University of Washington
Department of Chemistry
Chemistry 453
Winter Quarter 2014
Lecture 10 1/31/14
A Non-Cooperative & Fully Cooperative Binding: Scatchard & Hill Plots
• Assume N binding sitesWe have derived two equations for the extremes of
behavior:
•
Non-cooperative (i.e. Independent) Binding: ν =
•
Fully-cooperative Binding:
ν =
•
•
Nk N [ L ]
N
1 + k N [ L]
N
=
NK [ L ]
Nk [ L ]
1 + k [ L]
(10.1)
N
1 + K [ L]
N
(10.2)
In the contexts of practical experiments, a researcher may have no
preconceived notion as to whether binding is non-cooperative or
cooperative. There are two types of data plots however that are diagnostic
for cooperative versus non-cooperative binding:
Equation 10.1 can be linearized into the Scatchard Equation
ν
[ L]
•
= Nk − k ν
Equation 10.3 means a plot of
ν
[ L]
(10.3)
(i.e. y) as a function of ν (i.e. x)
is a straight line with a slope of –N, a y-intercept of kN, and a x
intercept of N. In Figure 10.1 shows a Scatchard Plot for N=4 and
K=5.
•
•
If a ligand binds cooperatively to a protein and obeys equation (10.2), a
Scatchard plot will not be linear. Equation 10.2 can be linearized as follows.
Calculate the quantity 1-fB:
ν
1
= 1− fB =
(10.4)
1−
N
N
1 + k [ L]
•
Now calculate the ratio of fB to 1-fB:
fB
K [ L]N / (1 + K [ L]N )
=
= K [ L ]N
(10.5)
N
1 − fb
1/ (1 + K [ L] )
Equation 10.5 is a form of the Hill Equation. The version of the Hill equation
normally displayed as a plot is obtained by taking the logarithm of both sides
of equation 10.5:
⎛ f ⎞
(10.6)
ln ⎜ B ⎟ = ln K + N ln[ L]
⎝ 1− fB ⎠
•
•
Equation 10.6 means that if a ligand binds to a protein with full cooperativity,
a plot of ln(fB/(1-fB) versus ln[L] yields a straight line with a slope of N and a
y intercept of lnK. Such a plot is called a Hill Plot.
B. Partial Cooperativity; Adair Equation
•
•
•
•
For non-cooperative binding a Hill plot will have slope = N=1. For fully
cooperative binding the slope will be N>1. For Hemoglobin which is believed
to bind 4 oxygen molecules cooperatively, we would expect a linear Hill plot
with slope N=4. But the appearance of hemoglobin’s Hill plot is shown in
Figure 10.2:
Myoglobin is an
oxygen storage
protein for which
N=1. Myoglobin has
a linear Hill plot with
slope=1 as expected.
Hemoglobin (Hb) has
a nonlinear Hill plot,
shown in red in
Figure 10.2.
The Hill plot of Hb
has three distinct
reagions, a situation
that results because
Hb binds oxygen with
partial cooperativity.
Figure 10.2: The oxygen storage protein myoglobin has N=1 and a linear Hill plot (black).
Hemoglobin is an oxygen transport protein which has N=4 and a non-linear Hill plot (red).
•
Partial cooperativity means a solution of Hb is a mixture of unbound Hb,
singly bound hemoglobin Hb-O2, doubly bound hemoglobin Hb-2O2, triply
bound hemoglobin Hb-3O2 and filled hemobglobin Hb-4O2, but the binding
constants for these Hb-O2 complexes are different. The binding affinity
between the various forms of Hb and O2 increases as Hb fills its binding sites
with oxygen, i.e. k1 < k2 < k3 < k4 . This means that Hb’s binding affinity is
regulated by the amount of O2 bound, an effect called allosterism.
The Adair equation was the first equation to quantify Hb-O2 binding.The
Adair equation is derived by writing out the binding polynomial for four
binding sites with affinity constants k1 < k2 < k3 < k4
•
(
Q = [ Hb ] 1 + 4k1 [O2 ] + 6k1k2 [O2 ] + 4k1k2 k3 [O2 ] + k1k2 k3 k4 [O2 ]
•
2
3
4
)
(10.7)
In equation 10.7 the first term in the parenthesis is the amount of free
hemoglobin [Hb]. The second term 4k1[L][Hb] is the amount of hemoglobin
with one oxygen site bound etc.
From equation 10.7 we obtain the fraction of oxygen sites bound in Hb for a
certain concentration of oxygen:
•
(
)
k1 [O2 ] + 3k1k2 [O2 ] + 3k1k2 k3 [O2 ] + k1k2 k3 k4 [O2 ]
1 [O2 ] ∂Q
(10.8)
fB =
=
=
4
4 Q ∂ [O2 ] 1 + 4k1 [O2 ] + 6k1k2 [O2 ]2 + 4k1k2 k3 [O2 ]3 + k1k2 k3 k4 [O2 ]4
ν
•
•
•
•
•
•
(
2
3
4
)
Equation 10.8 is the Adair equation and the constants k1 < k2 < k3 < k4 can be
adjusted to fit Hb’s non-linear Hill plot. This I smost easily seen by looking at
the binding limits.
In the weak binding limit where [O2]<<1 equation 10.8 is
⎛ f ⎞
f
(10.9)
[O2 ] << 1: B ≈ k1 [O2 ] or ln ⎜ B ⎟ = ln k1 + ln [O2 ]
1− fB
⎝ 1− fB ⎠
So in the WEAK binding limit the Hill plot is linear , slope=1, and the yintercept is lnk1.
In the strong binding limit
⎛ f ⎞
f
(10.10)
[O2 ] >> 1: B ≈ k4 [O2 ] or ln ⎜ B ⎟ = ln k4 + ln [O2 ]
1− fB
⎝ 1− fB ⎠
So in the STRONG binding limit the Hill plot is linear , slope=1, and the yintercept is lnk4.
In the intermediate region the Hill plot must be fit using the entire equation
10.8. The resulting slope is 2.9-3.5.
C. Protein Allostery &: Pauling’s Sequential Model
•
•
The Adair equation can fit the Hill plot for Hb but it has four adjustable
parameters and there is no physical insight as to why k1 < k2 < k3 < k4 .
Linus Pauling first proposed a sequential model for Hb allosterism where
in Hb, O2 binding was enhanced as a result of pair-wise interactions
between bound sites which are gradually increased in number by
sequential binding of oxygen.
Pauling assumed that the oxygen binding sites occupied the vertices of a
tetrahedron in Hb and thus are all equidistant. This allowed him to
increased the O2 binding affinity of Hb as a simple function of the number
of pair-wise interactions between occupied binding sites.
Assuming an equilibrium between Hb and Hb-O2, only a single site is
bound in the product so no pair-wise interactions are present. Therefore
k
ZZX
Hb ⋅ O2
k1=k: Hb + O2 YZZ
For the equilibrium between HbO2 and Hb-2O2, the product has one pairwise interaction so that the affinity constant is enhanced by
k2 = e−ε 0 / kBT k = fk where f = e −ε 0 / kBT is the enhancement factor from a
•
•
•
•
ZZZ
X
single pair-wise interaction with energy ε0: Hb ⋅ O2 + O2 YZZ
Z Hb ⋅ 2O2
For the equilibrium between Hb-2O2 and Hb-3O2
k3 = e −2ε 0 / kBT k = f 2 k reflecting the two additional pair-wise interactions in
•
f k
ZZZ
X Hb ⋅ 3O2
Hb3O2 versus Hb2O2: Hb ⋅ 2O2 + O2 YZZZ
For the equilibrium between Hb-3O2 and Hb-4O2
k4 = e −3ε 0 / kBT k = f 3 k reflecting the three more pairwise interactions in
fk
2
f 3k
ZZZX
Hb4O2 versus Hb3O2: Hb ⋅ 3O2 + O2 YZZ
Z Hb ⋅ 4O2
Figure 10.3: The pairwise interactions between oxygen bound sites that enhance oxygen
binding according to pauling’s Sequential Model.
• Using Paulings hypothesis the four adjustable parameters in Adair’s
equation are reduced to two adjustable parameters: k and f. The binding
polynomial is
(
Q = [ Hb ] 1 + 4k [O2 ] + 6 fk 2 [O2 ] + 4 f 3 k 3 [O2 ] + f 6 k 4 [O2 ]
2
3
4
)
(10.11)
o Note each term in the binding polynomial has f raised to the power of the
number of pair-wise interactions in the Hb-O2 complex. For example,
Figure 10.3 shows that in Hb where all four sites are filled with O2, there
are 6 pairwise interactions so the fifth term in Q has contains f6.
With equation 10.11 the Adair equation becomes
•
fB =
•
( k [O ] + 3 fk [O ] + 3 f k [O ] + f
2
2
ν
4
=
(
2
3
3 3
2
2
6
k 4 [O2 ]
4
)
1 + 4k [O2 ] + 6 fk 2 [O2 ] + 4 f 3 k 3 [O2 ] + f 6 k 4 [O2 ]
2
3
4
)
(10.12)
Equation 10.12 can be fitted to the Hill Plot in Figure 10.2 by adjusting f and
k.
D. Protein Allostery and Concerted Models
a. Sequential Models assume oxygen binding sites are driven from weak to
strong form by sequential addition of O2 to Hb.
b. An alternative to sequential models are concerted models. Concerted
models assume Hb exists in a form R where ALL binding sites are strong
and form T where ALL binding sites are weak. R and T exist in
equilibrium. All four O2 binding sites change together (i.e. in a concerted
fashion) when R changes to T. Addition of O2 shifts the equilibrium from
favoring T forms at low O2 levels to favoring R forms at high O2 levels.
c. Monod-Wyman-Changeaux (MWC) Theory is a concerted model that was
proposed as an explanation of cooperative oxygen binding in hemoglobin.
X-ray studies have identified some intermediates ppredicted by MWC
which indicate this model has validity.
d. According to MWC theory, in the absence of oxygen , Hb exists in two
[T ] ,
K
ZZZ
X
forms T and R that are in dynamic equilibrium R YZZ
ZT; K =
[ R]
•
•
•
In the T state, all sites bind O2 weakly
In the R state, all binding sites bind O2 tightly
o In the absence of oxygen , the T form is favored, i.e. K>>1
o As oxygen is added, the RL and RL2 forms are favored over
TL and TL2.
o MWC theory was demonstrated for 4 binding sites in Hb. For
simplicity we only show results for two binding sites.
o In the two binding site model there are
K
ZZZ
X
R
T
YZZ
Z
six protein forms:
T , TL, TL2 , R, RL, RL2 . The MWC
7 kR
7 kT
cK
model proposes a dynamic exchange RL ZZZ
X
YZZ
Z TL
between R and T forms as shown
7 kT
below for two oxygen binding sites: 7 k R
2
c K
ZZZ
X TL2
RL2 YZZZ
The equilibria between the various forms of bound and unbound T are
characterized by the equilibrium constant kT. The equilibria between the
•
•
•
various forms of bound and unbound R are characterized by the equilibrium
k
constant kR. The ratio C = T << 1 because R binds O2 more strongly than T.
kR
Note that as more and more oxygen is added more RL, RL2, TL, and TL2 are
formed. But because C<1 then C2K<CK<K, and the equilibria between T and
R forms shifts from favoring T over R to favoring RL2 over TL2.
Assume two binding sites on T and R we start with the fraction of sites bound:
[ RL ] + 2 [ RL2 ] + [TL ] + 2 [TL2 ]
ν =
(10.13)
[ R ] + [T ] + [ RL] + [ RL2 ] + [TL] + [TL2 ]
Substitute K =
[T ] and C = kT
kR
[ R]
to obtain after some algebra the Hill equation
for the MWC model:
⎛ 1 + Ck R [O2 ] ⎞
1 + KC ⎜⎜
⎟
1 + k R [O2 ] ⎟⎠
fB
⎝
= k R [O2 ]
1− fB
⎛ 1 + Ck R [O2 ] ⎞
1 + K ⎜⎜
⎟⎟
⎝ 1 + k R [O2 ] ⎠
•
(10.14)
Equation 10.14 seems complicated but it also explains the Hill Plot for Hb.
• Assume the weak binding limit where [O2 ] << 1 . Then:
fB
1 + KC
≈ k R [O2 ]
1− fB
1+ K
•
(10.15)
Note K>>1 because the T form is favored. Then
fB
≈ k R C [O2 ]
1− fB
(10.16)
or
•
•
•
⎛ f ⎞
(10.17)
ln ⎜ B ⎟ = ln ( k R C ) + ln [O2 ] = ln kT + ln [O2 ]
⎝ 1− fB ⎠
According to equation 10.17 the Hill Plot is linear at low [O2] with
slope = 1 and intercept ln kT
In the high oxygen concentration limit [O2 ] >> 1
fB
1 + KC 2
≈ k R [O2 ]
(10.18)
1− fB
1 + KC
R has a higher binding affinity so C<<1. Therefore the Hill equation
becomes in the limit [O2 ] >> 1 :
•
•
⎛ f ⎞
ln ⎜ B ⎟ = ln k R + ln [O2 ] , i.e. slope = 1 and intercept ln k R
⎝ 1− fB ⎠
• These limiting equations, together the general equation that is effective
at intermediate ligand concentrations, yields the Hill plot below.
Although the sequential and concerted models both explain the Hill Plot data for
Hb-O2 binding , the simple form of the MWC concerted theory and the fact that
many of its proposed intermediates have been identified by crystallography have
caused this theory to be favored over sequential theories.
By the late 1990’s some of the intermediates proposed on sequential models had
been detected with the result that the real model for binding between O2 and Hb is
likely a hybrid theory of the sequential and concerted models.