Dynamics of a membrane interacting with an active wall

# Dynamics of a membrane interacting with an active wall

Kento Yasuda    Shigeyuki Komura    Ryuichi Okamoto Department of Chemistry, Graduate School of Science and Engineering, Tokyo Metropolitan University, Tokyo 192-0397, Japan
July 15, 2019
###### Abstract

Active motions of a biological membrane can be induced by non-thermal fluctuations that occur in the outer environment of the membrane. We discuss the dynamics of a membrane interacting hydrodynamically with an active wall that exerts random velocities on the ambient fluid. Solving the hydrodynamic equations of a bound membrane, we first derive a dynamic equation for the membrane fluctuation amplitude in the presence of different types of walls. Membrane two-point correlation functions are calculated for three different cases; (i) a static wall, (ii) an active wall, and (iii) an active wall with an intrinsic time scale. We focus on the mean squared displacement (MSD) of a tagged membrane describing the Brownian motion of a membrane segment. For the static wall case, there are two asymptotic regimes of MSD ( and ) when the hydrodynamic decay rate changes monotonically. In the case of an active wall, the MSD grows linearly in time () in the early stage, which is unusual for a membrane segment. This linear-growth region of the MSD is further extended when the active wall has a finite intrinsic time scale.

## I Introduction

The random slow dynamics of fluid membranes visible as a flickering phenomenon in giant unilamellar vesicles (GUVs) or red blood cells (RBCs) has attracted many interests in the last few decades Lipowsky95 (). These thermally excited shape fluctuations can be essentially understood as a Brownian motion of a two-dimensional (2D) lipid bilayer membrane in a three-dimensional (3D) viscous fluid such as water. For spherically closed artificial GUVs, characteristic relaxation times for shape deformations were calculated analytically MS87 (); KS93 (); SK95 (); Komura96 (). Analysis of shape fluctuations can be used for quantitative measurements of surface tension and/or bending rigidity of single-component GUVs Popescu06 () or GUVs containing bacteriorhodopsin pumps FLPJPB05 ().

Historically, investigations on fluctuations of cell membranes have started with RBCs whose flickering can be observed under a microscope LB14 (). Brochard and Lennon were among the first to describe quantitatively membrane fluctuations as thermally excited undulations, mainly governed by the bending rigidity of the membrane Brochard75 (). Later experiments showed that flickering in RBCs is not purely of thermal origin but rather corresponds to a non-equilibrium situation because the fluctuation amplitude decreases upon ATP depletion Levin91 (); Tuvia98 (). Here ATP hydrolysis plays an important role to control membrane-spectrin cytoskeleton interactions AlbertsBook (). More advanced techniques have demonstrated that, at longer time scales (small frequencies), a clear difference exists between the power spectral density of RBC membranes measured for normal cells and those ATP depleted; the fluctuation amplitude turns out to be higher in the former Betz09 (); Park10 (). At shorter time scales, on the other hand, membranes fluctuate as in the thermodynamic equilibrium. It should be noted, however, that the role of ATP in flickering is still debatable because Boss et al. have recently claimed that the mean fluctuation amplitudes of RBC membranes can be described by the thermal equilibrium theory, while ATP merely affects the bending rigidity Boss12 ().

In order to understand shape fluctuations of RBCs, one needs to properly take into account the effects of spectrin cytoskeleton network that is connected to the membrane by actin, glycophorin, and protein 4.1R Lipowsky95 (); AlbertsBook (). Gov et al. treated the cytoskeleton as a rigid wall (shell) located at a fixed distance from the membrane, and assumed that its static and dynamic fluctuations are confined by the cytoskeleton Gov03 (); GovZilamSafran04 (). They further considered that the sparse connection of membrane and cytoskeleton gives rise to a finite surface tension for length scales larger than the membrane persistence length. The bending free energy for a membrane was extended to include a surface tension and a confinement potential with which the effects of ATP on the membrane fluctuations was described. However, since an active component of the membrane fluctuations also depend on the fluid viscosity Tuvia97 (), they cannot be solely attributed to the static parameters such as the surface tension or the potential. Gov and Safran later estimated the active contribution to the membrane fluctuations due to the release of stored tension in the spectrin filament and membrane in each dissociation event GovSafran05 (); Gov07 (). In contrast to static thermal fluctuations, they showed that the active cytoskeleton may contribute to the membrane fluctuations at intermediate length scales.

Effects of membrane confinement are important not only for shape fluctuations of RBCs but also for a hydrodynamic coupling between closely apposed lipid bilayer membranes Kaizuka04 (); Kaizuka06 (), and dynamical transitions occurring in lamellar membranes under shear flow Diat93 (); Lu08 (). After the seminal works by Kramer Kramer () and by Brochard and Lennon Brochard75 (), the wavenumber-dependent decay rate for the bending modes of a membrane bound to a wall was calculated by Seifert Seifert94 () and Gov et al. GovZilamSafran04 (). In particular, Seifert showed that the scale separation between the membrane-wall distance and the correlation length determined by the confinement potential can lead to various crossover behaviors of the decay rate. In these hydrodynamic calculations, however, the wall that interacts with the membrane was treated as a static object and does not play any active role.

Quite generally, active motions of a membrane can be induced by non-thermal fluctuations that occur in the outer environment of the membrane such as cytoskeleton or cytoplasm. In this paper, we consider the dynamics of a membrane interacting with an active wall that generates random velocities in the ambient fluid. These random velocities at the wall can be naturally taken into account through the boundary conditions of the fluid. We first derive a dynamic equation for the membrane fluctuation amplitude in the presence of hydrodynamic interactions. Then we calculate the membrane two-point correlation functions for three different cases; (i) a static wall, (ii) an active wall, and (iii) an active wall with an intrinsic time scale. We especially focus on the mean squared displacement (MSD) of a tagged membrane segment, and discuss its asymptotic time dependencies for the above cases. For the static wall case, the membrane fluctuates due to thermal agitations, and there are two asymptotic regimes of MSD ( and ) if the hydrodynamic decay rate changes monotonically as a function of the wavenumber. When the wall is active, there is a region during which the MSD grows linearly with time (), which is unusual for a membrane segment. If the active wall has a finite intrinsic time scale, the above linear-growth regime of the MSD is further extended. As a whole, active fluctuations at the wall propagate through the surrounding fluid and greatly affects the membrane fluctuations.

This paper is organized as follows. In the next section, we discuss the hydrodynamics of a bound membrane that interacts with an active wall. We also derive a dynamic equation for the membrane fluctuation amplitude in the presence of hydrodynamic interactions. In Sec. III, we calculate the membrane two-point correlation functions for three different cases of the wall as mentioned above. We investigate various asymptotic behaviors of the MSD of a tagged membrane both in the static and the active wall cases. Some further discussions are provided in Sec. IV.

## Ii Hydrodynamics of a bound membrane

### ii.1 Free energy of a bound membrane

As depicted in Fig. 1, we consider a fluid membrane bound at an average distance from a wall which defines the -plane. Within the Monge representation, which is valid for nearly flat surfaces, the membrane shape is specified by the distance between the membrane and the wall. The free energy of a tensionless membrane in a potential per unit area reads Safran (); Lipowsky86 ()

 F=∫d2ρ[κ2(∇2ℓ)2+V(ℓ)], (1)

where is the bending rigidity and . We use a harmonic approximation for fluctuations around the minimum of the potential at , and obtain the approximated form

 F≈κ2∫d2ρ[(∇2h)2+ξ−4h2], (2)

where is the correlation length due to the potential. Later we use a dimensionless quantity defined by in order to discuss different cases.

In the following, we introduce the 2D spatial Fourier transform of defined as

 h(q)=∫d2ρh(ρ)e−iq⋅ρ, (3)

where . Then the static correlation function can be obtained from Eq. (2) as

 ⟨h(q)h(−q)⟩=kBTκ(q4+ξ−4)=kBTE(q,ξ), (4)

where is the Boltzmann constant, the temperature, and we have introduced the notation .

In the present model, we assume that the wall is rigid and does not deform. Even when the wall, mimicking the cytoskeleton network, is deformable, the above free energy Eq. (1) would not be changed if we regard as a local distance between the membrane and the cytoskeleton. In this case, however, the bending rigidity should be replaced with an effective one which is also dependent on the bending rigidity of the cytoskeleton network itself LZ89 ().

### ii.2 Hydrodynamic equations and boundary conditions

The dynamics of a membrane is dominated by the surrounding fluid which is assumed to be incompressible and to obey the Stokes equation. We choose as the coordinate perpendicular to the wall located at as in Fig. 1. Then the velocity and the pressure for satisfy the following equations

 ∇⋅v=0, (5)
 η∇2v−∇p−f=0, (6)

where is the viscosity of the surrounding fluid and is any force acting on the fluid. The fluid velocity can be obtained from the above equations by supplementing them with proper boundary conditions. In Appendix A, we show a formal solution appropriate for the membrane/wall system, and obtain the fluid velocity in terms of the force . Without loss of generality, we can choose the - and -coordinates as the parallel (longitudinal) and the perpendicular (transverse) directions to the in-plane vector , respectively. Since the transverse -component of the velocity is not coupled to the other components, we are allowed to set in what follows.

Let us denote the fluid regions and with the superscripts “” and “”, respectively. In general, we consider time-dependent boundary conditions at and time-independent conditions at :

 v−x(q,z=0,t)=Vx0(q,t), (7) v−z(q,z=0,t)=Vz0(q,t), (8) v+x(q,z→∞,t)=v+z(q,z→∞,t)=0. (9)

The statistical properties of and will be discussed for different types of walls in the next Section. As described in Appendix, the -component of the velocity is then obtained as

 v−z(q,z,t)= A[sinh(qz)−qzcosh(qz)]+Bqzsinh(qz) −iqzVx0(q,t)e−qz+(1+qz)Vz0(q,t)e−qz, (10)
 v+z(q,z,t)=Ce−q(z−¯ℓ)+Dq(z−¯ℓ)e−q(z−¯ℓ), (11)

where , , , and are the coefficients determined by the other boundary conditions at the membrane . Note that both and can be also expressed in terms of these four coefficients.

At where the membrane exists, continuity of and yields

 v−x(q,z=¯ℓ,t)=v+x(q,z=¯ℓ,t), (12)
 v−z(q,z=¯ℓ,t)=v+z(q,z=¯ℓ,t), (13)

and incompressibility of the membrane requires that the in-plane divergence of vanishes

 iqv−x(q,z=¯ℓ,t)=0. (14)

Moreover, the forces are required to balance in the normal direction at . This condition is written as

 −T+zz+T−zz=−δFδh(q,t)=−E(q,ξ)h(q,t), (15)

where was defined in Eq. (4). In the above, is the -component of the fluid stress tensor

 Tij=−pδij+η(∂ivj+∂jvi), (16)

evaluated at and . The above four boundary conditions in Eqs. (12)–(15) at determine the solution of and in the entire region of the fluid.

### ii.3 Dynamic equation of a bound membrane

Next we derive a dynamic equation for the membrane fluctuation amplitude. The time derivative of the fluctuation amplitude (membrane velocity) should coincide with the normal velocity of the fluid at the membrane obtained from Eqs. (10) and (11) together with the four coefficients (see also Appendix). Using the result of the above hydrodynamic calculation, we can write the dynamic equation of as follows

 ∂h(q,t)∂t= −γ(q,¯ℓ,ξ)h(q,t) +Λx(q,¯ℓ)Vx0(q,t)+Λz(q,¯ℓ)Vz0(q,t) +ζ(q,t). (17)

In the above, is the hydrodynamic decay rate

 γ(q,¯ℓ,ξ)=Γ(q,¯ℓ)E(q,ξ), (18)

where the kinetic coefficient is given by

 Γ(q,¯ℓ)=12ηqsinh2(q¯ℓ)−(q¯ℓ)2sinh2(q¯ℓ)−(q¯ℓ)2+sinh(q¯ℓ)cosh(q¯ℓ)+(q¯ℓ). (19)

The same expression was obtained by Seifert Seifert94 (). The second and the third terms on the r.h.s. of Eq. (17) are due to the wall boundary conditions Eqs. (7) and (8). Our calculation yields

 Λx(q,¯ℓ)=−iq¯ℓsinh(q¯ℓ)sinh2(q¯ℓ)−(q¯ℓ)2+sinh(q¯ℓ)cosh(q¯ℓ)+(q¯ℓ), (20)
 Λz(q,¯ℓ)=sinh(q¯ℓ)+q¯ℓcosh(q¯ℓ)sinh2(q¯ℓ)−(q¯ℓ)2+sinh(q¯ℓ)cosh(q¯ℓ)+(q¯ℓ). (21)

The last term in Eq. (17) represents the thermal white noise; its average vanishes while its correlation is fixed by the fluctuation-dissipation theorem (FDT) KuboBook (); LandauBook ()

 ⟨ζ(q,t)ζ(−q,t′)⟩=2kBTΓ(q,¯ℓ)δ(t−t′). (22)

### ii.4 Hydrodynamic decay rate

We first introduce as a characteristic time. In Fig. 2, we plot the scaled decay rate (see Eq. (18)) as a function of the dimensionless wavenumber when and . For our later discussion, it is useful here to discuss its asymptotic behaviors. We first note that the kinetic coefficient in Eq. (19) behaves as

 Γ≈{¯ℓ3q2/12η,q≪1/¯ℓ1/4ηq,q≫1/¯ℓ. (23)

Depending on the relative magnitude between and , two different asymptotic behaviors of the decay rate can be distinguished Seifert94 (). For (corresponding to in Fig. 2), the decay rate increases monotonically as

 γ≈⎧⎪ ⎪⎨⎪ ⎪⎩κ¯ℓ3q2/12ηξ4,q≪1/ξκ¯ℓ3q6/12η,1/ξ≪q≪1/¯ℓκq3/4η,1/¯ℓ≪q. (24)

The small- behavior results from the conservation of the fluid volume between the membrane and the wall Marathe89 (). The dependence in the intermediate regime, where the effect of potential becomes irrelevant, was predicted by Brochard and Lennon Brochard75 (). For large , we recover the behavior of a free membrane . All these asymptotic behaviors are observed in Fig. 2.

For (corresponding to in Fig. 2), on the other hand, changes non-monotonically as Seifert94 ()

 γ≈⎧⎪⎨⎪⎩κ¯ℓ3q2/12ηξ4,q≪1/¯ℓκ/4ηξ4q,1/¯ℓ≪q≪1/ξκq3/4η.1/ξ≪q. (25)

While the small- and large- behaviors are unchanged from Eq. (24), here the decay rate decreases with increasing in the intermediate range. This unusual decrease of the decay rate clearly appears for in Fig. 2. Such an anomalous behavior occurs due to the fact that the potential confines the mean-square fluctuation amplitudes to independently of (see Eq. (4)), while hydrodynamic damping becomes less effective with increasing  Seifert94 (). We also note that the absolute value of in the small- region is sensitive to the value of , while it is independent of in the large- region.

## Iii Membrane two-point correlation functions

Using the result of the hydrodynamic calculation, we shall discuss in this section the two-point correlation functions of bound membranes ZG96 (); ZG02 (). We separately investigate the cases of (i) a static wall, (ii) an active wall, and (iii) an active wall with an intrinsic time scale.

### iii.1 Static wall

In the case of a static wall, the velocities at the wall vanish in Eqs. (7) and (8), i.e., . Hence Eq. (17) reduces to

 ∂h(q,t)∂t=−γ(q,¯ℓ,ξ)h(q,t)+ζ(q,t), (26)

and one can easily solve for as

 h(q,t)= h(q,0)e−γ(q,¯ℓ,ξ)t +∫t0dt1ζ(q,t1)e−γ(q,¯ℓ,ξ)(t−t1). (27)

Using the above solution and Eq. (22), we calculate the membrane two-point correlation function which can be separated into two parts ZG96 (); ZG02 ()

 ⟨[h(ρ,t)−h(ρ′,0)]2⟩ =Φ(ρ−ρ′)+ϕ(ρ−ρ′,t), (28)

where the translational invariance of the system has been assumed. In the above, the first term is a purely static correlator

 Φ(ρ−ρ′) =⟨[h(ρ)−h(ρ′)]2⟩ =2∫d2q(2π)2⟨h(q)h(−q)⟩[1−eiq⋅(ρ−ρ′)], (29)

describing the static membrane roughness, while the second term is a dynamical correlator

 ϕ(ρ−ρ′,t)= 2∫d2q(2π)2⟨h(q)h(−q)⟩eiq⋅(ρ−ρ′) ×[1−e−γ(q,¯ℓ,ξ)t], (30)

describing the propagation of fluctuations with a distance .

Using the static correlation function for in Eq. (4), we first calculate the static correlator

 Φ(ρ−ρ′)=kBTπκ∫∞0dqqq4+ξ−4[1−J0(q|ρ−ρ′|)] (31)

where is the zero-order Bessel function of the first kind, and the Meijer -function is used in the last expression mathematica (). In Fig. 3, we plot the scaled static correlator as a function of where . Only in this plot, we use to scale the length because the above static correlator is solely determined by the free energy in Eq. (2), and Eq. (31) does not depend on . In the large distance , the (route mean square) height difference between two points on the bound membrane is proportional to . It is interesting to note that changes non-monotonically and shows a maximum around . A similar overshoot behavior of the membrane profile was reported before KA00 ().

As for the dynamical correlator in Eq. (30), we perform the angular integration and obtain the expression

 ϕ(ρ−ρ′,t)= kBTπκ∫∞0dqqq4+ξ−4 ×[1−e−γ(q,¯ℓ,ξ)t]J0(q|ρ−ρ′|). (32)

We first set and discuss the mean squared displacement (MSD) of a tagged membrane segment given by ZG96 (); ZG02 ()

 ϕ0(t)=kBTπκ∫∞0dqqq4+ξ−4[1−e−γ(q,¯ℓ,ξ)t], (33)

where we have used . Instead of the correlation length , we hereafter use to scale the length. Note that the hydrodynamic effect is manifested by the appearance of the length . In Fig. 4, we plot the dimensionless MSD as a function of (recall that ) for (monotonic damping case) and (non-monotonic damping case), respectively. In order to find out the asymptotic behaviors clearly, we have also plotted an effective growth exponent defined by

 α(t)=dlnϕ0(t)dlnt. (34)

For both and , the MSD increases monotonically as a function of time. For (), there are three different asymptotic regimes of the time dependence. In the small time regime (), the MSD behaves as which corresponds to the diffusion of a free membrane ZG96 (); ZG02 (). This scaling behavior can be obtained by using the large- behavior of the decay rate in Eq. (24)

 ϕ0(t) ≈kBTπκ∫∞0dq1q3[1−e−(κq3/4η)t] ∼kBTκ1/3η2/3t2/3. (35)

In the intermediate time regime (), we have which stems from the intermediate- behavior of in Eq. (24)

 ϕ0(t) ≈kBTπκ∫∞0dq1q3[1−e−(κ¯ℓ3q6/12η)t] ∼kBT¯ℓκ2/3η1/3t1/3. (36)

In this regime, as discussed by Brochard and Lennon Brochard75 (), the conservation of the enclosed incompressible volume between the membrane and the wall is important, while the effect of the potential acting between them is irrelevant. The Fourier transform of the above expression, i.e., the power spectral density, was previously discussed by Gov et al. in Ref. Gov03 (). In the long time regime (), the MSD saturates at the value given by

 ϕ0(t→∞)≈[kBT¯ℓ2πκ]πΞ24∼kBTκξ2. (37)

For (), on the other hand, there are only two asymptotic regimes. The MSD increases as in the small time regime (), whereas in the long time regime (), it saturates at the value given by Eq. (37).

Let us consider then the case . In Fig. 5, we plot the scaled in Eq. (32) as a function of for different times when . For all the cases, the dynamic correlator changes non-monotonically and exhibits a typical undershoot behavior. The minimum of occurs for larger as time evolves. In the long time limit, , in Eq. (32) coincides with the second term in Eq. (31) and is given by the Meijer -function.

### iii.2 Active wall

We now investigate the case when the wall is active so that it exerts random velocities on the ambient fluid. The membrane dynamics in the presence of an active wall is described by Eq. (17). This equation can be also solved for as

 h(q,t)= h(q,0)e−γ(q,¯ℓ,ξ)t +Λx(q,¯ℓ)∫t0dt1Vx0(q,t1)e−γ(q,¯ℓ,ξ)(t−t1) +Λz(q,¯ℓ)∫t0dt2Vz0(q,t2)e−γ(q,¯ℓ,ξ)(t−t2) +∫t0dt3ζ(q,t3)e−γ(q,¯ℓ,ξ)(t−t3). (38)

The random velocities generated at the wall are assumed to have the following statistical properties

 ⟨Vx0(ρ,t)⟩=⟨Vz0(ρ,t)⟩=0, (39)
 ⟨Vx0(ρ,t)Vx0(ρ′,t′)⟩=2Sxδ(ρ−ρ′)δ(t−t′), (40)
 ⟨Vz0(ρ,t)Vz0(ρ′,t′)⟩=2Szδ(ρ−ρ′)δ(t−t′), (41)
 ⟨Vx0(ρ,t)Vz0(ρ′,t′)⟩=0, (42)
 ⟨Vx0(ρ,t)ζ(ρ′,t′)⟩=⟨Vz0(ρ,t)ζ(ρ′,t′)⟩=0, (43)

where we have introduced the amplitudes and in Eqs. (40) and (41), respectively. With these statistical properties, we can calculate the total two-point correlation function which consists of the static and the dynamical parts as before

 ⟨[h(ρ,t)−h(ρ′,0)]2⟩tot =Φtot(ρ−ρ′)+ϕtot(ρ−ρ′,t). (44)

In the above total correlation function, the static correlator in the presence of the active wall becomes

 Φtot(ρ−ρ′)=1π∫∞0dqq[kBTκ(q4+ξ−4) +Sx|Λx(q,¯ℓ)|2γ(q,¯ℓ,ξ)+Sz|Λz(q,¯ℓ)|2γ(q,¯ℓ,ξ)][1−J0(q|ρ−ρ′|)] ≡Φ(ρ−ρ′)+Φx(ρ−ρ′)+Φz(ρ−ρ′), (45)

where and were obtained in Eqs. (20) and (21), respectively, while was defined in Eq. (31) for the static wall case. In the above equations, we have defined two correlators and . On the other hand, the dynamical correlator in Eq. (44) is given by

 ϕtot(ρ−ρ′,t)=1π∫∞0dqq[kBTκ(q4+ξ−4) +Sx|Λx(q,¯ℓ)|2γ(q,¯ℓ,ξ)+Sz|Λz(q,¯ℓ)|2γ(q,¯ℓ,ξ)] ×[1−e−γ(q,¯ℓ,ξ)t]J0(q|ρ−ρ′|). (46)

By setting , the total MSD of a tagged membrane segment in the presence of the active wall becomes

 ϕtot(t)=1π∫∞0dqq[kBTκ(q4+ξ−4) +Sx|Λx(q,¯ℓ)|2γ(q,¯ℓ,ξ)+Sz|Λz(q,¯ℓ)|2γ(q,¯ℓ,ξ)][1−e−γ(q,¯ℓ,ξ)t] ≡ϕ0(t)+ϕx0(t)+ϕz0(t), (47)

where the first term was defined before in Eq. (33) for the static wall case, while and have been newly defined here.

Before showing the result of MSD, we first discuss the wavenumber dependencies of the quantities and appearing in Eqs. (45)–(47). These quantities originating from the active wall are plotted in Fig. 6 as a function of for and . Using the asymptotic behaviors of , as shown in Eqs. (24) and (25), we can obtain the limiting expressions for and as well. When (corresponding to ), we have

 |Λx|2/γ≈⎧⎪⎨⎪⎩3ηξ4/κ¯ℓ,q≪1/ξ3η/κ¯ℓq4,1/ξ≪q≪1/¯ℓ4η¯ℓ2e−2¯ℓq/κq,1/¯ℓ≪q, (48)
 |Λz|2/γ≈⎧⎪ ⎪⎨⎪ ⎪⎩12ηξ4/κ¯ℓ3q2,q≪1/ξ12η/κ¯ℓ3q6,1/ξ≪q≪1/¯ℓ4η¯ℓ2e−2¯ℓq/κq,1/¯ℓ≪q. (49)

For (corresponding to ), on the other hand, we obtain

 |Λx|2/γ≈⎧⎪ ⎪⎨⎪ ⎪⎩3ηξ4/κ¯ℓ,q≪1/¯ℓ4ηξ4¯ℓ2q3e−2¯ℓq/κ,1/¯ℓ≪q≪1/ξ4η¯ℓ2e−2¯ℓq/κq,1/ξ≪q, (50)
 |Λz|2/γ≈⎧⎪ ⎪⎨⎪ ⎪⎩12ηξ4/κ¯ℓ3q2,q≪1/¯ℓ4ηξ4¯ℓ2q3e−2¯ℓq/κ,1/¯ℓ≪q≪1/ξ4η¯ℓ2e−2¯ℓq/κq.1/ξ≪q. (51)

The static correlators and defined in Eq. (45) due to the active wall can now be obtained by performing numerical integrals. In Fig. 7, we plot the static correlators and as a function of when . Here and are scaled by and , respectively. We notice that behaves similarly to that of the static wall case given in Eq. (31) and plotted in Fig. 3. On the other hand, diverges logarithmically for large because the integral is found to be infrared divergent. Such a logarithmic divergence is avoided when we consider a finite membrane size which gives rise to a cutoff for small wavenumbers in the integral of Eq. (45). It should be noted that both and depend on and , while is solely determined by . This means that and include the geometrical as well as the hydrodynamic effects.

In Figs. 8 and 9, we plot the scaled membrane MSD and (see Eq. (47)), respectively, as a function of when and . For (corresponding to ), there are three different asymptotic regimes both for and . In the small time regime (), we have and , showing a normal diffusive behavior. This is because can be approximated as

 ϕx0(t) ≈4η¯ℓ2Sxπκ∫∞0dqe−2¯ℓq[1−e−(κq3/4η)t] ≈¯ℓ2Sxtπ∫∞0dqe−2¯ℓqq3∼Sx¯ℓ2t. (52)

Notice that only small- contributes to the integral, and the same holds for . In the intermediate time regime (), we have and which can be asymptotically obtained by Eqs. (36) and (35), respectively. In the long time regime (), saturates at the value

 ϕx0(t→∞)≈[4η¯ℓSxπκ]3Ξ28∼ηξ2Sxκ¯ℓ. (53)

On the other hand, diverges logarithmically for , which can be seen in Fig. 9(a) and also shown analytically. Such a divergence in time occurs for small and can be avoided when the membrane size is finite as mentioned before.

For (corresponding to ), on the other hand, there are only two asymptotic regimes. The MSDs increase both linearly as and in the small time regime (). In the long time regime (), saturates at the value

 ϕx0(t→∞)≈[4η¯ℓSxπκ]3Ξ48∼ηξ4Sxκ¯ℓ3, (54)

while also diverges logarithmically as above.

### iii.3 Active wall with an intrinsic time scale

Finally we consider a situation in which the activity of the wall occurs over a finite time scale . In this case, the statistical properties of random velocities which have been given in Eqs. (40) and (41) would be replaced by the following exponential correlation function in time Gov04 (); GovSafran05 (); Gov07 ()

 ⟨Vx0(ρ,t)Vx0(ρ′,t′)⟩=Sxτδ(ρ−ρ′)e−|t−t′|/τ, (55)
 ⟨Vz0(ρ,t)Vz0(ρ′,t′)⟩=Szτδ(ρ−ρ′)e−|t−t′|/τ, (56)

while the other velocity correlations remain the same. In general, the intrinsic time scale can be different between the - and -components. In the above relations, we have put a factor so that the physical dimension of and is the same as before.

Repeating the same procedure as before, we obtain the total two-point correlation function which can be also separated into the static and dynamics parts as in Eq. (44). The static correlators in the presence of the active wall now become

 Φx(ρ−ρ′)= 1π∫∞0dqqSx|Λx(q,¯ℓ