Model Mathematics

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Component: environment

Component: membrane

stim_period = \{\begin{matrix} 500 if time \geq 5000 \land time ≦ 10000 \\ 1000 otherwise \end{matrix} I_Stim = \{\begin{matrix} stim_amplitude if time - (⌊ \frac{time}{stim_period} ⌋ stim_period) \geq 0 \land time - (⌊ \frac{time}{stim_period} ⌋ stim_period) ≦ stim_duration \\ 0 otherwise \end{matrix} FVRT = \frac{F V}{R T} FVRT_Ca = 2 FVRT \frac{d}{d time} (V) = \frac{- (I_Na + I_t + I_ss + I_f + I_K1 + I_B_Na + I_B_K + I_NaK + I_Stim + I_CaB + I_NaCa + I_pCa + I_LCC)}{Cm}

Component: cell_geometry

Component: pandit_sodium_current

g_Na_endo = 1.33 g_Na E_Na = \frac{R T}{F} \ln (\frac{Na_o}{Na_i}) i_Na = g_Na_endo m^{3} h j (V - E_Na)

Component: sodium_current_m_gate

m_infinity = \frac{1}{1 + ⅇ^{\frac{V + 45}{- 6.5}}} tau_m = \frac{1.36}{\frac{0.32 (V + 47.13)}{1 - (ⅇ^{(- 0.1) (V + 47.13)})} + 0.08 ⅇ^{\frac{- V}{11}}} \frac{d}{d time} (m) = \frac{m_infinity - m}{tau_m}

Component: sodium_current_h_gate

h_infinity = \frac{1}{1 + ⅇ^{\frac{V + 76.1}{6.07}}} tau_h = \{\begin{matrix} 0.4537 (1 + ⅇ^{\frac{- (V + 10.66)}{11.1}}) if V \geq - 40 \\ \frac{3.49}{0.135 ⅇ^{\frac{- (V + 80)}{6.8}} + 3.56 ⅇ^{0.079 V} + 310000 ⅇ^{0.35 V}} otherwise \end{matrix} \frac{d}{d time} (h) = \frac{h_infinity - h}{tau_h}

Component: sodium_current_j_gate

j_infinity = \frac{1}{1 + ⅇ^{\frac{V + 76.1}{6.07}}} tau_j = \{\begin{matrix} \frac{11.63 (1 + ⅇ^{(- 0.1) (V + 32)})}{ⅇ^{(- 0.0000002535) V}} if V \geq - 40 \\ \frac{3.49}{\frac{V + 37.78}{1 + ⅇ^{0.311 (V + 79.23)}} ((- 127140) ⅇ^{0.2444 V} - (0.00003474 ⅇ^{(- 0.04391) V})) + \frac{0.1212 ⅇ^{(- 0.01052) V}}{1 + ⅇ^{(- 0.1378) (V + 40.14)}}} otherwise \end{matrix} \frac{d}{d time} (j) = \frac{j_infinity - j}{tau_j}

Component: pandit_Ca_independent_transient_outward_K_current

g_t_endo = 0.4647 g_t E_K = \frac{R T}{F} \ln (\frac{K_o}{K_i}) i_t = g_t_endo r (a_endo s + b_endo s_slow) (V - E_K)

Component: Ca_independent_transient_outward_K_current_r_gate

r_infinity = \frac{1}{1 + ⅇ^{\frac{V + 10.6}{- 11.42}}} tau_r = \frac{1000}{45.16 ⅇ^{0.03577 (V + 50)} + 98.9 ⅇ^{(- 0.1) (V + 38)}} \frac{d}{d time} (r) = \frac{r_infinity - r}{tau_r}

Component: Ca_independent_transient_outward_K_current_s_gate

s_infinity = \frac{1}{1 + ⅇ^{\frac{V + 45.3}{6.8841}}} tau_s_endo = 550 ⅇ^{- ({(\frac{V + 70}{25})}^{2})} + 49 \frac{d}{d time} (s) = \frac{s_infinity - s}{tau_s_endo}

Component: Ca_independent_transient_outward_K_current_s_slow_gate

s_slow_infinity = \frac{1}{1 + ⅇ^{\frac{V + 45.3}{6.8841}}} tau_s_slow_endo = 3300 ⅇ^{- ({(\frac{V + 70}{30})}^{2})} + 49 \frac{d}{d time} (s_slow) = \frac{s_slow_infinity - s_slow}{tau_s_slow_endo}

Component: pandit_steady_state_outward_K_current

i_ss = g_ss r_ss s_ss (V - E_K)

Component: steady_state_outward_K_current_r_ss_gate

r_ss_infinity = \frac{1}{1 + ⅇ^{\frac{V + 11.5}{- 11.82}}} tau_r_ss = \frac{10000}{45.16 ⅇ^{0.03577 (V + 50)} + 98.9 ⅇ^{(- 0.1) (V + 38)}} \frac{d}{d time} (r_ss) = \frac{r_ss_infinity - r_ss}{tau_r_ss}

Component: steady_state_outward_K_current_s_ss_gate

s_ss_infinity = \frac{1}{1 + ⅇ^{\frac{V + 87.5}{10.3}}} tau_s_ss = 2100 \frac{d}{d time} (s_ss) = \frac{s_ss_infinity - s_ss}{tau_s_ss}

Component: pandit_inward_rectifier

i_K1 = (\frac{48e-3}{ⅇ^{\frac{V + 37}{25}} + ⅇ^{\frac{V + 37}{- 25}}} + (10e-3)) \frac{0.001}{1 + ⅇ^{\frac{V - (E_K + 76.77)}{- 17}}} + \frac{g_K1 (V - (E_K + 1.73))}{(1 + ⅇ^{\frac{1.613 F (V - (E_K + 1.73))}{R T}}) (1 + ⅇ^{\frac{K_o - 0.9988}{- 0.124}})}

Component: pandit_hyperpolarisation_activated_current

f_K = 1 - f_Na i_f_Na = g_f y f_Na (V - E_Na) i_f_K = g_f y f_K (V - E_K) i_f = i_f_Na + i_f_K

Component: hyperpolarisation_activated_current_y_gate

y_infinity = \frac{1}{1 + ⅇ^{\frac{V + 138.6}{10.48}}} tau_y = \frac{1000}{0.11885 ⅇ^{\frac{V + 80}{28.37}} + 0.5623 ⅇ^{\frac{V + 80}{- 14.19}}} \frac{d}{d time} (y) = \frac{y_infinity - y}{tau_y}

Component: pandit_background_currents

i_B_Na = g_B_Na (V - E_Na) i_B_K = g_B_K (V - E_K)

Component: pandit_sodium_potassium_pump

sigma = \frac{ⅇ^{\frac{Na_o}{67.3}} - 1}{7} i_NaK = \frac{\frac{\frac{i_NaK_max 1}{1 + 0.1245 ⅇ^{\frac{(- 0.1) V F}{R T}} + 0.0365 sigma ⅇ^{\frac{(- V) F}{R T}}} K_o}{K_o + K_m_K} 1}{1 + {(\frac{K_m_Na}{Na_i})}^{4}}

Component: pandit_standard_ionic_concentrations

Component: hinch_CaRU

Component: CaRU_Transitions

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})}

Component: DS_Calcium_Concentrations

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}

Component: LCC_and_RyR_fluxes

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}

Component: CaRU_states

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

Component: CaRU_reduced_states

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

Component: RyR_current

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}

Component: LCC_current

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}

Component: hinch_Na_Ca_Exchanger

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})}

Component: hinch_SERCA

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

Component: hinch_Sarcolemmal_Ca_pump

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

Component: hinch_Background_Ca_current

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

Component: hinch_SR_Ca_leak_current

I_SR = g_SRl (Ca_SR - Ca_i)

Component: hinch_calmodulin_Ca_buffer

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

Component: convert_hinch

I_LCC = (- 1.5) hinch_I_LCC 2 V_myo_uL F I_NaCa = hinch_I_NaCa V_myo_uL F I_pCa = hinch_I_pCa 2 V_myo_uL F I_CaB = (- hinch_I_CaB) 2 V_myo_uL F I_RyR = 1.5 hinch_I_RyR

Component: niederer_thinfilaments

Component: niederer_tropomyosin

K_2 = \frac{alpha_r2 {z_p}^{n_Rel}}{{z_p}^{n_Rel} + {K_z}^{n_Rel}} (1 - (\frac{n_Rel {K_z}^{n_Rel}}{{z_p}^{n_Rel} + {K_z}^{n_Rel}})) K_1 = \frac{alpha_r2 {z_p}^{n_Rel - 1} n_Rel {K_z}^{n_Rel}}{{({z_p}^{n_Rel} + {K_z}^{n_Rel})}^{2}} z_max = \frac{\frac{alpha_0}{{(\frac{Ca_TRPN_50}{Ca_TRPN_Max})}^{n_Hill}} - K_2}{alpha_r1 + K_1 + \frac{alpha_0}{{(\frac{Ca_TRPN_50}{Ca_TRPN_Max})}^{n_Hill}}} Ca_50 = Ca_50ref (1 + beta_1 (lambda - 1)) Ca_TRPN_50 = \frac{Ca_50 Ca_TRPN_Max}{Ca_50 + \frac{k_Ref_off}{k_on} (1 - (\frac{(1 + beta_0 (lambda - 1)) 0.5}{gamma_trpn}))} alpha_Tm = alpha_0 {(\frac{Ca_b}{Ca_TRPN_50})}^{n_Hill} beta_Tm = alpha_r1 + \frac{alpha_r2 z^{n_Rel - 1}}{z^{n_Rel} + {K_z}^{n_Rel}} \frac{d}{d time} (z) = alpha_Tm (1 - z) - (beta_Tm z)

Component: niederer_troponin

k_off = \{\begin{matrix} k_Ref_off (1 - (\frac{Tension}{gamma_trpn T_ref})) if 1 - (\frac{Tension}{gamma_trpn T_ref}) > 0.1 \\ k_Ref_off 0.1 otherwise \end{matrix} I_TRPN = (Ca_TRPN_Max - TRPN) k_off - (Ca_i TRPN k_on)

Component: niederer_Myofilaments

ExtensionRatio = \{\begin{matrix} 1 if time > 3e5 \\ 1 otherwise \end{matrix} lambda_prev = ExtensionRatio dExtensionRatiodt = 0 lambda = \{\begin{matrix} ExtensionRatio if ExtensionRatio > 0.8 \land ExtensionRatio ≦ 1.15 \\ 1.15 if ExtensionRatio > 1.15 \\ 0.8 otherwise \end{matrix}

Component: niederer_filament_overlap

overlap = 1 + beta_0 (lambda - 1)

Component: niederer_length_independent_tension

T_Base = \frac{T_ref z}{z_max}

Component: niederer_isometric_tension

T_0 = T_Base overlap

Component: niederer_Cross_Bridges

Q = Q_1 + Q_2 + Q_3 Tension = \{\begin{matrix} \frac{T_0 (a Q + 1)}{1 - Q} if Q < 0 \\ \frac{T_0 (1 + (a + 2) Q)}{1 + Q} otherwise \end{matrix} \frac{d}{d time} (Q_1) = A_1 dExtensionRatiodt - (alpha_1 Q_1) \frac{d}{d time} (Q_2) = A_2 dExtensionRatiodt - (alpha_2 Q_2) \frac{d}{d time} (Q_3) = A_3 dExtensionRatiodt - (alpha_3 Q_3)

Component: intracellular_ion_concentrations

Ca_b = Ca_TRPN_Max - TRPN \frac{d}{d time} (Na_i) = \frac{(- (I_Na + I_B_Na + I_NaCa 3 + I_NaK 3 + I_f_Na)) 1}{V_myo_uL F} \frac{d}{d time} (K_i) = \frac{(- (I_Stim + I_ss + I_B_K + I_t + I_K1 + I_f_K + (- 2) I_NaK)) 1}{V_myo_uL F} \frac{d}{d time} (TRPN) = I_TRPN \frac{d}{d time} (Ca_i) = beta_CMDN ((I_RyR - I_SERCA) + I_SR + I_TRPN - (\frac{(- 2) I_NaCa + I_pCa + I_CaB + I_LCC}{2 V_myo_uL F})) \frac{d}{d time} (Ca_SR) = \frac{V_myo_uL}{V_SR_uL} ((- I_RyR) + I_SERCA - I_SR)