Hemoglobin
Cooperativity and Binding

It is often the case that biochemical systems require tighter control than can be achieved with the sort of simple affector system that is based on just a single linear binding event. Often one needs a response more like a switch -- little effect at low concentration then a rapid rise to maximal activity. A classic example has to do with the protein hemoglobin which binds oxygen in your lungs and carries it to other tissues of your body. The problem is simply this, in the lungs you want hemoglobin to bind oxygen tightly so that a large fraction of the hemoglobin will pick up an oxygen before cycling back through the other tissues. However, in the other tissues, one wants hemoglobin to bind oxygen loosely so that it can be easily relinquished for use in respiration. How do we control the binding? Obviously the oxygen concentration in the blood near the lungs is going to be higher than it is in tissues were oxygen is rapidly being used up. But the difference is not large enough to cause nearly complete binding in the lungs and nearly complete oxygen release in the other tissues. A curve describing the fraction of hemoglobin molecules bound to oxygen as a function of the oxygen concentration is calculated assuming the sort of simple equilibrium binding model we discussed before is as follows:

where Kd is the dissociation constant, [Hb] is the free hemoglobin concentration, and [Hb]t is the total hemoglobin concentration. The final equation should look very familiar to you. The dissociation constant, Kd, is just a backwards equilibrium constant.

Now if this was the whole picture, a plot of the fraction of Hb with oxygen bound to it would look like the solid line in Fig. 1 (see below -- this is just a plot of the equation above after dividing both sides by the total amount of Hb). What we can see is that the curve is not very sharp. That is, it takes a relatively large change in the oxygen concentration in the blood to cause a significant uptake or release of oxygen from Hb. We would like a finer switch, where a small change in oxygen concentration resulted in going from almost all Hb in the unbound state to almost all Hb in the bound state.
Nature has come up with an interesting solution to this problem which is called cooperativity. Hb is actually a tetramer of four subunits each of which can bind an oxygen. This binding is cooperative, that is, the binding of the first oxygen makes binding of the second more favorable which makes binding of the third even more favorable which makes binding of the fourth even more favorable. Lets draw a picture of this:

Four successive oxygen molecules are bound to hemoglobin and each time one binds the dissociation constant for the next one decreases by a factor a. Remember that a lower dissociation constant means tighter binding. What does this do to the fraction bound vs. oxygen concentration curve? Let's work through it first with a simpler case. Consider:

We want to know how many A's are bound per E (the occupancy of A on E).

Here N is the number of A's bound per enzyme. This is equal to all bound forms of A (remember there are two possible sites for A on E that are distinguishable resulting in AE and EA). The AEA term is multiplied by 2 because it contains two bound A molecules. Et is the total enzyme concentration. It equals all forms of both the free enzyme and enzyme bound to A. In this case AEA term is not multiplied by two since we are interested in the number of enzyme molecules represented in the AEA complex.

Remember that K defined in this way is a dissociation constant, the inverse of a normal equilibrium constant. It is often written as Ks or Kd to avoid confusion, but I will just refer to it as K here. Cooperativity is represented by multiplying the dissociation constant by the factor a when considering binding of the second A to E. If a=1 there is no cooperativity. If a<1 then binding of the second A is stronger than the first. If a>1 then binding of the second A is weaker than the first. We can see that

Solving for the occupancy of A on E:

Now if a=1 (no cooperativity) this reduces to

which is essentially the same as the simple equation for Hb binding to one O
2 except that it is multiplied by two because two A's can bind per E.

What the cooperativity does, however, is it makes the terms with [A]2 much more important.  For positive cooperativity, a<1 and therefore the terms with a in the denominator become larger.  This makes the curve in Fig. 1 look more like a parabola at low [A] instead of a straight line.  The result is a sigmoidal curve with a sharper dependence on the concentration of A.

If we now consider a more realistic situation with hemoglobin which has four total binding sites, the situation becomes a bit more complex. We must again remember that the four binding sites are distinguishable, and therefore we must consider all forms of singly, doubly, triply and quadruplely bound hemoglobin:

Note that there are four possible ways for one O2 to be bound, six ways for two, four ways for three and only one way for four O2's to be bound. As before, we can write the concentration of one bound, two bound, three bound or four bound species in terms of the concentration of Hb and O2 by using the dissociation constant, K, and the cooperativity factor, a:

This is more complicated than the case for two ligands binding, but you can see the pattern is the same.  The expression for N, the number of O2's bound per hemoglobin (the occupancy) is:

substituting the expressions for the different bound forms of hemoglobin into the above equation yields (after some simplification):

For a = 1 (no cooperativity) this factors into:

as one might expect for a four binding site molecule with no cooperativity. We can also consider what happens if a is significantly less than 1. In this case, only the higher order terms become significant and we have:

The resulting plot is shown as the dashed line in Fig. 1. As you can see, there is now a much smaller change in oxygen concentration required to go from mostly bound to mostly unbound. Positive cooperativity of substrate binding is a very common method for causing the control of a reaction or process to be much more sensitive to the substrate concentration.

As practice please do the following problem:

1) The repressor of lambda phage is a dimer in its active form (if you want to read about lambda repressor you can look it up in any biochem text book). This is true of a number of regulatory proteins. Consider the chemical interactions between a regulator and an operator diagrammed below (if you don't know what an operator is, it doesn't really matter, just consider it as something that R binds to):

Case 1:

Case 2:

where K, K1 and K2 are all equilibrium constants for the reactions as shown, R is the regulator which is functional as a monomer in case 1 and as a dimer in case 2, and O is the operator that the regulator binds to. For each of the two cases, derive an expression for the fraction of the operator sites that have regulator bound to them in terms only of the equilibrium constants shown and the concentration of the regulator. Remember that in case 2 only the dimer can bind to the operator. The monomer does not. To get you started, realize that the fraction of the operator with regulator bound in case 1 is just:

The definition of the fraction of the operator with dimer regulator bound in case 2 is:

Make a sketch of these two functions showing how the binding curve differs in the case of a dimer regulator from that of a monomer. Why is this difference important to the gene regulation of bacteriophage lambda?