- Question & Answer
- Open Access
© BioMed Central Ltd 2009
- Published: 16 June 2009
- Hill Equation
- Oxygen Binding
- Positive Cooperativity
- Binding Curve
- Negative Cooperativity
Cooperativity is a type of behavior where a number of seemingly independent components of a system act collectively, in unison or near-unison. Think of a school of fish, a flock of birds, or a pack of lemmings. Cooperativity implies some sort of communication among the system's seemingly independent components.
That depends on the system, but let's take hemoglobin. Hemoglobin's mission is to pick up a large amount of oxygen in the lungs, where the oxygen concentration (or partial pressure) is about 100 torr, and then drop off a good fraction of it in the peripheral tissues where the oxygen concentration is about 20 torr. Cooperativity helps make this transport efficient.
You would like to do better. Ideally, the binding curve should be higher than the green curve at 100 torr, so hemoglobin would pick up more oxygen in the lungs, and lower at 20 torr, so that hemoglobin would unload more completely. A sigmoidal curve would fit the bill, and the experimentally determined binding curve is in fact steeply sigmoidal, like the red curve shown in Figure 3. The sigmoidal shape of the oxygen binding curve helps hemoglobin to achieve a high oxygen-delivery throughput. Sigmoidal binding curves occur if binding is cooperative.
What sort of physical mechanism can give you cooperative binding and thus a sigmoidal binding curve?
where EC50 is the partial pressure of oxygen at which the binding is 50% of maximal, and n, the polynomial order of the binding reaction, is commonly referred to as the Hill coefficient.
This is the famous Hill equation, and it has many virtues. It is simple, not much more complicated than the Langmuir equation (which is equivalent to a Hill equation with n = 1). Its parameters, n and EC50, are easy to understand, empirically determinable quantities. And, perhaps most importantly, the equation fits the experimental data on hemoglobin's oxygen binding very well. Not perfectly, but very well. Moreover, whenever one encounters a sigmoidal response in biochemistry, chances are good that the Hill equation will fit the experimental data adequately.
One problem is that hemoglobin's oxygen binding is not fit by a Hill equation with a Hill coefficient of 4, but rather with a Hill coefficient of approximately 2.7 (and the red curve plotted above is, in fact, a Hill equation curve with n = 2.7). But the bigger problem, of course, is that the model's assumption that n molecules of ligand or stimulus simultaneously collide with the multimeric protein is patently unrealistic. And without this assumption, you do not get a Hill equation.
So suppose you assume the oxygens bind sequentially rather than simultaneously. What sort of curve can you come up with?
In 1966, Dan Koshland, George Némethy and David Filmer provided a simple rationalization, known as the Koshland-Némethy-Filmer (KNF) model, for these puzzling equilibrium constants. Koshland, Némethy and Filmer assumed that when the first oxygen bound to one of the hemes (with relatively poor affinity), the binding allosterically induced the globin subunits that had not yet bound oxygens to increase their affinity for oxygen. This mechanism requires a fairly complicated chain of events; the 'information' that the first oxygen has bound needs to be transmitted from one heme group out to the surface of that globin subunit; the information is relayed from that globin's surface to a neighboring globin; and then it is relayed from there to the neighboring globin's heme. Complicated or not, the conceptual framework fitted very well with Koshland's idea of induced fit as the basis of enzymatic catalysis, and it provided a credible, tangible physical picture to show why the oxygen binding curve of hemoglobin is sigmoidal.
In the KNF model, once one heme binds oxygen it becomes progressively easier for the other hemes to bind it. One heme leads and the others follow. Allosteric communication between the hemoglobin subunits allows the whole protein to behave collectively in a way that non-cooperative, truly independent subunits would not.
Could you get cooperativity without allosteric communication between the subunits of the binding protein?
Yes – you could have a multivalent ligand binding to a multi-subunit protein. The classic example is an antigen with repeated structural features, or epitopes (for example, the surface of a bacterium or a virus) interacting with the two arms of an antibody. The binding of one antigen epitope to the antibody makes the second binding event much more favorable by forcing the antigen's second epitope into close proximity of the antibody's remaining free antigen binding site. This type of cooperative interaction is often referred to as enforced proximity, or the avidity effect, and it is common in protein-protein interactions.
While the avidity effect is similar to KNF cooperativity in that one binding event promotes the next, it is different in terms of the consequences for the shape of the saturation curve. Because the first binding event promotes a zero-order second binding event (that is, it occurs within the complex rather than between the complex and a second ligand), the result here is a Langmuir/Michaelian-type binding curve, not a sigmoidal one.
No. You can have anti-cooperativity, or negative cooperativity, in which binding the first molecule makes it harder, not easier, for the second one to bind. The result is usually a binding curve that looks sort of like a Langmuir curve, but approaches maximal binding even more slowly than the Langmuir curve does.
And if the binding of one molecule of ligand has absolutely no effect on the binding of any of the others, the complicated Adair/KNF equation can be reduced to a simple Langmuir equation, and the binding is said to be non-cooperative.
Note that the KNF concepts of cooperativity (or positive cooperativity) and anti-cooperativity (or negative cooperativity) are most cleanly defined for a dimeric protein with two binding sites. If the first binding event increases the affinity of the second site, there is positive cooperativity. If the first binding event decreases the affinity of the second site, there is negative cooperativity. With a four subunit protein like hemoglobin the distinction can be a bit murkier. For example, what would you call it if the first binding event makes the second one weaker (as with negative cooperativity), which makes the third one stronger (as with positive cooperativity), and then the fourth one weaker? This sort of behavior has actually been inferred from fits of the Adair/KNF equation to (some) hemoglobin oxygen-binding datasets, and so technically you might consider the whole process to exhibit mixed positive and negative cooperativity. However, since the net effect is a sigmoidal binding curve, as with simple positive cooperativity, that is what it might as well be called.
Well, negative cooperativity is fairly common. For example, most G-protein coupled receptors probably function as dimers. For some, the binding curves are sigmoidal, indicating positive cooperativity. But for about as many, the binding curves are even more graded than Langmuir curves, indicating negative cooperativity. So negative cooperativity is common and therefore probably important.
One thought is that negative cooperativity occurs in cells when it is worth sacrificing the ability of a system to respond decisively to one particular range of ligand concentrations in favor of the ability to respond at least a little to a very wide range of concentrations. Positive cooperativity gives you a response that is decisive, but only over the limited range of ligand concentrations that correspond to the steep upslope of the binding curve. Negative cooperativity gives you a response that is less decisive but also less restricted with respect to the range of ligand concentrations.
No. The famous team of Jacques Monod, Jeffries Wyman and Jean-Pierre Changeux proposed a different model for oxygen binding by hemoglobin. They broke the binding of oxygen to hemoglobin down into four sequential steps, just as Adair and Koshland, Némethy and Filmer did. However, they assumed that the binding of the first oxygen had no effect on the affinities of the other globins. At this point there was nothing in the model that would make the binding curve different from a Langmuir curve.
Next they assumed that hemoglobin exists in two alternative conformations. They termed these conformations 'tense' (the blue states below) and 'relaxed' (the pink states). Furthermore, they assumed that if one hemoglobin monomer was relaxed, all of the hemoglobins in that complex would be, and that if one was tense, they all would be. Essentially, they replaced Hill's assumption of the simultaneous binding of four oxygens to one hemoglobin with the assumption of concerted conformation changes among the four hemoglobin monomers. This assumption seems much more reasonable. Think of four kittens sleeping cuddled up in a pile. For one kitten to shift position, perhaps all four will need to.
Like the Adair/KNF equation, the MWC equation is a ratio of two complicated nth order polynomials. And as with the Adair/KNF equation, it is possible to choose K values that yield sigmoidal curves and reproduce experimental oxygen binding data extremely well.
In the MWC model, the oxygen binding seems, at first glance, to be totally noncooperative; the binding of an oxygen to a globin within a tense hemoglobin complex is explicitly assumed to have no effect on the affinities of other globins for oxygen. And the same is true of the binding of an oxygen to a globin within a relaxed complex. Instead, the cooperativity here is embodied in the notion that the whole hemoglobin complex flips between the tense and relaxed states as a unit.
This concerted conformation change has the effect of allowing the binding of the first oxygen to indirectly promote the binding of the second, and the second to promote the binding of the third, and the third to promote the binding of the fourth. This is because the binding of each oxygen makes the flip to the tight-binding state more favorable, and the flip to the tight-binding state makes the binding of the next oxygen more favorable.
For most cooperative systems it is nearly impossible to choose between the two models simply on the basis of the shape of the binding curve – either model can usually be fitted to experimental binding data quite well. For that matter, even the Hill equation, based on a patently unrealistic physical scenario, usually fits experimental data satisfactorily. What is really needed is some other type of evidence that gets at the nature of the intermediates that are formed when the cooperative protein is partially saturated.
The most fruitful approaches in this regard have been studies on single molecules of cooperative, multimeric ion channels. For example, the nicotinic cholinergic receptor consists of five homologous subunits with two to five acetylcholine binding sites. When the receptor binds acetylcholine, it opens, allowing cations to flow through its central pore from one side of the plasma membrane to the other. In patch clamp experiments, in which ion flow through single nicotinic receptors can be monitored, one observes the channel flipping between two conductance states, consistent with an MWC-style symmetrical, concerted transition of all of the subunits between a closed and an open conformation, rather than the three or more conductance states that might be expected in a KNF-style mechanism. Moreover, crystal structures of open and closed nicotinic receptors show that the whole complex does seem to change conformation in concert.
On the other hand, there are many examples of negative cooperativity – receptors that are saturated by ligand even more gradually than a non-cooperative receptor would be. And although negative cooperativity is easy to account for with a KNF model, it cannot arise for any choice of parameters in an MWC model. Thus, both MWC and KNF types of mechanisms are probably found in nature.
Certainly. We have already mentioned a couple of examples from cell signaling: the cooperative nicotinic cholinergic receptors, and the cooperative or anti-cooperative activation of G-protein coupled receptors. These are both examples of cooperativity in signal reception. We suspect that sigmoidal responses will be at least as important in signal processing. One way of seeing why this might be is to think about how signals would propagate down a signal transduction pathway if none of the components of the pathway exhibited sigmoidal responses to their upstream activators.
And finally, suppose that the system is asked to respond to a whopping-big change in input stimulus (x), an 81-fold change. You get the largest change in A if you use the middle of the response curve, with x ranging from EC50 to 9 EC50; A then goes from 10% to 90% of its maximal activity. Thus, an 81-fold change in input stimulus has yielded a 9-fold change in output response. If you feed this 9-fold change in A into the response of B, the best you can get is to drive B from 25% to 75%. And if you drive this 3-fold change in B into the response of C, the best you can get is to drive C from 37% to 63%, a 1.7-fold change. So, in three steps, this Michaelian cascade has reduced a decisive, 81-fold change in input stimulus to a murky, gray, 1.7-fold change in output. Given that signaling pathways often contain even more than three successive signal relayers, this seemingly ineluctable descent into murkiness is a big problem.
This problem can be circumvented if some of the signaling proteins exhibit sigmoidal response curves; with a sigmoidal curve, the fold change in output can be as big as, or bigger than, the fold change in stimulus. For example, for a system whose response is given by a Hill curve with a Hill coefficient of 3, a 9-fold change in input can give you a 25-fold change in output. So sigmoidal response curves can restore or even amplify the 'contrast' of a signal propagating down a signaling pathway.
Isn't a sigmoidal binding curve inherently cooperative, irrespective of the mechanism that generates it?
In a sense, yes. With a Langmuir binding curve, every increment of ligand concentration gives you a little less binding than the previous increment did. Langmuir binding obeys the law of diminishing returns: every time a binding site is filled, it makes it a little harder for the next ligand molecule to find a binding site. However, with a sigmoidal binding curve, for a while each increment of ligand concentration results in a little more binding than the previous increment did. Regardless of what mechanism makes the curve bend upward, the upward bend itself means that the system is responding in a sort of collective, cooperative, all-or-none fashion. Or at least in a more cooperative fashion than a system with a Langmuir binding curve does.
However, there are a number of well explored mechanisms in cell signaling that can give rise to steeply sigmoidal response curves, but have nothing to do with multisubunit proteins and allosteric communication between binding sites. Probably the best examples are zero-order ultra-sensitivity, discovered by Goldbeter and Koshland in the course of theoretical studies of signaling cascades, and inhibitor ultrasensitivity, a simple stoichiometric buffering reaction. These non-cooperative mechanisms for generating sigmoidal response curves are probably at least as important as cooperativity in the overall scheme of cellular regulation. For more on this interesting topic, see the 1996 Trends in Biochemical Science paper referenced below.
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