PHYSICAL REVIEW E 85, 031913 (2012)

Effective tension and fluctuations in active membranes

Bastien Loubet,1 Udo Seifert,2 and Michael Andersen Lomholt11Department of Physics, MEMPHYS - Center for Biomembrane Physics, Chemistry and Pharmacy, University of Southern Denmark,

Campusvej 55, 5230 Odense M, Denmark2Universitat Stuttgart, II. Institut fur Theoretische Physik, Pfaffenwaldring 57/III, D-70550 Stuttgart, Germany

(Received 9 December 2011; published 23 March 2012)

We calculate the fluctuation spectrum of the shape of a lipid vesicle or cell exposed to a nonthermal source ofnoise. In particular, we take constraints on the membrane area and the volume of fluid that it encapsulates intoaccount when obtaining expressions for the dependency of the membrane tension on the noise. We then investigatethree possible origins of the nonthermal noise taken from the literature: A direct force, which models an externalmedium pushing on the membrane, a curvature force, which models a fluctuating spontaneous curvature, anda permeation force coming from an active transport of fluid through the membrane. For the direct force andcurvature force cases, we compare our results to existing experiments on active membranes.

DOI: 10.1103/PhysRevE.85.031913 PACS number(s): 87.16.dj

I. INTRODUCTION

Lipid membranes are an essential part of any biologicalcell, for example, as the interface between the cell andits surrounding environment. The cell actively is engagedin interactions with its environment, which means that themembrane is kept out of thermal equilibrium. One of theinteresting properties of the membrane is its shape fluctuations,which can influence, for instance, membrane adhesion [1–5].The shape fluctuations must be influenced by the activity sinceit is, for example, known that the fluctuations of red bloodcells depend on the concentration of adenosine triphosphate(ATP) [6]. Artificial vesicles are a valuable model for livingmembranes as well as an interesting physical object on itsown [7]. The simplest fluid vesicles are in thermal equilibrium.Their fluctuations are purely thermal and are characterizedby the well known Helfrich effective free energy [8]. Itcontains the bending rigidity, the Gaussian bending rigidity,and the tension. The bending rigidity and the Gaussian bendingrigidity are material parameters, which depend mainly on themembrane composition, while the tension can be interpretedas a Lagrange multiplier for a constraint on the membranearea [9]. Therefore, the tension is related to the excess areastored in the fluctuations of the membrane shape. Increasingthe excess area often leads to a response of decreasing tensionand vice versa. A number of theoretical and experimentalpapers on active membranes consider the active contributionin the membrane dynamical equations disregarding effects onthe interplay between tension and excess area of the membrane[10–14]. The correct physical treatment of the renormalizationof the tension is an important problem since, for instance,the membrane tension has been shown to affect membraneendocytosis [15] or mechanosensitive channels [16–18] as wellas influencing the motility of some cells [19]. In this paper,we use a simple model for an active force on the membrane toextend the idea of area conservation to active membranes. Todo so, we compare the passive and active fluctuation spectra ofthe same vesicle, and we calculate the tension self-consistentlythrough a constraint on the excess area. We will considerthree different physical scenarios for the forces that havebeen studied in the literature: direct force [12,20], curvatureforce [10–12], and a case we call permeation force [10,21].

For the direct force, we will consider the example of theATP activated cytoskeleton of red blood cells [22–24] andreferences therein. For the curvature force, we will comment onexperiments with giant unilamellar vesicles having active ionspumps included in their membranes [10,25,26]. We will giveorder of magnitude estimates of the effects for the permeationforce case.

The outline of the paper is as follows: In Sec. II, we recall thedynamical equations governing the fluctuations and the areaconservation for passive vesicles. In Sec. III, we add an activecontribution to the dynamic equations for the vesicle and givethe corresponding changes in the general expressions for thefluctuations and the area conservation equations. In Sec. IV,we discuss the direct force case, in Sec. V, we introduce thecurvature force case, and in Sec. VI, we discuss the permeationforce case. Finally, we give our conclusions and outlook inSec. VII.

II. EQUILIBRIUM FLUCTUATIONS

We begin by recalling the equations of motion for thepassive fluctuations of a quasispherical vesicle taking the areaand volume constraints into account [9]. We take the followingeffective free energy for our membrane:

F =∫

dA(2κH 2 + �). (1)

The first term in this equation is the classical Helfrich energywith κ being the bending rigidity, and H is the local meancurvature and the integration being over the whole area of themembrane. The second term is a Lagrange multiplier for thearea: The tension � is determined self-consistently such thatthe total area is conserved. For the quasispherical vesicle, wewrite the radial coordinate of the surface as

R(θ,φ,t) = R0[1 + u(θ,φ,t)], (2)

where R0 is the radius defined through the fixed volume V

of the vesicle by V = 4πR30/3 and θ and φ are the polar and

azimuthal angles. We expand u(θ,φ,t) in spherical harmonics

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LOUBET, SEIFERT, AND LOMHOLT PHYSICAL REVIEW E 85, 031913 (2012)

as

u(θ,φ,t) =lm∑

l=0

l∑m=−l

ulm(t)Yml (θ,φ), (3)

where lm is a cutoff value for l. The equation of motionfor ulm(t) is the normal component of the force balanceequation of the membrane. Detailed derivations have beengiven elsewhere, Ref. [11] or [27], for example. We only statethe result here

η

�l

ulm(t) = − κ

R30

El(σth)ulm(t) + ξ(th)lm (t), (4)

where

�l ≡ l(l + 1)

4l3 + 6l2 − 1, (5)

El(σ ) ≡ (l + 2)(l − 1)[l(l + 1) + σ ]. (6)

The left hand side of Eq. (4) is the friction of the surroundingbulk fluid on the membrane with η being the viscosity of thefluid and the dot denoting a time derivative. The first termon the right hand side is the elastic restoring force comingfrom the Helfrich free energy Eq. (1) where σth ≡ �R2

0/κ isa dimensionless tension. The last term is the thermal noisegiving rise to the thermal fluctuations. The properties of thisthermal noise are ⟨

ξ(th)lm (t)

⟩ = 0, (7a)⟨ξ

(th)lm (t)ξ (th)

l′m′(t ′)∗⟩ = 2η

kBT

R30

1

�l

δll′δmm′δ(t − t ′), (7b)

where the asterisk denotes complex conjugation, kB is theBoltzmann constant, T is the temperature of the surroundingfluid, δab is the Kronecker delta, and δ(t) is the Dirac deltafunction. Here and throughout the paper, the brackets mean thatwe average over noise realizations. From either these equationsor the free energy, we can calculate the ulm fluctuations to be

〈ulm(t)u∗lm(t)〉th = 1

κEl(σth), (8)

where κ ≡ κ/(kBT ) is a reduced bending rigidity. We nowcalculate the tension σth as in Ref. [9] through constraints onthe membrane area and enclosed volume. We start by writingdown the area and volume of the vesicle as a function of theulm’s,

A = R20

{4π

(1 + u00√

4π

)2

+lm∑

l=1

l∑m=−l

|ulm|2[1 + l(l + 1)/2] + O(u3

lm

)}, (9)

V = R30

[4π

3

(1 + u00√

4π

)3

+lm∑

l=1

l∑m=−l

|ulm|2 + O(u3

lm

)].

(10)

Having a fixed volume V = 4πR30/3 fixes the amplitude u00

as a function of the other ulm,

u00 = −lm∑

l=1

l∑m=−l

|ulm|2√4π

. (11)

The excess area, which we will define as � ≡ (A −4πR2

0)/R20, is then,

� =lm∑

l=2

l∑m=−l

|ulm|2 1

2(l + 2)(l − 1). (12)

Note that we did not include a factor of 4π in the definition ofthe excess area for convenience and that the l = 1 mode, whichcorresponds to translation, cancels. To obtain an equation thatgives the tension as a function of the excess area, we consider athermodynamic limit where we can replace |ulm|2 by 〈|ulm|2〉th

in Eq. (12). It then reads

2κ� =lm∑

l=2

2l + 1

l(l + 1) + σth. (13)

This gives a self consistent equation for the passive tension σth

that has been studied in Ref. [9].Note that, in a previous paper from two of the authors

[28], we have investigated the effect of an additional tensioncontribution for a compressible vesicle with a finite areaexpansion modulus. For simplicity, in this paper, we considerthe active contribution to the tension to be low enoughcompared with the area expansion modulus such that the areadoes not change. Also note that we are concerned about theso called internal tension (as opposed to a frame tension),which, within our approximations, is identical to the tensionmeasurable at low wave numbers in the passive fluctuationspectrum. The issue of how the internal tension and the frametension are related is subject to an ongoing debate, see, forinstance, Ref. [29].

III. FLUCTUATIONS IN THE ACTIVE CASE

To model the activity, we add an active random contributionin the force balance equation, Eq. (4). The equation of motionin the active case is then,

η

�l

ulm(t) = − κ

R30

El(σa)ulm(t) + ξ(th)lm (t) + ξ

(a)lm (t), (14)

where we introduced a new subscript in the symbol for thetension σa ≡ �aR

20/κ in the active case, indicating that it

can be different from the tension σth in the passive case. Thebending rigidity in the definition of the unitless active tensionκ is the same as for the passive vesicle since there is no activecontribution to it in our model. ξ (a)

lm (t) is the noise coming fromthe active processes. We take this noise to have the followingproperties: ⟨

ξ(a)lm (t)

⟩ = 0, (15a)⟨ξ

(a)lm (0)ξ (a)∗

l′m′ (t)⟩ = χaxlδll′δmm′ exp

[−|t |

τa

], (15b)

where τa is a characteristic correlation time of the activeprocess studied. It could be on the order of the average timefor a pumping cycle for an ion pump, for example. χa givesthe strength of the noise. It has the dimension of a force perunit area squared (N2 m−4). xl is a dimensionless quantity thatcarries the wave number dependency of the noise. By sphericalsymmetry, it can only be a function of l and not m. From theLangevin equation (14) and the noise properties Eqs. (7) and

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EFFECTIVE TENSION AND FLUCTUATIONS IN ACTIVE . . . PHYSICAL REVIEW E 85, 031913 (2012)

(15), the ulm fluctuations can be calculated as

〈|ulm|2〉a = 1

κEl(σa)

(1 + χa

xl

El(σa)

τa

τa + t(l)m

), (16)

where

χa = χaR6

0

kBT κ, (17)

t (l)m ≡ ηR3

0

κ�lEl(σa). (18)

χa is a nondimensional measure for the strength of the noise,and t (l)

m is the correlation time for mode l of the shape in thermalequilibrium. The fluctuation spectrum is similar to the passivefluctuation spectrum with the active tension σa replacing thepassive tension σth and an additional positive contributionproportional to the strength of the noise χa. The effectiveequation of area conservation (12) (with |ulm|2 replaced by〈|ulm|2〉a) is

2κ� =lm∑

l=2

2l + 1

l(l + 1) + σa

(1 + χa

xl

El(σa)

τa

τa + t(l)m

). (19)

Equation (19) is a central result of this paper. It relates theexcess area � and the tension σa to the parameters describingthe activity: χa, xl , and τa. We see that the effect of the activityis twofold: It gives a direct contribution to the fluctuationspectrum Eq. (16), and it indirectly renormalizes the tensionthrough the area constraint Eq. (19). Also, note that the activetension σa depends on the viscosity, whereas, the passivetension σth does not. This is a distinct experimentally testablesignature of the nonequilibrium activity.

A. Tension renormalization

Using Eq. (19), we can deduce some general properties ofthe tension in our model. Comparing the equations for the areaconservation in both the active case, Eq. (19), and the passivecase, Eq. (13), we can see that σa � σth with σa = σth onlyif χa = 0. Because we have fixed the area of the membrane,then the additional noise due to the active forces always leadsto a counterbalancing increase in the tension. Also, note that,in our model, the total tension that appears in the fluctuationspectrum, Eq. (16), is fixed self-consistently entirely throughEq. (19). This implies that any direct contribution to thetension in the force balance equation (14) has no effect on thefluctuation spectrum since it is canceled by a counterbalancingtension contribution to keep the excess area conserved. Inparticular, the direct contribution to the tension from the dipolemoment of a distribution of force due to an active proteinfound in Ref. [30] does not affect the fluctuation spectrumin the present model of an incompressible membrane (seeRef. [28] for the case of a compressible membrane). However,the quadrupole contribution found in Ref. [30] acts like thecurvature force discussed here in Sec. V, and thereby, itindirectly renormalizes the tension through the area constraint.

We now investigate the relation among the active tensionσa, the excess area �, and the other parameters of the systemusing Eq. (19) in different limiting cases. First, we distinguishtwo limits: long and short active correlation times. For long

correlation time, formally τa � t (2)m , Eq. (19) becomes

� = 5

6 + σa

(1 + χa

x2

E2(σa)

)

+lm∑

l=3

2l + 1

l(l + 1) + σa

(1 + χa

xl

El(σa)

), (20)

where � is the reduced excess area,

� ≡ 2 �κ = 2�κ

kBT. (21)

For a short correlation time, formally τa t (lm)m , we get

� = 5

σa + 6

(1 + χaτ x2

6

55

)

+lm∑

l=3

2l + 1

σa + l(l + 1)(1 + χaτ xl�l) , (22)

where τ reads

τ ≡ τaκ

ηR30

. (23)

Note that this case corresponds to a timewise Dirac deltacorrelated active noise, i.e., a white noise. It can be obtainedas τ → 0 while χa → ∞ such that χaτ is finite. In both cases,Eqs. (20) and (22), we have isolated the l = 2 term as this termdiverges when σa → −6. In the last term of both equations,we approximate the sum by an integral while the first termis kept as is. Using this approach, we can invert Eqs. (20)and (22) in order to obtain a relation between σa and thereduced excess area �. Similar to the treatment of the passivecase in Ref. [9], we also break up our results depending onthe value of σa: σa → −6, 1 σa l2

m, and 1 l2m σa.

After expanding in the appropriate limit of σa and lm, wekeep only the dominating term proportional to χa as well asthe dominating term that does not depend directly on χa inEqs. (20) and (22). This procedure gives the results presentedin Table I for each of the three cases of force that we motivateand consider in the following sections. These results are theessential findings of this paper. They give the behavior ofthe tension in terms of the active parameters χa and τa inthe different physical cases we consider here. These cases,classified by their dependency of xl on the wave number l, areas follows: xl ∼ 1 (direct force), xl ∼ l2 (permeation force),and xl ∼ l4 (curvature force).

IV. DIRECT FORCE

The case xl = 1 corresponds to a noisy force that is beingapplied locally on the membrane, i.e., a direct random forcecontribution in Eq. (14). This kind of noise has been studiedpreviously, mainly to model an active coupling between thecytoskeleton and the membrane in red blood cells [24,31,32].In this case, it represents the punctual force applied to themembrane (or more precisely, a release of tension) when ATPis used in the process of cutting a link between an actin filamentand the membrane. The amplitude of the active correlation canbe estimated as

χa ≈ ρ

(F0

R0

)2

, (24)

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LOUBET, SEIFERT, AND LOMHOLT PHYSICAL REVIEW E 85, 031913 (2012)

TABLE I. The different behaviors of the tension obtained for the direct force, curvature force, and permeation force in the limit of shortand long correlation times for the active noise.

σa → −6 1 σa l2m 1 l2

m σa

Curvature

τa � t (2)m σa = −6 + 5

2�

(1 +

√1 + 4

5 �χa

)σa = l2

m exp[− �+χa/4

1+χa/4

]σa = l2m

2�

(1 +

√1 + 1

2 �χa

)τa t (lm)

m σa = −6 + 5�

(1 + 4χaτ�2) σa = l2m exp

(−� + χa τ

4l3m6

)σa = l2m

�

(1 + χa τ

40 l3m

)Direct

τa � t (2)m σa = −6 + 5

2�

(1 +

√1 + 1

5 �χa

)� = ln

(l2mσa

)+ χa

σa2

[ln

(σa4

) − 38

]σa = l2m

2�

[1 +

√1 + 4 χa�

l4mln

(l2m4

)]τa t (lm)

m σa = −6 + 5�

(1 + χaτ�2) � = ln(

l2mσa

)+ π

4 χaτ1√σa

σa = l2m�

(1 + χa τ

2lm

)Permeation

τa � t (2)m σa = −6 + 5

2�

(1 +

√1 + 605

36 �χa

)� = ln

(l2mσa

)+ 16 χa

σaσa = l2m

2�

(1 +

√1 + 64�

χa

l2m

)τa t (lm)

m σa = −6 + 5�

(1 + 55

6 χaτ)

σa = l2m exp[−� + 8χaτ lm] σa = l2m

�

(1 + 8

3 χaτ lm)

where F0 is on the order of the force applied on the membraneby one active center and ρ is the concentration of active centers.In the cytoskeleton case, ρ will be related to the average densityof links activated by ATP, and hence, it will depend on the ATPconcentration [24]. The active correlation time τa should be onthe order of the release time of an actin filament. Note that thedirect force case requires an external source of momentumfor Newton’s third law to be obeyed; the case cannot beapplied to proteins included in the membrane without externalattachment, for example, ion pumps.

A. Tension renormalization

The activity will modify the tension through the areaconstraint as described in Table I. We begin our discussionof the application to the cytoskeleton by pointing out thatthe ATP induced activity, by attempting to increase the mem-brane fluctuations, increases the membrane tension throughthe area constraint. Hence, the tension increases when theATP concentration increases. In Fig. 1, we have shown therenormalization of the tension as a function of the strength ofthe activity with some typical parameter values for a red bloodcell. Here, we compare this result to another model [22,32].This model introduced a confining potential that acts on themembrane in order to model the attachment of the cytoskeletonto the membrane. In addition, the calculation was performedfor an almost flat membrane with an infinite frame area. In thespherical limit in this paper, we can get the flat case limit bykeeping only the highest power of l and then replacing l byqR0 where q is the Fourier transform variable. In this limit,El , including a uniform potential, becomes

κEl

R40

→ κq4 + σq2 + γ. (25)

The parameter γ is the strength of the confining potential.There were no constraints on the area, but an effective(increased) tension was considered to come from the con-finement of the membrane [22]. Here, we can give thefollowing two remarks, comparing to our approach. First, inour model, a laterally uniform harmonic confining potential

over the membrane decreases the tension compared to acase where there is no confining potential (no attachmentto the membrane) because the uniform potential tends todecrease the long wavelength fluctuations; hence, the absolutevalue of the tension is diminished by the confinement. Second,the increase in the tension in our model is due to the effectof the noisy activity through the area constraint. Note that,with increasing ATP concentration, the strength of the activityχa will increase while the confinement will decrease becausethe cytoskeleton will have less attachments to the membrane.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50

5

10

15

107 χa

ath

σσ

th

FIG. 1. (Color online) The relative renormalization of the tensionas a function of the strength of the activity in the direct force case. Thehorizontal axis represents χa defined by Eq. (24) where F0 go from0 to 0.2 pN. The big circles represent σa’s calculated from Eq. (19),while the dotted line is σa calculated from the formula in Table I inthe direct force case for 1 σa l2

m and τa � t (2)m . The parameters

used are as follows: R0 � 5 μm, η � 9 × 10−4 kg m−1 s−1, kBT �4 × 10−21 J, κ � 4kBT , and τa � 0.1 s, assuming a typical distancebetween two active centers of 100 nm, we have ρ � 1014 m−2, andtaking a membrane thickness of d � 5 nm, we get lm � R0/d �1000. We also chose �/4π � 3% from which we found σth � 529by numerically solving Eq. (13).

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EFFECTIVE TENSION AND FLUCTUATIONS IN ACTIVE . . . PHYSICAL REVIEW E 85, 031913 (2012)

Thus, taking the area constraint into account leads us toconclude that, in our model, when the ATP induced activityincreases, the tension increases. This contrasts the modelingof Refs. [22,23] where a nonhomogenous potential, due to thepresence of the cytoskeleton, induces a positive contribution tothe tension, which implies that the tension should decrease withincreasing ATP concentration. In the next section, we show thatour result of an activity induced increase in the tension is notin contradiction with the decrease in the fluctuations observedin ATP depleted red blood cells. Also, note that the tension wecalculated here is only partially related to the force needed topull tethers out of cells as the attachment of the membrane tothe cytoskeleton directly contributes to this force as discussedin Ref. [33].

B. Fluctuation renormalization

In Fig. 2, we compare the passive fluctuation spectrumχa = 0 with the corresponding active one for F0 � 0.2 pNand the same parameters as in Fig. 1. We can see that theactivity increases the large wavelength (small l) fluctuationswhile decreasing the small wavelength (large l) fluctuations;following the general argument laid out in the SupplementalMaterial [34], we can show that, in the direct force case forσth > 0, there exists a crossover l = lc such that the activityalways increases the fluctuations for l < lc and decreases thefluctuations for l > lc. In the inset of Fig. 2, we have plotted theamplitude of the fluctuations in real space 〈|u(θ,φ)|2〉, whichcan be evaluated as

〈|u(θ,φ)|2〉 = 〈|u|2〉 = 1

4π

lm∑l=0

l∑m=−l

〈|ulm|2〉. (26)

This amplitude is independent of θ and φ due to rotationalsymmetry. From the inset, we can see that the real spacefluctuation amplitudes can increase by a factor of 4 due tothe activity even though the tension is increased and the excessarea is conserved. Such an increase in the fluctuations has beenobserved in Ref. [35] by varying the ATP concentration in redblood cells. The effect of the direct force activity is to increasethe large wavelength fluctuations (even though the tension is

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

5

10

15

107 χa

u2

au

2th

5 10 50 100 500 100010 13

10 11

10 9

10 7

10 5

0.001

l

u lm

2

FIG. 2. (Color online) Two fluctuation spectra for the direct forcecase of the same vesicle with no activity (F0 = 0, continuous line)and with F0 = 0.2 pN (dashed line). Inset, the relative increase influctuations in real space as a function of the active amplitude. Theparameters used are the same as in Fig. 1.

increased) while decreasing the fluctuations in an intermediaterange of wavelengths. This increases the observed real spaceamplitude of the fluctuations while still keeping the total excessarea unchanged.

V. CURVATURE FORCE

The curvature case also has been studied previously. It ispresented as a random active contribution to the spontaneouscurvature in Ref. [31]. It also can be seen as a randomcontribution to the quadrupole moment of a microscopicforce density due to active transmembrane proteins [10,30].Note that any second order derivative of a field on themembrane surface, in the force expression in real space, wouldgive a “curvature force” noise in our linear theory. Sucha noise will lead to the scaling xl ∼ l4 at large l. For thesubdominant behavior, we follow the behavior found for thequadrupole contribution in Ref. [11] and, therefore, choosexl = (l + 2)2(l − 1)2/4 for this case and estimate the strengthof the noise by

χa ≈ ρ

(F2

R30

)2

, (27)

where F2 has the unit of force times length squared. ρ is theconcentration of active proteins per area of the membrane, andτa is a time scale on the order of the pumping cycle time for anion pump, for example. Note that, in order to derive Eq. (27),we have assumed, as a first approximation, that the activityof each protein is independent from the others. This impliesthat χa is linear in the active protein concentrations on themembrane.

A. Micropipette experiments

In Refs. [10,25], vesicles containing active ion pumps havebeen studied experimentally using a micropipette aspirationtechnique. In these experiments, a vesicle is sucked partiallyinto a micropipette while measuring the change in excess areaand the tension applied on the membrane. In both cases, theyactivated the pump externally (by shining light on them orby injecting ATP). They compared the relation between theexcess area and the applied tension for the same vesicle withand without activity. For a passive vesicle, the relation betweenthe tension and the excess area is given by Eq. (13). In thiscase, a small increment of the excess area d� is related to anincrement of the logarithm of the tension d(ln σth) in the case1 σth l2

m by

2κ

kBTd� = −d(ln σth). (28)

For the active membrane, they found that this linear relation-ship between the change in excess area and the logarithm ofthe tension still holds, albeit with a decreased prefactor. Thisdecreased prefactor was assigned to an increased effectivetemperature Teff . Using the equations in Table I, we can relatethe excess area and the tension in the curvature case forτa � t (2)

m and 1 σa l2m as

2κ

kBTd� = −

(1 + χa

4

)d (ln σa) . (29)

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LOUBET, SEIFERT, AND LOMHOLT PHYSICAL REVIEW E 85, 031913 (2012)

Thus, in our model, an effective temperature can be definedas Teff/T = 1 + χa/4. The difference between our modelpresented here and the one proposed in Ref. [10] is that welook at a noisy force distribution with a zero mean quadrupolemoment while Ref. [10] has a non-noisy quadrupole forcefor each protein, which influences the fluctuations througha coupling to the protein density and the curvature of themembrane. Both approaches lead to the linear relationship ofEq. (29) with the correspondence that F2 = 2wPa , where w

is defined as the membrane thickness in Ref. [10] and Pa islabeled the force dipole. In the case of bacteriorhodopsin, theyfound experimentally that Teff/T � 2. By taking κ � 10kBT

and ρ � 1016 m−2, we find, from this, the quadrupole momentF2 � 2.6 × 10−28 J m. It should be noted that the effectivetemperature Teff defined here for the curvature case is relatedto the increase in the fluctuations at small wavelengths ofthe fluctuation spectrum. Indeed, taking the limit of the wavenumber l → ∞ in Eq. (16), we get

〈|ulm|2〉a → kBT

κl4

(1 + χa

4

)= kBTeff

κl4, (30)

showing that the same effective temperature appears in thesmall wavelength region of the fluctuation spectrum.

B. Videomicroscopy experiment of fluctuating vesicle

Another experiment on artificial active membranes wasperformed in Ref. [26]. Here, both the active and the passivefluctuation spectra of a giant unilamellar vesicle containingthe proton pump bacteriorhodopsin was recorded by videomi-croscopy without the vesicle tension being constrained exter-nally. Before discussing their results, we lay out what we wouldexpect in terms of the present theory: The short wavelengtheffective temperature found in the micropipette experimentshould still be present in this free vesicle experiment. Thus, theactive contribution should still increase the short wavelength(large l) fluctuations compared to the passive ones. For themembrane to conserve its excess area, it needs to compensatefor the increase in the small wavelength fluctuations throughan increase in the tension such that the large wavelengthfluctuations are decreased. This can be seen in Fig. 3 wherethe expected active spectrum (dashed line) is compared tothe passive one (full line). In fact, for the curvature case, itcan be proven that the active fluctuation spectrum is alwayssmaller than the passive one for small l and then becomes largerat values of l above a critical l = lc (see the SupplementalMaterial [34] for a calculation of the lc’s for small χa).

If we take the limit of long correlation time τa � t (2)m of

Eq. (16) for the curvature force case, we get

〈|ulm|2〉a = kBT

κEl(σa)

(1 + χa

4

(l + 2)2(l − 1)2

El(σa)

). (31)

The flat limit of this equation, i.e., ignoring subdominant termsin l, e.g., (l + 2)(l − 1) ∼ l2, is the equation used to fit theexperimentally obtained fluctuation spectrum in Ref. [26] ifwe evaluate χa using Eq. (27). This is true even though wehave considered a shot noise with zero mean while they haveconsidered a mean contribution coupled to the proteins densitydifference (similar to the discussion in the last subsection).Or in other words, a very long correlation time for the

5 10 50 100 500 100010 14

10 12

10 10

10 8

10 6

10 4

l

u lm

2

FIG. 3. (Color online) Three fluctuation spectra for the curvaturecase. The full line corresponds to the passive case χa = 0, takingthe passive tension obtained in Ref. [26]: �th � 4 × 10−7 N m−1,leading to σth � 940. The dashed line corresponds to the activecase with χa = 4 and a tension σa = 3.8 × 104 (giving a conservedexcess area). The dot dashed line is the fluctuation spectrumusing values mimicking the experiments, χa = 4 and σa � 130,see the discussion in the text. The vertical dotted line denotes theapproximate maximum wave vector observed in Ref. [26]. We usedthe parameters: κ � 10 kBT , ρ � 1016 m−2, R0 � 10 μm, and lm =R0/d � 2000.

noise is equivalent to a constant mean contribution in thiscase. Experimentally, they found a significant increase in thefluctuation spectrum at low l values when the proteins wereactivated, which they interpreted as an active reduction in thetension from �th � 4 × 10−7 N m−1 in the passive case to�a � 5.3 × 10−8 N m−1 in the active case. This gives σth =940 and σa = 130 for our unitless tensions. The correspondingactive fluctuation spectrum calculated from Eq. (31), with thevalue χa = 4 taken from the fit to the micropipette experimentand σa = 130, is plotted in Fig. 3 as a dot dashed line.

C. Discussion

The lowering of the tension is necessary to explain theincrease in the large wavelength fluctuations observed inRef. [26], and the increase in effective temperature is necessaryto explain the micropipette experiment [10]. But this gives anactive fluctuation spectrum consistently larger than the passiveone for all wave numbers. These results suggest that the excessarea is not conserved in these experiments. We can suggest twodifferent mechanisms in order to explain this increase: The firstexplanation would be that the increase in the excess area is dueto the elastic properties of the lipid bilayer itself and a verylarge negative contribution to the tension in the active case.In this case, it is the lipidic part of the membrane’s area thatis expanded due to the additional negative active contributionto the tension. However, the active contribution to the tensionneeds to be very large in magnitude, at least on the order of10−3 N m−1, to give a visible effect on the fluctuation spectrumdue to the high compressional modules of a lipid membrane,see Ref. [28].

Another explanation could be, for these experiments usingbacteriorhodopsin, that the area the proteins take up in themembrane increases when they are active (at least, on average

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EFFECTIVE TENSION AND FLUCTUATIONS IN ACTIVE . . . PHYSICAL REVIEW E 85, 031913 (2012)

in time) since this protein is known to periodically expand dur-ing its active cycle [36]. In this case, the area per lipid head inthe lipid region of the membrane would be unchanged, and theincrease in the excess area would be due to the expansion of theproteins. This would allow the membrane tension to decrease.

Assuming χa � 4, we can evaluate the excess area �a

required to obtain σa = 130. Using σth = 940, lm = 2000, andκ = 10kBT , we find, for the passive case, �th/(4π ) � 0.03.From the result for the tension in Table I for the curvature casewith τa � t (2)

m and 1 σa l2m, we obtain

�a =(

1 + χa

4

)[ln

(σth

σa

)+ �th

]− χa

4. (32)

With σa = 130 and χa � 4, we find �a/(4π ) � 0.07. Onecan evaluate the corresponding increase in the area perprotein ai between the passive and active cases. For a proteinconcentration of ρ � 1016 m−2, we find ai � 4 × 10−18 m2.This corresponds to an ∼10% increase in the active radius ifwe take a protein to occupy a disk of radius rp � 2.5 nm inthe passive case.

VI. PERMEATION FORCE

Here, we show that a contribution of the form xl ∼ l2 canbe justified through a model of the activity with a transfer offluid between the two sides of the membrane. In Eq. (14), weused a no slip boundary condition to obtain the hydrodynamicstresses on the membrane from the surrounding fluid. Theradial component of this boundary condition reads

R(θ,φ,t) − vrfluid(θ,φ,R,t) = 0, (33)

where R is the radial position of the membrane and vrfluid is

the radial component of the fluid velocity evaluated at themembrane. To obtain the permeation force case, we add anactive contribution to the boundary condition. It then reads

R(θ,φ,t) − vrfluid(θ,φ,R,t) = vr

a(θ,φ,t), (34)

where vra is the random active contribution of zero mean and

with the following correlation after expanding in sphericalharmonics:⟨

vra,lm(0)vr

a,l′m′(t)⟩ = ρ( ˙V a)2 δll′δmm′

R20

exp

[−|t |

τa

]. (35)

Here, ρ is the mean density of active centers, and ˙V a is onthe order of the volume transferred through the membraneper active center per unit time. We have assumed that theactive centers are operating independently in Eq. (35). UsingEq. (34) instead of the no slip boundary condition, we obtaina dynamical equation of the form of Eq. (14) with ξ

(a)lm (t) =

ηvra,lm(t)/(�lR0). The active parameters χa and xl then read

χa = ρ

( ˙V aη

R20

)2

and xl = (�l)−2, (36)

which gives xl ∼ l2 at large l.In Ref. [10], a similarly active permeation term was

considered for the force dipole model using Darcy’s law ofpermeation. The membrane was taken to have a permeationcoefficient λp, and a force Fa was pushing fluid through

it. Our parameters are related to theirs as ρ ˙V2

a ∼ ρ(λpFa)2

where ρ = ρ is the average pump density on the membrane.They neglected this contribution in their effective temperatureexpression for bacteriorhodopsin because the water perme-ation coefficient through the lipid bilayer is very low, λp �10−12 m3 N−1 s−1; however, this case could give a significanteffect for pumps if they could provide a high enough flowthrough the membrane. Using Eq. (36) in Eq. (16), we foundthe same equation for the fluctuation spectrum as [21] fora zero mean active flow in the flat limit. Unfortunately, weare unaware of any experiment relating fluctuations to fluidpumping through the membrane. For a detailed paper on thefluctuation spectrum in this permeation force case, we referthe reader to the Supplemental Material [34].

As an example, we evaluate the flow ˙V a, required toobtain a significant renormalization of the tension using theparameters of the experiment described in Fig. 3. We haveplotted the renormalization of the tension, in the limit of longcorrelation time, as a function of the mean volume per secondper protein ˙V a in Fig. 4. With the chosen protein concentration,we start to have an effect for the renormalization of the tensionaround 5 × 10−20 m3/s, which roughly corresponds to 106

water molecules per millisecond. This order of magnitude forthe flow can be obtained for the passive channel aquaporin [37],for instance.

But, we believe it is unlikely that active pumps can attainthat level of flow through the membrane. Note, however, thatdecreasing the tension will increase the effect of the activityfor a fixed χa.

In principle, the volume constraint will be violated by theflow across the membrane. Although we assume equal amountsof proteins pumping in each direction, then the fluctuationsin pumping can create an asymmetry in the net transportedvolume of fluid, which we are neglecting in this paper. Thisis justified, in particular, for ion pumps for which a large

5 10 15 20 25 30

1 10 72 10 75 10 71 10 62 10 65 10 61 10 52 10 5

l

u lm

2

0 5 10 15 200.0

0.5

1.0

1.5

2.0

1020 V

σσ

ath

th

FIG. 4. (Color online) The relative increase in tension as afunction of the mean volume transferred through the membrane perunit time for the permeation case. The dotted line is calculated usingthe equation for τa � t (2)

m in Table I, while the big circles are numericalcalculations directly from Eq. (19). Inset, the first 30 modes of thefluctuation spectrum. The continuous line is for the passive case,while the dashed line is for ˙V a � 1.5 × 10−19 m3/s. We used theparameters κ � 10 kBT , ρ � 1016 m−2, R0 � 10 μm, lm = R0/d �2000, η � 10−3 kg m−1 s−1, and σth � 940.

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LOUBET, SEIFERT, AND LOMHOLT PHYSICAL REVIEW E 85, 031913 (2012)

asymmetry is suppressed since an asymmetry in pumpingresults in a buildup of an electric membrane potential thatcounteracts the asymmetry.

VII. CONCLUSION

In this paper, we have investigated the fluctuation spectrumof vesicles with noisy active inclusions. We used a simplermodel than previous papers [10–12] for the activity sinceour noise had vanishing mean, but instead, we took the highcompressional modulus of a lipid membrane into accountby an area constraint. We investigated the effect of the areaconstraint on three physical models for the activity. We showedthat our model for the direct force case agrees well withprevious experimental and theoretical papers on red blood

cells; albeit, with a modified interpretation for the increasein the tension. For the force curvature case, we comparedour model to the experiments of Refs. [10,26], and weshowed that, even though our model recovers the effectivetemperature found in the micropipette experiment, the data forthe free vesicle containing active bacteriorhodopsin seem atodds with the conservation of the excess area. This suggeststhat bacteriorhodopsin might increase the excess area of thevesicle when the proteins are active or that the force curvaturecase does not completely describe those experiments. We alsoinvestigated a permeation force case and gave an order ofmagnitude of the flow required to obtain a significant tensionrenormalization for a specific example. We hope that this paperwill motivate further theoretical and experimental studies onactive biomembranes.

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