Transport Across Membranes A Black-Box Approach

As a first step towards a model for transport across membranes, let us consider a very simple situation where just one kind of a neutral molecule is transported across the membrane under the action of its own concentration difference Ac = c - c, c and c being the concentrations of the molecule at the two sides of the membrane. The simplest ansatz for the flux J of the molecule, i.e., the number of moles penetrating through the membrane per time and area, would be Pick s first law. 

The ansatz (2.16) is a typical black-box approach in that it completely neglects all details of membrane structure and molecular transport mechanisms. Let us represent this ansatz by a very simple network where the transport process is expressed as a material resistance R upon which the chemical potentials y and y of the molecule at the two sides of the membrane are acting  

For this network, we can formulate a relation which is analogous to Ohm s law in an electric network, namely 

So far, our simple model only describes steady state transport across membranes, but no relaxation. In the network of Fig. 2, the flux J would adjust to a new steady state value without time delay after a change of the potential difference. In order to exhibit relaxation behaviour, the membrane must be capable of storing the molecules which are transported across the membrane. The network representation of such a storage phenomenon is a material capacitance which has to be added to the network as shown in Fig. 3  

In Fig. 3, we have assumed that the membrane is symmetric such that the material resistance is split into two equal parts R/2 between which the capacitance has been placed. The definition of the material capacitance follows that of an electrical capacitance, namely 

---