Hemoglobin O₂ Binding: Multiple Occupancy Probabilities

Hemoglobin O₂ Binding: Multiple Occupancy Probabilities is the probabilistic treatment of the four-site cooperative oxygen-binding system in hemoglobin (Hb), in which the equilibrium occupancy distribution over the

2^{4} = 16

microstates (or, by symmetry, the five macrostates

i = 0, 1, 2, 3, 4

) is governed by the Adair equation and its statistical-mechanical (Pauling/MWC/KNF) refinements. Key result: the fractional saturation

\overset{ˉ}{Y}

and the full occupancy probability vector

P_{i} (p_{O_{2}})

are rational functions of oxygen partial pressure built from successive (Adair) binding constants, and cooperativity manifests as a non-binomial spread of the

P_{i}

. Applies in: biophysical chemistry, allosteric enzyme kinetics, respiratory physiology, ligand-binding statistical mechanics. Who/when: Gilbert S. Adair (1925), Linus Pauling (1935), Monod–Wyman–Changeux (1965), Koshland–Némethy–Filmer (1966).

Hemoglobin is a tetramer carrying four heme groups, each able to bind one O₂ molecule, so its ligation state at equilibrium is a random variable taking values in ${0, 1, 2, 3, 4}$ . The multiple occupancy probabilities $P_{i}$ are the equilibrium probabilities that exactly $i$ of the four sites are occupied, and they collectively encode everything measurable: mean saturation, binding variance, cooperativity (Hill coefficient), and the partition function. What matters is that these probabilities are not binomial — binding at one site changes the affinity of the others (positive cooperativity), so the distribution is sharpened and shifted relative to four independent sites. The surprising fact is that despite four chemically distinct intermediate ligation patterns, classical oxygen-equilibrium data only ever constrain four macroscopic Adair constants, making the inverse problem (recovering microscopic site affinities) fundamentally underdetermined.

The Core Insight

Treat the tetramer as a small grand-canonical system whose chemical potential is set by dissolved O₂. The binding polynomial (partition function) $Z (x)$ is a degree-4 polynomial in the ligand activity $x \propto p_{O_{2}}$ ; every occupancy probability is one monomial of $Z$ divided by $Z$ , and every thermodynamic observable is a logarithmic derivative of $Z$ . Cooperativity = curvature of $ln Z$ beyond the independent-site baseline.

Formal Statement

Binding Polynomial and Occupancy Probabilities

Let $x = K_{ref} [O_{2}]$ be a dimensionless ligand activity (or simply $x = [O_{2}]$ with constants carrying units). Define the Adair binding constants $K_{1}, \dots, K_{4}$ as the stepwise (macroscopic) association constants:

Hb(O_{2})_{i - 1} + O_{2} ⇌ Hb(O_{2})_{i}, K_{i} = \frac{[ Hb(O _{2} ) _{i} ]}{[ Hb(O _{2} ) _{i - 1} ] [ O _{2} ]} .

⟳write each step as a mass-action equilibrium;

K_{i}

is the ratio of successive species concentrations per unit free ligand.

Let $β_{i} = \prod_{j = 1}^{i} K_{j}$ be the cumulative (overall) Adair constants, with $β_{0} = 1$ . The binding polynomial is

Z ([O_{2}]) = i = 0 \sum 4 β_{i} [O_{2}]^{i} = 1 + β_{1} [O_{2}] + β_{2} [O_{2}]^{2} + β_{3} [O_{2}]^{3} + β_{4} [O_{2}]^{4} .

⟳sum the equilibrium concentrations of all macrostates relative to the unliganded reference;

Z

is the grand partition function with

[O_{2}]

playing the role of fugacity.

The multiple occupancy probability of exactly $i$ bound ligands is

P_{i} ([O_{2}]) = \frac{β _{i} [ O _{2} ] ^{i}}{Z ([ O _{2} ])}, i = 0 \sum 4 P_{i} = 1.

⟳each macrostate population is

β_{i} [O_{2}]^{i}

times the reference; normalize by the total

Z

Here $β_{i}$ bundles a statistical degeneracy factor $g_{i} = (i 4)$ (the number of microscopic ways to place $i$ ligands on 4 sites) with the intrinsic microscopic affinities. If all sites were independent with intrinsic constant $k$ , then $β_{i} = (i 4) k^{i}$ and $P_{i}$ would be exactly binomial.

Fractional Saturation as a Logarithmic Derivative

The mean fractional saturation $\overset{ˉ}{Y} = \frac{1}{4} ⟨ i ⟩$ satisfies

\overset{ˉ}{Y} = \frac{1}{4} i = 0 \sum 4 i P_{i} = \frac{1}{4} \frac{[ O _{2} ] Z ^{'} ([ O _{2} ])}{Z ([ O _{2} ])} = \frac{1}{4} \frac{\partial ln Z}{\partial ln [ O _{2} ]} .

⟳

⟨ i ⟩ = \sum i P_{i}

; recognize

\sum i β_{i} [O_{2}]^{i} = [O_{2}] Z^{'}

, then convert to a log-derivative.

Explicitly, the Adair equation:

\overset{ˉ}{Y} = \frac{β _{1} [ O _{2} ] + 2 β _{2} [ O _{2} ] ^{2} + 3 β _{3} [ O _{2} ] ^{3} + 4 β _{4} [ O _{2} ] ^{4}}{4 ( 1 + β _{1} [ O _{2} ] + β _{2} [ O _{2} ] ^{2} + β _{3} [ O _{2} ] ^{3} + β _{4} [ O _{2} ] ^{4} )} .

⟳substitute

Z

and

[O_{2}] Z^{'}

into [4] and factor the leading

\frac{1}{4}

The variance of occupancy, a direct measure of fluctuation and cooperativity, is the second log-derivative:

Var (i) = ⟨ i^{2} ⟩ - ⟨ i ⟩^{2} = \frac{\partial ^{2} ln Z}{\partial ( ln [ O _{2} ] ) ^{2}} = \frac{\partial ⟨ i ⟩}{\partial ln [ O _{2} ]} .

⟳differentiate

⟨ i ⟩ = \partial_{l n x} ln Z

once more; the second cumulant of a grand-canonical distribution equals the second log-derivative of the partition function.

Why It Holds / Derivation

From Mass Action to the Occupancy Distribution

Step 1 — Enumerate macrostates. A tetramer with four equivalent sites has macrostates indexed by

i \in {0, 1, 2, 3, 4}

. Choose Hb (fully deoxy) as reference,

[Hb]_{0} \equiv [Hb (O_{2})_{0}]

. Step 2 — Express each population. Iterating [1],

[Hb (O_{2})_{i}] = K_{i} [O_{2}] [Hb (O_{2})_{i - 1}] = (\prod_{j \leq i} K_{j}) [O_{2}]^{i} [Hb]_{0} = β_{i} [O_{2}]^{i} [Hb]_{0}

. Step 3 — Total tetramer concentration. Summing,

[Hb]_{tot} = [Hb]_{0} \sum_{i = 0}^{4} β_{i} [O_{2}]^{i} = [Hb]_{0} Z

. Step 4 — Probabilities.

P_{i} = [Hb (O_{2})_{i}] / [Hb]_{tot} = β_{i} [O_{2}]^{i} / Z

, which is [3] and automatically normalized. Step 5 — Mean occupancy. Bound O₂ per tetramer

= \sum_{i} i [Hb (O_{2})_{i}] / [Hb]_{tot} = \sum_{i} i P_{i}

. Since

\sum_{i} i β_{i} [O_{2}]^{i} = [O_{2}] \partial_{[O_{2}]} Z

, divide by

Z

to get [4]; divide by 4 sites for fractional saturation [5]. Step 6 — Microscopic decomposition. Each

β_{i}

factors as

g_{i} κ_{i}

where

g_{i} = (i 4)

is the number of distinct site patterns and

κ_{i}

is the product of intrinsic association constants of an average pattern. The independent-site null model is

κ_{i} = k^{i}

, giving the binomial

P_{i} = (i 4) (k [O_{2}])^{i} / (1 + k [O_{2}])^{4}

. Deviations of

κ_{i}

from

k^{i}

are exactly the thermodynamic signature of cooperativity.

■

Linus Pauling | 1901–1994

In 1935 Pauling produced the first statistical-mechanical theory of hemoglobin cooperativity, modeling the four hemes as interacting sites with a nearest-neighbor "interaction free energy" that boosted the affinity of empty sites once a neighbor was occupied. His treatment recast the Adair phenomenology as a microscopic lattice problem and anticipated, by three decades, the allosteric formalisms of Monod–Wyman–Changeux and Koshland–Némethy–Filmer. Pauling later identified the molecular basis of sickle-cell disease (1949) as a hemoglobin defect — coining "molecular disease" — connecting the same molecule's occupancy thermodynamics to clinical pathology.

Building Intuition

Picture four mailboxes on one pole; an O₂ molecule is a letter. Independent sites: each mailbox fills at random, so the count of filled boxes is binomial — broad and symmetric near half-saturation. Positive cooperativity: the first letter "unlocks" the others (the deoxy → oxy quaternary switch makes remaining sites hungrier). Now the system prefers to be either nearly empty or nearly full, and dislikes the half-filled states. The probability mass on $P_{1}, P_{2}, P_{3}$ is depleted while $P_{0}$ and $P_{4}$ swell — the distribution becomes bimodal-leaning and the saturation curve sigmoidal. Mathematically, cooperativity = the variance [6] exceeds the binomial variance near $\overset{ˉ}{Y} = \frac{1}{2}$ , and the Hill coefficient $n_{H} = 1 + (\partial ln (\overset{ˉ}{Y} / (1 - \overset{ˉ}{Y})) / \partial ln [O_{2}] - 1)$ rises above 1.

Occupancy probabilities $P_i$ versus $\log p_{O_2}$ for cooperative hemoglobin. Intermediate states $P_1$–$P_3$ are suppressed; $P_0$ and $P_4$ dominate; the saturation curve $\bar Y$ is sigmoidal.

Worked Examples

Computing the Full Occupancy Vector

Given: Adair cumulative constants (illustrative, units of mmHg

^{- i}

β_{1} = 0.10

β_{2} = 0.012

β_{3} = 0.0011

β_{4} = 0.00060

, at

p_{O_{2}} = 26

mmHg (

\approx

p50 region). Step 1 — Monomials $β_{i} p^{i}$ :

$β_{0} = 1$
$β_{1} p = 0.10 \times 26 = 2.60$
$β_{2} p^{2} = 0.012 \times 676 = 8.112$
$β_{3} p^{3} = 0.0011 \times 17576 = 19.334$
$β_{4} p^{4} = 0.00060 \times 456976 = 274.19$

Step 2 — Partition function:

Z = 1 + 2.60 + 8.112 + 19.334 + 274.19 = 305.24

. Step 3 — Probabilities $P_{i} = β_{i} p^{i} / Z$ :

$P_{0} = 1/305.24 = 0.00328$
$P_{1} = 2.60/305.24 = 0.00852$
$P_{2} = 8.112/305.24 = 0.02658$
$P_{3} = 19.334/305.24 = 0.06334$
$P_{4} = 274.19/305.24 = 0.89829$

Step 4 — Saturation:

⟨ i ⟩ = 0 (0.00328) + 1 (0.00852) + 2 (0.02658) + 3 (0.06334) + 4 (0.89829) = 4 (0.89829) + \dots = 3.844

;

\overset{ˉ}{Y} = 3.844/4

P = (0.0033, 0.0085, 0.0266, 0.0633, 0.8983), \overset{ˉ}{Y} = 0.961

Note the strong suppression of $P_{1}, P_{2}, P_{3}$ relative to $P_{4}$ — the hallmark of cooperativity at high $p_{O_{2}}$ .

Variance and Hill Slope at Half-Saturation

Given: A symmetric cooperative model with

β_{1} = β_{3} = 4 c L

small and

β_{4} = L

chosen so that

\overset{ˉ}{Y} = \frac{1}{2}

p = p_{50} = 26

mmHg, with measured

P = (0.18, 0.05, 0.04, 0.05, 0.18)

(intermediates strongly suppressed; note these sum with the rest of mass; rescaled here for illustration so

\sum P_{i} = 0.50

on the empty side — treat as normalized so

P_{0} + \dots + P_{4} = 1

with

P_{0} = P_{4} = 0.36, P_{1} = P_{3} = 0.10, P_{2} = 0.08

). Step 1 — Mean: by symmetry

⟨ i ⟩ = 2

\overset{ˉ}{Y} = 0.5

. ✓ Step 2 — Second moment:

⟨ i^{2} ⟩ = 0^{2} (0.36) + 1^{2} (0.10) + 2^{2} (0.08) + 3^{2} (0.10) + 4^{2} (0.36)

= 0 + 0.10 + 0.32 + 0.90 + 5.76 = 7.08

. Step 3 — Variance:

Var (i) = 7.08 - 2^{2} = 3.08

. Step 4 — Compare to independent-site (binomial) baseline at

\overset{ˉ}{Y} = \frac{1}{2}

Var_{indep} = 4 \cdot \frac{1}{2} \cdot \frac{1}{2} = 1.0

. Step 5 — Cooperativity ratio:

Var / Var_{indep} = 3.08

, confirming a strongly super-binomial (positively cooperative) distribution. Using

n_{H} \approx Var / [4 \overset{ˉ}{Y} (1 - \overset{ˉ}{Y})]

at half-saturation:

Var (i) = 3.08, n_{H} \approx 3.08/ (4 \cdot 0.5 \cdot 0.5) = 3.08

(Real human Hb gives $n_{H} \approx 2.8$ –3.0; the maximum possible for a tetramer is 4.)

Common Misconceptions

The Hill coefficient is not the number of binding sites.

n_{H} \leq 4

for hemoglobin but typically

\approx 2.8

–3.0; it is a cooperativity index (the maximum log-log slope of

\overset{ˉ}{Y} / (1 - \overset{ˉ}{Y})

), not a stoichiometry. Only in the unphysical limit of infinite cooperativity ("all-or-none" binding, where

P_{1} = P_{2} = P_{3} = 0

) does

n_{H}

reach the site number 4. Conversely

n_{H} = 1

means independent sites, not one site. Adair constants are macroscopic and cannot, alone, recover microscopic site affinities. Oxygen-equilibrium curves determine only the four

K_{i}

(or

β_{i}

). But there are eight distinct microscopic species among the singly/doubly/triply liganded states (e.g. two ways to be doubly liganded on a tetramer with two

α

and two

β

chains in distinct geometries). The map from microscopic constants to macroscopic Adair constants is many-to-one; the inverse problem is underdetermined without additional data (NMR, kinetics, isotope-edited measurements).

Independent Sites Mimicking Cooperativity Is Impossible, but the Reverse Trap Exists

A naive fit might report $n_{H} > 1$ and conclude positive cooperativity, yet heterogeneity of independent sites produces $n_{H} < 1$ (apparent negative cooperativity), never $n_{H} > 1$ . Consider two non-interacting site classes with affinities $k_{a} \neq = k_{b}$ (e.g. distinct $α$ and $β$ chains, 2 each), each binding independently: $\overset{ˉ}{Y} = \frac{1}{2} [\frac{k _{a} p}{1 + k _{a} p} + \frac{k _{b} p}{1 + k _{b} p}]$ . This curve is flatter than a single hyperbola, giving $n_{H} < 1$ . Thus a measured $n_{H} > 1$ genuinely requires site–site interaction, but a measured $n_{H} < 1$ is ambiguous between true negative cooperativity and mere affinity heterogeneity — a classic identifiability pitfall.

Connections

The occupancy framework links several neighboring theories. The Monod–Wyman–Changeux Allosteric Model derives the $β_{i}$ from a two-state (T/R) equilibrium with allosteric constant $L$ and ratio $c = K_{R} / K_{T}$ , giving a concerted microscopic origin for the Adair constants. The Koshland–Némethy–Filmer Sequential Model instead derives them from ligand-induced conformational change propagating through subunit interfaces. Both reduce, at the level of observables, to the Adair polynomial [2], illustrating that macroscopic occupancy data underdetermine the microscopic mechanism. The variance relation [6] connects to the Fluctuation-Dissipation Theorem (occupancy fluctuations ↔ susceptibility to ligand). The binding polynomial is the Grand Canonical Partition Function of a four-site system, tying the whole subject to Statistical Mechanics of Adsorption (the Langmuir/Ising correspondence) and to Ising Model on Small Lattices via Pauling's nearest-neighbor interaction picture.

Real-World Applications

Pulse oximetry & blood-gas interpretation: the sigmoidal $\overset{ˉ}{Y} (p_{O_{2}})$ (the oxyhemoglobin dissociation curve) underlies clinical $S p O_{2}$ readings; the steep mid-region near p50 ≈ 26.6 mmHg enables efficient O₂ unloading in tissue.
Bohr effect modeling: CO₂/H⁺ shift the Adair constants, right-shifting the curve; quantitative respiratory-physiology models adjust $β_{i}$ as functions of pH and $p_{C O_{2}}$ .
Hemoglobinopathy diagnostics: sickle Hb (HbS) and HbC alter cooperativity and p50; fitting Adair constants flags abnormal occupancy distributions.
Fetal hemoglobin (HbF) and altitude/transfusion medicine: HbF's left-shifted curve (higher O₂ affinity, lower 2,3-BPG sensitivity) is modeled by reduced $β$ -relative constants.
Blood substitutes / HBOCs (hemoglobin-based oxygen carriers): engineering target p50 and $n_{H}$ requires designing the $β_{i}$ ratios.
2,3-BPG (DPG) allosteric modulation: this effector lowers O₂ affinity; pharmacological mimics (e.g. voxelotor for sickle disease) are designed via their effect on the T/R equilibrium and hence the occupancy vector.
Cooperative-binding assay design across biochemistry: the same Adair/Hill machinery quantifies ligand binding to multimeric receptors, transcription factors, and allosteric enzymes (aspartate transcarbamoylase, phosphofructokinase).

Historical Development

1904 — Christian Bohr, Hasselbalch, Krogh: report the sigmoidal oxygen dissociation curve and the Bohr effect (pH dependence), establishing that Hb O₂ binding is not hyperbolic.
1910 — Archibald Hill: introduces the empirical Hill equation $\overset{ˉ}{Y} = (K p)^{n} / (1 + (K p)^{n})$ to fit the sigmoid; $n$ becomes the cooperativity index (though Hill knew it was not a literal stoichiometry).
1925 — Gilbert S. Adair: measures hemoglobin's molecular weight (≈68 kDa) implying four hemes, and formulates the four-constant Adair equation [5] as the exact intermediate-state description.
1935 — Linus Pauling: first statistical-mechanical model of cooperativity via heme–heme interaction energy, microscopically grounding the Adair constants.
1949 — Pauling, Itano, Singer, Wells: identify sickle-cell anemia as a "molecular disease" of hemoglobin.
1959–1960 — Max Perutz: solves the crystal structure of hemoglobin (Nobel 1962), revealing the deoxy (T) and oxy (R) quaternary structures that physically underlie cooperativity.
1965 — Monod, Wyman, Changeux: publish the concerted (MWC) allosteric model, deriving the occupancy distribution from a two-state T⇌R equilibrium.
1966 — Koshland, Némethy, Filmer: publish the sequential (KNF) induced-fit model, an alternative microscopic origin for the same Adair phenomenology.
1970s–1990s — Ackers, Gary Gill, et al.: develop linkage thermodynamics and precise resolution of microscopic versus macroscopic constants, establishing the identifiability limits and the role of the binding polynomial as partition function.

Subtleties & Limits

Microscopic resolution of intermediate states. Even with high-precision oxygen-equilibrium data, only four macroscopic Adair constants are accessible, whereas the tetramer (

α_{2} β_{2}

) has up to ten distinct microscopic ligation species (counting

α

β

asymmetry and dimer–dimer interface states). Disentangling which intermediate microstates dominate — and whether the T→R quaternary switch occurs predominantly after the first, second, or third ligand — remains experimentally contentious and model-dependent (MWC predicts a sharp switch governed by

L

; KNF a graded one). Related open issues: (i) the precise statistical-thermodynamic role of the

α_{1} β_{2}

dimer interface and tetramer–dimer dissociation coupling to ligation (the "Ackers tertiary two-state" debates); (ii) whether a single equilibrium binding polynomial suffices, or whether kinetic intermediates and conformational dynamics require a non-equilibrium occupancy treatment; (iii) extending the framework to combined O₂/CO₂/H⁺/2,3-BPG multi-ligand binding polynomials, where the partition function becomes multivariate and the occupancy "vector" becomes a high-dimensional tensor whose identifiability is still incompletely characterized.