arXiv:q-bio/0702031v1 [q-bio.SC] 14 Feb 2007
Cooperative Gating and Spatial Organization of Membrane Proteins through Elastic
Tristan Ursell1, Kerwyn Huang2, Eric Peterson3and Rob Phillips1,4∗
Departments of1Applied Physics and3Physics, California Institute of Technology, Pasadena, CA 91125, USA
4Kavli Nanoscience Institute, Pasadena, CA 91125, USA
2Department of Molecular Biology, Princeton University, Princeton, NJ 08544, USA
Submitted to PLoS Computational Biology
February 7, 2008
Biological membranes are elastic media in which the presence of
a transmembrane protein leads to local bilayer deformation. The
energetics of deformation allow two membrane proteins in close
proximity to influence each other’s equilibrium conformation via
their local deformations, and spatially organize the proteins based
on their geometry. We use the mechanosensitive channel of large
conductance (MscL) as a case study to examine the implications
of bilayer-mediated elastic interactions on protein conformational
statistics and clustering. The deformations around MscL cost
energy on the order of 10kBT and extend ∼ 3nm from the protein
edge, as such elastic forces induce cooperative gating and we propose
experiments to measure these effects.Additionally, since elastic
interactions are coupled to protein conformation, we find that
conformational changes can severely alter the average separation
between two proteins. This has important implications for how
conformational changes organize membrane proteins into functional
groups within membranes.
Biological membranes are active participants in the
function and spatial organization of membrane proteins
[1, 2, 3]. At the simplest level, the membrane positions
proteins into a two-dimensional space, where they are of-
ten laterally organized into groups. These groups can serve
specific purposes on the cell surface and within organelles,
such as sensing, adhesion and transport [4, 5, 6, 7, 8, 9].
Electrostatic and van der Waals forces help drive lateral
organization , however there is an additional class of
purely bilayer-mediated elastic forces that can facilitate
the formation of complexes of membrane proteins.
Conformational changes of membrane proteins result
from a wide range of environmental factors including tem-
perature, pH, ligand and small molecule binding, mem-
brane voltage, and membrane tension. Likewise, confor-
mational state is often tightly coupled with function (e.g.
for ion channels) [11, 12, 13]. In this work, we demon-
strate how elastic interactions can communicate informa-
tion about protein conformation from one neighboring pro-
tein to another coupling their conformational state. Ad-
ditionally, we find that these interactions lead to spatial
∗corresponding author: email@example.com
organization within the bilayer that is strongly dependent
on protein conformation.
We suggest that elastic forces play a role in the func-
tion and spatial organization of many membrane proteins
across many cell types, given the generically high areal
density of membrane proteins  and the strength of
these interactions. We use the mechanosensitive channel
of large conductance (MscL) from E. coli as the model
protein for this study. MscL is a transmembrane homo-
pentamer found in the plasma membrane of E. coli (and
many other bacteria) serving as an emergency relief valve
under hypo-osmotic shock [11, 15, 16]. As membrane ten-
sion increases, this non-selective ion channel changes con-
formation from a closed state to an open state, releasing
water and osmolytes [17, 18]. Though several substates
have been identified in this gating transition, the relatively
short dwell-times in these substates as compared to the
fully open or fully closed states, allows us to approximate
the protein as a simple two-state system [17, 19]. Crys-
tal and electron-paramagnetic-resonance structures sug-
gest the bilayer-spanning region is nearly cylindrical in
both the open and closed conformations [15, 20, 21], mak-
ing MscL particularly amenable to mechanical modeling.
Electrophysiology of reconstituted channels allows mea-
surement of the state of one or more of these proteins with
excellent temporal and number resolution. Therefore, the-
oretical predictions for how elastic interactions change the
gating behavior of a MscL protein can be readily tested us-
ing electrophysiology and other experimental techniques.
Following earlier work, we use continuum mechanics to
break down the deformation caused by a cylindrical trans-
membrane protein into a term penalizing changes in bi-
layer thickness and a term penalizing bending of a bilayer
leaflet [22, 23, 24, 25, 26], and we introduce a third term
that preserves bilayer volume under deformation . Due
to its structural symmetry, MscL can be characterized by
its radius and bilayer-spanning thickness in its two con-
formations (i.e. open and closed), neglecting any specific
molecular detail (see Figure 1). As these geometric pa-
rameters change with conformation, the bilayer-mediated
interaction between two channels is altered. Using the in-
teraction potentials in each combination of conformations,
we explore how both the single-channel and interacting en-
ergetics affect the spatial and conformational behavior of
In the first section we cover the physical principles
behind bilayer deformation due to the presence of mem-
brane proteins.In the second section we explore the
differences in gating behavior of two MscL proteins when
held at a fixed separation. In the third section we explore
the conformational and spatial behavior of diffusing MscL
proteins as a function of areal density.
fourth section we discuss the relevance of these forces
as compared to other classes of bilayer-mediated forces
and support our hypotheses with results from previous
Finally, in the
Elastic Deformation Induced by Membrane Pro-
The bilayer is composed of discrete lipid molecules
whose lateral diffusion (D ∼ 10µm2/s)  is faster
than the diffusion of transmembrane proteins (D ∼ 0.1 −
1µm2/s) [29, 30, 31]. In the time it takes a transmem-
brane protein to diffuse one lipid diameter, many lipids
will have exchanged places near the protein to average
out the discreteness of the lipid molecules.
ally, the transition time for protein conformational change
(∼ 5µs)  is slow compared to lipid diffusion. Hence,
we argue the bilayer can be approximated as a continuous
material in equilibrium with well-defined elastic proper-
ties . Further, we choose to formulate our analysis
in the language of continuum mechanics, rather than lat-
eral pressure profiles . In particular, each leaflet of
the bilayer resists changes in the angle between adjacent
lipid molecules, leading to bending stiffness of the bilayer
[22, 35]. Likewise, the bilayer has a preferred spacing of
the lipid molecules in-plane and will resist any changes
in this spacing due to external tension . Finally, ex-
periments suggest that the volume per lipid is conserved
[37, 38] such that changes in bilayer thickness are accom-
panied by changes in lipid spacing [2, 33].
Transmembrane proteins can compress and bend a bi-
layer leaflet via at least two mechanisms. The protein can
force the bilayer to adopt a new thickness, matching the
hydrophobic region of the protein to the hydrophobic core
of the bilayer. Additionally, a non-cylindrical protein can
induce a slope in the leaflet at the protein-lipid interface
For transmembrane proteins like MscL, that can be ap-
proximated as cylindrical, symmetry dictates that the de-
formation energy of the bilayer is twice the deformation
energy of one leaflet. Presuming the protein does not de-
form the bilayer too severely, we can write the bending and
compression (thickness change) energies in a form analo-
gous to Hooke’s law, and account for external tension with
a term analogous to PV work. We denote the deformation
of the leaflet by the function u(r), which measures the de-
viation of the lipid head-group from its unperturbed height
as a function of the position r (see Figure 1). In all the
calculations that follow the physical parameters chosen are
representative of a typical phosphatidylcholine (PC) lipid
bilayer and the number of lipids in this model bilayer is
fixed. The energy penalizing compression of the bilayer is
where KA is the bilayer area stretch modulus (∼
58kBT/nm2, kBT is the thermal energy unit) and l is
the unperturbed leaflet thickness (∼ 1.75nm) . The
bending energy of a leaflet is
(∇2u(r) − co)2d2r, (2)
where κb(∼ 14kBT) is the bilayer bending modulus, cois
the spontaneous curvature of the leaflet [33, 36, 40], and
∇2= ∂2/∂x2+ ∂2/∂y2is the Laplacian operator.
Coupling external tension to bilayer deformations is
more subtle than the previous two energetic contributions.
We note that the bilayer is roughly forty times more resis-
tant to volume change than area change [37, 38], hence if a
transmembrane protein locally thins the bilayer, lipids will
expand in the area near the protein to conserve volume.
Likewise, if the protein locally thickens the bilayer, lipids
near the protein will condense (see Figure 1). Therefore,
the area change near the protein is proportional to the
compression u(r), and the work done on the bilayer is the
integrated area change multiplied by tension
where τ is the externally applied bilayer tension. Slightly
below bilayer rupture, and near the expected regime of
MscL gating, τ ≃ 2.6kBT/nm2[17, 36].
bilayer deformation energy is
In total, the
?∇2u − co
where we have made use of the constant bilayer area to
elucidate the interplay between tension and compression1.
To obtain the length and energy scales of these defor-
mations, we non-dimensionalize the bilayer deformation
energy, G. We scale both the position r and displacement
u(r) by λ = (κbl2/KA)1/4≃ 1nm, the natural length scale
of deformation, to give the new variables ρ and η(ρ) re-
spectively, where ρ = r/λ and η(ρ) = u(r)/λ. Then G
can be written as
(η + χ)2+?∇2η − νo
1Specifically, we added a constant proportional to membrane area
and τ2, which is identically zero when calculating differences in free
Mismatch between the hydrophobic regions of the lipid bilayer
and an integral membrane protein gives rise to bending and
compression deformations in each leaflet of the bilayer. The
largest deformations occur at the protein-lipid interface, and
over the scale of a few nanometers the bilayer returns to its
unperturbed state. MscL is shown schematically at zero tension
in its closed and open states with relevant dimensions. The
red region of the protein indicates the hydrophobic zone. The
hydrophobic mismatch at the protein-lipid interface is denoted
by uo. The deformation profile, denoted by u(r), is measured
with reference to the unperturbed leaflet thickness (l) from the
protein center at r = 0.
Schematic of Bilayer Deformations due to MscL.
where νo= λcois the dimensionless spontaneous curvature
and χ = τl/KAλ is the dimensionless tension, which is
≃ 0.09 in the regime of MscL gating . The energy scale is
set by the bending modulus, κb.
Using the standard Euler-Lagrange equation from the
calculus of variations , the functional for the deforma-
tion energy can be translated into the partial differential
∇4η + η + χ = 0.
The deformation profile u(r) that solves this partial dif-
ferential equation depends on four boundary conditions.
In the far-field, we expect the bilayer to be flat and
slightly thinner in accordance with the applied tension,
i.e. |∇u(∞)| = 0 and u(∞) = −τl/KA, respectively. At
the protein-lipid interface (r = ro) the hydrophobic re-
gions of the protein and the bilayer must be matched, i.e.
u(ro) = uo (see Figure 1), where uo is one-half the mis-
match between the hydrophobic region of the protein and
the hydrophobic core of the bilayer. Finally, the slope of
the bilayer at the protein-lipid interface is set to zero (i.e.
|∇u(ro)| = 0). The motivation for this last boundary con-
dition is subtle and will be examined in more detail in the
2With no loss of generality, the equation that governs deformation
can also be written in the parameter free form ∇4η +η = 0, with all
parametric sensitivity absorbed into the boundary conditions.
To understand how the deformation energy scales with
hydrophobic mismatch (uo), protein size (ro), and tension
(τ), we solve Eq. 6 analytically for a single cylindrical pro-
tein. The deformation energy is
(η + χ)2+?∇2η − νo
where ρo= ro/λ is the dimensionless radius of the protein.
The bilayer deformation around a single protein is a linear
combination of zeroth order modified Bessel functions of
the second kind (K0) [25, 26]. For proteins such as MscL
with a radius larger than λ (i.e. 1nm) the deformation
energy is well-approximated by
The deformation energy scales linearly with protein ra-
dius and depends quadratically on the combination of hy-
drophobic mismatch (uo) and tension (τ). This makes the
overall deformation energy particularly sensitive to the hy-
drophobic mismatch, and hence leaflet thickness l. The
deformation energy is fairly insensitive to changes in KA
(i.e. most terms in the energy are sublinear), and gener-
ally insensitive to changes in the bending modulus since
G ∝ κ1/4
Using our standard elastic bilayer parameters and the
dimensions of a MscL channel (see Figure 1), the change
in deformation energy between the closed and open states
is ∆Gsingle ≃ 50kBT. The measured value for the free
energy change of gating a MscL protein, including inter-
nal changes of the protein and deformation of surrounding
lipids is ≃ 51kBT . This close correspondence does
not indicate that bilayer deformation accounts for all of
the free energy change of gating , but does suggest
that it is a major contributor.
The gating energy of two channels in close proximity is a
complex function of their conformations and the distance
between them. As two proteins come within a few nanome-
ters of each other (i.e. a few λ), the deformations which
extend from their respective protein-lipid interfaces begin
to overlap and interact. The bilayer adopts a new shape
(i.e. a new u(r)), distinct from the deformation around
two independent proteins, and hence the total deforma-
tion energy changes as well. This is the physical origin
of the elastic interaction between two bilayer-deforming
proteins [23, 24].
Each protein imposes its own local boundary conditions
on the bilayer, that vary with conformation, hence the de-
formation around a pair of proteins is a function of their in-
dividual conformation and the distance between them. A
MscL protein has two distinct conformations, hence there
can be pairwise interactions between two closed channels,
an open and a closed channel, or two open channels (see
Figure 2). Tension also affects the deformations. The hy-
drophobic mismatch can be either positive or negative (i.e.
Figure 2: Elastic Potentials between Two MscL Proteins. To
minimize deformation energy, two transmembrane proteins ex-
ert elastic forces on each other. MscL has three distinct interac-
tion potentials between its two distinct conformations. Exter-
nal tension weakens the interaction between two open channels
(Voo) and strengthens the interaction between two closed chan-
nels (Vcc), but has almost no effect on the interaction between
an open and closed channel (Voc). The open-open and closed-
closed interactions are both more strongly attracting than the
open-closed interaction, indicating that elastic potentials favor
interactions between channels in the same state. The ‘hard
core’ distance is where the proteins’ edges are in contact.
the protein can be thicker or thinner than the bilayer), thus
tension will strengthen the interaction of proteins that are
thicker than the bilayer (e.g. the closed-closed interac-
tion of two MscL proteins) and weaken the interaction of
proteins that are thinner than the bilayer (e.g. the open-
open interaction). This effect is demonstrated in Figure
2. The interactions due to leaflet deformations have been
explored before [23, 24], but our model elucidates the role
that these interactions can play in communicating confor-
mational information between proteins. Additionally, in
our model, tension can play an important role in deter-
mining the overall deformation energy around a protein.
In a one-dimensional model, the interaction potentials
can be solved for analytically. For two identical proteins
in close proximity (e.g. closed-closed and open-open inter-
actions), the approximate shape of the potential is linearly
attractive κb(uo/λ)2(d/2λ −√2). Between two dissimilar
proteins in close proximity (e.g. open-closed interaction)
the potential is approximately κbλu2
cases d is measured from the edges of the proteins. This
illustrates the general principle that two similar proteins
attractively interact, while two dissimilar proteins tend to
repel each other. In a two-dimensional bilayer, however,
the geometry of the two proteins makes it difficult to solve
for the interaction analytically, thus numerical techniques
were used (see Materials and Methods).
This theoretical framework provides a strong foun-
oπ4/4d3, where in both
dation for understanding how protein geometries and
lipid properties give rise to elastic interactions.
this, we can investigate how elastic forces change the
conformational statistics of a two-state protein population.
Gating Behavior of Two Interacting Channels
To probe the range of separations over which elastic
interactions affect the gating of two MscL proteins, we
need to account for the non-interacting energetics of gat-
ing a single channel in addition to the interactions be-
tween two channels. The non-interacting energy is the
sum of three effects. First, there is some energetic cost
to deform the surrounding membrane, which we already
calculated as ∆Gsingle.Second, there is some cost to
change the protein’s internal conformation, independent
of the membrane. Together, these first two effects are
the gating energy ∆Ggate≃ 51kBT for MscL . Finally,
there is an energetic mechanism that overcomes these costs
and opens the channel as tension increases. This mecha-
nism is provided by the bilayer tension working in con-
cert with the conformational area change of the protein
(∆A ≃ 20nm2for MscL ). Given the experimentally
determined values for ∆Ggateand the area change during
gating, the critical tension, defined by ∆Ggate= τc∆A, is
In our thermodynamic treatment we need to keep track
of the conformations of each protein in a population in a
way that allows us to tabulate the non-interacting and in-
teracting contributions to the free energy. To this end, we
assign a state variable, si, to each channel indicating the
conformational state of a protein, where si= 0 indicates
that the ith channel is closed and si= 1 indicates that the
ith channel is open. The non-interacting energy for two
channels is then
Hnon(s1,s2;τ) = (∆Ggate− τ∆A)(s1+ s2). (9)
If both channels are closed (s1= s2= 0) the free energy
is defined to be zero. If one channel is open and the other
closed (s1 = 1,s2 = 0 or s1 = 0,s2 = 1) this counts as
the cost to gate one channel working against the benefit
at a particular tension to opening the channel. Likewise,
this counts twice if both channels are open (s1= s2= 1).
We will measure all energies that follow in units of kBT
(≃ 4.14 × 10−21J).
As we alluded to earlier, the interacting component of
the free energy between two channels is a function of their
states (s1and s2), their edge separation (d), and the ten-
sion. Using a numerical relaxation technique to minimize
the functional in Eq. 5 (see Materials and Methods), we
calculated the interaction potentials Hint(s1,s2,d;τ) for a
range of tensions and separation distances (see Figure 2).
The total energy, Hnon+ Hint, is used to derive the Boltz-
mann weight for the three possible configurations of the
z(s1,s2) = e−(Hnon(s1,s2;τ)+Hint(s1,s2,d;τ)).(10)
The probability that the system has two closed channels
where the partition function Z is the sum of the Boltzmann
weights for all possible two-channel configurations,
z(s1,s2) = z(0,0) + 2z(0,1) + z(1,1). (12)
Likewise, the probabilities for the system to have exactly
one or two open channels are
respectively. Finally, the probability for any one channel
in this two channel system to be open is
Popen(τ,d) =z(0,1) + z(1,1)
If the distance between two channels is much greater than
λ, they will behave independently. As the channels get
closer (d ? 5λ) they begin to interact and their conforma-
tional statistics are altered. Popenas a function of tension
for certain fixed separations is shown in Figure 3. The
open-open interaction is the most energetically favorable
for most separations, hence the transition to the open state
generally shifts to lower tensions as the distance between
the two proteins is decreased.
Interactions also affect channel ‘sensitivity’, defined as
the derivative of Popenwith respect to tension, which quan-
tifies how responsive the channel is to changes in tension.
The full-width at half maximum of this peaked function is
a measure of the range of tension over which the channel
has an appreciable response. The area under the sensitiv-
ity curve is equal to 1, hence increases in sensitivity are
always accompanied by decreases in range of response, as
demonstrated by the effects of the beneficial open-open
interaction on channel statistics (see Figure 3).
In summary, we find that elastic interactions between
two proteins have significant effects when the protein
edges are closer than ∼ 5nm. At these separations the
elastic interactions alter the critical gating tension and
change the tension sensitivity of the channel (see Figure
3).The critical gating tension and sensitivity are the
key properties which define the transition to the open
state, and are analogs to the properties which define the
transition of any two-state ion channel. Hence we have
shown that elastic interactions can affect channel function
at a fundamental level.
Interactions between Diffusing Proteins
With an understanding of how two proteins will inter-
act at a fixed distance, we now study the conformational
statistics of two freely-diffusing MscL proteins allowed to
Proteins. Interactions between neighboring channels lead to
shifts in the probability that a channel will be in the open
state (dashed lines). The sensitivity and range of response
to tension, dPopen/dτ, are also affected by bilayer deforma-
tions (solid lines). Popen and dPopen/dτ are shown for sepa-
rations of 0.5nm (red) and 1.5nm (green) with reference to
non-interacting channels at d = ∞ (blue). Interactions shift
the critical gating tension for the closest separation by ∼ 12%.
Additionally, the peak sensitivity is increased by ∼ 90% from
∼ 5nm2/kBT to ∼ 9.5nm2/kBT, indicating a Hill coefficient of
Conformational Statistics of Interacting MscL
interact via their elastic potentials. In biological mem-
branes, transmembrane proteins that are not rigidly at-
tached to any cytoskeletal elements are often free to dif-
fuse throughout the membrane and interact with various
lipid species as well as other membrane proteins. On av-
erage, the biological areal density of such proteins is high
enough (∼ 100 − 1000nm2/protein ) that elastic in-
teractions should alter the conformational statistics and
average protein separations.
We expect that if two MscL proteins are diffusing and
interacting, the open probability will be a function of their
areal density as well as the tension. It then follows that
for a given areal density, elastic interactions will couple
conformational changes to the average separation between
the proteins. To calculate the open probability of two
diffusing MscL proteins, the Boltzmann weight for these
proteins to be in the conformations s1 and s2 must be
summed at every possible position, giving
?z(s1,s2)? = e−Hnon
where ?...? indicates a sum over all positions. The dis-
tance between the proteins is measured center-to-center
as |r1− r2| and only the absolute distance between the
two proteins determines their interaction, hence we can
rewrite the integrand as a function of r = |r1− r2|. We
then change the form of the integrand to
e−Hint(s1,s2,r;τ)= 1 + f12(r), (16)
which allows us to separate the interacting effects from
the non-interacting effects (the function f12is often called
the Mayer-f function). Thus, the position-averaged Boltz-
mann weights are
?z(s1,s2)? = e−Hnon
where A is the total area occupied by the two proteins.
Following our previous calculations, the probability that
any one channel is open in this two-channel system is
Popen(τ,α) =?z(0,1)? + ?z(1,1)?
where α is the area per protein (i.e. α = A/2) and ?Z? =
areal density, from the area of ∼ 100 lipids up to ar-
eas on the whole-cell scale.
open interaction tends to shift the transition to the open
state to lower tensions, with the most pronounced ef-
fect when the two proteins are most tightly confined.
For the estimated biological membrane protein density of
∼ 102− 103nm2/protein , the gating tension is de-
creased by ∼ 13%, the sensitivity is increased by ∼ 85%
and the range of response is decreased by ∼ 55%. For
the in vivo expression of MscL of ∼ 105−106nm2/protein
 the gating tension is reduced by ∼ 7%, the sensitivity
is increased by ∼ 70% and the range of response is de-
creased by ∼ 40%. These changes in gating behavior are
accessible to electrophysiological experiments where MscL
proteins can be reconstituted at a known areal density
(∼ 105− 107nm2/protein), and the open probability can
be measured as a function of tension.
In addition to lowering the critical tension and augment-
ing channel sensitivity, the conformational states of chan-
nels are tightly coupled by their interaction. The proba-
bility that exactly one channel is open (P1) decreases dra-
matically as areal density increases. For tensions above the
critical tension, interacting channels (∼ 103nm2/protein)
are nearly three orders of magnitude less likely to gate
as single channels than their non-interacting counterparts
(∼ 109nm2/protein), as shown in Figure 4b. Additionally,
the tension at which it is more likely to have both channels
open, rather than a single channel, is significantly lower for
interacting channels, signaling that gating is a tightly cou-
pled process. In addition to altering the open probability
of two channels, the favorable open-open interaction pro-
vides an energetic barrier to leaving the open-open state.
Based on a simple Arrhenius argument, the average open
lifetime of two channels that are both open and interact-
ing will be orders of magnitude longer than two open but
Having shown conformational coupling over a range of
areal densities, it is reasonable to expect that elastic inter-
actions will affect the separation between the two proteins.
We ask, how do interactions affect the average separation
between proteins? How often will we find the two proteins
In Figure 4a, we plot Popen(τ,α) over a wide range of
The more beneficial open-
Figure 4: Elastic Interactions Lower Open Probability Tran-
sition and Couple Conformation Changes. Two MscL proteins
in a square box of area A diffuse and interact via their elas-
tic potentials. a) At high area per protein, the response to
tension is the same as an independent channel. As the area
per protein decreases, the more beneficial open-open interac-
tion (see Figure 2) shifts the open probability to lower tensions
and decreases the range of response (dashed lines) while in-
creasing the peak sensitivity, indicating that areal density can
alter functional characteristics of a transmembrane protein. b)
The probability for exactly one channel to be open (P1 - solid
lines) is shown at a high (red) and low (blue) area per protein.
For tensions past the critical tension, interacting channels are
∼ 1000 times less likely to gate individually. The probability
for both channels to be open simultaneously (P2 - dashed lines)
is shown for high (red) and low (blue) area per protein. The
tension at which two simultaneously open channels are favored
is significantly lower for interacting channels. Together these
facts signify a tight coupling of the conformational changes for
two interacting channels.
separated by a distance small enough that we can consider
From Eqs. 15 and 16 it follows that the Boltzmann
weight for the two proteins to be separated by a distance
z(s1,s2,r) = e−Hnon2π
A(1 + f12)r. (19)
The probability that the proteins are separated by a dis-
tance r, regardless of their conformation, is
from which we calculate the average separation
This equation is valid as long as the area does not confine
the proteins so severely that they are sterically forced to
interact. The constant δ is an order-one quantity that de-
pends on the actual shape of the surface3; for a square box,
δ ≃ 1, and for a circle, δ ≃√2. The average separation of
two MscL proteins as a function of tension is plotted for
various areal densities in Figure 5. Note that for certain
densities, elastic interactions couple the conformational
change from the closed to open state with a decrease in the
average separation by more than two orders of magnitude.
Our estimates of biological membranes yield fairly high
membrane-protein densities (∼ 100 − 1000nm2/protein)
 which corresponds to the most highly confined con-
ditions on the Figures 4, 5 and 6. In the native E. coli
plasma membrane, MscL, with a copy number of ∼ 5 ,
is present at a density of ∼ 105− 106nm2/protein, which
means that even membrane proteins expressed at a low
level are subject to the effects of elastic interactions.
To quantify the effects of interaction on the spatial or-
ganization of two channels, we define a ‘dimerized’ state
by the maximum separation below which two channels will
favorably interact with an energy greater than kBT (i.e.
Hint(s1,s2,τ,r) < −1). This defines a critical separation,
rc(s1,s2,τ), which depends on the conformations of each
protein and the tension. The probability that the two pro-
teins are found with a separation less than or equal to rc
This ‘dimerization probability’ is plotted as a function of
tension and areal density in Figure 6.
3On a surface, S(A), the average separation has an entropic com-
ponent given byR R
the surface applies non-trivial bounds on this integral.
√A. The shape of
nificantly due to Elastic Interactions. The average separation
between two diffusing MscL proteins in a box of area A is plot-
ted as a function of tension for a range of areal densities, each
shown as a different line color. The grey region roughly in-
dicates when gating is occurring. At low areal density (most
blue) the conformational change does not draw the proteins sig-
nificantly closer together. As the areal density increases, the
conformational change is able to draw the proteins up to ∼ 100
times closer than they would otherwise be. At the highest areal
density (most red) the steric constraint of available area intrin-
sically positions the proteins close to one another regardless of
their conformation. The average separation begins to increase
again as higher tension weakens the open-open interaction.
Average Separation between Proteins Drops Sig-
At low tension and low area per protein, the channels
are closed and near enough that the closed-closed inter-
action can dimerize them a fraction of the time. Keeping
the area per protein low, increasing tension strengthens
the closed-closed interaction and the dimerization proba-
bility increases until tension switches the channels to the
open state, where the significantly stronger open-open in-
teraction dimerizes them essentially 100 percent of the
time. When the area per protein increases to moderate
levels, as denoted by the white dashed lines in Figure 6,
the dimerization is strongly correlated with the conforma-
tional change to the open state. The zero tension sepa-
ration between the two proteins for this one-to-one cor-
relation is ∼ 40nm to ∼ 2.2µm. Finally, when the area
per protein is very large, entropy dominates, and neither
the closed-closed, nor the open-open interaction is strong
enough to dimerize the channels.
In summary, we have shown that over a broad range,
areal density plays a non-trivial role in allowing two chan-
nels to communicate conformational information.
communication can lead to large changes in the average
separation between two proteins and the probability that
they will be found together in a dimerized state. This
may have implications for how conformational changes of
transmembrane proteins in biological membranes are able
Figure 6: Elastic Interactions Tightly Couple Conformational
Change with Protein Dimerization. Diffusing MscL proteins
are considered dimerized when they are close enough that they
attract with an energy greater than kBT.
protein, the net attractive closed-closed interaction is sufficient
to dimerize the two channels part of the time. As the area per
protein increases, the closed-closed interaction is not strong
enough to dimerize the two channels − now dimerization only
happens at higher tensions after both channels have switched
to the open conformation. As the area per protein grows even
larger, the open-open interaction is no longer strong enough to
overcome entropy. This loss of dimerization is amplified by the
fact that the open-open interaction is weaker at higher tensions
(see Figure 2). The white dashed lines roughly indicate the
range of areal densities for which dimerization probability and
open channel probability are equal to each other (see Figure
At low area per
to facilitate the formation of functional groups of specific
In this section, we will perform a brief survey of other
bilayer-mediated forces between proteins and make a com-
parison of their relative length and energy scales. We will
also address some of the finer details of our model and how
boundary conditions can affect deformation energy around
a protein. Finally, we will suggest experiments using MscL
to observe the predicted changes in conformational statis-
tics, as well as provide evidence from previous experiments
that leaflet interactions lead to significant changes in con-
There are at least two other classes of purely bilayer-
mediated forces between membrane proteins.
is a different type of bilayer deformation that bends the
mid-plane of the bilayer. This arises from transmembrane
proteins with a trapezoidal shape that impose a bilayer
slope at the protein-lipid interface [39, 44]. If the protein
does not deform the bilayer too severely, the mid-plane
deformation energy of a bilayer is
where h(r) is the deviation of the height of the mid-plane
from a flat configuration [26, 35]. These kinds of interac-
tions have been calculated for a variety of bilayer curva-
ture environments and protein shapes at zero tension .
Using a bilayer bending modulus of ∼ 100kBT, attrac-
tive interactions of order ∼ 1 − 5kBT were found when
the proteins were separated by 1-2 protein radii (which we
estimate to be 5 − 10nm measured center-to-center for a
typical transmembrane protein). If we adjust the energy
scale to be consistent with a PC bilayer bending modu-
lus of ∼ 14kBT this lowers the interaction energetics to
∼ 0.4 − 2kBT. Hence, although the length scale of ap-
preciable interaction for mid-plane deformation is longer
than for leaflet deformation, the interaction energies from
leaflet deformation can be 10 times greater depending on
protein geometry. The deformation fields h(r) and u(r)
exert their effects independent of one another , sug-
gesting that while energetically weaker than leaflet defor-
mation, mid-plane deformation probably also contributes
to the spatial organization and conformational communi-
cation between transmembrane proteins. However, for the
resting tension of many biological membranes , the in-
teraction due to midplane deformation has a length-scale
(?κb/τ ≃ 50nm) longer than the nominal spacing of pro-
other proteins from feeling the deformation of a neighbor-
ing protein, and hence interactions are not (in general)
pairwise additive. In fact, this is a general feature for
both leaflet and midplane elastic interactions - they can
be shielded by the presence of other proteins, and non-
specific protein interactions can couple to conformation
and position within the membrane in the same manner as
the specific interactions we have explored in the previous
teins (≃ 10 − 30nm ). Thus, one protein can shield
The second class of bilayer-mediated forces is a prod-
uct of the thermal fluctuations of the bilayer. There is a
small thermal force due to the excluded volume between
two proteins, calculated via Monte Carlo methods to have
a favorable ∼ 2kBT interaction . This force only exists
when the proteins are separated by a fraction of the width
of a lipid molecule. There is also a long-range thermal
force, due to the surface fluctuations of the bilayer, which
tends to drive two rigid proteins closer together [10, 47].
This force is proportional to 1/r4and is generally attrac-
tive. Estimates using this power law indicate that the in-
teraction is ∼ 1kBT when the center-to-center separation
is roughly 2 protein radii. Though elegant, the derivation
of this force is only valid in the far-field, thus how this
force might contribute to conformational communication
between proteins in close proximity is not entirely clear.
To gauge the overall importance of leaflet interactions,
the virial coefficient used in Eq. 17,
CV = 2π
quantifies how the combination of length and energy scales
leads to a deviation from non-interacting behavior; it is ex-
ponentially sensitive to the energy but only quadratically
sensitive to the length-scale. One can interpret the virial
coefficient as the area per particle that makes the com-
peting effects of entropy and interaction equivalent. Using
this measure, we estimated the virial coefficients for all of
these bilayer-mediated forces and found that leaflet defor-
mations, while having a short length scale, actually lead to
the most significant deviation from non-interacting behav-
ior, due to their high energy scale. We estimate the virial
coefficients from leaflet interactions to be ∼ 104−106nm2,
while mid-plane bending interactions are ∼ 103nm2, and
the thermal forces ∼ 102nm2.
Examining our elastic model in greater detail, we have
assumed that the slope of the leaflet at the protein-lipid
interface is zero, which eliminates any dependence on the
spontaneous and Gaussian curvatures of the leaflet. In
a more general continuum-mechanical theory, the slope
would be left as a free parameter with respect to which the
energy could be minimized . We examined this possi-
bility and found that, at most, the energy was reduced
by a factor of two. Spontaneous curvature couples to
the slope of the leaflet at the protein-lipid interface, how-
ever the spontaneous curvature of bilayer forming lipids,
such as phosphatidylcholines, is small . In addition,
for proteins whose radius is larger than λ, if we assume
the modulus associated with Gaussian curvature is of the
same magnitude as the mean curvature modulus (κb) ,
the Gaussian contribution to the deformation energy is a
second-order effect. We also examined the possibility of a
term proportional to (∇u)2; using the interfacial tension
(∼ 5kBT/nm2) as a modulus for this term; these effects
were also second-order. Finally, we imposed the ‘strong
hydrophobic matching’ condition at the protein-lipid in-
terface, assuming that the interaction of lipids with the
hydrophobic zone of the protein is very favorable. Relax-
ing this condition would result in a decrease in the magni-
tude of the hydrophobic matching condition, uo, and hence
an overall decrease in energetics .
There are also experimental and mechanical reasons
to believe the boundary slope on a cylindrical protein
is small. The membrane protein gramicidin was used to
comment on this so-called ‘contact angle’ problem of lipid-
protein boundary conditions [22, 50]. It was found that in-
deed the slope was nearly zero. From a mechanical stand-
point, if the lipids are incompressible, a positive boundary
slope that deviates significantly from zero would corre-
spond to the creation of an energetically costly void at the
protein-lipid interface when the protein is shorter than the
bilayer. Conversely, lipid would have to penetrate the core
of the protein to produce a negative slope when the protein
is taller than the bilayer, again a very costly proposition.
We examined a roughly cylindrical protein and demon-
strated the interesting effects elastic interactions would
have in such cases.However, the scope of possible ef-
fects increases when non-cylindrical proteins are consid-
ered. Most notably, non-cylindrical cross-sectionsallow for
orientational degrees of freedom in the interaction, hence
such proteins do not just attract or repel each other, but
would have preferred orientations in the membrane with
respect to each other.
Measuring the changes in conformational statistics of
two MscL proteins held at a fixed separation would al-
low for quantitative verification of our predictions. Elec-
trophysiology is a common tool used to probe the con-
formation of ion channels, and is routinely used to mea-
sure the open probability of a single MscL protein in
vitro [17, 19, 51]. Cysteine point mutations on the outer
edges of two MscL proteins  could be covalently linked
[52, 53, 54, 55] by a polymer with a specific length (∼
0.5 − 10nm) to control the separation distance [56, 57].
Linking stoichiometry could be controlled genetically 
to ensure one channel interacts with only one other chan-
Similar experiments have been performed using grami-
cidin A channels . The conducting form of gramicidin
A is a cylindrical transmembrane protein which, like MscL,
tends to compress the surrounding bilayer [22, 33, 60] and
hence have a beneficial interaction. Electrophysiology of
polypeptide-linked gramicidin channels  qualitatively
supports our hypothesis that the beneficial interaction of
the deformed lipids around two gramicidin channels sig-
nificantly increases the lifetime of the conducting state.
As another example, recent FRET studies showed that
oligomerization of rhodopsin is driven by precisely these
kinds of elastic interactions, and exhibits a marked de-
pendence on the severity of the deformation as modulated
by bilayer thickness . Additionally, recent experimen-
tal work has shown that the bacterial potassium channel
KscA exhibits coupled gating and spatial clustering in ar-
tificial membranes .
In summary, we have demonstrated that leaflet defor-
mations are one of the key mechanisms of bilayer-mediated
our choice of boundary conditions at the protein-lipid
interface, and suggested that extensions of our model have
exciting possibilities for the specificity of elastic interac-
tion. Finally, we suggested how one might measure the
predicted changes in conformational statistics and drew
an analogy to previous gramicidin channel experiments.
We provided support for
We have described the important role of an elastic bi-
layer in the function of, and communication between,
membrane proteins. Over a wide range of areal densi-
ties, transmembrane proteins can communicate informa-
tion about their conformational state via the deformations
they cause in the surrounding bilayer. We demonstrated
with a model protein, the tension-sensitive channel MscL,
how deformations lead to elastic forces and result in coop-
erative channel gating. Additionally, we found that elastic
interactions strongly correlate conformational changes to
changes in spatial organization, aggregating two channels
even at low areal densities, bringing them together over
very large distances relative to their size.
The elastic theory presented here can be easily ex-
panded to include more complex deformation effects
(such as spontaneous curvature) and protein shapes. Our
calculations for the conformational statistics, average
separation, and dimerization are insensitive to the actual
stimulus triggering the conformational change.
we suggest that elastic interactions are likely to play a
role in the function and organization of many membrane
proteins which respond to environmental stimuli by
forming functional groups of multiple membrane proteins.
Recent work suggests chemotactic receptors in E. coli
function by precisely this kind of spatially clustered and
conformationally coupled modality .
Materials and Methods
To compute the pairwise elastic potentials in Figure 2, we
discretize the bilayer height, η(ρ), and minimize the deforma-
tion energy in Eq. 5 using a preconditioned conjugate gradi-
ent approach. A separate minimization with the aforemen-
tioned boundary conditions, including the zero-slope bound-
ary condition, was computed for each combination of chan-
nel configurations, protein-protein separation, and bilayer ten-
sion. Except in the regions of the bilayer nearby a protein
at position (xo,yo), we use a Cartesian grid with spacing
dx = dy = 0.1λ = 0.093nm.
in the bilayer are largest at the circular membrane-protein in-
terface, we interpolate between a polar grid at the interface
at r = ro and a Cartesian grid along the square S defined by
|x − xo| < ∆,|y − yo| < ∆, where ∆ is chosen to be an in-
tegral multiple of dx. This interpolation ensures an accurate
estimate of the elastic deformation energy of a single protein
and preserves the symmetry of the protein in its immediate
The lines connecting the grid points along S define nθ angu-
lar grid points θi (i = 1,...,nθ), and nr+1 grid points within
the interpolation region are defined by the polar coordinates
(rij,θi) = (ro+ δrij/nr,θi), where ro is the radius of the pro-
tein and the distance from the center of the protein to S along
θi is ro + δri (e.g., for θi = 0, δri = ∆ − ro; for θi = π/4,
δri = ∆√2−ro. For a protein in the open or closed configura-
tion, ∆ was chosen such that nθ = 320 or 224, respectively.
The deformation energy determined using this numerical
relaxation method is converged with respect to dx, ∆ and
the overall dimensions of the bilayer (18.5nm × 37.1nm), and
reproduces the analytic results for a single protein given by Eq.
8. The elastic potentials were determined over the relevant
range of channel separations from 0 to ∼ 8nm (measured
from protein edge to protein edge), and for a range of bilayer
tensions from 0 to 3.4kBT/nm2.
However, since deformations
The primary accession numbers (in parentheses) from the
Protein Data Bank (http://www.pdb.org) are: mechanosen-
sitive channel of large conductance (2OAR; formerly 1MSL),
gramicidin A ion channel (1GRM), bacterial potassium ion
channel KscA (1F6G), and bovine rhodopsin (1GZM).
We would like to thank Doug Rees, Olaf Andersen, Pierre
Sens, Sergei Sukharev, Nily Dan, Jennifer Stockdill, and Ned
Wingreen for their thoughtful comments on the manuscript,
Chris Gandhi for his input into possible experiments and Ben
Freund for useful discussion.
Author contributions. TU conceived of the experiment
and performed the analytical calculations. TU, EP and KH
performed numerical simulations. TU, KH, EP and RP ana-
lyzed the data and wrote the paper.
RP acknowledges the support of the National
Science Foundation Award No. CMS-0301657.
acknowledge the support of the NSF CIMMS Award No. ACI-
0204932 and NIRT Award No. CMS-0404031 as well as the Na-
tional Institutes of Health Director’s Pioneer Award. EP was
supported by the Department of Homeland Security Graduate
Fellowship program and the NIH Director’s Pioneer Award.
KH was supported by the NIH Award No. A1K25 GM75000.
Part of this work took place at the Kavli Institute for Theo-
retical Physics, Santa Barbara, CA and the Aspen Center for
Physics, Aspen, CO.
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