Thermodynamic Efficiency of a Proton Pump

Protons are pumped in living systems to establish a proton gradient, and the energy necessary for this pumping is frequently provided by the hydrolysis of ATP, in which ADP and phosphate are formed [7]. In this section, we study a model of a proton pump found in the plasma membrane of plants [8-12] and include the coupling of potassium and calcium ion transport. As in prior examples, we calculate the thermodynamic efficiency [13] of the proton pump with a constant influx of ATP and compare that to the thermodynamic efficiency with an oscillatory influx of ATP, the average of which is the same as the constant concentration of ATP. 

In formulating the mechanism of the proton pump, we require the presence of certain nonlinearities to find the possibility of changing the dissipation, or the efficiency, with an oscillatory input of ATP. The minimum elements of a proton pump, although nonlinear, lead only to monotone relaxation kinetics, and thus only to decreases in efficiency upon imposition of an oscillatory influx of ATP. However, by including the coupling of other ion transport processes, such as those of potassium and calcium, the mechanism of the proton pump behaves like a damped oscillator, which has been observed in experiments 

The reaction sequence of a model of the cyclic, ATP driven proton pump 

In step 1, on the inner surface of the membrane, the uncharged enzyme X is phosphorylated, and in step 2 a proton from the cytoplasma binds to the 

The rate coefficients 2 and k 2 in step 4 are potential dependent and assumed to be given by 

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