Ž . Bioelectrochemistry and Bioenergetics 45 1998 181–192 Modelling the interrelation between the transmembrane potential and pH difference across membranes with electrogenic proton transport upon build-up of the proton-motive force Stefan Schuster a, ) , Rachid Ouhabi b , Michel Rigoulet c , Jean-Pierre Mazat b a Max Delbruck Center for Molecular Medicine, Department of Bioinformatics, Robert-Rossle-Str. 10, 13125 Berlin-Buch, Germany ¨ ¨ b UniÕersite Bordeaux II, G.E.S.B.I., 146 rue Leo Saignat, F-33076 Bordeaux, France ´ ´ c C.N.R.S., Institut de Biochimie et Genetique Cellulaires, 1 rue Camille Saint-Saens, F-33077 Bordeaux, France ´´ ¨ Received 18 November 1997; revised 8 February 1998; accepted 23 February 1998 Abstract Ž . The wide-spread belief that the electric potential difference D f across membranes with electrogenic proton transport varies nearly m Ž . linearly with the pH difference D pH upon build-up of the proton-motive force is critically examined. First, we analyse experimental literature data concerning rat liver and yeast mitochondria, and E. coli. We then present a model describing the interrelation between the D pH and D f as the activities of the proton pumps or H q -ATPase or the influence of the proton leak vary. It is based on the m quasi-electroneutrality condition, the dissociation equilibrium of impermeant weak acids, a simple description of the cation–proton antiporters and cation leak, and the Nernst equation applied to all those ions subsisting in equilibrium. The model yields a nonlinear equation giving D f as a function of D pH. In various situations this function is quasi-linear in physiologically relevant ranges of D pH. m Thus, the linearity hypothesis can be substantiated theoretically, but is not necessarily justified under all circumstances. It is shown that Ž the slope of the D f vs. D pH curve is, in the quasi-linear regions, about y2.303 RTrF thus having the same value, but the opposite m . sign as in the Nernst equation when the cation–proton antiporters are absent or completely inhibited, and can be much higher in absolute value when these antiporters are operative. q 1998 Elsevier Science S.A. All rights reserved. Keywords: Proton transport; Transmembrane potential; pH difference; Electrochemical model; Mitochondria; Escherichia coli 1. Introduction Electrogenic proton transport across a great variety of Ž membranes e.g. the mitochondrial inner membrane, thyl- akoid membrane in chloroplasts, yeast and plant vacuolar . membranes, and lysosome membrane generates a proton- motive force, D m q, which consists of two parts, the pH ˜ H Ž . Ž . gradient D pH and the electric potential difference D f , m according to the formula D m qsy2.303 RT D pH q FD f . 1 Ž. ˜ H m Upon build-up of D m q by operation of the proton pumps, ˜ H both of its constituents usually increase in absolute value. On the other hand, the back-flow of protons through the H q -ATPase and proton leak tends to diminish both D pH and yD f . These quantities are therefore interrelated in a m ) Corresponding author. Fax: q49-30-94062834; e-mail: schuster@bp.biologie.hu-berlin.de monotonic way in that a change in D pH is accompanied with a change in yD f in the same direction and vice m versa, upon activation or inhibition of the proton pumps or the H q -ATPase activity or a change in passive proton permeability. We will call this relationship the covariant relation between the two constituents of D m q. Their ˜ H exact interdependence is not, however, immediately obvi- ous because it is affected by matrix buffering power and the presence of anions and cations on either side of the w x membrane 1,2 . There are situations where a change in D pH leads to a change in yD f in the opposite direction. This may be m achieved experimentally, for example, by addition of nigericin, thus increasing the potassium-proton exchange w x 3,4 , or by addition of valinomycin, thus decreasing the w x displacement of potassium from equilibrium 4–6 . A simi- lar behaviour results from variation of the acetate concen- wx tration in the medium 3 . In some cases, the two con- stituents of D m q are interconvertible virtually quantita- ˜ H 0302-4598r98r$19.00 q 1998 Elsevier Science S.A. All rights reserved.