Efficiency of cellular uptake of nanoparticles via receptor-mediated endocytosis
aa r X i v : . [ q - b i o . S C ] O c t Efficiency of cellular uptake of nanoparticles viareceptor-mediated endocytosis
Anand Banerjee, Alexander Berzhkovskii, and Ralph Nossal Program in Physical Biology, Eunice Kennedy Shriver National Institute of Child Health,and Human Development, National Institutes of Health, Bethesda, Maryland 20892, USA Mathematical and Statistical Computing Laboratory,Division of Computational Bioscience,Center for Information Technology,National Institutes of Health, Bethesda, Maryland 20892, USA (Dated: July 30, 2018)
Abstract
Experiments show that cellular uptake of nanoparticles, via receptor-mediated endocytosis,strongly depends on nanoparticle size. There is an optimal size, approximately 50 nm in diam-eter, at which cellular uptake is the highest. In addition, there is a maximum size, approximately200 nm, beyond which uptake via receptor-mediated endocytosis does not occur. By comparingresults from different experiments, we found that these sizes weakly depend on the type of cells,nanoparticles, and ligands used in the experiments. Here, we argue that these observations areconsequences of the energetics and assembly dynamics of the protein coat that forms on the cy-toplasmic side of the outer cell membrane during receptor-mediated endocytosis. Specifically, weshow that the energetics of coat formation imposes an upper bound on the size of the nanoparti-cles that can be internalized, whereas the nanoparticle-size-dependent dynamics of coat assemblyresults in the optimal nanoparticle size. The weak dependence of the optimal and maximum sizeson cell-nanoparticle-ligand type also follows naturally from our analysis.email - [email protected] NTRODUCTION
In recent years there has been great interest in using nanoparticles (NPs) for variousbiomedical applications including imaging, biosensing, and targeted gene/drug delivery (seereview articles [1–3] and references therein). Successful realization of these applicationsrequires efficient cellular uptake of the NPs. To this end, the NPs are coated with ligandsthat allow them to bind to specific cell surface receptors and be internalized via receptor-mediated endocytosis. An understanding of how the physical properties of NPs, like theirsize, shape, charge, etc., affect the internalization process is crucial for designing NPs forbiomedical purposes.Experiments show that the NP size is an important parameter that determines the mech-anism of their cellular uptake. In particular, NPs smaller than approximately 200 nm areinternalized typically via receptor-mediated endocytosis whereas, for larger NPs other mech-anisms are involved [4, 5]. Furthermore, the uptake rate of the NPs, that are internalizedvia receptor-mediated endocytosis, is strongly size dependent. There is an optimal NP size,approximately 50 nm in diameter, at which the uptake rate is highest [4–11]. In Table I wecollected data from various experiments designed to study size dependence of NP uptake.The experiments were performed using different kinds of cell lines, NPs, and ligands; yet theoptimal size was found to be approximately the same. This surprising observation suggeststhat the optimal size weakly depends on the above mentioned factors.Several theoretical models of receptor-mediated endocytosis of NPs have been proposedin the literature [12–16]. A common feature of these models is that they assume the uptakeis controlled by the formation of chemical bonds between receptors on the cell surface andligands attached to the NP. Using such an approach these studies conclude that the optimalNP size is a function of the receptor density on the cell membrane, ligand density on the NP,and the receptor-ligand binding energy. These parameters however can change significantlydepending on the cell line and ligands used in the experiment. Therefore, in the framework ofthese models, it is difficult to explain the same optimal size observed in different experiments.Furthermore, such an approach leads to the prediction that even micron sized NPs canbe internalized via receptor-mediated endocytosis [12, 13] which has never been observedexperimentally.Along with the formation of chemical bonds between the NP ligands and cell surface2eceptors, receptor-mediated endocytosis involves the assembly of a protein coat on thecytoplasmic side of the outer cell membrane. In the case of clathrin-mediated endocyto-sis - which is a form of receptor-mediated endocytosis - the coat contains several proteinsincluding clathrin, adaptor proteins, membrane bending proteins like epsin, amphiphysin,etc.[17, 18]. The coat assembly plays a vital role in internalization of cargo, and any inter-ference with this process drastically reduces the endocytic capacity of a cell. For example,cellular uptake of NPs is significantly reduced when the cells are pretreated with sucrose orpotassium-depleted medium [4, 19], which are known to disrupt coat formation. The abovementioned models do not take the coat assembly into explicit consideration. Therefore, howthis key aspect of the cellular endocytic machinery affects the uptake of NPs remains to beelucidated.In this paper we make a step in this direction. Our main purpose is to demonstratethat, contrary to the current understanding, the size-dependence of cellular uptake of NPsis determined by the coat assembly. To do so, we use a previously developed coarse-grainedmodel of the coat assembly that focuses on vesicle formation during clathrin-mediated endo-cytosis [20]. The model was developed to explain the fates and lifetimes of clathrin coated
TABLE I: Summary of experimental results on the size dependence of cellular uptake of NPs. NPssmaller than 200 nm are internalized via receptor-mediated endocytosis, whereas for larger NPsother internalization mechanisms are involved [4, 5]. Bold faced numbers indicate the NP size forwhich the cellular uptake is highest.Cells NP type Ligand NP size (diameter nm) Ref.B16-F10 Latex beads No ligand ,100,200,500 4MNNG/HOS Metal hydroxide Not specified ,100,200,350 5Hela Quantum dots Not specified 5,15, ,74,100 7CL1-0, Hela Gold single-stranded DNA ,75,110 8HeLa Mesoporous silica Not specified 30, ,110,170,280 9A549, HeLa, MDA Gold Transferrin 15,30, ,72,86,97,162 11 MODEL
We start by describing the main steps involved in NP internalization via clathrin-mediatedendocytosis. During clathrin-mediated endocytosis a ligand-coated NP first binds to a spe-cific receptor on the surface of the cell membrane (see Fig. 1). After binding, the NP-receptorcomplex binds adaptor proteins (typically AP-2), which recruit other endocytic proteins, andclathrin-coated pit (CCP) assembly begins. CCP assembly is a stochastic process which hastwo possible final outcomes [21, 22]. One is that the CCP grows in size and forms a vesi-cle, in which case the NP is completely wrapped and internalized. The other possibilityis that the CCP grows only up to a certain size and then disassembles. In this case the
Ligand clathrin-coated pit clathrin-coated vesicle cytoplasm coat proteins
FIG. 1: Schematic diagram of NP internalization via clathrin-mediated endocytosis. A NP firstbinds to a specific cell surface receptor forming a NP-receptor complex. The complex binds thecoat proteins and CCP assembly begins. The CCP either grows to form a vesicle, in which casethe NP is internalized, or grows only up to a certain size and then disassembles.
Quantifying internalization efficiency
Similar to other approaches [12, 15], we characterize the NP internalization efficiency bythe mean internalization time, τ , defined as the mean time between the binding of a NP tothe membrane and its internalization. As shown in Appendix A, τ = τ w + ( τ + P f τ f ) /P w , (1)where τ is the mean time required for the initiation of CCP assembly around a free NP-receptor complex, P w and P f = 1 − P w denote the probabilities of successful and unsuccessfulwrapping of a NP, and τ w and τ f denote the mean durations of these processes. Here w and f indicate successful and failed wrapping of the NP, respectively. Our assumption thatthe NP-receptor dissociation may be neglected is valid if P w is not too small. Otherwise, τ becomes very large, and the dissociation of the complex should be taken into consideration.In order to calculate the quantities appearing in Eq. 1 we use the model of CCP assemblydeveloped in Ref. 20. Here we briefly describe the model and list the underlying assumptions.As mentioned earlier the coat that forms during CCP assembly contains several proteinsand has a complex structure. Proteins like epsin and amphyphysin bind directly to the cellmembrane and impart a local curvature; whereas clathrin triskelions (three-legged, pinwheel-wheel shaped complexes) bind with other clathrin triskelions to form a three-dimensionalscaffold which is linked to the membrane through the adaptor proteins (typically AP-2).The clathrin scaffold imparts global curvature to the cell membrane. Incorporating thiscomplex structure of the coat into a model is an extremely complicated task. To overcomethis difficulty, in Ref. 20 we proposed a coarse-grained description of CCP assembly. Themain idea was to replace the real protein coat by a coat made up of identical units referredto as monomers (Fig. 2). We assumed that (1) the coat made up of monomers has its ownbending rigidity and a spontaneous curvature, (2) the shape of the model CCP (pit) is aspherical cap, (3) the monomers are structureless, which means that at the time of binding5he orientation of a monomer is not important. Due to these assumptions certain details ofCCP assembly were lost, but this is the price we had to pay for a tractable model whichstill contained the essential features of the assembly process. We validated this model byshowing that it was capable of explaining the experimentally measured lifetime distributionof CCPs [20]. Similar coarse-grained approaches for modeling the protein coat have beenused in Refs. 23 and 24 for studying endocytic vesicle formation in yeast and COP vesicleformation in the Golgi, respectively. Clathrin Adaptors Other proteins
Monomers Monomer coat Protein coat
FIG. 2: Coarse-grained model of a clathrin-coated pit (CCP). In a real CCP the protein coatcontains clathrin and several other proteins. The model coat is made of identical monomeric units.The shape of the model CCP is assumed to be a spherical cap, and the average area of a monomeris chosen to be the same as that occupied by a clathrin triskelion in a real CCP.
Using the coarse-grained model of CCP discussed above, the dynamics of CCP assemblyaround a NP-receptor complex can be described by the kinetic scheme shown in Fig. 3. Inthis kinetic scheme the symbol n is the number of monomers in a pit that forms around theNP-receptor complex, and N is the number of monomers needed for a complete vesicle. N is related to the NP diameter d NP by the relationship N = π ( d NP + 2 l b ) /λ, (2)where l b is the typical length of the receptor-ligand bond between the NP and the cell +1 α β α β … α n -1 β n n α n β n +1 … α N -1 β N N k N k β FIG. 3: Kinetic scheme of pit assembly. Symbols n , and N refer to the number of monomers ina pit and a vesicle, respectively. The rate constants α n and β n characterize the growth and decayrates of a pit of size n . k is the rate at which the first monomer binds to the NP-receptor complex,and k N is the rate of scission of a vesicle from the membrane. λ is the average area occupied by a monomer. The rate constants α n and β n characterize the growth and decay rates of a pit of size n , k = 1 /τ is the rate constantfor binding of the first monomer to the NP-receptor complex, and k N is the rate constantfor the scission of a vesicle from the membrane. We assume that the forward and backwardrate constants are related through detailed balance β n = α n − exp[ ˜ F ( n ) − ˜ F ( n − , n = 2 , ..., N, (3)where ˜ F ( n ) = F ( n ) / ( k B T ), F ( n ) is the formation free energy for a pit containing n monomeric units, k B is the Boltzmann constant, and T is the absolute temperature.We choose the forward rate constants to be of the form α n = γf ( n, N ), n = 1 , , ..., N − γ is a kinetic parameter proportional to the product of the free monomers concentra-tion and the bimolecular association rate constant between a free monomer and a pit. Thefunction f ( n, N ) gives the number of available binding sites on the edge of a pit of size n .Using that the shape of a pit is a spherical cap, this function can be written as f ( n, N ) = ρ p n ( N − n ) /N , (4)where ρ is a dimensionless parameter (see Appendix B).In terms of the kinetic scheme in Fig. 3, the quantities P w and P f (see Eq. 1) are theprobabilities that a random walk, starting from site n = 1, eventually reaches sites n = N and n = 0, respectively, and the times τ w and τ f are the mean durations of the two processes(which formally are conditional mean first-passage times [25, 26]). Analytical expressionsfor these quantities are well known [25] P w = Ψ Ψ + N P m =1 Φ m , P f = 1 − P w , (5) τ w = N P m =1 exp[ − ˜ F ( m )] m P l =1 Ψ l N P l = m Φ l Ψ + N P m =1 Φ m , (6) τ f = Ψ N P m =1 exp[ − ˜ F ( m )]( N P l = m Φ l ) N P l =1 Φ l (Ψ + N P m =1 Φ m ) , (7)where functions Ψ n and Φ n are given by Ψ n = exp[ ˜ F ( n )] /β n and Φ n = exp[ ˜ F ( n )] /α n .7 ree energy of pit formation The free energy of pit formation can be written as [20, 24] (see details in Appendix C) F ( n, N ) = E ( N ) n + σ p n ( N − n ) /N . (8)The free energy is mainly dominated by the first term, E ( N ) n , which is proportional to thenumber of monomers in the pit. It includes the costs of the membrane and protein coatdistortions, entropic cost of immobilizing the monomers, and the binding energy gained dueto coat formation. The second term is the line tension energy with σ being the edge-energyconstant. We use a Helfrich type expression [27] for the membrane and coat distortionenergy, and assume that the spontaneous curvature of the cell membrane is zero and that ofthe coat is finite. In addition, we assume that the binding energy and the entropic cost ofimmobilizing the monomers are proportional to the number of monomers in the pit. Basedon these assumptions we get E ( N ) = 8 πκ m N + 8 πκ p N − s NN p ! − ǫ b , (9)where κ m and κ p are the bending rigidities of the cell membrane and the coat, respectively, N p is the natural number of monomers in the coat - which is determined by the intrinsiccoat curvature, and ǫ b is the effective monomer binding energy, i.e., the difference betweenthe binding energy and entropic cost. Parameter values
The values of the parameters used in our calculation are summarized in Table II. Thesevalues, except for ǫ b , are identical to those in Ref. 20. The rationale behind the choices isas follows: The value of κ m typically lies between 10-25 k B T [28]; we choose κ m = 20 k B T .In vitro, clathrin triskelions assemble into baskets of different sizes. The size distribution ofbaskets is typically narrow and has a peak close to d p = 90 nm in diameter [29]. Using theaverage area occupied by a clathrin molecule, λ = 310 nm , and the relation λN p = πd p ,we find that a 90 nm basket would have approximately 80 clathrin triskelions. So we choosethe natural coat size to be N p = 80. The value of λ = 310 nm was estimated using therelation between diameters of clathrin baskets of different sizes and the number of clathrin8 ABLE II: Parameter valuesParameter Description Value κ m Membrane bending rigidity 20 k B Tκ p Protein coat bending rigidity 200 k B TN p Number of monomers in a typical vesicle 80 ǫ b Binding energy per monomer 10 k B Tσ Edge energy constant 2 k B Tγ Kinetic parameter 0.18 sec − τ Average time for initiation of a CCP 20 sec β Backward rate constant 0.1 sec − k N Vesicle scission rate ∞ λ Average area occupied by a monomer 310 nm l b Length of receptor ligand bond 15 nm ρ Dimensionless parameter 2 triskelions they contain [30]. In Ref. 20, using experimental data on lifetime distribution ofabortive CCPs (CCP with no cargo) we estimated the value of the effective binding energyper monomer, ǫ b , to be approximately 5 k B T . Experiments show that in the presence ofcargo (present case) the binding energy increases. An approximate range for its value canbe determined using the following argument: for a clathrin-coated vesicle to be energet-ically stable, the effective binding energy has to be greater than the membrane bendingenergy (8 πκ m ≈ k B T ). Since a typical vesicle has 80 monomers, the binding energyper monomer should be greater than 500 / ≈ k B T . To get an upper bound to the bind-ing energy we consider the electrostatic binding energy between proteins containing a BAR(Bin-Amphiphysin-Rvs) domain and the membranes, which is estimated to be around 15 k B T [31]. We choose a number in between these values and set ǫ b = 10 k B T . The parameter, κ p , captures the effective bending rigidity of the protein coat. In Ref. 20 we estimated itsvalue to be approximately κ p = 200 k B T ; here we use the same value. To the best of ourknowledge, the value of τ in the case of nanoparticles has never been measured. In the caseof a particular virus (canine parvovirus) entering via clathrin-mediated endocytosis, it wasfound to be approximately 20 sec [32]. Thus we choose τ = 20 sec. We choose the length of9 receptor ligand bond to be l b = 15 nm [33]. Parameters σ , ρ , and γ have the same valuesas in Ref. 20: σ = 2 k B T , ρ = 2, γ = 0 .
18 sec − . RESULTS
Figure 4 shows E ( N ), Eq. 9, as a function of the NP size. It attains a minimum at d ∗ E ≈
68 nm, which corresponds to the size of the NP whose carrier vesicle is energeticallymost stable. This size can be found by solving the equation ∂E ( N ) /∂N = 0, which leads to d ∗ E = (1 + κ m /κ p ) q λN p /π − l b . (10)For NPs larger than d ∗ E , E ( N ) increases due to the energetic cost of protein coat deformation,whereas for NPs smaller than d ∗ E , E ( N ) increases mainly due to the energetic cost of cellmembrane deformation. The vertical dash-dotted lines at d min ≈
46 nm and d max ≈
105 nmshow the NP sizes at which E ( N ) = 0. Analytical expressions of these sizes can be obtained
40 50 60 70 80 90 100 110−4−2024 E n e r g y ( k B T ) W r a pp i n g P r o b a b ili t y d NP = p N λ/π − l b (nm) P w ( N ) E ( N ) d min d max d ∗ E FIG. 4: Energy E ( N ), Eq. 9, (solid curve) and the wrapping probability, P w ( N ), (dashed curve),as functions of the NP size. P w ( N ) is high when E ( N ) <
0, and pit assembly is energeticallyfavorable. Dashed-dotted vertical lines at d min ≈
46 nm and d max ≈
105 nm correspond to the NPsizes at which E ( N ) = 0. The arrow at d ∗ E ≈
68 nm indicates the NP size whose carrier vesicle isenergetically most stable.
10y solving the equation E ( N ) = 0, which leads to d min d p = 1 − p (1 + k ′ m ) ǫ ′ b − k ′ m (1 − ǫ ′ b ) − l b d p , d max d p = 1 + p (1 + k ′ m ) ǫ ′ b − k ′ m (1 − ǫ ′ b ) − l b d p , (11)where κ ′ m = κ m /κ p , ǫ ′ b = N p ǫ b / πκ p , and d p = p λN p /π .Figure 4 also shows the plot of the wrapping probability, P w ( N ). In regions where the sumof membrane and coat distortion energies is greater than the binding energy ( E ( N ) > P w ( N ) is negligibly small. In contrast, inthe region where E ( N ) < P w ( N ) rises sharply and then remains high (about 0.8) andapproximately constant. Notably, sizes of several viruses which enter through clathrin-mediated endocytosis, including dengue virus (40-60 nm), semliki forest virus (50-70 nm),and reovirus (60-80 nm), fall within this range [34].Figure 5 shows plots of the mean wrapping time, τ w , Eq. 6, and the mean internalizationtime, τ , Eq. 1, as functions of the NP size. The mean wrapping time has a minimum at d ∗ w ≈
55 nm, which is different from d ∗ E ≈
68 nm. The difference between the two sizes is dueto the NP size-dependent dynamics of coat assembly. At d ∗ w , E ( N ) ≈ − k B T , and hence α n /β n ≈ ≫
1. This implies that coat assembly (hence NP wrapping) proceeds with a low
40 50 60 70 80 90 10050100150200250 T i m e ( s ec ) d NP = p N λ/π − l b (nm) d min d max d opt d ∗ w τ τ w FIG. 5: Mean wrapping time, τ w , Eq. 6, (dashed curve), and the mean internalization time, τ ,calculated using Eq. 1 with τ = 20 sec, as functions of the NP size. Arrows at d ∗ w and d opt indicatethe NP sizes at which τ w and τ are minimum, respectively. Our estimate d opt ≈
55 nm matcheswell with experimental observations presented in Table I. d ∗ w , up to approximately 85 nm,the inequality α n /β n ≫ d ∗ w , the free energy E ( N ) rises sharply,and, therefore, the rate constants α n and β n become comparable. Thus, even though thenumber of monomers needed to wrap a NP decreases, the wrapping time increases due tofrequent dissociation of the monomers. The mean time of failed attempts, τ f , given in Eq. 7,shows a trend very similar to that of τ w , but its values are almost an order of magnitudesmaller.In Fig. 5 we show the mean internalization time, τ , calculated using Eq. 1 with τ = 20 sec.The mean internalization time includes the mean wrapping time, τ w , and the mean timespent by the NP in failed attempts. In the region where P w is high and almost constant(see Fig. 4), we find P w τ w ≫ P f τ f . Equation 1 then simplifies to τ ≈ ( τ /P w ) + τ w , and τ is just the mean wrapping time, τ w , shifted by a constant τ /P w . When d NP is closeto d min or d max , P w drops sharply and, therefore, τ increases more rapidly than τ w . Themean internalization time has a minimum at d opt , which corresponds to the NP size atwhich the cellular uptake of the NPs is the fastest. Our analysis predicts d opt ≈
55 nm,which is close to the optimal NP size observed in different experiments (see Table 1). Inthe case of NPs, the size dependence of internalization times at a single NP level has neverbeen measured, therefore a direct comparison of our results with with experimental data isnot possible. However, internalization times of some viruses and virus-like particles enteringinto cells via clathrin-mediated endocytosis have been measured to be in the range 50-400 sec[21, 32, 35, 36], which agrees with our analysis.Finally, we look at sensitivity of our results to variations in coat parameter values. InFig. 6 we show plots of d min , d max , d opt , and d ∗ E as functions of the parameters κ p , ǫ b , and N p . The dependence of the optimal size, d opt , on the coat parameters has been determinednumerically, while other dependences are are given by Eqs. 10 and 11. Only the maximumNP size, d max , shows appreciable variation; the other quantities show weak dependences onthe coat parameters. This demonstrates that our main results are stable with respect tosmall variations in the coat parameters around their chosen values.12
00 150 200 250406080100120140160 κ p ( k B T ) S i ze ( n m ) Chosen κ p value d max d ∗ E d opt d min (a) ǫ b ( k B T ) S i ze ( n m ) Chosen ǫ b value d max d ∗ E d opt d min (b)
50 60 70 80 90 100406080100120140 N p S i ze ( n m ) Chosen N p value d max d ∗ E d opt d min (c) FIG. 6: Plots of d min , d max (Eq. 11), d ∗ E (Eq. 10), and d opt (calculated numerically), as functions of(a) κ p - the bending rigidity of the protein coat (b) ǫ b - effective monomer binding energy, and (c) N p - natural number of monomers in the coat. The vertical dash-dotted line in each case showsthe parameter value used in our calculations. The plot shows that our results for d min , d ∗ E , and d opt are stable with respect to small variations in the parameter values. Only the maximum NPsize d max shows significant variation. UMMARY AND DISCUSSION
In this study we investigated how the assembly of the protein coat on the cytoplasmicside of the plasma membrane affects the cellular uptake of NPs. To address this question wehave used a previously developed model of clathrin-coated vesicle formation. We have usedthe mean internalization time of a NP, τ , as the measure of its internalization efficiency, andcalculated the dependence of τ on the NP size. We found that the NP size has lower andupper boundaries ( d min and d max ) at which the internalization time becomes very large, i.e.,beyond these sizes, internalization via clathrin-mediated endocytosis is highly improbable.We also found that there is an optimal NP size d opt at which the internalization time is aminimum. All these sizes are determined by the parameters of the coat assembly process.As described earlier, d opt is determined by the dynamics of coat assembly. Since the coatparameters do not change appreciably between different cells and are also independent ofthe details of the NP design, an explanation for why the same optimal size was observedin different experiments (see Table I) follows naturally from our analysis. In contrast, thisobservation is difficult to rationalize using previous models which predict that optimal sizedepends on the ligand density on the NP, the density of the corresponding receptors on thecell membrane, and the receptor-ligand binding energy [12–14], since these parameters canvary significantly depending on the cell line and ligand used in the experiment.Our calculation of the smallest NP size, d min ≈
46 nm, is based on the assumption thatthe size of the NP is related to the size of its carrier vesicle through d minV = d min + 2 l b , whichgives d minV ≈
76 nm. In principle, however, NPs with diameter slightly smaller than d min can be internalized in a vesicle of size d minV . Also, it has been shown that NPs much smallerthan d min can be internalized in clusters (multiple particles per vesicle), in a vesicle of sizemuch larger than the size of individual NPs [19]. Therefore, a comparison of d min withexperimental data is meaningless. In this case it is more meaningful to compare d minV withthe size of the smallest clathrin-coated vesicles. Experimentally observed size of smallestclathrin-coated vesicles is approximately 70 nm in diameter [37], which agrees very well withour estimate.Our estimate of the largest NP size, d max ≈
105 nm, and the size of its carrier vesicle, d maxV ≈
105 + 30 = 135 nm, are smaller than their corresponding experimentally measuredvalues 200 nm [4, 5], and 200 nm [37], respectively. As shown in Fig. 6 the value of d max
14s sensitive to the coat parameter values. By changing their values slightly we can get d max close to experimentally observed values, while keeping d opt the same. For example, for κ p = 150 k B T and ǫ b = 12 k B T , we get d max = 140 nm, d maxV = 170 nm, d min = 40 nm, and d opt = 50 nm. However, in this paper our aim is not to match the different sizes precisely,but rather to see the extent to which our previously developed model can explain the sizedependence of NP uptake without changing the parameter values. Considering the fact thatmost of the parameters were determined in a completely different context (by fitting lifetimedistribution of abortive CCPs) we think that such a disparity is acceptable. As mentionedearlier, other models that do not take coat assembly into consideration, incorrectly predictthat very large (micron size) NPs can be internalized via receptor-mediated endocytosis[12, 13].Although in our model the density of ligands on a NP, the receptor density on the cellmembrane, and the receptor-ligand binding energy do not appear explicitly, these factors doenter our model implicitly. For example, our initial assumption that the dissociation of theNP from the membrane can be neglected would hold true only if either the receptor-ligandbinding energy is strong, or the ligand and receptor densities are large enough so that a NPquickly attaches to the cell membrane by multiple receptor-ligand bonds. Multiple bondslead to an increased lifetime of NPs on the cell membrane [38, 39]. Also, it has been shownthat CCP assembly is triggered when there is receptor clustering [40] which, for our model,implies that a few receptor-ligand bonds probably have to form before the first monomercan arrive. Thus, the time τ might be affected by the above mentioned parameters.Experimentally, the question of how the ligand density on a NP and receptor density oncell membrane affect cellular uptake is not clearly understood. It has been observed thatincreasing the ligand density increases cellular uptake due to an enhanced residence time ofthe NP on the cell membrane, and not due to an increase in the internalization rate [38]. Thisobservation is consistent with our hypothesis that the internalization time is determined bythe kinetics of coat assembly. The overall picture of NP internalization proposed in our modelcan be tested experimentally with total internal reflection fluorescence (TIRF) microscopy.Using dual color TIRF, where both NPs and CCPs are fluorescent, uptake of NPs at a singleparticle level can be monitored. This will allow simultaneous measurements of τ , τ w , and τ . To conclude we show that several experimental observations related to size dependent15ellular uptake of NPs, including the optimal NP size, can be understood to be consequencesof the protein coat assembly process. Therefore, future efforts on modeling endocytosis ofNPs and designing NPs for biomedical applications must take the effect of the protein coatassembly explicitly into consideration. APPENDICIESAppendix A: Mean internalization time τ Here we derive the expression for the mean internalization time τ given in Eq. 1. Thistime is defined as the average time between the NP binding to the cell membrane and itsinternalization, assuming that the binding is irreversible. Consider an ensemble of NP-receptor complexes formed on the cell membrane at time t = 0. The mean internalizationtime τ can be written as τ = P w ( τ + τ w ) + P f P w (2 τ + τ f + τ w ) + P f P w (3 τ + 2 τ f + τ w ) + ..... (12)The first term in right hand side of this equality is the contribution from the fraction ( P w ) ofcomplexes which are internalized on the first attempt, i.e., the complexes which bind to coatprotein and get internalized. The second term is the contribution from the fraction ( P f P w )of complexes that are internalized on the second attempt, i.e., they bind to coat proteins,dissociate from them, bind to coat protein for the second time, and then get internalized.Subsequent terms can be understood in the same way. Upon summing the series, we obtain τ = ( τ + P w τ w + P f τ f ) /P w . (13)This expression for the mean internalization time τ can be written in the form given in Eq. 1. Appendix B: Derivation of f ( n ) in Eq. 4 Using spherical coordinates (see Fig.7) the surface area of the pit can be written as A ( θ ) = 2 πR [1 − cos( θ )] = λn, (14)where R is the radius of a sphere having the same curvature as the pit. This leads to therelation between cos( θ ) and the number of monomers, n , in the pit,cos( θ ) = 1 − λn πR . (15)16rom the above equation, we thus infer that the radius, r ( n ), of the circular growing edgeof a pit is r ( n ) = R sin( θ ) = R p [1 + cos( θ )][1 − cos( θ )] = s(cid:18) λnπ (cid:19) (cid:18) − λn πR (cid:19) . (16)Introducing the average linear span of the monomer, denoted by L , we find that the numberof available binding sites on the periphery of the pit is f ( n ) = 2 πr ( n ) L = 2 πL s(cid:18) λnπ (cid:19) (cid:18) − λn πR (cid:19) . (17)By changing variables from R to N using the relation 4 πR = λN , we arrive at f ( n ) = ρ p n ( N − n ) /N , (18)where ρ is a dimensionless parameter given by ρ = p πλ/L . Appendix C: Derivation of F ( n ) in Eq. 8 The formation free energy of a pit made of n monomers and having a curvature c can bewritten as F ( n, c ) = 2 κ m λnc + 2 κ p λn ( c − c p ) − ǫ b n + σf ( n, c ) . (19)The first term is the Helfrich energy [27] describing the energetic cost of bending the cellmembrane assuming that its spontaneous curvature is zero. The second term represents the R r ( n ) q FIG. 7: Spherical cap model of a pit. c p being the spontaneous curvature of the coat.The third term represents the effective binding energy. The fourth term is the line tensionenergy. By changing variables from c to N and c p to N p using the relations 4 π/c = λN and 4 π/c p = λN p , we arrive at the expression for the free energy of the pit formation givenin Eq. 8. ACKNOWLEDGMENTS
This study was supported by the Intramural Research Program of the National Institutesof Health (NIH) –
Eunice Kennedy Shriver
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