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Hill equation.

H(x,k):=xx+kH(x, k) := \frac{x}{x + k}
(1)
H(x,k,n):=xnxn+knH(x, k, n) := \frac{x^n}{x^n + k^n}
(2)

Relative exponential function. The definition here is reciprocal to the Python one.

exprel(x):=xexp⁡(x)−1exprel(x) := \frac{x}{ \exp(x) - 1}
(3)

Logistic function

expit(x):=11+exp⁡(−x)expit(x) := \frac{1}{1 + \exp(-x)}
(4)

Thermal voltage.

VT=RTF≈26.7mVV_T = \frac{RT}{F} \approx 26.7 \text{mV}
(5)

GHK flux equation.

GHK(p,z,Vm,Si,So):=pzF⋅exprel(−zVm/VT)⋅(Si−Soexp⁡(−zVm/VT))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))
(6)

Ion concentrations:

nax=[Na+]xkx=[K+]xcax=[Ca2+]x\begin{align} na_x &= \mathrm{[Na^+]_x} \\ k_x &= \mathrm{[K^+]_x} \\ ca_x &= \mathrm{[Ca^{2+}]_x} \\ \end{align}
(7)

Reversal potentials:

ENa=VTln⁡naonaiEK=VTln⁡kokiECa=0.5VTln⁡caocaslEKr=VTln⁡(0.98ko+0.02nao0.98ki+0.02ai)\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}
(8)

General parameters

ParameterValueUnitsDescription
r_SR6μmRadius of SR
r_SL10.5μmRadius of sarcolemma
V_SR0.0903pLSR volume
V_NSR0.9V_SRpLNetwork SR volume
V_JSR0.1V_JSRpLJunctional SR volume
V_subSR0.046pLSub-SR volume
V_subSL0.137pLSub-sarcolemma volume
V_myo3.944pLCytosolic volume
A_cap1385.44μm²Cell membrane area
C_m1μFcm⁻²Cell membrane capacitance
ca_o1796μMExternal calcium concentration
na_o154578μMExternal sodium concentration
k_o5366μMExternal potassium concentration
[ATP]5000μMATP concentration

Cytosolic calcium diffusion

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

Calcium buffering in each compartment:

βCa,x=(1+ΣTrpn⋅KmTrpn,2(cax+KmTrpn,2)2+ΣCmdn⋅KmCmdn(cax+KmCmdn,2)2)−1KmTrpn,2=KmTrpnfPKATnIfPKATnI=1.61−0.611−TnIPKAp1−fracTnIp0\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}
(9)

Calcium diffusion space is divided into (rSL−rSR)/dx(r_{SL} - r_{SR}) / dx concentric compartments.

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

ddtcai=Dca⋅βCa,idx2⋅ji((ji+1)cai+1−2ji⋅cai+(ji−1)cai−1)ji=rSR/dx+i−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}
(10)

Otherwise,

ddtca1=Dca⋅βCa,1dx2⋅j1((j1+1)ca2−2j1⋅ca1+(j1−1)ca1+JCaSR)ddtcan=Dca⋅βCa,ndx2⋅jn((jn+1)can−2jn⋅can+(jn−1)can−1+JCaSL)j1=rSR/dxjn=rSL/dxcasr=ca1casl=can\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}
(11)
ParameterValueUnitsDescription
ΣTrpn35μMTotal troponin content
Km_Trpn0.5μMHalf-saturation Ca concentration
ΣCmdn30μMTotal calmodulin content
Km_Cmdn2.38μMHalf-saturation Ca concentration
D_ca7μm²ms⁻¹Calcium diffusion rate
dx0.1μmDiscretization distance
fracTnIp_00.062698-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).

JCaSR=VNSRVsubSR(Jleak−Jup)+JrelJrel=kRyR⋅PO1RyR⋅(caJSR−casr)Jtr=ktrCaSR(caJSR−caNSR)Jleak=0.5(1+5RyRCKp)kSRleakJup=VmaxSRfSR−rSR1+fSR+rSRddtPO1RyR=kaposRyR⋅H(casr,KmRyR,4)⋅PC1RyR−kanegRyR⋅PO1RyRddtcaJSR=βSR(−JrelVsubSR+JtrVNSR)/VJSRddtcaNSR=Jup−Jleak−JtrβSR=11+ΣCsqnKmcsqn(caJSR+Kmcsqn)2fSR=(casrKmfp)2rSR=(caNSRKmrSR)2KmRyR=3.51⋅expit(−caJSR−530200)+0.25PC1RyR=1−PO1RyRKmfp=min⁡(fCKIIPLB,fPKAPLB)fPKAPLB=(1−0.5531)1−fracPLBpfracPKAPLBo+0.5531fCKIIPLB=(1−0.5∗fracPLBCKp)\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}
(12)
ParameterValueUnitsDescription
k_RyR201/sRyR permeability
kapos_RyR10001/sRyR state transition rate
kaneg_RyR1601/sRyR state transition rate
Vmax_SR999.6μM/sSERCA reaction rate
Kmf_SR0.5μMCalcium affinity for SERCA
Kmr_SR7000KmfSRKmf_{SR}μMCalcium affinity for SERCA
kSR_leak0.0051/sSR leak rate
ktrCa_SR501/sCalcium diffusion rate from NSR to JSR
ΣCSQN$24750μMCalsequestrin concentration
Km_csqn800μMCalcium affinity for calsequestrin
fracPKA_PLBo00.920245-

Sarcolemmal ion channels

CmddtVm=−(INab+INaCa+ICaL+ICaT+If+Ito+IK1+IKs+IKr+INa+INaK+ICab+Istim)ddtnai=−(IfNa+INab+INa+3INaCa+3INaK)AcapCmFVmyoddtki=−(IfK+Ito+IK1+IKs+IKr+Istim−2INaK)AcapCmFVmyo\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}
(13)

Sodium channels

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

INa=GˉNa⋅mNa3⋅hNa⋅jNa(Vm−ENa)INa,b=GˉNa,b(Vm−ENa)dmNadt=αm−mNa(αm+βm)dhNadt=αh−hNa(αh+βh)djNadt=αj−mNa(αj+βj)αm=3.2ms−1exprel(−(Vm+47.13)/10)βm=0.08ms−1exp⁡(−Vm/11)αh=0.135ms−1exp⁡(−(Vm+80)/6.8)βh=7.6923ms−1expit((Vm+10.66)/11.1)αj=(−127140exp⁡(0.2444Vm)−3.474⋅10−5exp⁡(−0.04391Vm))Vm+37.781+exp⁡(0.311(Vm+79.23))/msβj=0.3ms−1exp⁡(−2.535⋅10−7Vm)expit(0.1(Vm+32))\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}
(14)
ParameterValueUnitsDescription
G_Na12.8mS/μFFast sodium channels conductance
G_Nab0.0026mS/μFBackground sodium channels conductance

Potassium currents

IK1=GK1⋅H(ko,210)(Vm−6.1373−EK)0.1653+exp⁡(0.0319(Vm−6.1373−EK))Ito=Gt⋅ir((1−fis)is,slow+fisis)(Vm−EK)IKs=2GKs⋅inKs2(0.68804+0.71283IKURPKAp)(Vm−EK)IKr=GKr⋅iOK(Vm−EKr)IfNa=fNa⋅Gf⋅iy⋅(Vm−ENa)IfK=(1−fNa)⋅Gf⋅iy⋅(Vm−EK)If=IfK+IfNadirdt=r∞−irτrdisdt=s∞−isτsdis,slowdt=slow∞−is,slowτs,slowdinKsdt=nks∞−inKsτnKsdiCK1dt=kb,IKriCK2−kf,IKriCK1+0.022348e0.01176VmiCK0−0.047002e−0.0631VmiCK1diCK2dt=−kb,IKriCK2+kf,IKriCK1−0.013733iCK2e0.038198Vm+6.89⋅10−5e−0.04178VmiOKdiOKdt=0.006497iIKe−0.03268Vm+0.013733iCK2e0.038198Vm−6.89⋅10−5e−0.04178VmiOK−0.090821e0.023391VmiOKdiIKdt=−0.006497iIKe−0.03268Vm+0.090821e0.023391VmiOKdiydt=y∞−iyτy\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}
(15)
s∞=expit((Vm+31.97156)/−4.64291)r∞=expit((Vm−3.55716)/14.61299)slow∞=s∞nks∞=αnksαnks+βnksαnks=0.00000481333/0.128∗exprel(−0.128∗(V+26.5))βnks=0.0000953333∗exp⁡(−0.038∗(V+26.5))τr=1000ms45.16exp⁡(0.03577(Vm+50))+98.9exp⁡(−0.1(Vm+38))τs=8.1ms+350ms⋅exp⁡(−1225(Vm+70)2)τs,slow=72.4ms+3700ms⋅exp⁡(−1900(Vm+70)2)1=iIK+iCK2+iOK+iCK0+iCK1y∞=expit(−0.15798(Vm+78.65))τy=1000ms0.56236exp⁡(−0.070472(Vm+75))+0.11885exp⁡(0.035249(Vm+75))\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}
(16)
ParameterValueUnitsDescription
G_K10.0515mS/μFPotassium channels conductance
G_t0.1mS/μFTransient outward potassium channels conductance
G_Ks0.05mS/μFPotassium channels conductance
τ_nKs750msPotassium channels time scale
G_Kr0.06mS/μFPotassium channels conductance
k_fIKr23.7611/sPotassium channels transition rate
k_bIKr36.7781/sPotassium channels transition rate
G_f0.021mS/μFFunny current conductance
f_Na0.021-Funny current sodium fraction
f_is0.706-Transient outward gating variable

Calcium currents

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

JCaSL=(2INaCa−ICaL−ICaT−ICab)ACAPCm2FVsubSLICaL=ICascale⋅GCaL⋅id⋅if⋅ifca⋅GHK(GCaL,2,Vm,casl,0.341cao)ICaT=gCaT⋅ib⋅ig(Vm+106.5−ECa)ICab=gCab(Vm−ECa)diddt=d∞−idτddifdt=f∞−ifτfdifcadt=(fca∞−ifca)(1−(fca∞>ifca)(Vm>−60mV))τfcadibdt=b∞−ibτbdigdt=g∞−igτgICascale=ICascale,0(1+0.561−fracLCCbpISOfracLCCbp0)INaCa=kNaCa⋅ICascalenai3caoexp⁡(gNaCaVm/VT)−nao3caslfNaCaexp⁡((gNaCa−1)VmF/RT)1+(nai3cao+nao3caslfNaCa)dNaCad∞=expit((V+11.1)/7.2)τd=(αdβd+γd)αd=1.4expit((Vm+35)/13)+0.25βd=1.4expit(−(Vm+5)/5)γd=expit((Vm−50)/20)f∞=expit(−(Vm+23.3)/5.4)τf=120+165∗expit((Vm−25)/10)+1125exp⁡(−(Vm+27)2/240)fca∞=(αfca+βfca+γfca+0.23)/1.46αfca=H(0.4875,casl,8)βfca=0.1expit(−(casl−0.5)/0.1)γfca=0.2expit(−(casl−0.75)/0.8)b∞=expit((Vm+37.49098)/5.40634)τb=0.6+5.4expit(−0.03(Vm+100))g∞=expit(−(Vm+66)/6)τg=1+40expit(−0.08(Vm+65))\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}
(17)
ParameterValueUnitsDescription
fNaCa1-
kNaCa2.268 * 10⁻¹⁶μAμF⁻¹μM⁻⁴
dNaCa10⁻¹⁶μM⁻⁴
gNaCa0.5-
G_CaL6.3 * 10⁻⁵m³s⁻¹F⁻¹
τ_fca10ms
g_CaT0.2mSμF⁻¹
g_Cab0.0008mSμF⁻¹
ICascale_00.95-
fracLCCbp_00.250657-
fracLCCbpISO0.525870-

Na-K pump

INaK=INaKmaxfNaKkoko+KmKoNaKnainNaKnainNaK+KmNaiNaKnNaKfNaK=(1+0.1245exp⁡(−0.1Vm/VT)+0.0365σNaKexp⁡(Vm/VT))−1σNaK=(exp⁡(nai/67.3mM)−1)/7\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}
(18)
ParameterValueUnitsDescription
Imax_NaK2.7μA/μFMaximal rate of Na-K pump
KmNai_NaK18600μM
KmKo_NaK1500μM
nNaK3.2-Hill coefficient for sodium of Na-K pump

Beta-adrenergic system

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

fPKACI=PKACI0+PKACIactH(ISO,PKACIKM)fPKACII=PKACII0+PKACIIactH(ISO,PKACIIKM)fPP1=PP10+PP1actH(PP1KI,ISO)fPLBp=PLBp0+PLBpactH(ISO,PLBpKM,PLBpn)fPLMp=PLMp0+PLMpactH(ISO,PLMpKM,PLMpn)TnIPKAp=TnIp0+TnIpactH(ISO,TnIpKM,TnIpn)LCCaPKAp=LCCap0+LCCapactH(ISO,LCCapKM)LCCbPKAp=LCCbp0+LCCbpactH(ISO,LCCbpKM)KURPKAp=KURp0+KURpactH(ISO,KURpKM)RyRPKAp=RyRp0+RyRpactH(ISO,RyRpKM)\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}
(19)
ParameterValueUnitsDescription
PKACI_00.0734-Basal PKACI activity
PKACI_act0.1995-Activated PKACI activity
PKACI_KM0.0139μMPKACI sensitivity to ISO
PKACII_00.1840-Basal PKACII activity
PKACII_act0.3444-Activated PKACII activity
PKACII_KM0.0103μMPKACII sensitivity to ISO
PP1_00.8927-Basal PP1 activity
PP1_act0.0492-Activated PP1 activity
PP1_KI0.00637μMPP1 sensitivity to ISO
PLBp_00.0824-Basal PLB phosphorylation
PLBp_act0.7961-Activated PLB phosphorylation
PLBp_KM0.00597μMPLB phosphorylation sensitivity to ISO
PLBp_n1.8167-Hill coefficient for ISO
PLMp_00.1172-Basal PLMp phosphorylation
PLMp_act0.6645-Activated PLMp phosphorylation
PLMp_KM0.00823μMPLM phosphorylation sensitivity to ISO
PLMp_n1.35784-Hill coefficient for ISO
TnIp_00.0669-
TnIp_act0.7524-
TnIp_KM0.007913μM
TnIp_n1.6736-
LCCap_00.2205-
LCCap_act0.2339-
LCCap_KM0.00726μM
LCCbp_0$0.2517-
LCCbp_act0.2461-
LCCbp_KM0.00695μM
KURp_00.4390-
KURp_act0.2563-
KURp_KM0.00557μM
RyRp_00.2054-
RyRp_act0.2399-
RyRp_KM0.0075135μM

CaMKII system

caavg=∑i=1NcaiNCaMKact=1−CaMKCaMK=1−(CaMKB+CaMKBOX+CaMKP+CaMKPOX+CaMKA+CaMKA2+CaMKAOX+CaMKOX)ddtCaMKB=−vIB−vBP−vBBoddtCaMKP=vAP+vBP−vPPoddtCaMKA=−vAP−vAI−vA1A2+vAoAddtCaMKA2=vA1A2ddtCaMKBOX=vIoBo−vBoPo+vBBoddtCaMKPOX=vAoPo+vBoPo+vPPoddtCaMKAOX=−vAoPo−vAoA−vAoIoddtCaMKOX=−vIoBo+vAoIo−vIoIvIB=kf⋅CaMK−kb⋅CaMKBvIoBo=kf⋅rCaMKO⋅CaMKOX−kb⋅CaMKBOXvAP=kf⋅rCaMKP⋅CaMKA−kb⋅CaMKPvAoPo=kf⋅rCaMKP⋅CaMKAOX−kb⋅CaMKPOXcamkb∞=kfa2CaMKcaavg2caavg2+kmCa2CaMK2+kfa4CaMKcaavg4caavg4+kmCa4CaMK4+kfbCaMKkf=vCaMK⋅camkb∞kb=vCaMK⋅(1−camkb∞)kph=kphosCaMK⋅aMKactvBP=kph⋅CaMKB−kdephCaMK⋅CaMKPvBoPo=kph⋅CaMKBOX−kdephCaMK⋅CaMKPOXvA1A2=kP1P2⋅CaMKA−kP2P1⋅CaMKA2vAI=kdephCaMK⋅CaMKAvAoIo=kdephCaMK⋅CaMKAOXvBBo=koxCaMK⋅ROS⋅CaMKB−krdCaMK⋅CaMKBOXvPPo=koxCaMK⋅ROS⋅CaMKP−krdCaMK⋅CaMKPOXvIoI=krdCaMK⋅CaMKOXvAoA=krdCaMK⋅CaMKAOX\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}
(20)
ParameterValueUnitsDescription
v_CaMK3HzCaMK-CaM binding rate
r_CaMKO0-Oxidized CaMK-CaM binding ratio
r_CaMKP0-Phosphorylated CaMK-CaM binding ratio
kb_CaMKP1/3HzDissociation rate of CaMKP
kfa2_CaMK0.2650-Maximal CaM-Ca2 binding ratio
kfa4_CaMK0.1636-Maximal CaM-Ca4 binding ratio
kfb_CaMK0.001-Basal CaMK-CaM binding ratio
kmCa2_CaMK0.7384μMHalf-saturation calcium concentration for CaM-Ca2 binding
kmCa4_CaMK1.2513μMHalf-saturation calcium concentration for CaM-Ca4 binding
kphos_CaMK5HzAutophosphorylation rate
kdeph_CaMK1/6HzDephosphorylation rate
k_P1P21/60HzSecond autophosphorylation rate
k_P2P11/15HzSecond dephosphorylation rate
kox_CaMK291Hz/mMOxidation rate
krd_CaMK1/45HzReduction rate

Initial conditions

See the Initial conditions page.

NRVM CaMKII model
NRVM CaMKII model
NRVM CaMKII model
CaMKII system simplification