429 lines
14 KiB
Python
429 lines
14 KiB
Python
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 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)
|