Curved fluid membranes behave laterally as an effective viscoelasticmedium

Mohammad Rahimi,a, Antonio DeSimone,b and Marino Arroyo,a∗

The lateral mobility of membrane inclusions is essential in biological processes involving membrane-bound macromolecules,which often take place in highly curved geometries such as membrane tubes or small organelles. Probe mobility is assisted bylateral fluidity, which is thought to be purely viscous for lipid bilayers and synthetic systems such as polymersomes. In previoustheoretical studies, the hydrodynamical mobility is estimated assuming fixed membrane geometry. However, fluid membranesare very flexible out-of-plane. By accounting for the deformability of the membrane and in the presence of curvature, we showthat the lateral motion of an inclusion produces a normal force, which results in a nonuniform membrane deformation. Sucha deformation mobilizes bending elasticity, produces extra lateral viscous and elastic forces, and results in an effective lateralviscoelastic behavior. The coupling between lateral and out-of-plane mechanics is mediated by the interfacial hydrodynamicsand curvature. We analyze the frequency and curvature dependent rheology of flexible fluid membranes, and interpret it with asimple four-element model, which provides a background for microrheological experiments. Two key technical aspects of thepresent work are a new formulation for the interfacial hydrodynamics, and the linearization of the governing equations around acylindrical geometry.

1 Introduction

Lipid membranes are highly flexible and malleable interfaces,which behave as in-plane viscous fluids in physiological con-ditions. The interfacial fluidity and bending flexibility are cru-cial for many cellular functions involving membrane shapetransformations, such as vesicular or tubular trafficking,1 orshaping the cell organelles, and also for variety of dynamicsobserved in biomimetic systems.2–6 The interfacial fluidity isalso essential to the mobility of inclusions such as membraneproteins7 or fluid domains,8 the transport of lipids betweencells through membrane tubes,9 and lateral reorganizations,such as the formation of lipid rafts.10 Other amphiphilic mem-branes such as polymersomes11 also exhibit in-plane fluidity,with a hyper-viscous behavior.12 Our goal here is to under-stand if and how shape deformations affect the lateral mobilityof inclusions in fluid membranes.

The lateral motion of membrane probes subjected tostochastic or external forces has a long and rich history, dat-ing back to the work of Saffman and Delbruck (SD).7 Thistheory estimates the diffusion coefficient through the hydro-

† Electronic Supplementary Information (ESI) available: [details of anysupplementary information available should be included here]. See DOI:10.1039/b000000x/a LaCaN, Universitat Politecnica de Catalunya-BarcelonaTech (UPC),Barcelona 08034, Spain. b SISSA–International School for Advanced Stud-ies, Via Beirut 2-4, 34014 Trieste, Italy. ∗ Corresponding author,Tel: +34-93-401-1805; E-mail: marino.arroyo@upc.edu

dynamic mobility of an inclusion of size a in an inextensi-ble membrane of interfacial viscosity µs embedded in bulkfluid of viscosity µb. For inclusions much smaller than theSD length scale `SD = µs/µb, such as proteins in cell mem-branes, the mobility is inversely proportional to the interfacialviscosity µs, with a weak logarithmic dependence on `SD/afor infinite planar membranes. While the dependence on a isstill controversial in some regimes,13 it has been recently pre-dicted theoretically14 and verified experimentally with mem-brane tubes15 that the mobility of probes is significantly modi-fied by the membrane curvature. In short, highly curved mem-branes are geometrically confined and the correction in themobility is ln(R/a) instead, where R is the radius of curvatureof the curved membrane, in close correspondence with the SDestimate for membranes of finite size.

The SD theory and subsequent refinements, either consid-ering planar16–19 or curved membranes,14,15 assume that themembrane geometry is fixed. Since the membrane rheology isassumed to be Newtonian, which is consistent with various ex-periments,20,21 these theories predict a purely viscous lateralbehavior. Here, by lifting this approximation and allowing themembrane to deform out-of-plane, we find that curved fluidmembranes with bending elasticity behave laterally as an ef-fective viscoelastic medium, and consequently exhibit muchricher rheology, including frequency-dependent mobility, andnon-zero storage modulus. At the root of this behavior, weshow that the purely lateral motion of an inclusion produces

1

Membrane free energy

The free energy of an inextensible membrane, including thebending energy and the work of pressure and surface tensioncan be written as

Π =κ

2

∫Γ

(2H)2dS+Σ0

∫Γ

dS− p0

∫V

dV, (15)

where V is the volume enclosed by the membrane. By intro-ducing the expansion in Eq. (11), retaining up to second orderterms, and noting that linear terms vanish since we linearizeabout an equilibrium state, we obtain the harmonic expansionof the free energy (see Appendix C)

Π = Π0 +12 ∑

n,mu∗nmEnmunm, (16)

where ()∗ denotes complex conjugation and conjugate trans-position for matrices, Enm = 2πL

(κqnm/r3

0− p+Σλnm/r0),

and qnm = −λnm/2+ λ 2nm−m2 + 1. It follows immediately

that

Π =12 ∑

n,m(u∗nmEnmvr

nm +unmEnmvr∗nm) , (17)

Note that the stability of the equilibrium state holds if Enm > 0for all n and m. This is always the case if p0 = 0, as consideredin all the examples of the paper.

Inextensibility

The dynamical equations of inextensible fluid membranesshould satisfy the local mass conservation constraint inEq. (2). The surface tension deviation from Σ0

Σ(θ ,z) = Σ(θ ,z)−Σ0 = ∑n,m

ΣnmYnm(θ ,z), (18)

acts a Lagrange multiplier for this constraint, which is en-forced variationally with the expression

Cinext =∫

Γ

Σ(∇s ·v−2vnH)dS. (19)

With the identities in Appendix B, the linearized form of theequation above is

Cinext = 2πr0L∑nm

Σnm

(−λnmv(1)nm/r0− vr

nm/r0

)= 2πr0L∑

nmΣnm

(imvθ

nm/r0 + iknvznm− vr

nm/r0

),

(20)

where we have used Eq. (14) in the last step.

Interfacial dissipation

Replacing the vector cylindrical harmonic expansion of theinterfacial velocity in Eq. (13) into Eq. (4), taking advantage ofthe expressions in Appendix B, and mapping back to the usualFourier expansion of the velocity with Eq. (14), we express themembrane interfacial dissipation potential at the cylindricalequilibrium state as

Wµs =12 ∑

n,mV∗nmDmem

nm Vnm, (21)

where

Dmemnm

2µsπr0L= B∗nm

2/r20 2m2/r2

0 −2knm/r02m2/r2

0 2λ 2nm/r2

0 0−2knm/r0 0 λ 2

nm/r20

Bnm.

Bulk dissipation

A general solution to the 3D Stokes equations in cylindricalcoordinates is given by,39

V b±(r,θ ,z) =∇ f±(r,θ ,z)+∇×[g±(r,θ ,z)ez

]+ r∂r

[∇h±(r,θ ,z)

]+∂zh±(r,θ ,z)ez,

p±(r,θ ,z) =−2µb∂2z h±(r,θ ,z),

(22)

where ∇ is the bulk nabla operator, and f±(r,θ ,z), g±(r,θ ,z),h±(r,θ ,z) are cylindrical harmonic functions given by f±(r,θ ,z)

g±(r,θ ,z)h±(r,θ ,z)

= ∑n,m

F±nmG±nmH±nm

P±m (kn,r)Ynm(θ ,z). (23)

Here, P±m (kn,r) denote modified Bessel functions of the sec-ond and first kind, i.e. P+

m = Km(|kn|r) and P−m = Im(|kn|r).Since limx→0 Km(x) = limx→∞ Im(x) = +∞, the modifiedBessel functions of the first and second kind are appropriatesolutions for the exterior and interior bulk fluid, respectively.We can calculate the coefficients of the harmonic functions(F , G, and H) by imposing the non-slip boundary conditionson the surface. Expressing the bulk fluid velocity surround-ing the cylindrical surface as V b±=V b±

r er+V b±θeθ +V b±

z ez,and recalling Eq. (22), we have

V b±r =∂r f±+

1r

∂θ g±+ r∂2rrh±,

V b±θ

=1r

∂θ f±−∂rg±−1r

∂θ h±+∂2θrh±,

V b±z =∂z f±+ r∂

2zrh±+∂zh±.

(24)

Combining Eqs. (23), (24) and (12), the bulk and surface ve-locities can be related through the coefficients of the harmonic

5

functions as F±nmG±nmH±nm

=(Q±nm

)−1 Vnm, (25)

where the components of Q±nm are given in Appendix D.The traction vector acting on the surface of a tube in cylin-

drical coordinates for an incompressible Newtonian fluid is,39

T b± =±er ·σb± =±

−p±+2µb∂rV b±r

µbr∂r(V b±

θ/r)+µb∂θ (V b±

r )/rµb(∂zV b±

r +∂rV b±z)

.Replacing Eqs. (24,25) into the above relation, we find

T b± =±µb ∑nm

YnmS±nm(Q±nm

)−1Vnm, (26)

where the nonzero components of S±nm are given in Ap-pendix D. From Eq. (26), the dissipation potential for the bulkfluid in the absence of body forces can be written as

Wµb =−12

∫Γ

V ·T b± dS =12 ∑

nmV∗nmDbulk±

nm Vnm, (27)

where Dbulknm = 2πr0Lµb

[−S+

nm (Q+nm)−1

+S−nm (Q−nm)−1]. The

final expressions for the components of the dissipation matrixDbulk

nm , which is Hermitian, are given in Appendix D.

Linearized governing equations

As in Section 2, we obtain the dynamics of the system byminimizing the rate of change of free energy, plus the to-tal dissipation potential, plus the power of the external forceswith respect to the variables expressing the rate of change ofthe system and subject to constraints (here inextensibility).For this purpose, we form the Lagrangian L (Vnm,Σnm) =Π[Vnm] + W tot[Vnm] + Πext[Vnm] + Cinext[Vnm,Σnm] collect-ing Eqs. (17,21,27,20). Expressing the external force withFourier expansion f ext(θ ,z, t) = ∑nmf

extnm (t)Ynm(θ ,z), we can

express the external power as Πext =∑nm V∗nmfextnm, where fext

nm =2πr0Lf ext

nm . Making the Lagrangian stationary, we obtain a setof algebraic-differential equations[

Dtotnm Lnm

LTnm 0

]{VnmΣnm

}=

{fnm + fext

nm0

}, (28)

where Dtotnm = Dbulk

nm +Dmemnm , fT

nm =[−Enmunm 0 0

], and

LTnm = 2πr0L

[−1/r0 im/r0 ikn

]. Inverting the matrix in

the left-hand-side and recalling that vrnm = unm, the governing

equations in Eq. (28) can be expressed asunmvθ

nmvz

nmΣnm

=

kr

nmkθ

nmkz

nmkΣ

nm

unm +

gr

nmgθ

nmgz

nm0

, (29)

where the coefficients denoted by k and g follow from simplealgebraic calculations. The first row can be easily be inte-grated to an exponential function of time, and the remainingalgebraic equations provide the tangential velocities and sur-face tension disturbances.

4 Probe mobility and membrane rheology

The SD theory assumes a purely viscous drag to motion, andrelates through the Stokes-Einstein relation the diffusion co-efficient to the viscosity of the embedding medium. Modernmicrorheology generalizes the Stokes-Einstein relations to lin-ear viscoelastic (LVE) media,30 and provides a link betweenthe statistical properties of the motion of probes beyond sim-ple diffusion, e.g., the mean squared displacement (MSD(t) =〈∆r2(t)〉), and the rheology of the medium. The hydrody-namical mobility, which can be probed under either an exter-nal force (active microrheology), or stochastic thermal forces(passive microrheology), becomes frequency-dependent for ageneral LVE medium.31 The theory summarized below pro-vides a precise mapping between the measurements of activeand passive microrheology. We exploit this correspondence inSection 5, where we calculate the lateral response to an ap-plied force, evaluate the effective viscoelastic behavior, andthen reconstruct a typical measurement of passive microrheol-ogy, the MSD.

The hydrodynamic mobility M is generalized for LVE ma-terials as40

vP(t) =∫ t

−∞

M(t− t ′) f P(t ′) dt ′,

where f P is the force applied on the probe, here the membraneinclusion, and vP(t) is the velocity of the probe. For a purelyviscous medium, the response to a force is instantaneous. Bytaking the Laplace transform of the previous equation, we cancompute the frequency-dependent mobility of the probe fromthe applied force and the response as

M(s) = vP(s)/ f P(s), (30)

where s is the Laplace frequency, and the Laplace-transformedfunctions are denoted by a tilde. From fluctuation-dissipation,the translational mobility of a probe in N dimensions and theMSD are related by31

MSD(s) =2NkBT

s2 M(s). (31)

Given the frequency-dependent mobility, we introduce an ef-fective lateral viscoelastic modulus. The theory behind in-terfacial microrheology is not fully established. To account for the fact that we deal with a two-dimensional medium, we

6 | 1–15

6

where ? is the Hodge star operator. Adding these two equa-tions and rewriting in the language of vector calculus, we ob-tain

−v ·∆Rs v =|∇s×v|2 +(∇s ·v)2

−∇s× [(n ·∇s×v)v− (∇s ·v)?v] ,(40)

where ?v is a tangent vector field on Γ orthogonal to v. Sum-marizing the equations above, we obtain

∇sv :∇sv = |∇s×v|2 +(∇s ·v)2−K|v|2

+12

∆s|v|2−∇s× [(n ·∇s×v)v− (∇s ·v)?v] ,(41)

where we point out the important fact that the terms in thesecond line of this equation are null-Lagrangians, i.e. whenintegrated over Γ they can be expressed by integration by parts(Stoke’s theorem) as an integral over the boundary of Γ. Thus,these terms do not contribute to the Euler-Lagrange equations,and for a closed surface their integral vanishes identically.

We examine now the second term in Eq. (36). Recallingthe identity relating the Kronecker delta and the permutationsymbol in two dimensions, δ

ji δ l

k = εikε jl +δ li δ

jk , we have

∇sv : ∇svT = vi

| jvj|i = δ

ji δ

lk vi|lv

k| j

= εikεjl vi|lv

k| j +δ

li δ

jk vi|lv

k| j

=(

εikεjl vivk

| j)|l− εikε

jl vivk| jl +(∇s ·v)2.

(42)

Manipulating the second term in the last line we find

εikεjl vivk

| jl = (δ ji δ

lk−δ

li δ

jk )v

ivk| jl = vi

(vk|ik− vk

|ki

)= K|v|2,

(43)

where in the last equality we have used the fact that the secondcovariant derivative does not commute. Thus, we obtain

∇sv : ∇svT =(∇s ·v)2−K|v|2 +

(εikε

jl vivk| j)|l, (44)

where the last term is a divergence, and therefore a null La-grangian. Collecting all the terms, we have

d : d=12|∇s×v|2 +(∇s ·v)2−K|v|2 +N (v), (45)

where N (v) is a null Lagrangian collecting the last two termsin Eq. (41) and the last term in Eq. (44). Integrating N (v)over a surface with boundary, it is easy to find the resultingboundary terms. For a closed surface, we find the form of thedissipation potential in Eq. (4), or recalling local inextensibil-ity the alternative form in Eq. (5). The derivation of the tan-gential and normal Euler-Lagrange equations (7,8) follows di-rectly by taking variations, integrating by parts, and recallingthat ∇s ·(k−2Hg) = 0 due to the Codazzi-Mainardi relations.

B Calculations in vector cylindrical harmonic

Using the vectorial form of cylindrical harmonics for the sur-face velocity field Eq. (13), it is easy to derive the followingidentities

∇s×vnm =−λnm

r0v(2)nmYnm, ∇s ·vnm =−λnm

r0v(1)nmYnm,

∇svnm:k =− 1r2

0

(m2v(1)nmYnm−mknr0v(2)nmYnm

),

where λnm = r20k2

n +m2. We also have the orthogonality rela-tions

∫Γ

Y ∗nm(θ ,z)·Yn′m′(θ ,z) dS = 2πr0Lδnn′δmm′ ,∫Γ

Ψ∗nm(θ ,z)·Ψn′m′(θ ,z) dS = 2λnmπr0Lδnn′δmm′ ,∫Γ

Φ∗nm(θ ,z)·Φn′m′(θ ,z) dS = 2λnmπr0Lδnn′δmm′ .

C Elastic energy calculations: cylindrical coor-dinates

Recalling Eq. (11), lengthy but direct calculations lead to thefollowing expressions38

H2 dA =1

4r0

[1− u

r0+

u2

r20−2r0∆su+ r2

0 (∆su)2

− 12(u,z)

2 +3

2r20

(u,θ)2

+4uu,θθ

r20

]dθdz,

2H dA =

(u2,θ +uu,θθ − r0u,θθ

r20

− r0u,zz−uu,zz +1

)dθdz,

dA =1

2r0

(r2

0u2,z +2r2

0 +2ur0 +u2,θ

)dθdz,

where ∆su = u,θθ +u,zz/r20.

12

D Bulk fluid traction: tubular membranes

The components of the matrix that maps the bulk and surfacerepresentations of the velocity field are

Q±11 =[∓|kn|r0P±m+1 +mP±m

]/r0,

Q±12 = imP±m /r0,

Q±13 =[(λnm−m)P±m ±|kn|,r0P±m+1

]/r0,

Q±21 = imP±m /r0,

Q±22 =[±|kn|r0P±m+1−mP±m ,

]/r0,

Q±23 = i[(

m2−m)

P±m ∓m|kn|r0P±m+1]/r0,

Q±31 = iknr0P±m /r0,

Q±32 = 0,Q±33 = ikn

[(m+1)P±m ∓|kn|r0P±m+1

],

where we have dropped the subindices nm. We note that inthese expressions P±m = P±m (Knr0) are scalars that depend on nand m alone, since the radius is fixed on the cylinder.

In Eq. (26), the nonzero components of S±nm are given by

S±11 = 2[±|kn|r0P±m+1 +(λnm−m)P±m

]/r2

0

S±13 = 2[∓|kn|r0 (1+λnm)P±m+1 +(m−1)(λnm−m)P±m

]/r2

0

S±22 = −[±2|kn|r0P±m+1 +

(λnm +m2−2m

)P±m]/r2

0

S±32 = −knmP±m /r0

S±12 = 2mi[∓|kn|r0P±m+1 +(m−1)P±m

]/r2

0

S±21 = 2mi[(m−1)P±m ∓|kn|r0P±m+1

]/r2

0

S±23 = 2mi[(λnm +1−2m)P±m ±2|kn|r0P±m+1

]/r2

0

S±31 = 2kni[mP±m ∓|kn|r0P±m+1

]/r0

S±33 = 2iλnmknP±m /r0,

where we have dropped the subindices nm.Recalling the final expression for the bulk dissipation ma-

trices, and introducing some notation

Dbulknm = 2πr0Lµb

[−S+

nm(Q+

nm)−1︸ ︷︷ ︸

D+nm

+S−nm(Q−nm

)−1︸ ︷︷ ︸D−nm

]

we have the following expressions for the components of thesematrices

D+11 =

−2|kn|(k2

nr2−|kn|r(3m+2)α++(4m+2m2−2k2nr2)α2

++2|kn|rmα3+

)|k3

n|r3− (3k2nr2m+2k2

nr2)α++ |knr|(2m2 +4m− k2nr2)α2

++mk2nr2α3

+

D+12 =

−2im|kn|(k2

nr2− (2|knr|+3|knr|m)α++(2m2 +4m− k2nr2)α2

++(1+m)|knr|α3+

)|k3

n|r3− (3m+2)k2nr2α++(4m+2m2− k2

nr2)r|kn|α2++ k2

nr2mα3+

D+12 = −D+

21

D+13 =

−2i|kn|knr(k2

nr2−|knr|(3m+1)α++(2m+2m2− k2nr2)α2

++ |knr|mα3+

)|k3

n|r3− k2nr2(3m−2)α++ |kn|r(2m2 +4m− k2

nr2)α2++ k2

nr2mα3+

D+13 = −Dbulk+

31

D+22 =

|kn|(−2k2

nr2 + |knr|(4− k2nr2 +6m)α++(4k2

nr2−8m−4m2 +2k2nr2m)α2

++(|k3n|r3−4|kn|rm)α3

+

)|k3

n|r3− k2nr2(3m+2)α++ |kn|r(2m2 +4m− k2

nr2)α2++ k2

nr2mα3+

D+23 =

−m|kn|knrα+(−|knr|+2mα++α2+|knr|)

|k3n|r3− k2

nr2(3m+2)α++ |knr|(2m2 +4m− k2nr2)α2

++mk2nr2α3

+

D+32 = D+

23

D+33 =

|kn|(2k2nr2−|knr|(6m+m2)α++(4m2 +2m3)α2

++ |knr|m2α3+)

|k3n|r3− (2k2

nr2 +3k2nr2m)α++ |knr|(2m2 +4mα2

+− k2nr2)α2

++ k2nr2α3

+m

13

D−11 =−2|kn|

(−k2

nr2−|kn|r(2+3m)α−− (2m2−4m+2k2nr2)α2

−+2|kn|rmα3−)

−|k3n|r3− k2

nr2(3m+2)α−+ |kn|r(k2nr2−2m2−4m)α2

−+ k2nr2mα3

−

D−12 =−2im|kn|

(−k2

nr2−|kn|r(3m+2)α−− (2m2 +4m+ k2nr2)α2

−+ |kn|r(m+1)α3−)

−|k3n|r− k2

nr2(3m+2)α−|kn|r(−2m2−4m+ k2nr2)α2

−+ k2nr2mα3

−D−21 = −D−12

D−13 =−2ikn|kn|r

(−k2

nr2 +(k2nr2−2m−2m2)α2

−−|kn|r(3m+1)α−+ |kn|rmα3−)

−|k3n|r3− k2

nr2(3m+2)α−+ |kn|r(k2nr2−2m2−4m)α2

−+ k2nr2mα3

−D−31 = −D−13

D−22 =|kn|(2k2

nr2 + |kn|r(6m+4− k2nr2)α−+(−2k2

nr2m+4m2 +8m−4k2nr2)α2

−+ |kn|r(k2nr2−4m)α3

−)

−|k3n|r3− k2

nr2(3m+2)α−+ |kn|r(k2nr2−2m2−4m)α2

−+ k2nr2mα3

−

D−23 =−mα−kn|kn|r

(−|kn|r−2mα−+α2

−|kn|r)

−|k3n|r3− k2

nr2(3m+2)α−+ |kn|r(k2nr2−2m2−4m)α2

−+ k2nr2mα3

−D−32 = D−23

D−33 =|kn|(2k2

nr2 + |kn|rm(6+m)α−−|kn|rm2α3−+(4m2 +2m3)α2

−)

|k3n|r3 + k2

nr2(3m+2)α−+ |kn|r(2m2 +4m− k2nr2)α2

−− k2nr2α3

−m

where we have dropped the subindices nm and α± =P±m /P±m+1.

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