Introduction:

For the study of the binding of gaseous molecules to hemoglobin and myoglobin, we use partial pressures.

The partial pressure of oxygen in air is 150 torr; this value is calculated based on the fact that air is 20% oxygen (0.20 x 750 torr (mm Hg) = 150 torr).

Partial oxygen pressures of interest:

pO₂ = 150 torr (air)
pO₂ = 100 torr (lungs)
pO₂ = 90 torr (blood)

Derivation of binding equation:

Mb + O₂ <==> Mb-O₂ written as an association (or binding) reaction

equilibrium expression: K_a = [Mb-O₂ ]/[Mb][O₂ ]

definition of a new term: (fractional saturation) = [Mb-O₂]/([Mb-O₂] + [Mb])

rearrangement of terms: (want theta in terms of [O₂] and a constant)

a. substitute [Mb-O₂] = Ka [Mb][O₂] in the theta expression

b. this gives: = K_a [Mb][O₂]/(K_a [Mb][O₂] + [Mb])

c. cancel [Mb]'s and divide numerator and denominator by Ka -->

= [O₂]/([O₂] + 1/K_a)

or = p O₂/(p O₂ + P₅₀), where P₅₀ = oxygen partial pressure at 1/2 saturation

This yields the expression K_a = 1/O₂ or 1/K_a = [O₂]

These equations are of the form y = x/(x + c)}, which yields a rectangular hyperbola.

The Hill equation - O₂ binding to Hemoglobin

Binding of oxygen to myoglobin produces a hyperbolic curve, but the binding curve for hemoglobin is very different. How do we describe the unusual curvature in the hemoglobin binding curve?

Archibald Hill (1913) produced the following analysis:

rearrange the equation above:
/(1 - ) = (P _O2 /P₅₀)n

take the log of each side:
log{ /(1 - Q\ )} = n{log P _O2 - log P₅₀}

Plot log{ /(1 - )} vs. log P₅₀;n is the slope of this line (y = mx + b)

The value of "n" the Hill Coefficient is an index of the degree of co-operativity.

Non-cooperative systems have a value of 1.0; this number could be as high as the number of subunits, but that is usually not observed.

The n values for hemoglobin range from 3.0 to 3.5, depending on physiological conditions.

Models of Allostery:

Allostery in hemoglobin is exhibited in three ways:

a. early molecules of oxygen bind with great difficulty
b. Hb's affinity for O₂ is affected by H⁺and CO₂
c. bisphosphoglycerate lowers Hb's affinity for O₂

Two biochemical models have been proposed to explain the co-operative behavior of multi-subunit proteins and enzymes: the sequential model and the concerted model.

The Sequential Model: Koshland, Nemethy, Filmer (U.C. Berkeley)

TT + L ---> RT + L ---> RR

It postulates:

1. there are two states: R (Relaxed) and T (Taut or Tense)
2. that ligand (L) binding changes the shape of only the subunit to which it binds
3. that the conformational change elicited by binding of ligand to the one subunit facilitates adjacent subunit to undergo the change from T --> R.

The Concerted model: Monod, Wyman, Changeux (France)

The French model called for conservation of symmetry in the subunits.

It postulates:

1. that one would find only T T or RR forms of the protein molecule
2. that ligand binding causes the transition from tense to relaxed (TT + L ---> RR)

Allosteric regulation can be of two types, either activating or inhibiting:

T_o + activator ---> R_o
or: R_o + inhibitor ---> T_o

At the molecular level, the hybrid species (R-T) is a prominent in the sequential model but is absent in concerted model.