Model Mathematics

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Component: cell

Component: wal_environment

I_T = \frac{1000}{1} \frac{vS - vT}{R_a}

\frac{d}{d time} (vS) = - (\frac{I_ionic_s + I_T}{C_m})

\frac{d}{d time} (vT) = - (\frac{I_ionic_t - (\frac{I_T}{gam})}{C_m})

I_ionic_s = I_Cl + I_IR + I_DR + I_Na + I_NaK + (- I_HH)

I_ionic_t = I_Cl_t + I_IR_t + I_DR_t + I_Na_t + I_NaK_t

\frac{d}{d time} (K_i) = (- f_T) \frac{I_IR_t + I_DR_t + I_K_rest + (- 2) I_NaK_t}{\frac{1000}{1} FF tsi} - (\frac{I_IR + I_DR + I_K_rest + (-2) I_NaK}{\frac{1000}{1} FF tsi2})

\frac{d}{d time} (K_t) = \frac{I_IR_t + I_DR_t + I_K_rest + (- 2) I_NaK_t}{\frac{1000}{1} FF tsi} - (\frac{K_t - K_e}{tau_K})

\frac{d}{d time} (K_e) = \frac{I_IR + I_DR + I_K_rest + (- 2) I_NaK}{\frac{1000}{1} FF tsi3} + \frac{K_t - K_e}{tau_K2}

\frac{d}{d time} (Na_i) = (- f_T) \frac{I_Na_t + I_Na_rest + 3 I_NaK_t}{\frac{1000}{1} FF tsi} - (\frac{I_Na + I_Na_rest + 3 I_NaK}{\frac{1000}{1} FF tsi2})

\frac{d}{d time} (Na_t) = \frac{I_Na_t + I_Na_rest + 3 I_NaK_t}{\frac{1000}{1} FF tsi} - (\frac{Na_t - Na_e}{tau_Na})

\frac{d}{d time} (Na_e) = \frac{I_Na + I_Na_rest + 3 I_NaK}{\frac{1000}{1} FF tsi3} + \frac{Na_t - Na_e}{tau_Na2}

E_K = \frac{RR TT}{FF} \ln (\frac{K_e}{K_i})

E_K_t = \frac{RR TT}{FF} \ln (\frac{K_t}{K_i})

Cl_i = \frac{156.5}{5 + ⅇ^{\frac{(- FF) E_K}{RR TT}}}

Cl_o = 156.5 - (5 Cl_i)

Cl_i_t = \frac{156.5}{5 + ⅇ^{\frac{(- FF) E_K_t}{RR TT}}}

Cl_o_t = 156.5 - (5 Cl_i_t)

J_K = vS \frac{K_i - (K_e ⅇ^{\frac{(-1) FF vS}{RR TT}})}{1 - (ⅇ^{\frac{(-1) FF vS}{RR TT}})}

J_K_t = vT \frac{K_i - (K_t ⅇ^{\frac{(-1) FF vT}{RR TT}})}{1 - (ⅇ^{\frac{(-1) FF vT}{RR TT}})}

I_HH = \{\begin{matrix} 150 if time \geq 0 \land time < 0.5 \\ 150 if time \geq 50 \land time < 50.5 \\ 150 if time \geq 100 \land time < 100.5 \\ 150 if time \geq 150 \land time < 150.5 \\ 150 if time \geq 200 \land time < 200.5 \\ 150 if time \geq 250 \land time < 250.5 \\ 150 if time \geq 300 \land time < 300.5 \\ 150 if time \geq 350 \land time < 350.5 \\ 150 if time \geq 400 \land time < 400.5 \\ 0 otherwise \end{matrix}

Component: sarco_Cl_channel

a = \frac{1}{1 + ⅇ^{\frac{vS - V_a}{A_a}}}

J_Cl = vS \frac{Cl_i - (Cl_o ⅇ^{\frac{FF vS}{RR TT}})}{1 - (ⅇ^{\frac{FF vS}{RR TT}})}

g_Cl = g_Cl_bar a^{4}

I_Cl = g_Cl \frac{J_Cl}{45}

Component: sarco_IR_channel

K_R = K_e ⅇ^{(- del) E_K \frac{FF}{RR TT}}

g_IR_bar = G_K \frac{{K_R}^{2}}{K_K + {K_R}^{2}}

y = 1 - ({(1 + \frac{K_S (1 + \frac{{K_R}^{2}}{K_K})}{{S_i}^{2} ⅇ^{\frac{2 (1 - del) vS FF}{RR TT}}})}^{-1})

g_IR = g_IR_bar y

I_IR = g_IR \{\begin{matrix} 1 if J_K > 0 \\ 0 otherwise \end{matrix} \frac{J_K}{50}

Component: sarco_DR_channel

alpha_n = alpha_n_bar \frac{vS - V_n}{1 - (ⅇ^{- (\frac{vS - V_n}{K_alpha_n})})}

beta_n = beta_n_bar ⅇ^{- (\frac{vS - V_n}{K_beta_n})}

h_K_inf = \frac{1}{1 + ⅇ^{\frac{vS - V_h_K_inf}{A_h_K_inf}}}

tau_h_K = 1000 ⅇ^{- (\frac{vS + 40}{25.75})}

\frac{d}{d time} (n) = alpha_n (1 - n) - (beta_n n)

\frac{d}{d time} (h_K) = \frac{h_K_inf - h_K}{tau_h_K}

g_DR = g_K_bar n^{4} h_K

I_DR = g_DR \frac{J_K}{50}

Component: sarco_Na_channel

alpha_h = alpha_h_bar ⅇ^{- (\frac{vS - V_h}{K_alpha_h})}

beta_h = \frac{beta_h_bar}{1 + ⅇ^{- (\frac{vS - V_h}{K_beta_h})}}

alpha_m = alpha_m_bar \frac{vS - V_m}{1 - (ⅇ^{- (\frac{vS - V_m}{K_alpha_m})})}

beta_m = beta_m_bar ⅇ^{- (\frac{vS - V_m}{K_beta_m})}

S_inf = \frac{1}{1 + ⅇ^{\frac{vS - V_S_inf}{A_S_inf}}}

tau_S = \frac{8571}{0.2 + 5.65 {(\frac{vS + V_tau}{100})}^{2}}

J_Na = vS \frac{Na_i - (Na_e ⅇ^{\frac{(-1) FF vS}{RR TT}})}{1 - (ⅇ^{\frac{(-1) FF vS}{RR TT}})}

\frac{d}{d time} (m) = alpha_m (1 - m) - (beta_m m)

\frac{d}{d time} (h) = alpha_h (1 - h) - (beta_h h)

\frac{d}{d time} (S) = \frac{S_inf - S}{tau_S}

g_Na = g_Na_bar m^{3} h S

I_Na = g_Na \frac{J_Na}{75}

Component: sarco_NaK_channel

sig = \frac{1}{7} (ⅇ^{\frac{Na_e}{67.3}} - 1)

f1 = {(1 + 0.12 ⅇ^{(-0.1) vS \frac{FF}{RR TT}} + 0.04 sig ⅇ^{- (vS \frac{FF}{RR TT})})}^{-1}

I_NaK_bar = FF \frac{J_NaK_bar}{{(1 + \frac{K_m_K}{K_e})}^{2} {(1 + \frac{K_m_Na}{Na_i})}^{3}}

I_NaK = I_NaK_bar f1

Component: t_Cl_channel

a_t = \frac{1}{1 + ⅇ^{\frac{vT - V_a}{A_a}}}

J_Cl_t = vT \frac{Cl_i_t - (Cl_o_t ⅇ^{\frac{FF vT}{RR TT}})}{1 - (ⅇ^{\frac{FF vT}{RR TT}})}

g_Cl_t = g_Cl_bar {a_t}^{4}

I_Cl_t = eta_Cl g_Cl_t \frac{J_Cl_t}{45}

Component: t_IR_channel

K_R_t = K_t ⅇ^{(- del) E_K_t \frac{FF}{RR TT}}

g_IR_bar_t = G_K \frac{{K_R_t}^{2}}{K_K + {K_R_t}^{2}}

y_t = 1 - ({(1 + \frac{K_S (1 + \frac{{K_R_t}^{2}}{K_K})}{{S_i}^{2} ⅇ^{\frac{2 (1 - del) vT FF}{RR TT}}})}^{-1})

g_IR_t = g_IR_bar_t y_t

I_IR_t = eta_IR g_IR_t \frac{J_K_t}{50}

Component: t_DR_channel

alpha_n_t = alpha_n_bar \frac{vT - V_n}{1 - (ⅇ^{- (\frac{vT - V_n}{K_alpha_n})})}

beta_n_t = beta_n_bar ⅇ^{- (\frac{vT - V_n}{K_beta_n})}

h_K_inf_t = \frac{1}{1 + ⅇ^{\frac{vT - V_h_K_inf}{A_h_K_inf}}}

tau_h_K_t = 1 ⅇ^{- (\frac{vT + 40}{25.75})}

\frac{d}{d time} (n_t) = alpha_n_t (1 - n_t) - (beta_n_t n_t)

\frac{d}{d time} (h_K_t) = \frac{h_K_inf_t - h_K_t}{tau_h_K_t}

g_DR_t = g_K_bar {n_t}^{4} h_K_t

I_DR_t = eta_DR g_DR_t \frac{J_K_t}{50}

Component: t_Na_channel

alpha_h_t = alpha_h_bar ⅇ^{- (\frac{vT - V_h}{K_alpha_h})}

beta_h_t = \frac{beta_h_bar}{1 + ⅇ^{- (\frac{vT - V_h}{K_beta_h})}}

alpha_m_t = alpha_m_bar \frac{vT - V_m}{1 - (ⅇ^{- (\frac{vT - V_m}{K_alpha_m})})}

beta_m_t = beta_m_bar ⅇ^{- (\frac{vT - V_m}{K_beta_m})}

S_inf_t = \frac{1}{1 + ⅇ^{\frac{vT - V_S_inf}{A_S_inf}}}

tau_S_t = \frac{8571}{0.2 + 5.65 {(\frac{vT + V_tau}{100})}^{2}}

J_Na_t = vT \frac{Na_i - (Na_t ⅇ^{\frac{(-1) FF vT}{RR TT}})}{1 - (ⅇ^{\frac{(-1) FF vT}{RR TT}})}

\frac{d}{d time} (m_t) = alpha_m_t (1 - m_t) - (beta_m_t m_t)

\frac{d}{d time} (h_t) = alpha_h_t (1 - h_t) - (beta_h_t h_t)

\frac{d}{d time} (S_t) = \frac{S_inf_t - S_t}{tau_S_t}

g_Na_t = g_Na_bar {m_t}^{3} h_t S_t

I_Na_t = eta_Na g_Na_t \frac{J_Na_t}{75}

Component: t_NaK_channel

sig_t = \frac{1}{7} (ⅇ^{\frac{Na_t}{67.3}} - 1)

f1_t = {(1 + 0.12 ⅇ^{(-0.1) vT \frac{FF}{RR TT}} + 0.04 sig_t ⅇ^{- (vT \frac{FF}{RR TT})})}^{-1}

I_NaK_bar_t = FF \frac{J_NaK_bar}{{(1 + \frac{K_m_K}{K_t})}^{2} {(1 + \frac{K_m_Na}{Na_i})}^{3}}

I_NaK_t = eta_NaK I_NaK_bar_t f1_t

Component: sternrios

k_C = 0.5 alpha1 ⅇ^{\frac{vT - Vbar}{8 K}}

k_Cm = 0.5 alpha1 ⅇ^{\frac{Vbar - vT}{8 K}}

\frac{d}{d time} (C_0) = (- k_L) C_0 + k_Lm O_0 + (-4) k_C C_0 + k_Cm C_1

\frac{d}{d time} (O_0) = k_L C_0 + (- k_Lm) O_0 + \frac{(-4) k_C O_0}{f} + f k_Cm O_1

\frac{d}{d time} (C_1) = 4 k_C C_0 + (- k_Cm) C_1 + \frac{(- k_L) C_1}{f} + f k_Lm O_1 + (-3) k_C C_1 + 2 k_Cm C_2

\frac{d}{d time} (O_1) = \frac{k_L C_1}{f} + (- k_Lm) f O_1 + \frac{4 k_C O_0}{f} + (- f) k_Cm O_1 + \frac{(-3) k_C O_1}{f} + 2 f k_Cm O_2

\frac{d}{d time} (C_2) = 3 k_C C_1 + (-2) k_Cm C_2 + \frac{(- k_L) C_2}{f^{2}} + f^{2} k_Lm O_2 + (-2) k_C C_2 + 3 k_Cm C_3

\frac{d}{d time} (O_2) = \frac{3 k_C O_1}{f} + (-2) f k_Cm O_2 + \frac{k_L C_2}{f^{2}} + (- k_Lm) f^{2} O_2 + \frac{(-2) k_C O_2}{f} + 3 f k_Cm O_3

\frac{d}{d time} (C_3) = 2 k_C C_2 + (-3) k_Cm C_3 + \frac{(- k_L) C_3}{f^{3}} + k_Lm f^{3} O_3 + (- k_C) C_3 + 4 k_Cm C_4

\frac{d}{d time} (O_3) = \frac{k_L C_3}{f^{3}} + (- k_Lm) f^{3} O_3 + \frac{2 k_C O_2}{f} + (-3) k_Cm f O_3 + \frac{(- k_C) O_3}{f} + 4 f k_Cm O_4

\frac{d}{d time} (C_4) = k_C C_3 + (-4) k_Cm C_4 + \frac{(- k_L) C_4}{f^{4}} + k_Lm f^{4} O_4

\frac{d}{d time} (O_4) = \frac{k_C O_3}{f} + (-4) f k_Cm O_4 + \frac{k_L C_4}{f^{4}} + (- k_Lm) f^{4} O_4

Component: razumova

V_o = 0.95 L_x π {R_R}^{2}

V_1 = 0.01 V_o

V_2 = 0.99 V_o

V_SR = 0.05 L_x π {R_R}^{2}

V_SR1 = 0.01 V_SR

V_SR2 = 0.99 V_SR

T_0 = T_tot + (- Ca_T_2) + (- Ca_CaT2) + (- D_0) + (- D_1) + (- D_2) + (- A_1) + (- A_2)

\frac{d}{d time} (Ca_1) = (i2 (O_0 + O_1 + O_2 + O_3 + O_4) \frac{Ca_SR1 - Ca_1}{V_1} - (nu_SR \frac{\frac{Ca_1}{Ca_1 + K_SR}}{V_1})) + L_e \frac{Ca_SR1 - Ca_1}{V_1} + (- tau_R) \frac{Ca_1 - Ca_2}{V_1} + (- (k_P_on Ca_1 (P_tot + (- Ca_P1) + (- Mg_P1)) + (- k_P_off) Ca_P1)) + (- (k_CATP_on Ca_1 ATP1 + (- k_CATP_off) Ca_ATP1))

\frac{d}{d time} (Ca_SR1) = (- (i2 (O_0 + O_1 + O_2 + O_3 + O_4))) \frac{Ca_SR1 - Ca_1}{V_SR1} + nu_SR \frac{\frac{Ca_1}{Ca_1 + K_SR}}{V_SR1} + (- L_e) \frac{Ca_SR1 - Ca_1}{V_SR1} + (- tau_SR_R) \frac{Ca_SR1 - Ca_SR2}{V_SR1} + (- (k_Cs_on Ca_SR1 (Cs_tot - Ca_Cs1) + (- k_Cs_off) Ca_Cs1))

\frac{d}{d time} (Ca_2) = (- nu_SR) \frac{\frac{Ca_2}{Ca_2 + K_SR}}{V_2} + L_e \frac{Ca_SR2 + (- Ca_2)}{V_2} + tau_R \frac{Ca_1 - Ca_2}{V_2} + (- (k_T_on Ca_2 T_0 + (- k_T_off) Ca_T_2 + k_T_on Ca_2 Ca_T_2 + (- k_T_off) Ca_CaT2 + k_T_on Ca_2 D_0 + (- k_T_off) D_1 + k_T_on Ca_2 D_1 + (- k_T_off) D_2)) + (- (k_P_on Ca_2 (P_tot + (- Ca_P2) + (- Mg_P2)) + (- k_P_off) Ca_P2)) + (- (k_CATP_on Ca_2 ATP2 + (- k_CATP_off) Ca_ATP2))

\frac{d}{d time} (Ca_SR2) = nu_SR \frac{\frac{Ca_2}{Ca_2 + K_SR}}{V_SR2} + (- L_e) \frac{Ca_SR2 + (- Ca_2)}{V_SR2} + tau_SR_R \frac{Ca_SR1 + (- Ca_SR2)}{V_SR2} + (- (k_Cs_on Ca_SR2 (Cs_tot + (- Ca_Cs2)) + (- k_Cs_off) Ca_Cs2)) - (\frac{1000}{1} (A_p (P_SR \frac{0.001}{1} Ca_SR2 - PP) \{\begin{matrix} 1 if P_SR \frac{0.001}{1} Ca_SR2 - PP > 0 \\ 0 otherwise \end{matrix} \frac{0.001}{1} P_SR Ca_SR2 - (B_p P_C_SR (PP - (P_SR \frac{0.001}{1} Ca_SR2)) \{\begin{matrix} 1 if PP - (P_SR \frac{0.001}{1} Ca_SR2) > 0 \\ 0 otherwise \end{matrix})))

\frac{d}{d time} (Ca_T_2) = k_T_on Ca_2 T_0 + (- k_T_off) Ca_T_2 + (- k_T_on) Ca_2 Ca_T_2 + k_T_off Ca_CaT2 + (- k_0_on) Ca_T_2 + k_0_off D_1

\frac{d}{d time} (Ca_P1) = k_P_on Ca_1 (P_tot + (- Ca_P1) + (- Mg_P1)) + (- k_P_off) Ca_P1

\frac{d}{d time} (Ca_P2) = k_P_on Ca_2 (P_tot + (- Ca_P2) + (- Mg_P2)) + (- k_P_off) Ca_P2

\frac{d}{d time} (Mg_P1) = k_Mg_on (P_tot + (- Ca_P1) + (- Mg_P1)) Mg1 + (- k_Mg_off) Mg_P1

\frac{d}{d time} (Mg_P2) = k_Mg_on (P_tot + (- Ca_P2) + (- Mg_P2)) Mg2 + (- k_Mg_off) Mg_P2

\frac{d}{d time} (Ca_Cs1) = k_Cs_on Ca_SR1 (Cs_tot + (- Ca_Cs1)) + (- k_Cs_off) Ca_Cs1

\frac{d}{d time} (Ca_Cs2) = k_Cs_on Ca_SR2 (Cs_tot + (- Ca_Cs2)) + (- k_Cs_off) Ca_Cs2

\frac{d}{d time} (Ca_ATP1) = k_CATP_on Ca_1 ATP1 + (- k_CATP_off) Ca_ATP1 + (- tau_ATP) \frac{Ca_ATP1 + (- Ca_ATP2)}{V_1}

\frac{d}{d time} (Ca_ATP2) = k_CATP_on Ca_2 ATP2 + (- k_CATP_off) Ca_ATP2 + tau_ATP \frac{Ca_ATP1 + (- Ca_ATP2)}{V_2}

\frac{d}{d time} (Mg_ATP1) = k_MATP_on Mg1 ATP1 + (- k_MATP_off) Mg_ATP1 + (- tau_ATP) \frac{Mg_ATP1 + (- Mg_ATP2)}{V_1}

\frac{d}{d time} (Mg_ATP2) = k_MATP_on Mg2 ATP2 + (- k_MATP_off) Mg_ATP2 + tau_ATP \frac{Mg_ATP1 + (- Mg_ATP2)}{V_2}

\frac{d}{d time} (ATP1) = (- (k_CATP_on Ca_1 ATP1 + (- k_CATP_off) Ca_ATP1)) + (- (k_MATP_on Mg1 ATP1 + (- k_MATP_off) Mg_ATP1)) + (- tau_ATP) \frac{ATP1 + (- ATP2)}{V_1}

\frac{d}{d time} (ATP2) = (- (k_CATP_on Ca_2 ATP2 + (- k_CATP_off) Ca_ATP2)) + (- (k_MATP_on Mg2 ATP2 + (- k_MATP_off) Mg_ATP2)) + tau_ATP \frac{ATP1 + (- ATP2)}{V_2}

\frac{d}{d time} (Mg1) = (- (k_Mg_on (P_tot + (- Ca_P1) + (- Mg_P1)) Mg1 + (- k_Mg_off) Mg_P1)) + (- (k_MATP_on Mg1 ATP1 + (- k_MATP_off) Mg_ATP1)) + (- tau_Mg) \frac{Mg1 + (- Mg2)}{V_1}

\frac{d}{d time} (Mg2) = (- (k_Mg_on (P_tot + (- Ca_P2) + (- Mg_P2)) Mg2 + (- k_Mg_off) Mg_P2)) + (- (k_MATP_on Mg2 ATP2 + (- k_MATP_off) Mg_ATP2)) + tau_Mg \frac{Mg1 + (- Mg2)}{V_2}

\frac{d}{d time} (Ca_CaT2) = k_T_on Ca_2 Ca_T_2 + (- k_T_off) Ca_CaT2 + (- k_Ca_on) Ca_CaT2 + k_Ca_off D_2

\frac{d}{d time} (D_0) = (- k_T_on) Ca_2 D_0 + k_T_off D_1 + k_0_on T_0 + (- k_0_off) D_0

\frac{d}{d time} (D_1) = k_T_on Ca_2 D_0 + (- k_T_off) D_1 + k_0_on Ca_T_2 + (- k_0_off) D_1 + (- k_T_on) Ca_2 D_1 + k_T_off D_2

\frac{d}{d time} (D_2) = k_T_on Ca_2 D_1 + (- k_T_off) D_2 + k_Ca_on Ca_CaT2 + (- k_Ca_off) D_2 + (- f_o) D_2 + f_p A_1 + g_o A_2

\frac{d}{d time} (A_1) = f_o D_2 + (- f_p) A_1 + h_p A_2 + (- h_o) A_1

\frac{d}{d time} (A_2) = (- h_p) A_2 + h_o A_1 + (- g_o) A_2

\frac{d}{d time} (P) = \frac{0.001}{1} (h_o A_1 - (h_p A_2)) + (-1) b_p P + (-1) k_p \frac{P - P_SR}{V_2}

\frac{d}{d time} (P_SR) = k_p \frac{P - P_SR}{V_SR2} - (1 (A_p (P_SR \frac{0.001}{1} Ca_SR2 - PP) \{\begin{matrix} 1 if P_SR \frac{0.001}{1} Ca_SR2 - PP > 0 \\ 0 otherwise \end{matrix} \frac{0.001}{1} P_SR Ca_SR2 - (B_p P_C_SR (PP - (P_SR \frac{0.001}{1} Ca_SR2)) \{\begin{matrix} 1 if PP - (P_SR \frac{0.001}{1} Ca_SR2) > 0 \\ 0 otherwise \end{matrix})))

\frac{d}{d time} (P_C_SR) = 1 (A_p (P_SR \frac{0.001}{1} Ca_SR2 - PP) \{\begin{matrix} 1 if P_SR \frac{0.001}{1} Ca_SR2 - PP > 0 \\ 0 otherwise \end{matrix} \frac{0.001}{1} P_SR Ca_SR2 - (B_p P_C_SR (PP - (P_SR \frac{0.001}{1} Ca_SR2)) \{\begin{matrix} 1 if PP - (P_SR \frac{0.001}{1} Ca_SR2) > 0 \\ 0 otherwise \end{matrix}))