Numerical analysis of viscoelasticity of two-dimensional fluid membranes under oscillatory loadings

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arXiv:2210.11074v5 [physics.bio-ph] 30 Mar 2024

Numerical analysis of viscoelasticity of two-dimensional ﬂuid

membranes under oscillatory loadings

Naoki Takeishia,b,∗,Masaya Santob,Naoto Yokoyamac,∗and Shigeo Wadab

aDepartment of Mechanical Engineering, Kyoto Institute of Technology, Goshokaido-cho, Matsugasaki, Sakyo-ku, 606-8585, Kyoto, Japan

bGraduate School of Engineering Science, Osaka University, 1-3 Machikaneyama, Toyonaka, 560-8531, Osaka, Japan

cDepartment of Mechanical Engineering, Tokyo Denki University, 5 Senju-Asahi, Adachi, 120-8551, Tokyo, Japan

A R T I C L E I N F O

Keywords:

Lipid bilayer, 2D ﬂuid membrane, Vis-

coelasticity, Computational Biomechan-

ics

A B S T R A C T

Biomembranes consisting of two opposing phospholipid monolayers, which comprise the so-called

lipid bilayer, are largely responsible for the dual solid-ﬂuid behavior of individual cells and viruses.

Quantifying the mechanical characteristics of biomembrane, including the dynamics of their in-plane

ﬂuidity, can provide insight not only into active or passive cell behaviors but also into vesicle design

for drug delivery systems. Despite numerous studies on the mechanics of biomembranes, their dynam-

ical viscoelastic properties have not yet been fully described. We thus quantify their viscoelasticity

based on a two-dimensional (2D) ﬂuid membrane model, and investigate this viscoelasticity under

small amplitude oscillatory loadings in micron-scale membrane area. We use hydrodynamic equa-

tions of bilayer membranes, obtained by Onsager’s variational principle, wherein the ﬂuid membrane

is assumed to be an almost planar bilayer membrane. Simulations are performed for a wide range

of oscillatory frequencies 𝑓and membrane tensions. Our numerical results show that as frequencies

increase, membrane characteristics shift from an elastic-dominant to viscous-dominant state. Fur-

thermore, such state transitions obtained with a 1-𝜇m-wide loading proﬁle appear with frequencies

between 𝑂(𝑓) = 101–102Hz, and almost independently of surface tensions. We discuss the formation

mechanism of the viscous- or elastic-dominant transition based on relaxation rates that correspond to

the eigenvalues of the dynamical matrix in the governing equations.

1. Introduction

Lipid bilayer membranes, consisting of a series of op-

posing phospholipids arranged in a two-dimensional (2D)

ﬂuid crystalline assembly with ∼ 5-nm thickness [49], are

a common and fundamentally important structure in all liv-

ing cells and many viruses [28]. Each lipid covers a sur-

face area of approximately 0.7nm2(= 70 Å2) [13]. The

membrane structure separates the inside and outside of the

cell, and assumes various function-related shapes [58]. In

addition, membrane mechanical properties aﬀect cell and

membrane dynamics, such as active cell migration [26] and

endocytosis [2,10]. From a mechanical viewpoint, while

phospholipids in membranes can move in the planar direc-

tion, their displacement in their vertical displacement is re-

stricted, and thus the bilayers can behave as a 2D ﬂuid mem-

brane. Such ﬂuid deformable surfaces exhibit a solid-ﬂuid

duality, resulting in unique and complex mechanical charac-

teristics wherein in-plane ﬂuidity and elasticity can emerge

simultaneously. Thus, quantifying the dynamical mechani-

cal properties of biomembranes can provide insight not only

into the aforementioned cell behaviors but also into the de-

velopment of drug delivery systems with vesicles [22], which

are closed biomembranes that typically consist of only lipids

and cholesterol without any proteins.

Although the mechanics of biomembranes have been well

studied using various experimental strategies such as atomic

∗Corresponding author

ntakeishi@kit.ac.jp (N. Takeishi); n.yokoyama@mail.dendai.ac.jp (N.

Yokoyama)

ORCID(s): 0000-0002-9568-8711 (N. Takeishi); 0000-0003-1460-1002

(N. Yokoyama)

force microscopy (AFM), micropipette aspiration, and op-

tical tweezers, as reviewed in Ref. [52], the dynamical vis-

coelasticity of (pure) lipid bilayers under oscillatory load-

ings has not yet been fully understood. Recent experimen-

tal techniques have successfully quantiﬁed dynamical vis-

coelasticity by complex moduli in lipid monolayers or bilay-

ers, assuming a linear mechanical response of membranes

to oscillatory shear strains [1,7,20,27]. For instance, for

diﬀerent concentrations of cholesterol [i.e., mixtures of 1,2-

dipalmitoyl-𝑠𝑛-glycero-3-phosphocholine(DPPC) and choles-

terol], Al-Rekabi et al. used AFM to produce a map of the

viscoelastic properties of a lipid bilayer composed of DPPC[1],

which is one of the primary lipids in lung surfactant [34] and

is ubiquitous in cell membranes. Despite these eﬀorts, there

is still no consensus on the transition mechanism between

viscous- and elastic-dominant states in lipid bilayers.

Along with these experimental studies, various theoret-

ical frameworks have been proposed to describe ﬂuid mem-

brane dynamics [4,45,46], and some have been applied

to problems regarding the spontaneous conformation of hu-

man red blood cells (RBCs) [32] and vesicles [11]. In these

works, the lipid bilayer is modeled as a continuous elastic

surface [21], considering the scale diﬀerence between the

micrometer system size and nanometer membrane thickness.

Lipowsky described spontaneous curvature by an approxi-

mation of solid shells that store elastic energy during stretch-

ing or bending [33]. In terms of soft matter physics, Seifert

and Langer successfully described bilayer hydrodynamics

for almost planar membranes [47], where coupling of the

membrane dynamics with the surrounding ﬂuid was taken

into account by modeling curvature, density-diﬀerence elas-

N. Takeishi & N. Yokoyama et al.: Preprint submitted to Elsevier Page 1 of 12

Viscoelasticity of two-dimensional ﬂuid membrane

ticity, intermonolayer friction, monolayer 2D viscosity, and

solvent three-dimensional (3D) viscosity.

To further provide a precise mechanical background in

membrane dynamics, some numerical models based on the

aforementioned results of [47] have been proposed, which

make it possible to deal with the complex interplay between

membrane elasticity and hydrodynamic forces acting at mi-

croscopic scales. Fournier use Onsager’s variational prin-

ciple to derive the governing equations describing the dy-

namics of an almost planar bilayer membrane [17], and nu-

merically investigated the eﬀect of membrane tension on the

relaxation rate. The principle here is an established, uniﬁed

framework for the dissipative dynamics of a soft matter sys-

tem [12,18]. It provides hydrodynamical equations pertain-

ing to bilayer membranes by minimizing a Rayleighian con-

sisting of potential power for dynamical changes and of dis-

sipated power, to resist the change. More recently, Torres et

al. proposed new computational methods that build on On-

sager’s formalism and arbitrarily Lagrangian-Eulerian for-

mulations [56]. Their methodologies were successfully ap-

plied not only to dynamic lipid bilayers, but also to adhesion-

independent cell migration. A molecular dynamic (MD) ap-

proach has been applied to the oscillatory behavior of the

lipid bilayer membrane [31] and to membrane ﬂuctuations

of RBCs [44]. It is expected that the membrane viscoeastic-

ity would actually be observed in all-atom MD simulations

as an emergent property if viscoelastic eﬀect are innate to

the bilayer. However, this has not yet achieved due to mas-

sive computational load. Thus, despite these eﬀorts, the dy-

namical viscoelastic nature of lipid bilayers, especially with

regard to tensile loadings, has not yet been fully described.

Therefore, the objective of this study is to clarify whether

the lipid bilayer characterized by a 2D ﬂuid membrane fea-

turing lipid bilayers exhibits a transition between the viscous-

and elastic-dominant states depending on the oscillatory load-

ing frequencies. We have made use of the AFM measure-

ments in DPPC bilayers [1,43] in our simulations. In order

to limit the study to linear mechanical response, small ampli-

tude oscillatory tensile loadings are considered. If frequency-

dependent viscoelasticity exhibits in the 2D ﬂuid membrane,

we also discuss whether the formation mechanism of the

viscous- or elastic-dominant transition can be explained based

on relaxation rates that correspond to the eigenvalues of the

dynamical matrix in the governing equations. The theoreti-

cal framework of 2D ﬂuid membrane in the present study fol-

lows the previous study by [17]. To quantify the dynamical

viscoelasticity of the 2D ﬂuid membrane model, we propose

metrics evaluated by scaled mass density and stress in the

membrane: the complex moduli 𝐸∗(𝜔) = 𝐸′(𝜔) + 𝑖𝐸′′(𝜔),

where 𝑖=−1 is the imaginary unit, 𝐸′is the storage mod-

ulus representing the elastic component of the stress, and 𝐸′′

is representingthe viscous dissipation. In this study, in terms

of the way of quantiﬁcation, we distinguish 𝐸∗and classical

viscoelastic metrics as complex “shear” moduli, 𝐺∗(𝜔) =

𝐺′(𝜔) + 𝑖𝐺′′(𝜔)although these may be potentially equiva-

lent. Simulations are performed for wide range of loading

frequencies 𝜔(= 2𝜋𝑓 ) and surface tensions 𝜎.

2. Methods

2.1. Model of lipid bilayer membrane

Following the previous theoretical and numerical study

by Fournier [17], we consider a lipid bilayer membrane made

of only one lipid type in an unbounded ﬂow ﬁeld. The mem-

brane shape is therefore characterized by the height 𝑧=

ℎ(𝒓, 𝑡)from the plane at 𝑧= 0 to the membrane mid-surface,

where 𝒓is the membrane coordinate projected onto the 𝑥-𝑦

plane. Position of a point on the membrane in this coordi-

nate system is thus denoted as 𝑹= (𝒓, ℎ(𝒓)). Two mono-

layers in the membrane possess a lipid mass density 𝑛±(𝒓, 𝑡)

in deformed state ℎ(𝒓, 𝑡), where superscript “±” represents

the upper monolayer (𝑧 > ℎ) and lower monolayer (𝑧 < ℎ),

respectively. Using mass density 𝑛0in the tensionless state

as the reference, the membrane state can be described by its

shape (or height) ℎ(𝒓, 𝑡)and scaled mass density 𝜌±(𝒓, 𝑡)as:

𝜌±(𝒓, 𝑡) = 𝑛±(𝒓, 𝑡) − 𝑛0

𝑛0

.(1)

We also consider the 3D solvent velocities 𝑉±

𝛼(𝑹, 𝑡)(𝛼=𝑥,

𝑦,𝑧) on either side of the membrane, and the 2D lipid veloc-

ities 𝑣±

𝑖(𝒓, 𝑡), (𝑖=𝑥,𝑦) in both monolayers. A schematic of

the 2D ﬂuid membrane is shown in Fig. 1.

Figure 1: Geometrical description of the membrane shape

ℎ(𝒓, 𝑡)and membrane coordinate 𝑹(𝒓, ℎ(𝒓)) with scaled densi-

ties 𝜌±, monolayer velocities 𝑣±, bulk solvent velocities 𝑉±, and

the surface distance away from the membrane mid-surface 𝑒.

The bulk solvent is assumed to be an incompressible ﬂuid,

and hence 𝑉±

𝛼satisﬁes the following mass conservation law:

𝜕𝛼𝑉±

𝛼= 0,(2)

where 𝜕𝛼=𝜕∕𝜕𝑅𝛼. The lipid mass conservation equation

is 𝜕𝑡𝑛±+𝜕𝑖(𝑛±𝑣±

𝑖) = 0, where 𝜕𝑖=𝜕∕𝜕𝑟𝑖. The ﬁrst order

approximation leads to the following equation:

𝜕𝑡𝜌±+𝜕𝑖𝑣±

𝑖= 0.(3)

The Einstein summation convention is adopted, wherein re-

peating the same index twice in a single term implies sum-

mation over all possible values of that index. Because of

N. Takeishi & N. Yokoyama et al.: Preprint submitted to Elsevier Page 2 of 12

Viscoelasticity of two-dimensional ﬂuid membrane

no-slip boundary conditions on the membrane surface, the

external ﬂuid must satisfy the following equations:

𝑉±

𝑖𝑧=ℎ=𝑣±

𝑖,(4)

𝑉±

𝑧𝑧=ℎ=𝜕𝑡ℎ. (5)

Equation (4) gives

𝜕𝑖𝑣𝑖=𝜕𝑖𝑉±

𝑖𝑧=ℎ+𝜕𝑉𝑖

𝜕𝑧 𝑧=ℎ

𝜕𝑖ℎ, (6)

and the ﬁrst term of the right-hand side is equal to −𝜕𝑉 ±

𝑧∕𝜕𝑧𝑧=ℎ

due to Eq. (2). The ﬁrst order approximation as well as the

Stokes approximation allows us to neglect the second term

in Eq. (6). Then, Eq. (3) gives

𝜌±=∫𝜕𝑉 ±

𝑧

𝜕𝑧 𝑧=ℎ

𝑑𝑡. (7)

Because 𝜌++𝜌−relaxes almost instantly to zero, the up-

per and lower monolayers satisfy 𝜌++𝜌−≃ 0 [17]. More

precise derivations of Eqs. (6) and (7) are described in Ap-

pendix §A.

2.2. Rayleighian and power components

In the Stokes approximation (i.e., all forces balance with-

out all inertial eﬀects), the dynamical equations for the mo-

tion of the membrane in the bulk solvent can be given by

minimizing the total Rayleighian of the system with respect

to all the dynamical variables [3,12,41]:

=1

2𝑊+̇

(8)

=±

𝑏+±

𝑠+±

𝑖+̇

𝑖𝑛𝑡 +̇

𝑒𝑥𝑡.(9)

Here, 𝑊∕2(= ±

𝑏+±

𝑠+±

𝑖)is the resistive power against

dynamical changes, the so-called dissipated power, which

consists of three sources [18]: 𝑏, the viscous dissipation

in the bulk solvent above and below the membrane; 𝑠, the

viscous dissipation in the lipid ﬂuids of the two monolay-

ers due to 2D viscosity; and 𝑖, the dissipation associated

with the intermonolayer friction [15,47]. Moreover, ̇

(=

𝑖𝑛𝑡 +̇

𝑒𝑥𝑡) is the driving power for dynamical changes and

consists of the intrinsic elastic power ̇

𝑖𝑛𝑡 and the external

elastic power ̇

𝑒𝑥𝑡.

As described in Ref. [17], the three dissipated power

sources in Eq. (9) can be written as:

±

𝑏=∫𝐵±

𝑑𝑹𝜂𝐷±

𝛼𝛽 𝐷±

𝛼𝛽 ,(10)

±

𝑠=∫𝑑𝒓𝜂2𝑑±

𝑖𝑗 𝑑±

𝑖𝑗 +𝜆2

2𝑑±

𝑖𝑖 𝑑±

𝑗𝑗 ,(11)

±

𝑖=∫𝑑𝒓𝑏

2𝒗+−𝒗−2,(12)

where 𝐵±is the volume deﬁned by 𝑧 > ℎ or 𝑧 < ℎ,𝐷±

𝛼𝛽 (𝒓) =

𝜕𝛼𝑉±

𝛽+𝜕𝛽𝑉±

𝛼∕2 is the rate-of-deformation tensor in the

bulk solvent, 𝑑±

𝑖𝑗 (𝒓𝑠) = 𝜕𝑖𝑣±

𝑗+𝜕𝑗𝑣±

𝑖∕2 is the rate-of-deformation

tensor in the monolayer ﬂuids, 𝜂is the bulk solvent viscosity,

𝜂2is the 2D viscosity, 𝜆2is the dilational viscosity, and 𝑏is

the intermonolayer friction coeﬃcient.

Assuming a small level of interdigitation between the

lipids, and considering a small area of the membrane mid-

surface 𝑑𝑆 =1 + (∇ℎ)2∕2𝑑𝒓+𝑂(ℎ4)and curvature 𝑐=

∇2ℎ+𝑂(ℎ3), the density ﬁelds are essentially uncoupled.

Hence, the internal elastic power of the membrane can be

written as described in Ref. [47]:

𝑖𝑛𝑡 =∫𝑆

𝑑𝑆 𝜎

2(∇ℎ)2+𝜅

2∇2ℎ2+

𝑘

2𝜌++𝑒∇2ℎ2+𝑘

2𝜌−−𝑒∇2ℎ2,(13)

where 𝜎is the membrane tension, 𝜅is the membrane bend-

ing rigidity, 𝑘is the monolayer stretching coeﬃcient, and 𝑒

is the surface distance away from the membrane mid-surface

[see Fig. 1].

The external elastic energy representing oscillatory load-

ings and the loading proﬁle are deﬁned as:

𝑒𝑥𝑡 =∫𝑆𝑝

𝑑𝒓ℎ𝑝(𝒓, 𝑡),(14)

𝑝(𝒓, 𝑡) = 𝑝0exp −12𝒓2

𝑤2sin (𝜔𝑡),(15)

where 𝑝0is the loading amplitude, and 𝑤is the width of the

loading proﬁle characterized by the Gaussian function. The

integration is performed in the area 𝑆𝑝, which is the projec-

tion onto the reference plane. Considering the previous mi-

cropipette aspiration test in blood granulocytes [14], where

experiments were carried out with pipette sizes of 2–2.75

𝜇m and suction pressures of ≥1 Pa, the loading amplitude

𝑝0and the width of the loading proﬁle 𝑤was set as 𝑝0= 0.5

Pa and 𝑤= 1 𝜇m, respectively. The loading area is equal to

or smaller than the scan sizes in the AFM experiment (≥2.0

𝜇m× 2.0𝜇m) [1]. In in vivo, the shear stress varies from 1

to 60 dynes/cm2(0.1–6Pa) depending in particular on ves-

sel types [35]. Thus, the loading amplitude 𝑝0corresponds

to the physiologically relevant stress in the microcirculation.

The scale of force applied to area 𝑤corresponds to gaps in

the endothelial barrier (∼ 1 𝜇m) during the initial stages of

transmigration of cancer cells [6], and can be found, for ex-

ample, in RBC-platelet [54] or RBC-microparticle [53] hy-

drodynamic interactions. Representative snapshots of ex-

tending membranes are shown in Figs. 2(a) and 2(b). We

also conﬁrm that the given amplitude of oscillatory tensile

loadings is small, and hence the linear mechanical responses

of the membrane to weak oscillations should be investigated,

see panel (c) of Fig. 2.

We have a dynamical equations of the membrane based

on the Stokes approximation and the diﬀerentiation of the

aforementioned Rayleighian (9). The precise derivation of

the equation is described in Appendix §A.

N. Takeishi & N. Yokoyama et al.: Preprint submitted to Elsevier Page 3 of 12

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摘要：

arXiv:2210.11074v5[physics.bio-ph]30Mar2024Numericalanalysisofviscoelasticityoftwo-dimensionalﬂuidmembranesunderoscillatoryloadingsNaokiTakeishia,b,∗,MasayaSantob,NaotoYokoyamac,∗andShigeoWadabaDepartmentofMechanicalEngineering,KyotoInstituteofTechnology,Goshokaido-cho,Matsugasaki,Sakyo-ku,606-8585,...

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Numerical analysis of viscoelasticity of two-dimensional fluid membranes under oscillatory loadings

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