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Downloaded from http://rsif.royalsocietypublishing.org/ on June 17, 2017 rsif.royalsocietypublishing.org 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. Downloaded from http://rsif.royalsocietypublishing.org/ on June 17, 2017 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. rsif.royalsocietypublishing.org 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 Downloaded from http://rsif.royalsocietypublishing.org/ on June 17, 2017 (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) rsif.royalsocietypublishing.org @ j(s, t) @ j(s, t) ¼ @t @s 3 Downloaded from http://rsif.royalsocietypublishing.org/ on June 17, 2017 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 rsif.royalsocietypublishing.org Liu et al. [30] solution based on linear elasticity Downloaded from http://rsif.royalsocietypublishing.org/ on June 17, 2017 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 rsif.royalsocietypublishing.org (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. Downloaded from http://rsif.royalsocietypublishing.org/ on June 17, 2017 (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 rsif.royalsocietypublishing.org 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 Downloaded from http://rsif.royalsocietypublishing.org/ on June 17, 2017 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 rsif.royalsocietypublishing.org 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 rsif.royalsocietypublishing.org 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 Downloaded from http://rsif.royalsocietypublishing.org/ on June 17, 2017 (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). References 1. 2. 3. 4. 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