Cited passages: Thread-Safety; Overwrite Operation; Writing to a Full Buffer; Iterator Invalidation; Caveats. The documented overwrite-by-assignment and synchronization behavior belongs to Boost.Circular Buffer.

Transverse relaxation describes loss of ensemble transverse magnetization. It must not automatically be identified with pure dephasing. In a simple completely positive two-level relaxation model,where $T_{\varphi }$ describes additional pure dephasing. Thus population recovery itself contributes to transverse decay. A spread of static magnetic fields can further cancel ensemble phases without the same microscopic relaxation; an observed free signal may have an effective $T_2^{\ast }$ rather than homogeneous $T_2$. The simple exponential examples below assume a uniform field and a single homogeneous $T_2$.

With no transverse drive, the Bloch equations giveLongitudinal nuclear spin relaxation restores the population imbalance and thus the magnetization parallel to the static magnetic field. It corresponds to population relaxation for the two-level model. Starting from $M_z(0)=0$, waiting $T_1$ recovers $1-e^{-1}$ of $M_0$, whereas waiting $5T_1$ recovers about $99.3\%$. This is why repeated measurements require a recovery-time decision.

Online text: openstax.org/books/biology-2e/pages/10-1-cell-division

Section of Biology 2e by Mary Ann Clark and others, whose whole-book authors are Mary Ann Clark, Matthew Douglas and Jung Choi. OpenStax publishes the book; no separate section author is credited. The citation panel gives whole-book publication on 28 March 2018, rather than a section revision date.

Cited passages: §10.1, Genomic DNA; Eukaryotic Chromosomal Structure and Compaction, Figure 10.4 and final paragraph; citation information identifying the authors and publication date.

Section of Biology 2e by Mary Ann Clark and others. Whole-book authors: Mary Ann Clark, Matthew Douglas and Jung Choi; no separate section byline is established. OpenStax publishes and hosts the text. The book was published on 28 March 2018; no separate section revision date is established.

Online text: openstax.org/books/biology-2e/pages/8-1-overview-of-photosynthesis

Inspected locators: Basic Photosynthetic Structures, Figure 8.6; The Two Parts of Photosynthesis, Figure 8.7; citation information.

By William Hayward Pickering, credited as William H. Pickering. Memoir pp. 24–38; institutional issuer: National Academy of Sciences.

In the aqueous model here, a strong base supplies hydroxide essentially completely under the conditions considered. Strength and concentration are different properties.

In 1997, Hiroyuki Noji and colleagues attached a fluorescent actin filament to the central $\gamma$ shaft of isolated $F_1$. With ATP supplied, the marker made sustained rotations. This exposed rotation directly, but used ATP hydrolysis: there was no membrane $F_0$ motor producing ATP from a proton-motive force in this assay.^[ref]

A subsequent primary experiment made the optical readout much faster. Purified Escherichia coli $F_1$ was immobilized on glass in a $30\mu \mathrm{m}$-deep flow cell, with a $60\mathrm{nm}$ gold bead attached to $\gamma$. With $2\mathrm{mM}$ ATP and magnesium supplied, laser illumination and dark-field microscopy recorded bead positions at $4000$ or $8000$ frames per second. The position trajectory resolved turns and intervening dwells. This apparatus follows a shaft-attached marker, rather than imaging the shaft's chemical atoms.^[ref]

Controls addressed perturbation and chemical activity: comparison of $40$–$200\mathrm{nm}$ probes found similar rates for the two smallest sizes, supporting low drag; a separate coupled-enzyme assay followed ATP hydrolysis through NADH absorbance. But the rotational analysis excluded pauses longer than $0.1\mathrm{s}$, so it characterizes continuously active intervals, not total output including long inhibited periods. It neither counts transported protons nor measures simultaneous single-molecule ATP synthesis.^[ref]

The experimental distinction is essential. Sustained directional marker motion tests a rotary mechanism. Demonstrating proton-driven synthesis additionally requires intact $F_0{F}_1$ in a membrane with a controlled proton-motive force and a chemical ATP-production readout; dependence on that driving force and inhibition or respiratory uncoupling controls then test the proposed energy coupling. A rotation movie alone cannot establish the complete material and free energy ledger.

Define side $P$ as the higher proton electrochemical potential reservoir and side $N$ as the synthesis-side reservoir. For transfer $P\to N$ at common temperature $T$, define the positive downhill proton-motive force magnitude $\Delta p_{\mathrm{drive}}$ byHere $a$ is thermodynamic activity, $\psi$ is electric potential, $R$ is the molar gas constant and $F$ is the Faraday constant. Thus both a higher proton thermodynamic activity and a more positive electric potential on side $P$ contribute to the driving magnitude. This definition fixes the sign; conventions for a signed proton-motive force differ. The calculation under chemiosmosis separates these two contributions numerically.

Assume fixed reservoirs, tight mechanical energy coupling, one transported proton per $c$ subunit per revolution, and three ATP produced per revolution. Neglect leaks, slipping and membrane transport of the chemical reagents. Let $g_{\mathrm{ATP}}>0$ be the local molar Gibbs free energy cost of ATP formation at the stipulated composition, pH and magnesium-binding convention. It is not necessarily the standard state value. Adding changes in the state function gives, per mole of complete turns,For this model net synthesis is thermodynamically favoured when $\Delta G_{\mathrm{turn}}<0$. Equality is a reversible balance, not a promise of a useful reaction rate. A negative value establishes an available driving margin; it does not specify how fast the enzyme runs.

Choose $n=10$, $g_{\mathrm{ATP}}=50.0\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$, $F=96485\mathrm{C}\mathrm{mo}{\mathrm{l}}^{-1}$ and $\Delta p_{\mathrm{drive}}=0.180\mathrm{V}$. These are illustrative inputs, not measurements of the preparation in the experiment below. Thenand $\Delta G_{\mathrm{turn}}=-23.673\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$. Per ATP, the ideal input is $57.891\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$, leaving a driving margin of $7.891\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$. The balance point isAt $0.140\mathrm{V}$ the same synthesis cycle instead has $\Delta G_{\mathrm{turn}}=+14.921\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$: the chosen reservoirs favour its reverse.

The machine is reversible: sufficient ATP hydrolysis can drive opposite rotation and pump protons uphill. Actual direction also depends on inhibition and available substrates; a weak proton-motive force need not make every intact enzyme immediately run backward. Respiratory uncoupling supplies a different failure mode: a parallel proton leak dissipates the gradient without productive shaft rotation.^[ref]

The motor budget $10/3$ is not the whole cost of ATP delivered outside a mitochondrion. Import of ADP and inorganic phosphate, and export of ATP, also draw on the electrochemical gradient.^[ref] For a deliberately simplified system budget, assign these transports together one additional downhill-proton equivalent per delivered ATP, valued at the same $F\Delta p_{\mathrm{drive}}$. This is a chosen accounting convention, not a claim of one extra literal proton through $F_0$, nor a universal transport stoichiometry. The total becomes $10/3+1=13/3$ equivalents and $75.258\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$ of gradient expenditure per delivered ATP at $0.180\mathrm{V}$. The ratio of stored chemical free energy to that expenditure is $50/75.258\approx 66.4\%$, compared with $50/57.891\approx 86.4\%$ for the motor alone. This system budget changes the resource expenditure, not the motor's balance point $\Delta p_{\ast }$. Real transport terms require their own electrochemical potentials, and leaks add further cost.

The membrane motor $F_0$ contains a rotating ring of $c$ subunits beside stationary subunit $a$. Two offset water-accessible half-channels connect opposite membrane sides to the ring. A proton enters one half-channel and protonates a glutamic acid residue on a $c$ subunit; neutralization permits that site to enter the hydrophobic lipid bilayer. Rotation carries it around to the other half-channel, where it releases the proton. A conserved arginine in subunit $a$ helps separate the routes and promotes release. Downhill proton transfer biases rotation; this is not a freely open transmembrane pore.^[ref]

The central shaft transmits torque to $F_1$; a peripheral stalk prevents the catalytic head from simply co-rotating. Its three $\beta$ catalytic sites, within an ${\alpha }_3{\beta }_3$ head, cycle through different binding shapes. In the simplified binding-change picture, a loose site captures ADP and inorganic phosphate, a tight site favours bound ATP, and opening releases ATP. Rotation changes binding and release as well as chemistry.^[ref]

One complete, tightly coupled turn cycles each catalytic site once, producing three ATP molecules. If the ring has $n$ proton-binding subunits, the ideal motor budget is $n$ transferred protons per turn, or $n/3$ per ATP. Ring size varies between organisms; the worked example chooses $n=10$, as in the studied bacterial systems, rather than assigning ten to every ATP synthase.^[ref]

For this example the ring's repeat angle is $360^{\circ }/10=36^{\circ }$, whereas the three-site catalytic cycle repeats after $120^{\circ }$. Their ratio is $10/3$, not an integer. Counting complete turns avoids inventing a fixed integer proton count for every catalytic event. The schematic does not require both parts to advance simultaneously as rigid gears; elasticity allows temporary twisting while their cycles proceed.^[ref]

Cited passages: equations (6)–(8). Page created 3 October 2016, updated 15 August 2025. The contact is not an author.