Lecture 11: Comprehensive Biochemistry Study Notes on Myoglobin, Hemoglobin, the Hill Equation, and Allosteric Regulation

Oxygen Binding in Myoglobin and Noncooperative Binding Models

• Fractional Saturation Equation for Myoglobin:
  • $Y = \frac{p\text{O}_2}{p_{50} + p\text{O}_2}$
  • $Y$ represents fractional saturation (plotted on the $y$-axis).
  • $p\text{O}_2$ represents the partial pressure of oxygen (plotted on the $x$-axis).
  • $p_{50}$ represents the dissociation constant ($K_d$).
• Properties of the Dissociation Constant ($p_{50}$):
  • Smaller $p_{50}$ value: Higher oxygen affinity, tighter oxygen binding.
  • Larger $p_{50}$ value: Lower oxygen affinity, weaker oxygen binding.
  • $p_{50}$ explicitly defines the partial pressure of oxygen required to achieve half-saturation ($Y = 0.5$ or $50\%$ bound protein, $50\%$ free protein).
  • For myoglobin, the $p_{50}$ value is exactly $1\,\text{torr}$.
• Characteristics of the Fractional Saturation Curve for Myoglobin:
  • Shape: Hyperbolic curve.
  • Asymptote: Features a theoretical horizontal asymptote corresponding to $100\%$ saturation ($Y = 1.0$).
  • Asymptotic behavior: The hyperbolic curve never actually touches the $100\%$ horizontal theoretical asymptote. Reaching full saturation requires an infinite partial pressure of oxygen ($p\text{O}_2 = \infty$). Higher oxygen concentrations bring the curve progressively closer to the asymptote, but physical convergence requires infinite pressure.
• Absence of Allosteric Regulation in Myoglobin:
  • Altering the pH (raising or lowering pH) or changing the concentrations of carbon dioxide ($\text{CO}_2$), chloride anions ($\text{Cl}^-$), or 2,3-bisphosphoglycerate ($\text{BPG}$) does not alter or shift this curve.
  • Myoglobin is a monomeric protein and is not allosterically regulated.
• Noncooperative Binding Dynamics:
  • A hyperbolic saturation curve directly signifies noncooperative binding.
  • Noncooperative binding means oxygen binding to one myoglobin molecule does not affect oxygen binding to an adjacent myoglobin molecule.
  • Every myoglobin molecule binds oxygen independently of all other myoglobin molecules.

Derivation and Linearization via the Hill Equation

• Historical Context and Purpose:
  • Developed by Archibald Hill prior to the availability of computers or smartphones capable of non-linear regression.
  • Designed to extract $p_{50}$ values, Hill coefficients, and standard deviations by mapping hyperbolic and sigmoidal binding curves onto a straight line.
• Mathematical Derivation for Noncooperative/Monomeric Binding:
  • Ratio of bound to free protein:
    • $\frac{Y}{1 - Y}$
    • $Y$ represents the fraction bound; $(1 - Y)$ represents the fraction free (where $1 = 100\%$).
  • Substituting fractional saturation definitions yields:
    • $\frac{Y}{1 - Y} = \frac{p\text{O}_2}{K_d} = \frac{p\text{O}_2}{p_{50}}$
  • Taking the logarithm of both sides generates the linear Hill equation:
    • $\log\left(\frac{Y}{1 - Y}\right) = \log(p\text{O}_2) - \log(p_{50})$
• Properties of the Noncooperative Hill Plot:
  • $y$-axis variable: $\log\left(\frac{Y}{1 - Y}\right)$
  • $x$-axis variable: $\log(p\text{O}_2)$
  • Slope of the line: Represents the Hill coefficient ($n_H$).
  • For myoglobin, the plot yields a straight line with a slope equal to $1$ ($n_H = 1$), confirming noncooperative binding.
• Extracting $p_{50}$ from the Hill Plot:
  • At half-saturation ($Y = 0.5$):
    • $\frac{0.5}{1 - 0.5} = 1$
    • $\log(1) = 0$
  • The horizontal line at $y = 0$ corresponds to the half-saturated state.
  • Projecting down to the $x$-axis at $y = 0$ yields $\log(p_{50})$.
  • For myoglobin, $y = 0$ corresponds to $x = 0$ ($\log(1) = 0$), yielding $p_{50} = 10^0 = 1\,\text{torr}$.

Hemoglobin Oxygen Saturation and Physiological Function

• Sigmoidal Binding Curve:
  • Hemoglobin exhibits an $S$-shaped or sigmoidal fractional saturation curve.
  • A sigmoidal curve signifies two essential properties:
    • The protein contains more than one binding site (hemoglobin possesses 4 binding sites).
    • There is positive cooperativity between the binding sites.
• Physiological Oxygen Transport Dynamics:
  • Lungs: Partial pressure of oxygen is high enough that hemoglobin essentially saturates with oxygen.
  • Respiring Tissues: Partial pressure of oxygen drops based on tissue metabolic demand.
  • Resting conditions: Only a small fraction of bound oxygen is released to respiring tissues.
  • Vigorous exercise: Tissue $p\text{O}_2$ in muscles can drop down to around $20\,\text{torr}\text{ to } 24\,\text{torr}$, triggering a substantially higher fraction of oxygen release.
• Biological Necessity of Positive Cooperativity:
  • The sigmoidal binding profile is essential for human life.
  • Tightly binding oxygen in the lungs without releasing it in respiring tissues is biologically useless.
  • Indiscriminately dumping all bound oxygen is equally dangerous; oxygen generates highly reactive oxygen species (ROS) that damage proteins, DNA, and lipid bilayers, leading to cell death.
  • Hemoglobin delivers oxygen proportionally as needed to respiring tissues across physiological pressure ranges.

Generalized Hill Equation and Cooperativity Models

• Biochemical Convention vs Physical Sciences:
  • Physical sciences write protein binding equations starting with free reactants going to complex.
  • Biochemistry convention places the substrate-saturated protein on the reactant side and free protein plus free substrate on the product side:
    • $E S_n \rightleftharpoons E + n S$
  • The corresponding equilibrium constant is defined as a dissociation constant ($K_d$).
• The Infinite Cooperativity Assumption:
  • Hill derived his simplified model using the infinite cooperativity assumption.
  • Assumes the protein exists strictly in two states: completely free of substrate ($E$) or completely saturated with substrate ($E S_n$), in direct equilibrium with each other.
  • Ignores intermediate partially bound species ($E S_1, E S_2, \dots, E S_{n-1}$).
• Comparison to Sequential / Rigorous Binding Models:
  • A rigorous model accounts for step-wise binding: $E + S \rightleftharpoons E S_1 \rightleftharpoons E S_2 \dots \rightleftharpoons E S_n$, where each step has its own equilibrium constant ($K_1, K_2, \dots, K_n$).
  • Disadvantage of rigorous models: Yields $n$ distinct equilibrium constants ($K_d^n$). As parameters multiply, the meaningfulness and interpretability of each individual parameter decreases.
  • Advantage of the Hill Equation: Simplifies complex systems into a single parameter—the Hill coefficient—which is straightforward to interpret.
• General Mathematical Formulation:
  • $K_d = \frac{[E][S]^n}{[E S_n]} \implies [E S_n] = \frac{[E][S]^n}{K_d}$
  • Fractional saturation ($Y$):
    • $Y = \frac{[E S_n]}{[E]_{total}} = \frac{[S]^n}{K_d + [S]^n}$
  • Linearized ratio of bound to free protein:
    • $\frac{Y}{1 - Y} = \frac{[S]^n}{K_d}$
    • $\ln\left(\frac{Y}{1 - Y}\right) = n \ln([S]) - \ln(K_d)$
    • Slope of this line equals $n$ (the Hill coefficient, $n_H$).
• Interpretation of Hill Coefficient Values ($n_H$):
  • Maximum limit: $n_H \le N$, where $N$ is the total number of physical binding sites.
  • $n_H = N$: Occurs only under true infinite cooperativity (which does not exist in real biological systems; thus $n_H < N$ in practice).
  • $n_H > 1$: Positive cooperativity. Increasing saturation increases binding affinity; decreasing saturation decreases affinity. Yields sigmoidal binding curves. Very common in key regulatory metabolic enzymes.
  • $n_H = 1$: Noncooperative binding. Each site operates independently. Standard for monomeric proteins.
  • $0 < n_H < 1$: Negative cooperativity. Increasing saturation decreases affinity; decreasing saturation increases affinity. Rare in nature compared to positive cooperativity.

Hill Plot Analysis of Hemoglobin T and R States

• Application to Oxygen Binding:
  • Substituting partial pressures and $K_d = (p_{50})^n$ into the general Hill equation:
    • $Y = \frac{(p\text{O}_2)^n}{(p_{50})^n + (p\text{O}_2)^n}$
    • $\log\left(\frac{Y}{1 - Y}\right) = n \log(p\text{O}_2) - n \log(p_{50})$
• Log-Log Hill Plot Coordinates:
  • Plotted on a log-log scale.
  • $y$-axis values: $\log\left(\frac{Y}{1 - Y}\right)$ spanning values such as $-2, -1, 0, 1, 2$
  • $x$-axis values: $\log(p\text{O}_2)$ spanning values such as $-2, -1, 0, 1, 2, 3$
• Deviations from Linearity in Hemoglobin:
  • Hemoglobin does not produce a single straight line on a Hill plot because it lacks infinite cooperativity.
• Low Affinity State (T State / Deoxygenated):
  • Observed at very low $p\text{O}_2$.
  • Slope of the Hill plot at low oxygen concentrations equals $1$.
  • Extrapolating this linear low-affinity segment to $y = 0$ ($Y = 0.5$) reveals $p_{50} = 34\,\text{torr}$.
• High Affinity State (R State / Oxygenated):
  • Observed at very high $p\text{O}_2$.
  • Slope of the Hill plot at high oxygen concentrations equals $1$.
  • Extrapolating this linear high-affinity segment to $y = 0$ ($Y = 0.5$) reveals $p_{50} = 0.3\,\text{torr}$ (or $0.34\,\text{torr}$).
• Affinity Difference Between States:
  • Hemoglobin in the high-affinity state binds oxygen with a $100$-fold greater affinity than in the low-affinity state ($p_{50}$ of $0.34\,\text{torr}$ vs $34\,\text{torr}$).
• Transition Region and Hemoglobin Hill Coefficient:
  • In the intermediate region between low and high $p\text{O}_2$, the plot forms a continuous curve with a changing slope as hemoglobin transitions from T to R state.
  • The Hill coefficient ($n_H$) for hemoglobin is defined as the slope/derivative of the curve evaluated precisely at half-saturation ($Y = 0.5$, $y = 0$).
  • For hemoglobin, $n_H = 3$ (compared to theoretical maximum $N = 4$).

Structural Mechanism of Allosteric Cooperativity in Hemoglobin

• Spatial Separation of Hemes:
  • The closest distance between any two heme irons in hemoglobin is $45\,\text{\AA}$.
  • Direct physical or electronic interaction between hemes is impossible due to this large distance.
• Conformational States and Key Distances:
  • Low affinity state: T (tense / deoxygenated) state ($p_{50} = 34\,\text{torr}$).
  • High affinity state: R (relaxed / oxygenated) state ($p_{50} = 0.34\,\text{torr}$).
• Sequential Structural Changes Triggered by Oxygen Binding:
  1. Oxygen binds to the iron atom of the porphyrin ring.
  2. The iron atom moves $0.3\,\text{\AA}$ into the plane of the porphyrin ring toward planarity.
  3. The attached proximal histidine moves along with the iron atom.
  4. The iron-nitrogen bond shortens by $0.1\,\text{\AA}$, producing a total proximal histidine displacement of $0.4\,\text{\AA}$.
  5. Because van der Waals packing between porphyrin leucines/valines prevents simple local displacement, the proximal histidine movement forces the F helix to translate linearly by $1\,\text{\AA}$.
  6. Translation of the F helix by $1\,\text{\AA}$ shifts the attached FG corner by $1\,\text{\AA}$ at its base, causing an outer displacement of $6\,\text{\AA}$ at the subunit interface.
  7. The $6\,\text{\AA}$ movement of the FG corner disrupts T-state quaternary interactions with the adjacent subunit, initiating the conversion to the R state.

Quaternary Structure, Symmetry, and Sliding Contacts

• Symmetry Characteristics of Hemoglobin:
  • Heterotetramer consisting of two identical $\alpha\beta$ dimers ($\alpha_1\beta_1$ and $\alpha_2\beta_2$).
  • Contains a $C_2$ axis of symmetry running through the central solvent-filled channel.
  • Symmetry is strictly preserved when transitioning between T and R states.
  • General features of all cooperative proteins: Multimeric subunits, internal symmetry preserved across low and high activity states, states in equilibrium.
• Crystallographic Evidence:
  • Diffraction of deoxygenated T-state crystals perfused with oxygen causes the crystals to shatter.
  • Shattering provides direct physical evidence of a massive quaternary structural rearrangement upon ligand binding.
• Classification of Quaternary Subunit Interfaces:
  • Static Contacts ($\alpha_1\beta_1$ and $\alpha_2\beta_2$ interfaces):
    • Form first during subunit folding and assembly.
    • Contain $35$ non-covalent quaternary interactions.
    • Remain completely unchanged during the T $\rightarrow$ R transition.
  • Sliding Contacts ($\alpha_1\beta_2$ and $\alpha_2\beta_1$ interfaces):
    • Located between the FG corner of one subunit and the C helix of an adjacent subunit.
    • Contain $19$ specific quaternary interactions in the T state.
    • During T $\rightarrow$ R transition, all $19$ T-state quaternary contacts break, and a completely distinct set of $19$ quaternary contacts forms to stabilize the R state.
• Rotational Movement:
  • Transitioning from T to R state involves one $\alpha\beta$ dimer rotating $15^\circ$ relative to the other $\alpha\beta$ dimer.
  • Reverting from R back to T state involves a $15^\circ$ rotation in the exact opposite direction.
• Molecular Mechanics of Sliding Contacts:
  • In the T state: Tyrosine ($7\text{th}$ residue of the C helix) hydrogen bonds to Aspartate G1 on the FG corner.
  • During transition: The FG corner slides $6\,\text{\AA}$—corresponding to exactly one helical turn (helix pitch = $5.4\,\text{\AA}$)—farther up the C helix.
  • In the R state: The original hydrogen bond breaks, and a new hydrogen bond forms between Aspartate G1 and Aspartate G4.

Allosteric Regulation of Hemoglobin

• Definition of Allosteric Regulation:
  • Allosteric regulators are small molecules that bind to sites physically distinct from the substrate/ligand binding site.
  • Regulators do NOT bind to the heme iron; molecules binding to heme iron (such as carbon monoxide, nitrous oxide, hydrogen sulfide) act as poisons.
• Absence of Positive Allosteric Effectors:
  • Hemoglobin does not require positive allosteric effectors because high $p\text{O}_2$ in pulmonary alveoli is sufficient to drive full saturation.
• Negative Allosteric Effectors of Hemoglobin:
  1. Hydrogen ions ($\text{H}^+$)
  2. Carbon dioxide ($\text{CO}_2$)
  3. Chloride anions ($\text{Cl}^-$)
  4. 2,3-Bisphosphoglycerate ($2,3\text{-BPG}$)
• Mechanism of Action:
  • All four act as negative allosteric effectors, stabilizing the low-affinity T state and lowering oxygen affinity.
  • As metabolic demand in respiring tissues increases, concentrations of these regulators rise, driving oxygen release exactly where needed.

Bicarbonate Buffer System and Physiological pH Homeostasis

• Plasma Buffering Equilibria:
  • Equilibrium 1: $\text{H}^+ + \text{HCO}_3^- \rightleftharpoons \text{H}_2\text{CO}_3 \quad (K_1)$
  • Equilibrium 2: $\text{H}_2\text{CO}_3 \rightleftharpoons \text{CO}_2(aq) + \text{H}_2\text{O} \quad (K_2)$
  • Equilibrium 3 (Lungs): $\text{CO}_2(aq) \rightleftharpoons \text{CO}_2(g)$
  • Net Reaction: $\text{CO}_2 + \text{H}_2\text{O} \rightleftharpoons \text{HCO}_3^- + \text{H}^+$
• Henderson-Hasselbalch Equation for Plasma:
  • $\text{pH} = \text{p}K_a + \log\left(\frac{[\text{HCO}_3^-]}{[\text{CO}_2]}\right)$
  • Net reaction $\text{p}K_a = 6.1$.
  • Average physiological plasma pH = $7.2$.
• Open System Dynamics:
  • If closed, a buffer with $\text{p}K_a = 6.1$ would be ineffective at pH $7.2$.
  • Because it is an open system, gaseous $\text{CO}_2$ can be exhaled or absorbed through the lungs, allowing dynamic pH control.
  • Acidosis response (high $[\text{H}^+]$): Shifts equilibrium left, converting $\text{H}^+$ and $\text{HCO}_3^-$ into $\text{CO}_2(g)$, which is exhaled, removing $\text{H}^+$ and elevating pH back to 7.2$.\n * Alkalosis response (low [\text{H}^+]$): Lungs absorb gaseous$\text{CO}_2(g)$and convert it to$\text{H}^+$and$\text{HCO}_3^-$, lowering pH back to$7.2$.
• Clinical Significance of Hyperventilation:
  • Pathophysiology example: In traumatic mountain biking accidents (e.g., handlebar injury in Park City, Utah at $>10{,}000\,\text{ft}$ elevation), acute pain and panic induce hyperventilation.
  • Breathing rapidly/heavily expels excessive gaseous $\text{CO}_2(g)$ from lungs.
  • Depleting gaseous $\text{CO}_2$ shifts equilibria left, depleting dissolved $\text{CO}_2(aq)$, carbonic acid, and hydrogen ions ($[\text{H}^+]$ drops).
  • Plasma pH rises significantly above normal.
  • With depleted $[\text{H}^+]$ (a negative allosteric effector), hemoglobin's oxygen affinity becomes abnormally high.
  • Even though hemoglobin remains $100\%$ saturated in blood, it refuses to release oxygen to respiring tissues, causing tissue hypoxia and a severe sensation of suffocation.