the Bourne-Stryer mechanism, PLC acts * as a protein activating G-alpha-GTP. PLC is an effector which * is activated when bound to G-alpha-GTP. The exchange of GTP * for bound GDP is catalysed by the hormone-receptor complex R*, * with the help of the G-beta-gamma subunit. Hydrolysis of the * bound GTP brings the G-protein back to the inactive state. GTP * hydrolysis by the G-alpha_GTP complexed with PLC is much faster * than that of G-alpha-GTP alone. The model is based on these * four reaction steps. * * A diagram of the G-protein signaling pathway * * [[Image file: chay_1995b.png]] * * The Ca2+-phosphatidylinositol signalling pathway plays a major * role in transmembrane signalling in a large number of different * cell types. In this pathway hormonal stimuli lead to the activation * of G-proteins as an effector system. This initiates PLC activation * and the subsequent formation of DAG and IP3 which triggers the * generation of repetitive [Ca2+]i spikes varying in frequency, * amplitude and duration depending on the strength and the type * of the extracellular agonist. */ import nsrunit; // Warning: unit conversion turned off due to unit errors in 6 equation(s) unit conversion off; // unit nanomolar predefined unit fifth_order_rate_constant=1E24 meter^12*second^(-1)*mole^(-4); unit first_order_rate_constant=1 second^(-1); unit second_order_rate_constant=1E6 meter^3*second^(-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: G_alpha_GTP, DAG, G_alpha_GDP, PLC_active, // Ca_i realDomain time second; time.min=0; extern time.max; extern time.delta; real G_alpha_GTP(time) nanomolar; //Warning: Assuming zero initial condition; nothing provided in original CellML model. when(time=time.min) G_alpha_GTP=0; real kp(time) first_order_rate_constant; real KD nanomolar; KD=25.0; real kg first_order_rate_constant; kg=0.005; real kp_ fifth_order_rate_constant; kp_=2.0E-7; real hg first_order_rate_constant; hg=0.0; real PLC(time) nanomolar; real DAG(time) nanomolar; //Warning: Assuming zero initial condition; nothing provided in original CellML model. when(time=time.min) DAG=0; real G_alpha_GDP(time) nanomolar; //Warning: Assuming zero initial condition; nothing provided in original CellML model. when(time=time.min) G_alpha_GDP=0; real kd(time) first_order_rate_constant; real kd_ first_order_rate_constant; kd_=700.0; real hd first_order_rate_constant; hd=100.0; real ld second_order_rate_constant; ld=250.0; real PLC_active(time) nanomolar; //Warning: Assuming zero initial condition; nothing provided in original CellML model. when(time=time.min) PLC_active=0; real Ca_i(time) nanomolar; //Warning: Assuming zero initial condition; nothing provided in original CellML model. when(time=time.min) Ca_i=0; real rho_kc second_order_rate_constant; rho_kc=90000.0; real rho_hc first_order_rate_constant; rho_hc=1.0; real rho_lc second_order_rate_constant; rho_lc=200.0; real IP3(time) nanomolar; real Ks nanomolar; Ks=300.0; real hp(time) first_order_rate_constant; real hp_ first_order_rate_constant; hp_=0.5; real PLC_total nanomolar; PLC_total=10.0; real G_total nanomolar; G_total=200.0; // // G_alpha_GTP:time=(kg*G_alpha_GDP-(4*kp*G_alpha_GTP^4*PLC+hg*G_alpha_GTP)); kp=(kp_*(DAG^2/(KD^2+DAG^2))); // DAG:time=(kd*PLC_active-hd*DAG+ld); kd=(kd_*(DAG^2/(KD^2+DAG^2))); // Ca_i:time=(rho_kc*(IP3^3/(Ks^3+IP3^3))-rho_hc*Ca_i+rho_lc); IP3=DAG; // PLC_active:time=(kp*G_alpha_GTP^4*PLC-hp*PLC_active); hp=(hp_*(DAG^2/(KD^2+DAG^2))); // PLC=(PLC_total-PLC_active); // G_alpha_GDP:time=(G_total-(G_alpha_GTP+4*PLC_active)); }