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Research
Cite this article: Wang J, Li L. 2015 Coupled
elasticity – diffusion model for the effects of
cytoskeleton deformation on cellular uptake of
cylindrical nanoparticles. J. R. Soc. Interface 12:
20141023.
http://dx.doi.org/10.1098/rsif.2014.1023
Received: 12 September 2014
Accepted: 27 October 2014
Subject Areas:
biophysics
Keywords:
cylindrical nanoparticles, cellular uptake,
diffusive receptor, cytoskeleton deformation
Author for correspondence:
Jizeng Wang
e-mail: [email protected]
Coupled elasticity– diffusion model for
the effects of cytoskeleton deformation
on cellular uptake of cylindrical
nanoparticles
Jizeng Wang and Long Li
Key Laboratory of Mechanics on Disaster and Environment in Western China, Ministry of Education, College of
Civil Engineering and Mechanics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China
Molecular dynamic simulations and experiments have recently demonstrated how cylindrical nanoparticles (CNPs) with large aspect ratios
penetrate animal cells and inevitably deform cytoskeletons. Thus, a coupled
elasticity– diffusion model was adopted to elucidate this interesting biological phenomenon by considering the effects of elastic deformations of
cytoskeleton and membrane, ligand –receptor binding and receptor diffusion. The mechanism by which the binding energy drives the CNPs with
different orientations to enter host cells was explored. This mechanism
involved overcoming the resistance caused by cytoskeleton and membrane
deformations and the change in configurational entropy of the ligand –receptor bonds and free receptors. Results showed that deformation of the
cytoskeleton significantly influenced the engulfing process by effectively
slowing down and even hindering the entry of the CNPs. Additionally,
the engulfing depth was determined quantitatively. CNPs preferred or
tended to vertically attack target cells until they were stuck in the cytoskeleton as implied by the speed of vertically oriented CNPs that showed much
faster initial engulfing speeds than horizontally oriented CNPs. These results
elucidated the most recent molecular dynamics simulations and experimental observations on the cellular uptake of carbon nanotubes and
phagocytosis of filamentous Escherichia coli bacteria. The most efficient
engulfment showed the stiffness-dependent optimal radius of the CNPs.
Cytoskeleton stiffness exhibited more significant influence on the optimal
sizes of the vertical uptake than the horizontal uptake.
1. Introduction
Nanoparticles (NPs) usually enter cells via endocytosis driven by the binding
energy between diffusive receptors on the cell membranes and ligands on the
surface of the NPs. This remarkable capability has resulted in the proposition
of NPs as potential candidates for site-specific drug-delivery systems [1]. However, incomplete phagocytosis of long, rigid biopersistent asbestos fibres and
carbon nanotubes (CNTs) with large aspect ratios has been hypothesized to
cause the release of inflammatory mediators and even cell death [2,3]. Therefore, understanding how cells interact with cylindrical nanoparticles (CNPs)
is greatly significant to advance our understanding of fundamental biological
and cellular immunological recognition as well as improve various practical
applications of drug delivery, pharmacology and nano-toxicology [4,5].
Numerous experiments and simulations have recently explored the mechanics and reasons for the penetration of one-dimensional nano-materials and
micro-materials into living cells, including macrophages. Shi et al. [6] experimentally and theoretically demonstrated that CNPs, such as CNTs, initially enter the
cells through the tip. This entry is assumed to be caused by tip recognition following rotation initiated by the asymmetric elastic strain at the tube–bilayer
interface. To determine whether such shape- and orientation-dependent
& 2014 The Author(s) Published by the Royal Society. All rights reserved.
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2. Theoretical model
An elastic half space covered by a cell membrane embedded
with diffusive mobile receptors was used to illustrate the proposed elasticity –diffusion model (figure 1a). The membrane
engulfed horizontally and vertically oriented elastic CNPs
coated with compatible ligands with a depth, h, below the
initially undeformed position of the membrane. Each CNP
with length, L, and radius, R, were capped by two hemispheres with radius, R, at its two ends. The ligands with a
fixed constant density, jL, were assumed to be immobile
and uniformly distributed on the particle surface. By contrast,
the receptors with an initial uniform density of j0 were
mobile and underwent diffusive motion within the plane of
the cell membrane. When the CNP with either orientation
came into contact with the cell, the binding energy of the
receptor–ligand complex drove the engulfment of the CNP
to overcome the resistance caused by the change in the configuration entropy in the receptors by diffusion and the
deformations of the membrane and cytoskeleton. Moreover,
the free energy of interaction was lowered further to maintain
the engulfing process. Thus, the receptors diffused to the
contact site and bound with the ligands on the surface of
the particle.
2
J. R. Soc. Interface 12: 20141023
and thermodynamic interaction of the NPs and the absorbing
membrane. In addition, Sun & Wirtz [23] and Li et al. [24]
developed an elastic model to investigate the resistance
during the process of viral particle entry into host cells by
neglecting receptor diffusion but considering a balance
between energetic forces. They concluded that the resistance
to engulfment is dominated by the membrane or cytoskeleton
deformation depending on the size of the virus, engulfing
stages and Young’s modulus of the cell. Most recently, Yi
et al. [25] investigated the adhesive wrapping of a soft elastic
lipid vesicle by a lipid membrane analogous to the cellular
uptake of elastic particles using variational methods and
establishing free energy functionals. They identified five distinct wrapping phases. Zhang et al. [26] and Yuan et al. [27]
analysed the equilibrium interaction between a group of
NPs and the membrane. They also showed that although
non-equilibrium processes of each individual NP are not considered, the predicted optimal particle size for maximal
cellular uptake is close to that for the shortest wrapping
time as provided by Gao et al. [21].
In spite of the aforementioned development, the understanding of cellular uptake of CNPs is still limited. Existing
models have not investigated how cytoskeleton deformation
may affect the equilibrium state and dynamic process of the cellular uptake behaviour. In this study, an elasticity–diffusion
model is proposed to examine the engulfment of CNPs, considering the coupled effects on cytoskeleton and membrane
deformations, receptor diffusion, ligand–receptor binding
and CNP orientations. The majority of previous theories on
the cellular uptake of NPs has focused on NP–membrane
interaction. By contrast, the present model consists of a
membrane-covered elastic solid with mobile molecular binders
diffused along the membrane under the ligand binding
action of NPs. The effects of the sizes and orientations of
NPs, as well as stiffness and receptor densities of host cells,
on the dynamic engulfing process of CNPs was investigated
according to the proposed model.
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mechanisms are also manifested by the phagocytosis of
bacterial filaments, Möller et al. [7] experimentally investigated the effects of shape and micro-environments on the
phagocytosis of filamentous Escherichia coli bacteria by macrophages. They found that complete uptake occurs only if one of
the terminal bacteria filament poles enters the cell first.
Abdolahad et al. [8] used an array of vertically aligned multiwalled CNTs (radius ¼ 32.5 nm) to distinguish healthy and
cancerous cells by evaluating the entrapment of different
cells. They found that the entrapment fraction of cancer cells
with higher metastatic grades was significantly greater than
those with lower metastatic grades, because the former are
less stiff than the latter. These previous studies have shown
that cellular uptake of one-dimensional nano-materials or
micro-materials with high-aspect ratios can be dependent on
orientation and cytoskeleton deformation.
The entry of long CNPs into cells inevitably deforms the
cytoskeleton. The long CNPs can penetrate very deeply into
cells, therefore, their interaction with the cytoskeleton is unavoidable. Obataya et al. [9] used an atomic force microscope
with a long nanoneedle (radius ¼ 1002150 nm) to indent
living cells. They demonstrated that the loading force curve
for an indentation depth of up to 2 mm is still consistent with
the classic Hertz model in contact mechanics. Similarly, Beard
et al. [10] used a nanoneedle probe with a radius of only
20 nm and recorded the loading force curve during the indentation of a corneocyte. The corneocyte is usually a target of many
viruses including the herpes simplex virus type 1 [11]. Likewise,
they confirmed that the Hertz model can fit the loading force
curve very well. The Hertz model is derived based on the contact problem of two elastic bodies. Thus, these results clearly
prove that during indentation the living cell behaves as a
deformable elastic solid instead of a membrane, implying
that deformation of the cytoskeleton plays a dominant role in
resisting CNP intrusion.
The cellular uptake of CNP processes via endocytosis for
long, short or spherical NPs, regardless of size, accompany
cytoskeleton remodelling, considering that many endocytic
proteins can be potentially linked to the actin cytoskeleton,
either directly or indirectly [12–15]. This phenomenon eventually leads to a close functional connection between the actin
cytoskeleton and the internalization step of endocytosis.
Therefore, many virologists and physiologists [16–18] have
pointed out that the physical barrier imposed by the cortical
actin meshwork on the endocytosis process should be overcome following the stimulation of actin cytoskeleton
remodelling. This phenomenon is observed in HIV-1 viral
particles such that the ERM protein family supplies functional linkage between integral membrane proteins and the
cytoskeleton in mammalian cells to regulate membrane
protein dynamics and cytoskeleton rearrangement [19,20].
After an HIV-1 Env protein binds to a host cell, a cytoskeleton-dependent clustering of infection receptors (CD4,
CXCR4 and CCR5) occurs. These receptors are essential for
efficient membrane fusion and subsequent entry of HIV-1
into the target cells [20]. A number of theoretical models
have been proposed to understand the underlying biophysical mechanism for the cellular uptake of NPs based on
these recognitions. Regarding diffusion kinetics, although
neglecting the effect of cytoskeleton deformation, Gao et al.
[21] and Shi et al. [22] showed that in the endocytosis of
cylindrical and spherical NPs, optimal sizes exist because of
the power balance between the kinetics of receptor diffusion
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(a)
Within the contact region, s , a(t), j (s, t) ¼ jL and j(s, t) ¼ 0.
The continuity equation
vertical engulfment
(2:3)
L
horizontal engulfment
can be substituted into equation (2.1) to yield [21,28]
R
(jL jþ )_a þ jþ ¼ 0 on s ¼ a(t),
h
(b)
x
xL
diffusion
a(t)
x(s, t)
x0
x+
contact edge
s
Figure 1. Schematic diagram of the entry of CNPs into a host cell.
(a) Remote mobile receptors diffuse to the binding site to engulf the NPs,
where the solid and dash lines describe cellular uptake of horizontally and
vertically oriented CNPs, respectively. (b) Receptor density distribution
along the membrane. (Online version in colour.)
Figure 1b schematically shows the coordinate system at the
centre of the CNP–cell contact zone. The half size of the contact
area was s ¼ a(t). At the contact zone, the receptor density was
assumed to reach the ligand density, i.e. j (s, t) ¼ jL, s , a(t).
During the engulfing process, the edge of the CNP–cell contact
zone, s ¼ a(t), and the distribution of membrane receptors outside the contact zone, j (s, t), continued to vary. The difference
was determined by solving the dynamic problem influenced
by ligand–receptor binding and elastic deformations of the
membrane and the cytoskeleton.
2.1. Horizontally oriented cylindrical nanoparticles
The engulfment of the horizontally oriented CNP with highaspect ratio, i.e. L R was considered first. In this case,
receptor diffusion along the axial direction was neglected
and mainly maintained along the direction perpendicular to
the axial direction. The capped CNP with total surface area,
2pR(L þ 2R), was also approximately replaced by a noncapped cylinder with the same diameter, surface area and
length, 2R þ L. In this situation, the CNP, which may or
may not be capped, only had a negligible effect on receptor
diffusion and deformation energy of contact because of its
high-aspect ratio.
During the engulfing process, a global number of conservation conditions for the receptors could be expressed in
terms of the receptor density, j (s, t), as [21,28]
ð
ð1
d a(t)
j ds þ
j(s, t)ds ¼ 0,
(2:1)
dt 0 L
a(t)
where a(t) is the half-width of the contact region. Outside
the contact region, i.e. s a(t), the receptor diffusion flux is
given by
j(s, t) ¼ D
@ j(s, t)
,
@s
where D is the diffusion coefficient.
(2:2)
where jþ ; j (a þ, t) and jþ ; j(a þ, t) denote receptor density and
flux in front of the contact edge, respectively. Boundary conditions at s ! 1 were assumed to be j (s, t) ! j0 and j(s, t) ! 0.
Upon initial CNP–cell contact, the formation of ligand–
receptor complexes drove the engulfing process to overcome
the resistance caused by membrane and cytoskeleton deformation and the change in the configuration entropy of
receptors. During this process, the binding energy of ligand–
receptor complexes in the contact region could be expressed
Ð a(t)
as 2(L þ 2R) 0 jL eRL kB Tds, where eRLkBT is the binding
energy of each single ligand–receptor bond. The deformation
Ð a(t)
energy of the membrane can be written as 4R 0 (2kkB THs2 þ
Ð a(t)
gkB T)ds þ 2L 0 (2kkB THc2 þ gkB T)ds,
where
Hs ¼ 1/R
and Hc ¼ 1/2R are the mean curvatures of the spherical and
cylindrical surfaces, respectively. kkBT and gkBT are the
bending modulus and surface tension of the membrane,
respectively. The free energy caused by the configurational
change of all receptors on the membrane can be estimated as
Ð a(t)
Ð a(t)
[21,28] 2(L þ 2R)[ 0 jL kB T ln jL =j0 ds þ 0 jkB T ln j=j0 ds].
Here kBT ln jL/j0 and kBT ln j/j0 are the energy per receptor
associated with the loss of configuration entropy of the
bonds and free receptors, respectively.
Cytoskeleton deformation also significantly contributes to
the energy of the system [18]. Recently, Liu et al. [30]
further addressed this issue by performing living cell indentation with a long nanoneedle (radius 90 nm). As shown in
figure 2, when we compare their experimental loading force
curve to the theoretical prediction based on linear elasticity,
good agreement is found until cell membrane penetration.
These experimental studies clearly prove that during deep
indentation by nanoneedles living cells can somehow behave
as linear elastic solids, implying that the Hertzian contact
model based on linear elasticity might be an appropriate framework to describe the interaction between the CNPs and
the cells, and under this situation, the complicated finite
deformation formulation may not be necessary.
Therefore, by treating the cell as an elastic solid and
within the limit where the cell is much larger than the NP,
the energy of cytoskeleton deformation during the interaction
between the CNP and the cell, based on the contact theory
[29], can be estimated as
"
#
p2
pE (L þ 2R)3 1
Fe ¼
ln
þ ,
2
2pE (L þ 2R)
Rp
where 1/E* ¼ (1 2 m2c /Ec) þ (1 2 m2n/En) is the combined
elastic modulus. mc and Ec are Poisson’s ratio and Young’s
modulus of the cell, respectively. The corresponding values
for the NP are mn and En. Variable p is an effective contact
force related to the engulfing depth by [29]
"
#
p
pE (L þ 2R)3
h¼
1 þ ln
,
(2:5)
pE (L þ 2R)
Rp
where the contact depth, h, as shown in figure 1a, is geometrically
J. R. Soc. Interface 12: 20141023
contact
(2:4)
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@ j(s, t)
@ j(s, t)
¼
@t
@s
3
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Differentiation of equation (2.7) with respect to time yields
_
jL
F(t)
¼ (L þ 2R) jL eRL g jL ln þ jL jþ a_ (t)
jþ
2kB T
0.6
force (nN)
0.5
0.4
þ 2ka_ (t)(2RHs2 þ LHc2 )
0.3
þ
cell membrane penetration
1 @ Fe @ p
a_ (t) (L þ 2R)
2kB T @ p @ a
ð1
a(t)
Dj
4
@ [ln(j=j0 ) þ 1] 2
ds:
@s
(2:8)
0.2
0.1
0
0.2
0.3
0.4
0.5
0.6
indentation depth (mm)
0.7
0.8
g
j
j
2k
ln L þ 1 þ (2RHs2 þ LHc2 )
jL
jþ
jL (L þ 2R)jL
p(t)
a(t)
¼ 0,
sin
2(L þ 2R)jL kB T
R
eRL Figure 2. Comparison between experiments on cell indentation [30] and
theoretical formula p ¼ 2EcRh/(1 2 m 2) [31] based on linear elastic contact
theory for indenter of radius R ¼ 90 nm [30], where Possion’s ratio of the
cell m ¼ 0.5 and Young’s modulus of the cell Ec ¼ 3.8 kPa are considered.
(Online version in colour.)
which was obtained by inserting equations (2.5) and (2.6) into
equation (2.8).
related to the half-width of the contact region a(t) as
2.2. Vertically oriented cylindrical nanoparticles
Rh
:
a(t) ¼ R arccos
R
(2:6)
Based on the above analysis, the free energy functional for the
cellular uptake of the CNP can be integrated as
ð a(t) F(t)
j
jL eRL þ g þ jL ln L ds
¼ 2(L þ 2R)
kB T
j0
0
ð a(t)
ð a(t)
þ 4R
2kHs2 ds þ 2L
2kHc2 ds
0
0
"
#
p2 (t)
pE (L þ 2R)3 1
þ
ln
þ
2pE kB T(L þ 2R)
2
Rp(t)
ð1
j
þ 2(L þ 2R)
j ln ds:
(2:7)
j
a(t)
0
8
A(t) 2pR2
< pkB Ta2 (t)(2kHs2 þ g)
Fb ¼ 4pkkB TR2 (Hs2 Hc2 ) þ pkB Ta2 (t)(2kHc2 þ g)
:
4pkkB TRL(Hc2 Hs2 ) þ pkB Ta2 (t)(2kHs2 þ g)
The deformation energy of the cytoskeleton was
pffiffiffiffi estimated
based on the contact theory [29,31] using 8E Rh5=2 =15. In
addition, the energy contribution caused by the loss of
configurational entropy of receptors can be given by
Ð a(t)
Ð1
0 2psjL kB T ln jL =j0 ds þ a(t) 2psjkB T ln j=j0 ds.
Eventually, the total free energy functional was obtained as
F (t)
¼
kB T
ð a(t)
0
þ
j
2ps jL eRL þ jL ln L ds
j0
ð1
a(t)
2psjL ln
pffiffiffi 5
j
2 E a (t)
ds þ Fb (t) þ
:
j0
15kB T R2
(2:9)
In the engulfment of vertically oriented CNPs, the associated
receptor diffusion was considered as an axisymmetric diffusion problem with the following governing equation
outside the contact region
2
@j
@ j 1 @j
þ
¼D
:
(2:10)
@t
@ s2 s @ s
The receptor conservation condition in this case was similar
to that in equation (2.4). The time-dependent contact area
can be expressed as A(t) ¼ 2pRh(t) ¼ pa 2(t). The binding
energy caused by the formation of ligand –receptor bonds
Ð a(t)
can be described as 0 2psjL eRL kB Tds. The deformation
energy of the membrane as a function of a(t) can be expressed
in a sectional type as
2pR2 , A(t) 2pR(L þ R)
2pR(L þ R) , A(t) 2pR(L þ 2R):
(2:11)
where the mean curvature is identified as
Hs A(t) 2pR2 and 2pR(L þ R) , A(t) 2pR(L þ 2R)
H¼
Hc 2pR2 , A(t) 2pR(L þ R):
(2:14)
Similarly, the power balance on the boundary of contact region
can be given by
pffiffiffi 3
j
2kH 2 þ g
j
2E a (t)
þ1 þ
¼ 0: (2:15)
eRL ln L jL
jþ
jL 6pkB T jL R2
(2:12)
Differentiating equation (2.12) with respect to time leads to
j
F_ (t)
¼ 2pa(t)_a(t) jL eRL jL ln L 2kH 2 g þ jL jþ
kB T
jþ
pffiffiffi 4
ð1
2 a (t)
@ [ ln (j=j0 ) þ 1] 2
_
þ
2
p
sD
j
ds:
a
(t)
3kB T R2
@s
a(t)
(2:13)
3. Results and discussion
3.1. Procedure for numerical solutions of governing
equations
Obtaining the analytical solutions of equations (2.3) and (2.10)
is a challenging task. In this section, these equations were
J. R. Soc. Interface 12: 20141023
A power balance equation for the horizontally oriented CNP can
be obtained by equating the decrease rate of the free energy to the
energy dissipated during receptor diffusion.
0.1
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Liu et al. [30]
solution based on linear elasticity
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numerically solved using the finite difference method similar to
that described in previous studies [21,28]:
(3:1)
parameter
value
reference
k
g, 1 nm – 2
eRL
D, nm2 s – 1
jL, 1 mm – 2
j0, 1 mm – 2
mc
20
[32]
0.005
15
[23]
[34]
104
5 103
[35]
[21]
50
0.5
[21]
[23]
and
eRL 2kHs2 þ g
j
j
ln L þ 1 þ ¼ 0:
jL
jþ
jL
(3:2)
In this situation, solutions of equations (2.3) and (2.10) at
t ¼ Dt could be obtained analytically as [21,28]
s
s
p
ffiffiffiffiffiffiffiffi
ffi
p
ffiffiffiffiffiffiffiffi
ffi
þ j0 Erf
(3:3)
j(s, Dt) ¼ Ah Erfc
2 DDt
2 DDt
and
j(s, Dt) ¼ j0 þ Av E1
s2
4DDt
(3:4)
Ð 1 u
where E1 (z) ¼ z eu du, Ah and Av are unknown
constants of integration. Inserting
(3.3
pffiffiffiffiffiffiffiffiequations
ffi
and (3.4), as well as a(Dt) ¼ 2a DDt [21,28], into
equation (2.5) yields
2 pffiffiffiffi
2 pffiffiffiffi
ea paErf(a)j0 þ j0 ea pajL
(3:5)
Ah ¼
pffiffiffiffi
1 ea2 paErfc(a)
and
Av ¼
a2 (jL j0 )
a2 E1 (a2 ) ea2
(3:6)
where a is the speed factor [21,28]. Then, substituting
equations (3.3)– (3.6) into the power balance relations of
equations (3.1) and (3.2) yields
"
#
2k(2RHs2 þ LHc2 )
j~g(a)
eRL þ ln
[1g(a)]þ1 j~ ¼ 0
1g(a)
jL (Lþ2R)
(3:7)
and
eRL 2kHs2 þ g
f1 (a) þ ln f1 (a) þ 1 ¼ 0,
jL
(3:8)
where
g(a) ¼
pffiffiffiffi a2
pae Erfc(a),
a2 (1 j~)E1 (a2 )
f1 (a) ¼ j~þ 2
a E1 (a2 ) ea2
and j~ ¼ j0 =jL . The speed factor, a, was obtained by solving equations (3.7) and (3.8). Once a was determined,
then the receptor density, j (s, Dt), at this time step was
obtained from equations (3.3) –(3.6). Furthermore, the diffusion flux, j(s,Dt), and subsequently the engulfing
speed, a_ (Dt), was derived from equations (2.2) and (2.4),
respectively.
(3) At time step 2, i.e. t ¼ 2Dt, the engulfing boundary was
expressed as a(2Dt) ¼ a(Dt) þ a_ (Dt)Dt. Inserting a(2Dt)
into the power balance relations (equations (2.9) and
(2.15)) provided another boundary condition of receptor
density jþ (2Dt) at s ¼ a(2Dt). Using this boundary condition and that specified in step 1, the receptor density,
j (s, 2Dt), was obtained by solving the diffusion equations
(2.3) and (2.10) using the finite difference method. The diffusion flux, j(s,2 Dt), and subsequently the engulfing
speed, a_ (2Dt), were derived from equations (2.2) and (2.4).
(4) Step 3 was repeated until the prescribed time step N, i.e.
t ¼ NDt, was reached. Finally, j (s, t) was obtained, where
t NDt.
3.2. Faster engulfment of vertically oriented cylindrical
nanoparticles
The bending modulus, k, of a typical cell membrane range
from 10 to 20 kBT [32], and the surface tension is approximately 0.005 kBT/nm2 [23]. Young’s modulus of the cell is
mostly on the order of 10 kPa or less [33]. In this study,
Poisson’s ratio of the cell was considered to be 0.5 [23]. The
binding energy of a single closed ligand –receptor bond eRL
is approximately 15 kBT at T ¼ 300 K [34]. The diffusion constant of a receptor on the membrane is approximately
104 nm2 s21 [34,35]. A previous study [21] has shown that
the ratio ~j ¼ j0 =jL should range from 0.01 to 0.1. In general,
Young’s modulus of viral particles and most artificial NPs are
much larger than that of a cell. Young’s modulus of CNTs is
on the order of 1 TPa [36]. Hence, the CNPs were assumed to
behave like rigid bodies during cellular uptake. Ligands distributed on the surface of each CNP had typical density
values of 5 103 mm22 [21].
The relevant parameters adopted in this study on cellular
uptake of NPs are summarized in table 1.
Figure 3 plots the numerically determined normalized
engulfment depth, h/R, as a function of engulfing time for
CNPs with radius, R ¼ 50 nm, and length, L ¼ 500 nm,
under horizontal (figure 3a) and vertical (figure 3b) orientations. Figure 3 shows that the vertically oriented CNP
enters the cell with an initial uptake speed close to
0.0133 s21, which is much faster than that of the horizontally
oriented CNP at 2.9 1024 s21.
In the authors’ previous study [24], based on a continuum
model, for the enveloped virus to enter the host cells, the binding
energy of the receptor–ligand complexes drove the engulfment
of NPs to overcome the resistance alternatively dominated by
membrane deformation and cytoskeleton deformation at different engulfing stages. In this study, the coupling effect between
receptor diffusion and elastic deformation was considered.
J. R. Soc. Interface 12: 20141023
2k(2RHs2 þ LHc2 ) g
j
j
eRL ln L þ 1 þ ¼ 0
jL
jL (L þ 2R)
jþ
jL
5
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(1) At time step 0, i.e. t ¼ 0, the initial and boundary conditions on receptor distribution and diffusion flux
were set to j (s, 0) ¼ j0 and j (1, t) ¼ j0, j(1, 0) ¼ 0,
respectively.
(2) At time step 1, i.e. t ¼ Dt, where Dt was small enough so
that the effect of cytoskeleton deformation was negligible. The power balance relations in equations (2.9)
and (2.15) were respectively reduced to
Table 1. Model parameters.
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(a)
2.0
1.0
60
Ec = 10 kPa
40
Ec = 12 kPa
Ec = 10 kPa
x+/x0
4t
.9e
~2
h/R
Ec = 8 kPa
20
0.6
1.0
80
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0.8
1.5
6
100
h (nm)
(a)
0
5000
t (s)
10 000
h/R
0.4
Ec = 8 kPa
0.5
Ec = 10 kPa
0.2
0
2000
4000
6000
t (s)
8000
10 000
0
20
40
(b) 1.5
(b)
0.7
40
20
Ec = 10 kPa
~0
.01
33
t
x+/x0
0.6
h/R
100
60
h (nm)
0.8
1.0
0.5
0
50
100
150
t (s)
h/ R
0.4
0.3
Ec = 6 kPa
0.2
Ec = 8 kPa
0.1
0
80
1.0
0.9
0.5
60
h (nm)
50
100
t (s)
150
Ec = 10 kPa
200
Figure 3. Normalized engulfing depth as a function of engulfing time for
Young’s moduli of cytoskeleton, 8, 10 and 12 kPa, and CNPs of radius,
R ¼ 50 nm and length, L ¼ 500 nm, under different orientations: (a) horizontal with initial engulfing speed of 2.9 1024 s21 and (b) vertical with
initial engulfing speed of 0.0133 s21. (Online version in colour.)
The resistance to CNP engulfment was still found to be initially
dominated by membrane deformation, and only later by cytoskeleton deformation. Figure 3 clearly demonstrates that cell
stiffness almost did not affect the initial uptake speed, which
started to significantly influence the speed only after a certain
engulfing depth. This phenomenon may imply that for the
engulfment of long CNPs or large NPs, the cytoskeleton process
can be crucial if the penetrations are sufficiently deep. By contrast, in tiny spherical NPs, the membrane process rather than
cytoskeleton remodelling plays the dominant role.
Figure 3 also shows that a stiffer cell or a stiffer part of a cell
can more easily resist the engulfing process after a certain
engulfing depth. As has been shown in figure 3b, the entry of
vertical CNPs can even be completely stopped after a certain
engulfing time. This phenomenon mainly depends on the
competition between different energy contributions. At
higher cell stiffness, the elastic deformation energy of the cytoskeleton will balance more binding energy of ligand–receptor
bonds. This behaviour decreases the effective energetic driving
force, causing receptor diffusion for engulfment to slow down.
3.3. Faster cylindrical nanoparticle engulfment by softer
cells or softer parts of a cell
To differentiate cell stiffness, the normalized receptor density
on the boundary of the contact region, jþ/j0, was
0
10
20
30
40
50
h (nm)
60
70
80
90
Figure 4. Normalized receptor density jþ/j0 on the boundary of the contact
region as a function of engulfment depth under different cell stiffness and NP
orientations: (a) horizontal and (b) vertical. The insets show the relationship
between engulfment depth and engulfing time at cell stiffness, Ec ¼ 10 kPa.
(Online version in colour.)
numerically determined and plotted in figure 4 as a function
of engulfment depth. This normalized receptor density
reflects the maximum inward diffusion flux at time t
(figure 5). Figure 4 also shows that for a small engulfing
depth, the normalized receptor density is small and is
almost not influenced by cell stiffness. This observation
implies that the engulfing process at this stage possesses
large inward receptor diffusion flux, fast engulfing speed
and is not influenced by cytoskeleton deformation. However,
for a relatively large engulfing depth, the normalized receptor
density at the contact edge or the maximum inward diffusion flux, starts to be strongly influenced by cell stiffness
(figure 4). At this stage of the engulfing process, the larger
cell stiffness corresponds to the larger normalized receptor
density at the contact edge, or the smaller maximum
inward diffusion flux, implying a slower engulfing process.
When Young’s modulus of the cell becomes large enough,
the diffusion flux eventually becomes zero, corresponding
to jþ/j0 ¼ 1 (figure 4). This condition also indicates that
when a virus comes into contact with a stiff part of the cell,
endocytosis may be stopped since receptors can no longer
be recruited by diffusion. Hence, viruses may seek soft
parts of a cell to attack, or viruses that attack soft parts of a
cell could more successfully infect the cell.
For the cellular uptake of horizontally oriented CNPs,
critical engulfment depth, hcr, corresponding to the maximum
receptor density on the engulfing boundary exists (figure 4).
J. R. Soc. Interface 12: 20141023
Ec = 12 kPa
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1.0
(a)
x0/xL = 0.01
0.9
t = 10 s
t = 50 s
t = 100 s
0.5
0.4
R = 50 nm Ec = 2 kPa
80
x0/xL= 0.01
70
h (nm)
j (1 nm s–1) (×10–3)
0.7
0.6
0.3
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90
0.8
7
100
x0/xL= 0.05
60
x0/xL= 0.1
50
40
30
0.2
20
0.1
J. R. Soc. Interface 12: 20141023
10
0
102
103
0 –1
10
s (nm)
Figure 5. Receptor diffusion flux, j, as a function of coordinate, s, along the
membrane’s contour at t ¼ 10, 50 and 100 s, during a horizontally oriented
CNP’s entry into the cell. (Online version in colour.)
1
103
102
10
time (s)
(b) 100
90
80
3.4. Effects on cellular uptake by initial receptor density
Figure 6 shows the engulfing depth as a function of time for
various initial receptor densities. For both horizontal and vertical uptakes of CNPs, the receptor density significantly
influenced the overall uptaking process. A larger receptor
density provided a faster uptaking process. However, when
the receptor density theoretically approaches infinity, an
unrealistic ultrafast uptaking process occurs. This result
reveals that the well-accepted receptor–diffusion-mediated
mechanics model of endocytosis might not be applicable in
the case when receptor density is high enough such that
receptor recruiting through diffusion is energetically
unfavourable during the engulfing process. Essentially, the
process of cellular uptake of CNPs is rather complicated.
This process cannot be fully understood based on theoretical
models employing only the membrane process. For example,
as shown in figure 7, at least three different regimes were
hypothesized for the uptaking process of CNPs in terms of
different initial receptor densities on the membrane.
3.4.1. Regime 1
If the normalized initial receptor density, j0/jL, is much
smaller than 1, the uptaking process of CNPs will be
h (nm)
70
The deformation energy of the cytoskeleton increases along
with the engulfing depth. When the engulfment depth is smaller than the critical value, the increased energy leads to
a decreased energetic driving force. When the engulfment
depth becomes larger than the critical value, the increasing
rate of binding energy of ligand–receptor complex becomes
larger than that of the deformation energy of cytoskeleton.
This phenomenon results in an increased driving force to
hasten the engulfing process as shown in the inset of figure 3a.
Such critical hcr can be numerically calculated by minimizing the
seventh term, f ¼ ( p(h) sin a(h)/R)/(2kBT(L þ 2R)), in equation
(2.10) with respect to h, where df/dh ¼ 0. The critical depth,
hcr ¼ 76.8 nm, for the CNP with radius, R ¼ 50 nm, only
depends on the shape of the CNP.
For the vertical uptake of CNPs, abrupt changes were
observed on the normalized receptor edge density when
the engulfing depth reached R, because of the sharp change
in the surface curvature of CNPs (figure 3b).
60
50
40
30
20
10
10–1
1
10
102
time (s)
Figure 6. Engulfment depth as a function of time under the influence of
different initial receptor densities for cellular engulfments of (a) horizontally
and (b) vertically oriented CNPs at R ¼ 50 nm and Ec ¼ 2 kPa. (Online
version in colour.)
dominated by receptor diffusion. This juncture determines
whether adhesion is sufficient to overcome the configurational entropy change of receptors and membrane bending
at the initial stage.
3.4.2. Regime 2
If j0/jL becomes larger but still smaller than 1, the uptaking process will be controlled by both receptor diffusion
and the deformations of the membrane and cytoskeleton.
In the case of intermediately large j0/jL, the initial
adhesion can easily overcome the configurational entropy
change of the receptors during diffusion to challenge membrane bending at the initial stage and later cytoskeleton
deformation. This phenomenon also indicates that the
uptaking process involves the full interplay between the
receptor diffusion, membrane deformations and cytoskeleton deformations. The theoretical model in this regime
should consider these three factors. Additionally, a pure
membrane model or a coupled membrane–diffusion
model may be valid only at the initial stage of the
uptake or only for the uptake of small NPs with negligible
cytoskeleton deformations. For the complete uptake of sufficiently long CNPs, a fully coupled model accounting for
receptor diffusion is necessary for both membrane and
cytoskeleton deformations.
Downloaded from http://rsif.royalsocietypublishing.org/ on June 17, 2017
receptor diffusion-limited
receptor diffusion and cytoskeleton
deformation-limited
8
cytoskeleton deformation-limited
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diffusion
diffusion
NPs
NPs
NPs
clathrin
creep
creep
cytoskeleton
1
x0/xL
Figure 7. Different regimes on cellular uptake of NPs recognized in terms of normalized initial receptor density. (Online version in colour.)
3.5. Effect of cytoskeleton stiffness on the optimal size
and maximum engulfing depth of cylindrical
nanoparticles
Similar to the previous studies [21], optimal sizes for both
horizontal and vertical uptake of NPs exist. Small CNPs
have large surface curvatures, so engulfing them implies
the need to overcome high membrane bending energies. By
contrast, engulfing large NPs implies the need to overcome
large deformation energies of the cytoskeleton.
Figure 8 shows the inverse of the engulfing time at engulfing depth, h ¼ R, as a function of Young’s modulus of
cytoskeleton and the radius of the CNPs. For the horizontal
uptake of CNPs, optimal sizes corresponding to the fastest
engulfing process exist (figure 8a). These optimal sizes are
almost independent of Young’s modulus. However, for the
vertical uptake of NPs, Young’s modulus of the cytoskeleton
obviously influences the optimal sizes (figure 8b). A relatively fast increase in the optimal sizes along with Young’s
modulus can be seen in figure 8b.
Figure 9 shows the engulfing time, h ¼ R, as a function of
the radius of the CNPs under different Young’s modulus of
the cytoskeleton. These optimal sizes correspond to the smallest engulfing times and are influenced by cell stiffness.
Interestingly, these optimal sizes are also approximately
20 and 30 nm, close to those previously predicted by the
receptor-mediated membrane model [21] and experimental
observations [38].
Figure 10 displays the maximum engulfing depth as a function of Young’s modulus of cells for the vertical engulfment of
CNPs with R ¼ 50 nm. It can be seen from figure 10 that, for
softer parts of cells, the maximum engulfing depth is larger.
When Young’s modulus is below 2 kPa, this final engulfing
depth can even reach 400 nm, eight times R.
(a)
Ec (kPa)
If j0/jL is larger than 1, receptor recruitment through diffusion is no longer energetically favourable. Adhesion will
easily overcome membrane deformation and cytoskeleton
deformation. In this case, the dynamic uptaking process
will be controlled by the creeping of the cytoskeleton (remodelling). The membrane model is no longer valid in this
regime. A detailed analysis of this regime has recently been
provided by Wang et al. [37].
(b)
1/t (h = R) s–1 × 10–3
10
9
4.5
8
4.0
7
3.5
6
3.0
5
2.5
4
2.0
3
1.5
2
1.0
1
0.5
0
10
15
20
25
R (nm)
30
35
40
1/t (h = R) s–1
5.0
0.030
4.5
4.0
0.025
3.5
Ec (kPa)
3.4.3. Regime 3
0.020
3.0
2.5
0.015
2.0
0.010
1.5
1.0
0.005
0.5
0
30
35
40
R (nm)
45
50
Figure 8. Inverse of engulfing time as a function of cell stiffness and radius
of cross section of (a) horizontally and (b) vertically oriented CNPs. (Online
version in colour.)
4. Conclusion
A coupled elasticity–diffusion model was established to
study the cellular uptake of CNPs with horizontal and vertical
orientations while considering the effects of ligand–receptor
binding, receptor diffusion, membrane deformation and cytoskeleton deformations. The proposed model showed that the
effect of cytoskeleton deformation should be considered to elucidate the uptaking processes of long CNPs or large spherical
NPs. Membrane models could be considered a good substitute
only when NPs are small enough. A CNP with vertical
J. R. Soc. Interface 12: 20141023
0
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(a) 1600
1400
t (h = R) (s)
1200
1000
800
600
Gao et al. [21]
Ec = 5 kPa
400
Ec = 10 kPa
15
20
25
30
R (nm)
35
40
80
t (h = R) (s)
300
250
200
150
100
50
0
2
4
6
8
Young¢s modulus of cells (kPa)
10
12
Figure 10. Maximum engulfing depth as a function of Young’s modulus of
cells for vertical CNPs with radius 50 nm. (Online version in colour.)
(b) 100
60
40
Gao et al. [21]
Ec = 2 kPa
Ec = 5 kPa
20
25
350
30
35
40
45
50
R (nm)
Figure 9. Comparison of the engulfing time, t, at h ¼ R versus the radius, R,
of the CNPs from previous model by Gao et al. [21] in the absence of cytoskeleton deformation and the coupled elasticity – diffusion model for different
cell stiffness, where j0/jL ¼ 0.01. The CNPs enter into cells (a) horizontally
and (b) vertically, respectively. (Online version in colour.)
orientation exhibited a much faster initial uptaking speed than
that with horizontal orientation at the initial stage of the uptaking process, where deformation of the membrane was more
important than that of the cytoskeleton. Cytoskeleton stiffness
significantly influenced the engulfing process only after a
certain engulfing depth. Larger cytoskeleton stiffness corresponded to a slower engulfing process. Optimal sizes of
CNPs engulfed at different orientations were observed. Cytoskeleton stiffness showed more significant influence on the
optimal size for vertical uptake than for horizontal uptake.
Based on the proposed model, the phenomenon identified
by Shi et al. [6] in which CNPs, such as CNTs, enter the cells
through the tip first, was hypothesized to be true only at the
initial stage of the uptaking process. In most cases, vertically
oriented CNPs probably stuck at a certain length, which may
have been engulfed by the soft part of the cytoskeleton. However, this attachment was subsequently stopped by the stiff
part of the cytoskeleton.
Funding statement. This research is supported by grants from the
National Natural Science Foundation of China (11472119, 11032006,
11121202), National Key Project of Magneto-Constrained Fusion
Energy Development Program (2013GB110002), the Fundamental
Research Funds for the Central Universities (lzujbky-2013-1).
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