Carrier-Mediated Diffusion

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Facilitated or Carrier-Mediated Diffusion is generally described in terms of a diffusing Substrate S and a co-diffusing Carrier C. There must be a reversible chemical reaction, formally expressed here as:

  C + S ⇄ CS
where CS is the carrier-substrate complex. The process is described by the respective diffusion coefficients D and the equilibrium parameter K:

  K[C][S] = [CS]
where K is not a constant if the actual reaction is more complicated than stated above, and the reaction is in terms of concentrations []. But also the rate parameters k', k can be important, even crucially important:

∂[CS] = k'[C][S] − k[CS]
∂t
 As with K, the k' and k need not be constants, but may be apparent.
Since there is co-diffusion, for net species transport Fick's laws have to be applied for all the species, C, S and CS:

   First law:   J S+CS = − DS∇[S] − DCS∇[CS]   but also:   J C+CS = − DC∇[C] − DCS∇[CS]

Second law:   ∂([S]+[CS]) = DS∇2[S] + DCS∇2[CS]
∂t
but also: ∂([C]+[CS]) = DC∇2[C] + DCS∇2[CS]
∂t
where J and ∇ are 3-dimensional flux and gradient operator respectively. Often, the carrier is confined to a certain space and the change in [C]+[CS] is zero, which causes mathematical problems in solving for the describing equations. A general treatment can be found in the reference (1). As, a general mathematical solution (2) for flat layers.

Focus here will be on the oxygen (O2) carriers hemoglobin (Hb) and myoglobin (Mb). Hb is in the red blood cell, Mb in muscle tissue.

(1) Goddard JD Schultz JS Suchdeo SR: Facilitated transport via carrier-mediated diffusion in membranes. II. Mathematical aspects and analyses. AIChE J 20: 625-645 (1974).
(2) Hoofd L Kreuzer F: A new mathematical approach for solving carrier-facilitated steady state diffusion problems. J Math Biol 8(1): 1-13 (1979).

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