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sampson-2010.mmt
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[[model]]
name: sampson-2010
version: 20240904
mmt_authors: Michael Clerx
display_name: Sampson-Iyer et al., 2010
desc: """
The 2010 model of the human Purkinje cell AP by Sampson et al. [1].
This implementation is based on the original fortran code published by the
authors as supplement to [1]. It was checked against the original code by
comparing the calculated derivatives (which matched to within machine
precision).
The stimulus was set to 0.5ms and approximately half the depolarisation
threshold.
[1] Sampson*, K. J., Iyer*, V., Marks, A. R., & Kass, R. S. (2010). A
computational model of Purkinje fibre single cell electrophysiology:
implications for the long QT syndrome. Journal of Physiology, 588(14),
2643-2655. (Sampson & Iyer contributed equally)
https://doi.org/10.1113/jphysiol.2010.187328
"""
# Initial values
membrane.V = -8.03864711399999976e+01
sodium.Nai = 1.20687050500000002e+01
potassium.Ki = 1.22483450199999993e+02
calcium.Cai = 7.56225546699999958e-05
calcium.CaNSR = 2.59898837400000027e-01
calcium.CaSS = 1.24655751999999995e-04
calcium.CaJSR = 2.59679515799999983e-01
ryr.C1 = 4.65590390899999984e-01
ryr.O1 = 6.34345983199999957e-04
ryr.C2 = 5.33775261499999987e-01
ryr.O2 = 2.58066283699999986e-09
ical.C0 = 8.74379838900000039e-01
ical.C1 = 2.55268330899999993e-02
ical.C2 = 2.79463949400000000e-04
ical.C3 = 1.35979496400000004e-06
ical.C4 = 2.48115507400000008e-09
ical.Open = 1.85837559400000008e-10
ical.CCa0 = 8.89536543300000065e-02
ical.CCa1 = 1.03878563100000005e-02
ical.CCa2 = 4.54899679899999998e-04
ical.CCa3 = 8.85360913299999938e-06
ical.CCa4 = 6.46181672899999939e-08
ical.yCa = 9.98998351100000015e-01
caflux.HTRPNCa = 9.74145534599999974e-01
caflux.LTRPNCa = 7.05428108299999967e-02
ito.C0 = 9.12446021800000007e-01
ito.C1 = 5.57395224500000022e-02
ito.C2 = 1.27681665400000005e-03
ito.C3 = 1.29938504799999993e-05
ito.O = 4.95957872400000005e-08
ito.CI0 = 1.41240034599999995e-02
ito.CI1 = 1.10576340999999998e-02
ito.CI2 = 4.37517185700000023e-03
ito.CI3 = 9.10523457199999969e-04
ito.OI = 5.71714766299999999e-05
ikr.C1 = 7.89638103000000036e-01
ikr.C2 = 7.51317227500000002e-04
ikr.C3 = 1.31802129699999990e-04
ikr.O = 2.70143786700000017e-05
ikr.I = 7.34187676599999975e-06
iks.xs_wt = 1.95694799400000008e-01
iks.xf_wt = 7.40047161399999998e-04
nav15.C3 = 6.13484026399999993e-01
nav15.C2 = 1.93896494799999999e-02
nav15.C1 = 2.47138734400000005e-04
nav15.O = 5.92094839499999983e-07
nav15.IC3 = 2.88855704800000002e-01
nav15.IC2 = 9.13117952200000020e-03
nav15.IF = 1.17674846899999997e-04
nav15.IS1 = 1.38841821899999998e-03
nav15.IS2 = 6.62152358299999966e-02
nav15.BC3 = 5.17784203000000017e-05
nav15.BC2 = 1.63649800299999998e-06
nav15.BC1 = 2.08587473400000016e-08
nav15.BO = 5.01137449999999989e-11
ihcn.hcn1 = 3.45377576399999997e-01
ihcn.hcn2 = 4.08547201099999979e-01
ihcn.hcn3 = 1.80893426200000013e-01
ihcn.hcn4 = 3.54868422399999967e-02
ihcn.hcn5 = 2.59639364999999980e-03
ihcn.hcn6 = 1.13417300900000000e-03
ihcn.hcn7 = 5.28665430399999966e-03
ihcn.hcn8 = 9.66413904500000066e-03
ihcn.hcn9 = 8.13729668700000075e-03
ihcn.hcn10 = 2.64074099199999987e-03
nav11.C3 = 4.00644227700000022e-01
nav11.C2 = 2.21514457499999992e-03
nav11.C1 = 4.69401528400000025e-06
nav11.O = 9.08194443799999939e-09
nav11.IC3 = 5.99583493199999998e-02
nav11.IC2 = 3.35246803199999981e-04
nav11.IF = 3.65674332999999992e-06
nav11.IS1 = 8.87492239200000053e-03
nav11.IS2 = 5.26928426500000047e-01
nav11.BC3 = 3.12749104299999976e-05
nav11.BC2 = 1.72916501500000004e-07
nav11.BC1 = 3.66022597099999997e-10
nav11.BO = 6.60499340399999993e-13
icat.n = 2.08213191599999990e-03
icat.l = 5.48343768800000020e-01
#
# Physical constants
# Appendix page 1, table 1
#
[phys]
F = 96.5 [C/mmol] : Faraday constant
in [C/mmol]
T = 310 [K] : Absolute temperature
in [K]
R = 8.315 [J/mol/K] : Ideal gas constant
in [J/mol/K]
RTF = R * T / F
in [mV]
VFRT = membrane.V / RTF
VFFRT = F * VFRT
in [C/mmol]
#
# Simulation engine variables
#
[engine]
time = 0 [ms]
in [ms]
bind time
pace = 0
bind pace
#
# Membrane potential
# Appendix page 24
#
[membrane]
use stimulus.i_stim
dot(V) = -(i_ion + i_diff + i_stim)
in [mV]
desc: Membane potential
label membrane_potential
i_ion = a1 + a2 + a3
in [A/F]
label cellular_current
a1 = nav15.INa + nav11.INa1 + ical.ICa + icat.I + ical.ICaK + ikr.IKr + iks.IKs + ihcn.IHCN
in [A/F]
a2 = ik1.IK1 + inaca.INaCa + inak.INaK + ito.Ito1 + isus.Isus
in [A/F]
a3 = ipca.IpCa
in [A/F]
i_diff = 0 [A/F]
in [A/F]
bind diffusion_current
#
# Stimulus current
#
[stimulus]
i_stim = engine.pace * amplitude
in [A/F]
amplitude = -30 [A/F] * 2
in [A/F]
#
# Nav1.5 current
# Appendix page 5
#
[nav15]
desc: """
Encodes a Markov model for the Nav1.5 current through SCN5A.
The paper [1] refers to [2] but the code given in the appendix (page 5) has
different values for mu1, mu2 and GNa. In the code, an extra state "BIF" is
given, which corresponds to the model used in [4], but the transition rates to
and from this extra state are set to zero.
As a result, the model given below is similar in form (but not in parameters)
to the mutation variant described in [2], but with the parameters from the WT
model and a lower transition rate from and to the "burst" states (mu1, mu2) and
a higher open conductivity GNa.
The intermediate paper [3] also uses the mutant model form (with changed
parameters) for the WT formulation. In [3] the value mu2 is equal to that used
in [1].
Used parameter values:
[1] Sampson & Kass (Appendix and code)
mu1 = 4.3e-8 [1/ms] # Transition rate to burst states
mu2 = 3e-4 [1/ms] # Transition rate out of burst states
GNa = 35 [mS/uF] # Open channel conductivity
[2] Clancy & Rudy 2002 (Data supplement)
mu1_wt = 0 # WT doesn't have this state
mu2_wt = 0 # WT doesn't have this state
mu1_1795insD = 1e-7
mu2_1795insD = 9.5e-4
GNa = 23.5
[3] Clancy & Kass 2003 (Data supplement)
mu1_wt = 1e-7
mu2_wt = 3e-4
GNa = 10.5
[4] Clancy & Kass 2004
No data supplement :(
The paper doesn't mention any of this so it's unclear why the WT version of [2]
was not used.
[1] Sampson, Iyer, Marks, Kass (2010) A computational model of Purkinje fibre
single cell electrophysiology: implications for the long QT syndrome
Journal of Physiology
http://dx.doi.org/10.1113/jphysiol.2010.187328
[2] Clancy, Rudy (2002) A single mutation underlies Brugada and LQT1
Circulation
* Equations given in data supplement
[3] Clancy, Tateyama, Liu, Wehrens, Kass (2003) Non-Equilibrium Gating in
Cardiac Na+ Channels: An Original Mechanism of Arrhythmia
Circulation
* Equations given in data supplement
[4] Clancy, Kass (2004) Theoretical Investigation of the Neuronal Na1 Channel
SCN1A: Abnormal Gating and Epilepsy
Biophysical Journal
* No data supplement, but the paper itself does contain a very clear
explanation of the markov model (but no equations).
"""
use membrane.V
a11 = 3.802 [1/ms] / (0.1027 * exp(-V / 17 [mV]) + 0.20 * exp(-V / 150 [mV]))
in [1/ms]
a12 = 3.802 [1/ms] / (0.1027 * exp(-V / 15 [mV]) + 0.23 * exp(-V / 150 [mV]))
in [1/ms]
a13 = 3.802 [1/ms] / (0.1027 * exp(-V / 12 [mV]) + 0.25 * exp(-V / 150 [mV]))
in [1/ms]
b11 = 0.1917 [1/ms] * exp(-V / 20.3 [mV])
in [1/ms]
b12 = 0.2 [1/ms] * exp(-(V - 5 [mV]) / 20.3 [mV])
in [1/ms]
b13 = 0.22 [1/ms] * exp(-(V - 10 [mV]) / 20.3 [mV])
in [1/ms]
a2 = 9.178 [1/ms] * exp(V / 29.68 [mV])
in [1/ms]
b2 = (a3 * a2 * a13) / (b3 * b13)
in [1/ms]
a3 = 3.7933e-7 [1/ms] * exp(-V / 7.7 [mV])
in [1/ms]
b3 = 0.0084 [1/ms] + 0.00002 [1/ms/mV] * V
in [1/ms]
a4 = a2 / 100
in [1/ms]
b4 = a3
in [1/ms]
a5 = a2 / 9.5e4
in [1/ms]
b5 = a3 / 50
in [1/ms]
# CX to BCX transitions, BCX to CX transitions
mu1 = 4.3e-8 [1/ms]
in [1/ms]
mu2 = 3.0e-4 [1/ms]
in [1/ms]
dot(C3) = IC3 * a3 + BC3 * mu2 + C2 * b11 - C3 * (a11 + b3 + mu1)
dot(C2) = IC2 * a3 + BC2 * mu2 + C1 * b12 + C3 * a11 - C2 * (a12 + b3 + mu1 + b11)
dot(C1) = IF * a3 + BC1 * mu2 + O * b13 + C2 * a12 - C1 * (a13 + b3 + mu1 + b12)
dot(O) = IF * b2 + BO * mu2 + C1 * a13 -O * (a2 + b13 + mu1)
dot(IC3) = IC2 * b11 + C3 * b3 - IC3 * (a11 + a3)
dot(IC2) = IF * b12 + C2 * b3 + IC3 * a11 - IC2 * (a12 + a3 + b11)
dot(IF) = IS1 * b4 + O * a2 + IC2 * a12 + C1 * b3 - IF * (a4 + b2 + b12 + a3)
dot(IS1) = IS2 * b5 + IF * a4 - IS1 * (a5 + b4)
dot(IS2) = IS1 * a5 - IS2 * b5
dot(BC3) = BC2 * b11 + C3 * mu1 - BC3 * (a11 + mu2)
dot(BC2) = BC1 * b12 + BC3 * a11 + C2 * mu1 - BC2 * (a12 + b11 + mu2)
dot(BC1) = BO * b13 + BC2 * a12 + C1 * mu1 - BC1 * (a13 + b12 + mu2)
dot(BO) = BC1 * a13 + O * mu1 - BO * (b13 + mu2)
GNa = 35 [mS/uF] : peak INa conductance
in [mS/uF]
INa = GNa * (O + BO) * (V - nernst.ENa)
desc: Na+ current
in [A/F]
#
# TTX sensitive Na current (Nav1.1)
# Appendix page 8
#
[nav11]
desc: """
Encodes a Markov model for the Nav1.1 current through SCN1A.
"""
use membrane.V
a11 = 2.802 [1/ms] / (0.21 * exp(-V / 17 [mV]) + 0.23 * exp(-V / 150 [mV]))
in [1/ms]
a12 = 2.802 [1/ms] / (0.23 * exp(-V / 15 [mV]) + 0.25 * exp(-V / 150 [mV]))
in [1/ms]
a13 = 2.802 [1/ms] / (0.25 * exp(-V / 12 [mV]) + 0.27 * exp(-V / 150 [mV]))
in [1/ms]
b11 = 0.4 [1/ms] * exp(-V / 20.3 [mV])
in [1/ms]
b12 = 0.4 [1/ms] * exp(-(V - 5 [mV]) / 20.3 [mV])
in [1/ms]
b13 = 0.4 [1/ms] * exp(-(V - 10 [mV]) / 20.3 [mV]) / 4.5
in [1/ms]
a3 = 3.7933e-7 [1/ms] * exp(-V / 7.6 [mV]) * 3
in [1/ms]
b3 = 0.0084 [1/ms] + 0.00002 [1/ms/mV] * V
in [1/ms]
a2 = 9.178 [1/ms] * exp(V / 29.68 [mV]) / 4.5
in [1/ms]
a4 = a2 / 100 * 1.5 * 0.285
in [1/ms]
b4 = a3 / 5
in [1/ms]
a5 = a2 / 95000 * 80
in [1/ms]
b5 = a3 / 30 / 10
in [1/ms]
b2 = (a3 * a2 * a13) / (b3 * b13)
in [1/ms]
# CX to BCX transitions, BCX to CX transitions
mu1 = 4.3e-8 [1/ms]
in [1/ms]
mu2 = 3.0e-4 [1/ms]
in [1/ms]
dot(C3) = (IC3 * a3 + BC3 * mu2 + C2 * b11) - C3 * (a11 + b3 + mu1)
dot(C2) = (IC2 * a3 + BC2 * mu2+C1 * b12+C3 * a11) - C2 * (a12 + b3 + mu1 + b11)
dot(C1) = (IF * a3 + BC1 * mu2+O*b13+C2 * a12) - C1 * (a13+b3+mu1+b12)
dot(O) = (IF * b2 + BO*mu2+C1 * a13) - O * (a2 + b13 + mu1)
dot(IC3) = (IC2 * b11 + C3 * b3)- IC3 * (a11 + a3)
dot(IC2) = (IF * b12 + C2 * b3 + IC3 * a11) - IC2 * (a12 + a3 + b11)
dot(IF) = (IS1 * b4 + O * a2 + IC2 * a12 + C1 * b3) - IF * (a4 + b2 + b12 + a3)
dot(IS1) = (IS2 * b5 + IF * a4) - IS1 * (a5 + b4)
dot(IS2) = (IS1 * a5)-(IS2 * b5)
dot(BC3) = (BC2 * b11 + C3 * mu1) - BC3 * (a11 + mu2)
dot(BC2) = (BC1 * b12 + BC3 * a11 + C2 * mu1) - BC2 * (a12 + b11 + mu2)
dot(BC1) = (BO * b13 + BC2 * a12 + C1 * mu1) - BC1 * (a13 + b12 + mu2)
dot(BO) = (BC1 * a13 + O * mu1) - BO * (b13 + mu2)
GNa1 = 9 [mS/uF]: peak Nav1.1 conductance
in [mS/uF]
INa1 = GNa1 * (O + BO) * (V - nernst.ENa)
desc: Nav1.1 current
in [A/F]
#
# IHCN sustained component of transient outward current / Funny current
# Appendix page 14
#
[ihcn]
use membrane.V, phys.VFRT
use nernst.ENa, nernst.EK
h_f = 2.2361
h_gamma = 0.04025 [1/ms] * exp(-1.242 * phys.VFRT) / 1000
in [1/ms]
h_alpha = 0.0001712 [1/ms] * exp(-1.465 * phys.VFRT) / 1000
in [1/ms]
h_beta = 26.17 [1/ms] * exp(1.465 * phys.VFRT) / 1000
in [1/ms]
h_delta = 287.5 [1/ms] * exp(1.242 * phys.VFRT) / 1000
in [1/ms]
dot(hcn1) = -(4*h_gamma+h_alpha) * hcn1 + h_delta*hcn2 + h_beta*hcn6
dot(hcn2) = -(h_delta + 3 * h_gamma + h_alpha * h_f) * hcn2 + 4*h_gamma*hcn1 + 2 * h_delta*hcn3 + h_beta/h_f*hcn7
dot(hcn3) = -(2 * h_delta+2 * h_gamma+h_alpha*h_f^2) * hcn3 + 3*h_gamma*hcn2 + 3*h_delta*hcn4 + h_beta/h_f^2 * hcn8
dot(hcn4) = -(3*h_delta+h_gamma+h_alpha*h_f^3) * hcn4 + 2 * h_gamma*hcn3 + 4*h_delta*hcn5 + h_beta/h_f^3*hcn9
dot(hcn5) = -(4*h_delta+h_alpha*h_f^4) * hcn5 + h_gamma*hcn4 + h_beta/h_f^4*hcn10
dot(hcn6) = -(h_beta+4*h_gamma*h_f) * hcn6 + h_alpha*hcn1 + h_delta/h_f*hcn7
dot(hcn7) = -(h_beta/h_f+h_delta/h_f+3*h_gamma*h_f) * hcn7 + h_alpha*h_f*hcn2 + 4*h_gamma*h_f*hcn6 + 2 * h_delta/h_f*hcn8
dot(hcn8) = -(h_beta/h_f^2+2 * h_delta/h_f+2 * h_gamma*h_f) * hcn8 + h_alpha*h_f^2 * hcn3 + 3*h_gamma*h_f*hcn7 + 3*h_delta/h_f*hcn9
dot(hcn9) = -(h_beta/h_f^3+3*h_delta/h_f+h_gamma*h_f) * hcn9 + h_alpha*h_f^3*hcn4 + 2 * h_gamma*h_f*hcn8 + 4*h_delta/h_f*hcn10
dot(hcn10) = -(h_beta/h_f^4+ 4*h_delta/h_f) * hcn10 + h_alpha*h_f^4*hcn5 + h_gamma*h_f*hcn9
IHCNmax = 0.3225 [mS/uF]
in [mS/uF]
IHCN = IHCNmax * (hcn6 + hcn7 + hcn8 + hcn9 + hcn10) * (V - (ENa / 3 + (2 * EK / 3)))
in [A/F]
#
# IKr current
# Appendix page 10
#
[ikr]
use membrane.V
T_Const = 5.320000001 # T constant from 23 to 37C
A0 = 0.017147641733086 [1/ms]
in [1/ms]
B0 = 0.03304608038835 [1/mV]
in [1/mV]
A1 = 0.03969328381141 [1/ms]
in [1/ms]
B1 = -0.04306054163980 [1/mV]
in [1/mV]
A2 = 0.02057448605977 [1/ms]
in [1/ms]
B2 = 0.02617412715118 [1/mV]
in [1/mV]
A3 = 0.00134366604423 [1/ms]
in [1/ms]
B3 = -0.02691385498399 [1/mV]
in [1/mV]
A4 = 0.10666316491288 [1/ms]
in [1/ms]
B4 = 0.00568908859717 [1/mV]
in [1/mV]
A5 = 0.00646393910049 [1/ms]
in [1/ms]
B5 = -0.04536642959543 [1/mV]
in [1/mV]
A6 = 0.00008039374403 [1/ms]
in [1/ms]
B6 = 0.00000069808924 [1/mV]
in [1/mV]
C2H_to_C3H = T_Const * 0.02608362043337 [1/ms]
in [1/ms]
C3H_to_C2H = T_Const * 0.14832978132145 [1/ms]
in [1/ms]
C1H_to_C2H = T_Const * A0 * exp(B0 * V)
in [1/ms]
C2H_to_C1H = T_Const * A1 * exp(B1 * V)
in [1/ms]
C3H_to_OH = T_Const * A2 * exp(B2 * V)
in [1/ms]
OH_to_C3H = T_Const * A3 * exp(B3 * V)
in [1/ms]
OH_to_IH = T_Const * A4 * exp(B4 * V)
in [1/ms]
IH_to_OH = T_Const * A5 * exp(B5 * V)
in [1/ms]
C3H_to_IH = T_Const * A6 * exp(B6 * V)
in [1/ms]
IH_to_C3H = (OH_to_C3H*IH_to_OH*C3H_to_IH)/(C3H_to_OH*OH_to_IH)
in [1/ms]
dot(C1) = C2H_to_C1H * C2 - C1H_to_C2H * C1
desc: HERG channel state C1 (closed)
dot(C2) = a1 - a2
a1 = C1H_to_C2H * C1 + C3H_to_C2H * C3
in [1/ms]
a2 = (C2H_to_C1H + C2H_to_C3H) * C2
in [1/ms]
desc: HERG channel state C2
dot(C3) = a1 - a2
a1 = C2H_to_C3H*C2 + OH_to_C3H*O + IH_to_C3H*I
in [1/ms]
a2 = (C3H_to_IH + C3H_to_OH + C3H_to_C2H) * C3
in [1/ms]
desc: HERG channel state C3
dot(O) = a1 - a2
a1 = C3H_to_OH * C3 + IH_to_OH * I
in [1/ms]
a2 = (OH_to_C3H + OH_to_IH) * O
in [1/ms]
desc: HERG channel state O (Open)
dot(I) = a1 - a2
a1 = C3H_to_IH * C3 + OH_to_IH * O
in [1/ms]
a2 = (IH_to_C3H + IH_to_OH) * I
in [1/ms]
desc: HERG channel state I1(Inactivated)
GKr = 0.0383724 [mS/uF] : peak IKr conductance
in [mS/uF]
label g_Kr
fKo = sqrt(extra.Ko / 4 [mM])
IKr = GKr * fKo * O * (V - nernst.EK)
desc: rapid activating delayed rectifier K+ current
in [A/F]
#
# IKs
# Appendix page 11
#
[iks]
use membrane.V
dot(xs_wt) = alpha * (1 - xs_wt) - beta * xs_wt
alpha = if(V == -11 [mV],
33e-6 [1/ms/mV] / 0.13 [1/mV],
33e-6 [1/ms/mV] * (V + 11 [mV]) / (1 - exp(-0.13 [1/mV] * (V + 11 [mV]))))
in [1/ms]
beta = 1e-4 [1/ms] * (exp(-0.015 [1/mV] * V))
in [1/ms]
desc: IKs chanel state C0 (closed)
dot(xf_wt) = alpha * (1 - xf_wt) - beta * xf_wt
alpha = if(V == 21 [mV],
1.46e-4 [1/ms/mV] / 0.078 [1/mV],
1.46e-4 [1/ms/mV] * (V - 21 [mV]) / (1 - exp(-0.078 [1/mV] * (V - 21 [mV]))))
in [1/ms]
beta = 9.1e-4 [1/ms] * (exp(-0.028 [1/mV] * V))
in [1/ms]
desc: IKs chanel state C1 (closed)
GKs = 0.02808190224571 [mS/uF] : peak IKs conductance
in [mS/uF]
IKs = GKs * xs_wt * xf_wt * (V - nernst.EK)
desc: Slow activating delayed rectifier K+ current
in [A/F]
#
# L-type calcium current (ICaL)
# Appendix page 21
#
[ical]
use membrane.V
use extra.Cao, extra.Ko
bL = 2 : mode transition parameter
aL = 2 : mode transition parameter
alpha = 4 * 1.2 * 0.416 [1/ms] * exp( 0.012 [1/mV] * (V - 35 [mV]))
in [1/ms]
beta = 4 * 0.45 * 0.049 [1/ms] * exp(-0.065 [1/mV] * (V - 22 [mV]))
in [1/ms]
alpha_prime = aL * alpha
in [1/ms]
beta_prime = beta / bL
in [1/ms]
gamma = 0.6 * 0.09233 [1/ms/mM] * calcium.CaSS
in [1/ms]
C0_to_C1 = 4 * alpha
in [1/ms]
C1_to_C2 = 3 * alpha
in [1/ms]
C2_to_C3 = 2 * alpha
in [1/ms]
C3_to_C4 = alpha
in [1/ms]
CCa0_to_CCa1 = 4 * alpha_prime
in [1/ms]
CCa1_to_CCa2 = 3 * alpha_prime
in [1/ms]
CCa2_to_CCa3 = 2 * alpha_prime
in [1/ms]
CCa3_to_CCa4 = alpha_prime
in [1/ms]
C1_to_C0 = beta
in [1/ms]
C2_to_C1 = 2 * beta
in [1/ms]
C3_to_C2 = 3 * beta
in [1/ms]
C4_to_C3 = 4 * beta
in [1/ms]
CCa1_to_CCa0 = beta_prime
in [1/ms]
CCa2_to_CCa1 = 2 * beta_prime
in [1/ms]
CCa3_to_CCa2 = 3 * beta_prime
in [1/ms]
CCa4_to_CCa3 = 4 * beta_prime
in [1/ms]
C0_to_CCa0 = gamma
in [1/ms]
C1_to_CCa1 = aL*C0_to_CCa0 # gamma*aL
in [1/ms]
C2_to_CCa2 = aL*C1_to_CCa1 # gamma*aL^2
in [1/ms]
C3_to_CCa3 = aL*C2_to_CCa2 # gamma*aL^3
in [1/ms]
C4_to_CCa4 = aL*C3_to_CCa3 # gamma*aL^4
in [1/ms]
CCa0_to_C0 = omega
omega = .0025 [1/ms] : mode transition parameter
in [1/ms]
in [1/ms]
CCa1_to_C1 = CCa0_to_C0/bL # omega/bL
in [1/ms]
CCa2_to_C2 = CCa1_to_C1/bL # omega/bL^2
in [1/ms]
CCa3_to_C3 = CCa2_to_C2/bL # omega/bL^3
in [1/ms]
CCa4_to_C4 = CCa3_to_C3/bL # omega/bL^4
in [1/ms]
fL = 0.3 [1/ms] : transition rate into open state
in [1/ms]
gL = 4 [1/ms]: transition rate out of open state
in [1/ms]
dot(C0) = a2 - a1
a1 = (C0_to_C1 + C0_to_CCa0) * C0
in [1/ms]
a2 = C1_to_C0 * C1 + CCa0_to_C0 * CCa0
in [1/ms]
desc: L-type Ca++ channel state C0
dot(C1) = a2 - a1
a1 = (C1_to_C0 + C1_to_C2 + C1_to_CCa1) * C1
in [1/ms]
a2 = C0_to_C1 * C0 + C2_to_C1 * C2 + CCa1_to_C1 * CCa1
in [1/ms]
desc: L-type Ca++ channel state C1
dot(C2) = a2 - a1
a1 = (C2_to_C1 + C2_to_C3 + C2_to_CCa2) * C2
in [1/ms]
a2 = C1_to_C2 * C1 + C3_to_C2 * C3 + CCa2_to_C2 * CCa2
in [1/ms]
desc: L-type Ca++ channel state C2
dot(C3) = a2 - a1
a1 = (C3_to_C2+C3_to_C4+C3_to_CCa3) * C3
in [1/ms]
a2 = C2_to_C3*C2 + C4_to_C3*C4 + CCa3_to_C3*CCa3
in [1/ms]
desc: L-type Ca++ channel state C3
dot(C4) = a2 - a1
a1 = (C4_to_C3+fL+C4_to_CCa4) * C4
in [1/ms]
a2 = C3_to_C4*C3 + gL*Open + CCa4_to_C4*CCa4
in [1/ms]
desc: L-type Ca++ channel state C4
dot(Open) = fL*C4 - gL*Open
desc: L-type Ca++ channel state Open
dot(CCa0) = a2 - a1
a1 = (CCa0_to_CCa1+CCa0_to_C0) * CCa0
in [1/ms]
a2 = CCa1_to_CCa0 * CCa1 + C0_to_CCa0 * C0
in [1/ms]
desc: L-type Ca++ channel state CCa0
dot(CCa1) = a2 - a1
a1 = (CCa1_to_CCa0+CCa1_to_CCa2+CCa1_to_C1) * CCa1
in [1/ms]
a2 = CCa0_to_CCa1 * CCa0 + CCa2_to_CCa1 * CCa2 + C1_to_CCa1 * C1
in [1/ms]
desc: L-type Ca++ channel state CCa1
dot(CCa2) = a2 - a1
a1 = (CCa2_to_CCa1+CCa2_to_CCa3+CCa2_to_C2) * CCa2
in [1/ms]
a2 = CCa1_to_CCa2 * CCa1 + CCa3_to_CCa2 * CCa3 + C2_to_CCa2 * C2
in [1/ms]
desc: L-type Ca++ channel state CCa2
dot(CCa3) = a2 - a1
a1 = (CCa3_to_CCa2+CCa3_to_CCa4+CCa3_to_C3) * CCa3
in [1/ms]
a2 = CCa2_to_CCa3*CCa2 + CCa4_to_CCa3*CCa4 + C3_to_CCa3*C3
in [1/ms]
desc: L-type Ca++ channel state CCa3
dot(CCa4) = a2 - a1
a1 = (CCa4_to_CCa3+CCa4_to_C4) * CCa4
in [1/ms]
a2 = CCa3_to_CCa4*CCa3 + C4_to_CCa4*C4
in [1/ms]
desc: L-type Ca++ channel state CCa4
dot(yCa) = (yCa_inf-yCa) / tau_yCa
tau_yCa = 1 / (0.00336336209452 [1/ms] / (0.5 + exp(V / -5.53899874036055 [mV])) +
0.00779046570737 [1/ms] * exp(V / -49.51039631160386 [mV]))
in [ms]
yCa_inf = a1 / (1 + exp((V + 28.5 [mV]) / 7.8 [mV])) + (1 - a1)
a1 = 0.82
desc: L-type Ca++ channel inactivation
Pscale = 1.8 : scaling factor for both PK and PCa
PCa = Pscale * 2.469e-1 [L/F/s] : L-type Ca++ channel permeability to Ca++
in [L/F/s]
ICamax = PCa * 4 * phys.VFFRT * (a1 / a2)
a1 = 1e-3 [mM] * exp(2 * phys.VFRT) - Cao * 0.341
in [mM]
a2 = exp(2 * phys.VFRT) - 1
in [A/F]
ICa = ICamax * yCa * Open
desc: L-type channel Ca++ current
in [A/F]
# Potassium component
PK = ical.Pscale * 4.574e-4 [L/F/s] : L-type Ca++ channel permeability to K+
in [L/F/s]
ICahalf = -0.265 [A/F] : ICa level that reduces PK by half
in [A/F]
PKprime = PK / (1 + (imax / ICahalf))
imax = if(ICamax > 0 [A/F], 0 [A/F], ICamax)
in [A/F]
in [L/F/s]
ICaK = PKprime * Open * yCa * phys.VFFRT * (a1 / a2)
a1 = potassium.Ki * exp(phys.VFRT) - Ko
in [mM]
a2 = exp(phys.VFRT) - 1
desc: L-type channel K+ current
in [A/F]
#
# T-type calcium current (ICaT)
# Appendix page 23
#
[icat]
use membrane.V
Ttypescale = 7.5614e-3 [L/F/s]: fraction of Pscale that is Ttype
in [L/F/s]
dot(n) = (inf - n) / tau
inf = 1 / (1 + exp(-(V + 48.4 [mV]) / 5.2 [mV]))
tau = if(V < -56 [mV],
2.44 [ms] * exp((V + 120 [mV]) / 40 [mV]),
1.34 [ms] + 0.035 [ms] * exp(-V / 11.8 [mV]))
in [ms]
desc: T-type calcium current gate
dot(l) = (inf - l) / tau
inf = 1/(1+exp((V + 75.6 [mV]) / 6.2 [mV]))
tau = if(V < -60 [mV],
500 [ms],
18.3 [ms] + 0.005 [ms] * exp(-V / 6.2 [mV]))
in [ms]
desc: T-type calcium current gate
I = ical.ICamax / ical.PCa * Ttypescale * n * n * l
desc: T-type channel Ca++ current
in [A/F]
#
# Ito1 current, Kv4.3
# Appendix page 13
#
[ito]
use membrane.V
# Rate scaling factors
f1 = 1.8936
f2 = 14.224647456
f3 = 158.574378389
f4 = 142.936645351
b1 = 6.77348
b2 = 15.6212705152
b3 = 28.7532603313
b4 = 524.576206679
# Voltage dependent rate parameters
alphaa0 = 0.543708 [1/ms]
in [1/ms]
aa = 0.028983 [1/mV]
in [1/mV]
betaa0 = 0.080185 [1/ms]
in [1/ms]
ba = 0.0468437 [1/mV]
in [1/mV]
alphai0 = 0.0498424 [1/ms]
in [1/ms]
ai = 0.000373016 [1/mV]
in [1/mV]
betai0 = 0.000819482 [1/ms]
in [1/ms]
bi = 0.00000005374 [1/mV]
in [1/mV]
alpha_act43 = alphaa0 * exp(aa * V)
in [1/ms]
beta_act43 = betaa0 * exp(-ba * V)
in [1/ms]
alpha_inact43 = alphai0 * exp( - ai * V)
in [1/ms]
beta_inact43 = betai0 * exp(bi * V)
in [1/ms]
C0_to_C1 = 4*alpha_act43
in [1/ms]
C1_to_C2 = 3*alpha_act43
in [1/ms]
C2_to_C3 = 2 * alpha_act43
in [1/ms]
C3_to_O = alpha_act43
in [1/ms]
CI0_to_CI1 = 4*b1*alpha_act43
in [1/ms]
CI1_to_CI2 = 3*b2*alpha_act43/b1
in [1/ms]
CI2_to_CI3 = 2 * b3*alpha_act43/b2
in [1/ms]
CI3_to_OI = b4*alpha_act43/b3
in [1/ms]
C1_to_C0 = beta_act43
in [1/ms]
C2_to_C1 = 2 * beta_act43
in [1/ms]
C3_to_C2 = 3*beta_act43
in [1/ms]
O_to_C3 = 4*beta_act43
in [1/ms]
CI1_to_CI0 = beta_act43/f1
in [1/ms]
CI2_to_CI1 = 2 * f1*beta_act43/f2
in [1/ms]
CI3_to_CI2 = 3*f2*beta_act43/f3
in [1/ms]
OI_to_CI3 = 4*f3*beta_act43/f4
in [1/ms]
C0_to_CI0 = beta_inact43
in [1/ms]
C1_to_CI1 = f1*beta_inact43
in [1/ms]
C2_to_CI2 = f2*beta_inact43
in [1/ms]
C3_to_CI3 = f3*beta_inact43
in [1/ms]
O_to_OI = f4*beta_inact43
in [1/ms]
CI0_to_C0 = alpha_inact43
in [1/ms]
CI1_to_C1 = alpha_inact43/b1
in [1/ms]
CI2_to_C2 = alpha_inact43/b2
in [1/ms]
CI3_to_C3 = alpha_inact43/b3
in [1/ms]
OI_to_O = alpha_inact43/b4
in [1/ms]
dot(C0) = a2 - a1
a1 = (C0_to_C1+C0_to_CI0) * C0
in [1/ms]
a2 = C1_to_C0*C1 + CI0_to_C0*CI0
in [1/ms]
dot(C1) = a2 - a1
a1 = (C1_to_C2+C1_to_C0+C1_to_CI1) * C1
in [1/ms]
a2 = C2_to_C1*C2 + CI1_to_C1*CI1 + C0_to_C1*C0
in [1/ms]
dot(C2) = a2 - a1
a1 = (C2_to_C3+C2_to_C1+C2_to_CI2) * C2
in [1/ms]
a2 = C3_to_C2*C3 + CI2_to_C2*CI2 + C1_to_C2*C1
in [1/ms]
dot(C3) = a2 - a1
a1 = (C3_to_O+C3_to_C2+C3_to_CI3) * C3
in [1/ms]
a2 = O_to_C3*O + CI3_to_C3*CI3 + C2_to_C3*C2
in [1/ms]
dot(O) = a2 - a1
a1 = (O_to_C3+O_to_OI) * O
in [1/ms]
a2 = C3_to_O*C3 + OI_to_O*OI
in [1/ms]
dot(CI0) = a2 - a1
a1 = (CI0_to_C0+CI0_to_CI1) * CI0
in [1/ms]
a2 = C0_to_CI0*C0 + CI1_to_CI0*CI1
in [1/ms]
dot(CI1) = a2 - a1
a1 = (CI1_to_CI2+CI1_to_C1+CI1_to_CI0) * CI1
in [1/ms]
a2 = CI2_to_CI1*CI2 + C1_to_CI1*C1 + CI0_to_CI1*CI0
in [1/ms]
dot(CI2) = a2 - a1
a1 = (CI2_to_CI3+CI2_to_C2+CI2_to_CI1) * CI2
in [1/ms]
a2 = CI3_to_CI2*CI3 + C2_to_CI2*C2 + CI1_to_CI2*CI1
in [1/ms]
dot(CI3) = a2 - a1
a1 = (CI3_to_OI+CI3_to_C3+CI3_to_CI2) * CI3
in [1/ms]
a2 = OI_to_CI3*OI + C3_to_CI3*C3 + CI2_to_CI3*CI2
in [1/ms]
dot(OI) = a2 - a1
a1 = (OI_to_O+OI_to_CI3) * OI
in [1/ms]
a2 = O_to_OI*O + CI3_to_OI*CI3
in [1/ms]
G = 0.15213816 [mS/uF] : Maximum conductance of Kv4.3 channel
in [mS/uF]
Ito1 = G * O * (V - nernst.EK)
desc: Transient outward K+ current
in [A/F]
#
# Sodium-Potassium pump current (INaK)
# Appendix page 16
#
[inak]
use phys.VFRT
use extra.Ko, extra.Nao, sodium.Nai
KmNai = 20 [mM] : Na+ half sat. constant for Na+/K+ pump
in [mM]
KmKo = 1.5 [mM] : K+ half sat. constant for Na+/K+ pump
in [mM]
fNaK = 1 / (a1 + a2)
a1 = 1 + 0.1245 * exp(-0.1 * VFRT)
a2 = 0.0365 * sigma * exp(-1.33 * VFRT)
sigma = (exp(Nao / 67.3 [mM]) - 1) / 7
INaKmax = 1.5 [A/F] : maximum Na+/K+ pump current
in [A/F]
INaK = INaKmax * fNaK * (a1 / a2)
a1 = Ko / (Ko + KmKo)
a2 = 1 + (KmNai / Nai)^1.5
desc: Na+/K+ pump current
in [A/F]
#
# Sodium-Calcium exchange current (INaCa)
# Appendix page 16
#
[inaca]
use phys.VFRT
use extra.Cao, calcium.Cai
use extra.Nao, sodium.Nai
nao3 = Nao ^ 3
in [mM^3]
a1 = exp(eta * VFRT) * Nai^3 * Cao
in [mM^4]
a2 = exp((eta - 1) * VFRT) * nao3 * Cai
in [mM^4]
a3 = 1.0 + ksat * exp((eta - 1) * VFRT)
a4 = KmCa + Cao
in [mM]
a5 = 5000 / (KmNa^3 + nao3)
in [1/mM^3]
eta = 0.35 : controls voltage dependence of Na+/Ca++ exch.
ksat = 0.2 : Na+/Ca++ exch. sat. factor at negative potentials
KmNa = 87.5 [mM] : Na+ half sat. constant for Na+/Ca++ exch.
in [mM]
KmCa = 1.38 [mM] : Ca++ half sat. constant for Na+/Ca++ exch.
in [mM]
kNaCa = 0.44 [A/F] : scaling factor of Na+/Ca++ exchange (uA/uF)
in [A/F]
INaCa = kNaCa * a5 * (a1 - a2) / (a4 * a3)
in [A/F]
#
# Sarcolemmal Ca pump
# Appendix page 18
#
[ipca]
use calcium.Cai
IpCamax = 0.05 [A/F] : maximum sarcolemmal Ca++ pump current
in [A/F]
KmpCa = 0.0005 [mM] : half sat. constant for sarcolemmal Ca++ pump
in [mM]
IpCa = IpCamax * Cai / (KmpCa + Cai)
in [A/F]
#
# Sustained component of transient outward current (Isus)
# Appendix page 14
#
[isus]
use membrane.V
Isusmax = 0.0919908 [mS/uF]
in [mS/uF]
Isus = Isusmax * (1 / (1 + exp(-(V - 20 [mV]) / 12 [mV]))) * (V - nernst.EK)
desc: new sustained outward current, TEA sensitive
in [A/F]
#
# Inward rectifying current (IK1)
# Appendix page 16
#
[ik1]
use membrane.V
GK1 = 0.0226 [mS/uF] : peak IK1 conductance
in [mS/uF]
IK1 = GK1 * sqrt(extra.Ko * 1 [1/mM]) * inf * (V - nernst.EK)
inf = 1 / (0.9681 + exp((V + 82.18623 [mV]) / 15.88642 [mV]))
desc: time independent K+ current
in [A/F]
#
# Intracellular Sodium
# Appendix page 24
#
[sodium]
use cell.a1, cell.a2
INa_tot = nav15.INa + nav11.INa1 + ihcn.IHCN / 3 + 3*(inaca.INaCa + inak.INaK)
in [A/F]
dot(Nai) = - a1 * INa_tot
desc: intracellular Na+ conc
in [mM]
#
# Intracellular Potassium
# Appendix page 24
#
[potassium]
use cell.a1, cell.a2
IK_tot = (ikr.IKr + iks.IKs + ik1.IK1 + ical.ICaK + (2 * ihcn.IHCN / 3)
+ isus.Isus + stimulus.i_stim - 2 * inak.INaK + ito.Ito1)
in [A/F]
dot(Ki) = - a1 * IK_tot
desc: intracellular K+ concentration
in [mM]
#
# Ryanodyne Receptor current
# Appendix page 18
#
[ryr]
use calcium.CaSS, calcium.CaJSR
dot(C1) = -kaplus * a2 * C1 + kaminus * O1
a2 = (CaSS * 1000 [uM/mM])^4
in [uM^4]
kaplus = 0.01215 [1/uM^4/ms] : RyR PC1 > PO1 rate constant
in [1/uM^4/ms]
kaminus = 0.576 [1/ms] : RyR PO1 > PC1 rate constant
in [1/ms]
desc: RyR channel state C1
dot(O2) = kbplus * a1 * O1 - kbminus * O2
a1 = (CaSS * 1000 [uM/mM])^3
in [uM^3]
kbplus = 0.00405 [1/uM^3/ms] : RyR PO1 > PO2 rate constant
in [1/uM^3/ms]
kbminus = 1.930 [1/ms] : RyR PO2 > PO1 rate constant
in [1/ms]
desc: RyR channel state O2
dot(C2) = kcplus * O1 - kcminus * C2
kcplus = 0.1 [1/ms] : RyR PO1 > PC2 rate constant (1/ms)
in [1/ms]
kcminus = 0.0008 [1/ms] : RyR PC2 > PO1 rate constant (1/ms)
in [1/ms]
desc: RyR channel state C2
dot(O1) = -(dot(C1) + dot(O2) + dot(C2))
desc: RyR channel state O1
Jrel = v1 * (O1 + O2) * (CaJSR - CaSS)
v1 = 1.8 [1/ms] : max RyR channel Ca++ flux
in [1/ms]
desc: JSR Ca++ release flux
in [mM/ms]
#
# SERCA2a Pump
# Appendix page 19
#
[serca]
fb = (calcium.Cai / Kfb) ^ Nfb
rb = (calcium.CaNSR / Krb) ^ Nrb
Kfb = 0.000168 [mM] : foward half sat. constant for Ca++ ATPase
in [mM]
Krb = 3.29 [mM] : backward half sat. constant for Ca++ ATPase