Probing conformational changes in membrane proteins by measuring diffusion

Richard G. Morris and Matthew S. TurnerAug 4th 2015

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 moleculesare arranged in typical bilayer fashion, with “tails” pointing inwards. A represen-tative amphiphile is shown which has an ionic sodium sulphate head group anda hydrocarbon tail of the form CnH2n+1. Sodium dodecyl sulphate, mentioned inSection 1.1, corresponds to n = 12.

Golgi apparatus of eukaryotic cells [11]. Moreover, a selectively permeable bilayer

typifies the underlying construction of almost all biological membranes [12], and

therefore the study of vesicles is relevant to a very wide range of problems. Of

particular interest here is the role of vesicles in pre-biotic chemistry and the

study of primitive life. One conjecture is that vesiculation, or budding, is related

to early forms of cell reproduction [13, 14]. Vesiculation is the process by which

shape changes give rise to the production of a small daughter vesicle [15].

1.1 Amphiphilic molecules, aggregates, and vesi-

cles

As stated above, an amphiphilic molecule is one which is both hydrophilic and

lipophilic [3]. The hydrophilic behaviour is produced by either charged chemical

groups (such as sulphates, phosphates, and amines) or uncharged polar groups

(such as alcohols). By contrast, the lipophilic behaviour is typically associated

with large hydrocarbon chains.

A simple example of an amphiphile is sodium dodecyl sulfate (C12H25SO4Na).

This is a common surfactant found in many household cleaning products and can

T ⇠ 37.5�C =)

Membranes (by Biologists)

• Behaviour of ion-channels in the membrane?

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.

in KvAP exhibit high lipid exposure according to the EPR data(Fig. 5 b and c, residues 124 and 127). In addition, a face of S1in Kv1.2 that would be expected to exhibit high lipid exposure(because it is on the lipid-facing surface) appears in KvAP to bein a protein environment (i.e., neither lipid nor water) accordingto the EPR data (Fig. 5 b and c, residues 31, 35, and 39). Thesedifferences between expected lipid exposure on the basis of theKv1.2 crystal structure and observed lipid exposure of corre-sponding amino acids in KvAP suggest that the voltage sensor inKvAP might be rotated slightly with respect to the pore,compared with the sensor in Kv1.2. The rotation of the voltagesensor that would be required to account for the EPR data is verymodest and in fact has already been made in the model shownin Fig. 4b, relative to the Kv1.2 voltage sensor in Fig. 4a. Thatsmall variations in the orientation of the voltage sensors mightoccur between KvAP and Kv1.2 is not surprising, especiallybecause the constraint imposed by the linker to the T1 domainin Shaker family Kv channels (e.g., Kv1.2) does not exist in KvAP.

We carried out disulfide cross-bridge studies to furtherexamine the relationship between KvAP and Kv1.2. Severalstudies have shown that a cysteine residue introduced at theposition of the first arginine residue on S4 (the extracellular-most arginine, corresponding to position 117 in KvAP) in theShaker K! channel (which is very similar to Kv1.2) readilyforms a cross-bridge with a cysteine introduced near theextracellular end of S5 of a neighboring subunit (19). Weobserve a qualitatively similar result with KvAP channels inmembranes (Fig. 5d). In KvAP, however, we do not observethe same quantitative degree of complete covalent tetramerformation that is observed in the Shaker K! channel (19). Thedata on KvAP are thus consistent with the voltage sensor beingin a similar location as in Shaker (or Kv1.2), but the lowerefficiency of tetramer formation is consistent with the possi-bility that S4 may be slightly further away from S5, as the EPRdata suggest.

We note additional aspects of the disulfide cross-bridge stud-ies on KvAP channels that parallel previous data on Shaker K!

channels. Cross-bridge formation between cysteine residues on

the voltage sensor paddle and the extracellular side of S5 do notdepend highly on the precise location of the cysteines (19–22),and a single cysteine residue near the extracellular ‘‘tip’’ of thevoltage sensor paddle can result in the formation of covalentsubunit dimers, presumably through linkage of voltage sensorpaddles from adjacent subunits (20). The nonspecificity ofcross-bridge formation and covalent subunit dimerization me-diated by single cysteine residues on the voltage sensor paddlesuggests that the paddle is a highly mobile unit (Fig. 5d).

DiscussionTwo additional crystal structures of KvAP show the voltagesensors within the plane of the lipid membrane, but the domainsstill deviate from their expected vertical orientation relative tothe pore. Antibody fragments per se are not the cause of thisdeviation because it is observed in the presence of Fab fragmentsand Fv fragments and in the absence of antibody fragments.Therefore, it is incorrect to conclude that antibody fragmentsshould be avoided in the crystallization of membrane proteins.One important conclusion of this study, and the answer to thesecond question raised in the Introduction, is that KvAP appearsto require an intact lipid membrane to maintain a nativeconformation, because the voltage sensors do not adhere tightlyto the pore. The sensors, like the pore, have their own intrinsicstructural and chemical properties to match the membrane’shydrophobic core, interface, and water layers, ensuring that they‘‘f loat’’ properly in the lipid membrane. Kv channels might be thefirst examples of membrane proteins with separate, weaklyattached membrane-spanning domains, but other importantmembrane proteins with this property will likely be identifiedand studied in the future.

For proteins like KvAP the detergent micelle may be animperfect mimic of the lipid bilayer, but detergent-mediatedcrystallization is still a valuable approach for understanding suchproteins. A non-native protein structure simply means thatadditional information is required to help interpret what we see.In the present analysis we have combined crystal structures ofthe KvAP channel with biochemical cross-bridge studies in

Fig. 6. A model of the KvAP tetramer in the open conformation. The top-down view (a) and the side view (b) of the proposed model of the KvAP tetramerin the open conformation. Each subunit 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 positionof the voltage sensor relative to the pore.

Lee et al. PNAS ! October 25, 2005 ! vol. 102 ! no. 43 ! 15445

BIO

PHYS

ICS

Induced membrane deformation

a

↵

KvAP

z

↵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/aHeight(nm)

h(r) ⇠ K0

rp/�

!

• Surface tension controls both height and extent of membrane deformation

�

Quemeneur et al. PNAS 2014

generally quantified by its spontaneous curvature Cp, whichreflects the preferred value of local membrane curvature. ForKvAP, we have measured Cp = 0:04 nm−1 (13). In contrast,AQP0 does not exhibit any preference for curved membranes,and thus has a zero spontaneous curvature (13). To investigatethe lateral mobility of these proteins in membranes, they werereconstituted at low density (∼40 inclusions per μm2) in fluidphase GUVs, and a small protein fraction was then labeled withquantum dots (QDs). Although both protein insertions are presentin the GUV, only one insertion was labeled with the QDs(13). Micropipette aspiration was then used to hold the GUV inplace and to control its membrane tension Σ (16) (Fig. 1A). Tomeasure protein displacements, the QD position was detectedwith SPT near the bottom of the vesicle using epifluorescencemicroscopy equipped with a fast and sensitive camera (17). Welimited our analysis to a small rectangular membrane area (Fig.1B). We find that the mean-square displacement (MSD) of theproteins exhibits a linear time dependence at short time anda crossover to a constant value dependent on the size of theobservation window at larger time (Materials and Methods andFig. S1). Details of our protocols and analysis are described inthe Supporting Information.We determined the diffusion coefficient Deff for each applied

membrane tension (Fig. 2 and Fig. S2). In the high-tension limit,the diffusion coefficients of AQP0 and KvAP are comparableand correspond to the value predicted by the SD model (Eq. 1),namely D0 = 2:5 μm2=s for ap = 4 nm. When the tension dropsfrom 10−3 to 10−6 N/m, less than a 5% variation of Deff is foundfor AQP0, whereas a drastic decrease of about 40% is revealedfor KvAP. In any case, such a tension dependence is incompatiblewith the standard SD approach.

To address this issue, we have developed an analytical modeland numerical simulations. SD theory implicitly assumes that theprotein diffuses in a membrane that remains flat and unaffectedby the presence of the protein. In contrast, we take into accountthat the protein strongly affects its environment. The back actionon the diffusing object, translates into a modified drag force.This phenomenon is general in physics and is known as polaroneffect (18). A polaron is a charge carrier that deforms a sur-rounding lattice and moves in it with an induced polarizationfield. Similarly in liquids, an isolated ion recruits nearby coun-terions in a process that affects its mobility. Here, we considera single protein diffusing on a membrane patch of size L ! apdescribed by a height function hðrÞ. We use the modified HelfrichHamiltonian:

H0½h;R%=κ2

Zd2r

!"∇2h

#2+Σκð∇hÞ2 −ΘGðr−RÞ∇2h

$; [2]

where the first two terms represent the energy of elastic bendingof the bilayer with modulus κ and tension Σ, and the last termmodels the membrane curvature induced at the location of theprotein R, which is time dependent. The strength of the inducedcurvature scales linearly with the protein spontaneous curvatureCp, Θ= 4πa2pCp, similarly to refs. 6 and 8. The range of influenceof the protein on the membrane is modeled by the weight func-tionG, which is normalized to 1 and is nonzero over a distance ofthe order of ap. This Hamiltonian carries with it a cutoff length a,which corresponds to the bilayer thickness (∼5 nm). From thisapproach, we obtain the membrane profile around the inclusiongiven in Eq. S37. The lateral characteristic width of this mem-brane profile is the crossover length between the tension and thebending regime for the fluctuations, namely ξ=

ffiffiffiffiffiffiffiffiκ=Σ

p, whereas

the characteristic height of the membrane deformation at zerotension scales as Θ (Eq. S31). The geometry of the local defor-mation from the membrane midplane induced by KvAP whensubjected to various tensions is shown in Fig. 3 (Fig. S3). Usingthe method of refs. 6, 8, and 10, we have carried out simulations(with parameters shown in Fig. S4), which also confirm theexpected theoretical membrane profile as shown in Fig. 3.

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 (p

ixel

)

QD1�m

B 30

Fig. 1. Experimental approach to diffusion measurements in fluctuatingmembrane. (A) Schematic of experimental setup: a GUV containing tracermolecule (lipid, AQP0, or KvAP) labeled with a QD is aspirated in a micro-pipette. In a typical sequence, 100–1,000 individual QDs explore the bottompole of the GUV. Single QD displacements are measured as a function of theapplied membrane tension Σ. (B) Example of QD trajectory. The truncatedcircle corresponds to the boundary of the area explored by QDs within thedepth of field of the microscope. Only the green part of the trajectorycontained in a centered square region of interest is considered for the single-QD tracking analysis. One pixel is 160 nm.

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. Semilogarithmicplot 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. Eachpoint represents a median diffusion coefficient obtained from hundreds ofindividual trajectories for a GUV at a given tension; the error bars correspond toSE. KvAP data adjusted by Eq. 3 (solid line) yields a protein coupling coefficientΘ= 3:5×10−7 m considering a= 5 nm, κ= 20 kBT and D0 ’ 2:5 μm2=s. Simu-lations of the protein diffusion on a membrane subject to thermal fluctuationsð■Þ agree well with the experimental data and theory. (Insets) Sketches ofmembrane deformation near proteins.

2 of 5 | www.pnas.org/cgi/doi/10.1073/pnas.1321054111 Quemeneur et al.

Quemeneur et al. PNAS 2014

generally quantified by its spontaneous curvature Cp, whichreflects the preferred value of local membrane curvature. ForKvAP, we have measured Cp = 0:04 nm−1 (13). In contrast,AQP0 does not exhibit any preference for curved membranes,and thus has a zero spontaneous curvature (13). To investigatethe lateral mobility of these proteins in membranes, they werereconstituted at low density (∼40 inclusions per μm2) in fluidphase GUVs, and a small protein fraction was then labeled withquantum dots (QDs). Although both protein insertions are presentin the GUV, only one insertion was labeled with the QDs(13). Micropipette aspiration was then used to hold the GUV inplace and to control its membrane tension Σ (16) (Fig. 1A). Tomeasure protein displacements, the QD position was detectedwith SPT near the bottom of the vesicle using epifluorescencemicroscopy equipped with a fast and sensitive camera (17). Welimited our analysis to a small rectangular membrane area (Fig.1B). We find that the mean-square displacement (MSD) of theproteins exhibits a linear time dependence at short time anda crossover to a constant value dependent on the size of theobservation window at larger time (Materials and Methods andFig. S1). Details of our protocols and analysis are described inthe Supporting Information.We determined the diffusion coefficient Deff for each applied

membrane tension (Fig. 2 and Fig. S2). In the high-tension limit,the diffusion coefficients of AQP0 and KvAP are comparableand correspond to the value predicted by the SD model (Eq. 1),namely D0 = 2:5 μm2=s for ap = 4 nm. When the tension dropsfrom 10−3 to 10−6 N/m, less than a 5% variation of Deff is foundfor AQP0, whereas a drastic decrease of about 40% is revealedfor KvAP. In any case, such a tension dependence is incompatiblewith the standard SD approach.

To address this issue, we have developed an analytical modeland numerical simulations. SD theory implicitly assumes that theprotein diffuses in a membrane that remains flat and unaffectedby the presence of the protein. In contrast, we take into accountthat the protein strongly affects its environment. The back actionon the diffusing object, translates into a modified drag force.This phenomenon is general in physics and is known as polaroneffect (18). A polaron is a charge carrier that deforms a sur-rounding lattice and moves in it with an induced polarizationfield. Similarly in liquids, an isolated ion recruits nearby coun-terions in a process that affects its mobility. Here, we considera single protein diffusing on a membrane patch of size L ! apdescribed by a height function hðrÞ. We use the modified HelfrichHamiltonian:

H0½h;R%=κ2

Zd2r

!"∇2h

#2+Σκð∇hÞ2 −ΘGðr−RÞ∇2h

$; [2]

where the first two terms represent the energy of elastic bendingof the bilayer with modulus κ and tension Σ, and the last termmodels the membrane curvature induced at the location of theprotein R, which is time dependent. The strength of the inducedcurvature scales linearly with the protein spontaneous curvatureCp, Θ= 4πa2pCp, similarly to refs. 6 and 8. The range of influenceof the protein on the membrane is modeled by the weight func-tionG, which is normalized to 1 and is nonzero over a distance ofthe order of ap. This Hamiltonian carries with it a cutoff length a,which corresponds to the bilayer thickness (∼5 nm). From thisapproach, we obtain the membrane profile around the inclusiongiven in Eq. S37. The lateral characteristic width of this mem-brane profile is the crossover length between the tension and thebending regime for the fluctuations, namely ξ=

ffiffiffiffiffiffiffiffiκ=Σ

p, whereas

the characteristic height of the membrane deformation at zerotension scales as Θ (Eq. S31). The geometry of the local defor-mation from the membrane midplane induced by KvAP whensubjected to various tensions is shown in Fig. 3 (Fig. S3). Usingthe method of refs. 6, 8, and 10, we have carried out simulations(with parameters shown in Fig. S4), which also confirm theexpected theoretical membrane profile as shown in Fig. 3.

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 (p

ixel

)

QD1�m

B 30

Fig. 1. Experimental approach to diffusion measurements in fluctuatingmembrane. (A) Schematic of experimental setup: a GUV containing tracermolecule (lipid, AQP0, or KvAP) labeled with a QD is aspirated in a micro-pipette. In a typical sequence, 100–1,000 individual QDs explore the bottompole of the GUV. Single QD displacements are measured as a function of theapplied membrane tension Σ. (B) Example of QD trajectory. The truncatedcircle corresponds to the boundary of the area explored by QDs within thedepth of field of the microscope. Only the green part of the trajectorycontained in a centered square region of interest is considered for the single-QD tracking analysis. One pixel is 160 nm.

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. Semilogarithmicplot 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. Eachpoint represents a median diffusion coefficient obtained from hundreds ofindividual trajectories for a GUV at a given tension; the error bars correspond toSE. KvAP data adjusted by Eq. 3 (solid line) yields a protein coupling coefficientΘ= 3:5×10−7 m considering a= 5 nm, κ= 20 kBT and D0 ’ 2:5 μm2=s. Simu-lations of the protein diffusion on a membrane subject to thermal fluctuationsð■Þ agree well with the experimental data and theory. (Insets) Sketches ofmembrane deformation near proteins.

2 of 5 | www.pnas.org/cgi/doi/10.1073/pnas.1321054111 Quemeneur et al.

Polaron-like model

Drag Forces in Classical Fields

Vincent Demery and David S. DeanUniversite de Toulouse, UPS, Laboratoire de Physique Theorique (IRSAMC), CNRS UMR5152, F-31062 Toulouse, France

(Received 20 December 2009; revised manuscript received 2 February 2010; published 23 February 2010)

Inclusions, or defects, moving at constant velocity through free classical fields are shown to be subject

to a drag force which depends on the field dynamics and the coupling of the inclusion to the field. The

results are used to predict the drag exerted on inclusions, such as proteins, in lipid membranes due to their

interaction with height and composition fluctuations. The force, measured in Monte Carlo simulations, on

a pointlike magnetic field moving through an Ising ferromagnet is also well explained by these results.

DOI: 10.1103/PhysRevLett.104.080601 PACS numbers: 05.70.Ln, 05.70.Jk, 87.16.dj, 87.16.dp

In quantum field theory forces between particles areinduced via their coupling to a quantum field [1]. Thesame phenomena arises for fields driven by thermal fluc-tuations, for example, interactions between inclusions influctuating membranes [2]. Similarly the Casimir force,both quantum and thermal, arises due to the impositionof boundary conditions on quantum or thermal fields [3].Casimir discovered his force in the quantum context butFisher and de Gennes [4] showed that a classical version ofthis effect should be expected for fluctuating thermal fields,such as those for the order parameter of systems near acritical point. This critical Casimir effect has only recentlybeen measured [5] and the technical progress involved inthis experiment may well open up the possibility of ex-ploring other aspects of the critical Casimir effect, notablydynamical phenomena. In all of the above, the effect of thefield is manifested by the interaction induced between twoor more particles or surfaces in the field. However, thepresence of the field can also be seen by looking at theforce exerted on a particle when it is not at rest. Forinstance, electrons moving in materials induce a localpolarization known as the polaron [6] which modifies theirdynamics. A frictional Casimir force is also induced by theuniform motion of a conductor in a volume of blackbodyradiation which is in equilibrium in the rest frame of acavity containing the radiation [7].

In this Letter we show that for classical fields, in alaboratory rest frame, a drag force is present on inclusionslinearly coupled to the field, and which move at constantvelocity. The underlying physics is caused by a polaronlikephenomena (see Fig. 1) which we generalize to a range ofstatistical field models arising in the study of soft con-densed matter systems. A key point in this analysis is thatwe examine the effect of the dynamical models commonlyused to study soft condensed matter systems on the dragforces on inclusions.

As a concrete example of the class of problems we willaddress we start by studying drag forces in the Isingferromagnet model on a d dimensional cubic lattice withHamiltonian

H ¼ "JX

ði;jÞSiSj þ hSi0 : (1)

Here J > 0 is a ferromagnetic coupling between nearestneighbor spins on a square lattice of spacing a, and h amagnetic field at the position of an inclusion i0 whichmoves in the direction z with a velocity v, so i0ðzÞ ¼vt=a (where we take the integer part). The system evolvesin the following manner, the underlying unit of time is aMonte Carlo sweep where N (the system size) randomlychosen spins are examined and flipped or not according tothe dynamical rules used: (i) dynamics not conserving thetotal magnetization—Glauber dynamics—a single spin ischosen and is flipped with probability pf ¼ 1=½1þexpð!!HÞ', where ! is the inverse temperature and !Hthe energy change associated with the spin flip; (ii) a form

5 15 25 35 45x

5

15

25

35

45

z

v = 0 v = 0.25

v = 0.5 v = 1

FIG. 1 (color online). Contour plot of the magnetization profile(polaron) for the 2D Ising model about a local magnetic field at asingle point moving with velocity v. (high temperature phase:! ¼ 1, J ¼ 0:4, h ¼ 6:66).

PRL 104, 080601 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending

26 FEBRUARY 2010

0031-9007=10=104(8)=080601(4) 080601-1 ! 2010 The American Physical Society

• Drag due to damping of surrounding fluid is negligible! • Other dissipative mechanisms are invoked (e.g., trans-

membrane shear)

H =

Z

SdA

⇥2H2 + � �⇥G (|r �R|)H⇤

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.

• 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:

Covariant hydrodynamics

⌘⇣vi;j

;j+Kvi

⌘� p,i = 0.

✓1

2�+K

◆� + hrK,r i = 0

vi;i = 0

vi =1p|g|"ij ,j

Euclidean Covariant

⌘r2v �rp = 0

r · v = 0

v = r⇥

r4 = 0

…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 = 12Gaussian curvature:

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

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)

CylinderRigid KvAP

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)

CylinderRigid KvAPElastic KvAP

k = 26.8 kBT

Results: angular strains

10-6 10-5 10-4 0.001 0.0100.0

0.1

0.2

0.3

0.4

� (N/m)

Strain

��

(Rad

)

�=0.42 Rad

�=0.22 Rad

Discussion����������� �����������

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• 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?

Thankyou

Problems I am interested in

Prolate

Oblate

20 40 60 80r

-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

o=80o=160o=240o=320o=400o=480o=560o=640

0 5 10 15

0

5

10

15