Monod-Wyman-Changeux (MWC): a molecular model of co-operativity.
Hill: Great but not so great.
Last class we went through the Hill model for co-operativity. In one way the Hill model
was very satisfying. Using this model we were able to make direct quantitative
predictions about the minimum number of ligands that bind to a receptor to produce a
binding curve that transitions between the full unbound form and the fully bound form
with a certain abruptness. I did not show this in class, but the model can be used to
describe the experimental behavior of many cooperative molecules with only two
adjustable parameters (K and n). We also saw that we can get to the Hill model from the
general binding polynomial by assuming that the macroscopic binding constants
dramatically decrease with each binding event.
On the other hand the Hill Model is clearly unrealistic from a molecular perspective. Hill
says that all ligands jump onto their binding sites simultaneously. In other words the
model predicts that the binding constant for the second, third and fourth ligand is
infinitely large once the first ligand is bound. This seems completely unreasonable from a
molecular perspective.
So while the Hill model is useful as a mathematical model to determine how co-operative
a receptor-ligand system is, it is pretty clear that it does not describe what is “really
happening” at the molecular level.
Molecular models of co-operativity
The challenge for any molecular model of positive co-operativity is to provide a
mechanism that generates binding sites with a higher effective affinity as a result of
binding a ligand to a binding site of lower apparent affinity. I.e. we need to generate the
high-affinity binding site in situ. If the high-affinity binding site was present from the
beginning our ligands would bind to this site first, generating a system that exhibits
binding behavior associated with negative cooperativity.
To understand what I mean by generating a high-affinity binding site in situ lets think of
some very simple examples of co-operativity. One such example is a nucleated
polymerization reaction. In these cases the first ligand is a physical part of the “binding
site” for the next ligand.
Think of a simple model of protein crystallization where the proteins are cubes that grow
on a cubic lattice. If the first two proteins associate, they only have one interaction
surface. The third momomer still makes only one contact, but the fourth makes two
contacts as a result the fourth monomer will bind more tightly than the first two
monomers. K3 is greater than both K1 and K2.
As we go to higher dimensions and add more and more monomers, the average number
of interaction surfaces per monomer goes up. This ultimately comes down to the fact that
the surface area of a cube growth slower (square dependence on cube dimensions) than
the volume (cubic dependence).
Similar mechanisms of co-operativity occur during the formation of helical filaments, or
the formation of a protein alpha helix. Once we completed the first turn of the helix and
we continue to add more residues/monomers, we can now make stabilizing interactions
with the residues/monomers that made up the first turn, thus increasing the affinity
between the new residue/monomer and the helix.
In these cases a “ligand”, once bound, forms a physical part of the binding site for
subsequent ligands. While such a mechanism is very elegant, I do not know of any
examples where such a model is utilized by a biological regulatory system. In none of the
cooperative binding proteins that I know of do the binding sites for the different ligands
physically overlap. Instead the binding sites are usually physically well separated, hence
the term allostery (other space).
If the ligands do not form a physical part of the binding site for subsequent ligands, then
we can only achieve lower apparent affinities, if binding of the first ligand causes some
form of structural change which switches the low binding form into the higher binding
form.
Lets see what a general model for such a binding scheme including structural changes
that switches the binding constants would look like.
First of all we would have five different states of our receptor in the complete absence of
a ligand. This would give us four structural equilibrium constants. For each ligand bound
we would have 4 more states.
So we end up with a square of 5x5 states and 52 equilibrium constants. This is nice as a
model, but it makes it very hard to grasp what is going on. Also, our binding curves are
pretty feature less, it would be very difficult to fit 52 parameters to them with any degree
of confidence.
If however we could fit all these binding constants we would be able to calculate how the
population of molecules changes as a function of the free ligand concentration.
This is sort of similar to the glycine example we had, where the conversion form the
doubly protonated form to the doubly deprotonated form went through the zwitter ion
form and not the neutral form.
There are two extreme “pathways” , through which a molecule with four ligand binding
sites can move from the completely non-bound form to the fully saturated state, as we
know it must.
The sequential model (induced fit)
This is how one might intuitively think about cooperativity. Lets say we have a receptor
that is made up of four identical monomers. Each monomer has a binding site and in the
absence of a ligand all monomers have the same structure. When the first ligand binds,
this binding event induces a structural change in the subunit it bound to. Somehow, this
structural change has the effect of increasing the binding constant for ligand binding to
the remaining 3 free binding sites. Binding of the second ligand then leads to a further
conformational change that again increases the affinity of the remaining free binding sites
and so on.
This sequential model, as reasonable as it seems, has serious problems. To make such a
system work, each monomer has to be able to switch between four different structural
states of increasing affinity. In each of these states the monomer has to be able to hold on
to its neighbors and also has to be able to transmit a signal to these neighbors should a
binding event occur. Doing all these things at the same time turns our to be an extremely
difficult thing to do for a molecule and there are very few molecular systems that seem to
employ this sequential / induced fit model.
The Monod Wyman Changeux (MWC) model of cooperativity
The MWC scheme is very simple to implement at the molecular level and there are many
natural molecular systems that employ this scheme. While the model is easy to
implement on the molecular level, understanding how this model leads to cooperative
behavior is rather tricky.
Here are the five assumptions that define the MWC model:
1) Identical subunits occupy equivalent positions in a protein. The contacts and
environment of each of these subunits is identical.
2) Each subunit contains a unique receptor site for a ligand.
3) At least two conformational states are reversibly accessible for the protein and the
microscopic binding constant for the two states differ. (To keep things simple we
will assume that only one of the two forms can bind ligands i.e. the binding
constant for the R state is infinitely small)
4) The conformational switching between these two states is concerted; the subunits
are either all in one conformation, or they are all in the other conformation i.e.
there are only two types of interfaces.
5) The microscopic binding constant for the ligand depends only on the
conformational state of the protein, but not on the binding state of the other
subunits.
You should be suspicious about how this can work. After all we just learned that the way
to get cooperativity is by having successively increasing binding constants for
consecutive binding events. And here these guys come and tell us that they can give us
cooperative behavior with binding constants that are independent of the ligation state of
the protein.
The model
First lets look at the model then we can work through the math and see if we get the
expected results. Finally we can then try to understand why the MWC model gives rise to
cooperative behavior. Our system has five equilibria. The first is between the R state
and the T state. R stands for relaxed and T for tense,
but you could give them any other name if you want.
The other four equilibira are between the ligand
bound forms. As the model states, the microscopic
binding constant k between each of the monomers
and the ligand is identical. However we have to
apply statistical prefactors. For example there are 3
open binding sites on TX. So for each receptor we
have 3 chances to go from TX to TX2, but there are
only two ligand-bound monomers in TX2 giving us
only 2 ways to go from TX2 back to TX. To correct
for this we have to apply the statistical prefactor of
3/2 to the binding constant k of the individual
monomer to obtain the equilibrium constant K2= 3/2
k = [TX2]/([TX][X]). After writing down our
reaction scheme and identifying the statistical
prefactors that relate our equilibrium constants to the
microscopic binding constant k we can now write
down all our equilibria. The goal as always is to
write end up with an equation that relates the
fraction receptor
sites.
[R]
L=
" [R] = [T]L
[T]
[TX]
4k =
" [TX] = [T]4k[X]
[T][X]
3
[TX 2 ]
3
3
k=
" [TX 2 ] = k[TX][X] = [T]# # 4 # k 2 [X]2 = [T]# 6 # k 2 [X]2
2
[TX][X]
2
2
2
[TX 3 ]
2
2 3
k=
" [TX 3 ] = k[TX 2 ][X] = [T]# # # 4 # k 3 [X]3 = [T]# 4 # k 3 [X]3
3
[TX 2 ][X]
3
3 2
1
[TX 4 ]
1
1 2 3
k=
" [TX 4 ] = k[TX 3 ][X] = [T]# # # # 4 # k 4 [X]4 = [T]# k 4 [X]4
4
[TX 3 ][X]
4
4 3 2
now lets write down the
!
"=
!
ligand sites occupied
[TX] + 2[TX 2 ] + 3[TX 3 ] + 4[TX 4 ]
=
total # of receptors [R] + [T] + [TX] + [TX 2 ] + [TX 3 ] + [TX 4 ]
the goal as always is to express our average number of ligands bound in terms of only the
free ligand concentration and the binding and conformation equilibrium constants. Once
again we do this by substituting our [TXi] by the appropriate expressions we derived
above.
"=
ligand sites occupied
[T]4k[X] + 2[T]# 6 # k 2 [X]2 + 3[T]# 4 # k 3 [X]3 + 4[T]# k 4 [X]4
=
total # of receptors [T]L + [T] + 4k[T][X] + [T]# 6 # k 2 [X]2 + [T]# 4 # k 3 [X]3 + [T]# k 4 [X]4
divide by [T]
" =4
k[X] + 3# k 2 [X]2 + 3# k 3 [X]3 + k 4 [X]4
L + 1+ 4 # k[X] + 6 # k 2 [X]2 + 4 # k 3 [X]3 + #k 4 [X]4
k[X](1+ 3# k[X] + 3# (k[X]) 2 + (k[X]) 3 )
k[X](1+ k[X]) 3
" =4
=4
L + 1+ 4 # k[X] + 6 # (k[X]) 2 + 4 # (k[X]) 3 + (k[X]) 4
L + (1+ k[X]) 4
!
Looking at the MWC binding curve
Lets see if this formula makes sense. As we always do lets play around with the numbers
and see if our result makes sense.
1) L=0 The first thing we want to check is the behavior if L=0, that is to say the
equilibrium between the R and the T state is entirely on the side of the T state, as if the R
state never really existed. This then is the case of the molecule with four independent
binding sites. We went through this model before and found that for this case the
behavior should that of a Langmuir times the number of binding sites on our molecule.
Lets look at the formula and indeed if we set L=0 then our formula turns into
4k[X]/1+k[X]
2) [X] is really large we get 4
Here is how the MWC model works. The first step of the binding reaction
Keff for the first binding step is
We have treated this one before. In our treatment of the Langmuir linked to a
conformational equilibrium.
The only difference here is how we define L (i.e. La =1/Lb). The result we got from this
was a Langmuir with a Kapp that was lower than the intrinsic/microscopic binding
constant of the binding site.
Here with the L defined as L =
K
1+
" (x) =
!
# [X]
K[X]
=
! =
1
1+ + K[X] 1+
# [X]
1
L
1+
L
defining K app =
" (x) =
1
L
K
[B0 ]
[A]
K
1
1+
L
we get
K app # [X]
[FX]
=
Pr otein total 1+ K app # [X]
So the only effect of adding the preceding equilibrium was to lower the apparent binding
constant for the first binding step. The reason for this was, that we needed to “pull” the
equilibrium across not only from the free to the bound form, but we also had to convert
the unfolded form to the folded form. So part of the price of binding a ligand was the
energetic cost for converting the unfolded form to the folded form.
The same holds for the first binding step in the MWC model. To get to the bound form,
we have to pay the energetic price for converting a molecule of the non-binding form A
into the binding competent form B.
If we now bind the second ligand, we already paid this price for converting A to B, so the
average binding energy (the energy released during binding) for binding two ligands is
much higher than the energy of binding the first one. Consequently our apparent
equilibrium constant becomes lower. For a third and a fourth ligand the fraction of the
energy of converting A to B becomes even lower.
So as a result our apparent binding constant is becoming lower and lower for additional
binding events, just the way we expect for a cooperative model.
Closing remarks
Based on binding curves ,which have a tendency to be noisy, it is almost impossible to
distinguish between the different possible models or to uniquely define the pathway the
molecule takes through our binding square. So we need to use additional experimental
data. For example, the sequential binding model predicts the presence of two squares and
two circles. The MWC model on the other hand predicts that such states are virtually
never populated. So one could use structural methods and hunt for such intermediates.
NMR spectroscopy is particularly good at picking out the presence of low concentrations
of alternative structural states. If MWC is correct, all we should see as a result of ligand
bininding is an increase in the relative population of the two states. In the sequential
model on the other hand we would see four distinct structural states for the overall
molecule.