Model Mathematics

FVRT = \frac{F V}{R T} FVRT_Ca = 2 FVRT V = \{\begin{matrix} 0 if time \geq 0 \land time ≦ 200 \\ - 80 otherwise \end{matrix}

expVL = ⅇ^{\frac{V - V_L}{del_VL}} t_R = 1.17 t_L alpha_p = \frac{expVL}{t_L (expVL + 1)} alpha_m = \frac{phi_L}{t_L} beta_poc = \frac{{C_oc}^{2}}{t_R ({C_oc}^{2} + {K_RyR}^{2})} beta_pcc = \frac{{Ca_i}^{2}}{t_R ({Ca_i}^{2} + {K_RyR}^{2})} beta_m = \frac{phi_R}{t_R} epsilon_pco = \frac{C_co (expVL + a)}{tau_L K_L (expVL + 1)} epsilon_pcc = \frac{Ca_i (expVL + a)}{tau_L K_L (expVL + 1)} epsilon_m = \frac{b (expVL + a)}{tau_L (b expVL + a)} mu_poc = \frac{{C_oc}^{2} + c {K_RyR}^{2}}{tau_R ({C_oc}^{2} + {K_RyR}^{2})} mu_pcc = \frac{{Ca_i}^{2} + c {K_RyR}^{2}}{tau_R ({Ca_i}^{2} + {K_RyR}^{2})} mu_moc = \frac{theta_R d ({C_oc}^{2} + c {K_RyR}^{2})}{tau_R (d {C_oc}^{2} + c {K_RyR}^{2})} mu_mcc = \frac{theta_R d ({Ca_i}^{2} + c {K_RyR}^{2})}{tau_R (d {Ca_i}^{2} + c {K_RyR}^{2})}

C_cc = Ca_i C_co = \frac{Ca_i + \frac{J_R}{g_D} Ca_SR}{1 + \frac{J_R}{g_D}} C_oc = \{\begin{matrix} \frac{Ca_i + \frac{\frac{J_L}{g_D} Ca_o FVRT_Ca ⅇ^{- FVRT_Ca}}{1 - (ⅇ^{- FVRT_Ca})}}{1 + \frac{\frac{J_L}{g_D} FVRT_Ca}{1 - (ⅇ^{- FVRT_Ca})}} if | FVRT_Ca | > 0.000000001 \\ \frac{Ca_i + \frac{J_L}{g_D} Ca_o}{1 + \frac{J_L}{g_D}} otherwise \end{matrix} C_oo = \{\begin{matrix} \frac{Ca_i + \frac{J_R}{g_D} Ca_SR + \frac{\frac{J_L}{g_D} Ca_o FVRT_Ca ⅇ^{- FVRT_Ca}}{1 - (ⅇ^{- FVRT_Ca})}}{1 + \frac{J_R}{g_D} + \frac{\frac{J_L}{g_D} FVRT_Ca}{1 - (ⅇ^{- FVRT_Ca})}} if | FVRT_Ca | > 0.000000001 \\ \frac{Ca_i + \frac{J_R}{g_D} Ca_SR + \frac{J_L}{g_D} Ca_o}{1 + \frac{J_R}{g_D} + \frac{J_L}{g_D}} otherwise \end{matrix}

J_Rco = \frac{J_R (Ca_SR - Ca_i)}{1 + \frac{J_R}{g_D}} J_Roo = \{\begin{matrix} \frac{J_R ((Ca_SR - Ca_i) + \frac{\frac{J_L}{g_D} FVRT_Ca}{1 - (ⅇ^{- FVRT_Ca})} (Ca_SR - (Ca_o ⅇ^{- FVRT_Ca})))}{1 + \frac{J_R}{g_D} + \frac{\frac{J_L}{g_D} FVRT_Ca}{1 - (ⅇ^{- FVRT_Ca})}} if | FVRT_Ca | > 0.00001 \\ \frac{J_R ((Ca_SR - Ca_i) + \frac{\frac{J_L}{g_D} 0.00001}{1 - (ⅇ^{- 0.00001})} (Ca_SR - (Ca_o ⅇ^{- 0.00001})))}{1 + \frac{J_R}{g_D} + \frac{\frac{J_L}{g_D} 0.00001}{1 - (ⅇ^{- 0.00001})}} otherwise \end{matrix} J_Loc = \{\begin{matrix} \frac{\frac{J_L FVRT_Ca}{1 - (ⅇ^{- FVRT_Ca})} (Ca_o ⅇ^{- FVRT_Ca} - Ca_i)}{1 + \frac{\frac{J_L}{g_D} FVRT_Ca}{1 - (ⅇ^{- FVRT_Ca})}} if | FVRT_Ca | > 0.00001 \\ \frac{\frac{J_L 0.00001}{1 - (ⅇ^{- 0.00001})} (Ca_o ⅇ^{- 0.00001} - Ca_i)}{1 + \frac{\frac{J_L}{g_D} 0.00001}{1 - (ⅇ^{- 0.00001})}} otherwise \end{matrix} J_Loo = \{\begin{matrix} \frac{\frac{J_L FVRT_Ca}{1 - (ⅇ^{- FVRT_Ca})} ((Ca_o ⅇ^{- FVRT_Ca} - Ca_i) + \frac{J_R}{g_D} (Ca_o ⅇ^{- FVRT_Ca} - Ca_SR))}{1 + \frac{J_R}{g_D} + \frac{\frac{J_L}{g_D} FVRT_Ca}{1 - (ⅇ^{FVRT_Ca})}} if | FVRT_Ca | > 0.00001 \\ \frac{\frac{J_L 0.00001}{1 - (ⅇ^{- 0.00001})} ((Ca_o ⅇ^{- 0.00001} - Ca_i) + \frac{J_R}{g_D} (Ca_o ⅇ^{- 0.00001} - Ca_SR))}{1 + \frac{J_R}{g_D} + \frac{\frac{J_L}{g_D} 0.00001}{1 - (ⅇ^{- 0.00001})}} otherwise \end{matrix}

denom = (alpha_p + alpha_m) ((alpha_m + beta_m + beta_poc) (beta_m + beta_pcc) + alpha_p (beta_m + beta_poc)) y_oc = \frac{alpha_p beta_m (alpha_p + alpha_m + beta_m + beta_pcc)}{denom} y_co = \frac{alpha_m (beta_pcc (alpha_m + beta_m + beta_poc) + beta_poc alpha_p)}{denom} y_oo = \frac{alpha_p (beta_poc (alpha_p + beta_m + beta_pcc) + beta_pcc alpha_m)}{denom} y_cc = \frac{alpha_m beta_m (alpha_m + alpha_p + beta_m + beta_poc)}{denom} y_ci = \frac{alpha_m}{alpha_p + alpha_m} y_oi = \frac{alpha_p}{alpha_p + alpha_m} y_ic = \frac{beta_m}{beta_pcc + beta_m} y_io = \frac{beta_pcc}{beta_pcc + beta_m} y_ii = 1 - y_oc - y_co - y_oo - y_cc - y_ci - y_ic - y_oi - y_io

r_1 = y_oc mu_poc + y_cc mu_pcc r_2 = \frac{alpha_p mu_moc + alpha_m mu_mcc}{alpha_p + alpha_m} r_3 = \frac{beta_m mu_pcc}{beta_m + beta_pcc} r_4 = mu_mcc r_5 = y_co epsilon_pco + y_cc epsilon_pcc r_6 = epsilon_m r_7 = \frac{alpha_m epsilon_pcc}{alpha_p + alpha_m} r_8 = epsilon_m z_4 = 1 - z_1 - z_2 - z_3 \frac{d}{d time} (z_1) = (- (r_1 + r_5)) z_1 + r_2 z_2 + r_6 z_3 \frac{d}{d time} (z_2) = (r_1 z_1 - ((r_2 + r_7) z_2)) + r_8 z_4 \frac{d}{d time} (z_3) = (r_5 z_1 - ((r_6 + r_3) z_3)) + r_4 z_4

J_R1 = y_oo J_Roo + J_Rco y_co J_R3 = \frac{J_Rco beta_pcc}{beta_m + beta_pcc} I_RyR = \frac{(z_1 J_R1 + z_3 J_R3) N}{V_myo}

J_L1 = J_Loo y_oo + J_Loc y_oc J_L2 = \frac{J_Loc alpha_p}{alpha_p + alpha_m} I_LCC = \frac{(z_1 J_L1 + z_2 J_L2) N}{V_myo}

I_NaCa = \frac{g_NCX (ⅇ^{eta FVRT} {Na_i}^{3} Ca_o - (ⅇ^{(eta - 1) FVRT} {Na_o}^{3} Ca_i))}{({Na_o}^{3} + {K_mNa}^{3}) (Ca_o + K_mCa) (1 + k_sat ⅇ^{(eta - 1) FVRT})}

I_SERCA = \frac{g_SERCA {Ca_i}^{2}}{{K_SERCA}^{2} + {Ca_i}^{2}}

I_pCa = \frac{g_pCa Ca_i}{K_mpCa + Ca_i}

E_Ca = \frac{R T}{2 F} \ln (\frac{Ca_o}{Ca_i}) I_CaB = g_CaB (E_Ca - V)

I_SR = g_SRl (Ca_SR - Ca_i)

I_TRPN = k_m_TRPN (B_TRPN - TRPN) - (k_p_TRPN TRPN Ca_i)

beta_CMDN = {(1 + \frac{k_CMDN B_CMDN}{{(k_CMDN + Ca_i)}^{2}})}^{- 1}

\frac{d}{d time} (TRPN) = I_TRPN \frac{d}{d time} (Ca_i) = beta_CMDN (((I_LCC + I_RyR - I_SERCA) + I_SR + I_NaCa - I_pCa) + I_CaB + I_TRPN) \frac{d}{d time} (Ca_SR) = \frac{V_myo}{V_SR} ((- I_RyR) + I_SERCA - I_SR) CaSR_plot = \frac{Ca_SR V_SR}{V_myo}