of 33
Supporting Information for
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
Phone: (626) 395-3374
S1
A Additional Ion Channel Data
In this section, we explore some of the additional experimental measurements available for the
nAChR and CNGA2 systems studied above and elaborate on several calculations mentioned
in the text. In A.1, we analyze the time scale required for an ion channel to reach equilibrium.
In A.2, we present data on additional L251S nAChR mutants. Using these mutants, we
examine the approximation made in the text that only the
total
number of mutations, and
not the identity of the subunits mutated, influences the resulting nAChR mutant behavior.
In A.3, we examine
p
open
(
c
)
for the classes of ion channels considered in the text and comment
on how this probability differs from the normalized current. In A.4, we examine data from
a similar class of L251T mutations and show that their qualitative behavior is similar to the
L251S mutants. In A.5, we discuss measurements of combinations of CNGA2 ion channels.
A.1 Dynamics Towards Equilibrium
k
+
e
k
-
e
O
OL
k
on
[
L
]
C
CL
k
on
[
L
]
k
off
(
O
)
k
off
(
C
)
k
+
o
k
-
o
Figure S1: Rates for an ion channel with one ligand binding site.
The ion channel tends
to transition from the closed (
C
) state to the open (
O
) state after binding to ligand (
L
). We assume
both ion channel states have the same diffusion-limited on-rate
k
on
= 10
9
1
M
·
s
. The remaining rates
of the bound states should satisfy
k
(
C
)
off
> k
(
O
)
off
and
k
o
+
> k
o
so that ligand binding drives the ion
channel to the open state
OL
.
In this section we derive an exact expression for the time constant for which an ion
channel with one ligand binding site will come to equilibrium. This analysis can be readily
extended numerically to include multiple ligand binding sites.
Fig S1 shows the rates between the four possible ion channel states: the unbound open
(
O
) and closed (
C
) states as well as the bound open (
OL
) and closed (
CL
) states. We
assume that there is a sufficient ligand
[
L
]
in the system so that when the ligand binds to
the ion channels its concentration does not appreciably diminish. Hence the rate equations
for the system can be written in matrix form (with bold denoting vectors and matrices) as
d
E
dt
=
KE
(S1)
where the right hand side represents the product of the transition matrix
K
=
(
k
e
+
+
k
on
[
L
])
k
(
C
)
off
k
e
0
k
on
[
L
]
(
k
o
+
+
k
(
C
)
off
)
0
k
o
k
e
+
0
(
k
e
+
k
on
[
L
])
k
(
O
)
off
0
k
o
+
k
on
[
L
]
(
k
o
+
k
(
O
)
off
)
(S2)
S2
and the vector representing the occupancy of each ion channel state
E
=
[
C
]
[
CL
]
[
O
]
[
OL
]
.
(S3)
The matrix
K
can be decomposed as
K
=
V
1
Λ
V
(S4)
where
V
’s columns are the eigenvectors of
K
and
Λ
is a diagonal matrix whose entries
are the eigenvalues of
K
. In general, it is known that the eigenvalues of such a matrix
K
representing the dynamics of any graph such as Fig S1 has one eigenvalue that is 0 while the
remaining eigenvalues are non-zero and have negative real parts.
1
(Indeed, because all of the
columns of
K
add up to zero,
K
is not full rank and hence one of its eigenvalues must be
zero.) Defining the vector
̃
E
V E
=
̃
E
1
̃
E
2
̃
E
3
̃
E
4
,
(S5)
Eq (S1) can be rewritten as
d
̃
E
dt
=
Λ
̃
E
.
(S6)
If the eigenvalues of
Λ
are
λ
1
,
λ
2
,
λ
3
, and 0, then
̃
E
j
=
c
j
e
λ
j
t
for
j
= 1
,
2
,
3
and
̃
E
4
=
c
4
where the
c
j
’s are constants determined by initial conditions. Since the
̃
E
j
’s are linear
combinations of
[
C
]
,
[
CS
]
,
[
O
]
,
and
[
OS
]
, this implies that the
1
λ
1
,
1
λ
2
,
and
1
λ
3
(or
1
<
(
λ
j
)
if the eigenvalues are complex) are the time scales for the normal modes of the system to
come to equilibrium, with the largest value representing the time scale
τ
for the entire system
to reach equilibrium,
τ
= max
(
1
λ
1
,
1
λ
2
,
1
λ
3
)
.
(S7)
Although the eigenvalues of this matrix can be calculated in closed form, as roots of a
cubic function, the full expression is complicated. Instead, we write the Taylor expansion of
λ
1
,
λ
2
, and
λ
3
in the limit
k
o
+
→∞
, since we suspect that the transition from
CS
OS
is
extremely fast. In this limit, the
λ
j
s take the forms
λ
1
=
(
k
(
O
)
off
+
k
on
[
L
]) +
O
(
1
k
o
+
)
(S8)
λ
2
=
(
k
(
C
)
off
+
k
o
+
k
o
+
) +
O
(
1
k
o
+
)
(S9)
λ
3
=
(
k
on
[
L
] +
k
e
+
k
e
+
) +
O
(
1
k
o
+
)
.
(S10)
S3
Fig S2 shows an example of how the system attains its equilibrium starting from a
random initial condition. The exact time scale Eq (S7) using the matrix eigenvalues leads to
τ
= 1
.
1
×
10
3
s, which is very close to the approximation using Eqs (S8)-(S10) which yields
τ
(
approx
)
= 1
.
0
×
10
3
s. The exact time scale is shown in Fig S2 as a dashed line, and states
achieve near total equilibrium by
t
= 10
2
s.
As a point of reference for this time scale described above for the system to come to
equilibrium, there are two other relevant times scales for an ion channel: (1) the time scale
for an ion channel to switch between the open and closed conformations and (2) the time
scale for an ion channel to stay in its open conformation before switching to the closed
conformation. The former occurs on the microsecond scale for nAChR,
2
while the latter
occurs on the millisecond scale.
3,4
Thus, the time to transition between the closed and open
conformations can be ignored, and the system reaches equilibrium after only a few transitions
between the open and closed states.
O
C
OL
CL
0.000
0.002
0.004
0.006
0.008
0.010
10
-
5
10
-
4
10
-
3
10
-
2
10
-
1
10
0
time
(
sec
)
probability
Figure S2: Kinetics of a system heading towards equilibrium.
The relative probabilities
of the four states are computed using Eqs (S1) and (S2) and the rate constants
k
on
[
L
] = 10
3
1
s
,
k
(
O
)
off
= 10
2
1
s
,
k
(
C
)
off
= 10
4
1
s
,
k
o
+
= 10
4
1
s
,
k
o
= 10
1
s
,
k
e
+
= 10
1
s
, and
k
e
= 10
3
1
s
. Note that the
rate constants must satisfy the cycle condition: the product of rates moving clockwise equals the
product of rates going counterclockwise.
5
The dashed line indicates the exact time scale Eq (S7) for
the system to reach equilibrium. Initial conditions were chosen randomly as
p
O
= 0
.
005
,
p
C
= 0
.
45
,
p
OL
= 0
.
54
, and
p
CL
= 0
.
005
.
Lastly, we compute the fractional occupancy of the four states ion channel states in steady
state,
d
E
dt
=
0
. We first make the standard assumption that the system is not expending
energy to drive a cyclic flux in the system. Formally, this implies that the rate constants
satisfy the cycle condition: the product of rates moving clockwise in Fig S1 equals the
product of rates going counterclockwise,
5
k
on
[
L
]
k
o
k
(
C
)
off
k
e
+
=
k
e
k
(
O
)
off
k
on
[
L
]
k
o
+
.
(S11)
S4
With this condition, the fractional occupancy of each state is given by
[
C
] =
k
e
k
e
+
(
1 +
k
on
[
L
]
k
(
O
)
off
)
+
k
e
k
e
+
(
1 +
k
on
[
L
]
k
(
C
)
off
)
(S12)
[
CL
] =
k
e
k
e
+
k
on
[
L
]
k
(
C
)
off
(
1 +
k
on
[
L
]
k
(
O
)
off
)
+
k
e
k
e
+
(
1 +
k
on
[
L
]
k
(
C
)
off
)
(S13)
[
O
] =
1
(
1 +
k
on
[
L
]
k
(
O
)
off
)
+
k
e
k
e
+
(
1 +
k
on
[
L
]
k
(
C
)
off
)
(S14)
[
OL
] =
k
on
[
L
]
k
(
O
)
off
(
1 +
k
on
[
L
]
k
(
O
)
off
)
+
k
e
k
e
+
(
1 +
k
on
[
L
]
k
(
C
)
off
)
,
(S15)
A system in steady state which satisfies the cycle condition must necessarily be in thermody-
namic equilibrium,
6
which implies that these fractional occupancies must be identical to the
result derived from the Boltzmann distribution (see Fig 2). And indeed, this correspondence
is made explicit if we define
e
β
=
k
e
k
e
+
(S16)
K
O
=
k
(
O
)
off
k
on
(S17)
K
C
=
k
(
C
)
off
k
on
.
(S18)
In this way, the MWC parameters can be defined through the ratios of the rate parameters
of the system.
A.2 Additional nAChR Mutants
In addition to the five constructs shown in Fig 3, namely
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
), Labarca
et al.
constructed multiple
other ion channel mutants listed in Table S1.
3
While complete dose-response curves are not
available for these other constructs, their
[
EC
50
]
values were measured. Using the
K
O
and
K
C
values for this entire class of mutants given in Fig 3, we can use the
[
EC
50
]
measurements
to fit the
β
value of each mutant, thereby providing us with a complete description of each
mutant.
In particular, we can predict the dose-response curves of each of these mutants, as shown
in Fig S3. We overlay the data from Fig 3 on top of these theoretical curves, where mutants
with the same total number
n
of mutations are drawn as shades of the same color. Note that
S5
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
Figure S3: Categorizing the full set of ion channel mutants.
Using the best-fit
K
O
and
K
C
values obtained from the five mutants in Fig 3, we can use the measured
[
EC
50
]
value for each
mutant in Table S1 to determine its
β
parameter. Thus, a single data point for each mutant
enables us to predict its complete dose-response curve. All mutants with the same total number
n
of mutations are plotted in shades of the same color, together with the complete dose-response
curves from Fig 3. Note that while each mutant family spans a range of
[
EC
50
]
values, the classes
are distinct and do not overlap.
there was some error in the original measurements, since the reported
[
EC
50
]
value for the
n
= 4
(
α
2
βγ
δ
) mutant shown in purple dots in Fig S3 should clearly be less than
10
9
M,
even though it was given as
(2
.
0
±
0
.
6)
×
10
9
M in Ref. 3.
Table S1: Dose-response relations for mouse muscle ACh receptors containing various
numbers of mutated L251S subunits (
n
).
Mutated subunits are indicated by an asterisk (
).
Standard error of the mean for
[
EC
50
]
was less than 10% of the mean, except where given. Responses
for the
α
2
β
γ
δ
mutant were too small for reliable measurements. Reproduced from Ref. 3.
n
subunits
[
EC
50
]
(nM)
0
α
2
βγδ
24,010
1
αα
βγδ
1,290
α
2
β
γδ
531
α
2
βγ
δ
1,910
α
2
βγδ
486
2
α
2
βγδ
202
α
2
β
γ
δ
49.7
α
2
β
γδ
208
±
69
α
2
βγ
δ
42.7
3
α
2
β
γδ
10.3
α
2
βγ
δ
15.1
α
2
βγδ
8
.
4
±
1
.
3
α
2
β
γ
δ
9
.
8
±
1
.
3
4
α
2
β
γδ
2.3
α
2
βγ
δ
2
.
0
±
0
.
6
5
α
2
β
γ
δ
<
1
S6
Fig S3 demonstrates that not all subunit mutations cause a tenfold decrease in
[
EC
50
]
,
but rather that there is a small spread in
[
EC
50
]
depending on precisely which subunit was
mutated. This variation is not unreasonable given that
α
2
βγδ
nAChR is a heteropentamer.
Indeed, such subunit-dependent spreading in
[
EC
50
]
values has also been seen in other het-
eromeric ion channels
7,8
but is absent within homomeric ion channels such as the CNGA2
ion channel explored in the text.
4
To explore this subunit-dependent shift in the dose-response curves, we now relax the
assumption that mutating any of the four nAChR subunits results in an identical increase
of roughly
5
k
B
T
to the allosteric gating energy

. Instead, we allow each type of subunit to
shift

by a different amount upon mutation. We begin by writing the

parameter of wild
type nAChR as

α
2
βγδ
= 2

α
+

β
+

γ
+

δ
,
(S19)
where

j
denotes the gating energy contribution from subunit
j
and we have assumed that
the five subunits independently contribute to channel gating. Upon mutation, we define the
free energy differences of each type of subunit as

α

α

α
(S20)

β

β

β
(S21)

γ

γ

γ
(S22)

δ

δ

δ
,
(S23)
where

j
denotes the gating energy from the mutated subunit
j
.
The allosteric energy of any nAChR mutant can be found using the wild type energy

α
2
βγδ
=
23
.
7
k
B
T
from the main text together with

α
,

β
,

δ
, and

γ
. For example,
the gating energy of
α
2
β
γδ
is given by

α
2
β
γδ
=

α
2
βγδ
+ ∆

β
while that of
α
2
βγδ
is given
by

α
2
βγδ
=

α
2
βγδ
+ 2∆

α
+ ∆

δ
.
Using the measured
[
EC
50
]
values of all the mutants in Table S1, we can fit the four

j
’s to determine how the different subunits increase the ion channel gating energy upon
mutation. We find the values

α
= 4
.
4
k
B
T
,

β
= 5
.
3
k
B
T
,

γ
= 5
.
4
k
B
T
, and

δ
= 5
.
2
k
B
T
, which show a small spread about the value of roughly
5
k
B
T
found in
the text by assuming that all four

j
’s are identical. To show the goodness of fit, we can
compare the
[
EC
50
]
values from this model to the experimental measurements in Table S1,
as shown in Fig S4.
A.3
p
open
(
c
)
Curves
Although the dose-response curves we analyze for nAChR were all presented using normalized
current, the underlying physical process - namely, the opening and closing of the ion channel
- is not required to go from 0 to 1. Fig S5 shows the normalized dose-response curves from
Fig 3A together with the average probability that each ion channel mutant will be open,
p
open
(
c
)
. Note that these
p
open
(
c
)
curves have exactly the same shape as the normalized
current curves but are compressed in the vertical direction to have the leakiness and dynamic
range specified by Fig 4A and B.
From the viewpoint of these
p
open
(
c
)
curves, various nuances of this ion channel class
S7
10
0
10
1
10
2
10
3
10
4
10
0
10
1
10
2
10
3
10
4
predicted
[
EC
50
]
measured
[
EC
50
]
Figure S4: Mutating different nAChR subunits changes the gating energy

by different
amounts.
Using a linear model where each subunit independently contributes to channel gating,
we fit all of the
[
EC
50
]
values in Table S1 to compute the increase of the gating energy

when each
subunit of
α
2
βγδ
nAChR is mutated (see Eqs (S20)-(S23)). Upon mutation, a subunit of type
j
increases the gating energy by

j
, where

α
= 4
.
4
k
B
T
,

β
= 5
.
3
k
B
T
,

γ
= 5
.
4
k
B
T
, and

δ
= 5
.
2
k
B
T
. For each mutant in Table S1, the
[
EC
50
]
from the model can be compared to the
corresponding experimental measurement, with the black dashed line denoting the line of equality
y
=
x
.
stand out more starkly. For example, the four mutant channels have
p
max
open
1
, noticeably
larger than the
p
max
open
0
.
9
value of the wild type channel. In addition, the
n
= 4
mutant is
the only ion channel with non-negligible leakiness, and Fig 4A suggests that an
n
= 5
mutant
with all five subunits carrying the L251S mutation would have an even larger leakiness value
greater than
1
2
. In other words, the
n
= 5
ion channel is open more than half the time even
in the absence of ligand, which could potentially cripple or kill the cell. This may explain
why Labarca
et al.
made the
n
= 5
strain but were unable to measure its properties.
3
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
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
)
p
open
Figure S5: Probability that an nAChR mutant will be open.
(A) Normalized current
curves of the five nAChR mutants from Fig 3A. (B) The probability that each ion channel will be
open is given by Eq (1). Note that the wild type ion channel has a smaller dynamic range and the
n
= 4
mutant has a noticeably larger leakiness than the other mutants.
Fig S6 repeats this same analysis for the CNGA2 dose-response curves from Fig 6A. In
this case, all of the ion channel mutants have uniformly small values of
p
min
open
0
.
03
and
S8
uniformly large
p
max
open
1
, as indicated by Fig 8A and B. Therefore, the
p
open
(
c
)
curves look
very similar to the normalized currents.
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
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
)
p
open
Figure S6: Probability that a CNGA2 mutant will be open.
(A) CNGA2 dose-response
curves from Fig 6A. (B) The probability that each ion channel will be open is given by Eq (1). Since
all of the channels have small leakiness (
0
.
03
) and large dynamic range, the
p
open
(
c
)
curves are
nearly identical to the normalized current curves.
A.4 nAChR L251T Mutation
In this section we consider a separate nAChR data from the one considered in the main
paper. Filatov and White constructed nAChR ion channel mutants closely related to those
of Labarca
et al.
but employing a L251T mutation.
9
They measured the
[
EC
50
]
of multiple
such constructs with the L251T mutation on different subsets of nAChR subunits, with the
results shown in Fig S7A as a function of the total number of mutated subunits
n
.
As in the case of the L251S mutations from Labarca (see Fig S3 and Table S1), there was
some variation in
[
EC
50
]
between different mutants with the same total number of muta-
tions
n
, but the entire class of mutants is well approximated as having
[
EC
50
]
exponentially
decrease with each additional mutation. Utilizing our analytical formula for the
[
EC
50
]
of
nAChR, Eq (11), and assuming that each mutation changes

by a fixed amount

, the
shift in
[
EC
50
]
due to
n
mutations is given by
[
EC
50
] =
e
β
(

(0)
+
n

)
/
2
K
O
.
(S24)
We can fit the logarithm of the
[
EC
50
]
values in Fig S7A to a linear function going through
the wild type (
n
= 0
) data point to obtain

= 3
.
64
k
B
T
from the slope of this line. This
value is comparable to that found for the L251S mutation (where

= 5
k
B
T
).
With the gating energy now fully determined for any number of mutations
n
, and using
the
K
O
and
K
C
parameters from Fig 3, we now have a complete theoretical model of the
L251T nAChR mutant class. For example, we can plot the predicted dose-response curves
for all such mutants. Fig S7B shows these predictions together with experimentally mea-
sured responses from the wild type channel and three mutant constructs. The dose-response
predictions should match the data on average for the entire class of mutants, although indi-
vidual channel responses may be slightly off. For example, Fig S7A indicates that the
[
EC
50
]
S9
A
0
1
2
3
4
5
10
-
8
10
-
7
10
-
6
10
-
5
n
[
EC
50
](
M
)
B
n
=
0
n
=
1
n
=
2
n
=
3
n
=
4
α
2
βγδ
α
2
βγ
*
δ
α
2
βγδ
*
α
2
βγ
*
δ
*
10
-
9
10
-
8
10
-
7
10
-
6
10
-
5
10
-
4
10
-
3
10
-
2
0.0
0.2
0.4
0.6
0.8
1.0
[
ACh
](
M
)
normalized current
Figure S7: Effects of L251T mutations on nAChR.
(A)
[
EC
50
]
values for another class of
L251T mutations introduced at different combinations of subunits.
9
This data set is separate from
the L251S mutation considered in the main text. The
[
EC
50
]
mainly depends on the total number of
mutations,
[
EC
50
]
e
1
.
82
n
, although there is slight variation depending upon which subunits are
mutated. From Eq (S24), we find that each mutation imparts

= 3
.
64
k
B
T
. (B) Once the MWC
parameters have been fixed from the
[
EC
50
]
measurements, we can predict the full dose-response
curves for the entire class of L251T nAChR mutants. Overlaid on these theoretical prediction are
four experimentally measured response curves for the wild type (
α
2
βγδ
), two
n
= 1
single mutants
(
α
2
βγ
δ
and
α
2
βγδ
), and the
n
= 2
double mutant (
α
2
βγ
δ
). We expect the predicted dose-
response curves to match the data on average for the entire class of mutants, but Part A shows that
the
[
EC
50
]
of the
n
= 1
and
n
= 2
mutants will be overestimated while that of the
n
= 4
and
n
= 5
mutants will be underestimated. Asterisks (
) in the legend denote L251T mutations.
of the
n
= 1
and
n
= 2
mutants will be lower than predicted while that of the
n
= 4
and
n
= 5
mutants (whose dose-response data was not provided) will be higher than predicted.
A.5 Combining Multiple Ion Channels
In this section, we consider the dose-response curve for the case in which the cell harbors
both wild type and mutant ion channels. Given
n
1
ion channels whose dose-response curves
are governed by
p
1
,
open
(
c
)
and
n
2
ion channels with a different response
p
2
,
open
(
c
)
, the current
produced by the combination of these two ion channels is given by
current
n
1
p
1
,
open
(
c
) +
n
2
p
2
,
open
(
c
)
.
(S25)
Experimental measurements are computed on a relative scale so that the data runs from 0
to 1. Analytically, this amounts to subtracting the leakiness and dividing by the dynamic
range,
(
normalized current
)
tot
=
n
1
p
1
,
open
(
c
) +
n
2
p
2
,
open
(
c
)
n
1
p
min
1
,
open
n
2
p
min
2
,
open
n
1
p
max
1
,
open
+
n
2
p
max
2
,
open
n
1
p
min
1
,
open
n
2
p
min
2
,
open
.
(S26)
Wongsamitkul
et al.
constructed cells expressing both the
n
= 0
wild type ion channels and
the
n
= 4
fully mutated ion channels in a ratio of 1:1 (i.e.
n
1
=
n
2
) as shown in Fig S8.
4
Recall from Fig 8 that these ion channels have very small leakiness (
p
min
1
,
open
p
min
2
,
open
S10
0
) and nearly full dynamic range (
p
max
1
,
open
p
max
2
,
open
1
). This implies that
p
1
,
open
(
c
)
(
normalized current
)
1
and
p
2
,
open
(
c
)
(
normalized current
)
2
, so that the total normalized
current due to the combination of ion channels is given by
(
normalized current
)
tot
=
(
normalized current
)
1
+ (
normalized current
)
2
2
.
(S27)
Fig S8 shows that this simple prediction compares well to the measured data.
n
=
4
n
=
0,4
(
1:1
)
n
=
0
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
Figure S8: Normalized currents for combinations of CNGA2 ion channels.
Channel
currents of cells producing equal amounts of wild type
n
= 0
and the
n
= 4
mutant ion channels.
As shown in Eq (S27), the resulting dose-response curve equals the average of the
n
= 0
and
n
= 4
individual response curves.
S11
B Computing nAChR and CNGA2 Characteristics
In B.1, we derive Eqs (9)-(12), the approximations for the leakiness, dynamic range,
[
EC
50
]
,
and the effective Hill coefficient
h
for the general MWC model Eq (1). We begin by Taylor
expanding the well known exact expressions from Ref. 10 in the limit
1

e
β

(
K
C
K
O
)
m
,
which we found to be appropriate for both the nAChR and CNGA2 ion channels, and find
the lowest order approximations.
Following that, in B.2 we consider how mutations in the ligand dissociation constants
K
O
and
K
C
affect these four properties. We show that ion channel dose-response curves are
robust to changes in
K
O
and
K
C
aside from left-right shifts dictated by
[
EC
50
] =
e
β/m
K
O
.
This discussion complements the nAChR section of the text where we considered mutations
of the
β
parameter.
Lastly, in B.3 we determine how ion channels comprised of a mix of wild type subunits
(with ligand dissociation constants
K
O
and
K
C
) and mutant subunits (with dissociation con-
stants
K
O
and
K
C
) influences the four properties. Specifically, we focus on the analytically
tractable case where half of the subunits are wild type and the other half are mutated (see
Eqs (18) and (19) in the text).
B.1 Characteristics of the MWC Model
Using
1

e
β
, the leakiness Eq (5) can be expanded as
leakiness
=
1
1 +
e
β
e
β
.
(S28)
Therefore, ion channels have a very small leakiness which scales exponentially with
β
.
Fig S9A shows that this is a good approximation across the entire range of parameters
within the class of nAChR mutants,
24
β
≤−
4
.
The dynamic range Eq (6) can be similarly expanded to obtain
dynamic range
=
1
1 +
e
β
(
K
O
K
C
)
m
1
1 +
e
β
1
e
β
(
K
O
K
C
)
m
e
β
.
(S29)
Keeping only the lowest order term yields the approximation Eq (10) that ion channels
have full dynamic range. Fig S9B shows that keeping the first order terms in Eq (S29)
also captures the behavior of the wild type channel (
β
(0)
=
23
.
7
) and the
n
= 4
mutant
(
β
(4)
=
4
.
0
).
We next turn to the
[
EC
50
]
Eq (7), whose exact analytic formula is given by
11
[
EC
50
] =
K
O
1
λ
1
m
λ
1
m
K
O
K
C
(S30)
where
λ
=
2
(
p
min
open
+
p
max
open
)
e
β
(
p
min
open
+
p
max
open
)
.
(S31)
S12
A
Exact expression
Approximate form
-
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
-
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 S9: Exact and approximate expressions for nAChR characteristics.
The approx-
imations Eqs (S28)-(S34) (dashed, teal) are valid in the limit
1

e
β

(
K
C
K
O
)
m
where they
closely match the exact expressions (purple). (A) Leakiness can be approximated as an exponen-
tially increasing function of
β
. (B) To lowest order, the dynamic range of an ion channel should
approach unity, with deviations only at very large and very small
β
values. (C) The
[
EC
50
]
is an
exponentially decreasing function of
β
. (D) The effective Hill coefficient is roughly constant for all
mutants, but as with the dynamic range it decreases for very large and very small
β
values.
The limit
1

e
β

(
K
C
K
O
)
m
suggests that we Taylor expand this formula to lowest order
about
e
β
(
K
O
K
C
)
m
0
and
e
β
≈∞
, which yields
[
EC
50
]
K
O
K
C
K
O
(
(
1
1
2+
e
β
)
1
/m
)
K
C
K
O
(
1
2+
e
β
)
1
/m
1
K
O
K
C
K
O
(
1
e
β/m
)
K
C
K
O
e
β/m
1
K
O
K
C
K
O
K
C
K
O
e
β/m
=
K
O
e
β/m
.
(S32)
S13