4.2.4.P7
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An important part of the functioning of a biological membrane is its ability to selectively pass either sodium ions (Na+) or potassium ions (K+) through channels in the membrane. Because these ions have the same charge (+e), the electric force exerted on them by the membrane will be similar. In addition, they are about the same size (RNa+ ~ 0.12 nm, RK+ ~ 0.15 nm). So how can channels in the membrane distinguish them? One mechanism that has been proposed to account for the fact that the membrane channels treat these two ions differently is the suggestion that the ions attract water molecules providing a sphere of water that magnifies the small size difference between the ions. This “coating of water” is illustrated in the figure at the right. |
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To actually calculate the size of the ion’s “water coat” is difficult. (for example, we would have to use quantum mechanics to include a repulsion force that keeps the atoms from getting too close to each other.) But we can get a first idea of what is happening by exploring the electric force between the ion and one water molecule as shown in the figure at the right.
We label the ion A, the oxygen in the water B, and the hydrogen atoms in the water C and D. |
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(a) To simplify the calculation, we’ll make a “simple physics model” – take the ion and the two hydrogens as each having a charge +e, while the oxygen has a charge -2e. Treat each of these as point charges. On the diagram below, draw arrows indicating the direction and relative magnitude of the force the ion exerts on each of the three atoms in water, B, C, and D.
(b) Assuming that atoms of the water are held together in the arrangement shown without collapsing onto a point (quantum mechanics prevents them from getting too close to each other!), is the net force that A exerts on the molecule (B+C+D) attractive or repulsive? Why do you say so?
(c) If the grid spacing in the figure above (between the darker lines) is 1 nm, calculate the magnitude of the force that the ion (labeled A) exerts on the water molecule in picoNewtons (10-12 N) by adding together the force it exerts on all three atoms of the water. (Remember that forces add as vectors!) How does the force that the water molecule exerts on the ion compare to this?
| (d) We would like to find not just the force on the water molecule for one particular position, but as a function of the distance between the two. Suppose the parameters of the size of the water molecule are a and b as shown in the figure at the right. We want to find the force between the water molecule and the ion as a function of x, depending on parameters a and b. First find r, the distance between the ion and the hydrogens as a function of x, a, and b. Then find the magnitude and components of the force the ion exerts on each of the water atoms. Then, taking components, find the net electric force that the ion, A exerts on the whole water molecule (B+C+D) as a function of x, a, and b. You might find it useful in taking components to define an angle and use sines and cosines to get components. But when you have put everything together, express your sines and cosines in terms of x, a, and b. |  |
Joe Redish 8/7/11