Download Kinetic models

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JH Steinbach - 2011
A brief introduction to the Hill equation and kinetic models.
I. The concentration-effect relationship
We often analyze a "dose-response" or "concentration-effect" relationship. This relationship describes
how much effect (receptor activation, for example) is produced by various concentrations of a drug.
Steady-state
Steady-state conditions mean that the system is not changing with time. Of course, in biological studies
a true (infinite) steady-state will not be present. Indeed, often it is the case that the system is far from
steady-state, perhaps over a limited time it is "pseudo-steady-state." However, we will be considering
steady-state responses.
The basic parameter for a dose-response curve is the fraction of the maximal activation which is
produced. The fraction can be expressed in two basic forms. Experimentally it is usually the response
relative to the largest response which can be measured. Theoretically, however, it is possible to
calculate the fraction of the maximal response which would be produced if all the receptors were fully
activated. The difference between these two fractions will be clear later on.
II. Kinetic models
A kinetic model is a short-hand description of a particular idea we might have for how a receptor works.
It shows "states" of the receptor (for example, a resting receptor with no ligands bound, or a receptor
with 2 ligands bound and an open channel). The states are connected by arrows that show what can
happen (a ligand can bind or unbind, a channel can open or close). The states show what we are
allowing the receptor to do, and the arrows show how the states are connected.
II.A. A single site occupancy model
This is the simplest type of model: binding directly results in gating. The simplest form is to say that a
receptor (R) has a single site to bind an activator (A), and when the site is occupied the receptor is
active (R*).
A+R
κ1[A][R]
⇄
λ1[AR*]
AR*
The association rate constant is κ1 and the dissociation rate constant λ1.
R
AR*
Rtot
A
AR*
Atot
κ1[A][R]
λ1[AR*]
κ1
λ1
K1
receptor with no drug bound
receptor with drug bound (defined as active for this model)
total receptor, Rtot = R + AR*
free drug
drug bound to receptor
total drug, Atot = A + AR*
association rate; units s-1
dissociation rate; units s-1
association rate constant; units M-2s-1
dissociation rate; units M-1s-1
dissociation constant, K1 = λ1/κ1 (units M)
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To calculate the steady-state concentration-response relationship for this model, we need to determine
the fraction of the total receptors (R + AR*) that is active (AR*). At steady state, there is no change in
AR* with time - the rate for entering AR* is the same as for leaving:
λ1[AR*] = κ1[A][R] = κ1([Atot] - [AR*])([Rtot] - [AR*])
In most experimental conditions there is a vast excess of free activator (that is [A] >>> [AR*]) available
for reaction, and so we assume that [A] is constant at Atot. (On some occasions this might not be true.
However, we will use this assumption for the rest of this handout.)
λ1[AR*] = κ1[A] ([Rtot] - [AR*])
[AR*]/[Rtot] = κ1[A]]/λ1 / {1+ κ1[A]/λ1}
This is the Michaelis-Menten or Langmuir equation. The dissociation constant (K1 = λ1/κ1) is usually
used so it looks like
[AR*]/[Rtot] = [A]/K1 / {1+ [A]/K1}
This is not a very useful equation for the receptors we study, but essential as the first step.
II.B. A two site occupancy model
In words, this model is:
- there are 2 binding sites for a drug on each receptor
- when a receptor has both sites occupied by drug the receptor is activated
- when the receptor is activated we see a response.
This is the simplest model for "cooperative" activation (a "two site occupancy model") in which an
occupied receptor is active.
The model looks like:
A+R
κ1[A][R]
⇄
λ1[AR]
AR + A
κ2[A][AR]
⇄
λ2[AR*A]
AR*A
The new rate constants are κ2 and λ2, giving dissociation constant K2..
The occupancies of the various states are:
[AR] = [R][A]/K1
[AR*A] = [AR][A]/K2 = [R][A]/K1 x [A]/K2
so
AR*A/Rtot = {[A]/K1 x [A]/K2 } / {1 + [A]/K1 + [A]/K2 + [A]/K1 x [A]/K2}
= {[A]2/(K1 K2 )}/ {1 + [A]/(K1 + K2) + [A]2/(K1 K2 )}
Numbers of binding site - A brief digression
The number of bound agonists necessary to produce an activated receptor determines the steepness
of the concentration-response curve. This is illustrated for a multi-site occupancy model in Fig. II-1.
Only receptors with all sites occupied are active.
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N site occupancy model
nA+R <-> ... <-> AnR*
1
n=1
n=2
n=3
n=5
n=50
0.1
AnR/Rtot
1.0
AnR/Rtot
N site occupancy model
nA+R <-> ... <-> AnR*n
0.5
0.01
n=1
n=2
n=3
n=5
n=50
0.001
0.0001
0.0
101
102
101
103
102
103
[drug]
[drug]
Fig. II-1. Increasing numbers of sites. In the left panel the fraction of receptors that are active is plotted against
the drug concentration for a number of values for the number of binding site. Remember that only receptors with
ALL sites occupied are active! The foot of the curve gets progressively more curved as n increases. On the log-log
plot (right panel), it can be seen that the steepness of the curve approaches the number of sites, for small responses
(the inset shows straight lines of increasing steepness). The curves differ less as the concentration gets higher and
higher.
II.C. Binding/gating models
A wealth of experiments has shown that occupancy models only are accurate for competitive
antagonists (drugs which act by binding to the binding site but having no other effect). An occupancy
model is not able to explain the actions of agonists (drugs which bind to the receptor and then activate
it). It is necessary to separate the binding step from the activation step. del Castillo and Katz proposed
the first version of this type of model (del Castillo and Katz. Proc R Soc Lond B Biol Sci. 1957;146:36981.).
In words, a binding/gating model might be:
- there are 2 binding sites for a drug on each receptor
- when a receptor has both sites occupied by drug then it can become activated
- a doubly-occupied receptor activates and inactivates
- when the receptor is activated we see a response.
A+R
κ1[A][R]
⇄
λ1[AR]
AR + A
κ2[A][AR]
⇄
λ2[AR*A]
ARA
β
⇄
α
AR*A
The new rate constants are the "activation" rate constant, β and "inactivation" rate constant, α. The
equilibrium opening constant is W 2 = β/α (W 2 because two molecules of A are bound).
[AR] = [R] [A]/K1 + [R] [A]/K2
[ARA] = [R] [A]/K1 x [A]/K2
[AR*A] = [R] [A]/K1 x [A]/K2 x W 2
so
AR*A/Rtot = {[A]/K1x[A]/K2xW 2} / {1 + [A]/K1 + [A]/K2 + [A]/K1x[A]/K2 + [A]/K1x[A]/K2xW 2}
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II.D. Inhibition of responses
II.D.1. Competitive inhibition - competition for the binding site
A one-site occupancy model is readily extended to include a competitive antagonist (a drug that binds
to the same site as A, but does not activate.
[B]/M
A+R+B

RB
[A]/K
⇄
AR*
(Note that rate constants have been replaced by equilibrium constants, to simplify the presentation.)
The steady state response is then described by the equation
AR*/ Rtot = [A]/K1 / {1+ [A]/K + [B]/M }
Two equations derived from this kinetic model are often used to derive an estimate for M. (Of course, it
may not be proven that this simple model truly applies!)
The Cheng-Prusoff equation
This simple equation (Cheng Y, Prusoff WH (1973). Biochem Pharmacol 22 : 3099–108) has been
used to estimate the dissociation constant for the antagonist, when K is known. This equation is derived
from the IC50 - the concentration of B that reduces AR*/ Rtot to one-half the control binding in the
presence of [A].
{AR*([B=IC50]/ Rtot } / {AR*([B=0]/ Rtot ) = 0.5 or
{[A]/K / {1+ [A]/K + IC50/M }}/{[A]/K1 / {1+ [A]/K}} = 0.5
Simplify and rearrange the equation to get the "Cheng-Prusoff equation"
M = IC50 /{1 + [A]/K}
The Schild equation
A similar relationship is the Schild equation, used in pharmacological studies of receptor activation (see
Colquhoun D. Trends Pharmacol Sci. (2007). 28:608-14).
Rather than using a single concentration of B, this relationship calculates a "dose ratio" at several [B].
The dose ratio is the ratio of the concentration of A required to give the control level of response when
B is present. Put another way, how much more [A] does it take to overcome the presence of [B]?
[A']/K / {1+ [A']/K +[B]/M } = [A0]/K / {1+ [A0]/K }
Simplify and rearrange the equation to get the "Schild" equation
dose ratio = [A'] / [A0] = 1+ [B]/M
The Schild approach is usually implemented by a Schild plot, of log({dose ratio}-1) against log([B]),
which has a slope of one and an intercept of -log(M) (log({dose ratio}-1) =0 or dose ratio = 2).
It is interesting to note that the Schild approach can work with more complex kinetic models. For
example, this is the result of a Schild analysis of the two-site binding/gating model with a competitive
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inhibitor:
A+R+B

A+BR+B

BRB
[B]/M
[B]/M
[A]/K
⇄
⇄
A+AR

BRA
[A]/K
⇄
ARA
W
⇄
AR*A
(for simplicity, assume the binding sites are identical)
AR*A/Rtot = ([A]/K)2 / {1 + 2[A]/K + ([A]/K)2 + ([A]/K)2 W + 2[B]/M + [B]/M x [A]/K}
2 site binding/gating model
Competitive Schild plot
2 site binding/gating model
Competitive inhibitor
[B]=0
[B]=10
[B]=30
[B]=60
[B]=100
[B]=300
[B]=1000
1.0
Data
Schild
Regression
0.5
log(DR-1)
Response/{max control)
1.0
0.5
0.0
-0.5
-1.0
0.0
100
101
102
103
104
1.0
1.5
2.0
2.5
3.0
log([B])
[A]
Fig. II-2. Competitive inhibition. The left panel shows the response to various concentrations of A, in the presence of
various concentrations of B. The curves were calculated with K = 100 units, M = 100 units and W = 20. An important
point to note is that the EC50 concentration increases with increasing [B], but the maximal response is constant - this is a
definition of competitive inhibition. The right hand panel shows a Schild plot, in which the target response was set to
40% of maximal (short dashed horizontal line in left panel) . The straight line shows the calculated Schild relationship.
The estimate from the "data" is that M = 107 units, close to the 100 units used in the calculations.
II.D.2. Noncompetitive inhibition - selective inhibition of the active state
There are a number of drugs that selectively act on a particular state. One particular example is a drug
that binds specifically to the active receptor (an "open channel blocker").
A+R+B
[A]/K
⇄
A+AR
[A]/K
⇄
ARA
W
⇄
AR*A+B

AR*AB
AR*A/Rtot = ([A]/K)2 / {1 + 2[A]/K + ([A]/K)2 + ([A]/K)2 W + [B]/M x ([A]/K)2 W }
[B]/M
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2 site binding/gating model
Open channel block
[B]=0
[B]=1
[B]=10
[B]=30
[B]=60
[B]=100
[B]=300
1.0
0.5
log(DR-1)
1.0
Response/{max control)
2 site binding/gating model
Open channel Schild plot
0.5
0.0
Data
Schild
Regression
Col 16 vs Col 17
Col 16 vs Col 17
-0.5
-1.0
0.0
100
101
102
[A]
103
104
1.0
1.5
2.0
2.5
3.0
log([B])
Fig. II-3 Selective active state block. The left panel shows the response to various concentrations of A, in the
presence of various concentrations of B. The curves were calculated with K = 100 units, M = 100 units and W = 20. In
contrast to a competitive inhibitor, there is little change in the EC50 concentration with increasing [B], but a significant
decrease in the maximal response. The right hand panel shows a Schild plot, in which the target response was set to
10%, 20% or 40% of maximal (lines in left panel). The straight line shows the calculated Schild relationship. The
estimate from the "data" is that M = 102 units, close to the 100 units used in the calculations when the target response
is small (10% of control), but gets progressively worse at higher targets. At the highest values for [B], it is not possible
for the response to reach the larger targets (see the left panel) and the values are not defined.
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III. The Hill equation
The Hill equation (also known as the logistic equation) has the basic form:
Y([A]) = ([A]/EC50)n / {1+ ([A]/EC50)n}
The response, Y, is a function of the drug concentration, [A]. The parameters are the Hill coefficient (n)
and the concentration of drug producing half of the maximal activation (EC50).
Sometimes the equation includes a maximum observed response (Ymax):
Y([A]) = Ymax ([A]/EC50)n / {1+ ([A]/EC50)n}
or an offset (a non-zero minimal value) (Ymin):
Y([A]) = Ymax ([A]/EC50)n / {1+ ([A]/EC50)n} + Ymin
For simplicity (and for the Hill plot) the values are normalized to cover the available range of values
(Ymax - Ymin):
y([A]) = (Y-Ymin) / (Ymax - Ymin) = ([A]/EC50)n / {1+ ([A]/EC50)n}
so the value of y runs from 0 to 1.
The Hill equation also can be used to describe inhibition (that is, a value of Y which decreases with
increasing X), simply by subtracting it from 1
Z([A]) = 1 – Y([A]) = 1 / {1+ ([A]/EC50)n}
(note that is the same thing as letting n take a negative value).
The Hill plot
The “Hill plot” was introduced as a graphical method for analyzing data in terms of the Hill equation.
Now that we can do non-linear fitting with computers, there is no reason for using a Hill plot (except to
show that the data do not really conform to the Hill equation!). The major difficulties with the Hill plot are
that it is extremely sensitive to errors in Ymax, and has a tendency to make no sense. Nonlinear fits of
untransformed data are much more robust. The Hill plot is described here just to provide some
historical context!
To make a Hill plot, the data values are normalized to give a new variable, h.
h = (y – 0) / (1 – y)
Using the definition of y given above,
h = ([A]/EC50)n
The Hill plot is made on log-log coordinates:
log(h) -vs.- log([A]) or
{n(log([A]) – log(EC50))} -vs.- log([A]).
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The slope gives n and the value of X at which log(h) = 0 (or h=1 or y = 0.5) gives EC50.
The Hill equation as a tool.
The Hill equation has no relationship to any actual physical situation – for example it does not
correspond to the predictions of any scheme for binding and gating. However, parameters estimated by
fitting data with the Hill equation provide a commonly used summary of the data. The reason why the
summary can be useful is that two critical features of the data are captured.
The first is an estimate of the “midpoint” of the amount/effect relationship of y on [A] given by EC50. This
gives a (rough) estimate of the affinity of the interaction between drug and site.
The second is an estimate of the steepness of the amount/effect relationship of y on [A]: n. This gives
an estimate of the minimal number of binding sites for A.
Comparing the Hill equation to the kinetic models discussed:
Two-site occupancy:
ARA/Rtot = {[A]/K1 x [A]/K2 } / {1 + [A]/K1 + [A]/K2 + [A]/K1 x [A]/K2}
Two-site binding/gating:
AR*A/Rtot = {[A]/K1x[A]/K2xW} / {1 + [A]/K1 + [A]/K2 + [A]/K1x[A]/K2 + [A]/K1x[A]/K2xW}
To simplify things, assume K1=K2=K (identical sites) then
Two-site occupancy:
ARA/Rtot = {([A]/K)2} / {(1 + 2[A]/K + ([A]/K)2)}
Two-site binding/gating:
AR*A/Rtot = {W([A]/K)2} / {(1 + 2[A]/K + ([A]/K)2)(1 + W)}
The normalized Hill equation (with n=2):
Y = ([A]/EC50)2 / {1+ ([A]/EC50)2}
The denominator of the Hill equation has fewer terms. The equation for two-site binding/gating gets
closer to the Hill equation when W >> 1. Two-site occupancy models only get closer if there is
cooperative binding, so that the dissociation constant at the first site bound is much greater (lower
affinity) than when the second drug molecule binds. These points will be illustrated in the next section.
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IV. Some simple models and plots.
In the rest of this Introduction I will discuss some specific examples of simple models for receptor
activation, and show how good (bad?) a job the Hill equation does in terms really describing the
(calculated) data.
Qualitatively, the amount of receptor activation is not necessarily directly related to the binding affinity
for a drug to any particular "state" of the receptor. Therefore, the value of EC50 cannot directly be
related to a molecular step (for example, binding of a drug to an inactive receptor).
Similarly, the steepness of the relationship (n) is not directly related to the number of drug molecules
which must bind to an effect to happen. However, it does provide an estimate of the smallest number of
drugs which must bind before the response occurs.
The best tool for getting a feeling of what your data might mean is puttering around with some
predictions of various possible schemes. I have done some of this, in the attached plots.
I calculated responses for 3 models which are simple but are applicable to the receptors we study.
- two-site occupancy model
- two-site binding model with only doubly- bound receptors able to be activated
- two-site binding model with both singly- and doubly-bound receptors able to be activated
In all of the plots, the points (connected by thin lines) are the predictions of the model and the thick
lines are the predictions of the fit Hill equations.
One objective of this handout is to indicate the value and shortcomings of using the Hill equation to
understand your data. Another is to illustrate the value of plotting your data several ways to look at
them.
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IV. A. Two-site occupancy model
A+R
κ1[A][R]
⇄
λ1[AR]
AR + A
κ2[A][AR]
⇄
λ2[AR*A]
AR*A
Assume: Initially sites 1 and 2 are identical. However, after binding to the first site the affinity of the
second site changes by a factor F:
K1 = λ1/κ1
K2 = λ2/κ2 = FK1
(Note that when F is smaller than 1, the second site has HIGHER affinity, and the converse.)
K1 was given the arbitrary value of 100, and predictions were made over a range of “concentrations”
around this midpoint.
The basic equation for ACTIVE receptors is for the equilibrium fraction of receptors which are doublyliganded (ARA):
AR*A / Rtot = {([A]/K1)2/F} / {1 + 2[A]/K1 + ([A]/(K1 × K1 × F)}
(Eqn IV.A.1)
(Notice that the factor of 2 in the denominator is there because there are initially 2 identical sites for
binding.) The first plot is one often used: the ordinate gives the "response" (fraction of receptors as
AR*A). Fig. IV.1 shows calculated concentration-effect curves for this model, with different values for F.
Note that the value shifts the curves along the concentration axis. It is also interesting that the
steepness of the curves changes, although the number of binding sites remains the same.
2 site occupancy model
2A+R <-> A+AR <-> A2R*
F=1
F=0.001
F=0.03
F=30
F=100
1.0
A2R/Rtot
Fig. IV.1. Two site occupancy. The first
plot is one you are probably used to seeing:
the ordinate gives the "response" (fraction of
receptors as AR*A). Data for a number of
assumed values for F are given. You might
be surprised that the midpoint of the curve is
NOT at 100 (=K1, dashed vertical line) when
the two sites are equal. This is because of
equation IV.A.1 - the value is 0.5 when
[A]/K1 = 1+√2. The solid lines show fits of
the Hill equation to the points.
0.5
0.0
10-2
10-1
100
101
102
103
104
105
106
[A]
I fit the predictions with the Hill equation, using SIGMAPLOT, with the following results:
F
n
EC50
0.001
1.92
3.28
0.03
1.67
21
1
1.19
254
30
100
1.02 1.006
6093 20090
For the linear plot the Hill equation (lines through the points) looks like a good description of the data.
Fig. IV.2 shows When the data are plotted in log-log form. This plot emphasizes the shape at low levels
of response. The slope of the lines approaches 2 for every plot, as would be expected because 2 A
molecules must be bound for the receptor to be active. However the curvature of the plots clearly differs
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between the various values.
2 site occupancy model
2A+R <-> A+AR <-> A2R*
Fig. IV-2. Two site occupancy. When F < 1, the sites
are “positively cooperative” in binding, so the second
site is occupied at a lower concentration than the first
site. As F > 1, the sites are “negatively cooperative” in
binding. This is formally equivalent to “non-identical,
non-cooperative sites” (although other experiments
can distinguish the cases). (For convenience, lines
with slopes of 1 and 2 on the log-log plot are shown.)
The Hill equation (solid lines) gives a rather poor
description of the smaller responses.
1
A2R/Rtot
0.1
0.01
F=1
F=0.001
F=0.03
F=30
F=100
0.001
0.0001
10-2
10-1
100
101
102
103
104
105
106
[A]
I also made a Hill plot of the data. As can be seen, the data clearly do not make straight lines on the
plot. As F << 1, then the plot is straight with slope 2 over a greater range. In this case, essentially you
look at titration of the second site being driven by binding to the first site. As F >> 1, the plot begins to
look straight with a slope of 1. In this case, you look at the independent titration of the second site
(since the first site is fully occupied at all of these concentrations). The plots indicate that the Hill
coefficient is the slope of the tangent to the data at X = EC50 (when the Hill variable, h, is 1).
2 site occupancy model
2A+R <-> A+AR <-> A2R*
Fig. IV-3. Two site occupancy Hill plot. One aspect to
remember with a Hill plot is the actual range of
experimentally accessible values; in physiological
experiments it is often hard to measure to an accuracy of
0.1%, while a Hill value of 0.001 or 999 requires this
accuracy. So, these plots are much less informative in the
real world.
1000
100
Hill variable
10
1
F=1
F=0.001
F=0.03
F=30
F=100
0.1
0.01
0.001
10-2
10-1
100
101
102
103
104
105
106
[A]
Occupancy of sites.
A second way to look at the data for this situation is to ask, what fraction of the available binding sites
are occupied?
Fraction of sites occupied = {[A]/K1 + 2 ([A]/K1)2/F} / {2 + 2 [A]/K1 + 2 ([A]/K1)2/F}
Fits of the Hill equation gave values:
F
n
EC50
0.001 0.03
1.963 1.814
3.162 17.3
1
1.29
99
30
100
0.652 0.508
590 1254
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Fig. IV-4. Two site occupancy model - binding curves.
To clarify the plots, I left out the intermediate values of
F. Two things are worth noting. First, when F = 0.001,
the binding seems to be "high affinity." This happens
because as soon as the first site is occupied, then so is the
second (K1=100 and K2=0.1). The second is that when
F=100 it is possible to see two steps in titration. The first
ligand binds with K1=100, but the second has a
dissociation constant of 10000, so the first site is
essentially saturated before any binding occurs to the
second. This "negative cooperativity" in binding also can
be produced by a receptor that has two distinct and noninteracting sites, of different affinities.
2 site occupancy model
2A+R <-> A+AR <-> A2R*
F=1
F=0.001
F=100
Sites occupied
1.0
0.5
0.0
0
0
1
10
100
[A]
1000
10000
2011
100000 1000000
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IV. B. Two-site binding/gating model
This is the next simplest model. In this case, after binding has occurred the receptor can be either
active or inactive. This is the simplest model which can explain the behavior of real receptors.
Two site binding/gating model:
A+R
κ1[A][R]
⇄
λ1[AR]
AR + A
κ2[A][AR]
⇄
λ2[AR*A]
ARA
β
⇄
α
AR*A
Here we will assume that sites 1 and 2 are identical and do not interact (K1 = K2 = 100; F=1). However,
after occupancy of both sites opening can occur with ratio W = β/α = (activation rate/inactivation rate).
The basic equation is for the fraction of receptors with open channels (AR*A):
(AR*A) / Rtot = {W[A]2/K2} / {1 + 2[A]/K + (1+W)[A] 2/ K2 }
Eqn IV.B
2 site bind/gate model
2A+R <-> A+AR <-> A2R <-> A2R*
W=1
W=0.01
W=10
W=100
W=1000
normalized response
1.0
0.5
0.0
10-2
10-1
100
101
102
103
104
105
106
Fig IV-5. Two-site binding/gating model. This plot is a
typical plot of experimental data, in which the response is
divided by the maximal response observed. A small value of
W (poor activation) shifts the concentration-effect curve to
the right, while a large value (strong activation) shifts it to
the left. In terms of equation IV.B a very small value for W
is negligible compared to 1 in the denominator, so in the
limit as W->0 the equation approaches the simple
occupancy model with F=1. Large W, however, dominates
over 1 and so this becomes equivalent to a very large value
for F.
[A]
Making W large is similar to making F small in case IV.A. Looking at the equations, you can see that
large W and small F enter into the predictions in similar ways.
The predicted shift in the EC50 values can be derived from Eqn IV.B. Define Q as the EC50
concentration of A.
0.5 = {WQ2/K2} / {1 + 2Q/K + (1+W)Q 2/ K2 } / {W/(1+W)}
Arranging and simplifying this gives the quadratic:
-(1+W) Q2/K2 + 2Q/K +1 = 0
The positive root for this is Q = K (1+√(2+W))/(1+W).
When W=0 (no gating) then Q =K(1+√(2) (the same as for the occupancy model with F=1).
When W gets very large, Q ~ K/(√(W)
I fit the non-normalized data with the Hill equation.
W
0.01
1
n
1.21
1.31
EC50
247
142
max
0.0098
0.5
10
1.53
42
0.9
100
1.81
11.1
0.99
1000
1.93
3.28
1
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page 14
2011
The values for EC50 are pretty close to the predicted values from the quadratic.
This plot is very deceptive. This is because it is normalized to the largest observed response. However,
in terms of receptor activation, the maximal activation is very different (for these values from <0.1% to
essentially 100%). Notice that this problem does not arise with an occupancy model, for which activity
equals activity, and occupancy becomes 1.0 at high [A]. How big would the response be if all the
receptors were actually activated? The next plot shows the predicted fraction of the total receptors
which are actually activated for the different values of W.
For small values of W only a small fraction of the receptors are activated, even when all the sites are
occupied. When W=1, then half the receptors would be active, when W=0. 01 only 1% and so forth.
This is a fundamental concern in studies of receptor activation: what is the activity of a single receptor
when the response is maximal? Various approaches can be taken to answering this question, including
recordings of single channel currents.
2 site bind/gate model
2A+R <-> A+AR <-> A2R <-> A2R*
1.0
A2R*/Rtot
Fig. IV-6. Two-site binding/gating model. When the data
are NOT normalized to the maximal observed response, the
graph looks quite different.
W=1
W=0.01
W=10
W=100
W=1000
0.5
0.0
10-2
10-1
100
101
102
103
104
105
106
[A]
2 site bind/gate model
2A+R <-> A+AR <-> A2R <-> A2R*
Fig. IV-7. Two-site binding/gating model. The
log-log plot shows that the slope of the
relationship approaches 2 at low concentrations,
as expected since two molecules of drug must
bind. The curvature depends on how large W is
compared to 1; large values give more rapid
approach to the slope of 2.
100
A2R*/Rtot
10-1
10-2
W=1
W=0.01
W=10
W=100
W=1000
10-3
10-4
10-2
10-1
100
101
102
[A]
103
104
105
106
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2011
IV.C. Gating with either one or two bound ligands.
This is the last model considered here. What happens if a receptor can be activated when either 1 or 2
drug molecules are bound? There are now 2 active states: AR* and AR*A (and in the diagram the
activation steps are now down).
A+R
κ1[A][R]
⇄
λ1[AR]
κ2[A][AR]
⇄
λ2[AR*A]
AR + A

AR*
y
ARA
α
z

AR*A
β
We assume that sites 1 and 2 are identical and not interacting (F=1). However, after occupancy of
either one or both sites opening can occur with ratio W = β/α with two sites occupied and V = z/y with
only one site.
The basic equation is for the fraction of receptors with open channels ([AR*A] + [AR*]):
([AR*A] + [AR*]) / Rtot =
{W x [A] 2/(K1K2) + V x 2[A]/K1]} / {1 + 2(1+V)[A]/K1 + (1+W) x [A] 2/(K1K2)}
In each case W was held constant at 100 (strong gating with two ligands bound). The results of fitting
with the Hill equation are:
V
n
EC50
0
1.81
11
2 site bind/gate model
either 1 (V) or 2 (W=100) open
Response
0.5
0.0
10-3
10-2
10-1
100
101
[A]
102
1
1.63
9.9
10
1.11
4.3
100
1.0
0.49
Fig. VI-8. Gating with either 1 or 2
molecules bound. When V=0, no gating of
monoliganded receptors occurs. As more
and more gating occurs of monoliganded
receptors, the curvature goes down and the
EC50 shifts to lower concentrations. When
gating is strong with only one bound
molecule, then the curve is the same as a
single site binding/gating model.
V=0
V=0.1
V=1
V=10
V=100
1.0
0.1
1.79
11
103
104
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2 site bind/gate model
either 1 (V) or 2 (W=100) open
Fig. IV-8. Gating with either 1 or 2 molecules bound.
The log-log concentration-effect plot is quite informative.
When V=W (=100) 99% of the receptors can be opened by
a single bound drug, the same fraction as when 2 are
bound. Accordingly, the curve is identical to one in which
there is only one site, and at low drug concentrations the
slope approaches 1. When V=0 there is no gating of
monoliganded receptors, and the low-concentration slope
is 2. At intermediate values of V, the slope at low
concentrations approaches 1 and the maximal slope lies
between 1 and 2.
1
Response
0.1
0.01
V=0
V=0.1
V=1
V=10
V=100
0.001
0.0001
10-3
10-2
10-1
100
101
102
103
2011
104
[A]
END