Model descriptions
Hill equation.
\[ H(x, k) := \frac{x}{x + k} \]
\[ H(x, k, n) := \frac{x^n}{x^n + k^n} \]
Relative exponential function. The definition here is reciprocal to the Python one.
\[ exprel(x) := \frac{x}{ \exp(x) - 1} \]
Logistic function
\[ expit(x) := \frac{1}{1 + \exp(-x)} \]
Thermal voltage.
\[ V_T = \frac{RT}{F} \approx 26.7 \text{mV} \]
GHK flux equation.
\[ GHK(p, z, V_m, S_i, S_o) := pzF \cdot exprel(-zV_m/V_T) \cdot (S_i - S_o \exp(-zV_m/V_T)) \]
Ion concentrations:
\[ \begin{align} na_x &= \mathrm{[Na^+]_x} \\ k_x &= \mathrm{[K^+]_x} \\ ca_x &= \mathrm{[Ca^{2+}]_x} \\ \end{align} \]
Reversal potentials:
\[ \begin{align} E_{Na} &= V_T \ln \frac{na_o}{na_i} \\ E_{K } &= V_T \ln\frac{k_o}{k_i} \\ E_{Ca} &= 0.5V_T \ln \frac{ca_o}{ca_{sl}} \\ E_{Kr} &= V_T \ln\left( \frac{0.98 k_o + 0.02na_o}{0.98 k_i + 0.02a_i} \right) \\ \end{align} \]
General parameters
| Parameter | Value | Units | Description |
|---|---|---|---|
| r_SR | 6 | μm | Radius of SR |
| r_SL | 10.5 | μm | Radius of sarcolemma |
| V_SR | 0.0903 | pL | SR volume |
| V_NSR | 0.9V_SR | pL | Network SR volume |
| V_JSR | 0.1V_JSR | pL | Junctional SR volume |
| V_subSR | 0.046 | pL | Sub-SR volume |
| V_subSL | 0.137 | pL | Sub-sarcolemma volume |
| V_myo | 3.944 | pL | Cytosolic volume |
| A_cap | 1385.44 | μm² | Cell membrane area |
| C_m | 1 | μFcm⁻² | Cell membrane capacitance |
| ca_o | 1796 | μM | External calcium concentration |
| na_o | 154578 | μM | External sodium concentration |
| k_o | 5366 | μM | External potassium concentration |
| [ATP] | 5000 | μM | ATP concentration |
Cytosolic calcium diffusion
Cytosolic calcium is diffused between sub-sarcolemma (SL) and sub-sarcoplasmic (SR) spaces.
Calcium buffering in each compartment:
\[ \begin{align} \beta_{Ca,x} &= \left(1 + \frac{\Sigma Trpn \cdot Km_{Trpn, 2}}{(ca_x + Km_{Trpn, 2})^2} + \frac{\Sigma Cmdn \cdot Km_{Cmdn}}{(ca_x + Km_{Cmdn, 2})^2} \right)^{-1} \\ Km_{Trpn, 2} &= \frac{Km_{Trpn}}{fPKA_{TnI}} \\ fPKA_{TnI} &= 1.61 - 0.61 \frac{1 - TnI_{PKAp}}{1 - fracTnIp_0} \end{align} \]
Calcium diffusion space is divided into \((r_{SL} - r_{SR}) / dx\) concentric compartments.
For i = 2 to \((r_{SL} - r_{SR}) / dx - 1\)
\[ \begin{align} \frac{d}{dt}ca_i &= \frac{D_{ca} \cdot \beta_{Ca,i}}{dx^2 \cdot j_i} ((j_i + 1)ca_{i+1} - 2j_i \cdot ca_{i} + (j_i - 1)ca_{i-1}) \\ j_i &= r_{SR} / dx + i - 1 \end{align} \]
Otherwise,
\[ \begin{align} \frac{d}{dt}ca_1 &= \frac{D_{ca} \cdot \beta_{Ca,1}}{dx^2 \cdot j_1} ((j_1 + 1)ca_{2} - 2j_1 \cdot ca_{1} + (j_1 - 1)ca_{1} + J_{CaSR}) \\ \frac{d}{dt}ca_n &= \frac{D_{ca} \cdot \beta_{Ca,n}}{dx^2 \cdot j_n} ((j_n + 1)ca_{n} - 2j_n \cdot ca_{n} + (j_n - 1)ca_{n-1} + J_{CaSL}) \\ j_1 &= r_{SR} / dx \\ j_n &= r_{SL} / dx \\ ca_{sr} &= ca_1 \\ ca_{sl} &= ca_n \\ \end{align} \]
| Parameter | Value | Units | Description |
|---|---|---|---|
| ΣTrpn | 35 | μM | Total troponin content |
| Km_Trpn | 0.5 | μM | Half-saturation Ca concentration |
| ΣCmdn | 30 | μM | Total calmodulin content |
| Km_Cmdn | 2.38 | μM | Half-saturation Ca concentration |
| D_ca | 7 | μm²ms⁻¹ | Calcium diffusion rate |
| dx | 0.1 | μm | Discretization distance |
| fracTnIp_0 | 0.062698 | - | Baseline effect of PKA on Troponin |
Endoplasmic reticulum
Including ryanodine receptor (RyR) flux (Jrel), SERCA flux (Jup), SR leakage (Jleak), and calcium diffusion from NSR to JSR (Jtr).
\[ \begin{align} J_{CaSR} &= \frac{V_{NSR}}{V_{subSR}} (J_{leak} - J_{up}) + J_{rel} \\ J_{rel} &= k_{RyR} \cdot PO1_{RyR} \cdot (ca_{JSR} - ca_{sr}) \\ J_{tr} &= ktrCa_{SR} (ca_{JSR} - ca_{NSR}) \\ J_{leak} &= 0.5 (1 + 5 RyR_{CKp}) kSR_{leak} \\ J_{up} &= Vmax_{SR} \frac{fSR - rSR}{1 + fSR + rSR} \\ \frac{d}{dt} PO1_{RyR} &= kapos_{RyR} \cdot H(ca_{sr}, Km_{RyR}, 4) \cdot PC1_{RyR} - kaneg_{RyR} \cdot PO1_{RyR} \\ \frac{d}{dt} ca_{JSR} &= \beta_{SR} (-J_{rel} V_{subSR} + J_{tr} V_{NSR}) / V_{JSR} \\ \frac{d}{dt} ca_{NSR} &= J_{up} - J_{leak} - J_{tr} \\ \beta_{SR} &= \frac{1}{1 + \frac{\Sigma Csqn Km_{csqn}}{(ca_{JSR} + Km_{csqn})^2}} \\ fSR &= \left( \frac{ca_{sr}}{Kmfp} \right)^2 \\ rSR &= \left( \frac{ca_{NSR}}{Kmr_{SR}} \right)^2 \\ KmRyR &= 3.51 \cdot expit(-\frac{ca_{JSR} - 530}{200}) + 0.25 \\ PC1_{RyR} &= 1 - PO1_{RyR} \\ Kmfp &= \min(fCKII_{PLB}, fPKA_{PLB}) \\ fPKA_{PLB} &= (1 - 0.5531) \frac{1 - fracPLBp}{fracPKA_PLBo} + 0.5531 \\ fCKII_{PLB} &= (1 - 0.5 * fracPLB_{CKp}) \\ \end{align} \]
| Parameter | Value | Units | Description |
|---|---|---|---|
| k_RyR | 20 | 1/s | RyR permeability |
| kapos_RyR | 1000 | 1/s | RyR state transition rate |
| kaneg_RyR | 160 | 1/s | RyR state transition rate |
| Vmax_SR | 999.6 | μM/s | SERCA reaction rate |
| Kmf_SR | 0.5 | μM | Calcium affinity for SERCA |
| Kmr_SR | 7000\(Kmf_{SR}\) | μM | Calcium affinity for SERCA |
| kSR_leak | 0.005 | 1/s | SR leak rate |
| ktrCa_SR | 50 | 1/s | Calcium diffusion rate from NSR to JSR |
| ΣCSQN$ | 24750 | μM | Calsequestrin concentration |
| Km_csqn | 800 | μM | Calcium affinity for calsequestrin |
| fracPKA_PLBo | 00.920245 | - |
Sarcolemmal ion channels
\[ \begin{align} C_m \frac{d}{dt} \mathrm{V_m} &= -(I_{Nab} + I_{NaCa} + I_{CaL} + I_{CaT} + I_{f} + I_{to} + I_{K1} + I_{Ks} + I_{Kr} + I_{Na} + I_{NaK} + I_{Cab} + I_{stim}) \\ \frac{d}{dt} \mathrm{na_i} &= -(I_{fNa} + I_{Nab} + I_{Na} + 3 I_{NaCa} + 3 I_{NaK}) \frac{A_{cap} C_m}{ F V_{myo}} \\ \frac{d}{dt} \mathrm{k_i} &= -(I_{fK} + I_{to} + I_{K1} + I_{Ks} + I_{Kr} + I_{stim} - 2 I_{NaK}) \frac{A_{cap} C_m}{ F V_{myo}} \\ \end{align} \]
Sodium channels
Including fast sodium (\(\mathrm{I_{Na}}\)) and background sodium (\(\mathrm{I_{Na,b}}\)) currents.
\[ \begin{align} \mathrm{I_{Na}} &= \bar G_{Na} \cdot m_{Na}^{3} \cdot h_{Na} \cdot j_{Na} (V_m-E_{Na}) \\ \mathrm{I_{Na,b}} &= \bar G_{Na,b} (V_m - E_{Na}) \\ \frac{dm_{Na}}{dt} &= \alpha_{m} - m_{Na}(\alpha_{m} + \beta_{m}) \\ \frac{dh_{Na}}{dt} &= \alpha_{h} - h_{Na}(\alpha_{h} + \beta_{h}) \\ \frac{dj_{Na}}{dt} &= \alpha_{j} - m_{Na}(\alpha_{j} + \beta_{j}) \\ \alpha_{m} &= 3.2 \mathrm{ms}^{-1} exprel(-(V_m + 47.13) / 10) \\ \beta_{m} &= 0.08 \mathrm{ms}^{-1} \exp(-V_m / 11) \\ \alpha_{h} &= 0.135 \text{ms}^{-1} \exp(-(V_m+80)/6.8) \\ \beta_{h} &= 7.6923 \mathrm{ms}^{-1} expit((V_m+10.66)/11.1) \\ \alpha_{j} &= (-127140 \exp(0.2444 V_m)-3.474 \cdot 10^{-5}\exp(-0.04391 V_m))\frac{V_m + 37.78}{1 + \exp(0.311( V_m + 79.23))} / \text{ms} \\ \beta_{j} &= 0.3 \mathrm{ms}^{-1} \exp(-2.535 \cdot 10^{-7}V_m) expit(0.1(V_m + 32)) \\ \end{align} \]
| Parameter | Value | Units | Description |
|---|---|---|---|
| G_Na | 12.8 | mS/μF | Fast sodium channels conductance |
| G_Nab | 0.0026 | mS/μF | Background sodium channels conductance |
Potassium currents
\[ \begin{align} \mathrm{I_{K1}} &= \mathrm{G_{K1}} \cdot H(\mathrm{k_o}, 210) \frac{ ( V_m - 6.1373 - E_K )}{0.1653 + \exp(0.0319 (V_m - 6.1373 - E_K))} \\ \mathrm{I_{to}} &= G_t \cdot i_r (( 1 - f_{is} ) i_{s,slow} + f_{is} i_s )( V_m - E_K) \\ \mathrm{I_{Ks}} &= 2 G_{Ks} \cdot i_{nKs}^{2}( 0.68804 + 0.71283 \mathrm{IKUR_{PKAp}}) (V_m - E_K) \\ \mathrm{I_{Kr}} &= G_{Kr} \cdot i_{OK} (V_m - E_{Kr}) \\ \mathrm{I_{fNa}} &= f_{Na} \cdot G_f \cdot i_y \cdot ( V_m - E_{Na}) \\ \mathrm{I_{fK}} &= (1 - f_{Na}) \cdot G_f \cdot i_y \cdot ( V_m - E_K ) \\ \mathrm{I_{f}} &= \mathrm{I_{fK}} + \mathrm{I_{fNa}} \\ \frac{d i_r }{dt} &= \frac{ r_∞ - i_r }{τ_r} \\ \frac{d i_s }{dt} &= \frac{ s_∞ - i_s }{τ_s} \\ \frac{d i_{s,slow} }{dt} &= \frac{ slow_∞ - i_{s,slow} }{τ_{s,slow} } \\ \frac{d i_{nKs} }{dt} &= \frac{ nks_∞ - i_{nKs} }{τ_{nKs}} \\ \frac{d i_{CK1} }{dt} &= k_{b, IKr} i_{CK2} - k_{f, IKr} i_{CK1} + 0.022348 e^{0.01176 V_m } i_{CK0} - 0.047002 e^{ - 0.0631 V_m } i_{CK1} \\ \frac{d i_{CK2} }{dt} &= - k_{b, IKr} i_{CK2} + k_{f, IKr} i_{CK1} - 0.013733 i_{CK2} e^{0.038198 V_m} + 6.89 \cdot 10^{-5} e^{ - 0.04178 V_m} i_{OK} \\ \frac{d i_{OK} }{dt} &= 0.006497 i_{IK} e^{-0.03268 V_m} + 0.013733 i_{CK2} e^{0.038198 V_m} - 6.89 \cdot 10^{-5} e^{-0.04178 V_m} i_{OK} - 0.090821 e^{0.023391 V_m} i_{OK} \\ \frac{d i_{IK} }{dt} &= - 0.006497 i_{IK} e^{ - 0.03268 V_m } + 0.090821 e^{0.023391 V_m } i_{OK} \\ \frac{d i_y}{dt} &= \frac{y_∞ - i_y}{τ_y} \\ \end{align} \]
\[ \begin{align} s_∞ &= expit((V_m + 31.97156) / -4.64291) \\ r_∞ &= expit((V_m - 3.55716) / 14.61299) \\ slow_∞ &= s_∞ \\ nks_∞ &= \frac{\alpha_{nks}}{\alpha_{nks} + \beta_{nks}} \\ \alpha_{nks} &= 0.00000481333 / 0.128 * exprel(-0.128 * (V + 26.5)) \\ \beta_{nks} &= 0.0000953333 * \exp(-0.038 * (V + 26.5)) \\ τ_r &= \frac{1000 \text{ms}}{45.16 \exp(0.03577( V_m + 50)) + 98.9 \exp( - 0.1 ( V_m + 38))} \\ τ_s &= 8.1\text{ms} + 350\text{ms} \cdot \exp( - \frac{1}{225}(V_m + 70)^{2}) \\ τ_{s,slow} &= 72.4\text{ms} + 3700\text{ms} \cdot \exp( - \frac{1}{900}( V_m + 70 )^{2}) \\ 1 &= i_{IK} + i_{CK2} + i_{OK} + i_{CK0} + i_{CK1} \\ y_∞ &= expit(-0.15798 (V_m + 78.65)) \\ τ_y &= \frac{1000 \text{ms}}{0.56236 \exp(-0.070472 (V_m + 75)) + 0.11885 \exp(0.035249 ( V_m + 75))} \\ \end{align} \]
| Parameter | Value | Units | Description |
|---|---|---|---|
| G_K1 | 0.0515 | mS/μF | Potassium channels conductance |
| G_t | 0.1 | mS/μF | Transient outward potassium channels conductance |
| G_Ks | 0.05 | mS/μF | Potassium channels conductance |
| τ_nKs | 750 | ms | Potassium channels time scale |
| G_Kr | 0.06 | mS/μF | Potassium channels conductance |
| k_fIKr | 23.761 | 1/s | Potassium channels transition rate |
| k_bIKr | 36.778 | 1/s | Potassium channels transition rate |
| G_f | 0.021 | mS/μF | Funny current conductance |
| f_Na | 0.021 | - | Funny current sodium fraction |
| f_is | 0.706 | - | Transient outward gating variable |
Calcium currents
L-type calcium channels, T-type calcium channels, and background calcium currents.
\[ \begin{align} J_{CaSL} &= (2 I_{NaCa} - I_{CaL} - I_{CaT} - I_{Cab}) \frac{A_{CAP} C_m}{2 F V_{subSL}} \\ \mathrm{I_{CaL}} &= \mathrm{ICa_{scale}} \cdot G_{CaL} \cdot i_d \cdot i_f \cdot i_fca \cdot GHK(G_{CaL}, 2, V_m, \mathrm{ca_{sl}}, 0.341 \mathrm{ca_o}) \\ \mathrm{I_{CaT}} &= \mathrm{gCaT} \cdot i_b \cdot i_g ( V_m + 106.5 - \mathrm{E_Ca}) \\ \mathrm{I_{Cab}} &= \mathrm{gCab} (V_m - \mathrm{E_Ca}) \\ \frac{d i_d }{dt} &= \frac{d_∞ - i_d}{τ_d} \\ \frac{d i_f }{dt} &= \frac{f_∞ - i_f}{τ_f} \\ \frac{d i_{fca} }{dt} &= \frac{(fca_∞ - i_{fca}) \left( 1 - \left( fca_∞ > i_{fca} \right) \left( V_m > -60 \text{mV} \right) \right)}{τ_{fca}} \\ \frac{d i_b }{dt} &= \frac{b_∞ - i_b}{τ_b} \\ \frac{d i_g }{dt} &= \frac{g_∞ - i_g}{τ_g} \\ \mathrm{ICa_{scale}} &= \mathrm{ICa_{scale, 0}} \left( 1 + \frac{0.56}{1 - \frac{\mathrm{fracLCCbpISO}}{\mathrm{fracLCCbp0}}} \right) \\ \mathrm{I_{NaCa}} &= \mathrm{kNaCa} \cdot \mathrm{ICa_{scale}} \frac{\mathrm{na_i}^{3} \mathrm{ca_o} \exp( \mathrm{gNaCa} V_m /V_T ) - \mathrm{na_o}^{3} \mathrm{ca_{sl}} \mathrm{fNaCa} \exp (( \mathrm{gNaCa} - 1 ) V_m F /RT )} {1 + \left(\mathrm{na_i}^3 \mathrm{ca_o} + \mathrm{na_o}^{3} \mathrm{ca_{sl}} \mathrm{fNaCa} \right) \mathrm{dNaCa}} \\ d_∞ &= expit((V + 11.1) / 7.2) \\ τ_d &= (\alpha_d \beta_d + \gamma_d) \\ \alpha_d &= 1.4 expit((V_m + 35) / 13) + 0.25 \\ \beta_d &= 1.4 expit(-(V_m + 5) / 5) \\ \gamma_d &= expit((V_m - 50) / 20) \\ f_∞ &= expit(-(V_m + 23.3) / 5.4) \\ τ_f &= 120 + 165 * expit((V_m - 25) / 10) + 1125 \exp( -(V_m + 27)^{2} / 240) \\ fca_∞ &= (\alpha_{fca} + \beta_{fca} + \gamma_{fca} + 0.23) / 1.46 \\ \alpha_{fca} &= H(0.4875, \mathrm{ca_{sl}}, 8) \\ \beta_{fca} &= 0.1 expit(-(\mathrm{ca_{sl}} - 0.5) / 0.1) \\ \gamma_{fca} &= 0.2 expit(-(\mathrm{ca_{sl}} - 0.75) / 0.8) \\ b_∞ &= expit((V_m + 37.49098) / 5.40634) \\ τ_b &= 0.6 + 5.4 expit(-0.03 (V_m + 100)) \\ g_∞ &= expit(-(V_m + 66) / 6) \\ τ_g &= 1 + 40 expit(-0.08 (V_m + 65)) \\ \end{align} \]
| Parameter | Value | Units | Description |
|---|---|---|---|
| fNaCa | 1 | - | |
| kNaCa | 2.268 * 10⁻¹⁶ | μAμF⁻¹μM⁻⁴ | |
| dNaCa | 10⁻¹⁶ | μM⁻⁴ | |
| gNaCa | 0.5 | - | |
| G_CaL | 6.3 * 10⁻⁵ | m³s⁻¹F⁻¹ | |
| τ_fca | 10 | ms | |
| g_CaT | 0.2 | mSμF⁻¹ | |
| g_Cab | 0.0008 | mSμF⁻¹ | |
| ICascale_0 | 0.95 | - | |
| fracLCCbp_0 | 0.250657 | - | |
| fracLCCbpISO | 0.525870 | - |
Na-K pump
\[ \begin{align} \mathrm{I_{NaK}} &= I_{NaK}^{max} fNaK \frac{\mathrm{k_o}}{ \mathrm{k_o} + KmKo_{NaK} } \frac{\mathrm{na_i}^{nNaK}}{ \mathrm{na_i}^{nNaK} + KmNai_{NaK}^{nNaK} } \\ fNaK &= (1 + 0.1245 \exp(-0.1 V_m/ V_T) + 0.0365 \sigma_{NaK} \exp(V_m/V_T))^{-1} \\ \sigma_{NaK} &= (\exp(\mathrm{na_i} / 67.3 \text{mM}) - 1) / 7 \end{align} \]
| Parameter | Value | Units | Description |
|---|---|---|---|
| Imax_NaK | 2.7 | μA/μF | Maximal rate of Na-K pump |
| KmNai_NaK | 18600 | μM | |
| KmKo_NaK | 1500 | μM | |
| nNaK | 3.2 | - | Hill coefficient for sodium of Na-K pump |
Beta-adrenergic system
Activities are fitted to the steady-state activities in the Morroti model.
\[ \begin{align} f_{PKACI} &= PKACI_0 + PKACI_{act} H(ISO, PKACI_{KM}) \\ f_{PKACII} &= PKACII_0 + PKACII_{act} H(ISO, PKACII_{KM}) \\ f_{PP1} &= PP1_0 + PP1_{act} H(PP1_{KI}, ISO) \\ f_{PLBp} &= PLBp_0 + PLBp_{act} H(ISO, PLBp_{KM}, PLBp_{n}) \\ f_{PLMp} &= PLMp_0 + PLMp_{act} H(ISO, PLMp_{KM}, PLMp_{n}) \\ TnI_{PKAp} &= TnIp_0 + TnIp_{act} H(ISO, TnIp_{KM}, TnIp_{n}) \\ LCCa_{PKAp} &= LCCap_0 + LCCap_{act} H(ISO, LCCap_{KM}) \\ LCCb_{PKAp} &= LCCbp_0 + LCCbp_{act} H(ISO, LCCbp_{KM}) \\ KUR_{PKAp} &= KURp_0 + KURp_{act} H(ISO, KURp_{KM}) \\ RyR_{PKAp} &= RyRp_0 + RyRp_{act} H(ISO, RyRp_{KM}) \\ \end{align} \]
| Parameter | Value | Units | Description |
|---|---|---|---|
| PKACI_0 | 0.0734 | - | Basal PKACI activity |
| PKACI_act | 0.1995 | - | Activated PKACI activity |
| PKACI_KM | 0.0139 | μM | PKACI sensitivity to ISO |
| PKACII_0 | 0.1840 | - | Basal PKACII activity |
| PKACII_act | 0.3444 | - | Activated PKACII activity |
| PKACII_KM | 0.0103 | μM | PKACII sensitivity to ISO |
| PP1_0 | 0.8927 | - | Basal PP1 activity |
| PP1_act | 0.0492 | - | Activated PP1 activity |
| PP1_KI | 0.00637 | μM | PP1 sensitivity to ISO |
| PLBp_0 | 0.0824 | - | Basal PLB phosphorylation |
| PLBp_act | 0.7961 | - | Activated PLB phosphorylation |
| PLBp_KM | 0.00597 | μM | PLB phosphorylation sensitivity to ISO |
| PLBp_n | 1.8167 | - | Hill coefficient for ISO |
| PLMp_0 | 0.1172 | - | Basal PLMp phosphorylation |
| PLMp_act | 0.6645 | - | Activated PLMp phosphorylation |
| PLMp_KM | 0.00823 | μM | PLM phosphorylation sensitivity to ISO |
| PLMp_n | 1.35784 | - | Hill coefficient for ISO |
| TnIp_0 | 0.0669 | - | |
| TnIp_act | 0.7524 | - | |
| TnIp_KM | 0.007913 | μM | |
| TnIp_n | 1.6736 | - | |
| LCCap_0 | 0.2205 | - | |
| LCCap_act | 0.2339 | - | |
| LCCap_KM | 0.00726 | μM | |
| LCCbp_0$ | 0.2517 | - | |
| LCCbp_act | 0.2461 | - | |
| LCCbp_KM | 0.00695 | μM | |
| KURp_0 | 0.4390 | - | |
| KURp_act | 0.2563 | - | |
| KURp_KM | 0.00557 | μM | |
| RyRp_0 | 0.2054 | - | |
| RyRp_act | 0.2399 | - | |
| RyRp_KM | 0.0075135 | μM |
CaMKII system
\[ \begin{align} ca_{avg} &= \frac{\sum^N_{i=1} ca_i}{N} \\ CaMK_{act} &= 1 - CaMK \\ CaMK &= 1 - (CaMKB + CaMKBOX + CaMKP + CaMKPOX + CaMKA + CaMKA2 + CaMKAOX + CaMKOX) \\ \frac{d}{dt} CaMKB &= -v_{IB} - v_{BP} - v_{BBo} \\ \frac{d}{dt} CaMKP &= v_{AP} + v_{BP} - v_{PPo} \\ \frac{d}{dt} CaMKA &= -v_{AP} - v_{AI} - v_{A1A2} + v_{AoA} \\ \frac{d}{dt} CaMKA2 &= v_{A1A2} \\ \frac{d}{dt} CaMKBOX &= v_{IoBo} - v_{BoPo} + v_{BBo} \\ \frac{d}{dt} CaMKPOX &= v_{AoPo} + v_{BoPo} + v_{PPo} \\ \frac{d}{dt} CaMKAOX &= -v_{AoPo} - v_{AoA} - v_{AoIo} \\ \frac{d}{dt} CaMKOX &= -v_{IoBo} + v_{AoIo} - v_{IoI} \\ v_{IB} &= k_f \cdot CaMK - k_b \cdot CaMKB \\ v_{IoBo} &= k_f \cdot r_{CaMKO} \cdot CaMKOX - k_b \cdot CaMKBOX \\ v_{AP} &= k_f \cdot r_{CaMKP} \cdot CaMKA - k_b \cdot CaMKP \\ v_{AoPo} &= k_f \cdot r_{CaMKP} \cdot CaMKAOX - k_b \cdot CaMKPOX \\ camkb_\infty &= kfa2_{CaMK} \frac{ca_{avg}^2}{ca_{avg}^2 + kmCa2_{CaMK}^2} + kfa4_{CaMK} \frac{ca_{avg}^4}{ca_{avg}^4 + kmCa4_{CaMK}^4} + kfb_{CaMK} \\ k_f &= v_{CaMK} \cdot camkb_\infty \\ k_b &= v_{CaMK} \cdot (1 - camkb_\infty) \\ kph &= kphos_{CaMK} \cdot aMK_{act} \\ v_{BP} &= kph \cdot CaMKB - kdeph_{CaMK} \cdot CaMKP \\ v_{BoPo} &= kph \cdot CaMKBOX - kdeph_{CaMK} \cdot CaMKPOX \\ v_{A1A2} &= k_{P1P2} \cdot CaMKA - k_{P2P1} \cdot CaMKA2 \\ v_{AI} &= kdeph_{CaMK} \cdot CaMKA \\ v_{AoIo} &= kdeph_{CaMK} \cdot CaMKAOX \\ v_{BBo} &= kox_{CaMK} \cdot \mathtt{ROS} \cdot CaMKB - krd_{CaMK} \cdot CaMKBOX \\ v_{PPo} &= kox_{CaMK} \cdot \mathtt{ROS} \cdot CaMKP - krd_{CaMK} \cdot CaMKPOX \\ v_{IoI} &= krd_{CaMK} \cdot CaMKOX \\ v_{AoA} &= krd_{CaMK} \cdot CaMKAOX \\ \end{align} \]
| Parameter | Value | Units | Description |
|---|---|---|---|
| v_CaMK | 3 | Hz | CaMK-CaM binding rate |
| r_CaMKO | 0 | - | Oxidized CaMK-CaM binding ratio |
| r_CaMKP | 0 | - | Phosphorylated CaMK-CaM binding ratio |
| kb_CaMKP | 1/3 | Hz | Dissociation rate of CaMKP |
| kfa2_CaMK | 0.2650 | - | Maximal CaM-Ca2 binding ratio |
| kfa4_CaMK | 0.1636 | - | Maximal CaM-Ca4 binding ratio |
| kfb_CaMK | 0.001 | - | Basal CaMK-CaM binding ratio |
| kmCa2_CaMK | 0.7384 | μM | Half-saturation calcium concentration for CaM-Ca2 binding |
| kmCa4_CaMK | 1.2513 | μM | Half-saturation calcium concentration for CaM-Ca4 binding |
| kphos_CaMK | 5 | Hz | Autophosphorylation rate |
| kdeph_CaMK | 1/6 | Hz | Dephosphorylation rate |
| k_P1P2 | 1/60 | Hz | Second autophosphorylation rate |
| k_P2P1 | 1/15 | Hz | Second dephosphorylation rate |
| kox_CaMK | 291 | Hz/mM | Oxidation rate |
| krd_CaMK | 1/45 | Hz | Reduction rate |
Initial conditions
See the Initial conditions page.