A Mathematical Model of the Human Cardiac Na⁺ Channel

Abstract

Sodium ion channel is a membrane protein that plays an important role in excitable cells, as it is responsible for the initiation of action potentials. Understanding the electrical characteristics of sodium channels is essential in predicting their behavior under different physiological conditions. We investigated several Markov models for the human cardiac sodium channel Na_V1.5 to derive a minimal mathematical model that describes the reported experimental data obtained using major voltage clamp protocols. We obtained simulation results for peak current–voltage relationships, the voltage dependence of normalized ion channel conductance, steady-state inactivation, activation and deactivation kinetics, fast and slow inactivation kinetics, and recovery from inactivation kinetics. Good agreement with the experimental data provides us with the mechanisms of the fast and slow inactivation of the human sodium channel and the coupling of its inactivation states to the closed and open states in the activation pathway.

Appendix

Functions Used in all Models

$${f_1}\left( {\text{V}} \right)=\frac{1}{{1.0+\exp \left( {\frac{{V+50.0}}{{5.0}}} \right)}}$$

(22)

$${f_2}\left( {\text{V}} \right)=\frac{1}{{1.0+\exp \left( {\frac{{V+90.0~}}{{5.0}}} \right)}}$$

(23)

$${\alpha _1}\left( {\text{V}} \right)=12.0 \cdot \exp \left( {\frac{V}{{50.0}}} \right)$$

(24)

$${\alpha _2}\left( {\text{V}} \right)=3.0 \cdot \exp \left( {\frac{V}{{30.0}}} \right)$$

(25)

$${\beta _1}\left( {\text{V}} \right)=0.2 \cdot \exp \left( { - \frac{V}{{20.0}}} \right)$$

(26)

$${\beta _2}\left( {\text{V}} \right)=0.025 \cdot \exp \left( { - \frac{V}{{20.0}}} \right)$$

(27)

Model 1

$$\frac{{{\text{d}}{{\text{C}}_2}}}{{{\text{d}}t}}=3\alpha {{\text{C}}_3} - \beta {{\text{C}}_2}+2\beta {{\text{C}}_1} - 2\alpha {{\text{C}}_2}$$

(28)

$$\frac{{{\text{d}}{{\text{C}}_1}}}{{{\text{d}}t}}=2\alpha {{\text{C}}_2} - 2\beta {{\text{C}}_1}+3\beta {\text{O}} - \alpha {{\text{C}}_1}$$

(29)

$$\frac{{{\text{dO}}}}{{{\text{d}}t}}=\alpha {{\text{C}}_1} - 3\beta {\text{O}}+{k_{{\text{ifo}}}}{\text{IF}} - {k_{{\text{oif}}}}{\text{O}}$$

(30)

$$\frac{{{\text{dIF}}}}{{{\text{d}}t}}={k_{{\text{oif}}}}{\text{O}} - {k_{{\text{ifo}}}}{\text{IF}}$$

(31)

$${{\text{C}}_3}=1 - \left( {{{\text{C}}_2}+{{\text{C}}_1}+{\text{O}}+{\text{IF}}} \right)$$

(32)

$$\alpha \left( {\text{V}} \right)={f_1}\left( {\text{V}} \right)\left[ {{\alpha _2}\left( {\text{V}} \right)+{\alpha _1}\left( {\text{V}} \right)\exp \left( {\frac{{V+50.0}}{{5.3}}} \right)} \right]$$

(33)

$$\beta \left( {\text{V}} \right)={f_1}\left( {\text{V}} \right)\left[ {{\beta _2}\left( {\text{V}} \right)+{\beta _1}\left( {\text{V}} \right)\exp \left( {\frac{{V+50.0}}{{5.0}}} \right)} \right]~$$

(34)

$${k_{{\text{oif}}1}}\left( {\text{V}} \right)=0.432 \cdot \exp \left( {\frac{{V+50.0}}{{30.0}}} \right)~$$

(35)

$${k_{{\text{oif}}2}}\left( {\text{V}} \right)=0.0345 \cdot \exp \left( {\frac{{V+80.0}}{{14.8}}} \right)$$

(36)

$${k_{{\text{ifo}}1}}\left( {\text{V}} \right)=0.01 \cdot \exp \left( { - \frac{{V+50.0}}{{30}}} \right)$$

(37)

$${k_{{\text{ifo}}2}}\left( {\text{V}} \right)=0.018 \cdot \exp \left( { - \frac{{V+100.0}}{{13.6}}} \right)$$

(38)

$${k_{{\text{oif}}}}={f_2}\left( {\text{V}} \right)\left( {{k_{{\text{oif}}2}}+{k_{{\text{oif}}1}} \cdot \exp \left( {\frac{{V+90.0}}{{5.0}}} \right)} \right)$$

(39)

$${k_{{\text{ifo}}}}={f_2}\left( {\text{V}} \right)\left( {{k_{{\text{ifo}}2}}+{k_{{\text{ifo}}1}} \cdot \exp \left( {\frac{{V+90.0}}{{5.0}}} \right)} \right)$$

(40)

Model 2

$$\frac{{{\text{d}}{{\text{C}}_2}}}{{{\text{d}}t}}=3\alpha {{\text{C}}_3} - \beta {{\text{C}}_2}+2\beta {{\text{C}}_1} - 2\alpha {{\text{C}}_2}$$

(41)

$$\frac{{{\text{d}}{{\text{C}}_1}}}{{{\text{d}}t}}=2\alpha {{\text{C}}_2} - 2\beta {{\text{C}}_1}+3\beta {\text{O}} - \alpha {{\text{C}}_1}+{k_{{\text{ifc}}1}}{\text{IF}} - {k_{{\text{c1if}}}}{{\text{C}}_1}$$

(42)

$$\frac{{{\text{dO}}}}{{{\text{d}}t}}=\alpha {{\text{C}}_1} - 3\beta {\text{O}}+{k_{{\text{ifo}}}}{\text{IF}} - {k_{{\text{oif}}}}{\text{O}}$$

(43)

$$\frac{{{\text{dIF}}}}{{{\text{d}}t}}={k_{{\text{oif}}}}{\text{O}} - {k_{{\text{ifo}}}}{\text{IF}}+{k_{{\text{c1if}}}}{{\text{C}}_1} - {k_{{\text{ifc1}}}}{\text{IF}}$$

(44)

$${{\text{C}}_3}=1 - \left( {{{\text{C}}_2}+{{\text{C}}_1}+{\text{O}}+{\text{IF}}} \right)$$

(45)

$$\alpha \left( {\text{V}} \right)={f_1}\left( {\text{V}} \right)\left[ {{\alpha _2}\left( {\text{V}} \right)+{\alpha _1}\left( {\text{V}} \right)\exp \left( {\frac{{V+50.0}}{{5.3}}} \right)} \right]$$

(46)

$$\beta \left( {\text{V}} \right)={f_1}\left( {\text{V}} \right)\left[ {{\beta _2}\left( {\text{V}} \right)+{\beta _1}\left( {\text{V}} \right)\exp \left( {\frac{{V+50.0}}{{5.0}}} \right)} \right]$$

(47)

$${k_{{\text{oif}}1}}\left( {\text{V}} \right)=0.432 \cdot \exp \left( {\frac{{V+50.0}}{{30.0}}} \right)~$$

(48)

$${k_{{\text{oif}}2}}\left( {\text{V}} \right)=0.00552 \cdot \exp \left( {\frac{{V+80.0}}{{30}}} \right)$$

(49)

$${k_{{\text{ifo}}1}}\left( {\text{V}} \right)=0.01 \cdot \mu \cdot \exp \left( { - \frac{{V+50.0}}{{30}}} \right)$$

(50)

$${k_{{\text{ifo}}2}}\left( {\text{V}} \right)=0.018 \cdot \rho \cdot \exp \left( { - \frac{{V+100.0}}{{13.6}}} \right)$$

(51)

$$\mu =0.000006,\,\rho =0.000006$$

(52)

$${k_{{\text{oif}}}}={f_2}\left( {\text{V}} \right)\left( {{k_{{\text{oif}}2}}+{k_{{\text{oif1}}}}\exp \left( {\frac{{V+90.0}}{{5.0}}} \right)} \right)~$$

(53)

$${k_{{\text{ifo}}}}={f_2}\left( {\text{V}} \right)\left( {{k_{{\text{ifo}}2}}+{k_{{\text{ifo}}1}} \cdot \exp \left( {\frac{{V+90.0}}{{5.0}}} \right)} \right)$$

(54)

$${k_{{\text{c1if}}}}=0.3$$

(55)

$${k_{{\text{ifc1}}}}=3.0 \cdot \beta \cdot {k_{{\text{ifo}}}} \cdot \frac{{{k_{{\text{c1if}}}}}}{{\alpha \cdot {k_{{\text{oif}}}}}}$$

(56)

Model 3

$$\frac{{{\text{d}}{{\text{C}}_2}}}{{{\text{d}}t}}=3\alpha {{\text{C}}_3} - \beta {{\text{C}}_2}+2\beta {{\text{C}}_1} - 2\alpha {{\text{C}}_2}+\frac{{{k_{{\text{ic}}}}}}{{{f^2}}}{\text{I}}{{\text{C}}_2} - {k_{{\text{ci}}}}{f^2}{{\text{C}}_2}$$

(57)

$$\frac{{{\text{d}}{{\text{C}}_1}}}{{{\text{d}}t}}=2\alpha {{\text{C}}_2} - 2\beta {{\text{C}}_1}+3\beta {\text{O}} - \alpha {{\text{C}}_1}+\frac{{{k_{{\text{ic}}}}}}{f}{\text{I}}{{\text{C}}_1} - {k_{{\text{ci}}}}f{{\text{C}}_1}$$

(58)

$$\frac{{{\text{dO}}}}{{{\text{d}}t}}=\alpha {{\text{C}}_1} - 3\beta {\text{O}}+{k_{{\text{ifo}}}}{\text{IF}} - {k_{{\text{oif}}}}{\text{O}}~~$$

(59)

$$\frac{{{\text{dIF}}}}{{{\text{d}}t}}={k_{{\text{oif}}}}{\text{O}} - {k_{{\text{ifo}}}}{\text{IF}}+\frac{\alpha }{f}{\text{I}}{{\text{C}}_1} - 3\beta f{\text{IF}}$$

(60)

$$\frac{{{\text{dI}}{{\text{C}}_1}}}{{{\text{d}}t}}=\frac{{2\alpha }}{f}{\text{I}}{{\text{C}}_2} - 2\beta f{\text{I}}{{\text{C}}_1}+3\beta f{\text{IF}} - \frac{\alpha }{f}{\text{I}}{{\text{C}}_1}+{k_{{\text{ci}}}}f{{\text{C}}_1} - \frac{{{k_{{\text{ic}}}}}}{f}{\text{I}}{{\text{C}}_1}$$

(61)

$$\frac{{{\text{dI}}{{\text{C}}_2}}}{{{\text{d}}t}}=\frac{{3\alpha }}{f}{\text{I}}{{\text{C}}_3} - \beta f{\text{I}}{{\text{C}}_2}+{k_{{\text{ci}}}}{f^2}{{\text{C}}_2} - \frac{{{k_{{\text{ic}}}}}}{{{f^2}}}{\text{I}}{{\text{C}}_2}+2\beta f{\text{I}}{{\text{C}}_1} - \frac{{2\alpha }}{f}{\text{I}}{{\text{C}}_{2~}}$$

(62)

$$\frac{{{\text{dI}}{{\text{C}}_3}}}{{{\text{d}}t}}={k_{{\text{ci}}}}{f^3}{{\text{C}}_3} - \frac{{{k_{{\text{ic}}}}}}{{{f^3}}}{\text{I}}{{\text{C}}_3}+\beta f{\text{I}}{{\text{C}}_2} - \frac{{3\alpha }}{f}{\text{I}}{{\text{C}}_3}$$

(63)

$${{\text{C}}_3}=1 - \left( {{{\text{C}}_2}+{{\text{C}}_1}+{\text{O}}+{\text{IF}}+{\text{I}}{{\text{C}}_1}+{\text{I}}{{\text{C}}_2}+{\text{I}}{{\text{C}}_3}} \right)$$

(64)

$$\alpha \left( {\text{V}} \right)={f_1}\left( {\text{V}} \right)\left[ {{\alpha _2}\left( {\text{V}} \right)+{\alpha _1}\left( {\text{V}} \right)\exp \left( {\frac{{V+50.0}}{{5.3}}} \right)} \right]$$

(65)

$$\beta \left( {\text{V}} \right)={f_1}\left( {\text{V}} \right)\left[ {{\beta _2}\left( {\text{V}} \right)+{\beta _1}\left( {\text{V}} \right)\exp \left( {\frac{{V+50.0}}{{5.0}}} \right)} \right]~$$

(66)

$${k_{{\text{oif}}1}}\left( {\text{V}} \right)=0.432 \cdot \exp \left( {\frac{{V+50.0}}{{30.0}}} \right)$$

(67)

$${k_{{\text{oif}}2}}\left( {\text{V}} \right)=0.0345 \cdot \exp \left( {\frac{{V+80.0}}{{14.8}}} \right)$$

(68)

$${k_{{\text{ifo}}1}}\left( {\text{V}} \right)=0.01 \cdot \mu \cdot \exp \left( { - \frac{{V+50.0}}{{30}}} \right)$$

(69)

$${k_{{\text{ifo}}2}}\left( {\text{V}} \right)=0.018 \cdot \rho \cdot \exp \left( { - \frac{{V+100.0}}{{13.6}}} \right)$$

(70)

$$\mu =0.00915\quad \rho =0.00915$$

(71)

$${k_{{\text{oif}}}}={f_2}\left( {\text{V}} \right)\left( {{k_{{\text{oif}}2}}+{k_{{\text{oif}}1}} \cdot \exp \left( {\frac{{V+90.0}}{{5.0}}} \right)} \right)$$

(72)

$${k_{{\text{ifo}}}}={f_2}\left( {\text{V}} \right)\left( {{k_{{\text{ifo}}2}}+{k_{{\text{ifo1}}}} \cdot \exp \left( {\frac{{V+90.0}}{{5.0}}} \right)} \right)$$

(73)

$${f=0.3}$$

(74)

$${k_{{\text{ci}}}}~=~3.0 \cdot {k_{{\text{oif}}}}$$

(75)

$${k_{{\text{ic}}}}={k_{\text{ci}}} \cdot \frac{{{k_{{\text{ifo}}}}}}{{{k_{{\text{oif}}}}}}$$

(76)

Model 4

$$\frac{{{\text{d}}{{\text{C}}_2}}}{{{\text{d}}t}}=3\alpha {{\text{C}}_3} - \beta {{\text{C}}_2}+2\beta {{\text{C}}_1} - 2\alpha {{\text{C}}_2}+\frac{{{k_{{\text{ic}}}}}}{{{f^2}}}{\text{I}}{{\text{C}}_2} - {k_{{\text{ci}}}}{f^2}{{\text{C}}_2}$$

(77)

$$\frac{{{\text{d}}{{\text{C}}_1}}}{{{\text{d}}t}}=2\alpha {{\text{C}}_2} - 2\beta {{\text{C}}_1}+3\beta {\text{O}} - \alpha {{\text{C}}_1}+\frac{{{k_{{\text{ic}}}}}}{f}{\text{I}}{{\text{C}}_1} - {k_{{\text{ci}}}}f{{\text{C}}_1}$$

(78)

$$\frac{{{\text{dO}}}}{{{\text{d}}t}}=\alpha {{\text{C}}_1}~ - 3\beta {\text{O}}+{k_{{\text{ifo}}}}{\text{IF}} - {k_{{\text{oif}}}}{\text{O}}+{k_{{\text{iso}}}}{\text{IS}} - {k_{{\text{ois}}}}{\text{O}}$$

(79)

$$\frac{{{\text{dIF}}}}{{{\text{d}}t}}={k_{{\text{oif}}}}{\text{O}} - {k_{{\text{ifo}}}}{\text{IF}}+\frac{\alpha }{f}{\text{I}}{{\text{C}}_1} - 3\beta f{\text{IF}}$$

(80)

$$\frac{{{\text{dI}}{{\text{C}}_1}}}{{{\text{d}}t}}=\frac{{2\alpha }}{f}{\text{I}}{{\text{C}}_2} - 2\beta f{\text{I}}{{\text{C}}_1}+3\beta f{\text{IF}} - \frac{\alpha }{f}{\text{I}}{{\text{C}}_1}+{k_{{\text{ci}}}}f{{\text{C}}_1} - \frac{{{k_{{\text{ic}}}}}}{f}{\text{I}}{{\text{C}}_1}$$

(81)

$$\frac{{{\text{dI}}{{\text{C}}_2}}}{{{\text{d}}t}}=\frac{{3\alpha }}{f}{\text{I}}{{\text{C}}_3} - \beta f{\text{I}}{{\text{C}}_2}+{k_{{\text{ci}}}}{f^2}{{\text{C}}_2} - \frac{{{k_{{\text{ic}}}}}}{{{f^2}}}{\text{I}}{{\text{C}}_2}+2\beta f{\text{I}}{{\text{C}}_1} - \frac{{2\alpha }}{f}{\text{I}}{{\text{C}}_2}$$

(82)

$$\frac{{{\text{dI}}{{\text{C}}_3}}}{{{\text{d}}t}}={k_{{\text{ci}}}}{f^3}{{\text{C}}_3} - \frac{{{k_{{\text{ic}}}}}}{{{f^3}}}{\text{I}}{{\text{C}}_3}+\beta f{\text{I}}{{\text{C}}_2} - \frac{{3\alpha }}{f}{\text{I}}{{\text{C}}_3}$$

(83)

$$\frac{{{\text{dIS}}}}{{{\text{d}}t}}={k_{{\text{ois}}}}{\text{O}} - {k_{{\text{iso}}}}{\text{IS}}$$

(84)

$${{C}_3}=1 - \left( {{{C}_2}+{{C}_1}+{O}+{IF}+{{IC}_1}+{{IC}_2}+{{IC}_3}+{IS}} \right)$$

(85)

$${\alpha}\left( {V} \right)={{{f}}_1}\left( {V} \right)\left[ {{{\alpha}_2}\left( {V} \right)+{{\alpha}_1}\left( {V} \right)\exp \left( {\frac{{{{V}}+50.0}}{{5.3}}} \right)} \right]$$

(86)

$$\beta \left( {\text{V}} \right)={f_1}\left( {\text{V}} \right)\left[ {{\beta _2}\left( {\text{V}} \right)+{\beta _1}\left( {\text{V}} \right)\exp \left( {\frac{{V+50.0}}{{5.0}}} \right)} \right]$$

(87)

$${k_{{\text{oif}}1}}\left( {\text{V}} \right)=0.432 \cdot \exp \left( {\frac{{V+50.0}}{{30.0}}} \right)$$

(88)

$${k_{{\text{oif}}2}}\left( {\text{V}} \right)=0.0345 \cdot \exp \left( {\frac{{V+80.0}}{{14.8}}} \right)$$

(89)

$${k_{{\text{ifo}}1}}\left( {\text{V}} \right)=0.01 \cdot \mu \cdot \exp \left( { - \frac{{V+50.0}}{{30}}} \right)$$

(90)

$${k_{{\text{ifo}}2}}\left( {\text{V}} \right)=0.018 \cdot \rho \cdot \exp \left( { - \frac{{V+100.0}}{{13.6}}} \right)$$

(91)

$$\mu =0.00915,\quad \rho =0.00915$$

(92)

$${k_{{\text{oif}}}}={f_2}\left( {\text{V}} \right)\left( {{k_{{\text{oif}}2}}+{k_{{\text{oif}}1}} \cdot \exp \left( {\frac{{V+90.0}}{{5.0}}} \right)} \right)$$

(93)

$${k_{{\text{ifo}}}}={f_2}\left( {\text{V}} \right)\left( {{k_{{\text{ifo}}2}}+{k_{{\text{ifo}}1}} \cdot \exp \left( {\frac{{V+90.0}}{{5.0}}} \right)} \right)$$

(94)

$$f=0.3$$

(95)

$${k_{{\text{ci}}}}=3.0 \cdot {k_{{\text{oif}}}}$$

(96)

$${k_{{\text{ic}}}}={k_{{\text{ci}}}} \cdot \frac{{{k_{{\text{ifo}}}}}}{{{k_{{\text{oif}}}}}}$$

(97)

$${k_{{\text{ois}}}}=1.0$$

(98)

$${k_{{\text{iso}}}}=0.05$$

(99)

Model 5

$$\frac{{{\text{d}}{{\text{C}}_2}}}{{{\text{d}}t}}=3\alpha {{\text{C}}_3} - \beta {{\text{C}}_2}+2\beta {{\text{C}}_1} - 2\alpha {{\text{C}}_2}+\frac{{{k_{{\text{ic}}}}}}{{{f^2}}}{\text{I}}{{\text{C}}_2} - {k_{{\text{ci}}}}{f^2}{{\text{C}}_2}$$

(100)

$$\frac{{{\text{d}}{{\text{C}}_1}}}{{{\text{d}}t}}=2\alpha {{\text{C}}_2} - 2\beta {{\text{C}}_1}+3\beta {\text{O}} - \alpha {{\text{C}}_1}+\frac{{{k_{{\text{ic}}}}}}{f}{\text{I}}{{\text{C}}_1} - {k_{{\text{ci}}}}f{{\text{C}}_1}$$

(101)

$$\frac{{{\text{dO}}}}{{{\text{d}}t}}=\alpha {{\text{C}}_1} - 3\beta {\text{O}}+{k_{{\text{ifo}}}}{\text{IF}} - {k_{{\text{oif}}}}{\text{O}}$$

(102)

$$\frac{{{\text{dIF}}}}{{{\text{d}}t}}={k_{{\text{oif}}}}{\text{O}} - {k_{{\text{ifo}}}}{\text{IF}}+{k_{{\text{isif}}}}{\text{IS}} - {k_{{\text{ifis}}}}{\text{IF}}$$

(103)

$$\frac{{{\text{dIS}}}}{{{\text{d}}t}}={k_{{\text{ifis}}}}{\text{IF}} - {k_{{\text{isif}}}}{\text{IS}}+\frac{\alpha }{f}{\text{I}}{{\text{C}}_1} - 3\beta f{\text{IS}}$$

(104)

$$\frac{{{\text{dI}}{{\text{C}}_1}}}{{{\text{d}}t}}=\frac{{2\alpha }}{f}{\text{I}}{{\text{C}}_2} - 2\beta f{\text{I}}{{\text{C}}_1}+3\beta f{\text{IS}} - \frac{\alpha }{f}{\text{I}}{{\text{C}}_1}+{k_{{\text{ci}}}}f{{\text{C}}_1} - \frac{{{k_{{\text{ic}}}}}}{f}{\text{I}}{{\text{C}}_1}$$

(105)

$$\frac{{{\text{dI}}{{\text{C}}_2}}}{{{\text{d}}t}}=\frac{{3\alpha }}{f}{\text{I}}{{\text{C}}_3} - \beta f{\text{I}}{{\text{C}}_2}+{k_{{\text{ci}}}}{f^2}{{\text{C}}_2} - \frac{{{k_{{\text{ic}}}}}}{{{f^2}}}{\text{I}}{{\text{C}}_2}+2\beta f{\text{I}}{{\text{C}}_1} - \frac{{2\alpha }}{f}{\text{I}}{{\text{C}}_2}$$

(106)

$$\frac{{{\text{dI}}{{\text{C}}_3}}}{{{\text{d}}t}}={k_{{\text{ci}}}}{f^3}{{\text{C}}_3} - \frac{{{k_{{\text{ic}}}}}}{{{f^3}}}{\text{I}}{{\text{C}}_3}+\beta f{\text{I}}{{\text{C}}_2} - \frac{{3\alpha }}{f}{\text{I}}{{\text{C}}_3}$$

(107)

$${{\text{C}}_3}=1 - ({{\text{C}}_2}+{{\text{C}}_1}+{\text{O}}+{\text{IF}}+{\text{IS}}+{\text{I}}{{\text{C}}_1}+{\text{I}}{{\text{C}}_2}+{\text{I}}{{\text{C}}_3})$$

(108)

$$\alpha \left( {\text{V}} \right)={f_1}\left( {\text{V}} \right)\left[ {{\alpha _2}\left( {\text{V}} \right)+{\alpha _1}\left( {\text{V}} \right)\exp \left( {\frac{{V+50.0}}{{5.3}}} \right)} \right]$$

(109)

$$\beta \left( {\text{V}} \right)={f_1}\left( {\text{V}} \right)\left[ {{\beta _2}\left( {\text{V}} \right)+{\beta _1}\left( {\text{V}} \right)\exp \left( {\frac{{V+50.0}}{{5.0}}} \right)} \right]$$

(110)

$${k_{{\text{oif}}1}}\left( {\text{V}} \right)=0.432 \cdot \exp \left( {\frac{{V+50.0}}{{30.0}}} \right)$$

(111)

$${k_{{\text{oif}}2}}\left( {\text{V}} \right)=0.0345 \cdot \exp \left( {\frac{{V+80.0}}{{14.8}}} \right)$$

(112)

$${k_{{\text{ifo}}1}}\left( {\text{V}} \right)=0.01 \cdot \mu \cdot \exp \left( { - \frac{{V+50.0}}{{30}}} \right)$$

(113)

$${k_{{\text{ifo}}2}}\left( {\text{V}} \right)=0.018 \cdot \rho \cdot \exp \left( { - \frac{{V+100.0}}{{13.6}}} \right)$$

(114)

$$\mu =0.288\,\rho =0.288$$

(115)

$${k_{{\text{oif}}}}\,=\,{f_2}\left( {\text{V}} \right)\left( {{k_{{\text{oif}}2}}+{k_{{\text{oif}}1}} \cdot \exp \left( {\frac{{V+90.0}}{{5.0}}} \right)} \right)$$

(116)

$${k_{{\text{ifo}}}}\,=\,{f_2}\left( {\text{V}} \right)\left( {{k_{{\text{ifo2}}}}+{k_{{\text{ifo}}1}} \cdot \exp \left( {\frac{{V+90.0}}{{5.0}}} \right)} \right)$$

(117)

$$f\,=\,0.3$$

(118)

$${k_{{\text{ci}}}}\,=\,3.0 \cdot {k_{{\text{oif}}}}$$

(119)

$${k_{{\text{ic}}}}\,=\,{k_{{\text{ci}}}} \cdot \frac{{{k_{{\text{ifo}}}} \cdot {k_{{\text{ifoi}}}}}}{{{k_{{\text{oif}}}} \cdot {k_{{\text{oiif}}}}}}~$$

(120)

$${k_{{\text{ifoi}}}}\,=\,0.0035$$

(121)

$$~{k_{{\text{oiif}}}}\,=\,0.1015$$

(122)

Model 6

$$\frac{{{\text{d}}{{\text{C}}_2}}}{{{\text{d}}t}}=3\alpha {{\text{C}}_3} - \beta {{\text{C}}_2}+2\beta {{\text{C}}_1} - 2\alpha {{\text{C}}_2}+\frac{{{k_{{\text{ic}}}}}}{{{f^2}}}{\text{I}}{{\text{C}}_2} - {k_{{\text{ci}}}}{f^2}{{\text{C}}_2}$$

(123)

$$\frac{{{\text{d}}{{\text{C}}_1}}}{{{\text{d}}t}}=2\alpha {{\text{C}}_2}~ - 2\beta {{\text{C}}_1}+3\beta {\text{O}} - \alpha {{\text{C}}_1}+\frac{{{k_{{\text{ic}}}}}}{f}{\text{I}}{{\text{C}}_1} - {k_{{\text{ci}}}}f{C_1}$$

(124)

$$\frac{{{\text{dO}}}}{{{\text{d}}t}}=\alpha {{\text{C}}_1} - 3\beta {{\text{O}}}+ {{{k}_{\text{ifo}}}}{\text{IF}}- {{{{k}}_{\text{oif}}}}{\text{O}}$$

(125)

$$\frac{{{\text{dIF}}}}{{{\text{d}}t}}={k_{{\text{oif}}}}{\text{O}}~ - {k_{{\text{ifo}}}}{\text{IF}}+{k_{{\text{isif}}}}{\text{IS}} - {k_{{\text{ifis}}}}{\text{IF}}$$

(126)

$$\frac{{{\text{dIS}}}}{{{\text{d}}t}}={k_{{\text{ifis}}}}{\text{IF}} - {k_{{\text{isif}}}}{\text{IS}}+{k_{{\text{imis}}}}{\text{IM}} - {k_{{\text{isim}}}}{\text{IS}}$$

(127)

$$\frac{{{\text{dIM}}}}{{{\text{d}}t}}={k_{{\text{isim}}}}{\text{IS}} - {k_{{\text{imis}}}}{\text{IM}}+\frac{\alpha }{f}{\text{I}}{{\text{C}}_1} - 3\beta f{\text{IM}}$$

(128)

$$\frac{{{\text{dI}}{{\text{C}}_1}}}{{{\text{d}}t}}=\frac{{2\alpha }}{f}{\text{I}}{{\text{C}}_2} - 2\beta f{\text{I}}{{\text{C}}_1}+3\beta f{\text{IM}} - \frac{\alpha }{f}{\text{I}}{{\text{C}}_1}+{k_{{\text{ci}}}}f{{\text{C}}_1} - \frac{{{k_{{\text{ic}}}}}}{f}{\text{I}}{{\text{C}}_1}$$

(129)

$$\frac{{{\text{dI}}{{\text{C}}_2}}}{{{\text{d}}t}}=\frac{{3\alpha }}{f}{\text{I}}{{\text{C}}_3} - \beta f{\text{I}}{{\text{C}}_2}+{k_{{\text{ci}}}}{f^2}{{\text{C}}_2} - \frac{{{k_{{\text{ic}}}}}}{{{f^2}}}{\text{I}}{{\text{C}}_2}+2\beta f{\text{I}}{{\text{C}}_1} - \frac{{2\alpha }}{f}{\text{I}}{{\text{C}}_2}$$

(130)

$$\frac{{{\text{dI}}{{\text{C}}_3}}}{{{\text{d}}t}}={k_{{\text{ci}}}}{f^3}{{\text{C}}_3} - \frac{{{k_{{\text{ic}}}}}}{{{f^3}}}{\text{I}}{{\text{C}}_3}+\beta f{\text{I}}{{\text{C}}_2} - \frac{{3\alpha }}{f}{\text{I}}{{\text{C}}_3}$$

(131)

$${C_3}=1 - \left( {{{\text{C}}_2}+{{\text{C}}_1}+{\text{O}}+{\text{IF}}+{\text{IS}}+{\text{IM}}+{\text{I}}{{\text{C}}_1}+{\text{I}}{{\text{C}}_2}+{\text{I}}{{\text{C}}_3}} \right)$$

(132)

$$\alpha \left( {\text{V}} \right)={f_1}\left( {\text{V}} \right)\left[ {{\alpha _2}\left( {\text{V}} \right)+{\alpha _1}\left( {\text{V}} \right)\exp \left( {\frac{{V+50.0}}{{5.3}}} \right)} \right]$$

(133)

$$\beta \left( {\text{V}} \right)={f_1}\left( {\text{V}} \right)\left[ {{\beta _2}\left( {\text{V}} \right)+{\beta _1}\left( {\text{V}} \right)\exp \left( {\frac{{V+50.0}}{{5.0}}} \right)} \right]$$

(134)

$${k_{{\text{oif}}1}}\left( {\text{V}} \right)=0.432 \cdot \exp \left( {\frac{{V+50.0}}{{30.0}}} \right)$$

(135)

$${k_{{\text{oif}}2}}\left( {\text{V}} \right)=0.0345 \cdot \exp \left( {\frac{{V+80.0}}{{14.8}}} \right)$$

(136)

$${k_{{\text{ifo1}}}}\left( {\text{V}} \right)\,=\,0.01 \cdot \mu \cdot {\text{exp}}\left( { - \frac{{V+50.0}}{{30}}} \right)$$

(137)

$${k_{{\text{ifo2}}}}\left( {\text{V}} \right)\,=\,0.018 \cdot \rho \cdot \exp \left( { - \frac{{V+100.0}}{{13.6}}} \right)$$

(138)

$$\mu \,=\,0.288\,\rho \,=\,0.288$$

(139)

$${k_{{\text{oif}}}}\,=\,{f_2}\left( {\text{V}} \right)\left( {{k_{{\text{oif2}}}}+{k_{{\text{oif1}}}} \cdot \exp \left( {\frac{{V+90.0}}{{5.0}}} \right)} \right)$$

(140)

$${k_{{\text{ifo}}}}\,=\,{f_2}\left( {\text{V}} \right)\left( {{k_{{\text{ifo2}}}}+{k_{{\text{ifo1}}}} \cdot \exp \left( {\frac{{V+90.0}}{{5.0}}} \right)} \right)$$

(141)

$$f\,=\,0.3$$

(142)

$${k_{{\text{ci}}}}\,=\,3.0 \cdot {k_{{\text{oif}}}}$$

(143)

$${k_{{\text{ic}}}}\,=\,{k_{{\text{ci}}}} \cdot \frac{{{k_{{\text{ifo}}}} \cdot {k_{{\text{imis}}}} \cdot {k_{{\text{isif}}}}}}{{{k_{{\text{oif}}}} \cdot {k_{{\text{isim}}}} \cdot {k_{{\text{ifis}}}}}}$$

(144)

$${k_{{\text{isif}}}}\,=\,0.0035$$

(145)

$$~{k_{{\text{ifis}}}}\,=\,0.1015$$

(146)

$${k_{{\text{isim}}}}\,=\,0.0125~$$

(147)

$$~{k_{{\text{imis}}}}\,=\,0.0125$$

(148)