/* * Calcium Waves in Differentiated Neuroblastoma Cells * * Model Status * * This is the original unchecked version of the model imported * from the previous CellML model repository, 24-Jan-2006. * * Model Structure * * Intracellular calcium dynamics are frequently the subject of * theoretical mathematical models (De Young and Keizer, 1992, * Li and Rinzel, 1994, Keizer and Levine, 1996, Jafri-Rice-Winslow, * 1998, and Snyder et al., 2000 are just a few examples of calcium * dynamic models that have been coded up into CellML). The physical * and chemical laws of calcium waves and oscillations can be expressed * in terms of differential equations describing reaction kinetics, * fluxes through membranes, and diffusion. * * In this study, Charles C. Fink et al. produce an image-based * model of a intracellular calcium wave in differentiated neuroblastoma * cells (see below). One important conclusion from their analysis * is that neuronal morphology plays a key role in controlling * and shaping the inositol-1,4,5-triphosphate (IP3) dynamics that * underlie the calcium wave. The model is comprised of several * components including: IP3 dynamics — which account for IP3 synthesis * at the plasma membrane, diffusion into the cytosol, and its * degradation. Calcium dynamics — which calculate the changing * intracellular calcium concentration. Channel kinetics — to describe * calcium release from the endoplasmic reticulum (ER) into the * cytosol through an IP3-sensitive channel. SERCA pump kinetics * — to describe calcium uptake into the ER via the sarcoplasmic * endoplasmic reticulum ATPase (SERCA) pumps. Leak — which models * calcium leak from the ER to the cytosol. and Calcium buffering * — with endogenous buffers. * * Their model is based on experimental data. The binding of bradykinin * (BK) to its receptor initiates a G-protein cascade, activation * of phospholipase C, and degradation of phosphatidylinositol * bisphosphate (PIP2) to IP3. IP3 then diffuses through the cytosol * from the plasma membrane to the ER where it activates Ca2+ release * through the IP3R channel. The concentration of cytosolic Ca2+ * rises, and is subsequently reduced as Ca2+ binds to calcium * buffers and is pumped back into the ER through the SERCA. This * Ca2+ wave was captured by Fink et al. through the use of fluorescent * microscopy. The model of this process was assembled using the * Virtual Cell, a computational system for integrating experimentally * recorded image, biochemical and electrophysiological data. The * model was tested by comparing several simulation results with * the real experimental data, and Fink et al. found that there * was good spatiotemporal agreement between the two data sets. * * It should be noted that the following CellML description (for * the raw CellML description of the model, see below) is not quite * true to the mathematical model published in the original paper * (referenced below). Currently CellML is unable to handle spatial * elements, but this will hopefully be possible in the near future * with the development of FieldML, an XML based language for spatially * variable models. This is important, as the relative positions * of the cellular components such as receptors, pumps, channels * and enzymes will determine the length of diffusion pathways * and therefore the rate of reactions. * * An Image-Based Model of Calcium Waves in Differentiated Neuroblastoma * Cells, Charles C. Fink, Boris Slepchenko, Ion I. Moraru, James * Watras, James C. Schaff, and Leslie M. Loew, 2000, Biophysical * Journal, 79, 163-183. (Full text and PDF versions of the article * are available to subscribers on the Biophysical Journal website.) * PubMed ID: 10866945 * * Schematic diagram of model * * [[Image file: fink_2000.png]] */ import nsrunit; // Warning: unit conversion turned off due to unit errors in 7 equation(s) unit conversion off; // unit micromolar predefined // unit micrometre predefined unit micromolar_micrometre_per_second=1E-9 meter^(-2)*second^(-1)*mole^1; unit micrometre_per_second=1E-6 meter^1*second^(-1); unit micrometre_2_per_second=1E-12 meter^2*second^(-1); unit flux=1E-3 meter^(-3)*second^(-1)*mole^1; unit first_order_rate_constant=1 second^(-1); unit second_order_rate_constant=1E3 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: CaB1, CaB2 realDomain time second; time.min=0; extern time.max; extern time.delta; real IP3(time) micromolar; when(time=time.min) IP3=3.0; real j_IP3(time) micromolar_micrometre_per_second; real J_IP3 micromolar_micrometre_per_second; J_IP3=20.86; real k_0 first_order_rate_constant; k_0=1.188; real k_degr first_order_rate_constant; k_degr=0.14; real IP3_0 micromolar; IP3_0=0.16; real D_IP3 micrometre_2_per_second; D_IP3=283.0; real Ca_ER micromolar; Ca_ER=400.0; real Ca(time) micromolar; when(time=time.min) Ca=0.05; real D_Ca micrometre_2_per_second; D_Ca=220.0; real alpha dimensionless; alpha=0.0; real J_channel(time) flux; real J_pump(time) flux; real J_leak(time) flux; real R_buffering flux; real J_max flux; J_max=3500.0; real h(time) dimensionless; when(time=time.min) h=0.8; real K_act micromolar; K_act=0.3; real K_IP3 micromolar; K_IP3=0.8; real K_inh micromolar; K_inh=0.2; real k_on second_order_rate_constant; k_on=2.7; real V_max flux; V_max=3.75; real K_p micromolar; K_p=0.27; real L flux; L=0.1; real R1 flux; R1=0.1; real R2 flux; R2=0.1; real B1(time) micromolar; when(time=time.min) B1=450.0; real B2(time) micromolar; when(time=time.min) B2=75.0; real CaB1(time) micromolar; //Warning: Assuming zero initial condition; nothing provided in original CellML model. when(time=time.min) CaB1=0; real CaB2(time) micromolar; //Warning: Assuming zero initial condition; nothing provided in original CellML model. when(time=time.min) CaB2=0; real k1_on(time) second_order_rate_constant; real k1_off(time) first_order_rate_constant; real k2_on(time) second_order_rate_constant; real k2_off(time) first_order_rate_constant; real K1 micromolar; K1=10.0; real K2 micromolar; K2=0.24; real D_buffer micrometre_2_per_second; D_buffer=50.0; real J_Ca(time) flux; real gamma micrometre_per_second; gamma=8.0; real Ca_c micromolar; Ca_c=0.2; // // j_IP3=(J_IP3*exp((-1)*k_0*time)); IP3:time=(D_IP3*IP3-k_degr*(IP3-IP3_0)); // // Ca:time=(D_Ca*Ca+alpha*(J_channel+(-1)*J_pump+J_leak)+R_buffering); // J_channel=(J_max*(IP3/(IP3+K_IP3)*(Ca/(Ca+K_act))*h)^3*(1-Ca/Ca_ER)); h:time=(k_on*(K_inh-h*(Ca+K_inh))); // J_pump=(V_max*(Ca^2/(Ca^2+K_p^2))); // J_leak=(L*(1-Ca/Ca_ER)); // R_buffering=(R1+R2); R1=((-1)*(k1_on*Ca*B1)+k1_off*CaB1); R2=((-1)*(k2_on*Ca*B2)+k2_off*CaB2); B1:time=R1; CaB1:time=((-1)*R1); B2:time=(D_buffer*B2+R2); CaB2:time=(D_buffer*CaB2-R2); K1=(k1_on/k1_off); K2=(k2_on/k2_off); // J_Ca=(if (Ca>Ca_c) gamma*(Ca-Ca_c) else (0 flux)); }