INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERINGInt. J. Numer. Meth. Biomed. Engng. 2012; 28:346–368Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/cnm.2475
Axisymmetric multicomponent vesicles: A comparison ofhydrodynamic and geometric models
Jinsun Sohn 1,*,†, Shuwang Li 2, Xiaofan Li 2 and John S. Lowengrub 3
1Department of Mathematics, University of California, Los Angeles, Los Angeles, CA, USA2Department of Applied Mathematics, Illinois Institute of Technology, 10 W. 32nd St., Chicago, IL 60616, USA
3Department of Mathematics, University of California, Irvine, Irvine, CA, USA
Received 10 August 2011; Revised 21 December 2011; Accepted 24 January 2012
KEY WORDS: multicomponent vesicle; line tension; bending stiffness; inextensibility; boundary integralmethod; small scale decomposition; stokes flow; geometric and hydrodynamic model
1. INTRODUCTION
Biological membranes are well-known for their critical role in cell functions, and thus have beenthe subject of many studies. Recent experiments on giant unilamellar vesicles demonstrate thatthere exists a rich variety of behavior of multicomponent vesicles. Spinodal decomposition intodistinct surface domains (e.g., liquid-ordered, liquid-disordered) known as rafts (or domains), raftcoarsening, viscous fingering, vesicle budding, fission and fusion are all observed with concomi-tant changes in membrane morphology [1–11]. Membrane heterogeneity and the formation ofrafts is thought to play an important role in many biological processes such as vesicle tracking[12] and signal transduction [13], adhesion [14, 15], protein targeting and regulation [16], bud-ding, endocytosis and exocytosis [17]. Rafts have also been implicated with diseases includingAlzheimer’s and Parkinson’s diseases [18], as well as infection by bacteria and viruses [19, 20].In addition, multicomponent vesicles have also been proposed as efficient vehicles for targeted drugdelivery [21].
*Correspondence to: Jinsun Sohn, Department of Mathematics, Biomedical Engineering, Chemical Engineering andMaterial Science, University of California, Irvine, CA, USA.
HYDRODYNAMIC AND GEOMETRIC MODELS OF AXISYMMETRIC MEMBRANES 347
Although there has been an increasing focus on modeling multicomponent vesicles, studies havebeen limited to equilibrium investigations [22–26]. However, many of the biological processes dis-cussed earlier involve membrane dynamics. Thus, there is a need to develop accurate and efficientnumerical methods to simulate the evolution of multicomponent vesicles. The dynamics of multi-component vesicles are characterized by a competition between membrane elastic bending energy,surface inextensibility, surface phase separation and nonlocal fluid interactions because of hydrody-namic effects. The nonlinear coupling among these components makes the design of accurate andefficient computational methods highly challenging.
In [27], we introduced an efficient algorithm for multicomponent vesicle evolution in two-dimensional fluids (e.g., the vesicle boundary is a closed curve). In this paper, we extend this work,consider axisymmetric multicomponent vesicles and compare the results using a hydrodynamicmodel and a geometric model, which does not take the fluid into account. Although there have beensome studies in which geometric and hydrodynamic models have been compared for homogeneousbiomembranes [28], to our knowledge, there have been no such comparisons for multicomponentvesicles. In addition, we present a comparison of results in which only the surface area is conservedwith those when the membrane is locally inextensible. Although such a comparison may be aca-demic, we feel it is important to present because several studies enforce global rather than localconstraints (e.g.,[25, 28, 29]) and other studies mimic local constraints by introducing evolutionequations for Lagrange multipliers (e.g., [30–32]).
The rest of the paper is organized as follows. In Section 2, we present the equations governingthe dynamics of inextensible, multicomponent three-dimensional (3D) axisymmetric vesicles.Section 2.4 describes the two types of inextensibility constraint approaches. We introduce the hydro-dynamic model in Section 3.1 and the geometric model in Section 3.2. In Section 4, the numericalmethods are discussed. Numerical results are given in Section 5. Conclusions are drawn, and futurework is discussed in Section 6.
2. MATHEMATICAL MODEL
2.1. Surface evolution
Let X D X.s, t / D .r.s, t /, ´.s, t // denote the position of an evolving axisymmetric membranewith velocity
dXdtD V nC T s, (1)
where V and T are the normal and tangential velocities in the .r , ´/-plane, and n and s are thecorresponding normal and tangential vectors. Let ˛ denote a parametrization of the membrane thatbounds the vesicle (0 6 ˛ < 2�), then we may take s D .r˛ , ´˛/=s˛ where s˛ D
pr2˛ C ´
2˛
and subscripts denote partial derivatives. The outward normal vector is then n D .´˛ ,�r˛/=s˛ .Introducing the tangent angle � D tan�1 ´˛=r˛ , the tangent and normal vectors are also given bysD .cos � , sin �/ and nD .sin � ,� cos �/. Using these relations, the equations for the tangent angle� and local arclength variation s˛ are as follows:
�t D�Vs C �sT , s˛t D .Ts C �sV / s˛ , (2)
The local surface area variation is
.rs˛/t D
�1
r.rT /s CHV
�rs˛ , (3)
where the total curvature is
H DH � CH r D�˛
s˛C
sin �
rD �s C
sin �
r(4)
The Gaussian curvature is K DH �H r D �s sin �=r .
We introduce the following nondimensional variables:
X0 DXl
, t 0 Dt
�, u0 D
�
lu
where l is a characteristic length of the membrane (e.g., radius of a (equivalent) sphere with the sameenclosed volume as the vesicle) and � D �l3= Nb is the characteristic time associated with bending,with Nb a characteristic bending stiffness and � the viscosity of the exterior fluid. We also introduceanother parameter �3 known as the reduced volume: �3 D Vol=..4�=3/R3/ D 6
p�Vol=S3=2,
where Vol is the volume, S is the surface area and R is the radius of the equivalent sphere.Note that �3 D 1 for a sphere. This parameter has been used to compare axisymmetric exper-imental and numerical results and to classify vesicle shapes [24]. Hereafter, we drop the primenotation and present only the nondimensional equations. The constitutive laws for the veloci-ties in the hydrodynamic and geometric models are derived succeedingly following an energyvariation argument.
2.2. Energy variation
Thermodynamically, consistent constitutive relations for the forces, fluxes and velocities are deter-mined by requiring that the dynamics of the membrane decreases the total membrane free energyEM D Eb C ET C EG . This is expected to be the case in experimental systems, as the evo-lution should tend to an energy minimum in the absence of external forces and influences. Here,Eb is the normal bending energy, ET is the excess (line) energy associated with the surfacephase (e.g., ordered/disordered) domain boundaries and EG is the Gaussian bending energy, Theenergies are given as follows, using a surface phase field formulation. Consider two surface phases(e.g., disordered and ordered states) and let u be the concentration of one of the lipid phases(e.g., disordered phase). Then the normal bending energy is given by
Eb D1
2
Z†
b.u/ .H �H0.u//2 r.s, t /ds, (5)
where b.u/ is the nondimensional normal bending stiffness, H is the total curvature and H0 isthe spontaneous curvature, which may take different values in each surface phase. The line energyET is
ET Da
�
Z†
�f .u/C
�2
2jr†uj
2
�r.s, t /ds, (6)
where f is a double-well potential, for example, f .u/D 14u2.1�u/2, a is a nondimensional param-
eter, r† D .I � nn/r is the surface gradient, and � is a non-dimensional small scaling parameter,where the scaling is chosen such that as � ! 0, ET converges to �L where � D a=6
p2 (e.g.,
� is the line tension) and L is the perimeter of the surface domain boundaries [33]. The Gaussianbending energy EG is
EG D
Z†
g.u/K r.s, t /ds, (7)
where g.u/ is the nondimensional Gaussian bending stiffness, K is the Gaussian curvature.Next, we consider the variational derivatives of each component of the energy. We carry this out
by taking the time derivative, which is equivalent to varying u and the membrane† simultaneously.Taking the time derivative of Equations (5), (6) and (7), using Equation (2), integrating by partsand using periodicity, (because † is assumed to be a closed surface) we obtain the following:
HYDRODYNAMIC AND GEOMETRIC MODELS OF AXISYMMETRIC MEMBRANES 349
PEb D
Z†
.ut � T us/
�b0.u/
2.H �H0/
2 � b .H �H0/H00.u/
�r.s, t /ds
C
Z†
V
��1
r.r.b.H �H0//s/s �
b
2H�H 2 �H 2
0
�C 2b.H �H0/K
�r.s, t /ds.
(9)
PEG D
Z†
�.ut � T us/ g
0K �V
r.rH r@sg/s
�r.s, t /ds, (10)
where the overdots denote time derivatives, and the prime notation denotes the derivative withrespect to u.
Defining the variational derivatives,
ıEM
ıuDa
�
�f 0.u/�
�2
r.rus/s
�Cb0.u/
2.H �H0/
2 � b .H �H0/@H0
@u.u, s/C g0K, (11)
ıEM
ı†nD�
1
r.r.b.H �H0//s/s �
b
2H�H 2 �H 2
0
�C 2b.H �H0/K �
1
r.rH r@sg/s
Ca
�
�f .u/H C
�2
2u2s
�H r �H �
��.
(12)
we obtain
PEM D
Z†
utıEM
ıur.s, t /dsC
Z†
VıEM
ı†nr.s, t /dsC
Z†
T
��us
ıEM
ıu
�r.s, t /ds. (13)
To make further progress, we need to consider mass conservation.
2.3. Mass conservation
During experimental processes such as raft coarsening and vesicle budding, the total mass of theconcentration of one of the lipid phases
M D
Z†
u r.s, t /ds (14)
should be conserved, that is, PM D 0 . Defining M� DR� u r.s, t /ds then PM� D J j� where is
any portion of †, J is the mass flux and J j� is the net mass flux into . Taking the time derivative,we obtain
PM� D
Z�
�ut C
u
r.rT /s C uHV
�r.s, t /ds. (15)
Thus, local mass conservation implies that
ut Cu
r.rT /s C uHV D
Js
r, (16)
Using Equation (16) in Equation(13) gives
PEM D
Z†
�Js
r� .
1
r.rT /s CHV /u
�ıEM
ıurdsC
Z†
VıEM
ı†nrdsC
Z†
T
��us
ıEM
ıu
�rds. (17)
We note that during other experimental processes such as endocytosis or exocytosis, there may beproduction of lipids. We do not consider this case here.
2.4.1. Local inextensibility. We begin by considering the case where the membrane is not allowedto stretch locally. This is expected to be a good approximation of experimental systems because theaverage spacing between lipid molecules may increase only slightly; a larger spacing typically leadsto membrane rupture. Mathematically, this means that the local surface area variation should be timeindependent, that is, r.˛, t /s˛.˛, t /D r.˛, 0/s˛.˛, 0/, which implies that from Equation (3),
1
r.rT /s CHV D 0. (18)
We also enforce volume conservation explicitly even in the presence of a divergence-free fluid. Thisallows us to prevent errors in volume that may accumulate in time and degrade the accuracy ofthe simulation, and model the experimental case in which there is no net gain or leakage of fluidinternal to the vesicle. Following a previous work [27], we impose this constraint via a time andspace-dependent Lagrange multiplierƒLL.˛, t / for the surface area and a time-dependent LagrangemultiplierƒV .t/ for the enclosed volume. We append the membrane energy considered earlier withnull Lagrangians ELL and EV , where
ELL D
Z†
ƒLL.˛, t / .r.˛, t /s˛.˛, t /� r.˛, 0/s˛.˛, 0// d˛, (19)
EV DƒV .Vol.t/� Vol.0//, (20)
where the volume Vol.t/ D 1=3R†
x � n rds. Note that ƒLL can also be interpreted as the tensileforce needed to enforce local inextensibility. Taking the time derivative, we obtain
PELL D
Z†
ƒLLt .r.˛, t /s˛.˛, t /� r.˛, 0/s˛.˛, 0// d˛C
Z†
ƒLL
�1
r.rT /s CHV
�rds, (21)
PEV DƒVt .Vol.t/� Vol.0//CƒV
Z†
V rds. (22)
Putting everything together and assuming that ƒLL is chosen such that 1r.rT /s D �HV (e.g.,
recall Equation (3)) and ƒV is chosen such that Vol.t/D Vol.0/, gives
PEM D PET C PEb C PEG � PELL � PEV D
Z†
JsıEM
ıudsC
Z†
V
�ıEM
ı†n�ƒLLH �ƒV
�rds
C
Z†
T
�ƒLLs � us
ıEM
ıu
�rds. (23)
Finally, local inextensibility makes Equation (16) become
ut DJs
r. (24)
2.4.2. Global surface area conservation. Next, we consider the case in which the membrane is con-strained to have a fixed total surface area. Although it is not clear whether there is an experimentalmechanism that enables local, but not global stretching, we present this formulation for two reasons.First, we actually use the globally conservative model as part of the numerical solution to the locallyconservative model (see Section 4.2.1); and second, as discussed in Section 1, several studies in theliterature use the global constraint rather than the local constraint.
We introduce time-dependent Lagrange multipliersƒGL.t/ andƒV .t/ for the vesicle surface areaand enclosed volume, respectively. As before, we append the membrane energy considered earlierwith null Lagrangians EV and EGL, where
HYDRODYNAMIC AND GEOMETRIC MODELS OF AXISYMMETRIC MEMBRANES 351
where the surface area S.t/ DR†r ds. Note that ƒGL can also be interpreted as a tensile stress
introduced to enforce inextensibility. Taking the time derivative, we obtain
PEGL DƒGLt .S.t/� S.0//CƒGL
Z†
�1
r.rT /s CHV
�rds, (26)
As before, lettingEM DET CEbCEG�EGL�EV , putting everything together, and assumingthat ƒGL is chosen such that S.t/D S.0/. We then obtain
PEM D PET C PEb C PEG � PEGL � PEV
D
Z†
�Js
r� .1
r.rT /s CHV/u
�ıEM
ıurdsC
Z†
V
�ıEM
ı†n�ƒGLH �ƒV
�rds
C
Z†
T
��us
ıEM
ıu
�rds,
D
Z†
JsıEM
ıudsC
Z†
V
�ıEM
ı†n�
�ƒGLC u
ıEM
ıu
�H �ƒV
�rds
C
Z†
T
�u
�ıEM
ıu
�s
�rds. (27)
3. GOVERNING EQUATIONS
3.1. Model in the presence of hydrodynamic forces
We next assume that there is a viscous fluid inside and outside the vesicle. We assume that the lengthand velocity scales are small so that Stokes flow governs the motion of the fluid. Defining the stresstensor as follows:
Pi D�pi IC iDi , Di D�rui CruTi
�(28)
where pi is the pressure, with i D 1, 2 denoting the interior and exterior fluids, i are the nondi-mensional viscosities (e.g., 2=1 and 1 is the viscosity ratio), Di is the deformation tensor and uiis the velocity. The Stokes equations are as follows:
r � Pi D 0, r � ui D 0, (29)
where a velocity u D u1 may be imposed in the far-field. Across the membrane, a stress jumpcondition holds ŒP � n�† � .P � n/1 � .P � n/2 is given by
ŒP � n�† D�F, (30)
where the stress F is obtained from energy variation and thermodynamic consistency as describedhereafter. The velocity is assumed to be continuous across the membrane:
Œu�† D 0. (31)
Let the vesicle surface † have outward normal n so that the flow is driven by a jump in stress Facross †. In the remainder of this section, we assume that the viscosities of the fluids interior andexterior to the vesicle are matched: 1 D 2 D 1, which implies the viscosity ratio �D 1=2 D 1.An analysis of the effect of viscosity contrast is currently under study. Defining the stress compo-nentwise to be F.s, t / D .f1,f2/, the velocity u D .u1,u2/ of the vesicle membrane is based on aboundary integral formulation (e.g., [34]) for solving the Stokes equations with a moving boundary.When �D 1, the velocity uj .x/ on † has the following boundary integral representation:
Gij are the Green’s functions for the velocity field, and we have used the summation convention inwhich repeated indices are summed.
Following [35], the axisymmetric form of the 3D Stokes operator is given by
u.x/DZ 2�
0
K.x,˛/f.˛/y1.˛/ky˛k d˛, (34)
where yD .y1,y2/ in axisymmetric coordinates, ky˛k2 D y21˛Cy22˛ and the kernelK is composed
of elliptic integrals of first and second kind.
3.1.1. System with local inextensible constraint
Constitutive relations. We are now in a position to pose thermodynamically consistent constitutiverelations for the flux J and the stress F using local inextensible constraint. Assuming that each termin Equation (23) is dissipative, we take the following constitutive relations for the J and F:
J Dr
Pe
�ıEM
ıu
�s
, (35)
and
FD��ıEM
ı†n�ƒLLH �ƒV
�n�
�ƒLLs � us
ıEM
ıu
�s. (36)
With these choices, we obtain the dissipation formula:
PEM D�1
Pe
Z†
��ıEM
ıu
�s
�2rds �
1
2
Z†
D W D rds (37)
where D is defined in Equation (28).
3.1.2. System with global surface area conservation
Constitutive relations. We next pose thermodynamically consistent constitutive relations for theflux J and the stress F using the global inextensible constraint. As before, assuming that each termin Equation (27) is dissipative, we may take the following:
J Dr
Pe
�ıEM
ıu
�s
, (38)
where Pe is a non-dimensional Peclet number, and
FD��ıEM
ı†n�
�ƒGLC u
ıEM
ıu
�H �ƒV
�n� u
�ıEM
ıu
�s
s. (39)
With these choices, we again obtain the dissipation formula in Equation (37).
3.2. Geometric model
Constitutive relations in the absence of hydrodynamic forces. We next pose thermodynamicallyconsistent constitutive relations for the flux J and the velocities V and T for a locally inextensiblevesicle evolving in the absence of hydrodynamic forces. Assuming that each term in Equation (23)
HYDRODYNAMIC AND GEOMETRIC MODELS OF AXISYMMETRIC MEMBRANES 353
is dissipative, we pose the following constitutive relations. The diffusion flux may still be given byEquation (35). The normal and tangential velocities may be taken to be
V D�ˇV
�ıEM
ı†n�ƒLLH �ƒV
�(40)
T D�ˇT
�ƒLLs � us
ıEM
ıu
�(41)
where the mobilities ˇV and ˇT are positive. With these choices, we obtain the dissipation formula:
PEM D�Pe
Z†
J 2
rds �
1
ˇV
Z†
V 2 rds �1
ˇT
Z†
T 2 rds 6 0. (42)
To determine the Lagrange multipliers ƒLL and ƒV , we have the two additional equations:Z†
V rds D 0, (43)
.rT /s C rHV D 0. (44)
Rewriting Equations (40) and (41) as
V D ˇV�V uCƒLLH CƒV
�, (45)
T D ˇT�T u �ƒLL
s
�, (46)
where V u D� ıEM
ı†nand T u D us
ıEM
ıu, we obtain the following equations for the Lag-
range multipliers:
ƒV D�R†ˇV V
u rds �R†ˇVƒ
LLH rdsR†ˇV rds
, (47)
and
�1
r
�ˇT rƒ
LLs
�sC ˇVƒ
LLH 2 � ˇVH
R† ˇVƒ
LLH rdsR†ˇV rds
!
D�1
r.ˇT rT
u/s � ˇVHVuC ˇVH
�R† ˇV V
u rdsR†ˇV rds
�.
(48)
3.3. Summary of governing equations
3.3.1. The model with flow. The velocity is governed by the Stokes equations
r � Pi D 0, r � ui D 0, (49)
where a velocity u D u1 may be imposed in the far-field. Across the membrane, the stress jumpcondition is
ŒP�n�†D
8<:�ıEM
ı†n�ƒLLH �ƒV
�nC
�ƒLLs � us
ıEM
ıu
�s, (local inextensibility)�
ıEM
ı†n��ƒGLC u ıE
M
ıu
�H�ƒV
�nC u
�ıEM
ıu
�s
s, (global surface area conservation)
(50)whereas the inextensibility constraint equations are(
These equations are solved simultaneously to determine the velocities of † together with theLagrange multipliers ƒLL, or ƒGL, and ƒV . Given the interface position, this is a linear sys-tem for the Lagrange multipliers and can be solved numerically using an iterative method such asGMRES [36]. The membrane position is updated by
dX†dtD V nC T s, (53)
where V D u � n and T D u � s. The surface phase evolution is given by
ut�1
Pe
1
r
�r
�ıEM
ıu
�s
�s
D
�0 (local constraint)�u
�1r.rT /s CHV
�(global surface area conservation/
(54)
Note that when the membrane is subject to global surface area conservation, it is convenientto change the frame of reference and evolve the membrane using the equal arclength tangentialvelocity [37]:
Teq.˛, t /D Teq.0, t /C2�
L
Z ˛
0
V�˛0 d˛0 �
˛
L
Z 2�
0
V�˛0 d˛0, (55)
where L is the length of the interface. In this case, Equations (53) and (54) are reformulated as
dX†dtD V nC Teqs, (56)
ut�1
Pe
1
r
�r
�ıEM
ıu
�s
�s
D
�0. (local constraint)�u
�1r.rT /sCHV
�C�Teq�T
�us , (global surface area conservation/
(57)
3.3.2. The geometric model. In the geometric model of a locally inextensible vesicle, the normaland tangential velocities are obtained explicitly via
V D ˇV�V uCƒLLH CƒV
�(58)
T D ˇT�T u �ƒLL
s
�, (59)
where
ƒV D�R† ˇV V
u rds �R† ˇVƒ
LLH rdsR† ˇV rds
, (60)
and
�1
r
�ˇT rƒ
LLs
�sC ˇVƒ
LLH 2 � ˇVH
R† ˇVƒ
LLH rdsR† ˇV rds
!
D�1
r.ˇT rT
u/s � ˇVHVuC ˇVH
�R† ˇV V
u rdsR† ˇV rds
�.
(61)
As before, Equations (60) and (61) must be solved simultaneously to determine ƒV and ƒLL.The surface phase evolution equation is still given by Equation (57).
HYDRODYNAMIC AND GEOMETRIC MODELS OF AXISYMMETRIC MEMBRANES 355
4. NUMERICAL METHOD
We solve the governing equations for the hydrodynamic and geometric models using spectrallyaccurate front-tracking/boundary integral methods. All spatial derivatives are obtained using thediscrete Fourier transform in the interface parametrization variable ˛. All spatial integration is per-formed either using the Fourier transform or the trapezoidal rule. The removable singularities atthe poles are evaluated using L’Hopital’s rule [38]. The temporal integration is performed using anon-stiff algorithm. The overall strategy is to
1. Solve the coupled velocity/Lagrange multiplier equations to determine the velocity of thevesicle;
2. Update the position of the interface X†;3. Update the surface phase concentration; and4. Repeat.
Building upon previous work [27, 35, 37, 38], we perform an analysis of the equations at smallspatial scales (small scale decomposition) to develop the following: (i) spectrally accurate spatialdiscretizations of the boundary integrals; (ii) an efficient preconditioner for the velocity/Lagrangemultiplier linear systems; and (iii) non-stiff time integration methods.
4.1. Small scale decomposition
4.1.1. Hydrodynamic model. Following [38] and [39], it can be seen that the kernels in Equation(34) have a logarithmic singularity and take the form
R 2�0 '.˛0/ ln jrj C .˛0/ d˛0, where ' and
are smooth functions, and r D X.˛, t / � X.˛0, t /. Next, we decompose the logarithmic term as asingular and a nonsingular term:
log jrj D log 2j sin
�˛ � ˛0
2
�j C log
jrj
2 sin j�˛�˛0
2
�j, (62)
where the second term in Equation (62) is smooth. Using Equation (62) in Equation (34), we obtain[35, 40, 41]:
u.˛, t /��Ns˛
8�
Z 2�
0
log
�2j sin
�˛ � ˛0
2
�j
�F.˛0, t / d˛0, (63)
where the notation f � g means that f �g is a smoothing operator, for example, an integral opera-tor with a smooth kernel. In addition, Ns˛.t/Dmin˛ s˛.t/ is constant. It follows that the normal andtangential velocities, V D u .˛, t / � n.˛, t / and T D u .˛, t / � s.˛, t /, satisfy
V.˛, t /��Ns˛
8�
Z 2�
0
log
�2j sin
�˛ � ˛0
2
�j
�F.˛0, t / � n.˛0, t / d˛0, (64)
T .˛, t /��Ns˛
8�
Z 2�
0
log
�2j sin
�˛ � ˛0
2
�j
�F.˛0, t / � s.˛0, t / d˛0. (65)
Locally inextensible model.An analysis of Equation (36) reveals that
where Nb is a characteristic value of the bending stiffness. Plugging Equation (66) into Equation (64),we obtain
V.˛, t /��Nb
8Ns2˛LŒ�˛˛˛�.˛, t /�
�˛
8LŒƒLL�, (68)
T .˛, t /�1
8LƒLL˛
, (69)
where the integral operator LŒg�.˛/D 1�
R 2�0
log�2j sin
�˛�˛0
2
�j�g.˛0/ d˛0 for a smooth function
g has the symbol OLŒg�.k/ D � Og.k/=jkj, for k ¤ 0 and OLŒg�.k/ D 0 for k D 0, where k is thewavenumber, for example, see [27] and [37]. It follows that the local inextensibility condition atsmall spatial scales is
1
r.rT /s CHV �
1
8Ns˛LƒLL˛˛
��2˛8Ns˛
LŒƒLL�. (70)
Replacing �˛ by a characteristic value N�˛ , one may define the operator
ALŒƒ�D1
8Ns˛LƒLL˛˛
�N�2˛8Ns˛
LŒƒLL�. (71)
This can be inverted in Fourier space using the following symbol:
OA�1L D1
Ns˛jkj
�k2C N�2˛
��1. (72)
Using A�1L as the preconditioner for the Lagrange multiplier equations (51) and (52) [27, 35, 38],we found that the best results are obtained by taking N�˛ D 0.
Global surface area conserved model.An analysis of Equation (39) reveals that
V.˛, t /��Nb
8Ns2˛LŒ�˛˛˛�.˛, t /�
ƒGL
8LŒ�˛�, (73)
T .˛, t /� 0. (74)
4.1.2. Geometric model. An analysis of Equations (40) and (59) at small spatial scales yields
V � NV
Nb
Ns3˛�˛˛˛ C
ƒLL
Ns˛�˛ Cƒ
V
!, (75)
T ��NT
Ns˛ƒLL˛ . (76)
where Ns˛.t/Dmin˛ s˛.t/ and NV and NT are characteristic mobility parameters. The inextensibilityconstraint then becomes
HYDRODYNAMIC AND GEOMETRIC MODELS OF AXISYMMETRIC MEMBRANES 357
which can be inverted in Fourier space using the following symbol:
OA�1G DNT
Ns2˛
jkj2C
NV
NT
N�2˛
!�1. (79)
Using A�1G as a preconditioner for the local Lagrange multiplier system, we found that as in thehydrodynamic model, the best results are obtained by taking N�˛ D 0.
4.2. Interface velocity and Lagrange multipliers
4.2.1. Hydrodynamic model
Locally inextensible model. The algorithm for obtaining the locally inextensible velocity field isobtained by applying the procedure developed in [27] to the axisymmetric equations. In particular,we decompose the velocity u as
uD uuCƒGLwCƒVw0C Nu, (80)
where uu D uu1 j† D uu2 j† is the solution of the unconstrained Stokes equation:
r � Pui D 0, Pui D�pui ICDui , Dui D
�ruui Cruui
T�
,
r � uui D 0,
ŒPun�† D�ıEM
ı†n�Hu
ıEM
ıu
�n, Œuu�† D 0,
(81)
and wD w1j† D w2j† satisfies
r � Pwi D 0, Pwi D�pwi ICDwi , Dwi D
�rwi CrwTi
�,
r �wi D 0,
ŒPwn�† D�Hn, Œw�† D 0.
(82)
and w0 D w01j† D w02j† satisfies
r � Pw0
i D 0, Pw0
i D�pw0
i ICDw0
i , Dw0
i D�rw0i Crw0Ti
�,
r �w0i D 0,hPw0
ni†D�n,
w0†D 0.
(83)
In addition,
r � NPi D 0, NPi D� Npi IC NDi , NDi D�r Nui Cr NuTi
�(84)
r � Nui D 0, (85)NPn†D� NƒHnC Nƒss, Œ Nu�† D 0, (86)
where NƒDƒLL �ƒGL. The Lagrange multipliers, ƒGL and ƒV are chosen such thatZ†
so that the velocity field u is surface area and volume conserving. Rewriting Equations (87) and(88) using the decomposition of u from Equation (80), we obtainZ
. The local inextensibility constraint (51) is rewritten as
1
r.r Nu � s/s CH Nu � nD
1
r.r Qu � s/s CH Qu � n, (93)
where QuD uuCƒGLwCƒVw0. This system is solved using GMRES with the preconditioner givenby AL.
Global surface area conserved model. The velocity u of local stretching model can be consideredas uuCƒGLwCƒVw0 in the procedure of locally inextensible model.
4.2.2. Geometric model. The constraint Equations (60) and (61) are solved using GMRES with thepreconditioner AG given earlier.
4.3. Updating the interface position
We may rewrite the evolution of the tangent angle from Equation (2) as follows:
�t DMŒ� �.˛, t /CN1.˛, t /, (94)
where M is a linear operator on � that arises from the small scale analysis and N1 contains theremaining nonlinear terms. In particular, we obtain
MŒ� �D
8<:
Nb
8Ns3˛LŒ�˛˛˛˛� .hydrodynamic models/
�NVNb
Ns4˛�˛˛˛˛ .geometric model/,
(95)
andN1 D�VsC�sT �MŒ� �. Let OM denote the symbol of M. In the equations earlier, Nb is constantand taking Nb D max b.u/ works well in practice. Then, following previous work [27, 37], we maydiscretize Equation (94) using a second order accurate linear propagator method
HYDRODYNAMIC AND GEOMETRIC MODELS OF AXISYMMETRIC MEMBRANES 359
The arclength parameter s˛.t/ can be updated using the Adams–Bashforth method:
snC1˛ D sn˛ C�t
2
�3M n
1 �Mn�11
�, (97)
where
M1 D1
2�
Z 2�
0
T˛0 C �˛0V.˛0, t / d˛0. (98)
To reconstruct the interface .r.˛, tnC1/, ´.˛, tnC1// from the updated �nC1.˛/ and NsnC1˛ , we firstupdate a reference point .r.0, tnC1/, ´.0, tnC1//. This is carried out by using a second-order explicitAdams–Bashforth method. Once we update the reference point, we obtain the configuration of theinterface from the �nC1.˛/ and snC1˛ .˛/ by [37]
r.˛, tnC1/D x.0, tnC1/C
�Z ˛
0
cos.�nC1.˛0//snC1˛ .˛/ d˛0 �˛
2�
Z 2�
0
cos.�nC1.˛0//snC1˛ .˛/ d˛0�
,
´.˛, tnC1/D y.0, tnC1/C
�Z ˛
0
sin.�nC1.˛0//snC1˛ .˛/ d˛0 �˛
2�
Z 2�
0
sin.�nC1.˛0//snC1˛ .˛/ d˛0�
,
where the integration is performed using the discrete Fourier transform. Finally, the local surfacearea variation rs˛.t/ can also be updated by the Adams–Bashforth method:
rsnC1˛ D rsn˛ C�t
2
�3M n
2 �Mn�12
�, (99)
where
M2 D1
2�
Z 2�
0
.rT /˛0 C rHV.˛0, t /Ns˛0 d˛
0. (100)
4.4. Updating the surface phase
In both the hydrodynamic and geometric models, an analysis of the diffusion flux J at small spatialscales yields [27]
J ��a�
Pe Ns3˛ru˛˛˛ . (101)
Hence, we may rewrite Equation (57) as follows:
ut D�a�
Pe Ns4˛u˛˛˛˛ CN2.˛, t /, (102)
whereN2 contains the remaining nonlinear terms. As in the interface evolution, we then use a linearpropagator method
In this section, we investigate the dynamics of multicomponent vesicles, the differential effects oflocal and global inextensibility constraints, and the presence and absence of hydrodynamic forces.We have implemented our code on a computer, which has two quad core Xeon 2.93 GHz processorsand 96 GB of ram. The simulations presented here for the hydrodynamic and geometric models takeapproximately 1 week to complete, whereas that for the global surface area conserved case takesabout one half of 1 week.
5.1. Ellipsoidal vesicle
We consider the evolution of initially ellipsoidal vesicle with long axis perpendicular to the axisof symmetry. We use the following parameter and initial configuration setting: a D 20, spon-taneous curvature H0.u/ D 0, Gaussian bending stiffness g.u/ D 0 and bending stiffnessb.u/ D 2.1 � u/ C u. Also, we use the double well potential function f .u/ D 1=4u2.1 � u/2.The initial ellipsoid is given by:
r.˛, 0/D 1.26 cos.2�˛/
´.˛, 0/D�0.49 sin.2�˛/(104)
where .0 6 ˛ < 1/. Correspondingly, the reduced surface area �3 D 0.7787. The initial surfacephase concentration u is assumed to be a mixture of both phases:
where the average concentration Nu is varied between 0 and 1 to account for different ratios of thesurface phases and the perturbation parameter ı D 0.001.
5.1.1. Locally inextensible vesicles in a fluid. We study the surface phase separation and evolutiontoward equilibrium of a locally inextensible ellipsoidal vesicle containing a 1 : 1 mixture of lipidphases ( Nu D 0.5) with Pe D 1 in an initially quiescent fluid (e.g., far-field velocity u1 D 0). Weuse the time step�t D 1�10�4 andN D 512 grid points. The results are shown in Figures 1 and 2.In Figure 1[a], the vesicle morphologies and the surface concentrations u are shown at the timesindicated (blue and red correspond to the uD 0 and uD 1 phases, respectively). In Figure 1[b], thevesicle is shown in 3D. At early times phase separation rapidly occurs yielding two large regions ofthe u� 1 phase at the vesicle tips where the curvature is larger and the u� 0 phase emerges at thevesicle top and bottom where the curvature is smallest. This models an experimental configurationwhere the temperature or pH is rapidly dropped such that a homogeneous mixture of lipid com-ponents becomes unstable, leading to phase separation of the components. We note that the phaseseparation at early times is consistent with the assigned bending stiffness b.u/ because the u D 1
phase has smaller bending stiffness. Correspondingly, at very early times, there is a large drop inthe total energy and as seen in Figures 1[a] and 2[a]. Both energy components (line energy andbending energy) decrease rapidly, with the line energy decrease being the dominant effect becauseof the phase decomposition. When the uD 1 phases are located at the vesicle tips, this is still a highenergy state for the system because of the line energy associated with the surface transitions betweenordered and disordered states. Accordingly, the system continues to evolve, primarily driven by thetendency of the system to lower its line energy. This is manifest by the transport of the uD 1 phasefrom the vesicle tips to the upper region of the vesicle, whereas the uD 0 phase is correspondinglytransported to the vesicle tips. This transport occurs between times t D 4.0 and t D 5.0 and is asso-ciated with only a small changes in the vesicle shape. Note that the bending energy increases duringthis surface phase transport. At later times, the vesicle adjusts its shape on a much slower time scaleto continue to lower both the line and bending energies. The top of the vesicle becomes negatively
HYDRODYNAMIC AND GEOMETRIC MODELS OF AXISYMMETRIC MEMBRANES 361
[a]0 10 20 30 40 50 60
7
8
9
10
11
12
13
14
15
16
17
18
Time
Tot
al E
nerg
y
0
1A: t = 0
0 10
1B: t = 4.0
0
1G: t = 20.0
0 10
1H: t = 40.0
0 10
1I: t = 55.4
0 10
1C: t = 4.7
0 10
1E: t = 4.9
0 10
1D: t = 4.8
0 10
1F: t = 5
A
B
D
C
FG H
E
I
[b]
0 1
0 1
Figure 1. Multicomponent vesicle evolution with a 1 : 1 mixture of lipid phases with local constraints:Nu D 0.5 with � D 0.1 and Pe D 1. [a] The evolution of the total energy EM . Insets: cross-section ofvesicle morphologies and surface phase concentration u at indicated times. (blue and red correspond to theu D 0 and u D 1 phases, respectively). The stars (*) on the vesicle morphologies indicate the south pole
where ˛ D 0. [b] The 3D vesicle morphologies at the corresponding times.
curved as the vesicle takes on an experimentally observed stomatocyte shape. The correspondingnormal velocities are shown in Figure 2[b]. At early times, the largest (in magnitude) velocitiesoccur near the vesicle tips and tend to round off the tip. Away from the tip, the negative normalvelocities create the negatively curved regions. At the current resolution N D 512, the simulationbreaks down shortly after time t D 55.4. A larger number of grid points is needed to simulate theevolution further in time. We did not use this finer resolution because of the computational expense.In this simulation, the error in the enclosed volume is 1� 10�4%, and error in the total surface areais 2� 10�8%.
5.1.2. Global surface area conserved vesicles subject to hydrodynamic forces. Next, we examinethe effect of global versus local constraints on the area evolution on the dynamics of the vesicleshown in Figure 1. In Figures 3–5, the dynamics for the locally and globally constrained vesiclesare compared. All the parameters are the same as that in Figure 1. At very early times, the phasesdecompose into regions of u � 0 and u � 1 at approximately the same times and locations for thetwo models. Once the surface phases have fully separated, the globally constrained vesicle evolvesmore rapidly to the stomatocyte shape. In particular, as seen in Figure 4[a], the surface phases tendto be transported rapidly longer than the vesicle membrane in the global surface area conserved case,producing the u � 1 phase located at the vesicle top earlier than that for the locally inextensible
A: t = 0B: t = 4.0F: t = 5.0G: t = 20H: t = 40I: t = 55.4
Figure 2. [a] The energies EM , Eb , ET as a function of time [b]: The normal velocity at different timesfor the multicomponent vesicle shown in Figure 1.
vesicle. Correspondingly, the magnitude of the normal velocity is larger for the globally inextensi-ble vesicle, as seen in Figure 4[b]. In Figure 5, the amount of local stretching that occurs during thedynamics of the globally constrained vesicle is plotted. In particular, the quantity 1
r.rT /s CHV
is shown at different times during the evolution. Note that this term is equal to zero for a locallyinextensible vesicle. The interface is compressed where 1
r.rT /s CHV < 0 and expands in regions
where this quantity is positive. Observe that at early times, the tips of the vesicle are compressed,whereas regions close to the tips expand to compensate. At later times, most of the expansion occursnear the top of the vesicle, whereas the vesicle sides experience compression. As expected, the mag-nitude of the stretching decreases as the vesicle tends toward equilibrium. Finally, the total error inthe enclosed volume of global surface area conservation model is 4%, and the error in the total sur-face area is 4� 10�2%. Compared with the errors from the locally inextensible model, the errors involume and surface area are much larger for the global area conserved model. This is because whenthe volume and surface area are calculated for the Lagrange multiplier constraints, the quantity rs˛is used. Unlike the locally constrained case, this quantity is not constant in the globally constrainedcase and thus additional errors arise in the calculation of this term, which lead to the observed errorsin the volume and surface area.
5.1.3. Comparison of geometric and hydrodynamic models. We next investigate the effect of hydro-dynamics on the dynamics of locally inextensible vesicles. We compare the results using thegeometric model described in Section 3.2 with the results from Section 5.1.1, where hydrodynamicforces are taken into account. We note that the geometric model introduces separate mobility param-eters ˇV , ˇT for the normal and tangential velocities. These are chosen to match the maximummagnitude of the initial velocity with that from the model with fluid flow. The results are given inFigures 6[a,b], which show the vesicle morphologies and normal velocities, respectively. As seen inFigure 6[a], at very early times, the evolution of the geometric and hydrodynamic models are quite
HYDRODYNAMIC AND GEOMETRIC MODELS OF AXISYMMETRIC MEMBRANES 363
0 5 10 15 20 250
2
4
6
8
10
12
14
16
18
Time
Ene
rgy
Local inextensibilityGlobal surface area conservation
[a]
−2 0 2−1
0
1T =0
−2 0 2−1
0
1T =3.5
−2 0 2−1
0
1T =5
−2 0 2−1
0
1T =25
−2 0 2−1
0
1
Local at T =55.4Global at T = 25
Local inextensibilityGlobal surface area
conservation
[b]
Figure 3. The effect of global surface area conservation. A comparison of the 1 : 1 multicomponent vesiclewith two differently constrained models. The global surface area conserved model (red-dashed) and thelocally inextensible model (blue-solid) from Figure 1. [a] The total energies EM , ET and Eb . [b] Slices of
the vesicles at the times indicated.
similar. However, at later times, vesicle subject to hydrodynamic forces evolves much more rapidlyto the stomatocyte shape than the vesicle evolving according to the geometric model. The geomet-ric model reaches equilibrium only around t � 100. A comparison of the corresponding normalvelocities at different times is shown in Figure 6[b]. At very early times, the geometric model haslarger (in magnitude) velocities near the vesicle tips, indicating a more rapid retraction. However, atlater times, the magnitude of the velocity in the geometric model becomes smaller than that in thehydrodynamic model. This results in the slower dynamics that is observed.
In Figure 7, the total energy EM , the line energy ET and the bending energy Eb are showntogether for the geometric and hydrodynamic models. The evolution of the energy and its compo-nents are very similar for the two models. Consistent with the evolution of the vesicle morphologies,the rate of decrease of the energy is faster for the hydrodynamic model than for the geometric model.That is, the presence of hydrodynamic forces enables more rapid energy dissipation. The errors inthe enclosed volume and the total surface area of the geometric model are 4�10�4% and 2�10�8%,respectively, which are quite similar to the errors incurred in the hydrodynamic model with locallyinextensible vesicles.
5.2. Experimental system
We next study an experimental system considered in [24]. This paper presented experimental vesi-cle shapes, estimates of bending stiffnesses, line tensions and comparisons of experimental and bestfit analytical results obtained by solving the Euler–Lagrange equations for equilibrated phase sep-arated binary axisymmetric vesicles with a single phase boundary on the vesicle interface. In theexperimental image shown in Figure 8[a] (which is adapted from Figure 2[A] of [24]), the best fitparameters were found to be �3 D 0.76, b.0/=b.1/ D 6.1, .g.1/ � g.0//=b.1/ D 4 � D 66˙ 13and Nu D 0.56. In Figure 8[a], the red region is the liquid disordered phase enriched with lipids
Figure 4. [a] A comparison of the surface phase concentrations u from Figure 3. [b] A comparison of thenormal velocities from Figure 3. In both [a] and [b]: red-dashed, global surface area conserved; blue-solid,
locally inextensible.
dioleoyl phosphatidylcholine (DOPC), and the blue region is the liquid ordered phase enriched withegg sphingomeylin (egg SM), and cholesterol (CH). Here, we investigate the dynamics with theseparameters using the following initial condition:
r.˛, 0/D 0.62 sin.2�˛/
´.˛, 0/D�1.97 cos.2�˛/
emerges at the vesicle top and bottom where the curvature is smallestfor which the reduced volume�3 D 0.76, which matches that in the experiment (where volume and surface area are V D 3.21and A D 12.67). We use the average concentration Nu D 0.56 and take a D 437.6 correspondsto � D 51.57, the bending coefficient b.u/ D 6.1 � u/ C u and Gaussian bending coefficientg.u/D�5.6.1� u/� 2u. We use the time step �t D 2� 10�4 and N D 512 grid points. Also, weuse the small scaling thickness parameter � D 0.1.
HYDRODYNAMIC AND GEOMETRIC MODELS OF AXISYMMETRIC MEMBRANES 365
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−6
−5
−4
−3
−2
−1
0
1
2
3
α
(1/r
) (r
T) s
+H
V: i
next
ensi
ble
term
T = 0T = 3.5T = 5.0T = 25.0
Figure 5. Local stretching of the globally surface-conserved vesicle interface. The term 1r.rT /s CHV is
plotted at different times, as indicated.
−2 0 2−1
0
1T =0
−2 0 2−1
0
1T =4
−2 0 2−1
0
1T =7
−2 0 2−1
0
1T =55.4
−2 0 2−1
0
1
Fluid model at T =55.4 Geometric model at T = 90.0
Fluid model
Geometric model
[a]
0 0.5 1−0.5
0
0.5T =0
0 0.5 1−0.5
0
0.5T =4
0 0.5 1−0.5
0
0.5T =7
0 0.5 1−0.5
0
0.5T =55.4
0 0.5 1−0.5
0
0.5
Fluid model at T =55.4Geometric model at T = 90.0
Fluid model
Geometric model
[b]
Figure 6. The effect of hydrodynamic forces. A comparison of [a] vesicle morphologies and [b] the normalvelocities for the geometric model (red), with mobilities ˇV D 0.0162 and ˇT D 0.0169, and the vesicle in
the presence of fluid flow from Figure 1 (blue).
There is a good agreement between the simulation and experimental results with the neckregions being more smoothed in the simulation than in the experiment because of the finite inter-face thickness �. By reducing the interface thickness �, the transition region can be made sharper(not shown here).
Figure 7. A comparison of the line energyET and bending energyEb for the geometric and hydrodynamicresults shown in Figure 6.
[a]
[b]−1 0 1
−3
−2
−1
0
1
2
3
t = 0
−1 0 1
−3
−2
−1
0
1
2
3
t = 20
−1 0 1
−3
−2
−1
0
1
2
3
t = 50.2
Figure 8. Comparison between model and experiment. [a] A photon microscopy image of an axially sym-metric vesicle with fluid phase coexistence, adapted from Figure 2[a] in [24] (Red, liquid disordered phases,fluorescence dye lissamine rhodamine DPPE; blue, liquid ordered phases, fluorescence dye perylene).
[b]: cross sections of computational vesicle morphologies at the times indicated.
6. CONCLUSIONS AND FUTURE WORK
We have presented a study of the dynamics of 3D axisymmetric multicomponent locally inextensiblevesicles with and without hydrodynamic forces. We also considered a case in which the global sur-face area is conserved. The models were derived using an energy variation approach that accountedfor different surface phases (e.g., ordered/disordered), the excess energy (line energy) associatedwith surface phase domain boundaries, bending energy, inextensibility and fluid flow via the Stokesequations. The equations are high-order (fourth order) owing to bending forces and nonlinear and
HYDRODYNAMIC AND GEOMETRIC MODELS OF AXISYMMETRIC MEMBRANES 367
nonlocal owing to the inextensibility of the membrane and the incompressibility of the fluid. Tosolve the equations numerically, we used a nonstiff, pseudo-spectral boundary integral method thatrelied on an analysis of the equations at small scales. The algorithm is an extension of Sohn et al.[27] for two dimensional multicomponent vesicles.
We have found that compared to a geometric model, hydrodynamic forces provide additionalpaths for relaxing inextensible vesicles. As a consequence, multicomponent inextensible vesicles ina fluid evolve more quickly to an equilibrium state than do vesicles evolving according to the geo-metric motion law. In addition, the combination of hydrodynamic forces and a global surface areaconstraint provides the most rapid evolution towards equilibrium, although it is not clear whether aphysical mechanism exists to permit such an evolution. Although the vesicles simulated here usingthe different models all tend to the same equilibrium state, it is known that multiple equilibriumstates exist [42]. Therefore, the presence of hydrodynamic forces or possibly local stretching mayenable the dynamics to overcome local energy barriers and reach lower energy states than thoseaccessible by geometric motion or energy minimization algorithms. Because of the intimate con-nection between morphology, surface phase distribution and biological function, these results haveimportant consequences in understanding the role vesicles play in biological processes.
There are many interesting directions in which this work may be extended in the future. Oneimportant direction concerns the development of more physical models that incorporate the effectof local stretching (e.g., by a local elasticity law), but allow global surface area variation. Anotherimportant direction involves modeling the effect of membrane-bound proteins on the dynamics. Forexample, membrane-bound proteins within the phases can also have an important influence on thevesicle shape, curvature and surface phase distribution, for example, [43,44]. The models presentedcan be extended to incorporate the concentration(s) of membrane proteins in the phase-field modeldescribed here via an additional protein energy and introducing coupling between the protein andthe bending rigidity and spontaneous curvature (e.g., [45, 46]). Because of its biological relevance,another interesting direction includes simulating multiple, multicomponent vesicles in a channelgeometry; see [47] for the dynamics of homogeneous vesicles in complex geometries.
ACKNOWLEDGEMENTS
The authors thank Qing Nie and Axel Voigt for valuable discussions. The authors also thank the anonymousreferees for their comments, which have improved the manuscript. The authors gratefully acknowledgesupport from the National Science Foundation, Division of Mathematical Sciences (DMS).
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