Gnaiger 2018 MiPschool Tromso A2

From Bioblast
Erich Gnaiger
The protonmotive force and respiratory control. 1. Coupling of electron transfer reactions to vectorial translocation of protons. 2. From Einstein’s diffusion equation on gradients to Fick’s law on compartments.

Link: MitoEAGLE

Gnaiger E (2018)

Event: MiPschool Tromso-Bergen 2018

Peter Mitchell's protonmotive force is one of the most fundamental concepts in biology [1]. The catabolic reactions of mitochondrial electron transfer (ET) are coupled to vectorial translocation of protons at three coupling sites, which are the proton pumps of the ET system (ETS): respiratory Complexes CI, CIII, and CIV. The driving force of the ETS in the catabolic (k) reaction expressed as O2 consumption is the Gibbs force of reaction, ΔkFO2, which is typically in the range of -460 to -480 kJ/mol (~ -1.2 V). The Gibbs force is an isomorphic force, also known as a generalized force (the negative affinity of chemical reactions in nonequilibrium thermodynamics) [2]. Confusion is caused by the failure of terminological distinction between Gibbs energy change of reaction, ΔrG [J], and Gibbs force equal to the partial Gibbs energy change per advancement of reaction [3]. For the protonmotive force the proton is the motive entity, which can be expressed in a variety of formats with different motive units, MU.

A problem in the bioenergetic literature is the confusion between proton gradients (vector analysis in continuous systems) and differences of proton concentrations (activities) between compartments separated by a semipermeable membrane (vectorial analysis in compartmental systems). Fundamental insights are gained by distinguishing between vectoral forces and flows, versus vectorial forces and flows. This is explained by (1) appreciation of Ludwig Bolzmann's impact on today's scientific world-view, and (2) an explanation of the relevance of Einstein's diffusion equation for understanding the relation between protonmotive force and metabolic flux. Boltzmann committed suizide in 1906, one year after Einstein applied his particle concept of physics successfully to explain diffusion on the basis of Brownian motion [4]. At steady-state the local concentration along a diffusion gradient changes as a function of the chemical potential gradient, whereas the concentration gradient is constant along the diffusion path (in a homogenous medium at steady state; Fick's law of diffusion). Therefore, the concentration gradient of a continous system can be replaced by the concentration difference in a discontinuous system. Here is where some simple equations help. Thermodynamics and ergodynamics are inherently mathematical, but relationships expressed in well defined terms rather than mathematical equations are the basis of understanding. Development of such understanding and immediate applications to plan and interpret experimental results on mitochondrial respiratory control, however, is greatly aided by making us familiar with a large number of fundamental physicochemical terms (or concepts) and their (mostly simple) mathematical relationships. But why should we be interested in the Gas law (the Gas equation)?

Keywords: Force, Protonmotive force, Flux Bioblast editor: Gnaiger E

Abstract continued

Consider the Boltzmann constant, k, times absolute temperature, T, as a factor that assigns an exergy to a single particle [J·x-1], where J is the symbol for the SI unit of energy or exergy, and x is the symbol for the dimensionless unit of a particle (or an object). Isomorphically, the Gas constant , R, times absolute temperature, T, assigns an exergy to an amount of particles (molecules) expressed in moles [mol], such that RT is expressed in chemical units [J·mol-1]. k and R = k·NA are thus fundamental constants expressed in different motive units, in the particle and chemical formats, respectively. If a concentration difference is expressed in the particle format [x·m-3] and multiplied by kT [J·x-1], we obtain a pressure difference [J·m-3 = Pa]. Equally, a pressure difference is obtained by multiplication of a chemical concentration difference [mol·m-3] with RT [J·mol-1]. The diffusion pressure difference at steady-state is the product of a concentration (activity; [x·m-3] or [mol·m-3]) times the driving force of diffusion, which is the partial Gibbs energy change per advancement of diffusion, in particle units [J·x-1] or chemical units [J·mol-1].
We can derive the equation for the stoichiometric chemical potential difference, ΔμH+ [J·mol-1] from Einstein's diffusion equation. This provides an amazing insight into the nature of the chemical component of the protonmotive force. More importantly, however, Einsteins's diffusion equation is the gateway towards the realization, that the non-ohmic (non-linear) flux-force relation, which remains an enigmatic experimental phenomenon in bioenergetics, is deeply rooted in the Boltzmann equation. It leads to the realization, that the flux-pressure concept (such as Fick's law of diffusion) rather than flux-force relationships of the thermodynamics of irreversible processes provides the key to understand mitochondrial respiratory control [5].
  • From protonmotive force to chemiosmotic pressure:
  1. Molecular motion in a diffusion gradient: from Fick’s law, the gas law and the van’t Hoff equation, to Einstein’s diffusion equation
  2. We do not measure gradients: from concentration and potential gradients to concentration and potential differences
  3. Electrochemical protonmotive force and chemiosmotic pressure: proton pumps and counterions
  4. Why are mitochondria small? Implications of the mitochondrial volume fraction on -200 mV, non-linear respiratory control by the protonmotive force explained by protonmotive pressure.

Affiliations and support

  1. D. Swarovski Research Lab, Dept. Visceral, Transplant Thoracic Surgery, Medical Univ Innsbruck
  2. Oroboros Instruments, Innsbruck, Austria
Contribution to COST Action CA15203 MitoEAGLE, supported by COST (European Cooperation in Science and Technology), and K-Regio project MitoFit.


  1. Mitchell P (1966) Chemiosmotic coupling in oxidative and photosynthetic phosphorylation. Glynn Research, Bodmin. Biochim Biophys Acta Bioenergetics 1807:1507-38.
  1. Gnaiger E (1993) Nonequilibrium thermodynamics of energy transformations. Pure Appl Chem 65:1983-2002. - »Bioblast link«
  1. Gnaiger E (2018) Gibbs energy or Gibbs force? Mitochondr Physiol Network 2018-08-07. - »Bioblast link«
  2. Einstein A (1905) Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann Physik 4, XVII:549-60. - »Bioblast link«
  3. Gnaiger E (2018) The protonmotive force under pressure: an isomorphic analysis. Gnaiger 2018 EBEC2018



Educational innovation: Bioblast quiz
  1. Convert the molar format of the Gibbs force of reaction, ΔkFnO2 [kJ/mol], into the electrical format, ΔkFeO2 [V]. Which physicochemical constant is required?
  2. What is the meaning of the symbol zO2?
  3. How are the units of electric energy [J] and electric force [V] related?
  4. Express -460 kJ/mol O2 as electrical force in units of volt [V].
  5. Why should we do that?

Labels: MiParea: Respiration 

Regulation: mt-Membrane potential 

HRR: Theory  Event: A2, Oral  MitoEAGLE 

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