VOLUME 92, N UMBER 16
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PHYSICA L R EVIEW LET T ERS
Internal States of Active Inclusions and the Dynamics of an Active Membrane
Hsuan-Yi Chen
Department of Physics and Center for Complex Systems, National Central University, Chungli, 32054 Taiwan
(Received 18 June 2003; published 19 April 2004)
A theoretical model of a two-component fluid membrane containing lipids and two-state active
inclusions is presented. This model predicts several nonequilibrium morphology transitions. (i) Active
pumping of the inclusions can drive a long-wavelength undulation instability. (ii) Active excitation of
the inclusions can induce aggregation of high-curvature excited inclusions. (iii) Active inclusion
conformation changes can produce finite-size domains. The resulting steady state domain size depends
on inclusion activities. For a stable membrane the height fluctuation spectrum in the long-wavelength
limit is similar to previous studies which neglected the inclusion internal states.
DOI: 10.1103/PhysRevLett.92.168101
A biological membranes is a multicomponent mixture
of lipids, proteins, carbohydrates, and other materials [1].
Since the active inclusions (e.g., proteins) in the membranes participate in physiological processes such as ion
transport, signal transduction, and cell locomotion, recent theoretical and experimental studies have focused on
the collective effect of these active inclusions on the
membranes [2 –5]. An important conclusion of these
studies is that, unlike equilibrium membranes [6], the
height fluctuation of a tensionless active membrane in
the long-wavelength regime depends on the inclusion
activities. Activity-induced instabilities are also predicted in previous theoretical models [3,7]. On the other
hand, although the fact that these active inclusions have
more than one internal conformation state has been discussed for a single transmembrane protein [8] and for a
single ion pump [9], the equally important subject of the
effects of inclusion internal states on the collective behavior of the lipid-inclusion system is seldom addressed
in previous studies of active membranes [10].
In this Letter I discuss the dynamics of a fluid membrane containing lipids and identical two-state active
inclusions, the simplest model which takes the inclusion
internal states into account. This is an extension of an
earlier work by Ramaswamy, Toner, and Prost (RTP) [3].
In RTP the inclusions were treated as point pumps with
no internal structure; in this Letter the internal states of
the inclusions are considered. As shown in Fig. 1, inclusions in different conformation states have different couplings to the membrane [11]. An inclusion can change its
conformation by external stimuli (active transition) or by
the lateral pressure exerted by the surrounding medium
(passive transition). During a conformation change, an
inclusion also exerts forces to the lipids and the solvent.
By taking these effects into account, the main predictions
of the model are as follows: (i) The long-wavelength
height variance of a stable membrane is similar to previous studies [3,7] which neglected the inclusion internal
states. (ii) The system has two types of long-wavelength
instabilities. They are pump-driven undulation instability
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0031-9007=04=92(16)=168101(4)$22.50
PACS numbers: 87.16.–b, 05.40.–a, 05.70.Np
due to the pumping-induced attraction between the inclusions and excitation-driven undulation instability due to
aggregation of many high-curvature excited inclusions.
(iii) Depending on the structural details of the inclusions,
the system can have a finite-wavelength instability which
leads to domains with activity-dependent size.
Let hx; y be the membrane height. The number density
of inclusions in state ( 1 or 2) is r? ; t where
r? x; y. It is useful to introduce 1 2 =2.
The Hamiltonian of the system H Hm Hi Hc includes the elastic energy of the membrane, the direct
interaction energy of the inclusions, and the membraneinclusion coupling. To lowest order
Hm 1Z 2
d r? r2? h2 r? h2 ;
2
(1)
where is the bending rigidity, and is the surface
tension of the membrane. In this Letter I consider the
regime where the system is close to a phase separation in
, and is noncritical; therefore Hi in the harmonic
theory is assumed to be [12,13]
Z
1
m
2
2
Hi d r? 2 r? 2 r2 2
0
2
2
;
(2)
0 is small R
and m r2 . is chosen so
where r2 > R
2 r = d2 r
that 0
d
? ? is fixed. > 0 is the
‘‘excitation energy’’ for an inclusion, thus ‘‘state 1 (2)’’
is the ‘‘ground (excited) state’’ of an inclusion. The
excitation
relaxation
FIG. 1. Schematics of an inclusion in different conformation
states. This figure provides an example when the ground state
inclusion prefers lower membrane curvature and the excited
state prefers higher membrane curvature.
 2004 The American Physical Society
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VOLUME 92, N UMBER 16
inclusion-membrane coupling is
Z
Hc d2 r? c1 1 c2 2 r2? h
Z
d2 r? c c
r2? h;
The solvent which embeds the membrane satisfies the
modified Stokes equation,
@H
@
r? ;
r2?
@
@t
H
r ; tz^z
h ?
e
u
Pa z w1 z wd1 ke 1 z^
Pra z wu2 z wd2 kr 2 z^ 'r2 v fv :
0 rpr; t (3)
where c c1 c2 . The equilibrium properties of
this system are summarized below. The membrane is
stable when c2 q? 2 c2 =m 2 c2
=q2? r2 q4? q2? > 0. For a tensionless membrane there is
a long-wavelength instability when eff 2 c2 =m 2 c2
=r2 < 0, where a phase separation
in occurs. Since r2 > 0, this phase separation is
induced by membrane-inclusion coupling. For a membrane under tension the system has a finite-wavelength
p
instability when eff < f2 =r2 2 2 =r2 g,
where 2 c2 =m. In this case, the long-wavelength instability is cut off by the surface tension at large
length scales. When the system is stable, the variance of
the membrane height in the long-wavelength limit is
hhq? h
q? i kB T=eff q4? for a tensionless membrane and hhq? h
q? i kB T=q2? for a membrane
under tension.
New variables are introduced in the following to
discuss the dynamics of the system. When c2 q? > 0,
are chosen to be the deviations from the equilibrium
solutions of . When c2 q? < 0, are chosen to be
0
and so that the nonequilibrium effects on the stability of the membrane can be
discussed conveniently. On experimental time scales the
membrane does not exchange inclusions with the solvent,
therefore obeys the conserved dynamics:
(4)
where the vector thermal noise has zero
mean and variance hr? r; tr? r0 ; t0 i 2kB T r2? 3 r r0 t t0 . In general satisfies
an equation of the following form:
H
@
k k
:
0 2 r2? @t
(5)
The 0 term is the passive transition of the inclusions
[14], the 2 term is the mutual diffusion, k k
comes from the active transitions [15], and is
the thermal noise. The solvent flow, the permeation of
solvent through the membrane, and the force exerted by
the inclusions during active transitions all contribute to
the dynamics of the membrane, thus
H
@h
vz p
Pea ke 1 Pra kr 2 h ; (6)
h
@t
where Pea (Pra ) is the active permeation during an active
excitation (relaxation) process. h is the thermal noise.
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PHYSICA L R EVIEW LET T ERS
(7)
Here p is the pressure, ' is the solvent viscosity, and
the third and fourth terms on the right-hand side
come from the active transitions of the inclusions.
To lowest order the force distribution of an active
inclusion-excitation (relaxation) event is approximated
by a force dipole [4]. Here wu1 , wd1 , wu2 , and wd2
are characteristic lengths associated with the inclusion conformation transition processes. These
force-dipole terms were not considered in RTP.
However, later work [4] found that both active and passive
permeation are negligible for q
1
? 1 cm. Therefore,
similar to Ref. [4], from now on I neglect permeation
and keep force-dipole terms. The vector thermal noise fv
has zero mean and variance hfvi r; tfvj r0 ; t0 i 2kB T'
ij r2? @i @j 3 r r0 t t0 .
Since the system is close to a phase separation in , it
is convenient to treat as a fast mode even though it is
conserved [16]. Solving the Stokes equation (7) and ,
the linearized equations of motion for h and are
obtained.
@hq? ;t @t
@
q? ;t
@t
Dh q? Dh q? Dh q? D q? fh q? ; t
hq? ; t
;
f q? ; t
q? ; t
(8)
where
1
2 c ca 4
2
q? q? ;
Dh q? 4'q?
m
D q? 0 2 q2? 2 q2? r2 k
;
1
ca q2 ;
Dh q? 4'q? ?
and
Dh q? 0 2 q2? c
q2? k
c 2
q :
m ?
(9)
ca c v , and v Pea ke wu1 2 wd1 2 =2 Pra kr wu2 2 wd2 2 =2 depend on structural details and
active transition rates. ca can be understood as the ‘‘renormalized’’ inclusion-membrane coupling in the presence of nonequilibrium activities. Dimensional analysis
[4,5] shows that v fl2 , where l 5 nm is a typical
length associated with an inclusion, and f has the dimension of a force. At high activities f 10
12 N [4,5],
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VOLUME 92, N UMBER 16
together with c lkB T gives v =c & O1. Therefore ca and c (ca
and c
) are of the same order of
magnitude and usually have the same sign. The noises
fh q? ; t and f q? ; t have zero mean and variances
hfh q? ;tfh q0? ;t0 i2kB T=4'q? 1v2 q? =4' m2 t
t0 2 q? q0? , and hf q? ;tf q0? ;t0 i2kB T0 2 q2? k2 = m2 q2? t
t0 2 q? q0? . They depend
on v and k ; thus they are not equilibrium thermal
noises.
The linear stability analysis of Eq. (8) for a tensionless
( 0) membrane shows that the system is unstable
in the long-wavelength region when 0 0 aeff k =r2 2 c ca
=m k
=r2 a < 0, where a 2 c ca =m 2 c2 =m is positive, and aeff 2 c ca =m 2 c
ca
=r2 is the renormalized
eff . When 2 0 2 =r2 a 2 aeff > 0, the system
is stable at positive 0 . When 2 < 0 and 0 < 0 <
22 =42 2 =r2 a max
0 , the system has a finite-wavelength instability. The unstable wavelengths are plotted
schematically against 0 in Fig. 2 for both 2 > 0 and
2 < 0. The physical mechanisms of these instabilities
are discussed in the following.
In the absence of activities, k 0, a , and
aeff eff . The long-wavelength instability 0 < 0 is
simply the equilibrium instability eff < 0. In the presence of activities, as shown in Fig. 3, two mechanisms can
induce long-wavelength instabilities even when eff > 0.
Figure 3(a) shows that if the pumping of the active
inclusions is associated with positive and sufficiently
large v c =m v
c
=r2 such that aeff is large
and negative, then pumping induces attraction between
the inclusions, and the system becomes unstable against a
long-wavelength undulation. A similar instability is discussed in RTP [3], where the inclusions act on the membrane directly through active permeation. In the present
model, the pumping is due to the solvent flow induced by
the active inclusion force dipoles. Figure 3(b) shows that
another type of long-wavelength instability occurs when
k 2 c ca
k 2 c c
k 2 c21 c22 is sufficiently
Γ0
large and negative, i.e., the excited inclusions prefer large
local membrane curvature. Intuitively, this condition corresponds to the situation where the system excites many
inclusions to a state which prefers higher membrane
curvature.
The finite-wavelength instability occurs when 0 is
small, eff < 0, and 0 > 0 when k =r2 2 c ca
=m k
=r2 a is sufficiently large. These conditions correspond to the case when 0 < 0 at small k and passive
conformation transitions are negligible such that 2 < 0,
and 0 becomes positive at sufficiently large k . Since r2
is small, the sign of 0 can be controlled by varying k ;
thus this finite-wavelength instability should be accessible for a typical experiment. 0 increases either when
k
kr ke increases or (when 2 c ca
> 0) when k
increases. In the first case, k
increases, the instability in
the long-wavelength region due to negative eff is suppressed by increasing the active exchange rates of the
inclusions: a macroscopic phase separation cannot occur
when the inclusions change states sufficiently fast, and a
finite-wavelength instability exists for a range of k
. At
very high k
, 0 > max
0 , the inclusions change their
conformations at extremely high rates such that the membrane feels only the time-averaged inclusion conformation, and therefore no domain is formed [17]. The physics
of this activity-tunable finite-wavelength instability is
similar to that occurring in chemically reactive phaseseparating mixtures [18]. In the second case, 2 c ca
2 c21 c22 > 0 and k increases, the excitation of low
curvature inclusions suppresses the long-wavelength instability which is induced by the presence of many
high-curvature ground state inclusions. When 0 < 0 <
max
the ground state inclusions form finite-size high0
curvature patches. At very large k , 0 > max
0 , the presence of many small curvature excited inclusions
Γ0
stable
stable
q
(a)
Γ0
max
stable
unstable
unstable
stable
q
(a)
excitation
(b)
Γ2 >0
Γ2 <0
FIG. 2. The unstable wavelengths versus 0 . (a) 2 > 0.
There is a long-wavelength instability when 0 < 0, and there
is no instability when 0 > 0. (b) 2 < 0. There is a longthe
wavelength instability when 0 < 0. When 0 < 0 < max
0
long-wavelength instability is suppressed and the system has a
finite-wavelength instability. The system is stable when 0 >
max
0 : all instabilities are suppressed by inclusion activities.
168101-3
(b)
FIG. 3. Two mechanisms for long-wavelength instabilities:
(a) shows that pumping-induced attraction between inclusions
induces inclusion aggregation. The arrows indicate the direction of the effective forces exerted by the inclusions to
the environment during their active conformation changes.
(b) shows the instability induced by the presence of many
high curvature excited state inclusions.
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completely suppress the instability due to high-curvature
ground state inclusions. Since it is of great interest to
know the characteristic length R of the steady state
finite-size domains, here I present a scaling analysis for
R [18,19]. When there is no budding, the growth of intramembrane domains in the absence of inclusion activities
corresponds to a two-dimensional phase separation dynamics, i.e., Rt t , 1=3 [20]. This growth eventually saturates due to the active transitions. Since
the time scales associated with active transitions are k ,
the steady state domain size R should obey R k
M ,
where kM k
when k 2 c ca
=m is greater (smaller)
than k
a .
Similar to RTP and Refs. [4,5], the steady state height
fluctuation for a stable membrane in the long-wavelength
limit has the form kB Teff =eff q4? . However, due to the
active transitions which are not considered in RTP and
Refs. [4,5], the effective temperature,
Teff 8
>
<
>
:
Teff
2 c ca
k 2 c ca
m k
m
when 0 2 l
2
max ;
;
Teff
2
a
2 c ca
2 c
ca
k c c
=0 m
m r2 k
=0
;
otherwise;
(10)
has a different form. For a membrane under tension, the
1=q4? behavior is cut off and replaced by kB T=q2? at long
wavelengths, where T is the temperature of the solvent.
In summary, the internal states of active inclusions can
have significant effects on the dynamics and stability of a
membrane. These include instabilities induced by active
pumping or inclusion excitation, and activity-controlled
finite-size domains. In the stable case, the membrane
height fluctuation is similar to previous theoretical and
experimental studies. Although current analysis is restricted to a specific region of the parameter space, the
rich dynamical behaviors shown in this Letter are believed to exist in more general situations. This will be
analyzed in a future work [21]. Finally, it is important to
point out the similarities between the activity-controlled
finite-size domains in this model and the dynamics of
nm-scale protein-rich domains in biomembranes [22].
This model might be of relevance to active conformational transitions in biological membranes. Experimental
work on artificial membranes can also be designed to
confirm the main ideas of this model.
I thank David Jasnow for his encouragement and
Peilong Chen for stimulating discussions. This work is
supported by the National Science Council of the
Republic of China under Grants No. NSC-91-2112-M008-052 and No. NSC-92-2112-M-008-019.
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[1] H. Lodish et al., Molecular Cell Biology (W. H.
Freeman, New York, 1995), 3rd ed.
[2] J. Prost and R. Bruinsma, Europhys. Lett. 33, 321 (1996).
[3] S. Ramaswamy, J. Toner, and J. Prost, Phys. Rev. Lett. 84,
3494 (2000).
[4] J.-B. Manneville, P. Bassereau, S. Ramaswamy, and
J. Prost, Phys. Rev. E 64, 021908 (2001).
[5] S. Sankararaman, G. I. Menon, and P. B. Sunil Kumar,
Phys. Rev. E 66, 031914 (2002).
[6] F. Brochard and J.-F. Lennon, J. Phys. (Paris) 36, 1035
(1975).
[7] M. Rao and R. C. Sarasij, Phys. Rev. Lett. 87, 128101
(2001).
[8] R. S. Cantor, J. Phys. Chem. B 101, 1723 (1997).
[9] C.-M. Ghim and J.-M. Park, Phys. Rev. E 66, 051910
(2002).
[10] M. C. Sabra and O. G. Mouritsen, Biophys. J. 74, 745
(1998).
[11] Recent experiments on the structures of several ion
channels have provided evidences of such inclusion conformation changes. See, for example, Y. Jiang et al.,
Nature (London) 417, 515 (2002); 417, 523 (2002);
Y. Jiang et al., Nature (London) 423, 33 (2003); 423,
42 (2003).
[12] In general there can be a term due to different
density susceptibilities of 1 and 2 . The effect of this
term will be discussed in a future work [21].
[13] P. M. Chaikin and T. C. Lubensky, Principles of
Condensed Matter Physics (Cambridge University
Press, Cambridge, 1995).
[14] S.-K. Ma, Modern Theory of Critical Phenomena
(Addison-Wesley Publishing Co., Redwood City, CA,
1976).
[15] The contribution of active inclusion conformation
changes to d12 =dt is ke 1 kr 2 . Thus k ke kr . Here ke , kr depend on the strength of external stimuli,
e.g., the concentrations of specific ligands. Since the
diffusion of the ligands in the solvent is fast compared
to the in-plane dynamics of the inclusions, I treat both ke
and kr as constants.
[16] The dynamics of is slow compared to that
2
of
when 0 2 l
2
max r2 k
mlmax , i.e.,
p
m=0 r2 k
lmax [because =2 O1],
where lmax is the longest wavelength in the experiments.
[17] A similar effect has been discussed in Ref. [10], where
two-state active inclusions with different hydrophobic
lengths were considered.
[18] S. C. Glotzer, E. A. Di Marzio, and M. Muthukumar,
Phys. Rev. Lett. 74, 2034 (1995).
[19] P. B. Sunil Kumar, G. Gompper, and R. Lipowsky, Phys.
Rev. Lett. 86, 3911 (2001).
[20] M. Seul, N. Y. Morgan, and C. Sire, Phys. Rev. Lett. 73,
2284 (1994).
[21] Hsuan-Yi Chen (to be published).
[22] G. Vereb et al., Proc. Natl. Acad. Sci. U.S.A. 100, 8053
(2003).
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