# A model of the sodium channel: Introducing CellML encapsulation and interfaces¶

The HH sodium channel has two types of gate, an \(m\) gate (of which there are 3) that is initially closed (\(m = 0\)) before activating and inactivating back to the closed state, and an \(h\) gate that is initially open (\(h = 1\)) before activating and inactivating back to the open state. The short period when both types of gate are open allows a brief window current to pass through the channel. Therefore,

where \(\bar{g}_{\text{Na}} = \ \)120
mS.cm^{-2}, and with
\(\left\lbrack \text{Na}^{+} \right\rbrack_{i}\)= 30mM and
\(\left\lbrack \text{Na}^{+} \right\rbrack_{o}\)= 140mM, the
Nernst potential for the sodium channel (z=1) is

The gating kinetics are described by

where the voltage dependence of these four rate constants is determined experimentally to be[1]

Before we construct a CellML model of the sodium channel, we first
introduce some further CellML concepts that help deal with the
complexity of biological models: first the use of *encapsulation groups*
and *public* and *private interfaces* to control the visibility of
information in modular CellML components. To understand encapsulation,
it is useful to use the terms ‘parent’, ‘child’ and ‘sibling’.

```
def group as encapsulation for
comp sodium_channel incl
comp sodium_channel_m_gate;
comp sodium_channel_h_gate;
endcomp;
enddef;
```

We define the CellML components **sodium_channel_m_gate** and
**sodium_channel_h_gate** below. Each of these components has its own
equations (voltage-dependent gates and first order gate kinetics) but
they are both parts of one protein – the sodium channel – and it is
useful to group them into one **sodium_channel** component as shown above:

We can then talk about the sodium channel as the parent of two children:
the m gate and the h gate, which are therefore siblings. A *private
interface* allows a parent to talk to its children and a *public
interface* allows siblings to talk among themselves and to their parents
(see Fig. 22).

The OpenCOR *CellML Text* for the HH sodium ion channel is given below.

```
def model sodium_ion_channel as
def unit millisec as
unit second {pref: milli};
enddef;
def unit per_millisec as
unit second {pref: milli, expo: -1};
enddef;
def unit millivolt as
unit volt {pref: milli};
enddef;
def unit per_millivolt as
unit millivolt {expo: -1};
enddef;
def unit per_millivolt_millisec as
unit per_millivolt;
unit per_millisec;
enddef;
def unit microA_per_cm2 as
unit ampere {pref: micro};
unit metre {pref: centi, expo: -2};
enddef;
def unit milliS_per_cm2 as
unit siemens {pref: milli};
unit metre {pref: centi, expo: -2};
enddef;
def unit mM as
unit mole {pref: milli};
enddef;
def comp environment as
var V: millivolt {pub: out};
var t: millisec {pub: out};
V = sel
case (t > 5 {millisec}) and (t < 15 {millisec}):
-20.0 {millivolt};
otherwise:
-85.0 {millivolt};
endsel;
enddef;
def group as encapsulation for
comp sodium_channel incl
comp sodium_channel_m_gate;
comp sodium_channel_h_gate;
endcomp;
enddef;
def comp sodium_channel as
var V: millivolt {pub: in, priv: out};
var t: millisec {pub: in, priv: out };
var m: dimensionless {priv: in};
var h: dimensionless {priv: in};
var g_Na: milliS_per_cm2 {init: 120};
var i_Na: microA_per_cm2 {pub: out};
var Nao: mM {init: 140};
var Nai: mM {init: 30};
var RTF: millivolt {init: 25};
var E_Na: millivolt;
var Na_conductance: milliS_per_cm2 {pub: out};
E_Na=RTF*ln(Nao/Nai);
Na_conductance = g_Na*pow(m, 3{dimensionless})*h;
i_Na= Na_conductance*(V-E_Na);
enddef;
def comp sodium_channel_m_gate as
var V: millivolt {pub: in};
var t: millisec {pub: in};
var alpha_m: per_millisec;
var beta_m: per_millisec;
var m: dimensionless {init: 0.05, pub: out};
alpha_m = 0.1{per_millivolt_millisec}*(V+25{millivolt})
/(exp((V+25{millivolt})/10{millivolt})-1{dimensionless});
beta_m = 4{per_millisec}*exp(V/18{millivolt});
ode(m, t) = alpha_m*(1{dimensionless}-m)-beta_m*m;
enddef;
def comp sodium_channel_h_gate as
var V: millivolt {pub: in};
var t: millisec {pub: in};
var alpha_h: per_millisec;
var beta_h: per_millisec;
var h: dimensionless {init: 0.6, pub: out};
alpha_h = 0.07{per_millisec}*exp(V/20{millivolt});
beta_h = 1{per_millisec}/(exp((V+30{millivolt})/10{millivolt})+1{dimensionless});
ode(h, t) = alpha_h*(1{dimensionless}-h)-beta_h*h;
enddef;
def map between environment and sodium_channel for
vars V and V;
vars t and t;
enddef;
def map between sodium_channel and sodium_channel_m_gate for
vars V and V;
vars t and t;
vars m and m;
enddef;
def map between sodium_channel and sodium_channel_h_gate for
vars V and V;
vars t and t;
vars h and h;
enddef;
enddef;
```

The results of the OpenCOR computation, with *Ending point* 40 and
*Point interval* 0.1, are shown in Fig. 23 with plots \(V\left( t \right)\), \(m\left( t \right)\),
\(h\left( t \right)\), \(g_{\text{Na}}\left( t \right)\) and
\(i_{\text{Na}}(t)\) for voltage steps from (a) -85mV to -20mV, (b) -85mV to 0mV and (c) -85mV to 20mV. There are several
things to note:

- The kinetics of the m-gate are much faster than the h-gate.
- The opening behaviour is faster as the voltage is stepped to higher values since \(\tau = \frac{1}{\alpha_{n} + \beta_{n}}\) reduces with increasing V (see Fig. 18).
- The sodium channel conductance rises (
*activates*) and then falls (*inactivates*) under a positive voltage step from rest since the three m-gates turn on but the h-gate turns off and the conductance is a product of these. Compare this with the potassium channel conductance shown in Fig. 21 which is only reduced back to zero by stepping the voltage back to its resting value – i.e.*deactivating*it. - The only time current \(i_{\text{Na}}\) flows through the sodium channel is during the brief period when the m-gate is rapidly opening and the much slower h-gate is beginning to close. A small current flows during the reverse voltage step but this is at a time when the h-gate is now firmly off so the magnitude is very small.
- The large sodium current \(i_{\text{Na}}\) is an inward current and hence negative.

Note that the bottom trace does not quite line up at t=0 because the values shown on the axes are computed automatically and hence can take more or less space depending on their magnitude.

Footnotes

[1] | The HH paper used \(\alpha_m\ =\ \frac{0.1(v+25)}{e^{\frac{(v+25)}{10}}-1}\); \(\beta_m\ =\ 4e^{\frac{v}{18}}\); \(\alpha_h\ =\ 0.07e^{\frac{v}{20}}\); \(\beta_h\ =\ \frac{1}{e^{\frac{(v+30)}{10}}+1}\);. |