Respiratory Biophysics

Derivation: Hill Equation for Cooperative Binding

Step 1 — The binding equilibrium: For a protein with apparent cooperativity, we write the binding as if all $n$ sites bind simultaneously:

$\text{Hb} + nO_2 \rightleftharpoons \text{Hb}(O_2)_n$

The fractional saturation is:

$Y = \frac{[O_2]^n}{K_d + [O_2]^n} = \frac{P_{O_2}^n}{P_{50}^n + P_{O_2}^n}$

where $P_{50}$ is the partial pressure at 50% saturation ($\approx 26.8$ mmHg) and $n$ is the Hill coefficient ($\approx 2.7$ for hemoglobin).

Step 2 — Hill plot linearisation: Taking the log:

$\log\left(\frac{Y}{1-Y}\right) = n\log(P_{O_2}) - n\log(P_{50})$

A plot of $\log(Y/(1-Y))$ vs $\log(P_{O_2})$ gives a straight line with slope $n$. For hemoglobin, $n \approx 2.7$, indicating strong positive cooperativity (the maximum possible for 4 sites would be $n = 4$).

MWC Model: T and R States

The Monod-Wyman-Changeux (MWC) model provides a physical basis for cooperativity. Hemoglobin exists in two conformational states:

• T state (tense): Low oxygen affinity, $K_T$ is large
• R state (relaxed): High oxygen affinity, $K_R$ is small

The equilibrium between states is characterised by the allosteric constant $L_0 = [T_0]/[R_0]$, and the ratio of affinities $c = K_R/K_T$. The fractional saturation from the MWC model is:

$Y = \frac{L_0 c\alpha(1+c\alpha)^3 + \alpha(1+\alpha)^3}{L_0(1+c\alpha)^4 + (1+\alpha)^4}$

where $\alpha = P_{O_2}/K_R$. When $L_0 \gg 1$ and $c \ll 1$, binding the first O$_2$ shifts the equilibrium towards the R state, dramatically increasing affinity for subsequent molecules — cooperativity.

Derivation: Bohr Effect and Oxygen Delivery

Bohr effect: Low pH (and high CO$_2$) stabilise the T state by protonating histidine residues that form salt bridges. This increases $P_{50}$, shifting the dissociation curve rightward:

$P_{50}(\text{pH}) \approx P_{50}^0 \times 10^{-0.48(\text{pH} - 7.4)}$

At tissue pH $\approx 7.2$, $P_{50}$ increases from 26.8 to ~33 mmHg, promoting O$_2$ release where it is most needed.

Oxygen delivery: The total oxygen content of blood is:

$C_{O_2} = 1.34 \times [\text{Hb}] \times Y + 0.003 \times P_{O_2}$

where 1.34 mL O$_2$/g is the Hufner constant and 0.003 mL/(dL$\cdot$mmHg) is the dissolved O$_2$ coefficient. The oxygen delivery rate is:

$\dot{V}_{O_2} = \dot{Q} \times (C_a - C_v)$

where $\dot{Q}$ is cardiac output, $C_a$ and $C_v$ are arterial and venous O$_2$ content. For $\dot{Q} = 5$ L/min,$C_a = 20$ mL/dL, $C_v = 15$ mL/dL:$\dot{V}_{O_2} = 50 \times 5 = 250$ mL/min.

Derivation: Alveolar Gas Equation

The alveolar oxygen partial pressure is determined by the balance between O$_2$inspired and O$_2$ consumed:

$P_AO_2 = F_IO_2(P_{\text{atm}} - P_{H_2O}) - \frac{P_ACO_2}{RQ}$

where $F_IO_2 = 0.21$ (fraction of inspired O$_2$),$P_{\text{atm}} = 760$ mmHg, $P_{H_2O} = 47$ mmHg (at 37$^\circ$C),$P_ACO_2 \approx 40$ mmHg, and $RQ \approx 0.8$ (respiratory quotient). This gives:

$P_AO_2 = 0.21 \times 713 - \frac{40}{0.8} = 149.7 - 50 = 99.7 \text{ mmHg}$

The A-a gradient is the difference between alveolar and arterial O$_2$:

$\text{A-a gradient} = P_AO_2 - P_aO_2 \approx 100 - 90 = 10 \text{ mmHg (normal)}$

An elevated A-a gradient (>20 mmHg) indicates either V/Q mismatch, shunt, or diffusion impairment.

Derivation: V/Q Ratios and the Shunt Equation

V/Q extremes: Consider the two limiting cases:

• $\dot{V}/\dot{Q} = 0$ (shunt): No ventilation, blood passes through without gas exchange. Alveolar gas = mixed venous ($P_{O_2} = 40$, $P_{CO_2} = 46$ mmHg).
• $\dot{V}/\dot{Q} = \infty$ (dead space): No perfusion, alveolar gas = inspired gas ($P_{O_2} = 150$, $P_{CO_2} = 0$ mmHg).
• $\dot{V}/\dot{Q} \approx 0.8$ (normal): Optimal matching for gas exchange.

Shunt equation: The fraction of cardiac output that bypasses gas exchange:

$\frac{\dot{Q}_s}{\dot{Q}_t} = \frac{C_c - C_a}{C_c - C_v}$

where $C_c$ is the O$_2$ content of end-capillary blood (= alveolar equilibrium), $C_a$ is arterial, and $C_v$ is mixed venous. The shunt fraction is normally 2–5% (bronchial and thebesian veins). In pneumonia or ARDS, it can exceed 30%, causing severe hypoxaemia that cannot be corrected by increasing $F_IO_2$alone.

In the upright lung, gravity creates a gradient: the base has high perfusion (high hydrostatic pressure) but relatively less ventilation, giving $\dot{V}/\dot{Q} \approx 0.6$. The apex has less perfusion and relatively more ventilation, giving $\dot{V}/\dot{Q} \approx 3$. This V/Q heterogeneity is the main reason the A-a gradient is not zero even in normal lungs.

Time Constants and Regional Lung Mechanics

Each lung unit has a time constant $\tau = R \times C$ that determines how quickly it fills and empties. For normal lung: $R \approx 2$ cmH$_2$O/(L/s) and$C \approx 0.2$ L/cmH$_2$O, giving $\tau \approx 0.4$ s.

During inspiration with constant pressure, the volume fills exponentially:

$V(t) = V_{\text{final}}\left(1 - e^{-t/\tau}\right)$

After 3$\tau$ ($\approx 1.2$ s), 95% of the tidal volume has been delivered. This is well within the normal inspiratory time (~2 s at rest), so all regions fill adequately.

Pathological heterogeneity: In disease, different lung regions develop different time constants. An emphysematous region might have $\tau = 2$ s (high compliance), while a fibrotic region might have $\tau = 0.1$ s (low compliance). During rapid breathing, the slow regions cannot fully inflate or deflate, leading to ventilation inhomogeneity and worsened V/Q mismatch. This explains why patients with COPD develop more hypoxaemia during exercise (increased respiratory rate, insufficient time for slow units to fill).

The closing volume is the lung volume at which small airways in dependent regions begin to close due to the weight of the lung above. In young adults, closing volume is well below FRC, so airways remain open during tidal breathing. With age, closing volume increases (loss of elastic recoil), eventually exceeding FRC, and airways close during normal breathing — creating shunt-like regions and reducing arterial oxygenation. This is a major reason why $P_aO_2$ declines with age:

$P_aO_2 \approx 104 - 0.27 \times \text{age (years)} \text{ mmHg}$

Oxygen Cascade: From Atmosphere to Mitochondria

The partial pressure of oxygen falls at every step from ambient air to the mitochondria, forming the “oxygen cascade”:

• Atmosphere: $P_{O_2} = 0.21 \times 760 = 160$ mmHg
• Humidified trachea: $P_{O_2} = 0.21 \times (760 - 47) = 150$ mmHg
• Alveolus: $P_AO_2 \approx 100$ mmHg (CO$_2$ dilution)
• Arterial blood: $P_aO_2 \approx 90-95$ mmHg (V/Q mismatch + shunt)
• Tissue capillary: $P_{O_2} \approx 40$ mmHg (extraction by tissues)
• Mitochondria: $P_{O_2} \approx 1-5$ mmHg (the functional sink)

The Krogh tissue cylinder model describes the radial diffusion of O$_2$ from a capillary into surrounding tissue:

$P_{O_2}(r) = P_c - \frac{\dot{V}_{O_2}}{4D_t K_t}\left(r^2 - r_c^2 - 2R_t^2 \ln\frac{r}{r_c}\right)$

where $r_c$ is the capillary radius, $R_t$ is the tissue cylinder radius,$D_t$ is the tissue diffusivity, and $K_t$ is the tissue solubility. The “lethal corner” at the venous end of the tissue cylinder ($r = R_t$, lowest $P_c$) is where hypoxic injury first occurs.

Applications of Respiratory Biophysics

• • Mechanical ventilation: The equation of motion$P = V/C + R\dot{V}$ directly governs ventilator settings. In pressure-controlled ventilation, the ventilator sets $P$ and $V$ depends on compliance; in volume-controlled, the ventilator sets $V$ and pressure rises as needed.
• • ARDS: Acute respiratory distress syndrome reduces compliance (surfactant dysfunction, alveolar flooding) and increases shunt fraction. Treatment uses PEEP (positive end-expiratory pressure) to recruit collapsed alveoli and low tidal volumes to avoid overdistension.
• • COPD: Chronic obstructive pulmonary disease increases airway resistance $R$ and compliance $C$ (emphysema), increasing the time constant $\tau = RC$ and causing air trapping.
• • Altitude physiology: At 5500 m, $P_{\text{atm}} = 380$ mmHg, so $P_AO_2 = 0.21 \times 333 - 50 = 20$ mmHg. Compensatory hyperventilation reduces $P_ACO_2$, improving $P_AO_2$ via the alveolar gas equation.
• • Diving physiology: At 30 m depth, $P_{\text{total}} = 4$ atm. Nitrogen narcosis occurs at $P_{N_2} > 3$ atm. Oxygen toxicity at $P_{O_2} > 1.6$ atm. Decompression sickness from bubble nucleation when $P_{N_2}$ in tissues exceeds ambient pressure.
• • Pulse oximetry: Exploits the different absorption spectra of oxygenated ($\text{HbO}_2$, red at 660 nm) and deoxygenated (Hb, infrared at 940 nm) hemoglobin. The ratio $R = (A_{660}/A_{940})$ gives $\text{SpO}_2$.

Physics of Mechanical Ventilation

Positive End-Expiratory Pressure (PEEP): PEEP maintains alveolar pressure above atmospheric at end-expiration, preventing collapse of recruitable alveoli. The optimal PEEP can be estimated from the lower inflection point of the static PV curve, where alveolar recruitment begins. The recruited volume adds to the effective gas exchange surface area:

$A_{\text{eff}} = A_{\max} \cdot \frac{V - V_{\text{collapsed}}}{V_{\text{TLC}} - V_{\text{collapsed}}}$

Driving pressure: The driving pressure $\Delta P = P_{\text{plat}} - \text{PEEP}$is the pressure that actually distends the alveoli during each breath. It equals $V_T / C_{\text{RS}}$(tidal volume over respiratory system compliance). Recent evidence suggests that driving pressure is the strongest ventilator-related predictor of mortality in ARDS. Keeping $\Delta P < 15$ cmH$_2$O is a key goal.

Stress and strain: Transpulmonary pressure represents lung stress ($\sigma = P_L = P_{\text{aw}} - P_{\text{pl}}$), while strain is the ratio of tidal volume to FRC ($\varepsilon = V_T / \text{FRC}$). The relationship follows:

$\sigma = E_L \times \varepsilon$

where $E_L$ is the specific lung elastance (~13 cmH$_2$O in both healthy and ARDS lungs). In ARDS, the “baby lung” concept means that the aerated lung volume is much smaller, so the same $V_T$ produces much higher strain in the remaining open lung.

• • Robert Boyle (1660): Demonstrated that animals die in a vacuum, establishing that air is essential for life. His gas law $PV = \text{const}$governs lung volume changes during breathing.
• • Antoine Lavoisier (1780s): Identified respiration as a form of combustion: O$_2$ is consumed and CO$_2$ is produced. Measured the “respiratory quotient” in humans.
• • Christian Bohr (1904): Discovered the pH-dependent shift of the O$_2$ dissociation curve (the Bohr effect). His son Niels Bohr became a quantum physicist; his grandson Aage won a Nobel Prize in physics.
• • J.S. Haldane: Pioneered respiratory physiology, including the Haldane effect (deoxygenated Hb carries more CO$_2$) and methods for measuring lung volumes.
• • John Clements (1957): Used a surface balance to demonstrate that lung extracts dramatically reduce surface tension and that this reduction is area-dependent — the key property of surfactant. This work eventually led to surfactant replacement therapy.