# 5.3: Reversible Binding of a Protein to a Ligand - Oxygen-Binding Proteins

### Myoglobin, Hemoglobin, and their Ligands

Almost all biochemistry textbooks start their description of the biological functions of proteins using the myoglobin and hemoglobin as exemplars. These are very rational approaches since they have become model systems to describe the binding of simple ligands, like dioxygen (O2), CO2, and H+, and how the structure of the protein determines and is influenced by binding of ligands.

Yet in a way these "ligands" are dissimilar to perhaps the majority of other proteins which bind small ligands such as substrates (for enzymes), inhibitors and activators or large "ligands" such as other proteins, nucleic acids, carbohydrates and lipids. These type of ligands are reversibly bound through noncovalent interactions described in detail in Chapter 2.4.

The classic ligands that reversibly bind to hemoglobin, dioxygen, carbon dioxide, and protons, are bound covalently. Dioxygen binds to a heme Fe2+ transition metal through a coordinate covalent or dative bond, protons obviously bind covalently to proton acceptors (like His), while CO2 binds covalently as if forms a carbamate with the N terminus of one of the hemoglobin chains.

In typical covalent bonds, each bonded atom contributes to and shares the two electron in the bond. In coordinate or dative covalent bonds, the ligand, a Lewis base, contributes both electrons in the bond. Both electrons are still considered to be "owned" by the ligand and not by the transition metal ion, a Lewis acid. Hence the ligand can readily dissociate from the metal ion, much as a ligand bound through classical noncovalent interactions can. This analogy can be extended to protons which are also Lewis acids (with no contributing electrons) as they react with Lewis bases (lone pair donors) on atoms such as N on a His side chain.

We'll start with myoglobin, a monomer protein containing Mb 8 α−helices, A-H) and end with hemoglobin, heterotetramer with two α -and two β−subunits, each which also contains 8 α−helices.  Both are oxygen binding proteins. Both contain heme (one in myoglobin, and 4 for the four subunits of hemoglobin.  Each heme has a central Fe2+ ion which forms a coordinate covalent bond with dioxygen.  Dioxygen is transported from lungs, gills, or skin of an animal to capillaries, where it can be delivered to respiring tissue. It has a low solubility in blood (0.1 mM). Whole blood, which contains 150 g Hb/L, can carry up to 10 mM dioxygen. Invertebrate can have alternative proteins for oxygen binding, including hemocyanin, which contains Cu and hemerythrin, a non-heme protein. On binding dioxygen, solutions of Hb change color to bright red. Solutions of hemocyanin and hemerythrin change to blue and burgundy colored, respectively, on binding dioxygen. Some Antarctic fish don't require Hb since dioxygen is more soluble at low temperature. Myoglobin is found in the muscle, and serves as a storage protein for oxygen transported by hemoglobin.

The structure of heme in myoglobin and hemoglobin, is shown in Figure $$\PageIndex{1}$$ below.

Figure $$\PageIndex{1}$$: : Heme

The heme group contains protoporphyrin IX, with four tetrapyrrole rings linked by methene bridges. Attached to the tetrapyrrole structure are four methyl, two vinyl, and two proprionate groups. These can be arranged in 15 ways, only one (IX) occurs in biological systems. Ferrous (Fe2+) ion bonded to the protoporphyrin IX constitutes the heme group.  The heme fits into a hydophobic crevice in the proteins with the proprionate groups exposed to solvent.

### Myoglobin (Mb)

Myoglobin is an extremely compact protein, and consists of 75% alpha helical structure. It has 8 α−helices labeled A-H.  Four are terminated by proline, a helix breaker.  The interior amino acids are almost entirely nonpolar. The only polar amino acids found completely buried are two histidines ligands.  One, the proximal one near the heme and which serves a  ligand to the heme Feand distal) which coordinate the heme Fe2+ where dioxygen binds.

Figure $$\PageIndex{2}$$ below shows an interactive iCn3D model of deoxymyoglobin from wild boar.  The heme is shown in sticks along with the proximal and distal hemoglobins.

Figure $$\PageIndex{2}$$: Deoxymyoglobin (1MWD) (Copyright; author via source). Click the image for a popup or use this external link:https://structure.ncbi.nlm.nih.gov/i...PTroKxtYLedjJ6

### Hemoglobin

Hemoglobin has an illustrious history.  It is the first protein whose molecular weight was determined and the first assigned a specific function (dioxygen transport). It was the first protein in which a mutation in a single amino acid cause by a single base pair change in the DNA coding sequence was shown to cause a disease (sickle cell trait and disease).  The mathematical theories developed to model dioxygen binding has been used to explain enzyme activity.  It also binds H+, CO2, and bisphosphoglcyerate which bind to sites (allosteric) distant from oxygen binding site which regulates it's dioxygen binding affinity.

As with myoglobin, the Fe2+ ion is coordinated to 4 Ns on the 4 pyrrole rings, The 5th ligand is a supplied by proximal His (the 8th amino acid on helix F) of the protein. In the absence of dioxygen, the 6th ligand is missing. and the geometry of the complex is somewhat square pyramidal, with the Fe slightly above (0.2 Å) the plane of the heme ring. A distal His (E7) is on the opposite side of the heme ring, but too far to coordinate with the Fe2+. When dioxgen binds, it occupies the 6th coordination site and pulls the Fe into the plane of the ring, leading to octahedral geometry.  These changes on oxygenation are shown in Figure $$\PageIndex{3}$$ below.

Figure $$\PageIndex{3}$$:  Changes in heme structure on binding of dioxygen.

The proximal histidine that provides the imidazole ligand is show.  Dioxygen is shown as red spheres.  Fe2+ ion is shown as a small orange sphere.  It's size has been dramatically reduced in this image so it movement can be more readily observed.

Carbon monoxide (CO), nitric oxide (NO), and hydrogen sulfide (H2S) also bind to the sixth coordination site, but with higher affinity than dioxygen, which can lead to CO poisoning for example. The distal histidine keeps these ligands (including dioxygen) bound in a bent, non-optimal geometry. This minimizes the chances of CO poisoning.

Figure $$\PageIndex{4}$$ below shows an interactive iCn3D model of human oxy-hemoglobin (2dn1)

Figure $$\PageIndex{4}$$: Human oxy-hemoglobin (2DN1) (Copyright; author via source). Click the image for a popup or use this external link:https://structure.ncbi.nlm.nih.gov/i...yBJD4p8EazRpU7

#### Fe2+ ion ligand interactions

When the 6th ligand, dioxygen, binds to heme Fe2+, the geometry of the complex becomes octahedral. The Fe2+ ion has 6 electrons in d orbitals. The electronic configuration of Fe is 3d64s2 while the Fe2+ ion is 3d6) as shown in Figure $$\PageIndex{5}$$ below.   Each of the orbitals would have the same energy except for the doubly occupied one which would have slightly higher energy due to the extra repulsion of the two electrons in the orbital.  The figure below shows them having the same energy

Figure $$\PageIndex{5}$$:  Electron atomic orbital diagram of  Fe2+, a d6 transition metal ion.

You will remember from introductory chemistry classes that transition metal complexes and their solutions are highly colored.  Since oxygenated hemoglobin (found in arteries) appears bright red/orangish, it must absorb blue/green light more than deoxyhemoglobin, which is darker red (but still reddish).  These absorbed wavelengths are removed from the spectrum, making hemoglobins shades or red.  Veins contained more deoxygenated blood returning to the heart to be reoxygenated.  The visible veins in your arms and legs appear blue not because of the spectral properties of deoxyhemoglobin.  Rather blue light doesn't penetrate into the tissue as far as red, so red is preferentially removed from the remaining light which is reflected, making veins appear blue.  Figure $$\PageIndex{6}$$ below shows a partial absorbance spectrum of deoxy- and oxy-hemoglobin from 280-1000 nm to 1000 nm.

Figure $$\PageIndex{6}$$:  Spectrum of deoxy- and oxyhemoglobin - log molar extinction coefficient vs wavelength.  The red line is the oxyHb spectra, and the dotted blue line shows the deoxyHb spectrum.  Modifed from  en:Bme591wikiproject Creative Commons Attribution-Share Alike 3.0 Unported .

The dashed light blue vertical line shows the approximate wavelength (around 450 nm) of greatest molar extinction coefficient difference between the two forms.  Note that the y axis is a log scale.  At 450 nm, oxyhemoglobin has an extinction coefficient of about 600,000 while that of deoxyhemoglobin is about 60,000, so much more blue light is removed from a solution of oxyHb, making it appear bright red.  Note also same differences appear in the red region of the spectra. but these extinction coefficients only vary from 3000 for oxyhemoglobin to 200 for deoxyHb, so the have little effect on the visible color of blood.

The spectrum shown in Figure $$\PageIndex{x}$$ also shows the near infrared region of the spectra.  Inexpensive pulse oximeters (some built into watches) have been increasingly used by people at home to their measure their oxygenation status during the Covid pandemic.  These use two pulsed LEDs, one at 660 nm, where oxyhemoglobin has a higher extinction coefficient, and one at 940 (infrared region), where deoxyhemoglobin has a higher extinction coefficient.

Most biochemistry books offer minimal coverage of bioinorganic chemistry, even though a large percent of proteins bind metal ions.  Since biochemistry is by definition an interdisciplinary field, it is important to bring up past learning in other biology and chemistry courses and show how it is applicable to biochemistry.  Most students study some transition metal chemistry in introductory chemistry courses and are familiar with transition metal ions, their electronic configuration, crystal field theory, high and low spin states, paramagnetism and diamagnetism.  Hence it is appropriate and important to bring these ideas into biochemistry and extend them when necessary.  The basis of the material below is modifed from Structure & Reactivity in Organic, Biological and Inorganic Chemistry by Chris Schaller (Creative Commons Attribution-NonCommercial 3.0 Unported License).

Let's look at the electronic structure of Fe2+ in oxyhemoglobin, in which the Fe2+ is coordinated to six ligands (4 pyrrole rings of the heme in a plane, one axial imidazole ring from the proximal histidine and dioxygen).  The geometry of the electron clouds around the Fe2+ is octahedral. Let's further assume that the six d orbitals are oriented in the same direction in the x, y, and z planes of the heme as the ligands.  This illustrated in Figure $$\PageIndex{7}$$ below.

Figure $$\PageIndex{7}$$: Fe2+ d orbtitals and their orientations

Two of the orbitals, dz2 and dx2-y2,  appear different in that they are oriented directly along the x, y and z axes while the other three are in-between the axes.  Now imagine six anionic ligands approaching along the axes.  The energy of the Fe electrons in the dz2 and dx2-y2 would be raised higher than the others due to electron-electron repulsion.  This is illustrated in Figure $$\PageIndex{8}$$ below.

Figure $$\PageIndex{8}$$: High spin and low spin electron energy level diagrams for Fe2+ ion in an octahedral complex

If the electrostatic interaction between the orbitals of the Fe ion and the incoming ligands is low, the energy of the dz2 and dx2-y2 orbitals would be a bit higher (by an amount Δ) as illustrated in the left part of Figure $$\PageIndex{8}$$.  When filling the electrons with six 3d6electrons for the the Fe2+ ion, you would add one electron to each of the 5 orbitals and then pair one for the sixth orbital.  In that case there would be 4 unpaired electrons and the complex would be paramagnetic.  We call this a high spin state. If however, the electrostatic interactions of the incoming ligands with the d orbital electrons is high, the  Δ  would be large.  When filling the orbitals in this case, the electron would be paired in the three lower energy orbitals and there would be no electrons unpaired and the complex is diamagnetic.

In either case, if light of energy equal to the Δ interacts with the complex, electrons would be promoted to the high energy levels.  For the low spin case (larger axial electrostatic interactions, Δ is large, the energy of the required photon is larger (more blue shifted) than in the high spin case. This could make a solution of the molecule in the low spin case appear redder than for the high spin case.  This is the case for dioxygen, which interactions strongly with the Fe2+ axially-oriented orbitals.  Hence oxy-Fe2+ heme complex is low spin and diamagnetic.

We've just described the basic idea of crystal field theory, which you probably discussed in introductory chemistry courses.  The theory is simplistic in several ways.  The ligands might not be anions.  More importantly modern chemistry tells us the bonding is best described by molecular orbitals.  A more comprehensive ligand field theory takes into account the effect of the donor electrons, orbitals and energies on the d orbitals of the transition metal ion and combines atomic orbitals on the metal with those of the ligands to produce molecular orbitals (MOs), which better describes bonding.

Take the simple case of a covalent bond between two singly occupied adjacent p atomic orbitals the two carbons of ethylene (C2H4).  Simplistically, you could image two carbon atoms with its two p orbitals approaching each other.  The two p orbitals (or more accurately) their wave function could combine constructively or destructively to form two new molecular orbitals (MO).   One is a pi bonding MO, π, which is lower in energy (promoting bond formation) that the atomic p orbitals.  The other is an pi antibonding MO, π*, which is higher in energy (antagonizing bond formation), as shown in Figure $$\PageIndex{9}$$ below.  Two atomic orbitals form two MOs!

Figure $$\PageIndex{9}$$: Formation of pi molecular orbtials for ethene

Let's use ligand field theory with its MOs to describe bonding of the ligands to the heme Fe2+ d orbitals.  This is illustrated in the MO diagram in Figure $$\PageIndex{10}$$ below.

Figure $$\PageIndex{10}$$: Ligand field molecular orbtials for the d6 Fe2+ ion for two axial ligands

For simplicity, for Fe2+  we will consider only d orbitals and focus on the dz2 and dx2-y2 (also called the eg) orbitals, since these are the ones that are most affect by the ligands as described in crystal field theory. Let's assume these interact with lone pairs (a simplistic assumption as well) in two ligand orbitals on the dioxygen and proximal histidine imidazole N.  From 7 atomic orbitals (5 Fe d2+ orbitals and 2 ligand orbitals), we produce 7 MOs.  Since the orbitals (in sp2- or sp3-like orbitals) on the ligands are closer in energy to the lower energy bonding MOs, their electrons would go there.  Since O2 interacts strongly with the Fe2+ d orbitals, the system is in a low spin state, with the unoccupied dz2 and dx2-y2 (eg orbitals) now considered antibonding orbitals.  The fully occupied dxy, dxz and dyz are considered nonbonding orbitals since they have the same relative energy atomic orbitals.

We will return to molecular orbitals occasionally throughout this book when they offer the best explanation for biological events.

Here are some important things to note.  When dioxygen binds to the heme iron, the oxidation state of the Fe2+ ion does not change, even though dioxygen is a great oxidizing agent. Hence the Fe2+ ion is a reversible carrier of dioxygen not of electrons. Free heme in solution is oxidized by dioxygen, forming a complex with water which occupies the 6th position, with the iron in the Fe3+ state. An intermediate in this process is the formation of a dimer of 2 hemes linked by 1 dioxygen. This can't occur readily when the heme is in Hb or Mb. Other heme proteins (like Cytochrome C) are designed to be carriers of electrons.  A small amount of the Fe2+ ion can get oxidized to Fe3+ ion. Myoglobin and hemoglobin in this state are called met-Hb and met-Mb. A enzyme is required to reduced the iron back to the Fe2+ state.

The difference between hemoglobin and myoglobin are also revealing.  Hemoglobin is a heterotetramer of two α and two β subunits held together by noncovalent interactions (an example of quarternary protein structure), with 4 bound hemes, each of which can bind a dioxygen. In a fetus, two other subunits make up hemoglobin (two zeta - ζ and two epsilon - ε subunits -analogous to the two α and two β subunits, respectively). This changes in development to two α and two γ subunits. Fetal hemoglobin has higher affinity for dioxygen than adult hemoglobin . Myoglobin is a single polypeptide chains which has higher affinity for dioxygen than hemoglobin.

The α and β chains of hemoglobin are similar to that of myoglobin, which is unexpected since only 24 of 141 residues in the α and β chains of Hb are identical to amino acids in myoglobin. This suggests that different sequences can fold to similar structures. The globin fold of myoglobin and each chain of hemoglobin is common to vertebrates and must be nature's design for dioxygen carriers. A comparison of the sequence of hemoglobin from 60 species show much variability of amino acids, with only 9 identical amino founds. These must be important for structure/function. All internal changes are conservative (e.g. changing a nonpolar for a nonpolar amino acid). Not even prolines are conserved, suggesting there are different ways to break helices. The two active site histidine are conserved, as is glycine B6 (required for a reverse turn). http://www.umass.edu/molvis/tutorials/hemoglobin/

### Normal and Cooperative Binding of Dioxygen - Structural Analyses

Plots of Y (fractional saturation) vs L (pO2) are hyperbolic for Mb, but sigmoidal for Hb, suggesting cooperative binding of oxygen to Hb (binding of the first oxygen facilitates binding of second, etc). Figure $$\PageIndex{11}$$ below shows fractional saturation (Y) binding curves vs dioxygen concentration (PO2) for both myoglobin and hemoglobin.

Figure $$\PageIndex{11}$$: Plots of Y (fractional saturation) vs L (pO2) are hyperbolic for Mb, but sigmoidal for Hb

In another difference, the affinity of Hb, but not of Mb, for dioxygen depends on pH. This is called the Bohr effect, after the father of Neils Bohr, who discovered it.  Figure $$\PageIndex{12}$$ shows binding curves for hemoglobin in the presence of increasing and decreasing concentrations of H+ (pH) as well as for CO2 and another ligand, 2,3-disphosphoglycerate (2,3-DPG).

Figure $$\PageIndex{12}$$:

Michał Komorniczak (Poland). https://commons.wikimedia.org/wiki/F...ohr_Effect.svgCreative Commons 3.0. Attribution-ShareAlike (CC BY-SA 3.0).

Protons (decreasing pH), carbon dioxide, and bisphosphoglycerate, all allosteric ligands which bind distal to the oxygen binding sites on the heme, shift the binding curves of Hb for oxygen to the right, lowering the apparent affinity of Hb for oxygen. The same effects do not occur for Mb. These ligands regulate the binding of dioxygen to Hb.

From these clues we wish to understand the molecular and mathematical bases for the sigmoidal binding curves, and binding that can be so exquisitely regulated by allosteric ligands. The two obvious features that differ between Mb and Hb are the tetrameric nature of Hb and its multiple (4) binding sites for oxygen. Regulation of dioxygen binding is associated with conformational changes in hemoglobin.

Based on crystallographic structures, two main conformational states appear to exists for Hb, the deoxy (or T - taut) state, and the oxy (or R -relaxed) state. The major shift in conformation occurs at the alpha-beta interface, where contacts with helices C and G and the FG corner are shifted on oxygenation. Figure $$\PageIndex{13}$$ shows conformation changes on dioxygen binding to Deoxy-Hemoglobin (files aligned with DeepView, displayed with Pymol). Dioxygen is shown as red spheres.

Figure $$\PageIndex{13}$$:  Conformational changes in deoxyhemoglobin on binding dioxygen (red spheres)

The deoxy or T form is stablized by 8 salt bridges which are broken in the transition to the oxy or R state. This is ilustrated in Figure $$\PageIndex{14}$$ below.

Figure $$\PageIndex{14}$$:: Salt Bridges in Deoxy Hb

6 of the salt bridges are between different subunits (as expected from the above analysis), with 4 of those involving the C- or N- terminus.

In addition, crucial H-bonds between Tyr 140 (alpha chain) or 145 (on the beta chain) and the carbonyl O of Val 93 (alpha chain) or 98 (beta chain) are broken. Crystal structures of oxy and deoxy Hb show that the major conformational shift occurs at the interface between the α and  subunits. When the heme Fe binds oxygen it is pulled into the plan of the heme ring, a shift of about 0.2 nm. This small shift leads to larger conformational changes since the subunits are packed so tightly that compensatory changes in their arrangement must occur. The proximal His (coordinated to the Fe) is pulled toward the heme, which causes the F helix to shift, causing a change in the FG corner (the sequence separating the F and G helices) at the alpha-beta interface as well as the C and G helices at the interface, which all slide past each other to the oxy-or R conformation.

Decreasing pH shifts the oxygen binding curves to the right (to decreased oxygen affinity). Increased [proton] will cause protonation of basic side chains. In the pH range for the Bohr effect, the mostly likely side chain to get protonated is His (pKa around 6), which then becomes charged. The mostly likely candidate for protonation is His 146 (on the βchain - HC3) which can then form a salt bridge with Asp 94 of the β(FG1) chain. This salt bridge stabilizes the positive charge on the His and raises its pKa compared to the oxyHb state. Carbon dioxide binds covalently to the N terminus to form a negatively charge carbamate which forms a salt bridge with Arg 141 on the alpha chain. BPG, a strongly negatively charged ligand, binds in a pocket lined with Lys 82, His 2, and His 143 (all on the beta chain). It fits into a cavity present between the β subunits of the Hb tetramer in the T state. Notice all these allosteric effectors lead to the formation of more salt bridges which stabilize the T or deoxy state. The central cavity where BPG binds between the β subunits become much smaller on oxygen binding and the shift to the oxy or R state. Hence BPG is extruded from the cavity.

The binding of H+ and CO2 helps shift the equilibrium to deoxyHb which facilitates dumping of oxygen to the tissue. It is in respiring tissues that CO2 and H+ levels are high. CO2 is produced from the oxidation of glucose through glycolysis and the Krebs cycle. In addition, high levels of CO2 increase H+ levels through the following equilibrium:

\mathrm{H}_{2} \mathrm{O}+\mathrm{CO}_{2} \leftrightarrow \mathrm{H}_{2} \mathrm{CO}_{3} \leftrightarrow \mathrm{H}^{+}+\mathrm{HCO}_{3}{ }^{-}

In addition, H+ increases due to production of weak acids such as pyruvic acid in glycolysis .

Hemoglobin, by binding CO2 and H+, in addition to O2, serves an additional function: it removes excess CO2 and H+ from the tissues where they build up. When deoxyHb with bound H+ and CO2 reaches the lungs, they leave as O2 builds and deoxyHb is converted to oxyHb.

### Mathematical Analysis of Cooperative Binding

How does the sigmoidal binding curve for Hb arise. Mathematics can offer clues that complement and extend structural information. At least three models (Hill, MWC, and KNF) have been developed that give rise to sigmoidal binding curves. Remember, sigmoidal curves imply cooperative binding of oxygen to hemoglobin.  As oxygen binds, the next oxygen seems to bind with higher affinity (lower KD). We will discuss the mathematics behind two of the models.  Both models are routinely applied to binding phenomena that give sigmoidal curves.

Previously we have shown that the binding of oxygen to myoglobin can be described by a chemical and mathematical equations.

\mathrm{M}+\mathrm{L} \leftrightarrow \mathrm{ML}

$Y = \dfrac{L}{K_D + L}$

The mathemamtical equations is that of a hyperbola where Y is fractional saturation.  Let's now explore two models that give sigmodial curves.

#### Hill Model

In this model, we base our mathematical analysis on the fact that the stoichiometry of binding is not one to one, but rather 4 to 1: Perhaps a more useful equation to express the equilibrium would be M + 4L ↔ ML4. For this equilibrium, we can derive an equation analogous to the equation 1 above. This equation is:

$Y = \dfrac{L^4}{K_D + L^4}$

For any given L and KD, a corresponding Y can be calculated. Using this equation, the plot of Y vs L is not hyperbolic but sigmoidal (see next link below). Hence we're getting closer to modeling that actual data. However, there is one problem. This sigmoidal curve does not give a great fit to the actual oxygen binding curve for Hb. Maybe a better fit can be achieved by altering the exponents in equation 2. A more general equation for binding might be M + nL ↔ MLn, which gives the following equation:

$Y = \dfrac{L^n}{K_D + L^n}$

If n is set to 2.8, the theoretical curve of Y vs L gives the best but still not perfect fit to the experimental data. It must seem arbitrary to change the exponent which seems to reflect the stoichiometry of binding. What molecular interpretation could you give to 2.8?

Consider another meaning of the equilibrium described above: M + 4L ↔ ML4.

One interpretation of this is that all 4 oxygens bind at once to Hb. Or, alternatively, the first one binds with some low affinity, which through associated conformational changes changes the remaining 3 sites to very high affinity sites which immediately bind oxygen if the oxygen concentration is high enough. This model implies what is described as infinite cooperative binding of oxygen.

(Notice that this equation becomes: Y = L/[KD + L], when n =1 (as in the case with myoglobin, and in any equilibrium expression of the form M+L↔ML. Remember plots of ML vs L or Y vs L gives hyperbolas, with KD = L at Y = 0.5.)

Does KD = L at Y = 0.5? The oxygen concentration at which Y = 0.5 is defined as P50. We can substitute this value into equation 3 which gives an operational definition of KD in terms of P50.

\begin{aligned} \mathrm{Y}=0.5=\frac{\mathrm{P}_{50}^{\mathrm{n}}}{\mathrm{K}_{\mathrm{D}}+\mathrm{P}_{50}^{\mathrm{n}}} & \text { multiply both sides by } 2 \\ 1=& \frac{2 \mathrm{P}_{50}^{\mathrm{n}}}{\mathrm{K}_{\mathrm{D}}+\mathrm{P}_{50}^{\mathrm{n}}} \\ \mathrm{K}_{\mathrm{D}}+\mathrm{P}_{50}^{\mathrm{n}}=2 \mathrm{P}_{50}^{\mathrm{n}} \end{aligned}

$\mathrm{K}_{\mathrm{D}}=\mathrm{P}_{50}^{\mathrm{n}}$

Note that for this equation, KD is not the ligand concentration at half-saturation as we saw in the case with hyperbolic binding curves.

Use the sliders in the interactive graph below to explore the effect of changes in KD and n on fractional saturation.

#### MWC Symmetry Model

In the model MWC (Monod, Wyman, and Changeux) model, in the absence of ligand (oxygen), hemoglobin is assumed to exists in two distinct conformations, the T state (equivalent to the crystal structure of deoxyHb) and the R state (equivalent to the crystal structure of oxyHb without the oxygen). In the absence of dioxygen, the T state (T0) is greatly favored over the unliganded R state (R0) at equilibrium. An constant (somewhat equivalent to a dissociation constant) can be defined.

\mathrm{L}=\mathrm{T}_{0} / \mathrm{R}_{0}

(Note: L is not the ligand concentration so don't get confused.) In addition, let us assume that hemoglobin can not exist with some of the monomers in the tetramer in the T state while others in the same tetramer are in the R state. Hence this model is often called the symmetry model. Finally lets assume that each oxygen can bind to either the T or R state with the dissociation constants KT and KR respectively. These constants do not depend on the number of oxygens already bound to the tetramer. Hence

\mathrm{K}_{\mathrm{R}}=\frac{\left[\mathrm{R}_{0}\right][\mathrm{S}]}{\left[\mathrm{R}_{1}\right]}=\frac{\left[\mathrm{R}_{1}\right][\mathrm{S}]}{\left[\mathrm{R}_{2}\right]}=\ldots \frac{\left[\mathrm{R}_{\mathrm{n}}\right][\mathrm{S}]}{\left[\mathrm{R}_{\mathrm{n}+1}\right]}

and

\mathrm{K}_{\mathrm{T}}=\frac{\left[\mathrm{T}_{0}\right][\mathrm{S}]}{\left[\mathrm{T}_{1}\right]}=\frac{\left[\mathrm{T}_{1}\right][\mathrm{S}]}{\left[\mathrm{T}_{2}\right]}=\ldots \frac{\left[\mathrm{T}_{\mathrm{n}}\right][\mathrm{S}]}{\left[\mathrm{T}_{\mathrm{n}+1}\right]}

where the subscript on R and T refers to the number of dioxygens bound to that form of R or T.  A cartoon representation of the T and R forms and accompanying dioxgen binding is shown in Figure $$\PageIndex{15}$$ below.

Figure $$\PageIndex{15}$$:  MWC model for cooperative binding of dioxygen to hemoglobin.

Now define two new parameters:

\alpha=\frac{\mathrm{pO}_{2}}{\mathrm{~K}_{\mathrm{R}}}=\frac{[\mathrm{S}]}{\mathrm{K}_{\mathrm{R}}}

where α is really a normalized ligand concentration describing how many times the KR the ligand concentration is, and

\mathrm{c}=\frac{\mathrm{K}_{\mathrm{R}}}{\mathrm{K}_{\mathrm{T}}}

the ratio of the dissociations constants for the R and T forms.

If oxygen binds preferentially to the R form of hemoglobin, c would be a small fractional number. In the limiting case, when oxygen didn't bind to the T form, KT would be infinite, and c =  0.

Using these definitions and equations, the following equation showing Y, fractional saturation vs α can be derived, with n, the number of binding sites per molecule, = 4 for Hb.

\mathrm{Y}=\frac{\alpha(1-\alpha)^{\mathrm{n}-1}+\operatorname{Lc} \alpha(1+\mathrm{c} \alpha)^{\mathrm{n}-1}}{(1+\alpha)^{\mathrm{n}}+\mathrm{L}(1+\mathrm{c} \alpha)^{\mathrm{n}}}

Figure $$\PageIndex{16}$$ below shows how fractional saturation (Y) vs alpha varies with L and c for the MWC model.

Figure $$\PageIndex{16}$$: fractional saturation (Y) vs alpha varies with L and c for the MWC model

When L is set at 9000 and c = 0.014, the Y vs α curve fits the experimental oxygen binding data well. Figure $$\PageIndex{17}$$ below shows the best experimental dioxygen binding data that we could find (obtained from a graph, not from a table),  the best fit of the Y vs L data using a Hill coefficient of n=2.8 (fitting equation 3 above), and the best fit of Y vs L using the MWC model, with L=9000, c=0.014, and Kr = 2.8 torr.

Figure $$\PageIndex{17}$$: Hb binding curves: Experimental vs Theoretical Hill and MWC Equations

Use the sliders in the interactive graph below to explore the effect of changes in L and c on fractional saturation.

Another way to think about the MWC Model

The MWC model assumes that oxygen binds to either the T or R form of Hb in a noncooperative fashion. That is KT and KR are constant, independent of the number of oxygens bound to that form. If that is so, what is the basis of the cooperative oxygen binding curves? The answer can be seen below. The cyan curve might reflect the binding of a ligand to the T form of a macromolecule, with KD = 100 uM (low affinity), for example. Notice that the binding curve looks linear but it is actual just the first "linear" part of a hyperbola.. Likewise, the magenta curve reflect the binding of a ligand to the R form of the macromolecule with KD = 10 uM. If the T and R form are linked through the T↔ R equilibrium, this equilibrium will be shifted to the tighter binding (lower KD) R form with increasing ligand concentration, assuming that the ligand binds preferentially to the R form. This shifts the actual binding curve from that resembling the T form at low ligand (cyan) to one resembling the R form (magenta) as ligand increases, imparting sigmoidal characteristics to the "observed" binding curve (gray).  Figure $$\PageIndex{18}$$ below shows how sigmoidal binding curve could arise from a switch from a low affinity to high affinity form.

Figure $$\PageIndex{18}$$: Cooperativity as transition between two hyperbolic binding curves with different Kd's

#### KNF Sequential Model

This KNF (Koshland, Nemethy, and Filmer) Sequential Model:model was developed to address concerns with the concerted model. One of the major problems with the concerted model is that it seemed unrealistic to expect all the subunits to change conformation together. Why shouldn't there be some difference in subunit conformation? The KNF model also fits the experimental data well.  Figure $$\PageIndex{19}$$ belows shown the linked equilibria in the KNF model.

Figure $$\PageIndex{19}$$: KNF model for cooperative binding

### Hemoglobin variants and disease

Many diseases have been associated with alternations in the amino acid sequence of hemoglobin. Around 300,000 babies are born each year with a genetic disorder causing an aberrant hemoglobin structure.  Over 80% of these are in low/middle income countries.  The worst is sickle cell anemia followed by β-thalassaemias and the less serious α-thalassaemias.  We will only focus on sickle cell disease. In this disease, red blood cells become distorted in share from a normal smooth discoid shape to a crescent-like shape under low oxygen concentrations found in capillaries.  These impede blood flow as shown in Figure $$\PageIndex{20}$$ and lead to symptoms ranging from anemia, episodic pain, swelling of hands and feet, and vision problem.  Complication often lead to premature death.

Figure $$\PageIndex{20}$$: Sickeling of red blood cells in sickle cell diseaes.  https://commons.wikimedia.org/wiki/File:Sickle_Cell_Disease_(27249799083).jpg.  National Human Genome Research Institute (NHGRI) from Bethesda, MD, USA.  Creative Commons Attribution 2.0 Generic license

Linus Paul and colleagues showed that the isoelectric point (pI) of oxy-Hb and deoxy-Hb in normal blood was 6.87 and 6.68, respectively, but where higher (7.09 and 6.91, respectively) in sickle cell disease.  This was the first evidence that a disease was cause by a molecular alteration of a protein.  Eventually we learned that a single negatively charged amino acid, Glu 6, on the β-chain of hemoglobin was mutated to nonpolar one, valine. This causes hemoglobin, which is present at a concentration of 150 g/L in blood to self-aggregate into a long polymer.  This distorts the cell in a sickle shape.  Humans have two genes for the beta chain of hemoglobin, one from the egg donor and the ohter from the sperm donor.  If only one mutated, the disease is called sickle cell trait. If both are mutated, sickle cell disease is observed.

The hydrophobic Val 6 in on the surface of both beta chains in sickle cell disease.  It can bind to a hydrophobic patch comprised of Ala 70, Phe 85 and Leu 88 on another β-chain on another hemglobin tetramer.  Hence hemglobin has two Val 6s and two hydrophobic patches, allowing first the formation of a sickle cell hemoglobin "dimer" of tetramers, followed by elongation of the growing fibril.  This is a disease of aberrant induced dipole-induced dipole interactions and the "hydrophobic effect".

Figure $$\PageIndex{21}$$ below shows a "dimer" aggregate of two hemoglobin S tetramers, in which each β-chain has the D6V mutation.  α chains are shown in grey, β chains in cyan,  Val 6 in red spacefill and the surface hydrophobic patch of A70, F85 and L88 in orange spacefill (1hbs).   Note the binding of the two tetramers is mediated by the interaction of the red Val 6 on the right tetramer with the orange surface hydrophobic patch on the left tetramer.  Note also that there are the "dimer" aggregate has three more exposed Val 6 (red sphere) and three more hydrophobic patches which would allow extension of the dimer into long fibril-like polymers in which binding is mediated by noncovalent interactions.

Figure $$\PageIndex{21}$$: Dimer of hemoglobin S tetramers showing the interaction of Val 6 (red sphere) with a surface hydrophobic path (orange) on a second tetramer.

Figure $$\PageIndex{22}$$ below shows an interactive iCn3D model of one tetramer of hemoglobin S.  As in Figure $$\PageIndex{21}$$ above, the α chains are shown in grey, the β chains are shown in cyan, and Val 6s are shown in red spacefill.  The surfaces of the hydrophobic pockets where the Val 6 another HbS tetramer binds, comprised of amino acids A70, F85 and L88, are shown in orange.

Figure $$\PageIndex{21}$$: Hemoglobin S tetramer (1HBS) (Copyright; author via source). Click the image for a popup or use this external link:    https://structure.ncbi.nlm.nih.gov/i...APyWAsDLnbAeR7

Sickle cell disease and trait are endemic in Sub-Saharan Africa, where over 4 million have the disease and over 40 million have the trait.  Its geographic distribution along with that of malaria is shown in Figure $$\PageIndex{22}$$ below.

Figure $$\PageIndex{22}$$:  Maps showing distribution of sickle cell trat and malaria.  https://commons.wikimedia.org/wiki/F...stribution.jpg.    CC-LAYOUT; CC-BY-SA-2.5,2.0,1.

The Plasmodium parasite reproduces in red bloods cells.  There ability to reproduced is compromised as red blood cells with sickle cell hemoglobin rupture more frequently.  Also the parasite uses hemoglobin as a source of amino acids.  The endocytose it and hydrolyze it to amino acids in digestive organelles. Sickle cell hemoglobin is more resistant to this process.   Hence evolution appears to have maintained the sickle cell gene in these areas as protection against malaria.

As sickle cell disease is a systemic problem, treatment of not just the cause but the multitude of symptoms is important.  Some of these are described in Figure $$\PageIndex{23}$$ below.

Figure $$\PageIndex{23}$$:  Treatment of sickle cell disease. Cisternos et al. Front. Physiol., 20 May 2020 |   https://doi.org/10.3389/fphys.2020.00435.  Creative Commons Attribution License (CC BY)

Of course the cure would be to use DNA editing to change the base pair for the mutated Val 6 gene back to the wild-type Glu 6 as shown in Figure $$\PageIndex{24}$$ below. The amino acid sequence encoded by the DNA and RNA shown below is Pro-Glu-Glu for the normal and Pro-Val-Glu for the sickle cell chain.

Figure $$\PageIndex{24}$$: Mutation in sickle cell disease.  https://www.biologycorner.com/worksh...icklecell.html.  Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

Crispr gene editing trials are now underway to attempt a cure of this dreadful diease.