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We apply the diagrammatic method developed by Hill (1977. Free Energy Transduction in Biology. Academic Press, New York) to analyze single-file water transport. We use this formalism to derive explicit expressions for the osmotic and diffusive permeabilities Pf and Pd of a pore. We first consider a vacancy mechanism of transport analogous to the one-vacancy pore model previously used by Kohler and Heckmann (1979. J. Theor. Biol. 79:381-401). (a) For the general one-vacancy case, we find that the permeability ratio can be expressed by Pf/Pd = (Pf/Pd)eqf(wA,wB), where the second factor is a function of the water activities in the two adjoining compartments A and B. As a consequence, the permeability ratio in general can effectively differ from its value at equilibrium. We also find that n - 1 less than or equal to (Pf/Pd)eq less than or equal to n, a result already proposed by Kohler and Heckmann (1979. J. Theor. Biol. 79:381-401). (b) When vacancy states are transient intermediates, the model can be reduced to a diagram consisting of only fully occupied states. Such a diagram resembles the one describing a no-vacancy mechanism of transport (c), but in spite of the similarity the expressions obtained for the permeability coefficients still retain the basic relationships of the original (a) nonreduced one-vacancy model. (c) We then propose a kinetic description of a no-vacancy mechanism of single-file water transport. In this case, the expressions derived for Pf and Pd are formally equivalent to those obtained by Finkelstein and Rosenberg (1979. Membrane Transport Processes. Vol. 3. C.F. Stevens and R.W. Tsien, editors, Raven Press, New York. 73-88.) A main difference with the vacancy mechanism is that here the permeability coefficients are independent of the water activities.
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