/* * Cooperative Effects Due to Calcium Binding by Troponin and Their * Consequences for Contraction and Relaxation of Cardiac Muscle * Under Various Conditions of Mechanical Loading * * Model Status * * This model runs in PCEnv and COR, but does not recreate the * published results. The units are continuous throughout. The * equations for F_PE and F_SE are wrong, but the F_CE equation * and all related equations (the bulk of the model) look good. * * Model Structure * * Abstract: A mathematical model for the regulation of mechanical * activity in cardiac muscle has been developed based on a three-element * rheological model of this muscle. The contractile element has * been modeled taking into account the results of extensive mechanical * tests that involved the recording of length-force and force-velocity * relations and muscle responses to short-time deformations during * various phases of the contraction-relaxation cycle. The best * agreement between the experimental and the mathematical modeling * results was obtained when a postulate stating two types of cooperativity * to regulate the calcium binding by troponin was introduced into * the model. Cooperativity of the first type is due to the dependence * of the affinity of troponin C for Ca2+ on the concentration * of myosin crossbridges in the vicinity of a given troponin C. * Cooperativity of the second type assumes an increase in the * affinity of a given troponin C for Ca2+ when the latter is bound * by molecules neighboring troponin. * * model diagram Schematic diagram of the Izakov et al model - * a classic rheological scheme of the heart muscle including contractile * element CE and two passive elastic elements PE (parallel one) * and SE (series one). The effects of calcium and troponin (Tn) * in facilitating actin-myosin binding is also highlighted. * * The complete original paper reference is cited below: * * Cooperative Effects Due to Calcium Binding by Troponin and Their * Consequences for Contraction and Relaxation of Cardiac Muscle * Under Various Conditions of Mechanical Loading, Valery Ya. Izakov, * Leonid B. Katsnelson, Felix A. Blyakhman, Vladamir S. Markhasin, * Tatyana F. Shklyar, 1991, Circulation Research, 69, 1171-1184. * PubMed ID: 1934350 */ import nsrunit; unit conversion on; unit um=1E-6 meter^1; // unit millisecond predefined unit per_um=1E6 meter^(-1); unit per_millisec=1E3 second^(-1); unit um_per_millisec=.001 meter^1*second^(-1); unit per_millisec2=1E6 second^(-2); math main { realDomain time millisecond; time.min=0; extern time.max; extern time.delta; real P_CE(time) dimensionless; real P_PE(time) dimensionless; real P_SE(time) dimensionless; real Ca(time) dimensionless; real C_2(time) per_millisec; real phi_A_1(time) dimensionless; real pi_n(time) dimensionless; real A(time) dimensionless; real A_1(time) dimensionless; when(time=time.min) A_1=0; real S(time) um; real n(time) dimensionless; real n_1(time) dimensionless; real n_2(time) dimensionless; when(time=time.min) n_2=0; real G_V dimensionless; real q_V per_millisec; real V um_per_millisec; real F_V dimensionless; real p_V dimensionless; real alpha_1 per_um; alpha_1=14.6; real alpha_2 per_um; alpha_2=14.6; real beta_1 dimensionless; beta_1=1; real beta_2 dimensionless; beta_2=0.0012; real lambda per_um; lambda=30; real V_max um_per_millisec; V_max=0.0043; real Ca_m dimensionless; Ca_m=45e-3; real t_d millisecond; t_d=170; real a_c per_millisec2; a_c=2.4e-4; real b_c per_millisec2; b_c=5e-4; real C_1 per_millisec; C_1=2.9e-2; real C_20 per_millisec; C_20=0.2; real q_k dimensionless; q_k=4; real V_1 um_per_millisec; real a dimensionless; a=0.25; real m_0 dimensionless; m_0=0.87; real g_1 per_um; g_1=0.4; real g_2 dimensionless; g_2=0.6; real pi_min dimensionless; pi_min=5e-2; real S_0 um; S_0=0.77; real q_1 per_millisec; q_1=0.017; real q_2 per_millisec; q_2=0.26; real q_3 per_millisec; q_3=0.03; real TnC dimensionless; TnC=1; real l_1(time) um; real l_2(time) um; real dl_1_dt um_per_millisec; // // V_1=(.1*V_max); Ca=(if (time<=t_d) Ca_m*(1-exp((-1)*a_c*time^2))^2 else Ca_m*((1-exp((-1)*a_c*time^2))*exp((-1)*b_c*(time-t_d)^2))^2); C_2=(C_20*pi_n*phi_A_1); pi_n=(if ((.75<=n) and (n<=1)) pi_min else if ((.25<=n) and (n<.75)) pi_min^(2*n-.5) else 1); phi_A_1=exp((-1)*q_k*A_1); S=(.5*l_1+S_0); A=(A_1*S/(1 um)); A_1:time=(C_1*Ca*(TnC-A_1)-C_20*pi_n*phi_A_1*A_1); n=(n_1*n_2); n_1=(if ((g_1*l_1+g_2)<0) 0 else if ((0<=(g_1*l_1+g_2)) and ((g_1*l_1+g_2)<=1)) g_1*l_1+g_2 else 1); n_2:time=(q_V*(m_0*G_V-n_2)); q_V=(if (V<=(0 um_per_millisec)) q_1-q_2*V/V_max else q_3); G_V=(1+.6*V/V_max); V=((-1)*dl_1_dt); F_V=(a*(1+V/V_max)/(a-V/V_max)); p_V=(F_V/G_V); P_CE=(lambda*S*A_1*n*p_V); P_PE=(beta_2*(exp(alpha_2*l_2)-1)); P_SE=(beta_1*(exp(alpha_1*(l_2-l_1))-1)); // dl_1_dt=(0 um_per_millisec); l_1=(if (((200 millisecond)<=time) and (time<=(201 millisecond))) (0 um) else (0 um)); l_2=(l_1+(1.87 um)); }