##### The basic model.

For the purpose of the model, activities are modelled as concentrations. This simplification holds true for low concentrations. At higher concentrations, the individual concentrations should be replaced by activities.

Consider two vessels, 1 and 2, containing weak acid, at equilibrium, from the definition of the equilibrium constant, the following holds:

- (1)

Consider now a situation where one of the vessels is the interior of a cell separated from the other by a semi-permeable membrane; Equation 1 must also be satisfied in an equilibrated system. Undissociated weak-acids are capable of diffusing freely through microbial membranes and do so until equilibrium is reached (Stein 1981; Stratford & Rose 1986). The equilibrium attained will satisfy Equation 1 and due to the free movement of the weak-acid across the membrane, [HA_{o}] _{=} [HA_{i}]. The dissociated anion is not freely permeable and is therefore trapped inside the cell when the weak acid dissociates. This means that any difference in the pH between the internal and extracellular fluids will also be reflected in the concentrations of the dissociated anion. The assumption is made that the dissociated anion cannot leave the cell, and that the attainment of [HA_{o}] _{=} [HA_{i}] is faster than any other process linked to the model.

From the definition of the equilibrium constant:

- -log [H
^{+}_{o}]-log [A^{-}_{o}]+log [HA_{o}] = -log [H^{+}_{i}]-log [A^{-}_{i}]+log [HA_{i}](2)

From the definition of pH:

- pH
_{o}-log [A^{-}_{o}]+log [HA_{o}] = pH_{i}-log [A^{-}_{i}]+log [HA_{i}](3)

For the situation where pH_{o =} pH_{i} and as, for a semi-permeable membrane, [HA_{o}] _{=} [HA_{i}], then [A_{o}^{−}] = [A_{i}^{−}]. If pH_{o}≠ pH_{i} then Equation 4 must be satisfied:

- (4)

With this model, a weak-acid has been added to a solution containing a microbe. The internal pH immediately falls and an equilibrium is reached such that the internal and external pH values are equal; this point is defined as time _{=} 0. It is assumed that the diffusion of weak-acid into the cell is infinitely fast compared with any active proton pumping that may occur. The model consists of calculating the accumulation of anion coupled to the rate of proton efflux, and then using this value to calculate the internal pH (Equation 4).

Within the cell HA H^{+} _{+} A^{−}.

Protons may be pumped from the cytoplasm by the H^{+}-ATPase. For every proton removed, one anion remains accumulated. HA then diffuses in through the membrane to immediately reset the equilibrium. However, as there are now ‘extra’ anions, the equilibrium concentrations required are slightly different and the internal pH alters. From Equation 4, at t _{=} 0, Equation 5 is obtained, where Q _{=} log [H_{o}^{+}][A_{o}^{−}].

- (5)

The rate of proton efflux is equal to the rate of anion accumulation. Thus, the change in internal pH can be obtained from Equation 6, where r _{=} rate of proton efflux, t _{=} time elapsed.

- (6)

Here, the rate of proton efflux is constant and independent of pH_{i} (anion accumulation is linear with time). On a longer time-scale, as the internal pH rises above 7, anion accumulation still occurs at the same rate. This is a system lacking feedback inhibition to the proton pump. As such this is not a realistic situation and the model requires adjustment. The modification involves limiting the rate of proton efflux with respect to the internal pH. A limiting factor, f, is introduced into Equation 6:

- (7)

The limiting factor must regulate the output of the proton pump. For this regulation a pH is defined, the nominal pH, pH_{n}, at which the effectiveness of the proton pump is zero (i.e. stops pumping) and the effectiveness of the proton pump is also defined at pH_{i}, t _{=} 0 (_{=}pH_{o}) to be 100%. In this scenario, the protons are pumped out as fast as possible to begin with and then, as the internal pH rises, the pumping slows down until pH_{n} is reached. In this model, change in internal pH is calculated over short time intervals (Equation 8), and the changes in pH summed to give the internal pH (Equation 9).

- (8)

- pH
_{i} = pH_{o}+ΣΔpH_{i}(9)

##### Modelling the H+-ATPase function.

To obtain a realistic model, the *in vivo* rate of H^{+}-ATPase activity with respect to pH should be used as the limiting factor. The efficiency of H^{+}-ATPase with respect to pH is known from experimental work (Willsky 1979; Eraso & Gancedo 1987). At low pH (<4·5), the enzyme was sluggish but achieved optimal performance at pH 5·5 (100% activity). At pH 7, it was shown to have 70% of optimum activity. Tests were carried out using isolated enzymes or membrane preparations. The work by Willsky (1979) gives activity at pH 10 which is obviously biologically unrealistic. In these tests, the enzyme lacked normal feedback inhibition mechanisms, and the operation of the H^{+}-ATPase would cease at some nominal pH because of feedback inhibition, except for enzyme used to maintain a pH to power active transport. The experimental data from low pH to optimum pH were fitted to half a Gaussian curve. The bold assumption was made that the feedback inhibition followed the other half of the Gaussian curve. This means that the efficiency of the H^{+}-ATPase approaches zero at low pH and also at the expected nominal pH (approximately pH _{=} 7). The fit to the experimental data is portrayed in Fig. 2. The Gaussian expression for the efficiency of the enzyme is described in Equation 10:

- efficiency = 10(-1/2(pH-pHp/Gw))
^{2}(10)

where pHp = peak pH of the Gaussian curve; G_{w =} measure of the width of the curve. A Gaussian function with pH_{p =} 5·5 and G_{w =} 0·487 (parameters from experimental data) was used as the enzyme factor in Equation 7 and modelled using the analogous form of Equation 8.