Journal of Computational Physics 273 (2014) 143–159
Contents lists available at ScienceDirect
Journal of Computational Physics
www.elsevier.com/locate/jcp
An immersed boundary method for endocytosis
Yu-Hau Tseng a , Huaxiong Huang b,∗
a
b
Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA
Department of Mathematics and Statistics, York University, Toronto, M3J 1P3, Canada
a r t i c l e
i n f o
Article history:
Received 12 December 2013
Received in revised form 5 May 2014
Accepted 6 May 2014
Available online 20 May 2014
Keywords:
Immersed boundary method
Endocytosis
Incompressible viscous fluids
Membrane
Canham–Helfrich Hamiltonian
a b s t r a c t
Endocytosis is one of the cellular functions for capturing (engulfing) vesicles or microorganisms. Understanding the biophysical mechanisms of this cellular process is essential from
a bioengineering point of view since it will provide guidance for developing effective
targeted drug delivery therapies. In this paper, we propose an immersed boundary (IB)
method that can be used to simulate the dynamical process of this important biological
function. In our model, membranes of the vesicle and the cell are treated as Canham–
Helfrich Hamiltonian interfaces. The membrane-bound molecules are modeled as insoluble
surfactants such that the molecules after binding are regarded as a product of a “chemical”
reaction. Our numerical examples show that the immersed boundary method is a useful
simulation tool for studying endocytosis, where the roles of interfacial energy, fluid flow
and viscous dissipation in the success of the endocytosis process can be investigated in
detail. A distinct feature of our IB method is the treatment of the two binding membranes
that is different from the merging of fluid–fluid interfaces. Another important feature of
our method is the strict conservation of membrane-borne receptors and ligands, which is
important for predicting the dynamics of the endocytosis process.
© 2014 Elsevier Inc. All rights reserved.
1. Introduction
Endocytosis [4], the process of the engulfing of molecules, proteins, and other particles into the host cells [1,18], is an
essential function for maintaining cellular homeostasis. Fig. 1 is a schematic of the basic steps involved in the process.
Endocytosis is used for the transport of neural, metabolic, and proliferative signals [13,25]; the uptake of many essential
nutrients; the regulated interaction with the external world; and the ability to mount an effective defence against invading
microorganisms. According to the size of the engulfed target, there are four categories of endocytosis pathways, namely,
clathrin-mediated endocytosis, caveolae, pinocytosis (cell drinking), and phagocytosis (cell eating). In more recent experiments, it has been suggested that endocytosis can be classified based on whether the process is clathrin-dependent.
It is well known that at the micrometer scale, resistance of water (due to its viscosity) could be significant to hinder the
motion of microorganisms. However, existing theoretical studies on endocytosis are based on the energy of the equilibrium
state of membranes (static problems) without the consideration of fluid flow [3,9,19,20,27]. Therefore, an important question
is how the endocytosis process [16,18] is affected by the presence of water. In this paper we develop a numerical method
that can be used to study the roles of fluid (water) as well as membrane properties in the engulfing stage of endocytosis
[6,7].
*
Corresponding author.
E-mail address: [email protected] (H. Huang).
http://dx.doi.org/10.1016/j.jcp.2014.05.009
0021-9991/© 2014 Elsevier Inc. All rights reserved.
144
Y.-H. Tseng, H. Huang / Journal of Computational Physics 273 (2014) 143–159
Fig. 1. Schematic diagram of endocytosis.
From the computational point of view, we solve the two-dimensional incompressible Navier–Stokes equations with an
immersed cell membrane and a vesicle. We assume that each individual membrane (lipid bilayer) can be represented by
an interface with zero thickness while the bound region is composed of an overlap of two interfaces, a “bilayer” or “dualinterface” structure. In our immersed boundary formulation, interfaces are represented by the usual singular delta functions,
centered at the immersed boundary points [22,23]. In the basic version of our method, when the immersed boundary points
from two different interfaces come close, a rigid connection forms and the two interfaces move with the same velocity. The
second version of our method assumes that the receptor molecules and ligands [1,9] on the cell and vesicle membranes
move along the interfaces due to molecular diffusion and fluid convection. They are modeled as insoluble reactive surfactants and the interface energy reduction due to the binding of receptors and ligands is proportional to the reacted quantity.
Physically the engulfing process is driven by the reduction of interface energy in the bound region [27]. This mechanism
induces changes of both normal and tangential stresses by a jump in the surface energy at the “triple junction”, where the
two interfaces join. Our numerical results show that for endocytosis to occur, this stress difference cannot be balanced by
the buildup of the membrane elastic forces. As a result, it leads to fluid motion and transfer of interface energy to kinetic
energy of the fluid and thermal energy (through viscous dissipation). In other words, when interfacial energy is reduced due
to binding, only part of it is stored as bending energy in the membranes due to increased curvature in the bound region and
near the triple junction. The remainder is utilized to drive the fluid flow in the form of a Marangoni force. Therefore, the
equilibrium energy argument only tells part of the story and a complete picture of endocytosis must include the dynamics
of the engulfing process. Our paper represents the first step towards the building of a more complete theory.
The rest of this paper is organized as follows. In Section 2, a mathematical model based on the Navier–Stokes equations
and molecule concentration equations is presented, and the interfacial forces are derived from the interfacial energy and
membrane model. The numerical aspect of the immersed boundary method is given in Section 3. Numerical examples are
presented in Section 4. Computation is also carried out to investigate the distribution of energy consumption. We finish our
paper with a short conclusion in Section 5.
2. Mathematical model
In this section, we present our mathematical model in an immersed boundary (IB) formulation [22] which consists of
fluid, fluid–interface interaction, and receptor and ligand concentration equations. We assume that the fluid is Newtonian
and incompressible. The interfacial forces are composed of tension-type forces (adhesion energy and membrane elasticity)
and bending forces. For membrane-bound (receptor and ligand) molecules, the reaction (binding) occurs on the edge of the
bound region and obeys the law of mass action.
The mathematical model is based on conservation laws, including conservations of mass and momentum of fluid, and
conservation of interfacial molecules. Since the membrane is treated as an elastic interface which is embedded in a Newtonian incompressible fluid, the corresponding governing equations are
∂u
+ (u · ∇)u + ∇ p = ∇ · (2μ D ) + f ,
∂t
∇ · u = 0,
ρ
(1)
(2)
where μ and ρ are fluid viscosity and density, respectively, u is the velocity field, p is the pressure, f is the forcing term,
and the rate of strain D = 12 (∇ u + ∇ u T ). For simplicity, we assume that the size of the engulfed target (vesicle) is much
smaller than that of the cell. Therefore the problem can be simplified as an initially flat membrane engulfing a circular
vesicle, see Fig. 2. The membranes are modeled as elastic interfaces that are immersed in and carried by the fluid. The
fluid velocity is continuous at these interfaces. The other condition at the interfaces (membranes) is the balance of the
Y.-H. Tseng, H. Huang / Journal of Computational Physics 273 (2014) 143–159
145
Fig. 2. Schematic setup of two-dimensional endocytosis with boundary conditions. (For interpretation of the references to color in this figure, the reader is
referred to the web version of this article.)
stresses in the tangential direction [2μ D · τ ]Σ = − F T , and in the normal direction [(− p I + 2μ D ) · n]Σ = − F N . Note that
these so-called jump conditions are not directly imposed in the immersed boundary formulation. The interfacial force F and
other details will be given below. Boundary conditions of the computational domain are also shown in Fig. 2, and a small
bound region (light blue area) is assumed at the beginning of the engulfing process.
From the energy point of view, the interfaces (vesicle and cell membranes) tend to evolve to a preferable state which
has a relative lower energy in the system. Due to the characteristics of lipid bilayers, the total interfacial energy E Σ consists
of an interfacial bending energy E b , a membrane adhesion energy − E a , and an interfacial elastic energy E λ . In this paper,
we adopt the Canham–Helfrich Hamiltonian [10,28,29]
E Σ = Eb − Ea + E λ,
(3)
where the negative sign in front of adhesion energy [27] plays an important role for the reduction of E Σ .
In two-dimensional space, interfaces of the vesicle and membrane are denoted respectively by Σ v and Σ m , and can be
parameterized as
Σ i = X i (s, t ) X i (s, t ) = X i (s, t ), Y i (s, t ) 0 ≤ s ≤ L i (t ) ,
i = v , m.
In each interface, s represents the arc-length which is a time-dependent
variable, we choose a time-independent variable
α ∈ [0, 1] for computational sake, the stretching factor S α (α , t ) = X α2 + Y α2 is defined as a Jacobian between s and α . The
corresponding unit tangent vector is evaluated by
1
τ (s, t )
=
τ= 1
τ2 (s, t )
Sα
X α (α , t )
Y α (α , t )
(4)
,
and the unit normal vector n is simply (τ2 , −τ1 )t which points outward for a simple closed curve in counterclockwise order.
The total interfacial energy of Σ = Σ v ∪ Σ m ∪ Σ a (Σ a represents the bound region) is expressed as
EΣ =
1
b(c )
2
2
κ − κ̃ (c ) ds −
Σa
Σ
γ (c )ds +
λ(s, t )ds,
(5)
Σ
where c is the molecular concentration which is a function of s and t. The adhesion energy density γ and bending rigidity b may depend on the molecule concentration. In general, the surface may have a naturally deformed state which can
be described by the effect of the spontaneous curvature [17], κ̃ = κ̃ (c ), a function which may depend on the molecular
concentration. Notice that adhesion energy is defined only on Σ a . For simplicity, we assume that both bending rigidity
and spontaneous curvature are independent of the molecular concentration in this paper. Due to the binding process between vesicle and cell membranes, interfacial energy inside the bound region is lower, which is reflected by the increase of
adhesion energy. In the last term, λ(s, t ) is the interfacial tension for elastic membranes.
The corresponding interfacial forces from (5) are derived by the variational method and the details are provided in
Appendix A. The bending force is given by
2
∂ 2 X sp
∂2
∂ X
.
F b c (s, t ), t = − 2 b(s)
−
∂s
∂ s2
∂ s2
(6)
Here, we assume that the effect of spontaneous curvature κ̃ is written as the second derivative with respect to the arc-length
s, of a presumed interface configuration X sp . Similarly, the adhesion force and the elastic tension are expressed by
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Y.-H. Tseng, H. Huang / Journal of Computational Physics 273 (2014) 143–159
∂X
∂
,
F γ c (s, t ), t =
γ c (s, t )
∂s
∂s
∂
∂X
λ(s, t )
,
F λ (s, t ) =
∂s
∂s
(7)
(8)
where the adhesion energy density, γ , is a concentration-dependent function. Instead of treating the membrane as inextensible with λ a Lagrange multiplier, we use an elastic model λ(s, t ) = λ0 ( S α (α , t ) − S α (α , 0)) J and a linear model ( J = 1)
is considered in this paper, while a comparison with a strain-hardening nonlinear model ( J = 3) is given in a subsequent
paper [26]. With this, the interfacial force can be written as f = f s + f b where
fs=
(− F γ + F λ )δ(x − X ) ds,
(9)
Σ
fb =
F b δ(x − X ) ds.
(10)
Σ
Here δ(x − X ) is the singular Dirac delta function.
Next, we consider the mass conservation of interfacial molecules (receptors and ligands). The function c (s, t ) is the
molecular concentration on the interface, and the total mass is given by M (t ) = Σ c (s, t )ds. Using the lemma in [12] and
the evolution of an interface element in two-dimensional space ∂t S α = (∇s · u ) S α , and following the derivation in [14], the
corresponding differential equation of c (s, t ) is
Dc
Dt
+ (∇s · u )c = 0.
(11)
Since there are three species in the system, we use superscripts to distinguish them. The concentration defined on the
vesicle is given by c v , the one on the cell membrane is denoted by cm , and ca is the concentration of the bound surface
molecules. The concept of molecular concentration on a membrane is similar to that of surfactant concentration, so we
borrow the idea from the reaction among several species of surfactant, and treat the binding process as a chemical reaction
between c v and cm . For simplicity, the law of mass action is considered, and the corresponding chemical reaction formulae
are
+
mk
a
cv + c c
⇒
k−
⎧d v
v
+ v m
− a
⎪
⎨ dt c = q = −k c c + k c
d m
m
+
v m
c = q = −k c c + k− ca
dt
⎪
⎩d a
c = qa = k + c v c m − k − c a ,
dt
(12)
where k+ and k− are the reaction rate and reverse reaction rate, respectively. Furthermore, the molecules can either be
carried by the fluid flow or move along the interface due to diffusion. Combining the reaction effect into Eq. (11), the
convection–diffusion–reaction equations of molecular concentrations are
Dc v
Dt
Dcm
Dt
Dca
Dt
+ (∇s · u )c v = ∇s · D v ∇s c v + q v ,
(13)
+ (∇s · u )cm = ∇s · D m ∇s cm + qm ,
(14)
+ (∇s · u )ca = ∇s · D a ∇s ca + qa ,
(15)
where D v , D m , and D a are the diffusivity in each region of the interfaces as follows:
D v (s) = D v ,o ,
s ∈ unbound region
D v (s) = D v ,a ,
s ∈ bound region
D (s) = D
m
m,o
, s ∈ unbound region
D m (s) = D m,a ,
s ∈ bound region
D a ( s ) = 0,
s ∈ unbound region
D a (s) = D a,a ,
s ∈ bound region.
(16)
Notice that the product produced by chemical reaction exists only in the bound region, so that the diffusivity D a is set be
zero in the unbound region.
Moreover, the constitutive relation to specify the adhesion energy density γ in terms of molecular concentration c is
needed in the energy reduction due to the molecular binding process. Due to fluid convection, interfacial molecules may
Y.-H. Tseng, H. Huang / Journal of Computational Physics 273 (2014) 143–159
147
aggregate on the interface. However, such an aggregation should be restricted due to the limitation of surface space area. In
this paper, we use a linear equation of state:
γ ca =
β ca , ca ≤ ca∞
β ca∞ , ca > ca∞ ,
(17)
where β is the effective coefficient and ca∞ is the maximum concentration.
3. Numerical method
In the membrane-bound receptor–ligand binding process, we use a two-layer structure which keeps the original interfacial configurations of both the membrane and the vesicle in the bound region. This strategy avoids the complexity
of removing immersed boundary points of the membrane portion and dealing with the boundary conditions at the triple
junction that separates the bound and unbound regions of the membranes. Under the two-layer structure, the interfacial
properties in the bound portion are determined by a linear combination of properties on each interface. Following the idea
from our previous work on surfactants [14], the ligand and receptor molecules are treated as insoluble surfactants and the
membrane-bound molecules are regarded as a product after a chemical reaction. We use a semi-implicit finite-difference
scheme that automatically satisfies discrete mass conservation.
In our computations, physical quantities such as characteristic vesicle length L̄, fluid viscosity μ̄, density ρ̄ , adhesion
energy density γ̄ , bending rigidity b̄, and molecule concentration c̄ are selected based on those relevant to endocytosis. The
characteristic velocity Ū is estimated by balancing the adhesion energy and viscous dissipation. Using these characteristic
quantities, we introduce the following transformation between dimensionless and dimensional variables
x = L̄x ,
u = Ū u ,
μ = μ̄μ ,
ρ = ρ̄ρ ,
γ = γ̄ γ ,
b = b̄b ,
c = c̄c ,
t = L̄ Ū −1 t and p = μ̄Ū L̄ −1 p . Substituting these into (1), (2), (13), (14), and (15), and omitting the superscript , the corresponding
dimensionless governing equations are as follows
∂u
1
1
1
1
+ (u · ∇)u + ∇ p = u +
fs+
f b,
∂t Re
Re
Re Ca
Re Bn
f s = (− F γ + F λ )δ(x − X ) ds,
(18)
(19)
Σ
fb =
F b δ(x − X ) ds,
(20)
Σ
Dc i
Dt
+ (∇s · u )c i =
1
Pes
∇s · D i ∇s c i + q i ,
for i = v , m, a,
(21)
where q v = K d ca − K r cm c v , qm = q v , and qa = −q v . The dimensionless numbers are
Re =
ρ̄ L̄ Ū
,
μ̄
Ca =
μ̄Ū
,
γ̄
Bn =
μ̄ L̄ 2 Ū
b̄
,
Pes =
L̄ Ū
D̄
,
Kd =
L̄k−
Ū
,
Kr =
L̄ c̄k+
Ū
.
(22)
Here, the Reynolds number Re represents the ratio between fluid inertia and viscous stress, capillary number Ca is the ratio
of viscous stress and interfacial tension, bending number Bn measures the stiffness of the interface, and the Péclet number
Pes measures the relative importance between convection and diffusion. Motion of the interface is determined by the fluid
flow with interfacial velocity U (a two-component vector (U , V )) which is related to bulk fluid velocity u (a vector field of
the form (u , v )) so that the equation of motion is
∂X
= U (s, t ) =
∂t
u (x, t )δ(x − X ) dx.
(23)
Ω
The IB method [22–24] is a hybrid method using the Eulerian and Lagrangian representations. In the fixed coordinates,
fluid variables are defined on a staggered marker-and-cell (MAC) mesh introduced by Harlow and Welch [11]. The pressure
is defined on the grid points labeled as (xi , y j ) = ((i − 1/2)h, ( j − 1/2)h) for i , j = 1, 2, . . . , N, the velocity components u and
v are defined at (xi +1/2 , y j ) = (ih, ( j − 1/2)h) and (xi , y j +1/2 ) = ((i − 1/2)h, jh), respectively, where h = x = y is chosen
on a domain Ω . In addition, the body force components are defined at the same locations as the velocity components. For
each immersed interface, we choose a fixed parameter α ∈ [0, 1] for convenience, and use a collection of discrete points
αk = kα , k = 0, 1, . . . , M such that the Lagrangian immersed boundary points are denoted by X k = X (αk ) = ( Xk , Y k ),
where α = 1/ M. In addition to immersed boundary points with integer indices, we also use points with half-integer
indices, X k+1/2 = X (αk+1/2 ) = ( X k+1/2 , Y k+1/2 ), on which molecule concentration c and adhesion energy density γ are
stored.
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Y.-H. Tseng, H. Huang / Journal of Computational Physics 273 (2014) 143–159
Let t be the time step size, and n be the time step index. At the beginning of each time step, the variables X nk =
X (αk , nt ), un = u (x, nt ), pn = p (x, nt ), and c (α , nt ) are given. The variables at the next time step (n + 1) are obtained
via numerical integration in six steps.
τ = (τ1 , τ2 ) of the interface as
2 2
= ( D α X )nk+ 1 = α −1 Xkn+1 − Xkn + Y kn+1 − Y kn ,
1. Compute the unit tangent
( S α )nk+ 1
2
(24)
2
τ nk+ 1 = ( S α )nk+ 1
2
− 1
2
which hold for
( D α X )nk+ 1 = 2
X nk+1
−
X nk
( Xkn+1 − Xkn )2 + (Y kn+1 − Y kn )2
,
(25)
αk+1/2 = (k + 1/2)α , k = 0, . . . , M − 1. The bending rigidity bkv+1/2 and bm
are constants in the
k+1/2
based on linear beam theory is used for the bound region. The
unbound region, while a simple addition bkv+1/2 + bm
k+1/2
adhesion energy density γ can be evaluated by (17) at half-integer points for each region of the interface: i = v , m, a
γ cki + 1 =
2
⎧ i i
i
⎨ β ck+ 1 , cki + 1 ≤ c ∞
2
2
i
⎩ β i c∞
,
ci
k+ 12
(26)
i
> c∞
.
Note that the superscript for the time step has been dropped here to avoid complicated notations. We now compute
the interfacial forces at the immersed boundary points X k , k = 1, . . . , M. First, the forces due to interfacial tension,
adhesion, and membrane elasticity are evaluated by
( F s )nk =
1 α
2
where F s = F λ − F γ and
points by
∂2 X
Gk = b 2
∂s
n
σkn+ 1 τ nk+ 1 − σkn− 1 τ nk− 1 S α−1 k ,
2
k
(27)
2
σ = −γ + λ0 ( S α − S α0 ). For bending forces, an intermediate variable G is evaluated at integer
=
2
bk − 1 + bk + 1
τ k+ 1 − τ k− 1
( S α )k− 1 + ( S α )k+ 1
α
2
2
2
2
2
2
,
(28)
and the forces are then computed by a central difference formula
n
2 n
−1 n
∂ G
∂ −1 ∂ G
( F b )nk = −
=
−
S
Sα k
α
2
∂α
∂α
∂s k
k
n
n
G nk − G nk−1 −1 n
−1 G k+1 − G k
=
−
Sα k .
( S α )n 1
α 2 ( S α )n 1
k+ 2
(29)
k− 2
2. Spread interfacial forces from the immersed boundary points to the fluid by
F=
1
Re Ca
f n (xi , j ) =
Fs +
M
1
Re Bn
(30)
F b,
F nk δh xi , j − X nk α ,
(31)
k =1
where a smooth delta function is given by
⎧
|r |
|r |
|r |2
1
⎪
(
3
−
2
+
1 + 4 h − 4 h2 ),
if 0 ≤ |r | ≤ h,
⎪
h
⎨ 8h
2
δh (r ) = 1
|r |
|r |
|r |
⎪
⎪ 8h (5 − 2 h − −7 + 12 h − 4 h2 ), if h ≤ |r | ≤ 2h,
⎩
0,
otherwise.
(32)
3. Solve the Navier–Stokes equations by the following second-order projection method [8]. The time derivative is approximated by a second-order backward difference formula and the nonlinear term is approximated by extrapolation using
the values from the previous time steps. The incremental pressure-correction projection method can be written as
follows
3u ∗ − 4un + un−1
h φ =
2t
3
2t
+ A h u n +1 = −
∇h · u ∗ ,
1
Re
∇h pn +
∂φ
= 0 on ∂Ω,
∂n
1
Re
h u ∗ + f n ,
(33)
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Y.-H. Tseng, H. Huang / Journal of Computational Physics 273 (2014) 143–159
u n +1 = u ∗ −
2t
3
p n +1 = p n + φ −
∇h φ,
1
Re
149
(35)
∇h · u ∗ .
(36)
Here, ∇h and h are discrete gradient and Laplace operators defined on a staggered grid, and the convection term is
given by A h un+1 = 2(un · ∇h )un − (un−1 · ∇h )un−1 . One can see that the above Navier–Stokes solver involves solving the
two Helmholtz equations for the velocity u ∗ = (u ∗ , v ∗ ) and one Poisson equation for the potential function φ . These
elliptic equations are solved by FISHPACK.
4. Interpolate the updated velocity from the fixed grid points onto the moving immersed boundary points to obtain interfacial velocity U k = (U k , V k ) and move immersed boundary points X k = ( X k , Y k ) to their new positions.
U nk +1 =
un+1 δh x − X nk h2 ,
x
X nk +1 = X nk + t U nk +1 + U A
k = 0, 1, . . . , M ,
n+1
k
(37)
τ nk , k = 0, 1, . . . , M .
(38)
Here an extra velocity U A is introduced in the tangential direction to maintain uniform distribution of the immersed
boundary points [15]
1
A
A
U (α , t ) − U (0, t ) = α
∂U
· τ dα −
∂ α
0
and
(U A )nk +1
UA
α
∂U
· τ dα ,
∂ α
(39)
0
is computed by the mid-point rule
n+1
= k α
k
M ∂U
∂α
i =0
·τ
n+1
α −
i
k ∂U
∂α
i =0
·τ
n+1
α .
(40)
i
5. Update the location of the triple junction between the bound and unbound membrane regions. First, we find the comof X kv on the membrane, and evaluate the distance H k = | X kv − X m
| between
plementary membrane-bound ligand X m
k
k
receptors and ligands. Let hm be the ligand–receptor binding distance and δm a tolerance parameter used to determine
the binding (reaction). The ligand–receptor pair bond when hm − δm < H k < hm + δm . In reality, these bonds can also
disassociate (break), especially under tension. For simplicity, we assume that once a ligand–receptor pair is formed, the
bond will not break.
6. Update interfacial ligand and receptor concentrations. We use the conservative scheme [14] to maintain strict mass
conservation (in the discrete sense) within round-off errors. Denote respectively c̄k = 12 (ck−1/2 + ck+1/2 ) and D h ck =
1
α (ck+1/2 − ck−1/2 ) as the concentration and the central difference of the concentration at index k. For the vesicle
portion Σ v , we assume that M J is the index of the triple junction and the numerical scheme is
(c v S αv )n+11 − (c v S αv )n
k+ 12
k+ 2
=
1
α
t
v ,n
D k +1
+1
( S αv )kn+
1
+1 v ,n+1
c̄
− (U A , v )kn+1 c̄kv ,n+1
(U A , v )kn+
1 k +1
−
α
v ,n+1
D h c k +1
−
v ,n
Dk
( S αv )kn+1
v ,n+1
D h ck
+ q̃n+11 ,
k+ 2
(41)
for k = 0, 1, . . . , M v − 1, where the source term is approximated by
q̃n+11 = k− c
k+ 2
a,n+1
k+ 12
− k+ cm,n1 c v ,n1+1 S αv ,n+1 ,
k+ 2 k+ 2
and for k = 0 to k = M J − 1, we have
(ca S αv )n+11 − (ca S αv )n
k+ 12
k+ 2
=
1
t
a,n
D k +1
−
+1 a,n+1
c̄
− (U A , v )kn+1 c̄ka,n+1
(U A , v )kn+
1 k +1
a,n+1
D c
−
n +1 h k +1
α ( S αv )
k +1
α
a,n
Dk
( S αv )kn+1
a,n+1
D h ck
− q̃n+11 .
k+ 2
(42)
The scheme for molecule concentration equations on the cell membrane is similar and will not be repeated here.
In the numerical schemes for the molecular concentration equations, mass conservation in the discrete form can be
verified by taking summation by part through (41) and (42),
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Y.-H. Tseng, H. Huang / Journal of Computational Physics 273 (2014) 143–159
Table 1
Parameters in the system.
k+
k−
D v ,a
D v ,o
D m,a
D m,o
D a,a
D a,o
β
o
x
o
o
o
o
o
x
o
v
M
−1
c
v
α=
v
M
−1
2
k =0
m
M
−1
k =0
n+1
+ ca k+ 1 ( S α )n+11 k+
2
c
m
k =0
n+1
+ ca k+ 1 ( S α )n+11 k+
2
α=
m
M
−1
2
k =0
n
c v + ca k+ 1 ( S α )n 1 α ,
k+ 2
2
n
cm + ca k+ 1 ( S α )n 1 α .
k+ 2
2
(43)
(44)
This is the discrete version of mass conservation
d
c v + ca dS = 0 and
dt
Σv
d
cm + ca dS = 0,
dt
(45)
Σm
which can be derived directly from integrating Eqs. (13)–(15).
Finally, we note that in addition to parameters related to the fluid flow, there are additional parameters in Eqs. (12)–(17).
However, not all parameter are used in this paper. In Table 1, parameters used in the numerical examples are marked as
“o”, otherwise they are marked by “x”.
4. Numerical examples
4.1. Parameter values
The characteristic length L̄ (vesicle diameter) is chosen to be 5 × 10−7 m (500 nm), which is in the range associated
with endocytosis. For fluid properties, we use those for water, i.e., density ρ̄ = 1000 kg/m3 and viscosity μ̄ = 0.001 Pa s.
The bending energy of the vesicle membrane in [27] varies from several kb T to a few hundreds kb T , where the Boltzmann
constant kb ∼ 1.38 × 10−23 J/K and T is the absolute temperature. For the regular temperature of the human body, T =
310 K, we have kb T ∼ 4.28 × 10−21 J. For endocytosis to occur, the adhesion energy must be greater than the bending
energy. As a base line case, we select a bending energy of magnitude around 5kb T (2.14 × 10−20 J) for the vesicle and
b̄ = 1kb T for the cell membrane, and the corresponding adhesion energy density is γ̄ = 4.28 × 10−7 N/m, which comes
from the balance γ̄ = 25b̄/ L̄ 2 . Due to this setup, the adhesion energy and fluid viscous dissipation dominate the dynamics
of the system so that Ca = μ̄Ū /γ̄ = 1 with a characteristic velocity Ū = 4.28 × 10−4 m/s. The corresponding Reynolds
number is Re = 2.14 × 10−4 and Bn = 25. For comparison purposes, we consider cases with different membrane stiffness,
by increasing or reducing the dimensionless bending rigidity of the membrane, bm (the vesicle has bending rigidity b v ).
Furthermore, the base line characteristic diffusivity of molecules is D̄ = 4.28 × 10−11 m2 /s which results in Pes = 5. Note
that the typical value of diffusivity can vary from 10−8 m2 /s (diffusion dominated) to 10−12 m2 /s (diffusion limited) [2,5].
We also consider several pairs of values of Pes and reaction rate K r to investigate how the diffusivity and reaction rate
affect the engulfing process. All results presented in this section are dimensionless, from which the corresponding physical
quantities can be recovered if needed.
The computations were done using the nondimensional equations and a dimensionless rectangular computational domain
Ω = [0, 2.5] × [−2.5, 2], the center of the initially circular vesicle being placed at (0, 0.3). The membrane is initially bound
with the vesicle at (0, 0.3 − r − hm ), where r is the radius of the vesicle and hm is the presumed bond activation distance. For
boundary conditions, we impose zero Neumann boundary conditions at x = 0 (due to symmetric property) and at x = 2.5,
and no-slip boundary condition on the other boundaries. The mesh size h of the Eulerian coordinates is 0.01, while s
of the interface is 0.75h; under this choice of s, relative error of the vesicle volume is controlled at O (10−3 ) in each
case. We choose the ligand–receptor binding length hm = 0.002 (corresponding to 10−9 m or 1 nm), and binding tolerance
δm = 0.1hm .
Cell membranes are either inextensible or hyper-elastic under normal circumstances. However, experiments suggest that
the binding of the vesicle may alter the structure of the skeleton of the cell and change membrane elastic properties. Since
the main objective of our study is to investigate the role of the fluid in endocytosis, we use a linear membrane elastic
model. A more detailed discussion, including a nonlinear strain-hardening model, will be given in a followup paper [26].
The computations presented in Sections 4.2–4.4 are for immobile surface molecules, while the additional computational
results investigating the roles of surface diffusion, reaction rate and fluid flow are given in Section 4.5.
4.2. Convergence test
Before presenting numerical results, we first examine the convergence properties of our numerical method using grid
refinement analysis. The numerical solution on a fine grid with h0 = 0.005 is selected as the reference, and solutions on
Y.-H. Tseng, H. Huang / Journal of Computational Physics 273 (2014) 143–159
151
Table 2
Convergence test using grid refinement analysis.
hi
uhi − uh0 2
Ratio
v hi − v h0 2
Ratio
| R h i − R h0 |
Ratio
h1
h2
h3
5.05 × 10−2
1.88 × 10−2
8.82 × 10−3
–
2.69
2.13
6.46 × 10−2
3.28 × 10−2
1.46 × 10−2
–
1.97
2.24
0.243
0.114
0.029
–
2.14
3.90
Fig. 3. Role of tangential stress in endocytosis process. (a) Evolution of the engulfing ratio. (b) Velocity field at t = 1.0 without Marangoni force (removing
tangential stress, ∂s σ = 0). (c) Velocity field at t = 1.0 (∂s σ = 0).
coarser grids h1 = 8h0 , h2 = 4h0 , and h3 = 2h0 are compared to the reference solution. Due to a large point-wise error near
the triple junction point, we use the L 2 -norm error of the velocity field in the convergence study. We also calculated the
engulfing ratio, R hi = L e / L v , between the length of the bound region L e to the entire vesicle length L v . Table 2 lists the
L 2 -norm errors of the velocity components u and v, and the corresponding error of R hi at t = 1.0. A ratio of 2 corresponds
to first-order convergence. The results show that the convergence is first order except for R hi .
4.3. Interfacial energy driven endocytosis
It has been proposed that in endocytosis, the reduction of interfacial energy due to the binding of receptors and ligands
is the main driving force [9,27], where the reduction of interfacial tension comes from the adhesion energy. This difference
of the interfacial energy across the bound and unbound regions leads to a Marangoni stress, which triggers the engulfing
process of endocytosis. In this paper, we use the engulfing ratio, which is defined as the fraction of the bound region on
the vesicle relative to the entire vesicle area as a numerical indicator of the endocytosis process.
To validate this argument, we first present a numerical example by turning on and off the Marangoni force. We assume
that the vesicle and the membrane are initially bound in a small region and that the dimensionless adhesion energy density
is γ = 1. The evolution of the engulfing ratio is shown in Fig. 3. The case without the tangential stress is given by the dashed
line in Fig. 3(a), while the solid line shows the evolution with a contribution from the tangential stress. The corresponding
snapshots of the velocity field at t = 1.0 are shown in Figs. 3(b) and 3(c), respectively. Notice that both quiver plots are
re-scaled (the original scale of Fig. 3(b) is two orders of magnitude smaller than that of Fig. 3(c)). The unbalanced normal
stress pulls the vesicle towards the membrane such that the direction of fluid drainage (caused by squeezing) between
vesicle and membrane is perpendicular to the interface movement. Since the interfacial tension is smaller in the bound
region (with value −γ ), the tangential stress points toward the region with a greater interfacial tension, which matches the
fluid drainage direction, as shown by the subplots of Figs. 3(b) and 3(c). In particular, we note the circulation occurring near
the triple junction region in Fig. 3(c).
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Y.-H. Tseng, H. Huang / Journal of Computational Physics 273 (2014) 143–159
Fig. 4. Interface configurations: each row presents four snapshots of the engulfing process up to t = 10.0 for a specific bm (cell bending rigidity) while b v
(vesicle bending rigidity) is fixed.
Fig. 5. Evolution of engulfing ratio. (a) Effect of cell membrane stiffness: b v = 5, bm = 0.1 (circular marker), bm = 0.5 (dashed line), bm = 1 (solid line),
and bm = 2 (dash-dotted line). (b) Effects of vesicle and cell membrane stiffness: b v = 5, bm = 1 (solid line) and b v = 10, bm = 1 (dot line). Note that the
engulfing process is not always smooth. This non-smooth behavior is the consequence of the modeling procedure used in this study. We model the binding
process by measuring the point-wise distance of the markers on both cell and vesicle membranes. When the distance between a pair of these points
reaches a critical value, these two points are considered as bonded. This procedure leads to in some sense a “randomness” and the non-smooth behavior
during the binding process in the current model.
The effect of the cell membrane stiffness is shown in Figs. 4 and 5. The engulfing process is shown in Fig. 4, in which
each row shows four snapshots of interface configuration for a given cell membrane rigidity. The result of the base case
(b v = 5, bm = 1) is presented as the solid line in Fig. 5(a), while the dashed line is for bm = 0.5 and dash-dotted line for
bm = 2. As expected, the smaller the bending rigidity, the higher the engulfing speed as well as the engulfing ratio. Since
the selected adhesion energy density γ = 1 is no longer sufficient for a complete engulfing process when bm = 2, it leads
to a 30% partially wrapped state. On the other hand, a ten times softer cell membrane (compared to the base case) leads to
a 93.8%-engulfing ratio, as shown by the line with circular marks in Fig. 5(b) and the fourth row in Fig. 4. In addition, we
Y.-H. Tseng, H. Huang / Journal of Computational Physics 273 (2014) 143–159
153
Fig. 6. Evolution of energies and rate of change of energies: (a) Surface adhesion energy (including the negative sign): solid line represents the evolution of
the vesicle and the dashed line is for the cell. (b) Bending energy: solid line for the vesicle and dashed line for the cell. (c) Rate of change of kinetic energy
(solid line) and viscous dissipation rate (dashed line). (d) Rate of total energy variation.
also consider a case with a more rigid vesicle which has b v = 10 and bm = 1, and the result shows only a small deviation
from the base case, which is shown by the dotted line in Fig. 5(b).
We remark that our computations are stopped either before or near complete engulfing states with one part of the cell
membrane almost in contact with another part, forming a neck region. To continue the engulfing process, membrane fusion
must occur. However, the physical mechanism of membrane fusion is complex, where the lipid bilayers rearrange. From
computational point of view, the immersed boundary method cannot handle interfacial topological changes readily. A standard approach is to perform a numerical “surgery”, i.e., by merging the interfaces manually. Since the main objective of this
paper is to investigate fluid dissipation and other factors on the engulfing process, we did not continue our computations
beyond these near complete engulfing states.
4.4. Energy balance
The numerical examples in the previous section showed that the engulfing speed slows down and the binding process
stops before reaching the completely wrapped stage. This is in contrast to the case of the equilibrium energy argument
which predicts the completion of the engulfing process [27]. In order to find the causes for this discrepancy between the
equilibrium theory prediction and our dynamic model simulation, we analyze the energy balance by computing the rate of
energy change
Ė (t ) =
d
dt
1
2
Ω
u : u dx +
1
∇ u : ∇ u dx −
Re
Ω
1
Re Ca
Ft +
1
Re Bn
Fb
· U ds.
(46)
Γ
On the right-hand-side of the equation are the rate of change of kinetic energy (first term), viscous dissipation (second
term), and interfacial energies (third term), respectively.
Fig. 6 shows the evolution of interfacial energies and the corresponding rates of change for the base line case (b v = 5 and
bm = 1), shown by the second row in Fig. 4. In Fig. 6(a), we observe that the magnitude of the adhesion energy increases for
both vesicle (solid line) and cell (dashed line) membranes. Since the binding process is assumed to proceed instantaneously
when the distance between the ligand–receptor pair reaches a given threshold, the energy decreases in a step-wise manner,
cf. the inserted subplot, an enlarged sketch of the energy evolution near t = 1.5. The dynamics of the bending energy is
shown in Fig. 6(b). The growth of the bending energy of the cell membrane comes mainly from the deformation of the
membrane in the bound region and at the vicinity of the triple junction.
In our model, due to the instantaneous binding process, the location of the triple junction moves instantly from one
position to another. The Marangoni stress acts at a new location and produces higher tangential velocity which increases
both kinetic energy and viscous dissipation, as shown by Fig. 6(c). The average magnitude of energy dissipation is of O (104 ),
while the magnitude for the rate of change of kinetic energy is only of O (102 ) at the binding time and three orders of
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Y.-H. Tseng, H. Huang / Journal of Computational Physics 273 (2014) 143–159
Fig. 7. Dynamics of molecule concentration for Pes = 5 and K r = 1: (a) Distribution of c v and the corresponding total mass (in inset) on the vesicle
membrane. (b) Distribution of cm and the corresponding total mass on the cell membrane. (c) Distribution of ca and the corresponding total mass in the
bound region of the membranes. (d) Total mass: solid line is for the vesicle membrane and dashed line is for the cell membrane.
magnitude smaller elsewhere in the domain. The relative error of the total energy change (relative to the rate of change of
the total interfacial energy) is depicted in Fig. 6(d). Notice that the change of interfacial energy is roughly balanced by the
fluid dissipation ( Ė (t ) ∼ O (10−3 )) except when the binding occurs. The relative error reaches its maximum value during
binding but stays within the 0.2% mark, which implies that our simplified numerical treatment during binding does not
induce significant error to affect the underlying physical process.
Our computational results on the energy balance show that the slowing of the engulfing process is mainly due to viscous
dissipation. The other minor factors of the slowdown comes from the increase of kinetic energy and membrane bending
energy. Therefore, fluid flow plays a significant role in the endocytosis process and its effect cannot be neglected.
4.5. Roles of convection, diffusion, and reaction
In this section, we investigate the effects of molecule diffusion, fluid flow and reaction rate of the binding molecules. We
assume that adhesion energy density is linearly dependent on ca (the concentration of the receptor–ligand pairs), given by
Eq. (17). Recall that the unbound interfacial molecules can move freely along the membranes, and bind only in the binding
region. The bounded receptor–ligand pairs, can also move freely inside the bound region. We assume that the diffusivity of
each reactant (unbounded molecules) in the bound region is half of the one in the outer region, that is, D v ,a = 0.5D v ,o and
D m,a = 0.5D m,o . In addition, the diffusivity of bound molecules is smaller, given by D a,a = 0.1( D v ,o + D m,o ).
For comparison purposes, we select the case of Pes = 5 and K r = 1 as the reference case. The corresponding dynamics
of the molecular concentrations c v (ligands on the vesicle surface), cm (receptors on the cell membrane), and ca (ligand–
receptor pairs in the bound region) are given in Fig. 7(a), (b), and (c), respectively. The inset in each figure shows the
evolution of the total mass of surface molecules in each region. From Fig. 7(d), it can be seen that discrete mass conservation is maintained (the relative error is of O (10−12 )).
To investigate the effects of convection, diffusion, and reaction rate on the engulfing speed, we vary the diffusivity so that
Pes = 0.05, 0.5, 5, 50 with the reaction constants K a = 0 (an irreversible reaction) and K r = 0.1, 1, and 10. Fig. 8(a) shows
the evolution of engulfing ratio before t = 12.5. Compared to Pes = 5, smaller Péclet numbers Pes = 0.5 and Pes = 0.05 show
almost the same engulfing rate, which implies Pes = 0.5 has reached the diffusion dominant threshold when K r = 1. For a
diffusion limited case, Pes = 50, the flow convection dominates the system and rapidly sweeps reactants out of the bound
region, and leads to a less successful engulfing process. For a constant Pes = 5, evolution of engulfing ratio of different K r
are presented in Fig. 8(b). As expected, we see that the greater the reaction constant, the faster the engulfing speed. On
the other hand, the increase of reaction rate in a diffusion limited system only slightly enhances the engulfing process, as
shown in Fig. 8(c).
The time sequences of the engulfing process (light blue represents the bound region) is given in Fig. 9. The corresponding
concentration distribution at t = 2.5 are shown in Fig. 10. The red bar of each line in Figs. 10(a) and 10(b) represents the
Y.-H. Tseng, H. Huang / Journal of Computational Physics 273 (2014) 143–159
155
Fig. 8. Engulfing rate affected by Péclet number Pes and reaction rate K r : (a) Fixing K r = 1 and varying Pes : Pes = 5 (solid line), Pes = 0.5 (dotted line),
Pes = 0.05 (dash-dotted line), and Pes = 50 (dashed line). (b) Fixing Pes = 5 and varying K r : K r = 1 (solid line), K r = 0.1 (dashed line), and K r = 10
(dash-dotted line). (c) Effect of reaction rate in a diffusion limited (Pes = 50) condition: K r = 1 (dashed line) and K r = 10 (solid line).
Fig. 9. The time sequences of the engulfing process. First row: Pes = 5 and K r = 1. Second row: Pes = 5 and K r = 10. Third row: Pes = 50 and K r = 1. Fourth
row: Pes = 50 and K r = 10. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
position of the boundary between bound (left of the red bar) and unbound (right of the red bar) regions. The second row
of Fig. 9 shows the engulfing process based on a reaction rate that is ten times greater than the reference case.
4.6. Discussion
Previous studies of the endocytosis processes are based on equilibrium energy arguments. By considering the dynamics
of the process and the influence of the fluid flow, our numerical results have shown that a large difference in adhesion and
bending energies does not necessarily lead to faster engulfing of the vesicles. Due to induced fluid flow, it may even have
an adverse effect (under a large Péclet number, for example for a smaller diffusivity). A more detailed parametric study,
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Y.-H. Tseng, H. Huang / Journal of Computational Physics 273 (2014) 143–159
Fig. 10. Comparison of concentrations at t = 2.5. Solid lines: Pes = 5, K r = 1; dashed lines: Pes = 50, K r = 1; and dash-dotted lines: Pes = 50, K r = 10.
(a) Concentration c v . (b) Concentration cm . (c) Concentration ca . (For interpretation of the references to color in this figure, the reader is referred to the
web version of this article.)
by systematically varying a variety of parameters, is needed before recommendations can be made for creating favorable
conditions for endocytosis for particles with different sizes and membrane properties.
We have assumed in this paper that membranes are impermeable to water. In reality, water can go through membrane
due to (slow) diffusion or through specialized channels under certain conditions. Therefore, one might ask whether water
permeation through the membranes have an effect on the endocytosis process. Our estimate (details in Appendix B) shows
that the energy dissipation through the membrane is several orders lower than the viscous dissipation in the bulk. This
suggests that even if we were to model water permeation through the membrane, the corresponding dissipation is expected
to be insignificant since water would flow along the membrane instead of through it because of the large resistance of the
membrane without the opening of water channels.
Finally, in this paper we have chosen a relatively large vesicle size so that the continuum approach used in this paper is
valid. For endocytosis of smaller vesicles on the order of tens of nanometers in diameter, the receptor size (around 10 nm)
is comparable to the vesicle size. In such cases a different model is needed for the binding process of the surface molecules
even though the continuum approximation for the fluid is still applicable. This will be considered in a future study.
5. Conclusion
In this paper, we have developed a numerical method for modeling endocytosis. The governing equations are formulated
in an immersed boundary framework where a mixture of Eulerian fluid and Lagrangian interfacial variables are linked by
the Dirac delta function, and the interfacial forces are derived from the interfacial and elastic energy of the membranes. By
tracking the interfaces in a Lagrangian manner, the convection–reaction–diffusion molecule concentration equations can be
solved easily by a mass-conserved finite difference scheme. The interfacial binding process keeps the two-layer structure in
the binding region: this strategy allows us to treat each interface individually and regain the interfacial forces by a simple
linear combination of the forces from interface.
Our immersed boundary method preserves the mass (receptors and ligands) strictly and the global conservation of
energy reasonably well. In the numerical examples, we have examined the roles of interfacial energy and fluid viscous
dissipation in the engulfing process. An analysis of the energy balance confirmed that energy dissipation plays a significant
role and that the equilibrium energy argument alone does not provide sufficient condition for endocytosis. The fluid (water)
consumes much more energy and makes it more difficult for the engulfing process to continue. A preliminary study of
the convection–diffusion–reaction interaction shows that a diffusion dominant environment and a high reaction rate are
essential to increase the success of endocytosis.
From the point of view of applications, an interesting question that has not been investigated is the effects of membrane
properties and vesicle size on the engulfing process. On the modeling side, there are several outstanding issues which have
not been addressed. For example, we have used a simple linear elastic model for the membrane. It will be interesting to
examine a more realistic nonlinear membrane model that takes the membrane remodeling process into account. The fluid
Y.-H. Tseng, H. Huang / Journal of Computational Physics 273 (2014) 143–159
157
inside and surrounding the cells usually has higher viscosity and might be non-Newtonian. In addition, we have used a
two-dimensional setting in this study while a three-dimensional spherical geometry might be more realistic, especially at
the initial and final stages of the engulfing process. Finally, the receptor–ligand binding mechanism was modeled as a simple
reaction. A more physically realistic model will need to consider the process of bond forming and dissociation rates. Some
of these issues will be addressed in a forthcoming paper [26] while others will be considered in future work.
As a final remark, we note that it will be interesting to compare our model prediction with experiments where the
number of successful events can be measured, especially when ligand density can be controlled on target vesicles. This will
be another topic for future research.
Acknowledgements
This research is supported in part by NSERC (HH) and the Centre for Mathematical Medicine at the Fields Institute (YT).
Appendix A. Derivation of interfacial forces from the interfacial energy
Instead of using normal and tangential forces directly in the computation, an alternative derivation based on a perturbation of the interface configuration is easier to implement using the immersed boundary method. First, we consider the
Frenet formula τ s = κ n and rewrite the bending energy as
Eb[ X ] =
1
b (s)
2
∂ 2 X sp
∂2 X
−
2
∂s
∂ s2
2
ds,
(47)
Σ
where X sp is the reference state of the membrane. A small perturbation of the energy is given by
E b, = E b [ X + Y ] =
1
2
2
2
∂ ( X + Y ) ∂ 2 X sp
b (s)
−
ds.
∂ s2
∂ s2
Y 1:
(48)
Σ
→ 0 gives
2
2
∂ 2 X sp
∂ Y
∂ X
E b, − E b
· 2 ds
lim
= b (s)
−
→0
∂ s2
∂ s2
∂s
Σ
2
2
∂ 2 X sp
∂
∂ X
· Y ds.
=
b
(
s
)
−
∂ s2
∂ s2
∂ s2
Taking the limit as
(49)
Σ
Therefore, in Cartesian coordinates the bending force can be expressed as
2
∂ 2 X sp
∂2
∂ X
.
F b (s, t ) = − 2 b (s)
−
∂s
∂ s2
∂ s2
(50)
For the surface (σ = γ − λ) energy
Et [ X ] =
∂X σ ds,
∂s
(51)
Σ
we consider the perturbation
E t , = E t [ X + Y ] =
∂ X + Y σ ds.
∂s
(52)
Σ
→ 0 yields
∂X E t , − E t
τ · ∂ Y ds
lim
= σ →0
∂s ∂s
Σ
∂X ∂
= −
σ τ · Y ds,
∂s
∂s
Taking the limit as
(53)
Σ
and thus we obtain
∂
∂X F t (s, t ) =
σ τ .
∂s
∂s
(54)
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Y.-H. Tseng, H. Huang / Journal of Computational Physics 273 (2014) 143–159
Fig. 11. Four snapshots of pressure along the membranes of the vesicle (upper row) and cell (lower row), respectively.
Appendix B. Membrane permeability
In this section we address the issue of water permeation rate and energy dissipation, when water can flow through
membrane. Note that this is a posterior estimate of the worse case scenario in the sense that results are based on the
pressure field obtained without water permeation through the membranes, which might reduce the pressure difference
across the membranes. We assume that Darcy’s law is valid and takes the form
vp = −
K
μ
∂n p
(55)
where v p and ∂n p are the normal velocity and the normal pressure gradient across the membrane, respectively. Here,
μ (Pa s) is the fluid viscosity, and K (m2 ) is the permeability of the membrane. The dimensional dissipation rate across a
membrane of thickness hθ is defined as
− v p ∂n p hθ dΣ =
Σ
K
μ
Σ
2
(∂n p )2 hθ dΣ = μU ∞
L
K ∂n p 2
hθ dΣ (56)
Σ
Therefore, the corresponding dimensionless form (consistent to the energy rate in Section 4.4) is given by
K ∂n p 2
hθ dΣ (57)
Σ
The permeability for lipid bilayer is based on the experimental measurement reported in [21], where the permeability
coefficient L i over a portion of a phospholipid vesicle with area A i is obtained. L i is related to the Darcy permeability in the
following way L i = K A i /μhθ . Based on the values of L i , A i , μ and hθ , we obtain the value of the dimensional permeability
K = 1.598 × 10−25 (m2 ), and the corresponding dimensionless K = 6.392 × 10−13 .
The pressure values along the membranes, p i for interior and p o for exterior, are obtained from the pressure on the
Eulerian grid, interpolated using one-side delta functions. The pressure drop in the unbound region is given by p i − p o .
The pressure drop in the bound region is evaluated by taking the difference of p iv and pm
. We assume that the membrane
i
has a constant thickness of 5 nm, at the unbounded region, while the thickness at the bounded region is sum of the two.
The dimensionless thickness in the unbounded and bounded regions are hθ = 0.01 and hθ = 0.02, respectively. The normal
derivative of the pressure is estimated by dividing the pressure difference with the membrane thickness.
Fig. 11 shows snapshots of pressure (both interior and exterior) along the membranes for the case in Section 4.4, corresponding to the second row in Fig. 4 at various engulfing stages. A large pressure difference occurs near the triple junction
region, as expected. The rate of dissipation due to water permeation across all the membranes is about O (10−8 ), as shown
in Fig. 12. It is much smaller than the rates reported in Fig. 6, Section 4.4. Therefore, we conclude that the dissipation of
water permeation is negligible compared to that in the bulk.
Y.-H. Tseng, H. Huang / Journal of Computational Physics 273 (2014) 143–159
159
Fig. 12. Time evolution of dissipation rate of water through the membranes: vesicle (dash) and cell membrane (solid).
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