import numpy as np import matplotlib.pyplot as plt from scipy.integrate import ode R = 8.314 # ideal gas constant (J/(mol*K)) F = 96.485 # coulomb_per_millimole, Faraday constant T = 298 # room temperature (K) RT = R * T class Model: def __init__(self): self.A_cap = 1.534e-4 # cm^2, Capacitive membrane area self.C_mem = 1.0 # uF/cm2 Specific membrane capacitance self.V_myo = 25.84e-6 # uL, Myoplasmic volume self.current2flux = self.A_cap * self.C_mem / 2 / F # self.period = 1000 # ms, pulse period self.V_mem_rest = -80.0 # mV, resting membrane potential self.Nai = 11000 # uM, Myoplasmic Na+ concentration self.Nao = 150000 # uM, Extracellular Na+ concentration self.eta = 0.35 # Controls voltage dependance of Na/Ca2+ exchange self.km_Na = 87500 # uM, Na+ half-saturation constant for Na+/Ca2+ exchange self.k_sat = ( 0.1 # Na+/Ca2+ exchange saturation factor at very negative potentials ) self.km_Ca = (1380,) # uM, Ca2+ half-saturation constant for Na+/Ca2+ exchange self.k_Na_Ca = 292.8 # pA/pF, Scaling factor of Na2+/Ca2+ exchange self.I_max_pCa = 1.0 # pA/pF, Maximum Ca2+ pump current # self.Cai = 0.1 # uM, Cytoplasmic Ca2+ concentration fixed at this point self.LTRPN_tot = ( 70.0 # uM, Total myoplasmic troponin low-affinity site concentration ) self.HTRPN_tot = ( 140.0 # uM, Total myoplasmic troponin high-affinity site concentration ) # self.LTRPNCa = 11.2684 # uM, Concentration Ca2+ bound low-affinity troponin-binding sites # self.HTRPNCa = 125.290 # uM, Concentration Ca2+ bound high-affinity troponin-binding sites self.k_htrpn_positive = ( 0.00237 # uM^(-1)/ms, Ca2+ on rate const. for troponin high-affinity sites ) self.k_htrpn_negative = ( 3.2e-5 # ms, Ca2+ off rate const. for troponin high-affinity sites ) self.k_ltrpn_positive = ( 0.0327 # uM^(-1)/ms, Ca2+ on rate const. for troponin low-affinity sites ) self.k_ltrpn_negative = ( 0.0196 # ms, Ca2+ off rate const. for troponin low-affinity sites ) self.CMDN_tot = 50 # uM, Total myoplasmic calmodulin concentration self.Km_CMDN = 0.238 # uM, Ca2 half-saturation constant for calmodulin self.nu_1 = 4.5 # 1/ms, Maximum RyR channel Ca2+ permeability Ca2+ leak rate constant from the NSR self.nu_2 = 1.74e-5 # ms^(-1), Ca2+ leak rate const. from the NSR self.nu_3 = 0.45 # uM/ms, SR Ca2+ -ATPase maximum pupmp rate self.Km_up = 0.5 # uM, Half-saturation constant for SR Ca2+ -ATPase pump self.km_p_ca = 0.5 # uM, Ca2+ half-saturation constant for Ca2+ pump current # self.Ca_NSR = 1299.50 # uM,NSR Ca2+ concentration self.tau_xfer = 8.0 # ms, Time constant for transfer from subspace to myoplasm self.K_pc_max = ( 0.23324 # 1/ms, Maximum time constant for Ca2+-induced inactivation ) self.K_pc_half = ( 20.0 # uM, Half-saturation constant for Ca2+-induced inactivation ) self.Kpcb = 0.0005 # 1/ms, Voltage-insensitive rate constant for inactivation self.km_Ca = 1380 # uM, Ca2+ half-saturation constant for Na+/Ca2+ exchange self.Gcab = 0.000367 # mS/uF self.Cao = 1130.0 # uM, Ca2+ outside the cell self.gCaL = ( 0.1729 # mS/uF, Specific maximum conductivity for L-type Ca2+ channel ) self.ECaL = 43.0 # mV, Reversal potential for L-type Ca2⫹ channel, kas arvutame voi jaab const? self.V_ss = 1.485e-9 # uL, Dyadic aka subspace volume # self.Ca_JSR = 1299.50 # uM, JSR Ca2+ concentration self.F_tot = 25 # uM, total concentration of Fluo-4 self.k_on = 0.1 # 1/uM * 1/ms, Fluo-4 reaction rate constant self.k_off = 0.11 # 1/ms, Fluo-4 reaction rate constant 2 self.V_NSR = 2.098e-6 # ul, Network SR volume self.tau_tr = 20 # ms, Time const for transfer from NSR to JSR self.CSQN_tot = 15000.0 # uM, total junctional SR calsequestrin concentration self.Km_CSQN = 800.0 # uM, Ca2 half-saturation constant for calsequestrin self.k_a_positive = 0.006075 # uM^(-4)/ms, RyR Pc1 - Po1 rate constant self.k_a_negative = 0.07125 # 1/ms, RyR Po1 - Pc1 rate constant self.k_b_positive = 0.00405 # uM^(-3)/ms, RyR Po1 - Po2 rate constant self.k_b_negative = 0.965 # 1/ms, RyR Po2 - Po1 rate constant self.k_c_positive = 0.009 # 1/ms, RyR Po1 - Pc2 rate constant self.k_c_negative = 0.0008 # 1/ms, RyR Pc2 - Po1 rate constant self.n_ryr = 4 # RyR Ca2+ cooperativity parameter Pc1 - Po1 self.m_ryr = 3 # RyR Ca2+ cooperativity parameter Po1 - Po2 self.I_CaL_max = 7.0 # pA/pF, normalization constant for L-type Ca2+ current self.V_JSR = 0.12e-6 # ul, Junctional SR volume def ode_system(self, t, states): ( Cass, Os, C2, C3, C4, I1, I2, I3, LTRPNCa, HTRPNCa, Ca_NSR, Ca_JSR, P_RyR, P_O1, P_O2, P_C2, Cai, FCa, ) = states V = self.mem_potential(t) VF = V * F Kpcf = 13.0 * (1 - np.exp(-((V + 14.5) ** 2) / 100)) alpha = ( # dimensionless parameter for L-type Ca2+ channel closed states 0.4 * np.exp((V + 12.0) / 10) * ( 1 + 0.7 * np.exp((-((V + 40.0) ** 2)) / 10.0) - 0.75 * np.exp(-((V + 20.0) ** 2) / 400.0) ) ) / (1 + 0.12 * np.exp((V + 12.0) / 10.0)) beta = 0.05 * np.exp( -(V + 12.0) / 13.0 ) # parameter for L-type Ca2+ channel closed states gamma = self.K_pc_max * Cass / (self.K_pc_half + Cass) C1 = 1 - (Os + C2 + C3 + C4 + I1 + I2 + I3) dC2dt = 4 * alpha * C1 - beta * C2 + 2 * beta * C3 - 3 * alpha * C2 dC3dt = 3 * alpha * C2 - 2 * beta * C3 + 3 * beta * C4 - 2 * alpha * C3 dC4dt = ( 2 * alpha * C3 - 3 * beta * C4 + 4 * beta * Os - alpha * C4 + 0.01 * (4 * self.Kpcb * beta * I1 - alpha * gamma * C4) + 0.002 * (4 * beta * I2 - Kpcf * C4) + 4 * beta * self.Kpcb * I3 - gamma * Kpcf * C4 ) dI1dt = ( gamma * Os - self.Kpcb * I1 + 0.001 * (alpha * I3 - Kpcf * I1) + 0.01 * (alpha * gamma * C4 - 4 * beta * self.Kpcb * I1) ) dI2dt = 0.001 * (Kpcf * Os - alpha * I2) +self.Kpcb * I3 - gamma * I2 + 0.002 * (Kpcf * C4 - 4 * beta * I2) dI3dt = ( 0.001 * (Kpcf * I1 - alpha * I3) + gamma * I2 - self.Kpcb * I3 + gamma * Kpcf * C4 - 4 * beta * self.Kpcb * I3 ) # Bi = (1 + (self.CMDN_tot * self.Km_CMDN)/(self.Km_CMDN+(Cai))**2)**(-1) # Bss = 1 / ((self.CMDN_tot * self.Km_CMDN) / (self.Km_CMDN + Cass) ** 2) Bi = 1 / ( 1 + (self.CMDN_tot * self.Km_CMDN) / (self.Km_CMDN + Cai) ** 2 ) Bss = 1 / ( 1 + (self.CMDN_tot * self.Km_CMDN) / (self.Km_CMDN + Cass) ** 2 ) J_xfer = (Cass - Cai) / self.tau_xfer dLTRPNCadt = self.k_ltrpn_positive * Cai * (self.LTRPN_tot - LTRPNCa) -self.k_ltrpn_negative * (LTRPNCa) dHTRPNCadt = self.k_htrpn_positive * Cai * (self.HTRPN_tot - HTRPNCa) -self.k_htrpn_negative * (HTRPNCa) J_trpn = ( self.k_htrpn_positive * Cai * (self.HTRPN_tot - HTRPNCa) - self.k_htrpn_negative * HTRPNCa + self.k_ltrpn_positive * Cai * (self.LTRPN_tot - LTRPNCa) - self.k_ltrpn_negative * LTRPNCa ) J_up = self.nu_3 * (Cai**2 / (self.Km_up**2 - Cai**2)) J_tr = (Ca_NSR - Ca_JSR) / self.tau_tr J_leak = self.nu_2 * (Ca_NSR - Cai) dCa_NSRdt = (J_up - J_leak) * (self.V_myo / self.V_NSR) -J_tr * (self.V_JSR / self.V_NSR) B_JSR = 1 / (1 + (self.CSQN_tot * self.Km_CSQN) / (self.Km_CSQN + Ca_JSR) ** 2) J_rel = 0 # self.nu_1*(P_O1+P_O2)*(Ca_JSR-Cass)*P_RyR dCa_JSRdt = B_JSR * (J_tr - J_rel) b = 1 / ( (self.km_Na**3 + self.Nao**3) * (self.km_Ca + self.Cao) * (1 + self.k_sat * np.exp((self.eta - 1) * VF / RT)) ) p = ( np.exp(self.eta * VF / RT) * self.Nai**3 * self.Cao - np.exp((self.eta - 1) * VF / RT) * self.Nao**3 * Cai ) I_NaCa = self.k_Na_Ca * b * p I_pCa = self.I_max_pCa * ((Cai) ** 2 / ((self.km_p_ca) ** 2 + (Cai) ** 2)) I_CaL = self.gCaL * Os * (V - self.ECaL) dP_RyRdt = -0.04 * P_RyR - 0.1 * (I_CaL / self.I_CaL_max) * np.exp( -((V - 5.0) ** 2) / 648 ) P_C1 = 1 - (P_C2 + P_O1 + P_O2) dP_O1dt = ( self.k_a_positive * (Cass) ** self.n_ryr * P_C1 ) -self.k_a_negative * P_O1 - self.k_a_positive * (Cass) ** self.m_ryr * P_O1 +self.k_b_negative * P_O2 - self.k_c_positive * P_O1 + self.k_c_negative * P_C2 dP_O2dt = ( self.k_b_positive * (Cass) ** self.m_ryr * P_O1 - self.k_b_negative * P_O2 ) dP_C2dt = self.k_c_positive * P_O1 - self.k_c_negative * P_C2 dCassdt = Bss * ( -J_xfer * self.V_myo / self.V_ss - I_CaL * self.current2flux / self.V_ss ) dOsdt = alpha * C4 - 4 * beta * Os + self.Kpcb * I1 -gamma * Os + 0.001 * (alpha * I2 - Kpcf * Os) ECaN = (R * T) / F * np.log(self.Cao / Cai) I_Cab = self.Gcab * (V - ECaN) """ dCaidt = Bi * (J_leak + J_xfer - J_up - J_trpn - (I_Cab - 2 * I_NaCa + I_pCa) * ((self.A_cap * self.C_mem)/(2 * self.V_myo * F))) """ # J_rel_caf = self.nu_1 * (self.Ca_JSR - Cass) #lisatud ette ennetavalt JFCa = self.k_on * (self.F_tot - FCa) * Cai - self.k_off * FCa dFCadt = JFCa dCaidt = Bi * ( J_leak + J_xfer - J_up - J_trpn - JFCa - (I_Cab - 2 * I_NaCa + I_pCa) * ((self.A_cap * self.C_mem) / (2 * self.V_myo * F)) ) # Cass, Os, C2, C3, C4, I1, I2, I3, LTRPNCa, HTRPNCa, Ca_NSR, Ca_JSR, P_RyR, P_O1, P_O2, P_C2, Cai, FCa = states return [ dCassdt, dOsdt, dC2dt, dC3dt, dC4dt, dI1dt, dI2dt, dI3dt, dLTRPNCadt, dHTRPNCadt, dCa_NSRdt, dCa_JSRdt, dP_RyRdt, dP_O1dt, dP_O2dt, dP_C2dt, dCaidt, dFCadt, ] def mem_potential(self, t): v1 = 0 t0 = 100 t1 = 350 n = (t // self.period) * self.period return v1 if t0 + n <= t < t1 + n else self.V_mem_rest def get_initial_values(self): Cai_0 = 0.11712 # Cai FCa_0 = (self.k_on * self.F_tot * Cai_0) / (self.k_on * Cai_0 + self.k_off) return [ Cai_0, # Cass 0.930308e-18, # Os, 0.124216e-3, # C2, 0.578679e-8, # C3, 0.119816e-12, # C4, 0.497923e-18, # I1, 0.345847e-13, # I2, 0.185106e-13, # I3, 11.2684, # LTRPNCa, 125.290, # HTRPNCa 1299.50, # Ca_NSR 1299.50, # Ca_JSR 0.0, # P_RyR 0.149102e-4, # P_O1 0.951726e-10, # P_02 0.167740e-3, # P_C2 Cai_0, # Cai FCa_0, # FCa ] def solve(self, initial_values, tspan, dt, times): times = np.arange(*tspan, dt) r = ode(self.ode_system) r.set_integrator( "lsoda", method="bdf", atol=1e-07, rtol=1e-07, max_step=0.1, nsteps=500 ) r.set_initial_value(initial_values, times[0]) states = np.array([[0.0] * times.size] * len(initial_values)) states[:, 0] = initial_values for i, t in enumerate(times[1:]): if r.successful(): r.integrate(t) states[:, i + 1] = r.y else: break V = np.array([self.mem_potential(t) for t in times]) # plt.plot(times, V) fig = plt.figure() ax1 = fig.add_subplot(141) ax2 = fig.add_subplot(142) ax3 = fig.add_subplot(143) ax4 = fig.add_subplot(144) ax1.plot(times, states[0, :], label="Cass") Cai = states[-2, :] ax1.plot(times, (states[0, :] - Cai) / self.tau_xfer, label="Jxfer") ax1.plot( times, -self.gCaL * states[1, :] * (V - self.ECaL) * self.current2flux / self.V_ss / 1000, label="JCaL mM/s", ) ax1.legend(frameon=False) ax1.set_xlabel("time [ms]") ax1.set_ylabel(r"$\mu mol/l$") ax2.plot(times, self.gCaL * states[1, :] * (V - self.ECaL), label="ICaL ") ax2.legend(frameon=False) ax2.set_xlabel("time [ms]") ax2.set_ylabel("pA/pF") ax3.plot(times, Cai, label="Cai ") ax3.legend(frameon=False) ax3.set_xlabel("time [ms]") ax3.set_ylabel(r"$\mu mol/3l$") ax4.plot(times, states[-1, :], label="Fca") ax4.legend(frameon=False) ax4.set_xlabel("time [ms]") ax4.set_ylabel(r"$\mu mol/3l$") plt.show() if __name__ == "__main__": # Cass, Os, C2, C3, C4, I1, I2, I3, LTRPNCa, HTRPNCa, Cai, FCa= states model = Model() Cai_0 = 0.11712 # Cai FCa_0 = (model.k_on * model.F_tot * Cai_0) / (model.k_on * Cai_0 + model.k_off) print(FCa_0) # Cass, Os, C2, C3, C4, I1, I2, I3, LTRPNCa, HTRPNCa, Ca_NSR, Ca_JSR, P_RyR, P_O1, P_O2, P_C2, Cai, FCa = states initial_values = [ Cai_0, # Cass 0.930308e-18, # Os, 0.124216e-3, # C2, 0.578679e-8, # C3, 0.119816e-12, # C4, 0.497923e-18, # I1, 0.345847e-13, # I2, 0.185106e-13, # I3, 11.2684, # LTRPNCa, 125.290, # HTRPNCa 1299.50, # Ca_NSR 1299.50, # Ca_JSR 0.0, # P_RyR 0.149102e-4, # P_O1 0.951726e-10, # P_02 0.167740e-3, # P_C2 Cai_0, # Cai FCa_0, # FCa ] model.solve(initial_values=initial_values, tspan=[0, 1000], dt=1.0, times=None)