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.

\[ V_T = \frac{RT}{F} \]

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:

\[\begin{split} \begin{align} na_x &= \mathrm{[Na^+]_x} \\ k_x &= \mathrm{[K^+]_x} \\ ca_x &= \mathrm{[Ca^{2+}]_x} \\ \end{align} \end{split}\]

Reversal potentials:

\[\begin{split} \begin{align} E_{Na} &= V_T \ln \frac{na_o}{na_i} \\ E_{K } &= V_T \ln\frac{k_o}{k_i} \\ E_{Ca} &= 0.5V_T \ln \frac{ca_o}{ca_{sl}} \\ E_{Kr} &= V_T \ln\left( \frac{0.98 k_o + 0.02na_o}{0.98 k_i + 0.02a_i} \right) \\ \end{align} \end{split}\]

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
\(V_{NSR}\)	0.9\(V_{SR}\)	pL	Network SR volume
\(V_{JSR}\)	0.1\(V_{JSR}\)	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
\(ca_o\)	1.796	mM	External calcium concentration
\(na_o\)	154.578	mM	External sodium concentration
\(k_o\)	5.366	mM	External potassium concentration
\(\mathrm{[ATP]}\)	5	mM	ATP concentration

Cytosolic calcium diffusion

Cytosolic calcium is diffused between sub-sarcolemma (SL) and sub-sarcoplasmic (SR) spaces.

Calcium buffering in each compartment:

\[\begin{split} \begin{align} \beta_{Ca,x} &= \left(1 + \frac{\Sigma Trpn \cdot Km_{Trpn, 2}}{(ca_x + Km_{Trpn, 2})^2} + \frac{\Sigma Cmdn \cdot Km_{Cmdn}}{(ca_x + Km_{Cmdn, 2})^2} \right)^{-1} \\ Km_{Trpn, 2} &= \frac{Km_{Trpn}}{fPKA_{TnI}} \\ fPKA_{TnI} &= 1.61 - 0.61 \frac{1 - TnI_{PKAp}}{1 - fracTnIp_0} \end{align} \end{split}\]

Calcium diffusion space is divided into \((r_{SL} - r_{SR}) / dx\) compartments.

For i = 2 to \((r_{SL} - r_{SR}) / dx - 1\)

\[\begin{split} \begin{align} \frac{d}{dt}ca_i &= \frac{D_{ca} \cdot \beta_{Ca,i}}{dx^2 \cdot j_i} ((j_i + 1)ca_{i+1} - 2j_i \cdot ca_{i} + (j_i - 1)ca_{i-1}) \\ j_i &= r_{SR} / dx + i - 1 \end{align} \end{split}\]

Otherwise,

\[\begin{split} \begin{align} \frac{d}{dt}ca_1 &= \frac{D_{ca} \cdot \beta_{Ca,1}}{dx^2 \cdot j_1} ((j_1 + 1)ca_{2} - 2j_1 \cdot ca_{1} + (j_1 - 1)ca_{1} + J_{CaSR}) \\ \frac{d}{dt}ca_n &= \frac{D_{ca} \cdot \beta_{Ca,n}}{dx^2 \cdot j_n} ((j_n + 1)ca_{n} - 2j_n \cdot ca_{n} + (j_n - 1)ca_{n-1} + J_{CaSL}) \\ j_1 &= r_{SR} / dx \\ j_n &= r_{SL} / dx \\ ca_{sr} &= ca_1 \\ ca_{sl} &= ca_n \\ \end{align} \end{split}\]

Parameter	Value	Units	Description
\(\Sigma Trpn\)	35	μM	Total troponin content
\(Km_{Trpn}\)	0.5	μM	Half-saturation Ca concentration
\(\Sigma Cmdn\)	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).

\[\begin{split} \begin{align} J_{CaSR} &= \frac{V_{NSR}}{V_{subSR}} (J_{leak} - J_{up}) + J_{rel} \\ J_{rel} &= k_{RyR} \cdot PO1_{RyR} \cdot (ca_{JSR} - ca_{sr}) \\ J_{tr} &= ktrCa_{SR} (ca_{JSR} - ca_{NSR}) \\ J_{leak} &= 0.5 (1 + 5 RyR_{CKp}) kSR_{leak} \\ J_{up} &= Vmax_{SR} \frac{fSR - rSR}{1 + fSR + rSR} \\ \frac{d}{dt} PO1_{RyR} &= kapos_{RyR} \cdot H(ca_{sr}, Km_{RyR}, 4) \cdot PC1_{RyR} - kaneg_{RyR} \cdot PO1_{RyR} \\ \frac{d}{dt} ca_{JSR} &= \beta_{SR} (-J_{rel} V_{subSR} + J_{tr} V_{NSR}) / V_{JSR} \\ \frac{d}{dt} ca_{NSR} &= J_{up} - J_{leak} - J_{tr} \\ \beta_{SR} &= \frac{1}{1 + \frac{\Sigma Csqn Km_{csqn}}{(ca_{JSR} + Km_{csqn})^2}} \\ fSR &= \left( \frac{ca_{sr}}{Kmfp} \right)^2 \\ rSR &= \left( \frac{ca_{NSR}}{Kmr_{SR}} \right)^2 \\ KmRyR &= 3.51 \cdot expit(-\frac{ca_{JSR} - 530}{200}) + 0.25 \\ PC1_{RyR} &= 1 - PO1_{RyR} \\ Kmfp &= \min(fCKII_{PLB}, fPKA_{PLB}) \\ fCKII_{PLB} &= (1 - 0.5531) \frac{1 - fracPLBp}{fracPKA_PLBo} + 0.5531 \\ fCKII_{PLB} &= (1 - 0.5 * fracPLB_{CKp}) \\ \end{align} \end{split}\]

Parameter	Value	Units	Description
\(k_{RyR}\)	20	Hz	RyR permeability
\(kapos_{RyR}\)	1000	Hz	RyR state transition rate
\(kaneg_{RyR}\)	160	Hz	RyR state transition rate
\(Vmax_{SR}\)	0.9996	μM/ms	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	Hz	SR leak rate
\(ktrCa_{SR}\)	50	Hz	Calcium dissusion rate from NSR to JSR
\(\Sigma Csqn\)	24.750	mM	Calsequestrin concentration
\(Km_{csqn}\)	0.8	mM	Calcium affinity for calsequestrin
\(fracPKA_{PLBo}\)	0.920245	-

Sarcolemmal ion channels

Sodium channels

Including fast sodium (\(\mathrm{I_{Na}}\)) and background sodium (\(\mathrm{I_{Na,b}}\)) currents.

\[\begin{split} \begin{align} \mathrm{I_{Na}} &= \bar G_{Na} \cdot m_{Na}^{3} \cdot h_{Na} \cdot j_{Na} (V_m-E_{Na}) \\ \mathrm{I_{Na,b}} &= \bar G_{Na,b} (V_m - E_{Na}) \\ \frac{dm_{Na}}{dt} &= \alpha_{m} - m_{Na}(\alpha_{m} + \beta_{m}) \\ \frac{dh_{Na}}{dt} &= \alpha_{h} - h_{Na}(\alpha_{h} + \beta_{h}) \\ \frac{dj_{Na}}{dt} &= \alpha_{j} - m_{Na}(\alpha_{j} + \beta_{j}) \\ \alpha_{m} &= 3.2 \mathrm{ms}^{-1} exprel(-(V_m + 47.13) / 10) \\ \beta_{m} &= 0.08 \mathrm{ms}^{-1} \exp(-V_m / 11) \\ \\ \text{For} \ V_m & \ge -40\text{mV} \\ \alpha_{h} &= \alpha_{j} = 0 \\ \beta_{h} &= 7.6923 \mathrm{ms}^{-1} expit((V_m+10.66)/11.1) \\ \beta_{j} &= 0.3 \mathrm{ms}^{-1} \exp(-2.535 \cdot 10^{-7}V_m) expit(0.1(V_m + 32)) \\ \\ \text{For} \ V_m & < -40\text{mV} \\ \alpha_{h} &= 0.135 \text{ms}^{-1} \exp(-(V_m+80)/6.8) \\ \alpha_{j} &= (-127140 \exp(0.2444 V_m)-3.474 \cdot 10^{-5}\exp(-0.04391 V_m))\frac{V_m + 37.78}{1 + \exp(0.311( V_m + 79.23))} / \text{ms} \\ \beta_{h} &= (3.56\exp(0.079 V_m) + 3.1 \cdot 10^{5} \exp(0.35 V_m)) / \text{ms} \\ \beta_{j} &= \frac{0.1212 \mathrm{ms}^{-1} \exp(-0.01052 V_m)}{1 + \exp(-0.1378(V_m + 40.14))} \\ \end{align} \end{split}\]

Parameter	Value	Units	Description
\(G_{Na}\)	12.8	mS/μF	Fast sodium channels conductance
\(G_{Na,b}\)	0.0026	mS/μF	Background sodium channels conductance

Potassium currents

\[\begin{split} \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} \\ 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} \end{split}\]

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
\(k_{f, IKr}\)	0.023761	1/ms	Potassium channels transition rate
\(k_{b, IKr}\)	0.036778	1/ms	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	-

Calcium currents

L-type calcium channels, T-type calcium channels, and background calcium currents.

\[\begin{split} \begin{align} J_{CaSL} &= (2 I_{NaCa} - I_{CaL} - I_{CaT} - I_{Cab}) \frac{A_{CAP} C_m}{2 F V_{subSL}} \\ \mathrm{I_{CaL}} &= \mathrm{ICa_{scale}} \cdot G_{CaL} \cdot i_d \cdot i_f \cdot i_fca \cdot GHK(G_{CaL}, 2, V_m, \mathrm{ca_{sl}}, 0.341 \mathrm{ca_o}) \\ \mathrm{I_{CaT}} &= \mathrm{gCaT} \cdot i_b \cdot i_g ( V_m + 106.5 - \mathrm{E_Ca}) \\ \mathrm{I_{Cab}} &= \mathrm{gCab} (V_m - \mathrm{E_Ca}) \\ \frac{d i_d }{dt} &= \frac{d_∞ - i_d}{τ_d} \\ \frac{d i_f }{dt} &= \frac{f_∞ - i_f}{τ_f} \\ \frac{d i_{fca} }{dt} &= \frac{(fca_∞ - i_{fca}) \left( 1 - \left( fca_∞ > i_{fca} \right) \left( V_m > -60 \text{mV} \right) \right)}{τ_{fca}} \\ \frac{d i_b }{dt} &= \frac{b_∞ - i_b}{τ_b} \\ \frac{d i_g }{dt} &= \frac{g_∞ - i_g}{τ_g} \\ \mathrm{ICa_{scale}} &= \mathrm{ICa_{scale, 0}} \left( 1 + \frac{0.56}{1 - \frac{\mathrm{fracLCCbpISO}}{\mathrm{fracLCCbp0}}} \right) \\ \mathrm{I_{NaCa}} &= \mathrm{kNaCa} \cdot \mathrm{ICa_{scale}} \frac{\mathrm{na_i}^{3} \mathrm{ca_o} \exp( \mathrm{gNaCa} V_m /V_T ) - \mathrm{na_o}^{3} \mathrm{ca_{sl}} \mathrm{fNaCa} \exp (( \mathrm{gNaCa} - 1 ) V_m F /RT )} {1 + \left(\mathrm{na_i}^3 \mathrm{ca_o} + \mathrm{na_o}^{3} \mathrm{ca_{sl}} \mathrm{fNaCa} \right) \mathrm{dNaCa}} \\ d_∞ &= expit((V + 11.1) / 7.2) \\ τ_d &= (\alpha_d \beta_d + \gamma_d) \\ \alpha_d &= 1.4 expit((V_m + 35) / 13) + 0.25 \\ \beta_d &= 1.4 expit(-(V_m + 5) / 5) \\ \gamma_d &= expit((V_m - 50) / 20) \\ f_∞ &= expit(-(V_m + 23.3) / 5.4) \\ τ_f &= 120 + 165 * expit((V_m - 25) / 10) + 1125 \exp( -(V_m + 27)^{2} / 240) \\ fca_∞ &= (\alpha_{fca} + \beta_{fca} + \gamma_{fca} + 0.23) / 1.46 \\ \alpha_{fca} &= H(0.4875, \mathrm{ca_{sl}}, 8) \\ \beta_{fca} &= 0.1 expit(-(\mathrm{ca_{sl}} - 0.5) / 0.1) \\ \gamma_{fca} &= 0.2 expit(-(\mathrm{ca_{sl}} - 0.75) / 0.8) \\ b_∞ &= expit((V_m + 37.49098) / 5.40634) \\ τ_b &= 0.6 + 5.4 expit(-0.03 (V_m + 100)) \\ g_∞ &= expit(-(V_m + 66) / 6) \\ τ_g &= 1 + 40 expit(-0.08 (V_m + 65)) \\ \end{align} \end{split}\]

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

Na-K pump

\[\begin{split} \begin{align} \mathrm{I_{NaK}} &= I_{NaK}^{max} fNaK \frac{\mathrm{k_o}}{ \mathrm{k_o} + KmKo_{NaK} } \frac{\mathrm{na_i}^{nNaK}}{ \mathrm{na_i}^{nNaK} + KmNai_{NaK}^{nNaK} } \\ fNaK &= (1 + 0.1245 \exp(-0.1 V_m/ V_T) + 0.0365 \sigma_{NaK} \exp(V_m/V_T))^{-1} \\ \sigma_{NaK} &= (\exp(\mathrm{na_i} / 67.3 \text{mM}) - 1) / 7 \end{align} \end{split}\]

Parameter	Value	Units	Description
\(I_{NaK}^{max}\)	2.7	μA/μF	Maximal rate of Na-K pump
\(KmNai_{NaK}\)	18.6	mM
\(KmKo_{NaK}\)	1.5	mM
\(nNaK\)	3.2	-

Beta-adrenergic system

Activities are fitted to the steady-state activities in the Morroti model.

\[\begin{split} \begin{align} f_{PKACI} &= PKACI_0 + PKACI_{act} H(ISO, PKACI_{KM}) \\ f_{PKACII} &= PKACII_0 + PKACII_{act} H(ISO, PKACII_{KM}) \\ f_{PP1} &= PP1_0 + PP1_{act} H(PP1_{KI}, ISO) \\ f_{PLBp} &= PLBp_0 + PLBp_{act} H(ISO, PLBp_{KM}, PLBp_{nHill}) \\ f_{PLMp} &= PLMp_0 + PLMp_{act} H(ISO, PLMp_{KM}, PLMp_{nHill}) \\ TnI_{PKAp} &= TnIp_0 + TnIp_{act} H(ISO, TnIp_{KM}, TnIp_{nHill}) \\ LCCa_{PKAp} &= LCCap_0 + LCCap_{act} H(ISO, LCCap_{KM}) \\ LCCb_{PKAp} &= LCCbp_0 + LCCbp_{act} H(ISO, LCCbp_{KM}) \\ KUR_{PKAp} &= KURp_0 + KURp_{act} H(ISO, KURp_{KM}) \\ RyR_{PKAp} &= RyRp_0 + RyRp_{act} H(ISO, RyRp_{KM}) \\ \end{align} \end{split}\]

Parameter	Value	Units	Description
\(PKACI_0\)	0.0734	-	Basal PKACI activity
\(PKACI_{act}\)	0.1995	-	Activated PKACI activity
\(PKACI_{KM}\)	0.0139	μM	PKACI sesitivity to iso
\(PKACII_0\)	0.1840	-	Basal PKACII activity
\(PKACII_{act}\)	0.3444	-	Activated PKACII activity
\(PKACII_{KM}\)	0.0103	μM	PKACII sesitivity to iso
\(PP1_0\)	0.8927	-	Basal PP1 activity
\(PP1_{act}\)	0.0492	-	Activated PP1 activity
\(PP1_{KI}\)	0.00637	μM	PP1 sesitivity to iso
\(PLBp_0\)	0.0824	-
\(PLBp_{act}\)	0.7961	-
\(PLBp_{KM}\)	0.00597	μM
\(PLBp_{nHill}\)	1.8167	-
\(PLMp_0\)	0.1172	-
\(PLMp_{act}\)	0.6645	-
\(PLMp_{KM}\)	0.00823	μM
\(PLMp_{nHill}\)	1.35784	-
\(TnIp_0\)	0.0669	-
\(TnIp_{act}\)	0.7524	-
\(TnIp_{KM}\)	0.007913	μM
\(TnIp_{nHill}\)	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

\[\begin{split} \begin{align} \mathrm{ca_{avg}} &= \frac{\sum^N_{i=1} \mathrm{ca_i}}{N} \\ CaMK_{act} &= 1 - CaMK \\ v_{IB} &= k_f \cdot CaMK - k_b \cdot CaMKB \\ v_{IoBo} &= k_f \cdot r_{CaMKO} \cdot CaMKOX - k_b \cdot CaMKBOX \\ v_{AP} &= k_f \cdot r_{CaMKP} \cdot CaMKA - k_b \cdot CaMKP \\ v_{AoPo} &= k_f \cdot r_{CaMKP} \cdot CaMKAOX - k_b \cdot CaMKPOX \\ kph &= kphos_{CaMK} \cdot aMK_{act} \\ v_{BP} &= kph \cdot CaMKB - kdeph_{CaMK} \cdot CaMKP \\ v_{BoPo} &= kph \cdot CaMKBOX - kdeph_{CaMK} \cdot CaMKPOX \\ v_{P1P2} &= k_{P1P2} \cdot CaMKA - k_{P2P1} \cdot CaMKA2 \\ v_{AI} &= kdeph_{CaMK} \cdot CaMKA \\ v_{AoIo} &= kdeph_{CaMK} \cdot CaMKAOX \\ v_{BBo} &= kox_{CaMK} \cdot \mathtt{ROS} \cdot CaMKB - krd_{CaMK} \cdot CaMKBOX \\ v_{PPo} &= kox_{CaMK} \cdot \mathtt{ROS} \cdot CaMKP - krd_{CaMK} \cdot CaMKPOX \\ v_{IoI} &= krd_{CaMK} \cdot CaMKOX \\ v_{AoA} &= krd_{CaMK} \cdot CaMKAOX \\ camkb_\infty &= kfa_{CaMK} H(\mathrm{ca_{avg}}, kmCa_{CaMK}, nCa_{CaMK}) + kfb_{CaMK} \\ CaMK &= 1 - (CaMKB + CaMKBOX + CaMKP + CaMKPOX + CaMKA + CaMKA2 + CaMKAOX + CaMKOX) \\ \end{align} \end{split}\]

Parameter	Value	Units	Description
\(r_{CaMK}\)	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
\(kfa_{CaMK}\)	0.4393	-	Activated CaMK-CaM binding ratio
\(kfb_{CaMK}\)	0.0056	-	Basal CaMK-CaM binding ratio
\(kmCa_{CaMK}\)	0.9716	μM	Calcium affinity to CaMK-CaM
\(nCa_{CaMK}\)	2.293	-	Hill coefficient for calcium
\(kphos_{CaMK}\)	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

ODE system

\[\begin{split} \begin{align} \frac{d\mathrm{na_i}}{dt} &= -(I_{Na} + 3I_{NaCa} + 3I_{NaK})\frac{A_{cap}}{V_{myo}F} + (V_{NHE} - 3V_{NaCa}) \frac{V_{mito}}{V_{myo}} \\ \frac{d[K^+]_i}{dt} &= -(I_{Ks} + I_{Kr} + I_{K1} + I_{Kp} + I_{Ca,K}-2I_{NaK})\frac{A_{cap}}{V_{myo}F} \\ C_m\frac{dV_m}{dt} &= -(I_{Na} + I_{CaL} + I_{Kr} + I_{Ks} + I_{K1} + I_{Kp} + I_{NaCa} + I_{NaK} + I_{pCa} + I_{Ca, b} + I_{K_{ATP}} + I_{stim}) \\ β_i &= \frac{(K_m^{CMDN} + [Ca^{2+}]_i)^2}{ (K_m^{CMDN} + [Ca^{2+}]_i)^2 + K_m^{CMDN} \cdot [CMDN]_{tot}} \\ β_{SR} &= \frac{(K_m^{CSQN} + [Ca^{2+}]_{SR})^2}{(K_m^{CSQN} + [Ca^{2+}]_{SR})^2 + K_m^{CSQN} \cdot [CSQN]_{tot}} \\ \frac{d[Ca^{2+}]_i}{dt} &= \beta_i(J_{xfer}\frac{V_{ss}}{V_{myo}} - J_{up} - J_{trpn} - (I_{Ca,b} -2I_{NaCa} + I_{pCa})\frac{A_{cap}}{2V_{myo}F} + (V_{NaCa} - V_{uni})\frac{V_{mito}}{V_{myo}}) \\ \frac{d[Ca^{2+}]_{SR}}{dt} &= \beta_{SR}(J_{up}\frac{V_{myo}}{V_{SR}} - J_{rel}\frac{V_{ss}}{V_{SR}}) \\ \end{align} \end{split}\]