3.3: Ion Transport
- Page ID
- 516015
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Ions moving around in batteries (galvanic cells) and across membranes in living cells are integral to energetic processes in both devices. There are significant dynamic (diffusion) contributions to these processes, but here we focus on the thermodynamics (energetics) of ions moving from one place to another, without accounting for things such as solvent viscosity or the size of the ions.
We will consider a simple model where there are two compartments separated by a membrane that is permeable to the ions in some way. For consistency with typical biochemical discussions of cell membranes we will refer to one side as the outside (extra-cellular) and the other as the inside (intra-cellular).
Concentration Gradient
Consider a single ion species that has a different concentration (activity, a) on the inside and outside. Using the convention of the inside as the "product", the chemical potential difference is then:
\[\Delta \mu = RT ln \frac{a_{in}}{a_{out}}\label{Dmugrad}\]
Potential Gradient
In living cells there is also a potential gradient due to a charge imbalance associated with the separation of the positive and negative ions that maintain overall charge balance onto opposite sides of the membrane (electrically similar to a capacitor). Figure \(\PageIndex{1}\) illustrates the situation including some of the most important contributors to both concentration gradients and charge imbalance across a cell membrane.
Figure \(\PageIndex{1}\): Although the system is electrically neutral overall, cells maintain a local charge imbalance across the cell membrane by separating counter ions on opposite sides of the membrane. In this figure the separated ions that cancel out each others' charges are the Na+ (blue) and the large A4- ions (orange). The large anions do not pass through the membrane (from https://commons.wikimedia.org).
The charge imbalance across the membrane leads to a potential difference across the membrane \(\Delta \phi = \phi_{in} - \phi_{out}\). Moving a mole of species with a charge of z down this potential gradient (in the diagram moving a positive charge from the outside to the inside with a \(\Delta \phi \lt 0\)) results in a
\[\Delta \mu = zF\Delta \phi < 0 \label{DmuEmf}\].
Total chemical potential change on crossing the membrane
If we add the potential changes in equations \(\ref{Dmugrad}\) and \(\ref{DmuEmf}\) we arrive at the total chemical potential change on transporting a particular ion species from outside to inside the membrane:
\[\Delta \mu = RT ln \frac{a_{in}}{a_{out}} + zF\Delta \phi \label{EQ:DmuMem}\]
If this is negative, motion of the ion into the cell is a spontaneous process and will happen as long as there is a way for the ion to pass through the membrane. Since the central section of the membrane is made up of primarily non-polar hydrocarbon tails, ions usually only pass through the membrane via ion channels primarily formed from protein complexes. These ion channels come in many forms.
Ions Crossing Cell Membranes
Ion channels can be divided into three very general categories:
- passive channels which allow ions to pass through them in the spontaneous direction. Many of these are selective toward particular ions. Some of them also only pass ions in one direction.
- channels that actively open and close switch either because of a change in the membrane potential (important in nerve signaling) or when a ligand attaches to or detaches from the protein structure. Closing actively switched channels decreases the rate at which ions travel down the potential gradient.
- ion pumps use an energy source to move ions in the nonspontaneous direction across the membrane. These actively maintain the concentration and charge gradient across the membrane. These are usually protein complexes that couple the conversion of \(\ce{ATP -> ADP}\) to the process of transport against the chemical potential.
An ion pump can effectively move an ion up the gradient as long as:
\[\Delta \mu = RT ln \frac{a_{in}}{a_{out}} + zF\Delta \phi + \Delta_r G(\ce{ATP -> ADP}) < 0 \label{EQ:DmuActive}\]