-
Notifications
You must be signed in to change notification settings - Fork 0
/
Copy pathheijman-2011.mmt
2947 lines (2870 loc) · 114 KB
/
heijman-2011.mmt
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
[[model]]
name: heijman-2011
version: 20230202
mmt_authors: Michael Clerx, Jordi Heijman
display_name: Heijman et al., 2011
desc: """
The 2011 model of the canine ventricular AP with beta-adrenergic signalling
by Heijman et al. [1].
An attempt was made to annotate the source of each component. Where section
numbers are given these refer to sections in the data supplement of [1],
the appendix to the thesis [2], or the previous model paper [3].
The implementation was checked by comparing the derivatives calculated for
several starting points with those calculated from the original matlab
code [4].
References :
[1] Heijman, J., Volders, P. G. A., Westra, R. L., & Rudy. Y. (2011). Local
control of beta-adrenergic stimulation: Effects on ventricular myocyte
electrophysiology and Ca2+-transient. Journal of Molecular and Cellular
Cardiology. 50(5) 863-871
https://doi.org/10.1016/j.yjmcc.2011.02.007
[2] Heijman, J. (2012). Computational analysis of beta-adrenergic
stimulation and its effects on cardiac ventricular electrophysiology.
https://doi.org/10.26481/dis.20120427jh
[3] Decker K. F., Heijman J., Silva J. R., Hund, T. J. & Rudy, Y. (2009).
Properties and Ionic Mechanisms of Action Potential Adaptation,
Restitution and Accomodation in Canine Epicardium. American Journal of
Physiology. Heart and Circulatory Physiology, 296(4), H1017-H1026.
https://doi.org/10.1152/ajpheart.01216.2008
[4] The matlab code published for [1] and [2], which was downloaded from
http://rudylab.wustl.edu
[5] Livshitz, L. M., & Rudy, Y (2007). Regulation of Ca2+ and electrical
alternans in cardiac myocytes: Role of CaMKII and repolarizing
currents. American Journal of Physiology. Heart and Circulatory
Physiology, 292(6), H2854-H2866.
https://doi.org/10.1152/ajpheart.01347.2006
"""
# Initial values
membrane.V = -87.491 # 1 Vm
calcium.uCa = 1.3394e-2 # 2 Ca_i
calcium.uCa_ss = 2.3413e-2 # 3 Ca_ss
calcium.uCa_CaL = 2.3413e-2 # 4 Ca_srCaL
calcium.uCa_jsr = 6.8659 # 5 Ca_JSR
calcium.Ca_nsr = 1.1910 # 6 Ca_NSR
camk.trap = 1.7546e-3 # 7 Ca_trap
sodium.Na = 6.8909 # 8 Na_i
sodium.Na_ss = 6.8909 # 9 Na_ss
potassium.K = 1.4562e2 # 10 K_i
chloride.Cl = 20.273 # 11 Cl_i
chloride.Cl_ss = 20.273 # 12 Cl_SR
irel.Irel_np = 3.6675e-9 # 13 Irel
ina_np.h = 6.8172e-3 # 14 H
ina_np.m = 9.0163e-1 # 15 m
ina_np.j = 9.9709e-1 # 16 J
inal.m = 7.0530e-4 # 17 mL
inal.h = 3.6003e-1 # 18 hL
ito.a_np = 1.7687e-5 # 19 to_a
ito.if_np = 9.9798e-1 # 20 to_if
ito.is_np = 9.8747e-1 # 21 to_is
ikr.ac = 1.2306e-8 # 22 xr (Ikr activation gate)
iclca.i2 = 9.9604e-1 # 23 AA (ITo2 inactivation gate)
ical_np.C = 1
ical_np.O = 0
ical_np.Cs = 0
ical_np.Os = 0
ical_np.CI = 0
ical_np.OI = 0
ical_np.CIs = 0
ical_np.OIs = 0
iks_np.C1 = 9.1141e-1
iks_np.C2 = 8.4395e-2
iks_np.C3 = 2.9306e-3
iks_np.C4 = 4.52285e-5
iks_np.C5 = 2.6175e-7
iks_np.C6 = 1.1424e-3
iks_np.C7 = 7.9337e-5
iks_np.C8 = 1.8366e-6
iks_np.C9 = 1.4172e-8
iks_np.C10 = 5.3696e-7
iks_np.C11 = 2.4861e-8
iks_np.C12 = 2.8776e-10
iks_np.C13 = 1.1217e-10
iks_np.C14 = 2.5967e-12
iks_np.C15 = 8.7874e-15
iks_np.O1 = 9.3722e-16
iks_np.O2 = 1.6595e-17
ical_p.C = 1
ical_p.O = 0
ical_p.Cs = 0
ical_p.Os = 0
ical_p.CI = 0
ical_p.OI = 0
ical_p.CIs = 0
ical_p.OIs = 0
iks_pka.C1 = 0.95624
iks_pka.C2 = 4.2127e-2
iks_pka.C3 = 6.9597e-4
iks_pka.C4 = 5.1101e-6
iks_pka.C5 = 1.4070e-8
iks_pka.C6 = 9.0269e-4
iks_pka.C7 = 2.9826e-5
iks_pka.C8 = 3.2850e-7
iks_pka.C9 = 1.2060e-9
iks_pka.C10 = 3.1956e-7
iks_pka.C11 = 7.0390e-9
iks_pka.C12 = 3.8763e-11
iks_pka.C13 = 5.0277e-11
iks_pka.C14 = 5.5374e-13
iks_pka.C15 = 2.9664e-15
iks_pka.O1 = 1.1201e-15
iks_pka.O2 = 1.6613e-18
ina_pka.h = 1.2360e-3 # 74 H (P)
ina_pka.m = 7.9472e-1 # 75 m (P)
ina_pka.j = 9.9123e-1 # 76 J (P)
ina_camk.m = 6.8127e-4 # 77 m (P, CAMKII)
ina_camk.h = 8.3805e-1 # 78 h (P, CAMKII)
ina_camk.j = 9.9281e-1 # 79 J (P, CAMKII)
irel.Irel_p = 7.3074e-9 # 80 IRel (P)
ito.if_camk = 1 # 81 to_if_CaMKII
ito.is_camk = 1 # 82 to_is_CaMKII
camk.f_plb = 0 # 83 fPLBP_CaMKII
camk.f_ryr = 0 # 84 fRyRP_CaMKII
camk.f_ito = 0 # 85 fIToP_CaMKII
camk.f_ina = 0 # 86 fINaP_CaMKII
camk.f_ik1 = 0 # 87 fIK1P_CaMKII
camk.f_ical = 0 # 88 fICaLP_CaMKII
beta_cav.Gs_aGTP = 0.00685041638458665 # 89 1 Gs_aGTP_CAV
beta_eca.Gs_aGTP = 0.0184627603007976 # 90 2 Gs_aGTP_ECAV
beta_cyt.Gs_aGTP = 0.000731420577213056 # 91 3 Gs_a_GTP_CYT
beta_cav.Gs_bg = 0.00745773247314215 # 92 4 Gs_bg_CAV
beta_eca.Gs_bg = 0.0191017987408719 # 93 5 Gs_bg_ECAV
beta_cyt.Gs_bg = 0.00115141243826747 # 94 6 Gs_bg_CYT
beta_cav.Gs_aGDP = 0.000607316088556676 # 95 7 Gs_aGDP_CAV
beta_eca.Gs_aGDP = 0.000639038440072507 # 96 8 Gs_aGDP_ECAV
beta_cyt.Gs_aGDP = 0.000419991861054322 # 97 9 Gs_aGDP_CYT
camp.cAMP_cav = 0.347102959606005 # 98 10 cAMP_CAV
camp.cAMP_eca = 9.62359241535767 # 99 11 cAMP_ECAV
camp.cAMP_cyt = 0.474081735738211 # 100 12 cAMP_CYT
beta_cav.Rb1_pka_tot = 0.0149041813757831 # 101 13 R_pkap_tot_CAV
beta_eca.Rb1_pka_tot = 0.203016833596288 # 102 14 R_pkap_tot_ECAV
beta_cyt.Rb1_pka_tot = 0.00944463350378086 # 103 15 R_pkap_tot_CYT
beta_cav.Rb1_grk_tot = 2.49592854373432e-10 # 104 16 R_grkp_tot_CAV
beta_eca.Rb1_grk_tot = 1.18055788874765e-9 # 105 17 R_grkp_tot_ECAV
beta_cyt.Rb1_grk_tot = 7.07824478944671e-11 # 106 18 R_grkp_tot_CYT
pka_cav.ARC = 0.0904820284659604 # 107 19 RLC_CAV
pka_cav.A2RC = 0.00276490711096605 # 108 20 L2RC_CAV
pka_cav.A2R = 0.225475702283053 # 109 21 L2R_CAV
pka_cav.C = 0.0326565916584703 # 110 22 C_CAV
pka_cav.PKIC = 0.192819110624505 # 111 23 PKI_CAV
pka_eca.ARC = 0.205444874210056 # 112 24 RLC_ECAV
pka_eca.A2RC = 0.174057375932567 # 113 25 L2RC_ECAV
pka_eca.A2R = 0.817161796756964 # 114 26 L2R_ECAV
pka_eca.C = 0.567249910261073 # 115 27 C_ECAV
pka_eca.PKIC = 0.249911886495890 # 116 28 PKI_ECAV
pka_cyt.ARC = 0.0646928309115710 # 117 29 RLC_CYT
pka_cyt.A2RC = 0.0664997605558791 # 118 30 L2RC_CYT
pka_cyt.A2R = 0.489063888619456 # 119 31 L2R_CYT
pka_cyt.C = 0.362113356111496 # 120 32 C_CYT
pka_cyt.PKIC = 0.126950532507959 # 121 33 PKI_CYT
pde.PDE3_P_cav = 0.0236821659448037 # 122 34 PDE3_P_CAV
pde.PDE3_P_cyt = 0.0128402905095188 # 123 35 PDE3_P_CYT
pde.PDE4_P_cav = 0.00637363047239019 # 124 36 PDE4_P_CAV
pde.PDE4_P_eca = 4.29171113639322e-5 # 125 37 PDE4_P_ECAV
pde.PDE4_P_cyt = 0.00917039986149184 # 126 38 PDE4_P_CYT
pp1.inhib1_p = 0.0282662056977524 # 127 39 Inhib1_P_CYT
akap_sig.ICaLp = 0.000673713947839317 # 128 40 fLCC_P not a fraction!
iks_sig.IKsp = 0.000765988420110534 # 129 41 fIKS_P not a fraction!
iup.f_plb = 0.592167467082831 # 130 42 fPLB_P
calcium.f_tni = 0.673518785672382 # 131 43 fTnI_P
ina.f_ina = 0.239479458960528 # 132 44 fINa_P
inak.f_inak = 0.126345311579566 # 133 45 fINaK_P
akap_sig.RyRp = 0.00410693810508171 # 134 46 fRyR_P not a fraction!
ikur.f_ikur = 0.0589379755147718 # 135 47 fIKur_P
beta_cav.Rb2_pka_tot = 0.0275455839709412 # 136 48 Rb2_pkap_tot_CAV
beta_cav.Rb2_grk_tot = 8.91799633266019e-10 # 137 49 Rb2_grkp_tot_CAV
beta_cav.Gi_aGTP = 0.00159632196638178 # 138 50 Gi_aGTP_CAV
beta_cav.Gi_bg = 0.00209911481235842 # 139 51 Gi_bg_CAV
beta_cav.Gi_aGDP = 0.000502792845976641 # 140 52 Gi_aGDP_CAV
beta_eca.Rb2_pka_tot = 0.0110248953370551 # 141 53 Rb2_pkap_tot_ECAV
beta_eca.Rb2_grk_tot = 1.13428924662652e-10 # 142 54 Rb2_grkp_tot_ECAV
beta_eca.Gi_aGTP = 0.000364315164237569 # 143 55 Gi_aGTP_ECAV
beta_eca.Gi_bg = 0.000705656306851923 # 144 56 Gi_bg_ECAV
beta_eca.Gi_aGDP = 0.000341341142614041 # 145 57 Gi_aGDP_ECAV
#
# Engine variables
#
[engine]
time = 0 [ms]
in [ms]
bind time
pace = 0
bind pace
#
# Total trans-membrane currents and voltage
# Section 1.26 of the thesis appendix (page 315)
#
[membrane]
dot(V) = -(i_ion + stimulus.i_stim) / 1 [uF/cm^2]
in [mV]
label membrane_potential
desc: The membrane potential
i_ion = sodium.INa_tot + potassium.IK_tot + calcium.ICa_tot + chloride.ICl_tot
in [uA/cm^2]
label cellular_current
#
# Stimulus current
# The stimulus current is taken to be a Potasisum current, corresponding to patch clamp conditions
#
[stimulus]
amplitude = -80 [uA/cm^2]
in [uA/cm^2]
i_stim = engine.pace * amplitude
desc: The stimulus current
in [uA/cm^2]
#
# Isoproterenol
# (used because it is very similar to adrenaline)
#
[iso]
L = 0.0 [uM]: concentration of isoproterenol
in [uM]
#
# Cell geometry
# Section 1.1 of the thesis appendix (page 261)
#
[cell]
pi = 3.141592653589793
length = 0.01 [cm] : Cell length # l = 0.01
in [cm]
radius = 0.0011 [cm] : Cell radius # a = 0.0011
in [cm]
volume = 1000 [uL/mL] * pi * radius * radius * length
in [uL]
desc: Cell volume # vcell = 3.801e-5 uL
geoArea = 2.0 * pi * radius * (radius + length)
in [cm^2]
desc: Geometric membrane area
# Matlab version uses two different versions of PI!
# Fix this before comparing...
capArea = 2.0 * geoArea
in [cm^2]
desc: Capacitive membrane area
AF = capArea / phys.F
in [mol * cm^2 / C]
desc: The capacitive membrane area divided by the Faraday constant (section Q)
# Compartment sizes
v_cav = 0.02 * volume
in [uL] # data.V_CAV = 0.02 * vcell;
desc: Volume of the caveolar subspace
v_eca = 0.04 * volume
in [uL] # data.V_ECAV = 0.04 * vcell;
desc: Volume of the extracaveolar subspace
v_cyt = volume * 0.678
in [uL] # data.V_CYT = data.vmyo = 0.678 * vcell;
desc: Volume of the Cytoplasm / Myoplasm
vr_cav = volume / v_cav
desc: Ratio of whole volume to caveolar subspace volume
vr_eca = volume / v_eca
desc: Ratio of whole volume to extracaveolar subspace volume
vr_cyt = volume / v_cyt
desc: Ratio of whole volume to cytoplasm volume
v_nsr = volume * 0.0552
in [uL]
desc: Volume of the network sarcoplasmic reticulum
v_jsr = volume * 0.0048
in [uL]
desc: Volume of the junctional sarcoplasmic reticulum
v_ss = volume * 0.02
in [uL]
desc: """
Dyadic subspace (SS) volume.
Note that ICaL does not flow into this space directly, but flows instead
into SS,CaL, a subspace embedded in this one.
"""
v_CaL = volume * 0.002
in [uL]
desc: Subspace within the dyadic subspace, into which ICaL flows.
#
# Physical constants
# Section 1.1 of the thesis appendix (page 261)
#
[phys]
F = 96487 [C/mol] : Faraday constant
in [C/mol]
R = 8314 [mJ/mol/K] : Gas constant
in [mJ/mol/K]
T = 310 [K] : Temperature
in [K]
RTF = R * T / F
in [mV]
FRT = 1 / RTF
in [1/mV]
FFRT = F * FRT
in[C/mol/mV]
#
# Extracellular constants.
# Section 1.1 of the thesis appendix (page 261)
#
[extra]
Nao = 140 [mM] : Extracellular sodium concentration
in [mM]
Ko = 5.4 [mM] : Extracellular potassium concentration
in [mM]
Cao = 1.8 [mM] : Extracellular calcium concentration
in [mM]
Clo = 100 [mM] : Extracellular chloride concentration
in [mM]
#
# Reversal potentials
# Section 1.1 of the thesis appendix (page 262)
#
[nernst]
use phys.RTF
ECl = -RTF * log(extra.Clo / chloride.Cl)
in [mV]
desc: Reversal potential for Chloride
ENa = RTF * log(extra.Nao / sodium.Na)
in [mV]
desc: Reversal potential for Sodium
EK = RTF * log(extra.Ko / potassium.K)
in [mV]
desc: Reversal potential for Potassium
EKs = RTF * log((extra.Ko + PNaK * extra.Nao) / (potassium.K + PNaK * sodium.Na))
desc: Reversal potential for IKs
in [mV]
PNaK = 0.01833 #: IKs sodium / potassium permeability ratio
#
# Calcium handling.
# Section 1.3 of the thesis appendix (page 265)
#
[calcium]
# Fraction of troponin phosphorylation. Section 2.9 of the thesis appendix.
ka_tni = 0.10408 [1/s]
in [1/s] # data.k_PKATnI = 1.0408e-001
desc: Rate of Troponin phosphorylation by PKA
Ka_tni = 2.7143e-5 [uM]
in [uM] # data.Km_PKATnI = 2.7143e-005
desc: Affinity of Troponin for phosphorylation by PKA
kp_tni = 5.2633e-2 [1/s]
in [1/s] # data.k_PPTnI = 5.2633e-002
desc: Rate of Troponin dephosphorylation by phosphatases
Kp_tni = 0.26714 [uM]
in [uM] # data.Km_PPTnI = 2.6714e-001
desc: Affinity of Troponin for dephosphorylation by phosphatases
dot(f_tni) = 1e-3 [s/ms] * (ka_tni * pka_cyt.C * (1 - f_tni) / (Ka_tni + (1 - f_tni) * 1 [uM]) - kp_tni * pp1.PP2A * f_tni / (Kp_tni + f_tni * 1 [uM]))
desc: Fraction of phosphorylated Troponin
# Intracellular calcium concentration
cbar = 0.05 [mM] : Maximum Calmodulin concentration
in [mM]
tbar = 0.07 [mM] : Maximum Troponin concentration
in [mM]
kc = 2.38e-03 [mM] : Affinity of Calmodulin for Ca2+
in [mM]
ktn = 5e-4 [mM] : Affinity of non-phosphorylated Troponin for Ca2+
in [mM]
ktp = 1.5 * ktn : Affinity of phosphorylated Troponin for Ca2+
in [mM]
fhat = if(val < 0, 0, val)
val = (f_tni - 0.6735188) / (0.9991797 - 0.6735188)
desc: Effective fraction of phosphorylated Troponin
kt = (1 - fhat) * ktn + fhat * ktp
desc: Combined affinity of phosphorylated and non-phosphorylated Troponin for Ca2+
in [mM]
Ca = -b / 3 + (2/3) * sqrt(b*b - 3*c) * cos(acos((9*b*c - 2*b*b*b - 27*d) / (2 * (b*b - 3*c)^1.5)) / 3)
desc: Intracellular calcium
in [mM]
ksum = kt + kc
in [mM]
kpro = kt * kc
in [mM^2]
b = ksum - uCa + cbar + tbar
in [mM]
c = kpro - uCa * ksum + tbar * kc + cbar * kt
in [mM^2]
d = -kpro * uCa
in [mM^3]
dot(uCa) = - r1 * (icab.ICab + ipca.IpCa - 2 * inaca.INaCa) \
- r2 * iup.Iup \
+ r3 * diff.Idiff_Ca
desc: Unbuffered intracellular Calcium
in [mM]
r1 = cell.AF / (2 * cell.v_cyt)
in [kmol/C/cm]
r2 = (cell.v_nsr / cell.v_cyt)
r3 = (cell.v_ss / cell.v_cyt)
# Ca_ss and Ca_CaL buffer
bsl_bar = 1.124 [mM] : Anionic Ca2+ binding sites on sarcolemma
in [mM]
bsr_bar = 0.047 [mM] : Anionic Ca2+ binding sites on SR
in [mM]
bar_sum = bsr_bar + bsl_bar
in [mM]
bsr_km = 0.00087 [mM] : Affinity of anionic Ca2+ binding sites on sarcolemma
in [mM]
bsl_km = 0.0087 [mM] : Affinity of anionic Ca2+ binding sites on SR
in [mM]
km_sum = bsr_km + bsl_km
in [mM]
km_pro = bsr_km * bsl_km
in [mM^2]
ss_sum = bar_sum + km_sum
in [mM]
ss_pro = bsr_bar * bsl_km + bsl_bar * bsr_km + km_pro
in [mM^2]
Ca_ss = -b/3 + 2/3 * sqrt(b*b - 3*c) * cos(acos((9*b*c - 2*b*b*b - 27*d) / (2*(b*b - 3*c)^1.5)) / 3)
desc: Calcium concentration in the SS subspace
in [mM]
b = ss_sum - uCa_ss
in [mM]
c = ss_pro - uCa_ss * km_sum
in [mM^2]
d = -km_pro * uCa_ss
in [mM^3]
dot(uCa_ss) = -(r1 * inaca.INaCaSR + r2 * irel.Irel + diff.Idiff_Ca + diff.Idiff_ss)
desc: Unbuffered Calcium concentration in the SS subspace
in [mM]
r1 = -cell.AF / cell.v_ss
in [kmol/C/cm]
r2 = -cell.v_jsr / cell.v_ss
Ca_CaL = -b/3 + 2/3 * sqrt(b*b - 3*c) * cos(acos((9*b*c - 2*b*b*b - 27*d) / (2*(b*b - 3*c)^1.5)) / 3)
desc: Calcium concentration in the SS,CaL subspace
in [mM]
b = ss_sum - uCa_CaL
in [mM]
c = ss_pro - uCa_CaL * km_sum
in [mM^2]
d = -km_pro * uCa_CaL
in [mM^3]
dot(uCa_CaL) = -r1 * ical.ICaL + r2 * diff.Idiff_ss
desc: Unbuffered Calcium concentration in the SS,CaL subspace
in [mM]
r1 = cell.AF / (2 * cell.v_CaL)
in [kmol/C/cm]
r2 = cell.v_ss / cell.v_CaL
# Junctional and network SR
csqn_km = 0.8 [mM] : Affinity of Calsequestrin for Ca2+
in [mM]
csqn_bar = 10 [mM] : Maximum Calsequestrin concentration
in [mM]
Ca_jsr = (sqrt(b*b + 4*c) - b) / 2
desc: Concentration of Ca in the JSR subspace
in [mM]
b = csqn_bar + csqn_km - uCa_jsr
in [mM]
c = uCa_jsr * csqn_km
in [mM^2]
dot(uCa_jsr) = diff.Itr - irel.Irel
desc: Unbuffered concentration of Ca2+ in the JSR subspace
in [mM]
dot(Ca_nsr) = iup.Iup - r1 * diff.Itr
desc: Concentration of Ca2+ in the NSR subspace
in [mM]
# Matlab code adds an extra term iup.Ileak which is already included in Iup in this implementation
r1 = cell.v_jsr / cell.v_nsr
# Total calcium current
ICa_tot = (ical.ICaL # Section 1.26 of the thesis appendix
+ icab.ICab
+ ipca.IpCa
- inaca.INaCa * 2
- inaca.INaCaSR * 2 )
desc: Total Ca2+ current through the membrane
in [uA/cm^2]
#
# Chloride concentrations
# Section 1.4 of the thesis appendix (page 268)
#
[chloride]
dot(Cl) = r1 * iclb.IClb + ctnacl.CTNaCl + ctkcl.CTKCl + r2 * diff.Idiff_Cl
r1 = cell.AF / cell.v_cyt
in [kmol/C/cm]
r2 = cell.v_ss / cell.v_cyt
desc: Intracullular Chloride concentration
in [mM]
dot(Cl_ss) = r1 * iclca.IClCa - diff.Idiff_Cl
r1 = cell.AF / cell.v_ss
in [kmol/C/cm]
desc: Intracullular Chloride concentration in the SR subspace (SS,SR)
in [mM]
ICl_tot = iclb.IClb + iclca.IClCa # Section 1.26 of the thesis appendix
desc: Total Chloride current through the membrane
in [uA/cm^2]
#
# Potassium concentration
# Section 1.5 of the thesis appendix (page 268)
#
[potassium]
dot(K) = r1 * (IK_tot + stimulus.i_stim) + ctkcl.CTKCl : Intracellular Potassium
r1 = -cell.AF / cell.v_cyt
in [kmol/C/cm]
in [mM]
IK_tot = ( ik1.IK1
+ ikr.IKr
+ iks.IKs
+ ikur.IKur
+ ito.ITo
- 2 * inak.INaK )
desc: Total Potassium current through the membrane
in [uA/cm^2]
#
# Sodium concentrations
# Section 1.6 of the thesis appendix (page 269)
#
[sodium]
dot(Na) = r1 * INa_cyt + r2 * diff.Idiff_Na + ctnacl.CTNaCl
desc: Intracellular Sodium concentration
in [mM]
r1 = -cell.AF / cell.v_cyt
in [kmol/C/cm]
r2 = cell.v_ss / cell.v_cyt
dot(Na_ss) = -(r1 * inaca.INaCaSR + diff.Idiff_Na)
desc: Sodium concentration in the SR subspace
in [mM]
r1 = 3 * cell.AF / cell.v_ss
in [kmol/C/cm]
INa_cyt = ( ina.INa
+ inab.INab
+ inal.INaL
+ inak.INaK * 3
+ inaca.INaCa * 3 )
in [uA/cm^2]
INa_tot = INa_cyt + inaca.INaCaSR * 3
desc: Total Na+ current through the membrane
in [uA/cm^2]
#
# CTNaCl :: Na-Cl Cotransporter
# Section 1.7.1 of the thesis appendix (page 269)
# Section H of [3]'s data supplement
#
[ctnacl]
NaClBar = 2.46108e-5 [mM/ms] : Maximum NaCl exchange
in [mM/ms]
CTNaCl = NaClBar * z1 / (z1 + z2)
z1 = (nernst.ENa - nernst.ECl) ^ 4
in [mV^4]
z2 = (87.8251 [mV])^4
in [mV^4]
in [mM/ms]
desc: Na+Cl- cotransporter
#
# CTKCl :: K-Cl Cotransporter
# Section 1.7.1 of the thesis appendix (page 269)
# Section H of [3]'s data supplement
#
[ctkcl]
KClBar = 1.77e-5 [mM/ms] : Maximum KCl exchange
in [mM/ms]
CTKCl = KClBar * z1 / (z1 + z2)
in [mM/ms]
desc: K+Cl- cotransporter
z1 = nernst.EK - nernst.ECl
in [mV]
z2 = 87.8251 [mV]
in [mV]
#
# ICab :: Background Calcium current
# Section 1.8 of the thesis appendix (page 269)
# Section M of [3]'s supplement
#
[icab]
ICab = pCab * 2 * phys.F * vfrt * (calcium.Ca * efrt - 0.341 * extra.Cao) / (efrt - 1)
in [uA/cm^2]
vfrt = 2 * membrane.V * phys.FRT
efrt = exp(vfrt)
pCab = 1.995e-7 [cm/s]
in [cm/s]
#
# ICaL :: L-type Calcium current
# Section 1.9.2 of the thesis appendix (page 271)
# Section 3.3.2 of [1]'s supplement (page 47)
#
[ical]
use akap_sig.fp_ICaL as fp
use akap_sig.ICaL_arn
use akap_sig.ICaL_tot
f_hat = if(val < 0, 0, val)
val = (fp - ratio) / (0.9273 - ratio)
ratio = 0.0269 + ICaL_arn / ICaL_tot
desc: Effective fraction of phosphorylated ICaL channels
ICaL = (1 - f_hat) * ical_np.ICaL + f_hat * ical_p.ICaL
desc: Total ICaL current
in [uA/cm^2]
#
# ICaL :: Non-PKA-phosphorylated channels
# Described on page 272 of the thesis appendix, page 48 of [1]'s supplement
#
[ical_np]
use membrane.V
# Which [Ca] to use is a setting in the original model:
use calcium.Ca_ss as Ca
# Activation: alpha, beta
# Activation stays the same regardless of Ca2+ level
ac_inf = 1 / ((1 + exp((13.56 [mV] - V) / 9.45 [mV])) * (1 + exp((25 [mV] + V) / -5 [mV])))
desc: Steady state value of activation
ac_tau = 0.59 [ms] + 0.8 [ms] * exp(0.052 [1/mV] * (V + 13 [mV])) / (1 + exp(0.132 [1/mV] * (V + 13 [mV])))
desc: Time constant of activation
in [ms]
alpha = ac_inf / ac_tau
desc: Transition rate from closed to open
in [1/ms]
beta = (1 - ac_inf) / ac_tau
desc: Transition rate from open to closed
in [1/ms]
# Inactivation: x, y (low Ca2+)
in_inf = 1 / (1 + exp((17.5 [mV] + V) / 3 [mV])) # Terms occurs twice
in_a = 1 / (70 * (1 - 0.5 * camk.f_ical) * (1 + exp((V + 49.1 [mV]) / 10.349 [mV]))) # Term (1 - 0.5 * y(88)) doesn't appear in paper!
in_b = 1 / (1 + exp((V + 0.213 [mV]) / -10.807 [mV]))
in_lo_inf = (0.2474 + in_inf) / 1.2474 # Supplement incorrectly writes 1.2472 here
desc: Steady state value for inactivation at low Ca2+ levels
in_lo_tau = 1 [ms] / (in_a + in_b / 26.553)
desc: Time constant for inactivation at low Ca2+ levels
in [ms]
x = in_lo_inf / in_lo_tau
desc: Transition rate from inactive to active, at low Ca2+ levels
in [1/ms]
y = (1 - in_lo_inf) / in_lo_tau
desc: Transition rate from active to inactive, at low Ca2+ levels
in [1/ms]
# Inactivation: xs, ys (high Ca2+)
delta_tau = 5 * camk.f_ical
ss_cal_4 = 1 + (1.1e-3 [mM] / Ca)^4
ss_cal_10 = 1 + (1.2e-2 [mM] / Ca)^10
inca = 13.825 - (6.3836 - delta_tau) / ss_cal_4 - 3.3696 / ss_cal_10
desc: Ca2+ dependent component of inactivation constant for ICaL non-phosphorylated channels and modulation by CaMKII
in_hi_inf = (0.001 + in_inf) / 1.001
desc: Steady state value for inactivation at high Ca2+ levels
in_hi_tau = 1 [ms] / (in_a + in_b / inca)
desc: Time constant for inactivation at high Ca2+ levels
in [ms]
xs = in_hi_inf / in_hi_tau
desc: Transition rate from inactive to active, at high Ca2+ levels
in [1/ms]
ys = (1 - in_hi_inf) / in_hi_tau
desc: Transition rate from active to inactive, at high Ca2+ levels
in [1/ms]
# Reaction to changes of [Ca2+] in SS_CaL, active channel: delta and theta
theta = 1 [1/ms]
desc: Transition rate from low [Ca2+] model to high [Ca2+] model, active channel
in [1/ms]
delta = 14.9186 [1/ms] / ss_cal_4
desc: Transition rate from high [Ca2+] model to low [Ca2+] model, active channel
in [1/ms]
# Reaction to changes of [Ca2+] in SS_CaL, inactive channel: delta1 and theta1
theta1 = 1e-6 [1/ms]
desc: Transition rate from low [Ca2+] model to high [Ca2+] model, inactive channel
in [1/ms]
delta1 = theta1 * (x * ys * delta) / (y_cor * xs_cor * theta)
desc: Transition rate from high [Ca2+] model to low [Ca2+] model, inactive channel
in [1/ms]
y_cor = if(abs(y) < 1e-12 [1/ms], 1e-12 [1/ms], y)
in [1/ms]
xs_cor = if(abs(xs) < 1e-12 [1/ms], 1e-12 [1/ms], xs)
in [1/ms]
# States
dot(C) = -(alpha + delta + y ) * C + beta * O + theta * Cs + x * CI
dot(O) = -(beta + delta + y ) * O + alpha * C + theta * Os + x * OI
dot(Cs) = -(alpha + theta + ys) * Cs + delta * C + beta * Os + xs * CIs
dot(Os) = -(beta + theta + ys) * Os + delta * O + alpha * Cs + xs * OIs
dot(CI) = -(alpha + delta1 + x ) * CI + y * C + theta1 * CIs + beta * OI
dot(OI) = -(beta + delta1 + x ) * OI + y * O + theta1 * OIs + alpha * CI
dot(CIs) = -(alpha + theta1 + xs) * CIs + ys * Cs + delta1 * CI + beta * OIs
dot(OIs) = -(beta + theta1 + xs) * OIs + ys * Os + delta1 * OI + alpha * CIs
# Current
PCa = 1.552e-4 [cm/s] * (1 + 0.4 * camk.f_ical)
desc: Permeability of non-PKA-phosphorylated channels
in [cm/s]
IBar = PCa * 4 * V * phys.FFRT * (Ca * vv - 0.341 * extra.Cao) / (vv - 1)
vv = exp(2 * V * phys.FRT)
desc: Maximum current through non-phosphorylated channels
in [uA/cm^2]
ICaL = IBar * (O + Os)
desc: ICaL current density through non-phosphorylated channels
in [uA/cm^2]
#
# ICaL :: PKA-Phosphorylated channels
# Described on page 274 of the thesis appendix, page 49 of [1]'s supplement
#
[ical_p]
use membrane.V
# Which [Ca] to use is a setting in the original model:
use calcium.Ca_ss as Ca
# Activation: alpha, beta
# Activation stays the same regardless of Ca2+ level
ac_inf = 1 / ((1 + exp((4.798 [mV] + V) / -7.5699 [mV])) * (1 + exp((25 [mV] + V) / -5 [mV])))
desc: Steady state value of activation
use ical_np.ac_tau
alpha = ac_inf / ac_tau
desc: Transition rate from closed to open
in [1/ms]
beta = (1 - ac_inf) / ac_tau
desc: Transition rate from open to closed
in [1/ms]
# Inactivation: x, y (low Ca2+)
in_inf = 1 / (1 + exp((29.979 [mV] + V) / 3.1775 [mV])) # Term occurs twice
use ical_np.in_a
use ical_np.in_b
in_lo_inf = (0.1 + in_inf) / 1.1
desc: Steady state value for inactivation at low Ca2+ levels
in_lo_tau = 1 [ms] / (in_a + in_b / 38.494) # Supplement uses 30
desc: Time constant for inactivation at low Ca2+ levels
in [ms]
x = in_lo_inf / in_lo_tau
desc: Transition rate from inactive to active, at low Ca2+ levels
in [1/ms]
y = (1 - in_lo_inf) / in_lo_tau
desc: Transition rate from active to inactive, at low Ca2+ levels
in [1/ms]
# Inactivation: xs, ys (high Ca2+)
delta_tau = 0.1 * camk.f_ical
ss_cal_4 = 1 + (0.002 [mM] / Ca)^4
ss_cal_10 = 1 + (0.01 [mM] / Ca)^10
inca = 32.5 - (18 - delta_tau) / ss_cal_4 - 10 / ss_cal_10
desc: Ca2+ dependent component of inactivation constant
in_hi_inf = (0.0001 + in_inf) / 1.0001
desc: Steady state value for inactivation at high Ca2+ levels
in_hi_tau = 1 [ms] / (in_a + in_b / inca)
desc: Time constant for inactivation at high Ca2+ levels
in [ms]
xs = in_hi_inf / in_hi_tau
desc: Transition rate from inactive to active, at high Ca2+ levels
in [1/ms]
ys = (1 - in_hi_inf) / in_hi_tau
desc: Transition rate from active to inactive, at high Ca2+ levels
in [1/ms]
# Reaction to changes of [Ca2+] in SS_CaL, active channel: delta and theta
theta = 1 [1/ms]
desc: Transition rate from low [Ca2+] model to high [Ca2+] model, active channel
in [1/ms]
delta = 6 [1/ms] / ss_cal_4
desc: Transition rate from high [Ca2+] model to low [Ca2+] model, active channel
in [1/ms]
# Reaction to changes of [Ca2+] in SS_CaL, inactive channel: delta1 and theta1
theta1 = 1e-6 [1/ms]
desc: Transition rate from low [Ca2+] model to high [Ca2+] model, inactive channel
in [1/ms]
delta1 = theta1 * (x * ys * delta) / (y_cor * xs_cor * theta)
y_cor = if(abs(y) < 1e-12 [1/ms], 1e-12 [1/ms], y)
in [1/ms]
xs_cor = if(abs(xs) < 1e-12 [1/ms], 1e-12 [1/ms], xs)
in [1/ms]
desc: Transition rate from high [Ca2+] model to low [Ca2+] model, inactive channel
in [1/ms]
# States
dot(C) = -(alpha + delta + y ) * C + beta * O + theta * Cs + x * CI
dot(O) = -(beta + delta + y ) * O + alpha * C + theta * Os + x * OI
dot(Cs) = -(alpha + theta + ys) * Cs + delta * C + beta * Os + xs * CIs
dot(Os) = -(beta + theta + ys) * Os + delta * O + alpha * Cs + xs * OIs
dot(CI) = -(alpha + delta1 + x ) * CI + y * C + theta1 * CIs + beta * OI
dot(OI) = -(beta + delta1 + x ) * OI + y * O + theta1 * OIs + alpha * CI
dot(CIs) = -(alpha + theta1 + xs) * CIs + ys * Cs + delta1 * CI + beta * OIs
dot(OIs) = -(beta + theta1 + xs) * OIs + ys * Os + delta1 * OI + alpha * CIs
# Current
PCa = 2.579e-4 [cm/s] * (1 + 0.1 * camk.f_ical)
desc: Permeability of PKA-phosphorylated channels
in [cm/s]
IBar = PCa * 4 * V * phys.FFRT * (Ca * vv - 0.341 * extra.Cao) / (vv - 1)
vv = exp(2 * V * phys.FRT)
desc: Maximum current through PKA-phosphorylated channels
in [uA/cm^2]
ICaL = IBar * (O + Os)
desc: ICaL current density through PKA-phosphorylated channels
in [uA/cm^2]
#
# IpCa :: The Sarcolemmal Calcium Pump
# Section 1.10.1 of the thesis appendix (page 276)
# Section L of [3]'s data supplement
#
[ipca]
IpCa_bar = 0.0575 [uA/cm^2] : Max. Ca current through sarcolemmal Ca pump
in [uA/cm^2]
Km_pCa = 5e-4 [mM] : Half-saturation concentration of sarcolemmal Ca pump
in [mM]
IpCa = IpCa_bar * calcium.Ca / (Km_pCa + calcium.Ca)
desc: Sarcolemmal Calcium pump current
in [uA/cm^2]
#
# IClb :: Background chloride current
# Section 1.11 of the thesis appendix (page 277)
# Section N of [3]'s data supplement
#
[iclb]
Gbar = 2.25e-4 [mS/cm^2]
in [mS/cm^2]
IClb = Gbar * (membrane.V - nernst.ECl)
desc: Background chloride current
in [uA/cm^2]
#
# IClCa :: Calcium-dependent chloride current (ITo2)
# Section 1.12 of the thesis appendix (page 277)
# Section E of [3]'s data supplement
#
[iclca]
use membrane.V
use phys.FRT
use phys.FFRT
kCaCl = 0.4 [mM/ms]
in [mM/ms]
PCl = 9e-7 [cm/s]
in [cm/s]
IClCa_bar = PCl * V * FFRT * (chloride.Cl - extra.Clo * vexp) / (1 - vexp)
in [uA/cm^2]
vexp = exp(V * FRT)
KClCa = 1 - (1 / (1 + (irel.Irel_pure / kCaCl)^2)) # Thesis says 1/(1 + (Irel / kCaCl)^2) here! (KCaIto2)
# Paper versions are unclear
# Matlab uses IRel without added Ileak_ryr term here, so followed that...
IClCa = IClCa_bar * KClCa * i2
desc: Calcium dependent chloride current (aka ITo2)
in [uA/cm^2]
tau = 8 [ms]
in [ms]
dot(i2) = (alpha / (alpha + beta) - i2) / tau
alpha = 0.025 / (1 + exp((V + 58 [mV]) / 5 [mV]))
beta = 0.200 / (1 + exp((V + 19 [mV]) / -9 [mV]))
desc: IClCa (ITo2) inactivation gate
#
# Diffusion between subcellular compartments
# Section 1.13.2 of the thesis appendix (page 278)
#
[diff]
tau = 0.2 [ms]
in [ms]
tau_tr = 75 [ms] # Version for beta-adrenergic model
in [ms]
tau_sr = 0.02 [ms]
in [ms]
Itr = (calcium.Ca_nsr - calcium.Ca_jsr) / tau_tr
in [mM/ms]
Idiff_ss = (calcium.Ca_ss - calcium.Ca_CaL) / tau_sr
in [mM/ms]
desc: Ca2+ diffusion between the CaL space (near the LCCs) and the SS space.
Idiff_Ca = (calcium.Ca_ss - calcium.Ca) / tau
in [mM/ms]
desc: Ca2+ diffusion between the SS space and bulk cytosol
Idiff_Na = (sodium.Na_ss - sodium.Na) / tau
in [mM/ms]
desc: Na+ diffusion between the SS space and bulk cytosol
Idiff_Cl = (chloride.Cl_ss - chloride.Cl) / tau
in [mM/ms]
desc: Cl- diffusion between the SS space and bulk cytosol
#
# IK1 :: Inward rectifier Potassium current
# Section 1.14.2 of the thesis appendix (page 279)
# Section 3.3.11 of [1]'s supplement (page 67)
#
[ik1]
Gbar = 0.5 [mS/cm^2] * sqrt(extra.Ko / 5.4 [mM])
in [mS/cm^2]
IK1_np = Gbar * (alpha / (alpha + beta)) * vv
desc: Non-phosphorylated component of IK1
in [uA/cm^2]
vv = membrane.V - nernst.EK
in [mV]
alpha = 1.02 / (1 + exp(0.2385 [1/mV] * (vv - 59.215 [mV])))
beta = (0.49124 * exp(0.08032 [1/mV] * (vv + 5.476 [mV])) + exp(0.06175 [1/mV] * (vv - 594.31 [mV]))) / (1 + exp(-0.5143 [1/mV] * (vv + 4.753 [mV])))
IK1_camk = IK1_np * 1.2
desc: CaMKII phosphorylated component of IK1
in [uA/cm^2]
IK1 = (1 - camk.f_ik1) * IK1_np + camk.f_ik1 * IK1_camk
desc: Inward rectifier Potassium current
in [uA/cm^2]
#
# IKr :: Rapid delayed rectifier Potassium current
# Section 1.15.1 of the thesis appendix (page 280)
#
[ikr]
use membrane.V as V
GKr = 0.0138542 [mS/cm^2] * sqrt(extra.Ko / 5.4 [mM])
desc: Maximum conductivity of IKr channel
in [mS/cm^2]
dot(ac) = (inf - ac) / tau
desc: Activation gate of IKr
inf = 1 / (1 + exp((V + 10.085 [mV]) / -4.25 [mV]))
tau = 1 [ms] / ( 6e-4 [1/mV] * (V - 1.73840 [mV]) / (1 - exp(-0.136 [1/mV] * (V - 1.73840 [mV])))
- 3e-4 [1/mV] * (V + 38.3608 [mV]) / (1 - exp(0.1522 [1/mV] * (V + 38.3608 [mV]))) )
in [ms]
inx = 1 / (1 + exp((V + 10 [mV]) / 15.4 [mV]))
desc: Inactivation gate of IKr
IKr = GKr * ac * inx * (V - nernst.EK)
desc: Rapid delayed rectifier Potassium current
in [uA/cm^2]
#
# IKS :: Slowly activating delayed rectifier Potassium current
# Section 1.16.2 of the thesis appendix (page 284)
# Section 3.3.3 of [1]'s data supplement (page 51)
#
[iks]
G = 0.19561 [mS/cm^2] * (1 + 0.6 / (1 + (3.8e-5 [mM] / calcium.Ca) ^ 1.4))
desc: Maximum conductivity of IKs
in [mS/cm^2]
IKs_np = G * (iks_np.O1 + iks_np.O2 ) * (membrane.V - nernst.EKs)
desc: Non-phosphorylated component of IKs
in [uA/cm^2]
IKs_pka = G * (iks_pka.O1 + iks_pka.O2) * (membrane.V - nernst.EKs)
desc: Phosphorylated component of IKs
in [uA/cm^2]
f_hat = if(val < 0, 0, val)
val = (iks_sig.fp_iks - ratio) / (0.785 - ratio)
ratio = 0.0306 + iks_sig.IKs_arn / iks_sig.IKs_tot
desc: Effective fraction of phosphorylated IKs channels
IKs = f_hat * IKs_pka + (1 - f_hat) * IKs_np
desc: Slowly activating delayed rectifier Potassium current
in [uA/cm^2]
#
# IKs :: Non-phosphorylated channels
# Described on page 284 of the thesis appendix, page 51 of [1]'s supplement
#
[iks_np]
use membrane.V
use phys.FRT
a = 7.3990e-3 [1/ms] / (1 + exp(FRT * (V - 3.1196e-2 [mV]) / -0.80019))
in [1/ms]
b = 5.6992e-3 [1/ms] / (1 + exp(FRT * (V - 4.1520e-2 [mV]) / 1.3489))
in [1/ms]
g = 3.8839e-1 [1/ms] / (1 + exp(FRT * (V + 0.15019 [mV]) / -0.60693))
in [1/ms]
d = 9.0654e-2 [1/ms] * exp(-0.11157 * V * FRT)
in [1/ms]
e = 3.1124e-3 [1/ms] + (2.833e-2 [1/ms] - 3.1124e-3 [1/ms]) / (1 + exp(FRT * (V + 5.166e-2 [mV]) / 1.5522))
in [1/ms]
t = 2.7304e-3 [1/ms]
in [1/ms]
o = 4.4198e-4 [1/ms] * exp(-1.2022 * V * FRT)
in [1/ms]
p = 4.0173e-4 [1/ms] * exp(2.0873e-4 * V * FRT)
in [1/ms]
# States
dot(C1) = b*C2 - C1 * (4*a)
dot(C2) = 4*a*C1 +2*b*C3 + d*C6 - C2 * (3*a + b + g)
dot(C3) = 3*a*C2 +3*b*C4 + d*C7 - C3 * (2*a +2*b +2*g)
dot(C4) = 2*a*C3 +4*b*C5 + d*C8 - C4 * ( a +3*b +3*g)
dot(C5) = a*C4 + d*C9 - C5 * ( 4*b +4*g)
dot(C6) = b*C7 + g*C2 - C6 * (3*a + d)
dot(C7) = 3*a*C6 +2*b*C8 +2*g*C3 +2*d*C10 - C7 * (2*a + b + g + d)
dot(C8) = 2*a*C7 +3*b*C9 +3*g*C4 +2*d*C11 - C8 * ( a +2*b +2*g + d)
dot(C9) = a*C8 +4*g*C5 +2*d*C12 - C9 * ( 3*b +3*g + d)
dot(C10) = b*C11 + g*C7 - C10 * (2*a +2*d)
dot(C11) = 2*a*C10 +2*b*C12 +2*g*C8 +3*d*C13 - C11 * ( a + b + g +2*d)
dot(C12) = a*C11 +3*g*C9 +3*d*C14 - C12 * ( 2*b +2*g +2*d)
dot(C13) = b*C14 + g*C11 - C13 * ( a +3*d)
dot(C14) = a*C13 +2*g*C12 +4*d*C15 - C14 * ( b + g +3*d)
dot(C15) = g*C14 - C15 * ( 4*d + t) + e*O1
dot(O1) = -(e + p) * O1 + o * O2 + t * C15
dot(O2) = p * O1 - o * O2
#
# IKs :: PKA phosphorylated channels
# Described on page 286 of the thesis appendix, page 53 of [1]'s supplement
#
[iks_pka]
use membrane.V
use phys.FRT
a = 9.9415e-3 [1/ms] / (1 + exp(FRT * (V - 4.4809e-2 [mV]) / -0.58172))
in [1/ms]
b = 3.3201e-3 [1/ms] / (1 + exp(FRT * (V - 9.4217e-2 [mV]) / 0.95364))
in [1/ms]
g = 5.6356e-1 [1/ms] / (1 + exp(FRT * (V + 0.17986 [mV]) / -0.58381))
in [1/ms]
d = 6.5700e-2 [1/ms] * exp(-0.11899* V * FRT)
in [1/ms]
e = 3.8525e-4 [1/ms] + (1.2406e-2 [1/ms] - 3.8525e-4 [1/ms]) / (1 + exp(FRT * (V + 6.4118e-2 [mV]) / 0.77992))
in [1/ms]
t = 4.6171e-3 [1/ms]
in [1/ms]
o = 2.3730e-4 [1/ms] * exp(-1.9742 * V * FRT)
in [1/ms]
p = 2.2652e-4 [1/ms] * exp(2.4689e-4 * V * FRT)