Model Mathematics

L_m = L_prime_m - L_b K_1 = k_1 Ca K_m = k_m Ca k_m = k_1 k_minus_1 = \frac{k_1}{K_max} k_minus_m = k_minus_1 g_prime_0 = g_0 g_prime_1 = g_1

Rbar_n = \frac{A_n k_minus_1}{1 + K_1} R_n = Rbar_n alpha L_m Abar_n = \frac{R_n K_1}{1 + k_minus_1} A_n = Abar_n alpha L_m

dRbar_s_dt = (- K_1) R_s + k_minus_1 A_s + (g_prime_0 + g_prime_1 V) U_s \frac{d}{d time} (Rbar_s) = dRbar_s_dt \frac{d}{d time} (R_s) = \{\begin{matrix} dRbar_s_dt alpha L_m - (Rbar_s L_m V) if V > 0 \\ dRbar_s_dt alpha L_m + (2 Rbar_d - Rbar_s) L_m | V | if V < 0 \\ dRbar_s_dt alpha L_m otherwise \end{matrix} dAbar_s_dt = K_1 R_s + (- f - k_minus_1) A_s + (g_0 + g_1 V) T_s \frac{d}{d time} (Abar_s) = dAbar_s_dt \frac{d}{d time} (A_s) = \{\begin{matrix} dAbar_s_dt alpha L_m - (Abar_s L_m V) if V > 0 \\ dAbar_s_dt alpha L_m + (2 Abar_d - Abar_s) L_m | V | if V < 0 \\ dAbar_s_dt alpha L_m otherwise \end{matrix} dTbar_s_dt = f A_s + (- g_0 - (g_1 V) - k_minus_m) T_s + K_m U_s \frac{d}{d time} (Tbar_s) = dTbar_s_dt \frac{d}{d time} (T_s) = \{\begin{matrix} dTbar_s_dt alpha L_m - (Tbar_s L_m V) if V > 0 \\ dTbar_s_dt alpha L_m + (2 Tbar_d - Tbar_s) L_m | V | if V < 0 \\ dTbar_s_dt alpha L_m otherwise \end{matrix} dUbar_s_dt = k_minus_m T_s + (- K_m - g_prime_0 - (g_prime_1 V)) U_s \frac{d}{d time} (Ubar_s) = dUbar_s_dt \frac{d}{d time} (U_s) = \{\begin{matrix} dUbar_s_dt alpha L_m - (Ubar_s L_m V) if V > 0 \\ dUbar_s_dt alpha L_m + (2 Ubar_d - Ubar_s) L_m | V | if V < 0 \\ dUbar_s_dt alpha L_m otherwise \end{matrix}

\frac{d}{d time} (Rbar_d) = (- K_1) R_d + k_minus_1 A_d + (g_prime_0 + g_prime_1 V) U_d R_d = Rbar_d alpha L_m \frac{d}{d time} (Abar_d) = K_1 R_d + (- f - k_minus_1) A_d + (g_0 + g_1 V) T_d A_d = Abar_d alpha L_m \frac{d}{d time} (Tbar_d) = f A_d + (- g_0 - (g_1 V) - k_minus_m) T_d + K_m U_d T_d = Tbar_d alpha L_m \frac{d}{d time} (Ubar_d) = k_minus_m T_d + (- K_m - g_prime_0 - (g_prime_1 V)) U_d U_d = Ubar_d alpha L_m

I_o = Q_o Ca I_s = Q_s ((1 - (ⅇ^{\frac{- time}{tau_SR}})) ⅇ^{\frac{- time}{tau_SF}} + I_l) Ca_0 I_i = Q_i ((1 - (ⅇ^{\frac{- time}{tau_iR}})) ⅇ^{\frac{- time}{tau_iF}} + I_l) Ca_r I_u = \frac{Q_u Ca}{K_mu + Ca} \frac{d}{d time} (Ca) = I_s + I_i - (I_o + I_u)

BCa_L = A_s + T_s + A_d + T_d + A_n \frac{d}{d time} (BCa_h) = (2 TRo - BCa_h) Ca k_h - (BCa_h k_minus_h)

F_CB = F_bar - (eta V) F_CE = (T_s + U_s) F_CB F_PE = \{\begin{matrix} E (ⅇ^{D (\frac{SL}{Sp_0} - 1)} - 1) + eta_PE V_SL if SL \geq Sp_0 \\ (- B) (1 - (\frac{SL}{Sp_0})) + eta_PE V_SL otherwise \end{matrix} F = F_CE + F_PE

V = d_alpha_dt