7. Oxidative Phosphorylation

Derivation 1: The Chemiosmotic Theory

Peter Mitchell's chemiosmotic hypothesis (1961, Nobel Prize 1978) proposed that the energy of electron transport is conserved not as a chemical intermediate but as an electrochemical proton gradient across the inner mitochondrial membrane.

The Proton-Motive Force (pmf)

The free energy stored in the proton gradient has two components: a chemical gradient ($\Delta\text{pH}$) and an electrical potential ($\Delta\psi$). For proton movement across the membrane, the electrochemical free energy is:

For protons ($z = +1$), the proton-motive force (pmf, denoted $\Delta p$) is defined by dividing by $F$:

Note: The sign convention is that $\Delta\text{pH} = \text{pH}_{\text{in}} - \text{pH}_{\text{out}}$ (matrix minus intermembrane space). Since the matrix is more basic (higher pH), $\Delta\text{pH}$ is positive, and the negative sign makes the chemical component contribute to a negative (proton-driving) pmf.

Numerical Evaluation

At $T = 310\;\text{K}$ (body temperature):

For mammalian mitochondria, typical values are $\Delta\psi \approx 150\text{-}170\;\text{mV}$ (matrix negative) and $\Delta\text{pH} \approx 0.5\text{-}1.0$ unit. Taking representative values:

The free energy available per proton returning to the matrix:

Derivation 2: The Electron Transport Chain

The ETC consists of four large protein complexes (I-IV) embedded in the inner mitochondrial membrane, plus two mobile electron carriers (ubiquinone and cytochrome c).

Thermodynamics of Electron Transfer

The free energy change for electron transfer between two redox couples is given by the Nernst relationship:

where $n$ = number of electrons transferred, $F$ = 96,485 C/mol (Faraday constant), and $\Delta E$ = difference in reduction potentials ($E_{\text{acceptor}} - E_{\text{donor}}$).

Complex I: NADH Dehydrogenase

Complex II: Succinate Dehydrogenase

Complex III: Cytochrome bc$_1$

Complex IV: Cytochrome c Oxidase

Overall: NADH to O$_2$

Total protons pumped per NADH: $4 + 4 + 2 = 10\;\text{H}^+$

Derivation 3: F$_1$F$_0$ ATP Synthase

ATP synthase is a remarkable rotary molecular motor that couples the flow of protons down their electrochemical gradient to the synthesis of ATP from ADP and P$_i$.

Structure

Proton Stoichiometry

The number of protons required to synthesize one ATP depends on the number of c subunits in the c-ring. In mammalian mitochondria, the c-ring has 8 subunits. One complete rotation of the c-ring:

• Translocates 8 protons
• Produces 3 ATP (one per $\beta$ subunit, three per 360$°$ rotation)

Therefore, the protons per ATP for synthesis alone:

However, transport of ATP$^{4-}$ out and ADP$^{3-}$ + P$_i^{2-}$ in via the ANT (adenine nucleotide translocase) and phosphate carrier costs an additional ~1 H$^+$ per ATP. The total is:

Thermodynamic Efficiency

The free energy required to synthesize ATP under cellular conditions:

The free energy available from $n$ protons flowing through ATP synthase:

The thermodynamic efficiency of ATP synthesis:

This remarkable efficiency reflects the tight coupling between proton translocation and ATP synthesis in the rotary mechanism. The remaining ~25% drives the reaction forward irreversibly.

Derivation 4: P/O Ratios and ATP Yield

P/O Ratio Calculation

The P/O ratio is the number of ATP synthesized per oxygen atom reduced (equivalent to per 2 electrons transferred). It depends on the number of protons pumped and the number required per ATP:

Total ATP Yield per Glucose

Complete oxidation of one glucose through glycolysis, PDC, TCA cycle, and oxidative phosphorylation:

*Cytoplasmic NADH enters mitochondria via shuttle systems: the malate-aspartate shuttle delivers electrons to mitochondrial NADH (2.5 ATP each, total 5), while the glycerol-3-phosphate shuttle delivers to FADH$_2$ (1.5 ATP each, total 3).

Overall Efficiency

Under cellular conditions, the efficiency of aerobic glucose oxidation is approximately 55-60%, which is far superior to most man-made engines. The remaining energy is released as heat, contributing to thermoregulation.

Derivation 5: Uncoupling of Oxidative Phosphorylation

Uncoupling refers to the dissipation of the proton gradient without ATP synthesis, converting the energy of electron transport entirely into heat.

Chemical Uncouplers

2,4-Dinitrophenol (DNP) is a classic protonophore. It is a weak acid ($\text{p}K_a \approx 4$) that can cross the inner mitochondrial membrane in both its protonated (neutral) and deprotonated (anionic) forms:

The delocalized electrons of the nitro groups stabilize the anion, allowing it to cross the hydrophobic membrane. The net effect: protons are shuttled from IMS to matrix, bypassing ATP synthase. Electron transport continues (even accelerates), but no ATP is made.

Biological Uncoupling: Thermogenin (UCP1)

Brown adipose tissue (BAT) contains thermogenin (Uncoupling Protein 1, UCP1), a regulated proton channel in the inner mitochondrial membrane:

This energy is entirely converted to heat for non-shivering thermogenesis. UCP1 is:

• Activated by: free fatty acids (released by norepinephrine-stimulated lipolysis)
• Inhibited by: purine nucleotides (ATP, GDP)

Quantitative Impact of Uncoupling

In a fully uncoupled state, all of the free energy of NADH oxidation ($\Delta G = -220\;\text{kJ/mol}$) is released as heat. Since no ATP is synthesized, the P/O ratio drops to zero:

Partial uncoupling (by basal proton leak or regulated UCPs) is physiologically important, accounting for approximately 20-25% of basal metabolic rate in mammals.

The Q Cycle: Mechanism of Complex III

Peter Mitchell proposed the Q cycle to explain how Complex III (cytochrome bc$_1$) pumps protons while transferring electrons from ubiquinol (CoQH$_2$) to cytochrome c. The key insight is that CoQH$_2$ carries two electrons, but cytochrome c accepts only one electron.

First Half of the Q Cycle

Second Half of the Q Cycle

Net Stoichiometry of the Q Cycle

For every pair of electrons transferred from CoQH$_2$ to 2 cytochrome c molecules, a total of 4 H$^+$ are translocated: 2 from the oxidation of CoQH$_2$ at Q$_o$ (released to IMS) and 2 taken up from the matrix for the reduction of CoQ at Q$_i$. This doubles the proton-pumping efficiency compared to a simple one-electron transfer.

Reactive Oxygen Species (ROS)

The semiquinone intermediate (CoQ$^{\bullet-}$) at the Q$_o$ site can occasionally donate its electron to O$_2$ instead of to cytochrome b$_L$, producing the superoxide radical:

Superoxide is rapidly dismutated by superoxide dismutase (SOD) to hydrogen peroxide:

Under normal conditions, approximately 0.1-2% of electrons leak to form superoxide. This ROS production increases when the ETC is highly reduced (e.g., high NADH/NAD$^+$, inhibition of downstream complexes). Complexes I and III are the major sites of mitochondrial ROS production, contributing to oxidative stress, aging, and disease.

ETC Inhibitors and Experimental Tools

Specific inhibitors of each ETC complex have been essential tools for elucidating the mechanism of oxidative phosphorylation. They also have clinical and toxicological significance.

Distinguishing ETC Inhibitors from Uncouplers

The key experimental distinction:

Mitochondrial Transport Systems

The inner mitochondrial membrane is impermeable to most metabolites and cofactors. Specialized transport systems are required to shuttle substrates, products, and reducing equivalents across this barrier.

The Malate-Aspartate Shuttle

This shuttle transfers cytoplasmic NADH electrons to mitochondrial NADH (preserving full energy), operating in liver, kidney, and heart:

Net result: cytoplasmic NADH $\rightarrow$ mitochondrial NADH. Each NADH yields 2.5 ATP.

The Glycerol-3-Phosphate Shuttle

A simpler but less energy-efficient shuttle, dominant in skeletal muscle and brain:

Net result: cytoplasmic NADH $\rightarrow$ mitochondrial FADH$_2$. Each FADH$_2$ yields only 1.5 ATP. This is why the ATP yield per glucose can be 30 or 32, depending on which shuttle is used.

Adenine Nucleotide Translocase (ANT)

ANT is the most abundant protein in the inner mitochondrial membrane. It exchanges ATP$^{4-}$ (out) for ADP$^{3-}$ (in) in an electrogenic antiport:

This exchange moves one net negative charge out of the matrix, which is driven by the membrane potential ($\Delta\psi$, matrix negative). The energy cost is approximately $\frac{1}{4}$ of a proton's worth of $\Delta p$, which is why the total cost per ATP is ~4 H$^+$ (including ANT + phosphate carrier).

ANT is inhibited by atractyloside (locks ANT in the cytoplasmic-facing conformation) and bongkrekic acid (locks it in the matrix-facing conformation). Both are lethal because they prevent ATP export from mitochondria.

Python Simulations

Mitochondrial Diseases and Evolutionary Perspectives

Mitochondrial DNA and Disease

Mitochondria contain their own circular DNA ($\approx 16.6\;\text{kb}$ in humans) encoding 13 protein subunits of the ETC, 22 tRNAs, and 2 rRNAs. The remaining ~1,000 mitochondrial proteins are encoded by nuclear DNA and imported. Key features:

• Maternal inheritance: mtDNA is inherited exclusively from the mother
• High mutation rate: ~10x higher than nuclear DNA (lacks histones, limited repair, proximity to ROS)
• Heteroplasmy: Cells contain thousands of mtDNA copies; mutations can be present in a fraction (threshold effect)
• Replicative segregation: Random distribution of mitochondria during cell division

The Threshold Effect

Mitochondrial diseases often show a threshold effect: symptoms appear only when the proportion of mutant mtDNA exceeds a critical level (typically 60-90%, depending on tissue):

where $f_{\text{mutant}}$ is the fraction of mutant mtDNA and $n$ reflects the number of mtDNA-encoded subunits in the complex. Tissues with the highest energy demands (brain, heart, skeletal muscle, retina) are most vulnerable.

Aging and Mitochondrial Dysfunction

The mitochondrial theory of aging proposes that accumulated mtDNA mutations and oxidative damage progressively impair ETC function, creating a vicious cycle:

Supporting evidence includes the observation that COX-negative (Complex IV-deficient) muscle fibers increase exponentially with age, and that mitochondrial DNA mutations accumulate in post-mitotic tissues. However, the relative contribution of mitochondrial dysfunction versus other aging mechanisms remains debated.

Endosymbiotic Origin of Mitochondria

Mitochondria arose approximately 1.5-2 billion years ago through endosymbiosis of an $\alpha$-proteobacterium by an ancestral archaeal host cell. Evidence includes:

• Double membrane structure (inner membrane from the bacterium, outer from the host)
• Circular DNA with bacterial-type promoters and ribosomes (70S)
• Bacterial-type division machinery (FtsZ-related GTPases)
• Phylogenetic analysis places mitochondrial genes within the $\alpha$-proteobacterial clade
• Cardiolipin (diphosphatidylglycerol) in the inner membrane, characteristic of bacterial membranes