# Gnaiger 1989 Energy Transformations

Gnaiger E (1989) Mitochondrial respiratory control: energetics, kinetics and efficiency. In: Energy transformations in cells and organisms. Wieser W, Gnaiger E (eds), Thieme, Stuttgart:6-17. |

Gnaiger Erich (1989) Thieme

*Abstract*: Mitochondrial respiratory control mechanisms are central in regulating the flux of ATP turnover which mediates between catabolism and anabolism. Such control can be exerted by the catabolic driving force for ATP production or by the load on the ATP system due to the ATP demand for biosynthesis, ion pumps or muscle activities. The signals of drive and load are integrated in the energy status of the ATP/ADP system, expressed by the phosphorylation potential. Is metabolic flux regulated by energy demand or energy supply? Do ATP/ADP ratios or ADP concentrations regulate the flux of mitochondrial oxidative phosphorylation? Different answers are obtained depending on metabolic conditions, but primarily depending on the particular model of flux control. Clearly, a reconciliation of divergent traits of kinetics and nonequilibrium thermodynamics ("ergodynamics") is of great theoretical and practical importance. Such a reconciliation is proposed here on the basis of Einstein's diffusion equation. The new concept accounts for the effects exerted on flux by both the Gibbs force of reaction and the total concentration of all substrates and products. Its application to the study of chemical reactions as complex as mitochondrial oxidative phosphorylation provides new insights into the regulatory mechanisms of energy transformation, emphasizing the effect of efficiency on the control of flux.
* • Keywords:* Gibbs force, free activity, reaction pressure

* • O2k-Network Lab:* AT Innsbruck Oroboros

## Force or pressure? - The linear flux-pressure law

- "For many decades the pressure-force confusion has blinded the most brilliant minds, reinforcing the expectation that Ohm’s linear flux-force law should apply to the hydrogen ion circuit and protonmotive force. .. Physicochemical principles explain the highly non-linear flux-force relation in the dependence of LEAK respiration on the
*pmF*. The explanation is based on an extension of Fick’s law of diffusion and Einstein’s diffusion equation, representing protonmotive pressure ― isomorphic with mechanical pressure, hydrodynamic pressure, gas pressure, and osmotic pressure ― which collectively follow the generalized linear*flux-pressure law*." - Gnaiger E (2020) Mitochondrial pathways and respiratory control. An introduction to OXPHOS analysis. 5
^{th}ed. Bioenerg Commun 2020.2. https://doi.org/10.26124/bec:2020-0002 - »
**pressure = force × free activity**

- "For many decades the pressure-force confusion has blinded the most brilliant minds, reinforcing the expectation that Ohm’s linear flux-force law should apply to the hydrogen ion circuit and protonmotive force. .. Physicochemical principles explain the highly non-linear flux-force relation in the dependence of LEAK respiration on the

## Einstein's diffusion equation and Fick's law: vectors

**J**_{d}= -*u*·*c*·**F**_{d}(Einstein 1905).

**J**_{d}is the flux of diffusion of a substance per unit area of a plane, in the direction perpendicular to the plane [mol·s^{-1}·m^{-2}]**F**_{d}= d*μ*/d*z*is the force of diffusion, which is the chemical potential gradient of the diffusing substance in the*z*direction perpendicular to the plane [J·mol^{-1}·m^{-1}].

*μ*is the chemical potential of the diffusing substance [J·mol^{-1}].*z*is the distance in the*z*direction perpendicular to the plane [m].

*P*_{V}=*J*_{d}·*F*_{d}[W·m^{-3}]; the product of flux and force is volume-specific power.*c*is the local concentration (*free activity*,*α*) of the diffusing substance, localized across the plane at a diffusion distance at which*F'***d****is measured [mol·dm**^{-3}].*c*=*n*/*V*; concentration is amount of substance,*n*, divided by volume,*V*.*u*is the mobility, or the reciprocal value of the frictional coefficient [mol·s^{-1}·kJ^{-1}·m].

- Diffusion pressure gradient: d
_{d}*Π*/d*z*=*c*·d*μ*/d*z*=*α*·*F*_{d}[kJ·m^{-3}·m^{-1}= kN·m^{-2}·m^{-1}= kPa·m^{-1}]

- Pressure of the ideal gas law (osmotic pressure, generalized by Van't Hoff):
*Π*=*RT*·*n*/*V*=*RT*·*c*

- Pressure of the ideal gas law (osmotic pressure, generalized by Van't Hoff):

- Diffusion pressure gradient: d

- Einstein's diffusion equation:
**J**_{d}= -*u*·d_{d}*Π*/d*z*= -*u*·*c*·d*μ*/d*z* - Fick's first law of diffusion:
**J**_{d}= -*D*·d*c*/d*z*

- Einstein's diffusion equation:

- Taken together:
*D*·d*c*/d*z*=*u*·*c*·d*μ*/d*z*=*u*·*RT*·d*c*/d*z*

*c*·d*μ*/d*z*=*RT*·d*c*/d*z*= d_{d}*Π*/d*z*; the diffusion pressure gradient is*RT*times the concentration gradient.*D*=*u*·*RT*; the diffusion coefficient equals the mobility times*RT*(Einstein 1905).

- Taken together:

- Fick's first law of diffusion describes a linear flux-pressure relation, but not a linear flux-force relation:
*J*_{d}= -*u*·*RT*·d*c*/d*z*= -*u*·d_{d}*Π*/d*z*

- Fick's first law of diffusion describes a linear flux-pressure relation, but not a linear flux-force relation:

## Compartmental description of diffusion: vectorial flux and force in a discontinuous system

## Chemical reaction, force of reaction, and reaction pressure

- Gibbs force of reaction:
*F*_{r}=*RT*·ln(*M*/*K*) = Σ_{B}(_{B}·*μ*_{B}[J·mol^{-1}]

- Reaction Gibbs energy: Δ
_{r}*G*=*F*_{r} - Equilibrium constant:
*K*=*M*_{eq}; the equilibrium constant equals the mass action ratio at equilibrium. - Mass action ratio:
*M*= Π_{B}(*a*_{B}^*ν*_{B}) = Π_{p}(*a*_{p}^*ν*_{p})/Π_{s}(*a*_{s}^-*ν*_{s});*M*is the product of the activities of all reactants, B, to the power of their stoichiometric numbers,*ν*_{B}.*ν*_{B}is positive for products, p, and negative for substrates, s. - Activity:
*a*_{B}=*c*_{B}/*c*°; division of concentrations by the standard concentration,*c*° = 1 mol.dm^{-3}), yields dimensionless quantities for activities,*K*, and*M*.

- Reaction Gibbs energy: Δ

- Gibbs force of reaction:

- Free activity,
*α*= (_{p}*a*-_{s}*a*)/ln(_{p}*a*/_{s}*a*) [mol·dm^{-3}]

_{p}*a*is the product activity, including all products._{s}*a*is the substrate activity, including all substrates.

- Free activity,

- Reaction pressure: Δ
_{r}*Π*=*α*·*F*_{r}[J·dm^{-3}= kJ·m^{-3}= kPa]

- Reaction pressure: Δ

- Reaction flux:
*J*_{r}= -*b*·*α*·*F*_{r}= -*b*·Δ_{r}*Π*(Gnaiger 1989)

- Reaction flux:

## Cited by

- Gnaiger E (2024) Addressing the ambiguity crisis in bioenergetics and thermodynamics. MitoFit Preprints 2024.3. https://doi.org/10.26124/mitofit:2024-0003

- Gnaiger E (2020) Mitochondrial pathways and respiratory control. An introduction to OXPHOS analysis. 5
^{th}ed. Bioenerg Commun 2020.2. https://doi.org/10.26124/bec:2020-0002

*Labels:* **MiParea:** Respiration

**Organism:** Rat
**Tissue;cell:** Liver
**Preparation:** Isolated mitochondria

**Regulation:** ADP, ATP, ATP production, Coupling efficiency;uncoupling, Flux control
**Coupling state:** OXPHOS

Pressure, BEC 2020.2, Gnaiger 2024 MitoFit