Mammalian cells express many types of Na⁺-linked symporters. The human genome encodes literally hundreds of different types of transporters that use the energy stored in the Na⁺ concentration gradient and in the inside-negative electric potential across the plasma membrane to transport a wide variety of molecules into cells against their concentration gradients. To see how such transporters allow cells to accumulate substrates against a considerable concentration gradient, we first need to calculate the change in free energy (ΔG) that occurs during Na⁺ entry. As mentioned earlier, two forces govern the movement of ions across a selectively permeable membrane: the voltage across the membrane and the ion concentration gradient across the membrane. The sum of these forces constitutes the electrochemical gradient. To calculate the free-energy change, ΔG, corresponding to the transport of any ion across a membrane, we need to consider the independent contributions from each of these forces to the electrochemical gradient.

For example, when Na⁺ moves from the outside to the inside of a cell, the free-energy change generated from the Na⁺ concentration gradient is given by

At the concentrations of Na_in and Na_out shown in Figure 11-25, which are typical for many mammalian cells, ΔG_c, the change in free energy due to the concentration gradient, is −1.45 kcal for transport of 1 mole of Na⁺ ions from the outside to the inside of a cell, assuming there is no membrane electric potential. Note that the free energy is negative, indicating spontaneous movement of Na⁺ into the cell down its concentration gradient.

The free-energy change generated from the membrane electric potential is given by

ΔG_m = FE (11-6)

where F is the Faraday constant [= 23,062 cal/(mol·V)]; and E is the membrane electric potential. If E = −70 mV, then ΔG_m, the free-energy change due to the membrane potential, is −1.61 kcal for transport of 1 mole of Na⁺ ions from the outside to the inside of a cell, assuming there is no Na⁺ concentration gradient. Since both forces do in fact act on Na⁺ ions, the total ΔG is the sum of the two partial values:

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ΔG = ΔG_c + ΔG_m = (−1.45) + (−1.61) = −3.06 kcal/mol

In this example, the Na⁺ concentration gradient and the membrane electric potential contribute almost equally to the total ΔG for transport of Na⁺ ions. Since ΔG is less than 0, the inward movement of Na⁺ ions is thermodynamically favored. As we will see next, the inward movement of Na⁺ is used to power the uphill movement of other ions and several types of small molecules into or out of animal cells. The rapid, energetically favorable movement of Na⁺ ions through gated Na⁺ channels is also critical in generating action potentials in neurons and muscle cells, as discussed in Chapter 22.