Chemiosmosis

The proton-motive force is derived from the Gibbs free energy. Let N denote the inside of a cell, and P denote the outside. Then[6]

${\displaystyle \Delta \!G=zF\Delta \!\psi +RT\ln {\frac {[\mathrm {X} ^{z+}]_{\text{N}}}{[\mathrm {X} ^{z+}]_{\text{P}}}}}$

where

• ${\displaystyle \Delta \!G}$ is the Gibbs free energy change per unit amount of cations transferred from P to N;
• ${\displaystyle z}$ is the charge number of the cation ${\displaystyle \mathrm {X} ^{z+}}$;
• ${\displaystyle \Delta \psi }$ is the electric potential of N relative to P;
• ${\displaystyle [\mathrm {X} ^{z+}]_{\text{P}}}$ and ${\displaystyle [\mathrm {X} ^{z+}]_{\text{N}}}$ are the cation concentrations at P and N, respectively;
• ${\displaystyle F}$ is the Faraday constant;
• ${\displaystyle R}$ is the gas constant; and
• ${\displaystyle T}$ is the temperature.

The molar Gibbs free energy change ${\displaystyle \Delta \!G}$ is frequently interpreted as a molar electrochemical ion potential ${\displaystyle \Delta \!\mu _{\mathrm {X} ^{z+}}=\Delta \!G}$.

For an electrochemical proton gradient ${\displaystyle z=1}$ and as a consequence:

${\displaystyle \Delta \!\mu _{\mathrm {H} ^{+}}=F\Delta \!\psi +RT\ln {\frac {[\mathrm {H} ^{+}]_{\text{N}}}{[\mathrm {H} ^{+}]_{\text{P}}}}=F\Delta \!\psi -(\ln 10)RT\Delta \mathrm {pH} }$

A diagram of chemiosmotic phosphorylation

where

${\displaystyle \Delta \!\mathrm {pH} =\mathrm {pH} _{\mathrm {N} }-\mathrm {pH} _{\mathrm {P} }}$.

Mitchell defined the proton-motive force (PMF) as

${\displaystyle \Delta \!p=-{\frac {\Delta \!\mu _{\mathrm {H^{+}} }}{F}}}$.

For example, ${\displaystyle \Delta \!\mu _{\mathrm {H} ^{+}}=1\,\mathrm {kJ} \,\mathrm {mol} ^{-1}}$ implies ${\displaystyle \Delta \!p=10.4\,\mathrm {mV} }$. At ${\displaystyle 298\,\mathrm {K} }$ this equation takes the form:

${\displaystyle \Delta \!p=-\Delta \!\psi +\left(59.1\,\mathrm {mV} \right)\Delta \!\mathrm {pH} }$.

Note that for spontaneous proton import from the P side (relatively more positive and acidic) to the N side (relatively more negative and alkaline), ${\displaystyle \Delta \!\mu _{\mathrm {H} ^{+}}}$ is negative (similar to ${\displaystyle \Delta \!G}$) whereas PMF is positive (similar to redox cell potential ${\displaystyle \Delta E}$).

It is worth noting that, as with any transmembrane transport process, the PMF is directional. The sign of the transmembrane electric potential difference ${\displaystyle \Delta \!\psi }$ is chosen to represent the change in potential energy per unit charge flowing into the cell as above. Furthermore, due to redox-driven proton pumping by coupling sites, the proton gradient is always inside-alkaline. For both of these reasons, protons flow in spontaneously, from the P side to the N side; the available free energy is used to synthesize ATP (see below). For this reason, PMF is defined for proton import, which is spontaneous. PMF for proton export, i.e., proton pumping as catalyzed by the coupling sites, is simply the negative of PMF(import).

The spontaneity of proton import (from the P to the N side) is universal in all bioenergetic membranes.[8] This fact was not recognized before the 1990s, because the chloroplast thylakoid lumen was interpreted as an interior phase, but in fact it is topologically equivalent to the exterior of the chloroplast. Azzone et al. stressed that the inside phase (N side of the membrane) is the bacterial cytoplasm, mitochondrial matrix, or chloroplast stroma; the outside (P) side is the bacterial periplasmic space, mitochondrial intermembrane space, or chloroplast lumen. Furthermore, 3D tomography of the mitochondrial inner membrane shows its extensive invaginations to be stacked, similar to thylakoid disks; hence the mitochondrial intermembrane space is topologically quite similar to the chloroplast lumen.:[9]

The energy expressed here as Gibbs free energy, electrochemical proton gradient, or proton-motive force (PMF), is a combination of two gradients across the membrane:

• the concentration gradient (via ${\displaystyle \Delta \!\mathrm {pH} }$) and
• electric potential gradient ${\displaystyle \Delta \!\psi }$.

When a system reaches equilibrium, ${\displaystyle \Delta \!\rho =0}$; nevertheless, the concentrations on either side of the membrane need not be equal. Spontaneous movement across the potential membrane is determined by both concentration and electric potential gradients.

The molar Gibbs free energy ${\displaystyle \Delta \!G_{\mathrm {p} }}$ of ATP synthesis

${\displaystyle \mathrm {ADP} ^{4-}+\mathrm {H} ^{+}+\mathrm {HOPO} _{3}^{2-}\rightarrow \mathrm {ATP} ^{4-}+\mathrm {H_{2}O} }$

is also called phosphorylation potential. The equilibrium concentration ratio ${\displaystyle [\mathrm {H} ^{+}]/[\mathrm {ATP} ]}$ can be calculated by comparing ${\displaystyle \Delta \!p}$ and ${\displaystyle \Delta \!G_{\mathrm {p} }}$, for example in case of the mammalian mitochondrion:[9]

H⁺ / ATP = ΔG_p / (Δp / 10.4 kJ·mol⁻¹/mV) = 40.2 kJ·mol⁻¹ / (173.5 mV / 10.4 kJ·mol⁻¹/mV) = 40.2 / 16.7 = 2.4. The actual ratio of the proton-binding c-subunit to the ATP-synthesizing beta-subunit copy numbers is 8/3 = 2.67, showing that under these conditions, the mitochondrion functions at 90% (2.4/2.67) efficiency.[9]

In fact, the thermodynamic efficiency is mostly lower in eukaryotic cells because ATP must be exported from the matrix to the cytoplasm, and ADP and phosphate must be imported from the cytoplasm. This "costs" one "extra" proton import per ATP,[6][7] hence the actual efficiency is only 65% (= 2.4/3.67).

A stepwise model for the emergence of chemiosmosis, a key element in the origin of life on earth, proposes that primordial organisms used thermal cycling as an energy source (thermosynthesis), functioning essentially as a heat engine:[11]

self-organized convection in natural waters causing thermal cycling →

added β-subunit of F₁ ATP Synthase

(generated ATP by thermal cycling of subunit during suspension in convection cell: thermosynthesis) →

added membrane and F_o ATP Synthase moiety

(generated ATP by change in electrical polarization of membrane during thermal cycling: thermosynthesis) →

added metastable, light-induced electric dipoles in membrane

(primitive photosynthesis) →

added quinones and membrane-spanning light-induced electric dipoles

(today's bacterial photosynthesis, which makes use of chemiosmosis).

Deep-sea hydrothermal vents, emitting hot acidic or alkaline water, would have created external proton gradients. These provided energy that primordial organisms could have exploited. To keep the flows separate, such an organism could have wedged itself in the rock of the hydrothermal vent, exposed to the hydrothermal flow on one side and the more alkaline water on the other. As long as the organism's membrane (or passive ion channels within it) is permeable to protons, the mechanism can function without ion pumps. Such a proto-organism could then have evolved further mechanisms such as ion pumps and ATP synthase.[10]

A proposed alternative source to chemiosmotic energy developing across membranous structures is if an electron acceptor, ferricyanide, is within a vesicle and the electron donor is outside, quinones transported by carbonaceous meteorites pick up electrons and protons from the donor. They would release electrons across the lipid membrane by diffusion to ferricyanide within the vesicles and release protons which produces gradients above pH 2, the process is conducive to the development of proton gradients.[12][13]