Model Mathematics

\frac{d}{d time} (Vm) = \frac{- (i_CaL + i_CaT + i_f + i_st + i_Kr + i_Ks + i_to + i_sus + i_NaK + i_NaCa + i_b_Ca + i_b_Na)}{Cm}

E_Na = \frac{R T}{F} \ln (\frac{Nao}{Nai}) E_K = \frac{R T}{F} \ln (\frac{Ko}{Ki}) E_Ks = \frac{R T}{F} \ln (\frac{Ko + 0.12 Nao}{Ki + 0.12 Nai})

i_CaL = Cm g_CaL (Vm - E_CaL) dL fL fCa

dL_infinity = \frac{1}{1 + ⅇ^{\frac{- (Vm + 13.5)}{6}}} tau_dL = \frac{1}{alpha_dL + beta_dL} alpha_dL = \frac{(- 0.02839) (adVm + 35)}{ⅇ^{\frac{- (adVm + 35)}{2.5}} - 1} - (\frac{0.0849 adVm}{ⅇ^{\frac{- adVm}{4.8}} - 1}) adVm = \{\begin{matrix} - 35.00001 if Vm = - 35 \\ 0.00001 if Vm = 0 \\ Vm otherwise \end{matrix} beta_dL = \frac{0.01143 (bdVm - 5)}{ⅇ^{\frac{bdVm - 5}{2.5}} - 1} bdVm = \{\begin{matrix} 5.00001 if Vm = 5 \\ Vm otherwise \end{matrix} \frac{d}{d time} (dL) = \frac{dL_infinity - dL}{tau_dL}

fL_infinity = \frac{1}{1 + ⅇ^{\frac{Vm + 35}{7.3}}} tau_fL = 44.3 + 257.1 ⅇ^{- ({(\frac{Vm + 32.5}{13.9})}^{2})} \frac{d}{d time} (fL) = \frac{fL_infinity - fL}{tau_fL}

fCa_infinity = \frac{Km_fCa}{Km_fCa + Ca_sub} tau_fCa = \frac{fCa_infinity}{alpha_fCa} \frac{d}{d time} (fCa) = \frac{fCa_infinity - fCa}{tau_fCa}

i_CaT = Cm g_CaT (Vm - E_CaT) dT fT

dT_infinity = \frac{1}{1 + ⅇ^{\frac{- (Vm + 26.3)}{6}}} tau_dT = \frac{1}{1.068 ⅇ^{\frac{Vm + 26.3}{30}} + 1.068 ⅇ^{\frac{- (Vm + 26.3)}{30}}} \frac{d}{d time} (dT) = \frac{dT_infinity - dT}{tau_dT}

fT_infinity = \frac{1}{1 + ⅇ^{\frac{Vm + 61.7}{5.6}}} tau_fT = \frac{1}{0.0153 ⅇ^{\frac{- (Vm + 61.7)}{83.3}} + 0.015 ⅇ^{\frac{Vm + 61.7}{15.38}}} \frac{d}{d time} (fT) = \frac{fT_infinity - fT}{tau_fT}

i_Kr = Cm g_Kr (Vm - E_K) (0.6 paF + 0.4 paS) pi_

pa_infinity = \frac{1}{1 + ⅇ^{\frac{- (Vm + 23.2)}{10.6}}} tau_paS = \frac{0.84655354}{0.0042 ⅇ^{\frac{Vm}{17}} + 0.00015 ⅇ^{\frac{- Vm}{21.6}}} tau_paF = \frac{0.84655354}{0.0372 ⅇ^{\frac{Vm}{15.9}} + 0.00096 ⅇ^{\frac{- Vm}{22.5}}} \frac{d}{d time} (paS) = \frac{pa_infinity - paS}{tau_paS} \frac{d}{d time} (paF) = \frac{pa_infinity - paF}{tau_paF}

pi_infinity = \frac{1}{1 + ⅇ^{\frac{Vm + 28.6}{17.1}}} tau_pi = \frac{1}{0.1 ⅇ^{\frac{- Vm}{54.645}} + 0.656 ⅇ^{\frac{Vm}{106.157}}} \frac{d}{d time} (pi_) = \frac{pi_infinity - pi_}{tau_pi}

i_Ks = Cm g_Ks (Vm - E_Ks) n^{2}

n_infinity = \frac{alpha_n}{alpha_n + beta_n} tau_n = \frac{1}{alpha_n + beta_n} alpha_n = \frac{0.014}{1 + ⅇ^{\frac{- (Vm - 40)}{9}}} beta_n = 0.001 ⅇ^{\frac{- Vm}{45}} \frac{d}{d time} (n) = \frac{n_infinity - n}{tau_n}

i_to = Cm g_to (Vm - E_K) q r i_sus = Cm g_sus (Vm - E_K) r

q_infinity = \frac{1}{1 + ⅇ^{\frac{Vm + 49}{13}}} tau_q = 6.06 + \frac{39.102}{0.57 ⅇ^{(- 0.08) (Vm + 44)} + 0.065 ⅇ^{0.1 (Vm + 45.93)}} \frac{d}{d time} (q) = \frac{q_infinity - q}{tau_q}

r_infinity = \frac{1}{1 + ⅇ^{\frac{- (Vm - 19.3)}{15}}} tau_r = 2.75352 + \frac{14.40516}{1.037 ⅇ^{0.09 (Vm + 30.61)} + 0.369 ⅇ^{(- 0.12) (Vm + 23.84)}} \frac{d}{d time} (r) = \frac{r_infinity - r}{tau_r}

i_f_Na = Cm 0.3833 g_if (Vm - E_Na) y^{2} i_f_K = Cm 0.6167 g_if (Vm - E_K) y^{2} i_f = i_f_Na + i_f_K

y_infinity = \frac{1}{1 + ⅇ^{\frac{Vm - VIf_half}{13.5}}} tau_y = \frac{0.7166529}{ⅇ^{\frac{- (Vm + 386.9)}{45.302}} + ⅇ^{\frac{Vm - 73.08}{19.231}}} \frac{d}{d time} (y) = \frac{y_infinity - y}{tau_y}

i_st = Cm g_st (Vm - E_st) qa qi

qa_infinity = \frac{1}{1 + ⅇ^{\frac{- (Vm + 57)}{5}}} tau_qa = \frac{1}{alpha_qa + beta_qa} alpha_qa = \frac{1}{0.15 ⅇ^{\frac{- Vm}{11}} + 0.2 ⅇ^{\frac{- Vm}{700}}} beta_qa = \frac{1}{16 ⅇ^{\frac{Vm}{8}} + 15 ⅇ^{\frac{Vm}{50}}} \frac{d}{d time} (qa) = \frac{qa_infinity - qa}{tau_qa}

qi_infinity = \frac{alpha_qi}{alpha_qi + beta_qi} tau_qi = \frac{6.65}{alpha_qi + beta_qi} alpha_qi = \frac{1}{3100 ⅇ^{\frac{Vm}{13}} + 700 ⅇ^{\frac{Vm}{70}}} beta_qi = \frac{1}{95 ⅇ^{\frac{- Vm}{10}} + 50 ⅇ^{\frac{- Vm}{700}}} + \frac{0.000229}{1 + ⅇ^{\frac{- Vm}{5}}} \frac{d}{d time} (qi) = \frac{qi_infinity - qi}{tau_qi}

i_b_Na = Cm g_b_Na (Vm - E_Na)

i_NaK = \frac{Cm i_NaK_max}{(1 + {(\frac{Km_Kp}{Ko})}^{1.2}) (1 + {(\frac{Km_Nap}{Nai})}^{1.3}) (1 + ⅇ^{\frac{- ((Vm - E_Na) + 120)}{30}})}

i_b_Ca = Cm g_b_Ca (Vm - E_CaL)

i_NaCa = \frac{Cm kNaCa (x2 k21 - (x1 k12))}{x1 + x2 + x3 + x4} x1 = k41 k34 (k23 + k21) + k21 k32 (k43 + k41) x2 = k32 k43 (k14 + k12) + k41 k12 (k34 + k32) x3 = k14 k43 (k23 + k21) + k12 k23 (k43 + k41) x4 = k23 k34 (k14 + k12) + k14 k21 (k34 + k32) k43 = \frac{Nai}{K3ni + Nai} RTOnF = \frac{R T}{F} k12 = \frac{\frac{Ca_sub}{Kci} ⅇ^{\frac{(- Qci) Vm}{RTOnF}}}{di} k14 = \frac{\frac{\frac{Nai}{K1ni} Nai}{K2ni} (1 + \frac{Nai}{K3ni}) ⅇ^{\frac{Qn Vm}{2 RTOnF}}}{di} k41 = ⅇ^{\frac{(- Qn) Vm}{2 RTOnF}} di = 1 + \frac{Ca_sub}{Kci} (1 + ⅇ^{\frac{(- Qci) Vm}{RTOnF}} + \frac{Nai}{Kcni}) + \frac{Nai}{K1ni} (1 + \frac{Nai}{K2ni} (1 + \frac{Nai}{K3ni})) k34 = \frac{Nao}{K3no + Nao} k21 = \frac{\frac{Cao}{Kco} ⅇ^{\frac{Qco Vm}{RTOnF}}}{do} k23 = \frac{\frac{\frac{Nao}{K1no} Nao}{K2no} (1 + \frac{Nao}{K3no}) ⅇ^{\frac{(- Qn) Vm}{2 RTOnF}}}{do} k32 = ⅇ^{\frac{Qn Vm}{2 RTOnF}} do = 1 + \frac{Cao}{Kco} (1 + ⅇ^{\frac{Qco Vm}{RTOnF}}) + \frac{Nao}{K1no} (1 + \frac{Nao}{K2no} (1 + \frac{Nao}{K3no}))

j_SRCarel = ks O (Ca_jsr - Ca_sub) kCaSR = MaxSR - (\frac{MaxSR - MinSR}{1 + {(\frac{EC50_SR}{Ca_jsr})}^{HSR}}) koSRCa = \frac{koCa}{kCaSR} kiSRCa = kiCa kCaSR \frac{d}{d time} (R) = kim RI - (kiSRCa Ca_sub R) - (koSRCa {Ca_sub}^{2} R - (kom O)) \frac{d}{d time} (O) = koSRCa {Ca_sub}^{2} R - (kom O) - (kiSRCa Ca_sub O - (kim I)) \frac{d}{d time} (I) = kiSRCa Ca_sub O - (kim I) - (kom I - (koSRCa {Ca_sub}^{2} RI)) \frac{d}{d time} (RI) = kom I - (koSRCa {Ca_sub}^{2} RI) - (kim RI - (kiSRCa Ca_sub R))

j_Ca_dif = \frac{Ca_sub - Cai}{tau_dif_Ca} j_up = \frac{P_up}{1 + \frac{K_up}{Cai}} j_tr = \frac{Ca_nsr - Ca_jsr}{tau_tr}

\frac{d}{d time} (fTC) = delta_fTC delta_fTC = kf_TC Cai (1 - fTC) - (kb_TC fTC) \frac{d}{d time} (fTMC) = delta_fTMC delta_fTMC = kf_TMC Cai (1 - (fTMC + fTMM)) - (kb_TMC fTMC) \frac{d}{d time} (fTMM) = delta_fTMM delta_fTMM = kf_TMM Mgi (1 - (fTMC + fTMM)) - (kb_TMM fTMM) \frac{d}{d time} (fCMi) = delta_fCMi delta_fCMi = kf_CM Cai (1 - fCMi) - (kb_CM fCMi) \frac{d}{d time} (fCMs) = delta_fCMs delta_fCMs = kf_CM Ca_sub (1 - fCMs) - (kb_CM fCMs) \frac{d}{d time} (fCQ) = delta_fCQ delta_fCQ = kf_CQ Ca_jsr (1 - fCQ) - (kb_CQ fCQ)

\frac{d}{d time} (Cai) = \frac{j_Ca_dif V_sub - (j_up V_nsr)}{V_i} - (CM_tot delta_fCMi + TC_tot delta_fTC + TMC_tot delta_fTMC) \frac{d}{d time} (Ca_sub) = \frac{j_SRCarel V_jsr}{V_sub} - (\frac{i_CaL + i_CaT + i_b_Ca - (2 i_NaCa)}{2 F V_sub} + j_Ca_dif + CM_tot delta_fCMs) \frac{d}{d time} (Ca_nsr) = j_up - (\frac{j_tr V_jsr}{V_nsr}) \frac{d}{d time} (Ca_jsr) = j_tr - (j_SRCarel + CQ_tot delta_fCQ)

V_cell = 0.001 π {R_cell}^{2} L_cell V_sub = 0.001 2 π L_sub (R_cell - (\frac{L_sub}{2})) L_cell V_jsr = V_jsr_part V_cell V_i = V_i_part V_cell - V_sub V_nsr = V_nsr_part V_cell