The function of membrane-embedded proteins such as ion channels depends crucially on their conformation. We demonstrate how conformational changes in asymmetric membrane proteins may be inferred from measurements of their diffusion. Such proteins cause local deformations in the membrane, which induce an extra hydrodynamic drag on the protein. Using membrane tension to control the magnitude of the deformations and hence the drag, measurements of diffusivity can be used to infer--- via an elastic model of the protein--- how conformation is changed by tension. Motivated by recent experimental results [Quemeneur et al., Proc. Natl. Acad. Sci. USA, 111 5083 (2014)] we focus on KvAP, a voltage-gated potassium channel. The conformation of KvAP is found to change considerably due to tension, with its `walls', where the protein meets the membrane, undergoing significant angular strains. The torsional stiffness is determined to be 26.8 kT at room temperature. This has implications for both the structure and function of such proteins in the environment of a tension-bearing membrane.

1. Probing conformational changes in
membrane proteins by measuring diffusion
Richard G. Morris and Matthew S. Turner
Aug 4th 2015

2. Membranes (by Physicists)
• Lipids are in a liquid ordered state.
1.1. AMPHIPHILIC MOLECULES, AGGREGATES, AND VESICLES 11
Figure 1.1: Diagram of a simple spherical vesicle. Here, amphiphilic molecules
are arranged in typical bilayer fashion, with “tails” pointing inwards. A represen-
tative amphiphile is shown which has an ionic sodium sulphate head group and
a hydrocarbon tail of the form CnH2n+1. Sodium dodecyl sulphate, mentioned in
Section 1.1, corresponds to n = 12.
T ⇠ 37.5 C =)

3. Membranes (by Biologists)
• Behaviour of ion-channels in the membrane?

4. KvAP: voltage-gated K+ channel
• Function requires precise positioning of functional
residues.
• Has a conical-like shape.
• Induces a deformation— or “dimple”— in planar
membranes.
sure according to the EPR data
d 127). In addition, a face of S1
d to exhibit high lipid exposure
surface) appears in KvAP to be
either lipid nor water) according
residues 31, 35, and 39). These
pid exposure on the basis of the
served lipid exposure of corre-
uggest that the voltage sensor in
tly with respect to the pore,
the voltage sensor paddle and the extracellular side of S5 do not
depend highly on the precise location of the cysteines (19–22),
and a single cysteine residue near the extracellular ‘‘tip’’ of the
voltage sensor paddle can result in the formation of covalent
subunit dimers, presumably through linkage of voltage sensor
paddles from adjacent subunits (20). The nonspecificity of
cross-bridge formation and covalent subunit dimerization me-
diated by single cysteine residues on the voltage sensor paddle
suggests that the paddle is a highly mobile unit (Fig. 5d).
Discussion
r in the open conformation. The top-down view (a) and the side view (b) of the proposed model of the KvAP tetramer
t is colored blue, green, gold, and red. This model is the same as in Fig. 4b but it is shown as a tetramer to show the position
ore.

5. Induced membrane deformation
a
↵
KvAP
ˆz
↵h(r)
R(r, ✓)
r
✓
=10-2 N/m
=5x10-4 N/m
=10-6 N/m
1 25 50
0
-1
-2
-3
r/a
Height(nm)
h(r) ⇠ K0
r
p
/
!
• Surface tension controls both height and extent of
membrane deformation

6. Quemeneur et al. PNAS 2014
the diffusion coefficients of AQP0 and KvAP are comparable
and correspond to the value predicted by the SD model (Eq. 1),
namely D0 = 2:5 μm2
=s for ap = 4 nm. When the tension drops
from 10−3
to 10−6
N/m, less than a 5% variation of Deff is found
for AQP0, whereas a drastic decrease of about 40% is revealed
for KvAP. In any case, such a tension dependence is incompatible
with the standard SD approach.
To address this issue, w
and numerical simulations
protein diffuses in a memb
by the presence of the pro
that the protein strongly af
on the diffusing object, t
This phenomenon is gene
effect (18). A polaron is
rounding lattice and mov
field. Similarly in liquids,
terions in a process that a
a single protein diffusing
described by a height funct
Hamiltonian:
H0½h; RŠ =
κ
2
Z
d2
r
À
∇2
where the first two terms r
of the bilayer with modul
models the membrane cur
protein R, which is time d
curvature scales linearly w
Cp, Θ = 4πa2
pCp, similarly t
of the protein on the mem
tion G, which is normalized
the order of ap. This Hami
which corresponds to the
approach, we obtain the m
given in Eq. S37. The lat
brane profile is the crossov
bending regime for the flu
objective
x100 oil
GUV
A
QD
tracer
pipette
PEG-linker
150% 130x30 pixels AQP0 2012-08-25_GUV03_80.15mm
x (pixel)
30 40 50 60 70 80 90 100
10
20
0
y(pixel)
QD
1 m
B 30
Fig. 1. Experimental approach to diffusion measurements in fluctuating
membrane. (A) Schematic of experimental setup: a GUV containing tracer
ð■Þ agree well with the exper
membrane deformation near pr

7. Quemeneur et al. PNAS 2014
ture Cp, which
curvature. For
). In contrast,
ed membranes,
To investigate
anes, they were
r μm2
) in fluid
en labeled with
ions are present
with the QDs
old the GUV in
) (Fig. 1A). To
n was detected
epifluorescence
amera (17). We
brane area (Fig.
(MSD) of the
short time and
the size of the
2.5
1.0
1.5
2.0
10-6
10-5
10-4
10-3
10-2
Membrane tension, Σ (N/m)
AQP0
KvAP
Fig. 2. Protein lateral mobility in fluctuating membranes. Semilogarithmic
plot of the diffusion coefficients (Deff) as a function of the membrane ten-
sion Σ, for AQP0 ð◆Þ and KvAP ð▲Þ labeled with streptavidin QDs. Each

8. Polaron-like model
meter of systems near a
effect has only recently
cal progress involved in
p the possibility of ex-
Casimir effect, notably
e above, the effect of the
on induced between two
he field. However, the
seen by looking at the
n it is not at rest. For
aterials induce a local
[6] which modifies their
ce is also induced by the
a volume of blackbody
in the rest frame of a
or classical fields, in a
is present on inclusions
which move at constant
caused by a polaronlike
generalize to a range of
the study of soft con-
nt in this analysis is that
the dynamical rules used: (i) dynamics not conserving the
total magnetization—Glauber dynamics—a single spin is
chosen and is flipped with probability pf ¼ 1=½1 þ
expð

9. ÁHÞŠ, where

10. is the inverse temperature and ÁH
the energy change associated with the spin flip; (ii) a form
5 15 25 35 45
x
5
15
25
35
45
z
v = 0 v = 0.25
• Drag due to damping of surrounding fluid is negligible!
• Other dissipative mechanisms are invoked (e.g., trans-
membrane shear)
H =
Z
S
dA
⇥
2  H2
+ ⇥ G (|r R|) H
⇤

11. What about hydrodynamics?
• A membrane is effectively a two-dimensional
incompressible fluid at low reynolds number.
• The protein has physical form, and cannot be
ignored.
• Boundary conditions between membrane and
surrounding fluids.

12. • Calculate drag , and use
Stokes-Einstein .
• Solve for p and using Stokes’
equation and incompressibility.
F = V
D = kBT/
v
r

⌧
⌘
µ
Outline of approach
• Equilibrium shape suffices!
• Saffman-Delbrück not needed:
Important points:

13. Covariant hydrodynamics
⌘
⇣
vi
;j
;j
+ Kvi
⌘
p,i
= 0.
✓
1
2
+ K
◆
+ hrK, r i = 0
vi
;i = 0
vi
=
1
p
|g|
"ij
,j
Euclidean Covariant
⌘r2
v rp = 0
r · v = 0
v = r ⇥
r4
= 0

14. …The effect of curvature
High tensions imply large Gaussian curvatures…
…large Gaussian curvatures imply large shear stresses…
…large shear stresses imply large drag and low mobility!
K = 12Gaussian curvature:

15. Remarks on the calculation
• Need to solve:
• Use perturbation theory: small angle
• Non-trivial corrections at
• Must recover Saffman-Delbrück as
✓
1
2
+ K
◆
+ hrK, r i = 0
↵
O ↵2
↵ ! 0

16. Results: rigid proteins
How can data and theory be reconciled? Elasticity!
10-6 10-5 10-4 0.001 0.010
1.0
1.5
2.0
2.5
(N/m)
D(10-12m2/s)
Cylinder
Rigid KvAP

17. Results: elastic deformation
Single parameter fit predicts torsion coefficient…
10-6 10-5 10-4 0.001 0.010
1.0
1.5
2.0
2.5
(N/m)
D(10-12m2/s)
Cylinder
Rigid KvAP
Elastic KvAP
k = 26.8 kBT

18. Results: angular strains
10-6
10-5
10-4
0.001 0.010
0.0
0.1
0.2
0.3
0.4
(N/m)
Strain(Rad)
=0.42 Rad
=0.22 Rad

19. Discussion
• Are proteins less rigid than first thought?
• What does this mean for the function of proteins in
membranes under tension?
• Do crystallographic methods do not tell the whole story?

20. Thankyou

21. Problems I am interested in
Prolate
Oblate
20 40 60 80
r
-0.010
-0.005
0.005
0.010
FHr L
-1
-0.5
0
0.5
1
0 50 100 150 200 250 300 350 400
m
z
=80
=160
=240
=320
=400
=480
=560
=640
0 5 10 15
0
5
10
15