Facilitated Diffusion in a Flat Layer
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Through a flat layer, there is only diffusion in one dimension, x:
where Y can be carrier C, substrate S or complex CS. All
carrier forms C and CS are confined to the layer, from x=0 to x=L.
First, steady state will be handled, no net change in concentrations. But that
has to consider, that two exchangeable forms of species and of carrier exist,
so that only [S]+[CS] and [C]+[CS] are constant, and Ficks's second law
results in constant values for JS+CS and JC+CS.
But nor C nor CS can pass the layer border so both JC and
JCS are zero at x=0; consequently, JC+CS is zero
everywhere, leading to:
DC[C] + DCS[CS] is constant.
For heavy carriers, like Hb and Mb, it is assumend that D of both forms is
the same so that total carrier concentration [C]+[CS] is constant; and thus known
for a given set-up.
Similarly, we find for the total substrate:
| DS[S] + DCS[CS] =
(DS[S] + DCS[CS])x=0 − JS+CS |
x |
| L |
In case of equilibrium between C, S and CS this equation yields concentrations
at any place x. But there is a fundamental problem. Close to the boundary, there
is a flux JS so a gradient in [S] but no flux JCS so no
gradient in [CS] and consequently no equilibrium; [CS] at x=0 cannot be taken
from the equilibrium relation. This leads to boundary zones that, even small,
have a significant effect.
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