Part VII: Medical Biophysics & Metabolism | Chapter 2

Energy Metabolism & Bioenergetics

Chemiosmotic theory, electron transport thermodynamics, ATP yield, metabolic control analysis, and basal metabolic rate

The Molecular Machinery of Energy Conversion

Peter Mitchell's chemiosmotic hypothesis, awarded the 1978 Nobel Prize, revolutionized our understanding of biological energy conversion. The central insight is breathtakingly elegant: the electron transport chain uses the free energy of redox reactions to pump protons across the inner mitochondrial membrane, creating an electrochemical gradient. This proton-motive force then drives ATP synthase, the world's smallest rotary motor, to synthesize ATP. The entire process couples electron transfer to phosphorylation through an intermediate of stored electrochemical potential energy.

This chapter develops the quantitative bioenergetics of oxidative phosphorylation, from the thermodynamics of individual electron transfer steps to the overall efficiency of ATP production, and extends to metabolic control analysis, which reveals that metabolic flux is not controlled by a single "rate-limiting step" but is distributed across all enzymes.

1. Chemiosmotic Theory (Mitchell)

The proton-motive force ($\Delta p$) is the electrochemical potential difference for protons across the inner mitochondrial membrane. It has two components: the membrane potential ($\Delta\psi$) and the pH gradient ($\Delta\text{pH}$).

Derivation: The Proton-Motive Force

The electrochemical potential of a proton on one side of the membrane is:

$$\tilde{\mu}_{\text{H}^+} = \mu^{\circ}_{\text{H}^+} + RT\ln[\text{H}^+] + F\phi$$

where $\phi$ is the electrical potential and $F$ is the Faraday constant. The difference across the membrane (matrix minus intermembrane space) is:

$$\Delta\tilde{\mu}_{\text{H}^+} = RT\ln\frac{[\text{H}^+]_{\text{in}}}{[\text{H}^+]_{\text{out}}} + F\Delta\psi$$

Since $\text{pH} = -\log_{10}[\text{H}^+]$, we have $\ln[\text{H}^+] = -2.303 \cdot \text{pH}$, so:

$$\Delta\tilde{\mu}_{\text{H}^+} = -2.303 RT \cdot \Delta\text{pH} + F\Delta\psi$$

where $\Delta\text{pH} = \text{pH}_{\text{in}} - \text{pH}_{\text{out}}$ (matrix is more alkaline, so $\Delta\text{pH} > 0$). Dividing by $F$ to express in volts:

$$\Delta p = \frac{\Delta\tilde{\mu}_{\text{H}^+}}{F} = \Delta\psi - \frac{2.303RT}{F}\Delta\text{pH}$$

At $T = 310$ K (body temperature):

$$\frac{2.303RT}{F} = \frac{2.303 \times 8.314 \times 310}{96485} = 0.0615 \text{ V} = 61.5 \text{ mV}$$

With typical values $\Delta\psi \approx -150$ mV (negative inside) and$\Delta\text{pH} \approx 0.5$ units:

$$\Delta p = -150 - 61.5 \times 0.5 = -150 - 30.75 = -180.75 \text{ mV}$$

The free energy available per proton flowing down this gradient is:

$$\Delta G_{\text{H}^+} = F \cdot \Delta p = 96485 \times 0.181 = 17.5 \text{ kJ/mol}$$

Protons per ATP: ATP synthase requires sufficient proton flux to drive ATP synthesis. The free energy needed for ATP synthesis under cellular conditions is $\Delta G_{\text{ATP}} \approx 50$ kJ/mol:

$$n = \frac{\Delta G_{\text{ATP}}}{\Delta G_{\text{H}^+}} = \frac{50}{17.5} \approx 2.9$$

Thus approximately 3–4 protons are needed per ATP synthesized. The F$_1$F$_0$ ATP synthase has a c-ring with 8–15 subunits (species-dependent); in mammals, 8 c-subunits mean 8 protons per full rotation producing 3 ATP, giving 8/3 $\approx$ 2.67 H$^+$/ATP. Including the additional proton needed for P$_i$ transport: 8/3 + 1 $\approx$ 3.67 H$^+$/ATP.

2. Electron Transport Chain Thermodynamics

The electron transport chain transfers electrons from NADH and FADH$_2$ to O$_2$through a series of redox carriers. The free energy released at each step is determined by the difference in reduction potentials.

Derivation: Free Energy from Redox Potentials

The relationship between free energy and reduction potential is derived from electrochemistry. For a reaction transferring $n$ electrons:

$$\Delta G = -nF\Delta E$$

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

For the overall reaction NADH $\to$ O$_2$:

$$E^{\circ\prime}(\text{NAD}^+/\text{NADH}) = -0.320 \text{ V}$$

$$E^{\circ\prime}(\text{O}_2/\text{H}_2\text{O}) = +0.816 \text{ V}$$

$$\Delta E^{\circ\prime} = +0.816 - (-0.320) = +1.136 \text{ V}$$

$$\Delta G^{\circ\prime} = -2 \times 96485 \times 1.136 = -219.2 \text{ kJ/mol}$$

This enormous free energy is released in steps at the three proton-pumping complexes:

Complex I (NADH $\to$ ubiquinone):$\Delta E \approx 0.36$ V, $\Delta G = -69.5$ kJ/mol, pumps 4 H$^+$

Complex III (ubiquinol $\to$ cytochrome c):$\Delta E \approx 0.19$ V, $\Delta G = -36.7$ kJ/mol, pumps 4 H$^+$

Complex IV (cytochrome c $\to$ O$_2$):$\Delta E \approx 0.58$ V, $\Delta G = -112$ kJ/mol, pumps 2 H$^+$

Total protons pumped per NADH: 4 + 4 + 2 = 10 H$^+$. The Nernst equation adjusts the redox potential for non-standard concentrations:

$$E = E^{\circ\prime} - \frac{RT}{nF}\ln Q = E^{\circ\prime} - \frac{0.0267}{n}\ln Q$$

at $T = 310$ K. The actual $\Delta E$ in vivo depends on the redox state of each carrier, which varies with metabolic demand.

3. ATP Yield Per Glucose

The theoretical and actual ATP yield from complete glucose oxidation is a central calculation in bioenergetics, involving contributions from glycolysis, the citric acid cycle, and oxidative phosphorylation.

Derivation: Theoretical Maximum ATP Yield

Step 1: Reducing equivalent production

Glycolysis: 2 NADH + 2 ATP + 2 pyruvate per glucose

Pyruvate dehydrogenase: 2 NADH (one per pyruvate)

Citric acid cycle (per turn): 3 NADH + 1 FADH$_2$ + 1 GTP

Two turns per glucose: 6 NADH + 2 FADH$_2$ + 2 GTP

Total reducing equivalents: 10 NADH + 2 FADH$_2$

Step 2: P/O ratios

Each NADH provides electrons to Complex I, pumping 10 H$^+$ total. With 3.67 H$^+$/ATP (mammalian ATP synthase + P$_i$ transport):

$$\text{P/O ratio (NADH)} = \frac{10}{3.67} \approx 2.73 \approx 2.5 \text{ (conventional)}$$

Each FADH$_2$ enters at Complex II, bypassing Complex I (only 6 H$^+$ pumped):

$$\text{P/O ratio (FADH}_2\text{)} = \frac{6}{3.67} \approx 1.64 \approx 1.5 \text{ (conventional)}$$

Step 3: Total ATP yield

$$\text{ATP}_{\text{total}} = \underbrace{10 \times 2.5}_{\text{NADH}} + \underbrace{2 \times 1.5}_{\text{FADH}_2} + \underbrace{2}_{\text{glycolysis}} + \underbrace{2}_{\text{GTP}} = 25 + 3 + 2 + 2 = 32$$

Corrections for actual yield:

• Glycolytic NADH must be shuttled into mitochondria. The malate-aspartate shuttle preserves full reducing power (2.5 ATP/NADH), but the glycerol-3-phosphate shuttle yields only 1.5 ATP/NADH (enters as FADH$_2$).

• Proton leak across the inner membrane dissipates about 20–25% of the proton gradient without producing ATP.

• ATP/ADP translocase uses 1 proton per exchange cycle.

The realistic ATP yield is approximately 30–32 ATP per glucose, depending on the NADH shuttle used and the extent of proton leak. Some recent estimates using updated H$^+$/ATP ratios give ~30 ATP for the malate-aspartate shuttle pathway.

4. Metabolic Control Analysis (MCA)

Metabolic control analysis, developed independently by Kacser & Burns (1973) and Heinrich & Rapoport (1974), provides a rigorous mathematical framework for understanding how control of metabolic flux is distributed among pathway enzymes.

Derivation: Flux Control Coefficients and Summation Theorem

The flux control coefficient of enzyme $i$ on pathway flux $J$ is defined as:

$$C_i^J = \frac{\partial \ln J}{\partial \ln e_i} = \frac{e_i}{J}\frac{\partial J}{\partial e_i}$$

where $e_i$ is the concentration (activity) of enzyme $i$. This measures how a small fractional change in enzyme $i$ affects the pathway flux.

Summation Theorem: Consider a linear pathway$S_0 \xrightarrow{E_1} S_1 \xrightarrow{E_2} S_2 \xrightarrow{E_3} S_3$. At steady state, all fluxes are equal: $J = v_1 = v_2 = v_3$. If we scale ALL enzyme concentrations by a factor $\alpha$: $e_i \to \alpha e_i$, then all $v_i \to \alpha v_i$ and the flux $J \to \alpha J$. Therefore:

$$\frac{d\ln J}{d\ln \alpha}\bigg|_{\alpha=1} = \sum_i \frac{\partial \ln J}{\partial \ln e_i} = \sum_i C_i^J = 1$$

This is the summation theorem: the sum of all flux control coefficients equals exactly 1. This has a profound implication: control isshared among enzymes. No single enzyme has complete control ($C_i^J = 1$) unless all others have zero control.

The elasticity coefficient describes the sensitivity of an individual enzyme rate to a metabolite concentration:

$$\varepsilon_S^{v_i} = \frac{\partial \ln v_i}{\partial \ln S} = \frac{S}{v_i}\frac{\partial v_i}{\partial S}$$

Connectivity Theorem: For an intermediate metabolite$S_j$ at steady state:

$$\sum_i C_i^J \cdot \varepsilon_{S_j}^{v_i} = 0$$

This connects the systemic property (flux control coefficients) to local enzyme kinetics (elasticities). For a two-enzyme pathway, the connectivity theorem gives$C_1^J / C_2^J = -\varepsilon_{S_1}^{v_2} / \varepsilon_{S_1}^{v_1}$. Combined with the summation theorem ($C_1^J + C_2^J = 1$), both control coefficients are uniquely determined from the elasticities alone. An enzyme with high elasticity to intermediates tends to have low flux control.

5. Basal Metabolic Rate

The basal metabolic rate (BMR) represents the minimum energy expenditure necessary to maintain life in a resting, fasted, thermoneutral state. It can be measured by indirect calorimetry from oxygen consumption.

Derivation: BMR from Oxygen Consumption and Respiratory Quotient

The respiratory quotient (RQ) is the ratio of CO$_2$produced to O$_2$ consumed:

$$\text{RQ} = \frac{\dot{V}_{\text{CO}_2}}{\dot{V}_{\text{O}_2}}$$

For carbohydrate (glucose):$\text{C}_6\text{H}_{12}\text{O}_6 + 6\text{O}_2 \to 6\text{CO}_2 + 6\text{H}_2\text{O}$

$$\text{RQ}_{\text{carb}} = \frac{6}{6} = 1.00$$

For fat (tripalmitin):$\text{C}_{51}\text{H}_{98}\text{O}_6 + 72.5\text{O}_2 \to 51\text{CO}_2 + 49\text{H}_2\text{O}$

$$\text{RQ}_{\text{fat}} = \frac{51}{72.5} = 0.703 \approx 0.70$$

For protein (alanine as representative):$2\text{C}_3\text{H}_7\text{NO}_2 + 6\text{O}_2 \to \text{CO(NH}_2\text{)}_2 + 5\text{CO}_2 + 5\text{H}_2\text{O}$

$$\text{RQ}_{\text{protein}} \approx 0.80$$

The caloric equivalent of oxygen depends on RQ. For a mixed diet (RQ $\approx 0.82$):

$$\text{Caloric equivalent} \approx 4.825 \text{ kcal/L O}_2$$

BMR is then calculated from measured $\dot{V}_{\text{O}_2}$:

$$\text{BMR} = \dot{V}_{\text{O}_2} \times \text{caloric equivalent}$$

For a typical adult consuming 250 mL O$_2$/min:

$$\text{BMR} = 0.250 \times 60 \times 24 \times 4.825 = 1737 \text{ kcal/day}$$

Harris-Benedict equation (revised, 1984):

Men: $\text{BMR} = 88.362 + 13.397W + 4.799H - 5.677A$

Women: $\text{BMR} = 447.593 + 9.247W + 3.098H - 4.330A$

where $W$ is weight in kg, $H$ is height in cm, and $A$ is age in years. These empirical equations derive from regression analysis of indirect calorimetry data. The decline with age (~5.7 kcal/year for men) reflects loss of metabolically active lean mass. The physical basis: BMR correlates most strongly with fat-free mass, reflecting the high metabolic rate of organs (brain ~20%, liver ~20%, muscle ~22% of BMR).

Derivation: Total Energy Expenditure and the Thermic Effect of Food

Total daily energy expenditure (TDEE) has three components:

$$\text{TDEE} = \text{BMR} + \text{TEF} + \text{AEE}$$

where TEF is the thermic effect of food (diet-induced thermogenesis) and AEE is activity energy expenditure. The TEF represents the metabolic cost of digestion, absorption, and processing of nutrients. It is derived from the obligatory cost of processing each macronutrient:

• Protein: TEF $\approx$ 20–30% of protein energy. The high cost comes from deamination, urea synthesis ($4$ ATP per urea), gluconeogenesis from amino acid skeletons, and protein synthesis ($4$ ATP per peptide bond).

• Carbohydrate: TEF $\approx$ 5–10%. Cost of glycogen synthesis: glucose $\to$ glucose-6-phosphate $\to$glycogen requires 1 ATP per glucose unit stored.

• Fat: TEF $\approx$ 0–3%. Dietary fat can be stored directly with minimal processing, hence the very low thermic effect.

For a mixed diet, TEF $\approx$ 10% of total energy intake. This has implications for weight management: a high-protein diet has a higher thermic effect, meaning less net energy is available for storage. The physical activity level (PAL) = TDEE/BMR ranges from 1.2 (sedentary) to 2.4 (very active athletes).

Oxygen Debt and Excess Post-Exercise Oxygen Consumption (EPOC)

After intense exercise, oxygen consumption remains elevated above resting levels. This excess post-exercise oxygen consumption (EPOC) reflects:

• Replenishment of phosphocreatine stores: PCr + ADP $\to$ Cr + ATP (re-phosphorylation requires O$_2$-dependent ATP synthesis)

• Lactate clearance and reconversion to glucose (Cori cycle): 6 ATP required per glucose resynthesized from 2 lactate molecules in the liver

• Elevated body temperature ($Q_{10}$ effect persists)

• Catecholamine-stimulated metabolic rate. The EPOC can account for 6–15% of the total exercise energy expenditure and can persist for hours after intense exercise.

6. Applications

Exercise Physiology

During exercise, metabolic rate can increase 10–20 fold above resting. At low intensity, fat oxidation dominates (RQ $\approx 0.7$). As intensity increases, carbohydrate becomes the primary fuel (RQ $\to 1.0$). Above the anaerobic threshold, RQ exceeds 1.0 due to excess CO$_2$ from lactate buffering. The crossover concept (Brooks, 1994) predicts the shift from fat to carbohydrate oxidation as a function of exercise intensity.

The Warburg Effect: Cancer Metabolism

Otto Warburg (1924, Nobel Prize 1931) observed that cancer cells preferentially use glycolysis even in the presence of oxygen ("aerobic glycolysis"). This is energetically wasteful: glycolysis produces only 2 ATP/glucose vs ~30 from oxidative phosphorylation. Modern understanding: the Warburg effect provides biosynthetic intermediates for rapid cell proliferation. Glycolytic intermediates are diverted into the pentose phosphate pathway (nucleotides), serine biosynthesis (one-carbon metabolism), and lipid synthesis. The flux advantage outweighs the efficiency penalty.

Obesity Thermodynamics and Energy Balance

The first law of thermodynamics applied to body mass change gives:$\Delta E_{\text{stored}} = E_{\text{in}} - E_{\text{out}}$. Fat tissue stores approximately 32.2 MJ/kg (7700 kcal/kg), accounting for the ~85% fat and ~15% water content of adipose tissue. A sustained energy surplus of 100 kcal/day would theoretically produce $100 \times 365 / 7700 \approx 4.7$ kg of weight gain per year. However, adaptive thermogenesis partially compensates: metabolic rate increases with body mass (larger body to maintain), and the thermic effect of increased food intake adds to expenditure. The dynamic energy balance model predicts that approximately 50% of a sustained caloric surplus is compensated by increased expenditure at the new steady state body weight.

Mitochondrial Diseases

Mitochondrial DNA mutations cause a spectrum of diseases affecting tissues with high energy demand: brain, muscle, heart. MELAS (mitochondrial encephalopathy, lactic acidosis, stroke-like episodes) results from impaired oxidative phosphorylation, forcing reliance on glycolysis and causing lactic acid accumulation. The threshold effect: symptoms appear when the proportion of mutant mtDNA exceeds a tissue-specific threshold (typically 60–90%), reflecting the distributed control predicted by MCA.

7. Computational Exploration

Energy Metabolism: Electron Transport, ATP Yield, and Metabolic Control Analysis

Python

script.py388 lines

import numpy as np

# ============================================================
# ENERGY METABOLISM & BIOENERGETICS: COMPREHENSIVE SIMULATION
# ============================================================

print("=" * 72)
print("ENERGY METABOLISM & BIOENERGETICS: COMPUTATIONAL ANALYSIS")
print("=" * 72)

# --------------------------------------------------
# 1. ELECTRON TRANSPORT CHAIN ENERGETICS
# --------------------------------------------------
print()
print("1. ELECTRON TRANSPORT CHAIN: REDOX THERMODYNAMICS")
print("-" * 55)
print()

F = 96485  # Faraday constant (C/mol)
R = 8.314  # Gas constant (J/(mol*K))
T = 310    # K (body temperature)

# Standard reduction potentials (V) at pH 7
redox_couples = [
    ("NAD+/NADH", -0.320),
    ("FMN/FMNH2 (Complex I)", -0.300),
    ("CoQ/CoQH2", +0.045),
    ("Cyt b (Fe3+/Fe2+)", +0.077),
    ("Cyt c1 (Fe3+/Fe2+)", +0.215),
    ("Cyt c (Fe3+/Fe2+)", +0.235),
    ("Cyt a (Fe3+/Fe2+)", +0.290),
    ("Cyt a3 (Fe3+/Fe2+)", +0.550),
    ("O2/H2O", +0.816),
]

print("Electron transport chain redox ladder:")
print(f"{'Redox couple':>30} {'E (V)':>8} {'cumulative dG':>15}")
print(f"{'':>30} {'':>8} {'(kJ/mol)':>15}")
E_prev = redox_couples[0][1]
cumulative_dG = 0
for name, E in redox_couples:
    dE = E - E_prev if name != redox_couples[0][0] else 0
    dG = -2 * F * dE / 1000  # kJ/mol, n=2 electrons
    cumulative_dG += dG
    print(f"{name:>30} {E:>+8.3f} {cumulative_dG:>15.1f}")
    E_prev = E

# Complex-by-complex analysis
print()
print("Free energy released at each complex (per NADH, n=2 electrons):")
complexes = [
    ("Complex I (NADH -> CoQ)", -0.320, 0.045, 4),
    ("Complex II (FADH2 -> CoQ)", -0.031, 0.045, 0),
    ("Complex III (CoQ -> Cyt c)", 0.045, 0.235, 4),
    ("Complex IV (Cyt c -> O2)", 0.235, 0.816, 2),
]

print(f"{'Complex':>30} {'dE (V)':>8} {'dG (kJ/mol)':>14} {'H+ pumped':>10} {'dG/H+':>10}")
total_pumped = 0
for name, E_donor, E_acceptor, h_pumped in complexes:
    dE = E_acceptor - E_donor
    dG = -2 * F * dE / 1000
    dG_per_H = abs(dG) / h_pumped if h_pumped > 0 else 0
    total_pumped += h_pumped
    print(f"{name:>30} {dE:>+8.3f} {dG:>14.1f} {h_pumped:>10} {dG_per_H:>10.1f}")

print(f"{'Total H+ pumped per NADH':>30} {'':>8} {'':>14} {total_pumped:>10}")

# --------------------------------------------------
# 2. ATP YIELD CALCULATOR
# --------------------------------------------------
print()
print()
print("2. ATP YIELD FROM GLUCOSE OXIDATION")
print("-" * 55)
print()

# Pathway accounting
print("Detailed ATP accounting per glucose molecule:")
print()
print(f"{'Source':>35} {'NADH':>6} {'FADH2':>6} {'ATP/GTP':>8}")

pathway_steps = [
    ("Glycolysis (substrate level)", 0, 0, 2),
    ("Glycolysis (NADH)", 2, 0, 0),
    ("Pyruvate dehydrogenase", 2, 0, 0),
    ("Citric acid cycle (NADH)", 6, 0, 0),
    ("Citric acid cycle (FADH2)", 0, 2, 0),
    ("Citric acid cycle (GTP)", 0, 0, 2),
]

total_NADH = 0
total_FADH2 = 0
total_substrate = 0
for name, nadh, fadh2, atp in pathway_steps:
    total_NADH += nadh
    total_FADH2 += fadh2
    total_substrate += atp
    print(f"{name:>35} {nadh:>6} {fadh2:>6} {atp:>8}")

print(f"{'TOTAL':>35} {total_NADH:>6} {total_FADH2:>6} {total_substrate:>8}")

# P/O ratios and ATP calculation
print()
print("ATP synthesis from oxidative phosphorylation:")
print()

# Different H+/ATP ratios
for c_subunits in [8, 10, 12, 15]:
    h_per_atp = c_subunits / 3.0 + 1  # +1 for Pi transport
    po_nadh = 10 / h_per_atp
    po_fadh2 = 6 / h_per_atp

# Glycolytic NADH via malate-aspartate shuttle
    atp_nadh_mito = (total_NADH - 2) * po_nadh  # 8 mitochondrial NADH
    atp_nadh_glyc = 2 * po_nadh  # 2 glycolytic NADH (malate-aspartate)
    atp_fadh2 = total_FADH2 * po_fadh2
    total_atp = atp_nadh_mito + atp_nadh_glyc + atp_fadh2 + total_substrate

# With glycerol-3-phosphate shuttle instead
    atp_nadh_gp3 = 2 * po_fadh2  # enters as FADH2
    total_atp_gp3 = atp_nadh_mito + atp_nadh_gp3 + atp_fadh2 + total_substrate

print(f"  c-ring = {c_subunits} subunits -> H+/ATP = {h_per_atp:.2f}")
    print(f"    P/O (NADH) = {po_nadh:.3f},  P/O (FADH2) = {po_fadh2:.3f}")
    print(f"    Total ATP (malate-aspartate shuttle): {total_atp:.1f}")
    print(f"    Total ATP (glycerol-3-P shuttle):     {total_atp_gp3:.1f}")
    print()

# Efficiency calculation
dG_glucose = 2870  # kJ/mol (magnitude)
dG_ATP_std = 30.5
dG_ATP_cell = 50

print("Thermodynamic efficiency of glucose oxidation:")
for atp_yield in [26, 28, 30, 32]:
    eff_std = atp_yield * dG_ATP_std / dG_glucose * 100
    eff_cell = atp_yield * dG_ATP_cell / dG_glucose * 100
    print(f"  {atp_yield} ATP: {eff_std:.1f}% (standard) / {eff_cell:.1f}% (cellular)")

# --------------------------------------------------
# 3. METABOLIC CONTROL ANALYSIS
# --------------------------------------------------
print()
print()
print("3. METABOLIC CONTROL ANALYSIS: FLUX CONTROL SIMULATION")
print("-" * 55)
print()
print("Simulating a 5-enzyme linear pathway: S0 -> S1 -> S2 -> S3 -> S4 -> S5")
print("Each enzyme follows Michaelis-Menten kinetics with product inhibition")
print()

# Simple linear pathway simulation
# v_i = Vmax_i * S_i / (Km_i + S_i) * Km_p_i / (Km_p_i + P_i)

n_enzymes = 5
Vmax = np.array([10.0, 15.0, 8.0, 12.0, 20.0])
Km = np.array([1.0, 2.0, 0.5, 1.5, 3.0])
Km_p = np.array([5.0, 3.0, 4.0, 2.0, 6.0])  # product inhibition constants
S0 = 10.0  # fixed substrate

def rate(S_sub, S_prod, Vmax_i, Km_i, Km_p_i):
    return Vmax_i * S_sub / (Km_i + S_sub) * Km_p_i / (Km_p_i + S_prod)

def find_steady_state(Vmax_vals, S0_val, n_iter=5000, dt=0.01):
    S = np.ones(n_enzymes + 1) * 1.0  # intermediate concentrations
    S[0] = S0_val  # fixed substrate
    for _ in range(n_iter):
        v = np.zeros(n_enzymes)
        for i in range(n_enzymes):
            prod = S[i + 1] if i < n_enzymes - 1 else 0.0
            v[i] = rate(S[i], prod, Vmax_vals[i], Km[i], Km_p[i])
        # Update intermediates (not S0 which is fixed, and not S_final)
        for i in range(1, n_enzymes):
            S[i] += dt * (v[i - 1] - v[i])
            if S[i] < 0:
                S[i] = 0.001
        S[n_enzymes] += dt * v[n_enzymes - 1]  # product accumulates
        if S[n_enzymes] > 100:
            S[n_enzymes] = 0  # reset product
    # Return steady-state flux (average of last rates)
    v_ss = np.zeros(n_enzymes)
    for i in range(n_enzymes):
        prod = S[i + 1] if i < n_enzymes - 1 else 0.0
        v_ss[i] = rate(S[i], prod, Vmax_vals[i], Km[i], Km_p[i])
    J = np.mean(v_ss)
    return J, S, v_ss

J_ref, S_ref, v_ref = find_steady_state(Vmax, S0)
print(f"Reference steady-state flux: J = {J_ref:.4f}")
print(f"Intermediate concentrations: {['%.3f' % s for s in S_ref]}")
print()

# Calculate flux control coefficients by perturbation
delta = 0.01  # 1% perturbation
FCCs = []
print(f"{'Enzyme':>10} {'Vmax':>8} {'C_i^J':>10} {'Interpretation':>30}")
for i in range(n_enzymes):
    Vmax_up = Vmax.copy()
    Vmax_up[i] *= (1 + delta)
    J_up, _, _ = find_steady_state(Vmax_up, S0)

Vmax_down = Vmax.copy()
    Vmax_down[i] *= (1 - delta)
    J_down, _, _ = find_steady_state(Vmax_down, S0)

C_i = ((J_up - J_down) / (2 * delta * Vmax[i])) * (Vmax[i] / J_ref)
    FCCs.append(C_i)

if C_i > 0.3:
        interp = "HIGH control"
    elif C_i > 0.1:
        interp = "moderate control"
    else:
        interp = "low control"
    print(f"{'E' + str(i + 1):>10} {Vmax[i]:>8.1f} {C_i:>10.4f} {interp:>30}")

print(f"{'SUM':>10} {'':>8} {sum(FCCs):>10.4f} {'(should be ~1.0)':>30}")

# --------------------------------------------------
# 4. RESPIRATORY QUOTIENT AND FUEL MIX
# --------------------------------------------------
print()
print()
print("4. RESPIRATORY QUOTIENT AND METABOLIC FUEL MIX")
print("-" * 55)
print()

# RQ for different substrates
substrates = [
    ("Glucose (C6H12O6)", 6, 6, 1.000),
    ("Palmitate (C16H32O2)", 16, 23, 0.696),
    ("Tripalmitin (C51H98O6)", 51, 72.5, 0.703),
    ("Oleic acid (C18H34O2)", 18, 25.5, 0.706),
    ("Alanine (protein)", 5, 6, 0.833),
    ("Leucine (ketogenic)", 6, 7.5, 0.800),
    ("Ethanol (C2H6O)", 2, 3, 0.667),
]

print(f"{'Substrate':>30} {'CO2':>6} {'O2':>6} {'RQ':>8} {'kcal/L O2':>10}")
for name, co2, o2, rq in substrates:
    # Approximate caloric equivalent
    cal_equiv = 3.815 + 1.232 * rq  # Weir's formula approximation
    print(f"{name:>30} {co2:>6.1f} {o2:>6.1f} {rq:>8.3f} {cal_equiv:>10.3f}")

# Non-protein RQ to fuel mix
print()
print("Non-protein RQ to carbohydrate/fat mix:")
print(f"{'RQ':>6} {'% Carb':>10} {'% Fat':>10} {'kcal/L O2':>12}")
for rq in np.arange(0.70, 1.01, 0.05):
    pct_carb = (rq - 0.707) / (1.000 - 0.707) * 100
    pct_fat = 100 - pct_carb
    if pct_carb < 0:
        pct_carb = 0
        pct_fat = 100
    cal_eq = 3.815 + 1.232 * rq
    print(f"{rq:>6.2f} {pct_carb:>10.1f} {pct_fat:>10.1f} {cal_eq:>12.3f}")

# --------------------------------------------------
# 5. HARRIS-BENEDICT BMR CALCULATIONS
# --------------------------------------------------
print()
print()
print("5. HARRIS-BENEDICT BMR PREDICTIONS")
print("-" * 55)
print()

def bmr_male(W, H, A):
    return 88.362 + 13.397 * W + 4.799 * H - 5.677 * A

def bmr_female(W, H, A):
    return 447.593 + 9.247 * W + 3.098 * H - 4.330 * A

print("Male BMR (kcal/day):")
print(f"{'Age':>5} {'60kg/170cm':>14} {'75kg/178cm':>14} {'90kg/185cm':>14} {'110kg/180cm':>14}")
for age in [20, 30, 40, 50, 60, 70, 80]:
    b1 = bmr_male(60, 170, age)
    b2 = bmr_male(75, 178, age)
    b3 = bmr_male(90, 185, age)
    b4 = bmr_male(110, 180, age)
    print(f"{age:>5} {b1:>14.0f} {b2:>14.0f} {b3:>14.0f} {b4:>14.0f}")

print()
print("Female BMR (kcal/day):")
print(f"{'Age':>5} {'50kg/160cm':>14} {'60kg/165cm':>14} {'70kg/170cm':>14} {'85kg/168cm':>14}")
for age in [20, 30, 40, 50, 60, 70, 80]:
    b1 = bmr_female(50, 160, age)
    b2 = bmr_female(60, 165, age)
    b3 = bmr_female(70, 170, age)
    b4 = bmr_female(85, 168, age)
    print(f"{age:>5} {b1:>14.0f} {b2:>14.0f} {b3:>14.0f} {b4:>14.0f}")

# --------------------------------------------------
# 6. PROTON MOTIVE FORCE
# --------------------------------------------------
print()
print()
print("6. PROTON-MOTIVE FORCE AND ATP SYNTHESIS")
print("-" * 55)
print()

R_kJ = 8.314e-3  # kJ/(mol*K)
T = 310
F_kJ = 96.485  # kJ/(mol*V)

print("Proton-motive force components:")
print(f"{'dpsi (mV)':>12} {'dpH':>8} {'dp (mV)':>10} {'dG_H+ (kJ)':>12} {'H+/ATP':>8} {'max ATP/gluc':>14}")
for dpsi_mV in [-120, -140, -150, -160, -180]:
    for dpH in [0.3, 0.5, 0.75, 1.0]:
        factor = 2.303 * R_kJ * T / (F_kJ * 1e-3)  # in mV
        dp = dpsi_mV - factor * dpH
        dG_H = abs(dp) * 1e-3 * F_kJ  # kJ/mol
        dG_ATP = 50  # kJ/mol cellular
        h_per_atp = dG_ATP / dG_H
        # 10 H+ per NADH, so ATP per NADH = 10/h_per_atp
        atp_per_nadh = 10 / h_per_atp
        max_atp = 10 * atp_per_nadh + 2 * (6 / h_per_atp) + 4  # from glucose
        if dpsi_mV == -150 and dpH == 0.5:
            print(f"{dpsi_mV:>12} {dpH:>8.1f} {dp:>10.1f} {dG_H:>12.2f} {h_per_atp:>8.2f} {max_atp:>14.1f}  <-- typical")
        elif dpsi_mV in [-140, -160] and dpH == 0.5:
            print(f"{dpsi_mV:>12} {dpH:>8.1f} {dp:>10.1f} {dG_H:>12.2f} {h_per_atp:>8.2f} {max_atp:>14.1f}")

# Summary
print()
print("=" * 72)
print("SUMMARY OF KEY RESULTS")
print("=" * 72)
print()
print("1. ETC releases ~219 kJ/mol from NADH to O2 (dE = 1.14 V)")
print("2. 10 H+ pumped per NADH across complexes I, III, IV")
print(f"3. Proton-motive force ~180 mV -> ~17.5 kJ per H+ -> ~3-4 H+/ATP")
print(f"4. Theoretical ATP yield: 30-32 per glucose (depends on shuttle, c-ring)")
print(f"5. Thermodynamic efficiency: 34-56% (standard vs cellular conditions)")
print(f"6. MCA: flux control is distributed (summation theorem sum C_i = 1)")
print(f"7. RQ identifies fuel mix: 0.7 (fat), 0.8 (protein), 1.0 (carb)")
print(f"8. BMR ~ 1500-1800 kcal/day, declining ~5-6 kcal/year with age")

# --------------------------------------------------
# 7. TOTAL ENERGY EXPENDITURE
# --------------------------------------------------
print()
print()
print("7. TOTAL DAILY ENERGY EXPENDITURE")
print("-" * 55)
print()

# Different activity levels
print("Energy expenditure for 75kg male, age 30, 178cm:")
bmr_ref = bmr_male(75, 178, 30)
print(f"BMR = {bmr_ref:.0f} kcal/day")
print()

activities = [
    ("Sedentary (office)", 1.2),
    ("Lightly active", 1.375),
    ("Moderately active", 1.55),
    ("Very active", 1.725),
    ("Extremely active", 1.9),
    ("Elite athlete", 2.4),
]

print(f"{'Activity Level':>25} {'PAL':>6} {'TDEE (kcal)':>14} {'TEF (kcal)':>12} {'AEE (kcal)':>12}")
for name, pal in activities:
    tdee = bmr_ref * pal
    tef = tdee * 0.10
    aee = tdee - bmr_ref - tef
    print(f"{name:>25} {pal:>6.2f} {tdee:>14.0f} {tef:>12.0f} {aee:>12.0f}")

# Thermic effect by macronutrient
print()
print("Thermic effect of food by macronutrient:")
print(f"{'Nutrient':>15} {'TEF %':>8} {'Intake (g)':>12} {'Gross (kcal)':>14} {'Net (kcal)':>12}")
macro_intake = [
    ("Protein", 0.25, 100),
    ("Carbohydrate", 0.07, 300),
    ("Fat", 0.02, 80),
]
total_gross = 0
total_net = 0
for name, tef_frac, grams in macro_intake:
    cal_per_g = 4 if name != "Fat" else 9
    gross = grams * cal_per_g
    net = gross * (1 - tef_frac)
    total_gross += gross
    total_net += net
    print(f"{name:>15} {tef_frac*100:>7.0f}% {grams:>12} {gross:>14.0f} {net:>12.0f}")
print(f"{'TOTAL':>15} {(1-total_net/total_gross)*100:>7.1f}% {'':>12} {total_gross:>14.0f} {total_net:>12.0f}")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Chapter Summary

• The proton-motive force $\Delta p = \Delta\psi - (2.303RT/F)\Delta\text{pH} \approx -180$ mV drives ATP synthesis via the chemiosmotic mechanism.
• The electron transport chain releases $\Delta G = -nF\Delta E = -219$ kJ/mol from NADH to O$_2$, pumping 10 H$^+$ per NADH.
• ATP yield per glucose is ~30–32, with P/O ratios of ~2.5 (NADH) and ~1.5 (FADH$_2$).
• Metabolic control analysis proves control is distributed: $\sum C_i^J = 1$ (summation theorem) and $\sum C_i^J \varepsilon_{S_j}^{v_i} = 0$ (connectivity theorem).
• BMR is measured from O$_2$ consumption; the respiratory quotient (RQ = CO$_2$/O$_2$) identifies fuel mix: 0.7 (fat), 0.8 (protein), 1.0 (carbohydrate).
• The Warburg effect in cancer cells trades thermodynamic efficiency for biosynthetic flux.

← Biological Thermodynamics Entropy & Aging →