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
‡
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 coef-
ficient, and explore the full spectrum of phenotypes that are accessible through muta-
tions. Specifically, we consider mutations 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 perspec-
tive offered by the MWC model allows us, perhaps surprisingly, to collapse the plethora
of dose-response data from different classes of ion channels into a universal family of
curves.
1
.
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1 Introduction
Ion channels are signaling proteins responsible for a huge variety of physiological functions
ranging 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 classification 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 acetylcholine receptor (nAChR) has two ligand binding sites for acetylcholine outside the
cytosol. (B) The homotetrameric 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 absence of ligand and open when bound to ligand.
The MWC model has long been used in the contexts of both nAChR and CNG ion
channels.
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 channel’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.
2
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Purohit and Auerbach used combinations of mutations to infer the nAChR gating energy,
finding that unliganded nAChR channels open up for a remarkably 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 coefficient),
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 dissociation 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 increase 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 framework with which we can explore the full expanse of ion channel phenotypes avail-
able through mutations. Using this methodology, we: (1) analytically compute important
ion channel characteristics, 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
3
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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 (orange). (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
between the closed and open conformations of the ion channel.
p
open
(
c
)
that the channel will be open at a ligand concentration
c
. For an ion channel with
m
identical ligand 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
(
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,
(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 (
<
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
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).
Current measurements are often reported as
normalized
current, implying that the cur-
4
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rent 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
,
(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
(
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
maximize 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 maximal 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 eval-
uated 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 concentrations for which the normalized current transitions from 0 to 1.
5
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In the limit
1
e
−
β
(
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
(9)
dynamic range
≈
1
(10)
[
EC
50
]
≈
e
−
β/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
βγδ
, 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
by a fixed
amount.
To test this idea, we analyze the nAChR data at various concentrations
c
of the ligand
acetylcholine using the MWC model Eq (1) with
m
= 2
ligand binding sites. Because the
L251S mutation 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
(
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
6
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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 mutation.
(A) Normalized currents of mutant nAChR ion channels at different concen-
trations of the agonist acetylcholine (ACh).
17
The curves from right to left show a receptor with
n
= 0
(wild type),
n
= 1
(
α
2
βγ
∗
δ
),
n
= 2
(
α
∗
2
βγδ
),
n
= 3
(
α
2
β
∗
γ
∗
δ
∗
), and
n
= 4
(
α
∗
2
βγ
∗
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
(
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
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.
the nAChR receptor is open as
p
open
(
c
)
≡
1
1 +
e
−
βF
(
c
)
,
(13)
where this last equation defines the
Bohr parameter
21
F
(
c
) =
−
k
B
T
log
e
−
β
(
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
). 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
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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
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
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
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; ac-
cording to the MWC parameters found here, the probability that the wild type channel
(
β
(0)
=
−
23
.
7
) will be open is less than
10
−
10
. Experimental measurements have shown
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that such spontaneous openings occur extremely infrequently in nAChR,
22
although direct
measurement is difficult for such rare events. Other mutational analysis has predicted gating
energies around
β
(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 precisely 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
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 logarithmic 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 allosteric 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
(
K
C
K
O
)
m
, which given the values of
K
C
and
K
C
for the nAChR mutant class translates to
−
22
.
β
.
−
5
. The
n
= 1
, 2, and
3 mutants all fall within this range, and hence each subsequent mutation exponentially in-
creases their leakiness and exponentially decreases their
[
EC
50
]
, while their dynamic range
and effective Hill coefficient remain indifferent to the L251S mutation. The
β
parameters of
the
n
= 0
and
n
= 4
mutants lie at the edge of the region of validity, so higher order approx-
imations can be used to more precisely fit their functional characteristics (see Supporting
Information section B).
9
.
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It is made available under a
(which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint
.
http://dx.doi.org/10.1101/102194
doi:
bioRxiv preprint first posted online Jan. 22, 2017;
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 Ex-
panded MWC Model
The nAChR mutant class discussed above had two equivalent ligand binding sites, and only
the gating free energy
β
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
binding site for the cyclic nucleotide ligands cAMP or cGMP.
26
Within the MWC model, the
probability 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
.
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
10
.
CC-BY 4.0 International license
It is made available under a
(which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint
.
http://dx.doi.org/10.1101/102194
doi:
bioRxiv preprint first posted online Jan. 22, 2017;
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
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
=
−
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.
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
(
1 +
c
K
C
)
m
−
n
(
1 +
c
K
∗
C
)
n
.
(15)
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
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 intermediary 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
11
.
CC-BY 4.0 International license
It is made available under a
(which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint
.
http://dx.doi.org/10.1101/102194
doi:
bioRxiv preprint first posted online Jan. 22, 2017;