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Active transport of the Ca2+-pump: introduction of the temperature difference as a driving force

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Abstract

We analyse a kinetic cycle of the Ca2+-ATPase molecular pump using mesoscopic non-equilibrium thermodynamics. The pump is known to generate heat, and by analysing the operation on the mesoscopic level, we are able to introduce a temperature difference and the corresponding heat flux in the description. Integration over the internal coordinates then results in non-linear flux–force relations describing the operation of the pump on the macroscopic level. Specifically, we obtain an expression for the heat flux associated with the active transport and the coupling of heat effects to the transport of ions and the rate of the ATP-hydrolysis.

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Notes

  1. Here, we have the possibility of choosing different boundary conditions. For instance, we could include the factor \(\exp(-4\mu_{{\rm H^+}}^{\rm cyt}/RT)\) in the boundary conditions for path two, rather than in the coefficient C XY,2. The solution given in Eq. (29) is however independent of this choice.

  2. In the entropy production, the thermal driving force is independent of γ ij while the corresponding heat flux is not independent. The contribution to the entropy production from this flux–force pair is due to the integrated heat flux multiplied by the driving force. Since only the integrated heat flux matters here, we can, without loss of generality, choose the heat flux independent of γ ij . For instance, if the heat flux does not change sign for \(\gamma_{ij}\in(0,1)\), then, by the mean value theorem for integration, we represent the integral of the heat flux by J q,ij * ij ) for some \(\gamma_{ij}^*\in(0,1).\)

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Acknowledgments

A.L. would like to thank The Faculty of Natural Sciences and Technology, Norwegian University of Science and Technology, for a PhD scholarship.

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Correspondence to Anders Lervik.

Appendix: Coefficients for flux–force equations

Appendix: Coefficients for flux–force equations

The coefficients for the flux–force relations in Eqs. (4850) are given by,

$$ \begin{aligned} D_{{\rm rr}} &= \frac{1}{\Upgamma} \left( \frac{\tilde{D}^{\prime\prime}}{\epsilon^2} +\tilde{D}^{*}\right) , \quad D_{{\rm rd}} = \frac{\tilde{D}^{\prime\prime}}{\Upgamma \epsilon^2} (1 + \epsilon), \quad D_{{\rm rq}} = \frac{D_3}{\Upgamma},\\ D_{{\rm dd}} &= \frac{2(\tilde{D}^\prime+\tilde{D}^{\prime\prime})}{\Upgamma\epsilon^2} (1 + \epsilon), \quad D_{{\rm dr}} = \frac{2\tilde{D}^{\prime\prime}}{\Upgamma\epsilon^2} , \quad D_{{\rm dq}} = \frac{-2 D_1}{\Upgamma},\\ D_{{\rm qq}} &= \frac{D_4}{\Upgamma}, \quad D_{{\rm qr}} = \frac{1}{\Upgamma} \left( \frac{D_{{\rm q}}^{\prime\prime}}{\epsilon^2} + D_{{\rm q}}^* \right), \quad D_{{\rm qd}} = \frac{D_{{\rm q}}^{\prime} + D_{{\rm q}}^{\prime\prime}}{\Upgamma\epsilon^2} (1+\epsilon), \end{aligned} $$
(52)

where \(\epsilon \equiv \hbox{exp}\left[\Updelta\left(\frac{\mu_{{\rm Ca/2H}}}{RT}\right)\right].\) This coefficient matrix is not symmetric.

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Lervik, A., Bedeaux, D. & Kjelstrup, S. Active transport of the Ca2+-pump: introduction of the temperature difference as a driving force. Eur Biophys J 42, 321–331 (2013). https://doi.org/10.1007/s00249-012-0877-6

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  • DOI: https://doi.org/10.1007/s00249-012-0877-6

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