Kinetic and mesoscopic non-equilibrium description of the Ca²⁺ pump: a comparison

Abstract

We analyse the operation of the Ca²⁺-ATPase ion pump using a kinetic cycle diagram. Using the methodology of Hill, we obtain the cycle fluxes, entropy production and efficiency of the pump. We compare these results with a mesoscopic non-equilibrium description of the pump and show that the kinetic and mesoscopic pictures are in accordance with each other. This gives further support to the mesoscopic theory, which is less restricted and also can include the heat flux as a variable. We also show how motors can be characterised in terms of unidirectional backward fluxes. We proceed to show how the mesoscopic approach can be used to identify fast and slow steps of the model in terms of activation energies, and how this can be used to simplify the kinetic diagram.

Appendix: Sum of directional diagrams for the kinetic cycle

For completeness, we show how to obtain the terms in the expression for the sum of the directional diagrams for the complete kinetic cycle, using the methodology of Hill (1989). The first 14 terms are

$$ \begin{aligned} \sum &= k_{\text{BC}} k_{\text{CD}} k_{\text{DE}} k_{\text{EF}} k_{\text{FA}} + k_{\text{BA}} k_{\text{CD}} k_{\text{DE}} k_{\text{EF}} k_{\text{FA}}\\ &+ k_{\text{CB}} k_{\text{BA}} k_{\text{DE}} k_{\text{EF}} k_{\text{FA}} + k_{\text{DC}} k_{\text{CB}} k_{\text{BA}} k_{\text{EF}} k_{\text{FA}}\\ &+ k_{\text{ED}} k_{\text{DC}} k_{\text{CB}} k_{\text{BA}} k_{\text{FA}} + k_{\text{FE}} k_{\text{ED}} k_{\text{DC}} k_{\text{CB}} k_{\text{BA}}\\ &+ k_{\text{DB}} k_{\text{CB}} k_{\text{BA}} k_{\text{EF}} k_{\text{FA}} + k_{\text{ED}} k_{\text{DB}} k_{\text{CB}} k_{\text{BA}} k_{\text{FA}}\\ &+ k_{\text{FE}} k_{\text{ED}} k_{\text{DB}} k_{\text{CB}} k_{\text{BA}} + k_{\text{CB}} k_{\text{BD}} k_{\text{DE}} k_{\text{EF}} k_{\text{FA}}\\ &+ k_{\text{DB}} k_{\text{CB}} k_{\text{BA}} k_{\text{EF}} k_{\text{FA}} + k_{\text{ED}} k_{\text{DB}} k_{\text{CB}} k_{\text{BA}} k_{\text{FA}}\\ &+ k_{\text{FE}} k_{\text{ED}} k_{\text{DB}} k_{\text{CB}} k_{\text{BA}} + k_{\text{CB}} k_{\text{BD}} k_{\text{DE}} k_{\text{EF}} k_{\text{FA}}\\ &+ 70\,\rm{other\,terms}.\\ \end{aligned} $$

(40)

We consider the kinetic cycle given in Fig. 3 and the sum of directional diagrams of state A. By removing the line connecting states B and D, there are now six ways to remove a second line to create a partial diagram. This gives the first six terms.

By removing the line connecting states D and C, there are four ways to remove a second line to create partial diagrams not already considered. This gives the four next terms.

By removing the line connecting states B and C, there are four ways to remove a second line to create partial diagrams not already considered. This gives the four next terms.

By considering the remaining states in the same fashion, the remaining 5 × 14 = 70 terms can be obtained.