Model Mathematics

\frac{d}{d time} (Vm) = \frac{- (I_tot + I_ext)}{Cm} I_tot = I_NaCa + I_NaK + I_PMCA + I_VDDR + I_AI + I_Ki + I_CaL

\frac{d}{d time} (Na_i) = \frac{- Inet_Na}{ZNa F Vi} \frac{d}{d time} (K_i) = \frac{- Inet_K}{ZK F Vi} Ca_i = \{\begin{matrix} 0.000000000001 if Cai_A ≦ 0.0 \\ Cai_A otherwise \end{matrix} \frac{d}{d time} (Cai_A) = \frac{- (Inet_Ca + I_up - (I_IP3R + I_leak))}{ZCa F Vi} Inet_Na = I_CaL_Na + I_AI_Na + 3.0 I_NaK + 3.0 I_NaCa Inet_K = I_Ki + I_CaL_K + I_AI_K - (2.0 I_NaK) Inet_Ca = I_CaL_Ca + I_VDDR + I_AI_Ca + I_PMCA - (2.0 I_NaCa) Na_CF = \frac{ZNa F Vm}{R T} \frac{Na_i - (Na_o ⅇ^{\frac{(- ZNa) F Vm}{R T}})}{1.0 - (ⅇ^{\frac{(- ZNa) F Vm}{R T}})} K_CF = \frac{ZK F Vm}{R T} \frac{K_i - (K_o ⅇ^{\frac{(- ZK) F Vm}{R T}})}{1.0 - (ⅇ^{\frac{(- ZK) F Vm}{R T}})} Ca_CF = \frac{ZCa F Vm}{R T} \frac{Ca_i - (Ca_o ⅇ^{\frac{(- ZCa) F Vm}{R T}})}{1.0 - (ⅇ^{\frac{(- ZCa) F Vm}{R T}})}

I_Ki = g_Ki {(\frac{K_o}{5.4})}^{0.62} (Vm - E_K) n fU E_K = \frac{R T}{1.0 F} \ln (\frac{K_o}{K_i}) fU = \frac{mu}{mu + lambda} mu = 3.3 ⅇ^{\frac{Vm - (E_K + 6.0)}{15.0}} lambda = 26.0 ⅇ^{\frac{Vm - (E_K + 6.0)}{52.0}}

n_1 = \frac{1.0}{1.0 + 0.1 ⅇ^{\frac{Vm - E_K}{15.0}} + 0.048 ⅇ^{\frac{Vm - E_K}{7.0}}} n_2 = n_1 AB n_3 = AB - n_2 AB = a + b a = \frac{2.5}{4.7 ⅇ^{\frac{Vm - E_K}{28.7}}} b = \frac{2.5}{6.0 ⅇ^{\frac{E_K - Vm}{25.8}}} n = \frac{n_2}{(n_2 + n_3) {(1.0 - fU)}^{3.0}}

I_CaL = I_CaL_Ca + I_CaL_K + I_CaL_Na I_CaL_Ca = PCaL Ca_CF m h I_CaL_K = 0.001 PCaL K_CF m h I_CaL_Na = 0.00005 PCaL Na_CF m h

alpha_m = \{\begin{matrix} 0.002175 \frac{Vm + 30.0}{1.0 - (ⅇ^{\frac{Vm + 30.0}{- 2.5}})} if Vm = -30.0 \\ 0.004189268 otherwise \end{matrix} beta_m = \{\begin{matrix} 0.001578959 if Vm = 0.0 \\ 0.0006315 \frac{Vm}{ⅇ^{\frac{Vm}{2.5}} - 1.0} otherwise \end{matrix} \frac{d}{d time} (m) = alpha_m (1.0 - m) - (beta_m m)

alpha_h = \{\begin{matrix} 0.000010001 if Vm = -34.0 \\ (1.775E-6) \frac{Vm + 34.0}{ⅇ^{\frac{Vm + 34.0}{5.633}} - 1.0} otherwise \end{matrix} beta_h = 0.427 Ca_i \frac{Vm + 64.0}{ⅇ^{\frac{Vm + 44.0}{- 4.16}} + 1.0} \frac{d}{d time} (h) = alpha_h (1.0 - h) - (beta_h h)

I_VDDR = PVDDR Ca_CF m h

alpha_m = \frac{m2}{m3 + {10.0}^{-20.0}} m2 = \frac{1.0}{1.0 + ⅇ^{\frac{- (Vm + 23.0)}{6.0999999}}} m3 = (4.4299998 - 0.660000003) ⅇ^{\frac{- ({(Vm + 150.0)}^{2.0})}{7118.0}} + 0.660000003 beta_m = \frac{1.0 - m2}{m3 + {10.0}^{-20.0}} m = \frac{alpha_m}{alpha_m + beta_m}

alpha_h = \frac{h2}{h3 + {10.0}^{-20.0}} h2 = \frac{1.0}{1.0 + ⅇ^{\frac{- (Vm + 75.0)}{- 6.5999999}}} h3 = (40.799999 - 0.83999997) ⅇ^{\frac{- ({(Vm + 106.0)}^{2.0})}{2292.0}} + 0.8399997 beta_h = \frac{1.0 - h2}{h3 + {10.0}^{-20.0}} h = \frac{alpha_h}{alpha_h + beta_h}

I_AI = I_AI_Ca + I_AI_K + I_AI_Na I_AI_Ca = PAI Ca_CF po I_AI_K = 0.36 PAI K_CF po I_AI_Na = 0.4 PAI Na_CF po po = 0.0017 + \frac{1.0 - 0.0017}{1.0 + \frac{Km_Ca_i}{Ca_i}}

I_NaCa = PNaCa (m y - (h (1.0 - y))) p_E1Na = \frac{1.0}{1.0 + \frac{1.0 + \frac{Ca_i}{Km_Ca_i}}{\frac{{Na_i}^{3.0}}{Km_Na_i}}} p_E2Na = \frac{1.0}{1.0 + \frac{1.0 + \frac{Ca_o}{Km_Ca_o}}{\frac{{Na_o}^{3.0}}{Km_Na_o}}} p_E1Ca = \frac{1.0}{1.0 + \frac{1.0 + \frac{{Na_i}^{3.0}}{Km_Na_i}}{\frac{Ca_i}{Km_Ca_i}}} p_E2Ca = \frac{1.0}{1.0 + \frac{1.0 + \frac{{Na_o}^{3.0}}{Km_Na_o}}{\frac{Ca_o}{Km_Ca_o}}}

alpha_y = 1.0 (h + p_E2Ca) beta_y = 1.0 (m + p_E1Ca) m = p_E1Na ⅇ^{0.32 \frac{Vm}{\frac{R T}{F}}} h = p_E2Na ⅇ^{(0.32 - 1.0) \frac{Vm}{\frac{R T}{F}}} \frac{d}{d time} (y) = alpha_y (1.0 - y) - (beta_y y)

I_NaK = \frac{pNaK}{1.0 + {(\frac{Km_Na_i}{Na_i})}^{1.36}} \frac{1.0 - ({(\frac{Vm + 50.0}{250.0})}^{2.0})}{1.0 + \frac{Km_K_o}{K_o}}

I_PMCA = \frac{pPMCA}{1.0 + {(\frac{Km_Ca_i}{Ca_i})}^{2.0}}

I_up = 1.5 Pup (p_E2Ca (1.0 - y) - (p_E1Ca y)) p_E1Ca = \frac{0.01}{1.0 + \frac{Km_Ca_i}{Ca_up}} p_E2Ca = \frac{1.0}{1.0 + \frac{Km_Ca_o}{Ca_i}} p_E1 = \frac{1.0}{1.0 + \frac{Ca_up}{Km_Ca_i}} p_E2 = \frac{0.01}{1.0 + \frac{Ca_i}{Km_Ca_o}}

alpha_y = p_E2Ca + p_E2 beta_y = p_E1Ca + p_E1 \frac{d}{d time} (y) = alpha_y (1.0 - y) - (beta_y y)

I_tr = 2.8 ptr (Ca_up - Ca_rel)

I_leak = pleak (Ca_up - Ca_i)

I_IP3R = PIP3R (Ca_rel - Ca_i) m kco = 0.147 {(\frac{Ca_i}{0.000057})}^{3} kic = 0.025 {(\frac{IP3}{0.001})}^{3} \frac{Ca_rel}{1.0} kab = koi kba = 0 kbc = kic kcb = kci kac = 0 kca = kco m_original = stO ⅇ^{(- SC_0) dt} + ndO ⅇ^{(- SC_1) dt} + SC_2 \frac{d}{d time} (m) = \frac{m_original - m}{dt_CellML} h_original = stA ⅇ^{(- SC_0) dt} + ndA ⅇ^{(- SC_1) dt} + SC_3 \frac{d}{d time} (h) = \frac{h_original - h}{dt_CellML} stO = \frac{OrgO_N4 + SC_1 m - (SC_2 SC_1)}{SC_1 - SC_0} ndO = \frac{OrgO_N4 + SC_0 m - (SC_2 SC_0)}{SC_0 - SC_1} stA = \frac{OrgA_N4 + SC_1 h - (SC_3 SC_1)}{SC_1 - SC_0} ndA = \frac{OrgA_N4 + SC_0 h - (SC_3 SC_0)}{SC_0 - SC_1} OrgO_N4 = kco h - (koi m) OrgA_N4 = kic (1.0 - (h + m)) - ((kci + kco) h) SC_0 = \frac{state_bf + {(| state_sbf |)}^{0.5}}{2.0} SC_1 = \frac{state_bf - ({(| state_sbf |)}^{0.5})}{2.0} SC_2 = \frac{state_c2 state_d3 - (state_c3 state_d2)}{state_c1 state_d2 - (state_c2 state_d1)} SC_3 = \frac{state_c1 state_d3 - (state_c3 state_d1)}{state_c2 state_d1 - (state_c1 state_d2)} state_bf = kab + kba + kac + kca + kbc + kcb state_sbf = {state_bf}^{2.0} - (4.0 ((kbc + kcb + kca) (kba + kab + kac) - ((kac - kbc) (kca + kba)))) state_c1 = kac - kbc state_c2 = - (kbc + kcb + kca) state_c3 = kbc state_d2 = kca - kba state_d1 = - (kba + kab + kac) state_d3 = kba

\frac{d}{d time} (Ca_up) = \frac{I_up - (I_tr + I_leak)}{ZCa Vup F} \frac{d}{d time} (Ca_rel) = \frac{I_tr - I_IP3R}{ZCa Vrel F}

\frac{d}{d time} (IP3) = PIP2_IP3 PIP2 - ((IP3_PIP2 + IP3_IP4) IP3) \frac{d}{d time} (PIP2) = IP3_PIP2 IP3 + IP4_PIP2 (IP3total - (IP3 + PIP2)) - ((PIP2_IP4 + PIP2_IP3) PIP2) PIP2_IP3 = 0.2 ⅇ^{\frac{Vm + 48.5}{18.1}} \frac{Ca_i}{Ca_i + Km_Ca_i} IP3_PIP2 = 0.5 ⅇ^{\frac{Vm + 100.0}{-28.5}} IP4_PIP2 = 0.0035 ⅇ^{\frac{Vm + 100.0}{-25.5}}

Vup = 0.1 Vi Vrel = 0.01 Vi