Monod-Wyman-Changeux Analysis of Ligand-Gated
Ion Channel Mutants
Tal Einav
†
and Rob Phillips
∗
,
‡
Department of Physics, California Institute of Technology, Pasadena, California 91125, United
States, and Department of Applied Physics and Division of Biology and Biological Engineering,
California Institute of Technology, Pasadena, California 91125, United States
E-mail: phillips@pboc.caltech.edu
Abstract
We present a framework for computing the gating properties of ligand-gated ion channel
mutants using the Monod-Wyman-Changeux (MWC) model of allostery. We derive simple
analytic formulas for key functional properties such as the leakiness, dynamic range, half-
maximal effective concentration (
[
EC
50
]
), and effective Hill coefficient, and explore the full
spectrum of phenotypes that are accessible through mutations. Specifically, we consider mu-
tations in the channel pore of nicotinic acetylcholine receptor (nAChR) and the ligand binding
domain of a cyclic nucleotide-gated (CNG) ion channel, demonstrating how each mutation
can be characterized as only affecting a subset of the biophysical parameters. In addition, we
show how the unifying perspective offered by the MWC model allows us, perhaps surpris-
ingly, to collapse the plethora of dose-response data from different classes of ion channels into
a universal family of curves.
∗
To whom correspondence should be addressed
†
Department of Physics, California Institute of Technology, Pasadena, California 91125, United States
‡
Department of Applied Physics and Division of Biology and Biological Engineering, California Institute of Tech-
nology, Pasadena, California 91125, United States
1
arXiv:1701.06122v1 [q-bio.SC] 22 Jan 2017
1 Introduction
Ion channels are signaling proteins responsible for a huge variety of physiological functions rang-
ing from responding to membrane voltage, tension, and temperature to serving as the primary
players in the signal transduction we experience as vision.
1
Broadly speaking, these channels are
classified based upon the driving forces that gate them. In this work, we explore one such clas-
sification for ligand-gated ion channel mutants based on the Monod-Wyman-Changeux (MWC)
model of allostery. In particular, we focus on mutants in two of the arguably best studied ligand-
gated ion channels: the nicotinic acetylcholine receptor (nAChR) and the cyclic nucleotide-gated
(CNG) ion channel shown schematically in Fig 1.
2,3
+
CYCLIC NUCLEOTIDE-
GATED CHANNEL
ACETYCHOLINE RECEPTOR
NICOTINIC
CYTOSOL
CYTOSOL
CYTOSOL
CYTOSOL
4 nm
acetylcholine
binding site
cyclic nucleotide
binding site
lipid
bilayer
–
–
–
–
+
+
+
acetylcholine
–
–
–
–
+
+
CLOSED
STATES
OPEN
STATES
cyclic nucleotide
+
A
B
Na
Figure 1: Schematic of nAChR and CNGA2 ion channels.
(A) The heteropentameric nicotinic acetyl-
choline receptor (nAChR) has two ligand binding sites for acetylcholine outside the cytosol. (B) The ho-
motetrameric cyclic nucleotide-gated (CNGA2) has four ligand binding sites, one on each subunit, for cAMP
or cGMP located inside the cytosol. Both ion channels have a higher probability of being closed in the ab-
sence of ligand and open when bound to ligand.
The MWC model has long been used in the contexts of both nAChR and CNG ion chan-
nels.
4–6
Although careful analysis of these systems has revealed that some details of ligand-gated
ion channel dynamics are not captured by this model (e.g. the existence and interaction of multiple
conducting states
7,8
), the MWC framework nevertheless captures many critical features of a chan-
nel’s response and offers one of the simplest settings to explore its underlying mechanisms. For
example, knowledge of both the nAChR and CNG systems’ molecular architecture and our ability
to measure their complex kinetics has only recently become sufficiently advanced to tie the effects
of mutations to key biophysical parameters. Purohit and Auerbach used combinations of mutations
2
to infer the nAChR gating energy, finding that unliganded nAChR channels open up for a remark-
ably brief 80
μ
s every 15 minutes.
9
Statistical mechanics has been used to show how changes to
the energy difference between conformations in allosteric proteins translates to different functional
behavior (i.e. how it modifies the leakiness, dynamic range,
[
EC
50
]
and the effective Hill coeffi-
cient),
10,11
and we extend this work to find simple analytic approximations that are valid within
the context of ion channels. Using this methodology, we systematically explore the full range of
behaviors that may be induced by different types of mutations. This analysis enables us to quantify
the inherent trade-offs between key properties of ion channel dose-response curves and potentially
paves the way for future biophysical models of evolutionary fitness in which the genotype (i.e.
amino acid sequence) of allosteric molecules is directly connected to phenotype (i.e. properties of
a channel’s response).
To this end, we consider two distinct classes of mutants which tune different sets of MWC
parameters – either the ion channel gating energy or the ligand-channel dissociation constants.
Previous work by Auerbach
et al.
demonstrated that these two sets of physical parameters can be
independently tuned within the nAChR ion channel; pore mutations only alter the channel gating
energy whereas mutations within the ligand binding domain only affect the ligand-channel dis-
sociation constants.
12
Utilizing this parameter independence, we determine the full spectrum of
nAChR phenotypes given an arbitrary set of channel pore mutations and show why a linear in-
crease in the channel gating energy leads to a logarithmic shift in the nAChR dose-response curve.
Next, we consider recent data from CNGA2 ion channels with mutations in their ligand binding
pocket.
13
We hypothesize that modifying the ligand binding domain should not alter the channel
gating energy and demonstrate how the entire class of CNGA2 mutants can be simultaneously
characterized with this constraint. This class of mutants sheds light on the fundamental differences
between homooligomeric channels comprised of a single type of subunit and heterooligomeric
channels whose distinct subunits can have different ligand binding affinities.
By viewing mutant data through its effects on the underlying biophysical parameters of the
system, we go well beyond simply fitting individual dose-response data, instead creating a frame-
work with which we can explore the full expanse of ion channel phenotypes available through
mutations. Using this methodology, we: (1) analytically compute important ion channel charac-
teristics, namely the leakiness, dynamic range,
[
EC
50
]
, and effective Hill coefficient, (2) link the
role of mutations with thermodynamic parameters, (3) show how the behavior of an entire family
of mutants can be predicted using only a subset of the members of that family, (4) quantify the
pleiotropic effect of point mutations on multiple phenotypic traits and characterize the correlations
between these diverse effects, and (5) collapse the data from multiple ion channels onto a single
master curve, revealing that such mutants form a one-parameter family. In doing so, we present a
unified framework to collate the plethora of data known about such channels.
Model
Electrophysiological techniques can measure currents across a single cell’s membrane. The current
flowing through a ligand-gated ion channel is proportional to the average probability
p
open
(
c
)
that
the channel will be open at a ligand concentration
c
. For an ion channel with
m
identical ligand
3
K
O
c
2
2
e
-
βε
K
C
c
2
K
O
c
1
+
2
K
C
c
1
+
e
-
βε
closed
open
ST
ATE
WEIGHT
STATE
WEIGHT
e
-
βε
1
2
K
C
c
e
-
βε
2
K
O
c
A
4
e
-
βε
3
K
C
c
6
e
-
βε
2
K
C
c
4
K
O
c
6
2
K
O
c
closed
open
ST
ATE
WEIGHT
STATE
WEIGHT
4
K
O
c
1
+
e
-
βε
4
K
C
c
1
+
e
-
βε
4
e
-
βε
K
C
c
4
K
C
c
e
-
βε
1
K
O
c
4
4
3
K
O
c
B
Figure 2: The probability that a ligand-gated ion channel is open as given by the MWC model.
(A)
Microscopic states and Boltzmann weights of the nAChR ion channel (green) binding to acetylcholine (or-
ange). (B) Corresponding states for the CNGA2 ion channel (purple) binding to cGMP (brown). The
behavior of these channels is determined by three physical parameters: the affinity between the receptor and
ligand in the open (
K
O
) and closed (
K
C
) states and the free energy difference
ε
between the closed and open
conformations of the ion channel.
binding sites (see Fig 2), this probability is given by the MWC model as
p
open
(
c
) =
(
1
+
c
K
O
)
m
(
1
+
c
K
O
)
m
+
e
−
β ε
(
1
+
c
K
C
)
m
,
(1)
where
K
O
and
K
C
represent the dissociation constants between the ligand and the open and closed
ion channel, respectively,
c
denotes the concentration of the ligand,
ε
(called the gating energy)
denotes the free energy difference between the closed and open conformations of the ion channel
in the absence of ligand, and
β
=
1
k
B
T
where
k
B
is Boltzmann’s constant and
T
is the temperature of
the system. Wild type ion channels are typically closed in the absence of ligand (
ε
<
0) and open
when bound to ligand (
K
O
<
K
C
). Fig 2 shows the possible conformations of the nAChR (
m
=
2)
and CNGA2 (
m
=
4) ion channels together with their Boltzmann weights.
p
open
(
c
)
is given by the
sum of the open state weights divided by the sum of all weights. Note that the MWC parameters
K
O
,
K
C
, and
ε
may be expressed as ratios of the experimentally measured rates of ligand binding
and unbinding as well as the transition rates between the open and closed channel conformations
(see Supporting Information section A.1).
4
Current measurements are often reported as
normalized
current, implying that the current has
been stretched vertically to run from 0 to 1, as given by
normalized current
=
p
open
(
c
)
−
p
min
open
p
max
open
−
p
min
open
.
(2)
p
open
(
c
)
increases monotonically as a function of ligand concentration
c
, with a minimum value in
the absence of ligand given by
p
min
open
=
p
open
(
0
) =
1
1
+
e
−
β ε
,
(3)
and a maximum value in the presence of saturating levels of ligand given as
p
max
open
=
lim
c
→
∞
p
open
(
c
) =
1
1
+
e
−
β ε
(
K
O
K
C
)
m
.
(4)
Using the above two limits, we can investigate four important characteristics of ion chan-
nels.
10,11
First, we examine the
leakiness
of an ion channel, or the fraction of time a channel
is open in the absence of ligand, namely,
leakiness
=
p
min
open
.
(5)
Next we determine the
dynamic range
, or the difference between the probability of the maximally
open and maximally closed states of the ion channel, given by
dynamic range
=
p
max
open
−
p
min
open
.
(6)
Ion channels that minimize leakiness only open upon ligand binding, and ion channels that maxi-
mize dynamic range have greater contrast between their open and closed states. Just like
p
open
(
c
)
,
leakiness and dynamic range lie within the interval
[
0
,
1
]
.
Two other important characteristics are measured from the normalized current. The
half max-
imal effective concentration
[
EC
50
]
denotes the concentration of ligand at which the normalized
current of the ion channel equals
1
⁄
2
, namely,
p
open
([
EC
50
]) =
p
min
open
+
p
max
open
2
.
(7)
The
effective Hill coefficient h
equals twice the log-log slope of the normalized current evaluated
at
c
= [
EC
50
]
,
h
=
2
d
d
log
c
log
(
p
open
(
c
)
−
p
min
open
p
max
open
−
p
min
open
)
c
=[
EC
50
]
,
(8)
which reduces to the standard Hill coefficient for the Hill function.
14
The
[
EC
50
]
determines how
the normalized current shifts left and right, while the effective Hill coefficient corresponds to the
slope at
[
EC
50
]
. Together, these two properties determine the approximate window of ligand con-
5
centrations for which the normalized current transitions from 0 to 1.
In the limit 1
e
−
β ε
(
K
C
K
O
)
m
, which we show below is relevant for both the nAChR and
CNGA2 ion channels, the various functional properties of the channel described above can be
approximated to leading order as (see Supporting Information section B):
leakiness
≈
e
β ε
(9)
dynamic range
≈
1
(10)
[
EC
50
]
≈
e
−
β ε
/
m
K
O
(11)
h
≈
m
.
(12)
Results
nAChR Mutants can be Categorized using Free Energy
Muscle-type nAChR is a heteropentamer with subunit stoichiometry
α
2
β γδ
, containing two ligand
binding sites for acetylcholine at the interface of the
α
-
δ
and
α
-
γ
subunits.
15
The five homologous
subunits have M2 transmembrane domains which move symmetrically during nAChR gating to
either occlude or open the ion channel.
16
By introducing a serine in place of the leucine at a key
residue (L251S) within the M2 domain present within each subunit, the corresponding subunit is
able to more easily transition from the closed to open configuration, shifting the dose-response
curve to the left (see Fig 3A).
17
For example, wild type nAChR is maximally stimulated with
100
μ
M of acetylcholine, while a mutant ion channel with one L251S mutation is more sensitive
and only requires 10
μ
M to saturate its dose-response curve.
Labarca
et al.
used L251S mutations to create ion channels with
n
mutated subunits.
17
Fig 3A
shows the resulting normalized current for several of these mutants; from right to left the curves
represent
n
=
0 (wild type) to
n
=
4 (an ion channel with four of its five subunits mutated). One
interesting trend in the data is that each additional mutation shifts the normalized current to the
left by approximately one decade in concentration (see Supporting Information section A.2). This
constant shift in the dose-response curves motivated Labarca
et al.
to postulate that mutating each
subunit increases the gating free energy
ε
by a fixed amount.
To test this idea, we analyze the nAChR data at various concentrations
c
of the ligand acetyl-
choline using the MWC model Eq (1) with
m
=
2 ligand binding sites. Because the L251S muta-
tion is approximately 4
.
5 nm from the ligand binding domain,
18
we assume that the ligand binding
affinities
K
O
and
K
C
are unchanged for the wild type and mutant ion channels, an assumption that
has been repeatedly verified by Auerbach
et al.
for nAChR pore mutations.
12
Fig 3A shows the
best-fit theoretical curves assuming all five nAChR mutants have the same
K
O
and
K
C
values but
that each channel has a distinct gating energy
ε
(
n
)
(where the superscript
n
denotes the number of
mutated subunits). These gating energies were found to increase by roughly 5
k
B
T
per
n
, as would
be expected for a mutation that acts equivalently and independently on each subunit.
One beautiful illustration of the power of the MWC model lies in its ability to provide a unified
perspective to view data from many different ion channels. Following earlier work in the context
of both chemotaxis and quorum sensing,
19,20
we rewrite the probability that the nAChR receptor
6
A
n
=
0
n
=
1
n
=
2
n
=
3
n
=
4
10
-
10
10
-
9
10
-
8
10
-
7
10
-
6
10
-
5
10
-
4
10
-
3
0.0
0.2
0.4
0.6
0.8
1.0
[
ACh
](
M
)
normalized current
B
-
6
-
4
-
2
0
2
4
6
0.0
0.2
0.4
0.6
0.8
1.0
Bohr parameter,
F
nAChR
(
k
B
T
units
)
p
open
Figure 3: Characterizing nicotinic acetylcholine receptors with
n
subunits carrying the L251S mu-
tation.
(A) Normalized currents of mutant nAChR ion channels at different concentrations of the agonist
acetylcholine (ACh).
17
The curves from right to left show a receptor with
n
=
0 (wild type),
n
=
1 (
α
2
β γ
∗
δ
),
n
=
2 (
α
∗
2
β γδ
),
n
=
3 (
α
2
β
∗
γ
∗
δ
∗
), and
n
=
4 (
α
∗
2
β γ
∗
δ
∗
) mutations, where asterisks (
∗
) denote a mutated
subunit. Fitting the data (solid lines) to Eqs (1) and (2) with
m
=
2 ligand binding sites determines the three
MWC parameters
K
O
=
0
.
1
×
10
−
9
M,
K
C
=
60
×
10
−
6
M, and
β ε
(
n
)
= [
−
4
.
0
,
−
8
.
5
,
−
14
.
6
,
−
19
.
2
,
−
23
.
7
]
from left (
n
=
4) to right (
n
=
0). With each additional mutation, the dose-response curve shifts to the left
by roughly a decade in concentration while the
ε
parameter increases by roughly 5
k
B
T
. (B) The probability
p
open
(
c
)
that the five ion channels are open can be collapsed onto the same curve using the Bohr parameter
F
nAChR
(
c
)
given by Eq (13). A positive Bohr parameter indicates that
c
is above the
[
EC
50
]
. See Supporting
Information section C for details on the fitting procedure.
is open as
p
open
(
c
)
≡
1
1
+
e
−
β
F
(
c
)
,
(13)
where this last equation defines the
Bohr parameter
21
F
(
c
) =
−
k
B
T
log
e
−
β ε
(
1
+
c
K
C
)
m
(
1
+
c
K
O
)
m
.
(14)
The Bohr parameter quantifies the trade-offs between the physical parameters of the system (in the
case of nAChR, between the entropy associated with the ligand concentration
c
and the gating free
energy
β ε
). When the Bohr parameters of two ion channels are equal, both channels will elicit the
same physiological response. Using Eqs (1) and (13) to convert the normalized current data into
the probability
p
open
(see Supporting Information section A.3), we can collapse the dose-response
data of the five nAChR mutants onto a single master curve as a function of the Bohr parameter for
nAChR,
F
nAChR
(
c
)
, as shown in Fig 3B. In this way, the Bohr parameter maps the full complexity
of a generic ion channel response into a single combination of the relevant physical parameters of
the system.
7
A
n
=
0
n
=
1
n
=
2
n
=
3
n
=
4
-
30
-
25
-
20
-
15
-
10
-
5
0
10
-
12
10
-
10
10
-
8
10
-
6
10
-
4
10
-
2
10
0
β
ε
leakiness
B
-
30
-
25
-
20
-
15
-
10
-
5
0
0.0
0.2
0.4
0.6
0.8
1.0
βε
dynamic range
C
Prediction
Dose-response data
-
30
-
25
-
20
-
15
-
10
-
5
0
10
-
10
10
-
9
10
-
8
10
-
7
10
-
6
10
-
5
10
-
4
β
ε
[
EC
50
](
M
)
D
-
30
-
25
-
20
-
15
-
10
-
5
0
0.0
0.5
1.0
1.5
2.0
βε
h
Figure 4: Theoretical prediction and experimental measurements for mutant nAChR ion channel
characteristics.
The open squares mark the
β ε
values of the five dose response curves from Fig 3A. (A)
The leakiness given by Eq (5) increases exponentially with each mutation. (B) The dynamic range from
Eq (6) is nearly uniform for all mutants. (C) The
[
EC
50
]
decreases exponentially with each mutation. (D)
The effective Hill coefficient
h
is predicted to remain approximately constant.
[
EC
50
]
and
h
offer a direct
comparison between the best-fit model predictions (open squares) and the experimental measurements (solid
circles) from Fig 3A. While the
[
EC
50
]
matches well between theory and experiment, the effective Hill
coefficient
h
is significantly noisier.
Full Spectrum of nAChR Gating Energy Mutants
We next consider the entire range of nAChR phenotypes achievable by only modifying the gating
free energy
ε
of the wild type ion channel. For instance, any combination of nAChR pore mutations
would be expected to not affect the ligand dissociation constants and thus yield an ion channel
within this class (see Supporting Information section A.4 for one such example). For concreteness,
we focus on how the
ε
parameter tunes key features of the dose-response curves, namely the
leakiness, dynamic range,
[
EC
50
]
, and effective Hill coefficient
h
(see Eqs (5)-(12)), although we
note that other important phenotypic properties such as the intrinsic noise and capacity have also
been characterized for the MWC model.
10
Fig 4 shows these four characteristics, with the open
squares representing the properties of the five best-fit dose-response curves from Fig 3A.
Fig 4A implies that all of the mutants considered here have negligible leakiness; according
to the MWC parameters found here, the probability that the wild type channel (
β ε
(
0
)
=
−
23
.
7)
will be open is less than 10
−
10
. Experimental measurements have shown that such spontaneous
openings occur extremely infrequently in nAChR,
22
although direct measurement is difficult for
8
such rare events. Other mutational analysis has predicted gating energies around
β ε
(
0
)
≈−
14
(corresponding to a leakiness of 10
−
6
),
12
but we note that such a large wild type gating energy
prohibits the five mutants in Fig 3 from being fit as a single class of mutants with the same
K
O
and
K
C
values (see Supporting Information section C.2). If this large wild type gating energy is correct,
it may imply that the L251S mutation also affects the
K
O
and
K
C
parameters, though the absence
of error bars on the original data make it hard to quantitatively assess the underlying origins of
these discrepancies.
Fig 4B asserts that all of the mutant ion channels should have full dynamic range except for
the wild type channel, which has a dynamic range of 0
.
91. In comparison, the measured dynamic
range of wild type nAChR is 0
.
95, close to our predicted value.
12
Note that only when the dynamic
range approaches unity does the normalized current become identical to
p
open
; for lower values,
information about the leakiness and dynamic range is lost by only measuring normalized currents.
We compare the
[
EC
50
]
(Fig 4C) and effective Hill coefficient
h
(Fig 4D) with the nAChR
data by interpolating the measurements (see Supporting Information section C.3) in order to pre-
cisely determine the midpoint and slope of the response. The
[
EC
50
]
predictions faithfully match
the data over four orders of magnitude. Because each additional mutation lowers the
[
EC
50
]
by
approximately one decade, the analytic form Eq (11) implies that
ε
increases by roughly 5
k
B
T
per mutation, enabling the ion channel to open more easily. In addition to the L251S mutation
considered here, another mutation (L251T) has also been found to shift
[
EC
50
]
by a constant log-
arithmic amount (see Supporting Information section A.4).
23
We also note that many biological
systems logarithmically tune their responses by altering the energy difference between two al-
losteric states, as seen through processes such as phosphorylation and calmodulin binding.
24
This
may give rise to an interesting interplay between physiological time scales where such processes
occur and evolutionary time scales where traits such as the
[
EC
50
]
may be accessed via mutations
like those considered here.
25
Lastly, the Hill coefficients of the entire class of mutants all lie between 1.5 and 2.0 except
for the
n
=
3 mutant whose dose-response curve in Fig 3A is seen to be flatter than the MWC
prediction. We also note that if the L251S mutation moderately perturbs the
K
O
and
K
C
values,
it would permit fits that more finely attune to the specific shape of each mutant’s data set. That
said, the dose-response curve for the
n
=
3 mutant could easily be shifted by small changes in
the measured values and hence without recourse to error bars, it is difficult to make definitive
statements about the value adopted for
h
for this mutant.
Note that the simplified expressions Eqs (9)-(12) for the leakiness, dynamic range,
[
EC
50
]
,
and effective Hill coefficient apply when 1
e
−
β ε
(
K
C
K
O
)
m
, which given the values of
K
C
and
K
C
for the nAChR mutant class translates to
−
22
.
β ε
.
−
5. The
n
=
1, 2, and 3 mutants all
fall within this range, and hence each subsequent mutation exponentially increases their leakiness
and exponentially decreases their
[
EC
50
]
, while their dynamic range and effective Hill coefficient
remain indifferent to the L251S mutation. The
β ε
parameters of the
n
=
0 and
n
=
4 mutants lie
at the edge of the region of validity, so higher order approximations can be used to more precisely
fit their functional characteristics (see Supporting Information section B).
9
4
K
C
c
1
+
e
-
βε
closed
open
ST
ATE
WEIGHT
STATE
WEIGHT
ST
ATE
WEIGHT
K
C
c
4
e
-
βε
e
-
βε
2
K
C
c
6
e
-
βε
4
K
C
c
e
-
βε
3
K
C
c
4
e
-
βε
4
K
C
c
1
+
e
-
βε
*
2
K
C
c
1
+
e
-
βε
2
K
C
c
1
+
*
3
K
C
c
1
+
e
-
βε
K
C
c
1
+
*
K
C
c
1
+
e
-
βε
3
K
C
c
1
+
*
4
K
O
c
1
+
4
K
O
c
1
+
*
2
K
O
c
1
+
2
K
O
c
1
+
*
3
K
O
c
1
+
K
O
c
1
+
*
K
O
c
1
+
3
K
O
c
1
+
*
= wild type subunit
= mutated subunit
n
0
1
2
3
4
Figure 5: States and weights for mutant CNGA2 ion channels.
CNGA2 mutants with
m
=
4 subunits
were constructed using
n
mutated (light red) and
m
−
n
wild type subunits (purple). The affinity between
the wild type subunits to ligand in the open and closed states (
K
O
and
K
C
) is stronger than the affinity of
the mutated subunits (
K
∗
O
and
K
∗
C
). The weights shown account for all possible ligand configurations, with
the inset explicitly showing all of the closed states for the wild type (
n
=
0) ion channel from Fig 2B. The
probability that a receptor with
n
mutated subunits is open is given by its corresponding open state weight
divided by the sum of open and closed weights in that same row.
Heterooligomeric CNGA2 Mutants can be Categorized using an Expanded
MWC Model
The nAChR mutant class discussed above had two equivalent ligand binding sites, and only the
gating free energy
β ε
varied for the mutants we considered. In this section, we use beautiful
data for the olfactory CNGA2 ion channel to explore the unique phenotypes that emerge from a
heterooligomeric ion channel whose subunits have different ligand binding strengths.
The wild type CNGA2 ion channel is made up of four identical subunits, each with one bind-
ing site for the cyclic nucleotide ligands cAMP or cGMP.
26
Within the MWC model, the proba-
bility that this channel is open is given by Eq (1) with
m
=
4 ligand binding sites (see Fig 2B).
Wongsamitkul
et al.
constructed a mutated subunit with lower affinity for ligand and formed
tetrameric CNGA2 channels from different combinations of mutated and wild type subunits (see
Fig 5).
13
Since the mutation specifically targeted the ligand binding sites, these mutant subunits
were postulated to have new ligand dissociation constants but the same free energy difference
β ε
.
We can extend the MWC model to compute the probability
p
open
that these CNGA2 constructs
will be open. The states and weights of an ion channel with
n
mutated subunits (with ligand
affinities
K
∗
O
and
K
∗
C
) and
m
−
n
wild type subunits (with ligand affinities
K
O
and
K
C
) is shown in
Fig 5, and its probability to be open is given by
p
open
(
c
) =
(
1
+
c
K
O
)
m
−
n
(
1
+
c
K
∗
O
)
n
(
1
+
c
K
O
)
m
−
n
(
1
+
c
K
∗
O
)
n
+
e
−
β ε
(
1
+
c
K
C
)
m
−
n
(
1
+
c
K
∗
C
)
n
.
(15)
10
A
n
=
0
n
=
1
n
=
2
n
=
3
n
=
4
10
-
7
10
-
6
10
-
5
10
-
4
10
-
3
10
-
2
0.0
0.2
0.4
0.6
0.8
1.0
[
cGMP
](
M
)
normalized current
B
-
6
-
4
-
2
0
2
4
6
0.0
0.2
0.4
0.6
0.8
1.0
Bohr parameter,
F
CNGA2
(
k
B
T
units
)
p
open
Figure 6: Normalized currents for CNGA2 ion channels with a varying number
n
of mutant subunits.
(A) Dose-response curves for CNGA2 mutants comprised of 4
−
n
wild type subunits and
n
mutated subunits
with weaker affinity for the ligand cGMP.
13
Once the free energy
ε
and the ligand dissociation constants of
the wild type subunits (
K
O
and
K
C
) and mutated subunits (
K
∗
O
and
K
∗
C
) are fixed, each mutant is completely
characterized by the number of mutated subunits
n
in Eq (15). Theoretical best-fit curves are shown using
the parameters
K
O
=
1
.
2
×
10
−
6
M,
K
C
=
20
×
10
−
6
M,
K
∗
O
=
500
×
10
−
6
M,
K
∗
C
=
140
×
10
−
3
M, and
β ε
=
−
3
.
4. (B) Data from all five mutants collapses onto a single master curve when plotted as a function
of the Bohr parameter given by Eq (13). See Supporting Information section C for details on the fitting.
Measurements have confirmed that the dose-response curves of the mutant CNGA2 channels only
depend on the total number of mutated subunits
n
and not on the positions of those subunits (for
example both
n
=
2 with adjacent mutant subunits and
n
=
2 with mutant subunits on opposite
corners have identical dose-response curves).
13
Fig 6A shows the normalized current of all five CNGA2 constructs fit to a single set of
K
O
,
K
C
,
K
∗
O
,
K
∗
C
, and
ε
parameters. Since the mutated subunits have weaker affinity to ligand (leading to the
larger dissociation constants
K
∗
O
>
K
O
and
K
∗
C
>
K
C
), the
[
EC
50
]
shifts to the right as
n
increases.
As in the case of nAChR, we can collapse the data from this family of mutants onto a single master
curve using the Bohr parameter
F
CNGA2
(
c
)
from Eqs (13) and (15), as shown in Fig 6B.
Although we analyze the CNGA2 ion channels in equilibrium, we can glimpse the dynamic
nature of the system by computing the probability of each channel conformation. Fig 7A shows
the ten possible states of the wild type (
n
=
0) channel, the five open states
O
j
and the five closed
states
C
j
with 0
≤
j
≤
4 ligands bound. Fig 7B shows how the probabilities of these states are all
significantly shifted to the right in the fully mutated (
n
=
4) channel since the mutation diminishes
the channel-ligand affinity. The individual state probabilities help determine which of the interme-
diary states can be ignored when modeling. One extreme simplification that is often made is to
consider the Hill limit, where all of the states are ignored save for the closed, unbound ion channel
(
C
0
) and the open, fully bound ion channel (
O
4
). The drawbacks of such an approximation are two-
fold: (1) at intermediate ligand concentrations (
c
∈
[
10
−
7
,
10
−
5
]
M for
n
=
0 and
c
∈
[
10
−
4
,
10
−
2
]
M
for
n
=
4) the ion channel spends at least 10% of its time in the remaining states which results in
fundamentally different dynamics than what is predicted by the Hill response and (2) even in the
limits such as
c
=
0 and
c
→
∞
where the
C
0
and
O
4
states dominate the system, the Hill limit
ignores the leakiness and dynamic range of the ion channel (requiring them to exactly equal 0 and
1, respectively), thereby glossing over these important properties of the system.
11
A
O
0
O
1
O
2
O
3
O
4
C
0
C
1
C
2
C
3
C
4
10
-
7
10
-
6
10
-
5
10
-
4
10
-
3
10
-
2
10
-
12
10
-
10
10
-
8
10
-
6
10
-
4
10
-
2
10
0
[
cGMP
](
M
)
n
=
0 state probabilities
B
10
-
6
10
-
5
10
-
4
10
-
3
10
-
2
10
-
1
10
-
12
10
-
10
10
-
8
10
-
6
10
-
4
10
-
2
10
0
[
cGMP
](
M
)
n
=
4 state probabilities
Figure 7: Individual state probabilities for the wild type and mutant CNGA2 ion channels.
(A) The
state probabilities for the wild type (
n
=
0) ion channel. The subscripts of the open (
O
j
) and closed (
C
j
)
states represent the number of ligands bound to the channel. States with partial occupancy, 1
≤
j
≤
3, are
most likely to occur in a narrow range of ligand concentrations
[
cGMP
]
∈
[
10
−
7
,
10
−
5
]
M, outside of which
either the completely empty
C
0
or fully occupied
O
4
states dominate the system. (B) The state probabilities
for the
n
=
4 channel. Because the mutant subunits have a weaker affinity to ligand (
K
∗
O
>
K
O
and
K
∗
C
>
K
C
),
the state probabilities are all shifted to the right.
Characterizing CNGA2 Mutants based on Subunit Composition
We now turn to the leakiness, dynamic range,
[
EC
50
]
, and effective Hill coefficient
h
of a CNGA2
ion channel with
n
mutated and
m
−
n
wild type subunits. Detailed derivations for the following
results are provided in Supporting Information section B:
leakiness
=
1
1
+
e
−
β ε
(16)
dynamic range
=
1
1
+
e
−
β ε
(
K
O
K
C
)
m
−
n
(
K
∗
O
K
∗
C
)
n
−
1
1
+
e
−
β ε
(17)
[
EC
50
]
≈
e
−
β ε
/
m
K
O
n
=
0
e
−
2
β ε
/
m
K
O
K
∗
O
K
O
+
K
∗
O
n
=
m
2
e
−
β ε
/
m
K
∗
O
n
=
m
(18)
h
≈
m
n
=
0
m
2
n
=
m
2
m
n
=
m
.
(19)
Note that we recover the original MWC model results Eqs (5)-(12) for the
n
=
0 wild type ion
channel. Similarly, the homooligomeric
n
=
m
channel is also governed by the MWC model with
K
O
→
K
∗
O
and
K
C
→
K
∗
C
. We also show the
[
EC
50
]
and
h
formulas for the
n
=
m
2
case to demonstrate
the fundamentally different scaling behavior that this heterooligomeric channel exhibits with the
MWC parameters.
Fig 8A shows that all of the CNGA2 mutants have small leakiness, which can be understood
from their small
ε
value and Eq (16). In addition, the first term in the dynamic range Eq (17) is
12