Biomedical Engineering Reference
In-Depth Information
the more complex dynamics of a Sodium channel. Each circle is a state that the channel may be in. The
α
and
β
terms are the rates of transition from one state to another and typically do not depend upon
V
m
.To
derive the differential equations for a state, we subtract the product of the state and all outflowing rates
(exiting arrows) from the product of the entering states and inflow rates (entering arrows). For example,
the equations for the model in Fig. 3.12 are:
dC
3
dt
=
−
β
23
C
2
α
32
C
3
(3.28)
dC
2
dt
=
β
12
C
1
+
α
32
C
3
−
(α
21
+
β
23
)C
2
(3.29)
dC
1
dt
=
β
01
O
+
α
21
C
2
−
(α
10
+
β
12
)C
1
(3.30)
dIF
dt
=
β
45
IS
+
α
40
O
−
(α
54
+
β
40
)I F
(3.31)
dIS
dt
=
α
54
IF
−
β
45
IS
(3.32)
dO
dt
=
β
40
IF
+
α
10
C
1
−
(α
40
+
β
10
)O .
(3.33)
The advantage of this type of model is that it is a more direct link between the genetic code,
expressed protein, structure of the channel, and electrophysiologic function. It is hoped that someday we
will be able to engineer sections of genetic code and simulate their function before they are made. In this
way, diseased channels could be repaired by making only the correction that is needed. Such a simulation
of the function will require a Markov model.
54
IF
IS
45
40
40
32
21
10
C3
C2
C1
O
23
12
01
Figure 3.12:
Markov model of Sodium channel.
3.4.4 Gating Dependence on pH,Temperature and Concentrations
It was mentioned in Sec. 3.1.4 that ion channels are often gated by
V
m
but could also be gated by pH,
ionic or molecular concentrations, temperature, or some combination of all of these gating mechanisms.
For example, the functions for
α
and
β
in the Hodgkin-Huxely model were for a particular temperature.
In general, these function may be much more complex, for example
α(V
m
,pH,
Mg
2
+
]
e
,
[
[
]
,T)
.
Although this may seem to be a daunting modeling task, there are some very important ion channels that
are not gated by
V
m
and are only sensitive to the concentration of some ion or molecule. These channels
are said to be
Ligand
-gated and typically play an important role in mutations, the impact of drugs and
the onset of diseases. In Ch. 6, the dynamics of the synapse will be modeled as gating by concentration
of neurotransmitter.
Atropine
