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Nanoparticle-decorated erythrocytes reveal that particle size controls the extent of adsorption, cell shape, and cell deformability Alexander Barbul, Karandeep Singh, Limor Horev-Azaria, Sabyasachi Dasgupta, Thorsten Auth, R. Korenstein, and Gerhard Gompper ACS Appl. Nano Mater., Just Accepted Manuscript • DOI: 10.1021/acsanm.8b00357 • Publication Date (Web): 02 Jul 2018 Downloaded from http://pubs.acs.org on July 9, 2018

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ACS Applied Nano Materials

Nanoparticle-Decorated Erythrocytes Reveal that Particle Size Controls the Extent of Adsorption, Cell Shape, and Cell Deformability Alexander Barbul1♣, Karandeep Singh2♣, Limor Horev–Azaria1, Sabyasachi Dasgupta2§, Thorsten Auth2, Rafi Korenstein1* and Gerhard Gompper2* 1

Department of Physiology and Pharmacology, Faculty of Medicine, Tel-Aviv University, Tel-Aviv 69978, Israel 2

Theoretical Soft Matter and Biophysics, Institute of Complex Systems and Institute for Advanced Simulation, Forschungszentrum Jülich, 52425 Jülich, Germany E-mail: [email protected]*, [email protected]* Key words: red blood cells, stomatocytes, Langmuir isotherms, multivalent binding, receptor free energy ♣ - contributed equally §- present address: Department of Physics, University of Toronto, Toronto ON M5S 1A7, Canada

Abstract Unraveling the interaction of nanoparticles with living cells is fundamental for nanomedicine and nanotoxicology. Erythrocytes are abundant and serve as model cells with wellcharacterized properties. Quantitative experiments addressing the binding of carboxylated polystyrene nanoparticles to human erythrocytes reveal saturated adsorption with only sparse (~2%) coverage of the cell membrane by partial-wrapped nanoparticles. The independence of the adsorbed area on particle size suggests a restricted number of adhesive sites on the membrane. Using a continuum membrane model combined with nanoparticle-membrane adhesion mediated by receptor-ligand bonds, we predict high bond energies and low receptor densities for partial-wrapped particles. With the help of computer simulations, we determine sets of receptor densities, receptor diffusion coefficients, minimal numbers of bound receptors required for multivalent binding, and maximal possible numbers of bound receptors that reproduce the experimental nanoparticle adsorption data. Nanoparticle decoration of

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erythrocytes leads to shape transformations and reduced cell deformability. We quantitatively characterize and interpret erythrocyte shape and deformability changes. The shape changes thus also offer insights into the modification of the mechanical properties of other mammalian cell membranes by adhered nanoparticles. A potential application of nanoparticle-loaded erythrocytes is retarded targeted drug delivery with a long lifetime of the particles in the blood circulation.

1. Introduction Adhesion of nanoparticles (NPs) to the plasma membrane is a crucial step for their initial interaction with cells. Unraveling the characteristics of this interaction is pivotal for advancements in nanotoxicology and nanomedicine. It has been shown that direct contact between NPs and the plasma membrane constitutes a primary event in nanotoxicology, which is followed by the endocytic uptake of the NPs with possible cytotoxic effects.1–4 The binding of NPs to cell membranes is also a cornerstone of targeted drug delivery for nanocarriers decorated with specific ligands that circulate in the blood.5–7 Experiments have demonstrated that the time that the NPs spend in circulation dramatically increases if they are bound to an erythrocyte.8 Nevertheless, besides potential applications, also a quantitative characterization of NP adhesion to cell membranes has been scarcely addressed,9,10 though a direct relation of the primary adhesion step and the secondary process of NP uptake has been observed.3 Studies of the interaction of NPs with lipid monolayers, supported lipid bilayers, and unilamellar vesicles have reported that the adsorption of small (~20 nm) amorphous silica NPs “freeze” the lipid bilayer. The NPs decrease the phospholipid lateral mobility, leading eventually to the release of membrane tension through stress-induced formation of a single microsized hole in a giant unilamellar vesicle. In contrast, larger particles (>80 nm) have been reported to increase lipid lateral mobility, probably due to defect formation in the membrane, eventually leading to collapse of the vesicles.11 In addition, the interaction of silica NPs with a 2 ACS Paragon Plus Environment

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dioleoyl phosphatidylcholine (DOPC) monolayer on a mercury film electrode has been shown to be inversely proportional to NP size when the NPs formed a monolayer on the DOPC bilayer, irrespective of their size.12 Theoretical and computer simulation studies of NPbilayer interactions range from molecular simulations for small NPs that penetrate the lipid bilayer13–19 to continuum models that can be solved analytically or numerically for large NPs with radii of 10 nm and above that get wrapped by the lipid bilayer.20–25 In the latter case, for wrapping a lipid-bilayer membrane around an NP, the deformation-energy costs due to bending rigidity and tension have to be overcome by the adhesion-energy gain for the contact between NP and membrane. In particular, continuum models allow for systematic studies of generic aspects of NP wrapping, such as membrane tension-stabilized partial-wrapped states for spherical NPs,23,26 continuous and discontinuous wrapping transitions due to different local curvatures of non-spherical NPs,22,27 membrane-mediated aggregation of NPs,23,28–30 and NP-induced tube formation.23,29,31,32 Recently, also stable partial-wrapped states and energy barriers for wrapping because of membrane spontaneous curvature and membrane curvature prior to wrapping have been predicted.20,21,31 Although lipid-bilayer model systems mimic some characteristics of cell membranes, they are devoid of the higher complexity of biological membranes with lipid-protein, protein-protein, and membrane-cytoskeleton interactions. In particular, the interaction of NPs with cell membranes is not limited to a homogeneous adhesion strength due to van der Waals attraction as for many model lipid-bilayer systems. A variety of adhesive sites, such as receptors in the membrane that interact with ligands on an NP,16 electrical charges,32 and domains or rafts that can have different mechanical properties and possibly different hydrophilicity than the surrounding membrane,33,34 lead to inhomogeneous NP-membrane interactions. For example, for receptor-ligand bonds both bond energy and receptor entropy within the membrane determine the effective adhesion strength. The maximal number of bound NPs is determined by receptor and ligand densities and the receptor diffusion coefficient within the 3 ACS Paragon Plus Environment


membrane.2,35–40

However, very few systematic experimental studies correlate pristine

physicochemical properties of NPs that are unmodified by proteins and lipids with NP adhesion to cells. We investigate the interaction of NPs and human erythrocytes using an interdisciplinary approach employing experiments, theoretical modeling, and computer simulations. Our study addresses the effect of NP size on the adhesion to erythrocytes and the consequent effect on the erythrocytes’ mechanical characteristics as reflected by erythrocyte shape and deformability changes. Erythrocytes do not show active endocytic uptake. This enables us to explore the adsorption of NPs to a complex cell membrane using a receptor model for the NPcell membrane interaction. Emphasis was put on preserving the physicochemical characteristics of the NPs, i.e. employing protein-free and lipid-free media, to prohibit the formation of a "corona" which coats NPs and masks their pristine surface characteristics.40 This allowed us to systematically explore the relationship between NP size and surface chemistry with their adhesion to the erythrocyte membrane. The binding of NPs was effectively characterized using Langmuir isotherms. However, the NPs strongly and irreversibly attach to cells as shown in the Supporting Information, which is not compatible with an isotherm. We therefore propose a binding mechanism dominated by kinetic processes. With the help of computer simulations and analytical calculations, we show that this hypothesis is consistent with the experimental observations. We model NP-membrane adhesion using receptor-ligand bonds and calculate the membrane deformation energy using a continuum membrane model. In order to obtain the wrapping state of an NP, we minimize the sum of membrane deformation energy and the receptor free energy. Our calculations predict partial wrapping, as observed in the experiments, if there are only few adhesive sites on the cell membrane. Adhered NPs of the smallest size of 27 nm were found to alter the shape distribution of erythrocytes, such that the fraction of discocytes is decreased. We attribute the experimentally observed shape changes to various mechanisms of NPs altering the 4 ACS Paragon Plus Environment

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spontaneous curvature of the membrane. These shape changes may be responsible for the decreased erythrocyte deformability.

2. Results and discussion 2.1. Experimental determination of the adsorption of polymeric nanoparticles of different sizes to human erythrocytes To quantitatively investigate the adhesion process of NPs to the cell membrane without contributions of active biological processes, we have chosen human erythrocytes as the cellular model system, since they are devoid of endocytosis. Our choice of erythrocytes also emerges from the many quantitative experiments, computer simulations, and analytical calculations that investigate shapes4–7,9 and potentially metabolically driven shape fluctuations of the cells.7,10,41–46 Moreover, erythrocytes are the most abundant cells in blood, and thus NPdecorated erythrocytes have a potential to serve as drug carriers in nanomedicine.5,7 The erythrocytes were exposed to a suspension of carboxylated surface-modified corefluorescent polystyrene NPs. To maintain a pristine NP surface devoid of protein or lipid corona, we incubated erythrocytes with NPs in a PBSG medium, containing salts and glucose solely. The medium thus provided an optimal chemical environment that interferes only minimally both with the surface of the NPs as well as with the erythrocyte membrane. In PBS medium, negative -potentials of about -30mV have been reported for the particles.47 While a bare polystyrene surface is hydrophobic, the—depending on the pH value—negatively charged carboxyl groups effectively make the NP surface more hydrophilic.48 Our lightscattering experiments show that the NP dispersions are stable for at least three hours, see section 4.1.

Polystyrene NPs are widely used for industrial products and biological assays.49 They are also convenient for investigating NP-erythrocyte interactions because of the commercial 5 ACS Paragon Plus Environment


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availability of a large variety of sizes and surface functionalizations.47 We have studied systems with NPs of three diameters: 27 nm, 45 nm, and 100 nm. The exposure was carried out under mild rotatory mixing of the NPs with cells for 15 min at 24ºC. After excessive washing, a combination of fluorescence and differential interference contrast (DIC) microscopy reveals NPs adhered to the erythrocyte membrane (Figure 1 A-D, Figure S1).

A

B

E

10 µm

0.5 µm

CD F

G

1 µm

1 µm

Figure 1. Adsorption of carboxylated polystyrene NPs to human erythrocytes. A-D Composition of differential interference contrast (DIC) and green fluorescence. Epifluorescent and DIC images of erythrocytes were collected using inverted Zeiss AxioObserverZ1 microscope equipped with X40 water immersion objective. E-G – Scanning electron microscope (SEM) micrographs of erythrocytes. For SEM imaging erythrocytes were attached to coverslips, fixed and viewed at Quanta 200 ESEM (FEI, USA). Incubation of erythroytes for 15 min at room temperature in suspension with ctot = 1012/ml: A – control erythrocytes without NPs; B,E – 27 nm; C,F – 45 nm; D,G – 100 nm..


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Visual inspection of the scanning electron microscope (SEM) images indicates that the NPs are adsorbed to the erythrocyte membrane with a significant contact area (Figure 1). For the 100 nm particles that are attached to the rims of the cells, we measure wrapping fractions of 0.1-0.15 and wrapped areas of 3,000 − 5,000 per NP (Figure 1 G, Figure S2). This finding is consistent with the observation that NPs were tightly adsorbed to the erythrocyte membrane, because they remained attached during the washing steps, and did not show significant desorption during a time period of 4 hours, as validated by flow cytometry (Figure S10). The erythrocyte population shows uneven labeling by the fluorescent NPs. Furthermore, some fluorescent spots on the erythrocyte membrane are brighter than others, suggesting NP aggregation. The presence of NP aggregates is further confirmed by SEM images (Figure 1 EG, Figure S2). Taking into account the absence of any significant NP aggregation in suspension (Figure S3), we assume that the clustering occurs on the cell membrane. Potential explanations for this process could be membrane-deformation mediated NP clustering, adhesion to existing lipid rafts, or the formation of lipid domains by NP adhesion.23,50,51 To quantify the adsorption of NPs to the cell membrane, we exposed erythrocytes to suspensions of fluorescent NPs with concentrations between ctot = 3.70 1010 particles/ml and ctot = 9.24 1013 particles/ml (0.4 g/ml to 1 mg/ml, volume fractions 4 10-4 to 10-3) for 27 nm NPs, between 9.98 109 particles/ml and 4.99 1013 particles/ml (0.5 g/ml to 2.5 mg/ml, volume fractions 5 10-4 to 2.5 10-3) for 45 nm NPs and between 1.82 109 particles/ml and 1.82 1012 particles/ml (1 g/ml to 1 mg/ml, volume fractions 10-6 to 10-3) for 100 nm NPs. Using flow cytometry, we measured the total fluorescence intensity per cell. The average number

of NPs adsorbed to a single erythrocyte (Figure 2, left) was calculated after calibration with beads that possess known amounts of dye molecules (Figure S3), taking into account the manufacturer data on the average amount of the fluorescent molecules in each batch of the studied NPs.



Figure 2. Adsorption isotherms of NPs to human erythrocytes and comparison of experimental adsorption data with computer simulations. Left panel - flow cytometry data in terms of the number of adsorbed NPs per erythrocyte after incubation with NPs of different sizes at various solution concentrations . The experiments have been performed at 5% hematocrit for 15 min at 24°C while shaking at 300 rpm. Amounts of adsorbed NPs per erythrocyte (blue lines) are compared to the amounts of adsorbed NPs obtained using Brownian dynamics simulations (red lines). The parameters used in the simulations are discussed in Section 2.2 and summarized in Table 2. Right panel – linear fitting of the 8 ACS Paragon Plus Environment

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adsorption data by Langmuir isotherms. The various values for the same concentrations are independent measurements with several thousand cells for each blood sample. Fitting parameters, goodness of fitting, as well as maximal adsorption and binding constants are provided in the figures. To examine the temporal dependence of NP adhesion on the incubation time, we have varied the duration of the exposure of erythrocytes to NP suspensions from less than one minute up to 90 minutes (Figure S5). Within this time frame, two regimes for the adsorption kinetics are observed. The fitting of the experimental data by an exponential association function (Figure S5) shows the existence of fast binding kinetics with a characteristic time of ~1 min, followed by much slower binding kinetics (Table S1). The slower process is represented by the almost linear gradual rise for 45 nm and 100 nm NPs, and an exponential accumulation with a characteristic time of 30-60 min, depending on the NP concentration, for 27 nm NPs. Due to the rise in erythrocyte shape volatility during prolonged incubations in the absence of the stabilizing effect of BSA, we restricted the incubation time with NPs to 15 min. Thus for studying the quantitative dependence of NP binding to erythrocytes on NP concentration, we focused our analysis on the first, fast-adsorption phase (Figure S5). Within the chosen time frame the erythrocyte viability is minimally compromised. We may relate the fast kinetics to the direct interaction of NP with receptors persisting on the cell surface and the slow kinetics to cell-membrane rearrangements, such as lipid flip-flop or clustering of membrane components, as a result of continued exposure to NPs. The characteristic time for enzyme-facilitated flip-flop and for scrambling of lipids in human erythrocytes indeed is of the order of tens of minutes, which resembles the time scale of the slow-binding process.52 Although we cannot precisely map the time scales of our coarsegrained simulations to the experiments, also our computer-simulation results clearly show a fast-binding and a slow-binding regime (Figures S6-S9). Here, these two regimes are expected due to multivalent NP binding, which has been shown to lead to a nonlinear relationship between the NP binding rate and the receptor density on the membrane.53 9 ACS Paragon Plus Environment


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Therefore, the observation of the fast-binding regime and the slow-binding regime further supports our hypothesis of multivalent binding. The adsorption of NPs varies between blood samples that have been taken and measured on different days. This variation is shown either by error bars (Figure 2, left) or by multiple data points for each NP concentration that represent individual measurements (Figure 2, right). Because several thousand cells have been analyzed in each measurement, the variation has to be attributed to the properties of the entire blood sample and not to inter-cell variation. The variation may thus reflect the conditions how the sample has been obtained and handled, such as the time of the day when the blood has been drawn and how fast the samples have been processed. The dependencies of the numbers of adsorbed NPs per cell on the bulk concentration of NPs for all three sizes of NPs were found to be fitted well by Langmuir-like adsorption model (Figure 2, right). The slope yields the maximal number of NPs adsorbed on a single erythrocyte (Slope=1/(maximal adsorption)), while the intercept with the y-axis gives the Langmuir adsorption constant KL for NPs to erythrocytes (Intercept=1/(KL x maximal

adsorption)) (Table 1). However, thermodynamic equilibrium and thus reversible binding of NPs is essential for a Langmuir-like adsorption model to apply. Table 1. Nanoparticle characterization and adsorption data from experiments and simulationsa NP size [nm] 27

45

100

NP surface area [nm2]

2.29 103

6.36 103

31.41 103

COOH groups per particle

4280

22000

107000

Langmuir model (exp.)

2500

714

172

Brownian dynamics simulations

2585

968

170


5.72

4.54

5.41


5.92

6.16

5.34


1.07 107

1.57 107

1.84 107


1.11 107

2.13 107

1.82 107

b

b

maximal adsorption of NPs per cell

c

total surface area of 2 adsorbed NPs [m ]

c

total number of adsorbed COOH groups


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8.01 10-14

4.67 10-13

1.16 10-11


1.25 1013

2.14 1012

8.62 1010


1.60 1013

9.50 1012

3.20 1010

Langmuir adsorption constant (exp.)

NP concentration at 50% occupancy [ml-1] a)

The parameters used in the simulations are discussed in Section 2.2 and summarized in Table 2. b) manufacturer data; c) at saturation

Because in our experiments irreversible binding has been observed (Figure S10), the Langmuir isotherms can only be used to characterize the experimental data effectively, while a different model is required to rationalize the physicochemical mechanism for NP adsorption. We propose a kinetic model based on receptor-ligand bond formation, fixed ligands on the NP, and mobile receptors in the membrane. The values for the NP concentrations, sizes, and diffusion coefficients directly correspond to the experimental values that are known precisely. Receptor density and diffusion coefficient are characteristic for the erythrocytes. Their values are not known because the molecules that act as receptors could not be identified; we have chosen fixed and equal values for all NP sizes and concentrations. The only parameters that are free and can be used to individually fit the simulation data to the experiments for each NP size are thus parameters that characterize bond formation and bond cooperativity. Using the independently determined values for the NP-related parameters and biologically reasonable values for receptor density and diffusion coefficient, and fitting the data with only two parameters per NP size, we find that most of the simulated adsorption data presented in Table 1 agree well with the experimental data. Comparing the dependence of the binding constant on the NPs size, we obtain higher binding constants for larger NPs. This observation is in line with receptor-ligand bonds used to interpret the experimental data, where larger NPs with a higher number of binding sites per NP can be attached stronger. We hypothesize that NP binding to the cell surface occurs between their negatively charged carboxyl groups and positively charged counterparts on the membrane, such as amino groups of lipids or proteins. A typical energy for a peptide bond is 11 ACS Paragon Plus Environment


4 . However, also many weak bonds that exert forces of only a few pN each can lead to high cluster stability. For short rebinding times, the cluster lifetime has been shown to grow exponentially with the cluster size.54

2.2. Simulation of NPs adsorption to erythrocytes We simulate NP adsorption to erythrocytes using a model based on receptor-ligand bonds (Figure 3) with a limited number of mobile receptors (binding sites) on the membrane. This assumption is in agreement with only 2% of the membrane being adhered to NPs at saturation. Furthermore, our model accounts for (i) NP radius R, (ii) a minimum number rmin of bound receptors required for an NP to irreversibly attach to an erythrocyte, (iii) a maximum number rmax of receptors that can bind to a single NP, (iv) NP concentration ctot in solution and receptor density σ on the membrane, and (v) NP and receptor diffusion coefficients, Dp and

Dr, respectively. Multivalent binding that requires a minimum number of receptors for stable NP attachment has also been found for receptor-ligand interaction in the case of supported lipid bilayers,55 while the maximum number of receptors that can bind is limited by NP size as well as the number of ligands on an NP. Our simulations aim at an interpretation of our experimental data and at testing the hypothesis of adhesion via receptor-ligand bond formation rather than a precise determination of all parameters.


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(a)

Figure 3. The receptor model. The erythrocyte membrane is discretized into sites on that mobile receptors diffuse. When an NP is in the vicinity of the membrane, its immobile ligands bind to these receptors, forming ligand-receptor bonds and initiating the membrane to curve and wrap around the NP. The computer simulations allow us to predict the dependence of the number of adhered NPs per cell on the parameters of the model. (i) Increasing rmin reduces the probability for NPs in the suspension to find a sufficient number of receptors at the same time and therefore reduces NP adsorption. (ii) Decreasing rmax decreases the number of receptors that can bind to each NP, which increases the total number of NPs that can bind to an erythrocyte. (iii) Increasing σ increases NP adsorption; complete wrapping becomes possible because more receptors are available to bind NPs. (iv) For small NP concentrations, increasing Dr/Dp increases NP adsorption. Higher values of Dr/Dp increase the probability that free NPs find rmin receptors to attach to erythrocytes while they are close to the membrane, which results in higher NP adsorption. For high NP concentrations and for small Dr/Dp, decreasing Dr/Dp increases NP adsorption because already bound NPs collect receptors from the system slower than for high ratios Dr/Dp, such that free receptors remain available to attach to NPs in suspension for a longer time. The NP concentrations in solution are controlled in our experiments (Figure 2) and the NP diffusion coefficients in water,

for 27 nm NPs, 13 ACS Paragon Plus Environment

for 45 nm


NPs, and

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for 100 nm NPs, are calculated using Stokes friction for spherical

particles. Experimentally measured diffusion coefficients for proteins and lipids the for Band 3,56 and

erythrocyte membranes are

for NBD-

PE.57 These measurements include hindering of the diffusion by the cytoskeleton;58–61 therefore the diffusion coefficients in the cell membrane are lower than those measured in for glycophorin,62

model lipid bilayers, such as rhodopsin,63

for

for NBD-POPE66 and NBD-DLPE,

and

respectively.64 Based on the range of diffusion coefficients provided by this published experimental data, we use ≈ 0.6 / for our simulations. This corresponds to / = 0.04 for 27 nm NPs, / = 0.07 for 45 nm NPs, and / = 0.15 for 100 nm NPs. Densities for potential receptors in the erythrocytes are " ≈ 7,300 # for Band 3, " ≈ 350 − 6,570 # for Glycophorin A (GPA), " ≈ 365 − 730 # for Glycophorin C (GPC), and " ≈ 252,000 # for sialic acid residues in the erythrocyte membrane.65–68 We assume a receptor density " = 114.2 # for our simulations. Systematic studies of the parameter dependence for the number of adsorbed NPs per erythrocyte are presented in the Supporting Information (Figure S11-S14). We choose the values of %&'( and %&)* in the simulations, such that the simulation results fit our experimental results for the three NP sizes (Figure 2 and Table 1), and thus extract parameter sets σ, Dr/Dp, rmin, and rmax. The parameter values are summarized in Table 2. Table 2. Values of simulation parameters that reproduce the experimental adsorption data NP size [nm] Parameter name

Parameter symbol 27 10

100

NP concentrations

3.70 10 ml - 9.24 1013 ml-1

9.98 10 ml - 4.99 1013 ml-1

1.82 109 ml-1 - 1.82 1012 ml-1

Receptor density

"

114.2 #

114.2 #

114.2 #

NP diffusion coefficient

16 /

9 /

4 /

Receptor diffusion coefficient

0.6 /

0.6 /

0.6 /

a

b

45 -1


9

-1

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a)

Minimal number of receptors required for binding an NP

%&'(

2

3

5

Maximal number of receptors that can bind to an NP

%&)*

60

75

80

in water;

b)

based on experimental values measured for cell membranes, see text

Our simulations cannot uniquely determine the sets of parameter values, several different combinations can reproduce the experimental data. In particular because the molecules that act as receptors in the experiments and therefore also the receptor density in the erythrocyte membrane are not known, matching the timescales for experiments and simulations is difficult (Table S1). The characteristic times for fast and slow binding kinetics predicted by the simulations are found to be two orders of magnitude shorter than those obtained from the experiments (Figure S6-S9). However, the receptor densities used in our coarse-grained simulations are most probably lower than typical experimental receptor densities. Therefore, we expect that also our expected values for

and

are smaller than those in the

experimental system. The characteristic times depend on the initial binding probability of an NP, which scales as the number

and therefore strongly depend on the receptor density " and

of receptors required for multivalent binding. However, the simulations

provide a systematic study of the dependence of the number of bound particles on receptor density, minimal and maximal number of receptors that can bind to an NP, and the ratio of receptor to NP diffusion coefficients (Figure S11-S14). In particular, our simulations show that multivalent binding is required to interpret the experimental data successfully ( that only a maximal number of receptors can bind to an NP (

, and

.

2.3. The energetics of receptor-limited NP adsorption to erythrocytes The energetics of NP adsorption to erythrocytes is described using a continuum model for the membrane deformation energy and using a lattice model for the receptor free energy (Figure



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3). The deformation energy of the lipid bilayer is calculated using Helfrich’s curvatureelasticity Hamiltonian, ,- = . /0122(4 − 5 ) + 289,

(1)

with bending rigidity , Gaussian saddle splay modulus , and spontaneous curvature mean curvature

and the Gaussian curvature

the two principal curvatures

and

. The

can be obtained from

at each point of the membrane. The most important

elastic parameters of the membrane in our calculations are bending rigidity and spontaneous curvature. The Gaussian saddle splay modulus is only relevant if topological changes occur, e.g., for detaching complete-wrapped NPs from the membrane, and is therefore neglected in the following. The spherical carboxylate surface-modified polystyrene NPs that we study have diameters that are similar to and smaller than the mesh size of the spectrin cytoskeleton, such that we do not expect a direct contribution of the shear modulus of the cytoskeleton to NP adhesion. The deformation-energy cost for the erythrocyte membrane has to be overcome by the adhesion-energy gain for the contact between the NPs and the membrane. For a homogeneous membrane with vanishing spontaneous curvature and tension, the deformation energy per NP is

, where

is the wrapped fraction of the NP surface area. The adhesion energy

gain depends on the physicochemical properties of the NP surface, the biochemical composition of the membrane, and the chemical environment provided by the solution. We calculate the entropy of the receptors by discretizing the membrane into which in total

receptors are distributed. The sites on the membrane are divided

into Sb sites with Nb receptors that bind to a ligand on a particle and membrane with

sites on

sites on the free

unbound receptors. The total free energy of the receptors is then

(2)


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with energy for

and

, where the first term is the total binding

receptor-ligand bonds with energy , and the second term is the free-energy

contribution due to receptor entropy. For our receptor free-energy calculations, the deformation energy per receptor Ed/N is negligible compared with the receptor-ligand bond energy U, see Materials and Methods. Minimization of the total free energy function of

for

as

gives the optimal wrapping state of an NP.

Wrapping fractions of NPs for various receptor densities and receptor-ligand bond energies are shown in Figure 4. For the case of receptor shortage, Nb as function of the wrapping fraction fw saturates for a partial-wrapped state once all receptors are bound (Figure 4 (a)). Therefore, the receptor free energy Fr remains constant for higher wrapping fractions, while the bending-energy cost Ed increases linearly with increasing wrapping fraction (Figure 4 (b)(c)). The minimal total energy thus often corresponds to a partial-wrapped state (Figure 4 (d)). Using receptor-ligand bond energies and numbers of sites that correspond to the systems studied in the experiments, we predict wrapping fractions for various receptor densities and receptor-ligand bond energies (Figure 4 (e)-(g)). To determine the number of sites on an NP, we assume for a receptor size a lipid head group area, : = 2.4 . The number of sites on the free membrane around an NP is 0;< /( => 0=> ) times larger than the number of sites on the NP, where => is the number of NPs that are bound at maximal solution concentration, and 0=> is the NP surface area. In general, the NPs remain unwrapped for small energies per receptor-ligand bond, which is in agreement with previous calculations for NP wrapping using a homogeneous adhesion strength.24,27 For very small receptor densities, the NPs attach to the membrane without wrapping. Stable partial-wrapped states that we observe in our experiments are found for high receptor-ligand bond energies and receptor shortage. Complete-wrapped states are stable mostly for high binding energies and high receptor densities. However, the parameter regime for complete-wrapped states extends also to low 17 ACS Paragon Plus Environment


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receptor densities for bond energies of few , where the energy gain for forming a receptor-ligand bond is comparable to the free-energy loss for clustering receptors on an NP. In this parameter regime receptors do not bind densely to the NP surface, such that a smaller number of receptors is sufficient to wrap the NP completely; thus a sufficient number of free receptors remains available until complete wrapping.

(a)

(b)

(c)

(e)

(d)

(f)

(g)

Figure 4. Receptor-mediated wrapping of NPs. (a) Number Nb of bound receptors as function of the wrapping fraction fw for = 2 10? , @A,&)* = 4.3 10? , @ = 4.3 10? , and U=15 . Corresponding (b) free energy Fr for the receptors, (c) membrane deformation energy Ed for κ=50 , and (d) total energy Ft. (e-g) Wrapping fractions fw for (e) 27 nm NPs (@A,&)* = 18 ACS Paragon Plus Environment

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3.8 10? , @ = 9.1 10C ), (f) 45 nm NPs (@A,&)* = 1.1 10C , @ = 3.2 10D ), and (g) 100 nm NPs (@A,&)* = 5.2 10D , @ = 1.3 10E ) as function of the binding energy U and the receptor area fraction "F obtained using Eqs. (1) and (2). 2.4. NP-induced shape changes of erythrocytes A typical shape distribution for erythrocytes from a healthy donor is 95.5 % discocytes, 1.5 % stomatocytes, 2.8 % echinocytes, and 0.2 % spherocytes (Figure 5, control).



Figure 5. Erythrocyte shape changes after interaction with NPs of different size. (A) Micrograph showing various erythrocyte shapes after adsorption of 27 nm NPs at a concentration of ctot = 1014 particles/ml. Simulated erythrocyte shapes are shown in Figure 6 A. (B) Fractions of erythrocyte shapes at various concentrations of 27 nm NPs. (C) Fractions of erythrocyte shapes at various concentrations of 45 and 100 nm NPs.


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The biconcave discocytic shape and the corresponding deformability of erythrocytes are essential for their biological function. This equilibrium shape is sensitive to various environmental and biochemical factors. Erythrocytes undergo shape transformations to spiculated echinocytes, for example, if they are exposed to anionic amphiphilic molecules, and to cup-shaped stomatocytes if they are exposed to cationic amphiphilic molecules.45 The examination of the consequences following exposure of erythrocytes to NPs reveals a change in the shape distributions of erythrocytes (Figure 5). The strongest effect is observed for 27 nm NPs: while for NP concentrations ctot = 9.2 1011 NPs/ml still most of the cells are discocytes, for ctot = 4.6 1012 NPs/ml about 64% of the erythrocytes are echinocytes and only 31% of the cells are discocytes. This sharp increase of the fraction of non-discocytic cells may be due to a threshold value of the spontaneous curvature required to induce a shape change to echinocytes, as shown in Figure 6 (A), in combination with the inhomogeneous distribution of NPs on the erythrocytes. Upon further increasing the NP concentration, the fraction of echinocytes decreases again, while the fraction of discocytes, stomatocytes, and spherocytes increases. At ctot = 9.2 1013 NPs/ml, 62.8% of the erythrocytes are discocytes, 21.8% are stomatocytes, 8.3% are spherocytes, and 7.1% are echinocytes. For 45 nm NPs and for 100 nm NPs, about 20-30% of the erythrocytes are non-discocytic at concentrations higher than ctot = 1013 NPs/ml and 9 1011 NPs/ml respectively. As for 27 nm NPs, the fraction of echinocytes for 45 nm NPs decreases while the fraction of stomatocytes increases with increasing NP concentration. We have identified three mechanisms by which NP adsorption changes erythrocyte shapes (Figure 6).



Figure 6. Mechanisms for shape changes of erythrocytes. (A) Shapes of erythrocytes and vesicles for various values of the spontaneous curvature c0 and reduced volume v. The lines mark the boundaries between prolate, oblate, and stomatocytic vesicles calculated in Ref. 72. The points indicate erythrocyte shapes reported in Ref. 41. The arrows b, c and d indicate the effect of (b,c) c0 and (d) v on the erythrocyte shapes and correspond to the subfigures (B-D), respectively. (B) NP-induced reduced spontaneous curvature Rc0 as function of the reduced NP membrane density R2σNP and fw from our area‐difference elasticity calculations. (C) NPinduced reduced spontaneous curvature Rc0 as function of R2σNP and fw,. (D) Effective reduced volume veff of the erythrocyte as function of r2σNP and fw, where R is the NP radius and σNP is the NP density on the erythrocyte. The horizontal lines correspond to the highest concentrations of the NPs used in the experiments for each NP size. The erythrocyte shapes in subfigure A are reproduced with permission from Ref. 41. Copyright (2002) National Academy of Sciences.


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(i) Area-difference elasticity. Any mechanism that expands or contracts the inner or outer monolayers of the lipid bilayer relative to each other changes the shapes of erythrocytes. Substances that increase the area of the outer monolayer are crenators that favor echinocytes, while substances that increase the area of the inner monolayer are cup-formers that favor stomatocytes.41,43,45 With the known relation between the area difference of the two layers and the membrane spontaneous curvature,42,69 the spontaneous curvature that is induced by the NP-decoration of the erythrocyte is (see Materials and Methods)

Here,

is the density of NPs on the membrane, and D ≈ 3 nm is the distance between the

neutral surfaces of the two monolayers.42,69 The NP-induced spontaneous curvature increases with increasing

and "=> for all NP sizes (Figure 6 (B)). NPs that attach to the outside of

erythrocytes take out more lipids from the inner monolayer than from the outer monolayer and are crenators. (ii) Attached spherical caps. Attached spherical caps and conical proteins induce an effective membrane spontaneous curvature, and can aggregate and induce bud formation.59–61 The partial-wrapped NPs in our experiments act similar to spherical caps and can therefore also induce erythrocyte shape transitions. NPs that attach to the outside of erythrocytes effectively curve the membrane towards the outside. For small wrapping fractions, the spontaneous curvature that is induced by the NPs is59 .

(4)

For larger wrapping fractions of NPs, when the membrane cannot anymore be considered to be almost flat, the induced spontaneous curvature has to be calculated numerically (see Materials and Methods). The NP-induced spontaneous curvature is maximal for half wrapping,

, and increases with increasing

and


(Figure 6 (C)). The plot is


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universal and independent of the NP size; we have marked the spontaneous curvature contributions for the three NP sizes used for our experiments at their saturation density. Via the attached spherical-cap mechanism, NPs that are partially attached to the outside of the erythrocyte act as cup-formers. (iii) Reduced volume. The reduced volume70 for discocytes is

. Both, shape

calculations that take into account the shear elasticity of the spectrin cytoskeleton and lipid bilayer-only calculations for vesicles, predict shape transitions for changes of the reduced volume, e.g., for osmotic swelling of erythrocytes.71–73 Upon NP adsorption, the effective reduced volume of an erythrocyte, ,

(5)

increases because the membrane area effectively decreases while the volume effectively increases when partial-wrapped NPs protrude into the cytosol. Here, NNP is the number of NPs adsorbed to an erythrocyte; the volume and the membrane area of discocytes are , respectively.69 NPs that attach to the outside of

and

erythrocytes effectively increase their reduced volume, mainly by reducing the area that is required for wrapping around the nanoparticles.31,74 This is expected to lead to spherocytes at sufficiently high densities (Figure 6 (D)). The large increase of the effective spontaneous curvature predicted for the NP-induced area difference that favors echinocytes, as well as the small decrease predicted for the sphericalcap effect that favors stomatocytes, are consistent with the spontaneous-curvature changes required to explain the observed shape changes in our experiments (Figure 5).69 The decrease of echinocytes and the increase of stomatocytes with increasing NP concentrations, therefore, hints that the area difference relaxes via different mechanisms. A possible such mechanism is the flip-flop of lipids, which may be facilitated by the distortion of the bilayers by the NPs75–77 or scramblases.52,78 A partial insertion of the 27 nm nanoparticles into the outer monolayer 24 ACS Paragon Plus Environment

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would also affect the area difference and shift the erythrocyte shape distribution towards stomatocytes. Furthermore, a redistribution of curved molecules inside the multi-component lipid bilayer can occur due to NP binding and affect the erythrocyte shape.79 Spherocytes occur because of an increase in the volume or a decrease in the membrane area, which may occur due to osmotic swelling,71,80,81 shedding of membrane,82 and endovesiculation.83 Furthermore, spherocytes are the endpoints of both stomatocytic as well as echinocytic deformations and may occur via one of these shape-deformation pathways.84

2.5. NP-induced deformability changes of erythrocytes Erythrocytes experience shear stresses during blood circulation with shear rates of the order of 100-1,000 Hz and have to squeeze through small capillaries to deliver oxygen to body tissues.85–88 Therefore, their ability to deform is crucial for proper physiological function. We have tested erythrocyte deformability before and after NP adsorption at different concentrations using a hemorheometer (RheoScan AnD-300). The erythrocyte suspension in a media containing polyvinylpyrrolidone (PVP) with a viscosity of about 33 mPa s is flown through a thin chamber under altered pressure. The erythrocyte deformation is estimated by measuring the geometry of the diffraction patterns of the erythrocyte suspension (Figure 7).



A

D

B

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C

E

Figure 7. Erythrocyte deformability accessed by RheoScan AnD-300 hemorheometer as a function of NP concentration: (A) measurement chamber; (B) typical diffraction image of cells at low pressure; (C) typical diffraction image for a pressure difference ΔH = 20 Pa. (D) Elongation indices measured for various NP concentrations as function of ΔH. The elongation index is defined as Ei=(a-b)/(a+b), where a and b are the lengths of major and minor axes of the fitting ellipse on the laser diffraction patterns (insets B and C). For each concentration data for least three independent experiments has been averaged. (E) Elongation indices Ei at ΔH =20 Pa. Significant erythrocyte stiffening was observed only for 27 nm NPs at very high concentrations of ctot = 5 1013/ml and 1014/ml. Larger NPs of 45 nm and 100 nm were almost without effect on erythrocyte deformability even at the highest concentrations studied. This correlates with the shape transformations of erythrocytes, where 27 nm NPs at high concentrations were most effective in inducing stomatocytes and spherocytes in erythrocyte populations. In agreement with our findings, stomatocytes have been reported to be much less deformable than discocytes in experiments with optical tweezers. 89,90

3. Conclusions Our combined experimental, theoretical, and numerical work suggests that NPs bind to adhesive sites on the erythrocyte membrane. The experimentally observed lack of detachment 26 ACS Paragon Plus Environment

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of NPs hints that a kinetic mechanism determines the concentration dependence of NP binding. We propose a receptor-ligand model with a requirement for multivalent binding and a maximal number of receptors that can bind to an NP. Using Brownian dynamics simulations, we extract sets of parameter values that reproduce the experimental data. These sets are not unique, because the molecules that act as receptors are unknown. Future experimental work should therefore aim at the determination of the exact interaction partners. The experimentally observed dependence of the number of bound NPs on the NP concentration in suspension can thus be described using both, effective Langmuir-like adsorption and—taking into account for the irreversible binding of the NPs observed in the experiments—our kinetic model based on receptor-ligand bonds. Our simulations show that mobile receptors and fixed ligands can lead to partial NP wrapping. Hindrance of NP wrapping by the spectrin cytoskeleton can further stabilize partial-wrapped NP states. Both, the sparse coverage of erythrocytes with NPs in the experiments and freeenergy minimization suggest that only a small number of adhesive sites, which strongly bind to NPs, is present in the membrane of erythrocytes. Because the mechanics of erythrocytes has been studied in detail in the past, our experiments and theory allow us to quantitatively predict spontaneous curvature changes of cell membranes by attached NPs. Partial-wrapped NPs that are attached to the outside of erythrocytes act as crenators via the area-difference elasticity mechanism, while they act as cup-formers via the spherical-cap mechanism. Furthermore, NPs that are attached to the outside of erythrocytes effectively decrease the area of the erythrocyte membrane and increase the volume of the cell, and thereby increase the reduced volume of the cell and make erythrocytes more spherical. Our results are of direct relevance for understanding mechanisms that can be applied in drug delivery and nanotoxicology, and they also provide a systematic basis to study more complex systems. Because NPs can be highly reactive in adsorbing small molecules and macromolecules from their immediate environment, the formation of a so-called protein 27 ACS Paragon Plus Environment


“corona” has to be taken into account for most applications involving biological systems.40,91 The chemical composition of an NP corona can be of temporal nature, as it is a function of the abundance and chemical affinities of competitive adsorbing entities to the NPs' surface.40 Within our model, a corona can be taken into account through an effective NP size and an effective NP-membrane adhesive interaction. Systematic experimental studies to identify these effective parameters for each system are required to connect our generic study with more complex biological environments. Because of the firm attachment of carboxylated polystyrene NPs to erythrocytes observed here and the fact that NPs bound to erythrocytes stay for much longer times in the blood circulation, NP-loaded nanoparticles may be exploited for sustained and targeted drug delivery.8 While erythrocytes do not have active uptake mechanisms, quantitative studies that differentiate between passive adhesion and uptake, and active, metabolic uptake processes for the interaction of NPs with other mammalian cells is a challenge for future studies.

4. Materials and Methods 4.1. Characterization of carboxylated polystyrene NPs FluoSpheres® Carboxylate surface-modified polystyrene NPs loaded with yellow-green fluorescent dye (Ex/Em 505/515), were purchased from Invitrogen. The pristine characteristics of the 27nm (F8787), 45nm (F8795) and 100nm (F8803) NPs are given in Table 1. NP size validation was performed by Dynamic Light Scattering (DLS) on Zetasizer ZS device (Malvern Instruments Ltd, UK). The backscattering angle was set to 173o with automatic selection for the optimal position and attenuation. Three 60s acquisitions were performed with the delay of 2 s between. A general-purpose analysis model was chosen. Concentrations ctot = 3.7 1012, 2.0 1012 and 7.3 1010/ml dispersions of 27 nm, 45 nm, and 100 nm NPs in PBSG solution, respectively, were analysed at 25°C (Figure S4). The dispersions were stable for at least 3 hours. Particle ζ-potentials were reported previously to be −27 ± 3 28 ACS Paragon Plus Environment

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mV, −24 ± 3 mV, and −34 ± 2 mV in PBS at pH 7.0 and 25°C for 27nm, 45nm, and 100 nm particles, respectively.47

4.2. Isolation of erythrocytes from blood Human blood donation was approved by the Helsinki Committee. About 100-150 µl of blood was drawn from a healthy donor (AB) by fingerpick, collected and diluted at a ratio of 1:10 (v/v) in cold phosphate buffered saline without Ca and Mg, and supplemented with glucose (PBSG solution: 137 mM NaCl; 2.7 mM KCl; 10 mM Na2HPO4, 2 mM KH2PO4, 10 mM glucose, pH 7.4). Blood cells were sedimented at 200g, 4 oC for 10 min. After removing of the buffy coat, the erythrocytes were washed thrice with PBSG at 1000g for 3 min and finally resuspended in PBSG solution supplemented with 0.9 mM CaCl2 and 0.5 mM MgCl2.

4.3. Adsorption of NPs to erythrocytes Erythrocytes at 5% hematocrit in PBSG solution with Ca and Mg were incubated with green fluorescently labeled carboxylate-modified spherical polystyrene NPs with diameters of 27 nm, 45 nm and 100 nm (Invitrogen) at various concentrations for 15 min at 24°C, while being shaken at 300 rpm. Immediately after incubation the samples were diluted by a factor 10 using PBSG and washed twice with PBSG by centrifugation at 1000g for 3 min. The erythrocytes were resuspended in PBSG for flow cytometry and supplemented with 1 mg/ml bovine serum albumin for other studies. The erythrocytes were analyzed by flow cytometry (FACSCalibur, BD Biosciences) for fluorescence and light-scattering. 10,000 cells were counted for one measurement typically. The correspondence of the fluorescence signal to the number of fluorophores was achieved employing calibrated fluorescent beads (Bangs Laboratories, Inc.), calibration curves are shown in the Supporting Information (Figure S3). Adsorbed NPs did not lead to hemolysis, as



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a significant concentration of free hemoglobin has not been detected in the supernatant and erythrocyte ghosts have not been observed by DIC microscopy.

4.4. Bright-field and fluorescence microscopy DIC microscopy and fluorescent images of erythrocytes were taken when the erythrocytes were suspended in PBSG with 0.9 mM CaCl2, 0.5 mM MgCl2 and 1 mg/ml BSA using an inverted Zeiss AxioObserverZ1 microscope equipped with a 40X water immersion objective. Green fluorescence was detected using an X-Cite light source and a FITC filter set. Cell-shape classifications of erythrocytes, based on DIC microscopy images, were carried out manually. Since the NP stocks are maintained in the presence of 2 mM azide, we examined whether there could be an induction of erythrocyte shape transition due to the presence of azide. No shape changes could be detected for erythrocytes at azide concentrations of 100µM, 50µM, 20µM or lower—corresponding to those implemented by our NP dilutions—after 15 min incubation at room temperature while shaking.

4.5. Imaging by Scanning Electron Microscopy For SEM imaging, erythrocytes were first attached to round coverslips by incubation in PBSG with 0.9 mM CaCl2, 0.5 mM MgCl2 and 1 mg/ml bovine serum albumin for 20 min at 37oC, washed by PBSG and fixed with PBSG solution containing 1mM EGTA and 2.5% glutaraldehyde. The samples were then dehydrated in a graded series of ethanol-water mixtures and dried using critical-point drying. Erythrocyte samples were coated with a 60/40 gold/palladium alloy and viewed by Quanta 200 ESEM (FEI, USA).

4.6. Measurements of cell deformability by ektacytometry Cell

deformability

was

accessed

using

a

RheoScan

AnD-300

hemorheometer

(RHEOMeditech, Korea). Samples with 6 µl of erythrocyte suspension at 50% hematocrit 30 ACS Paragon Plus Environment

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were thoroughly mixed with 0.6 ml of PBSG containing 1 mg/ml bovine serum albumin and 5.5% Polyvinylpyrrolidone (m.w. 360,000), and introduced in measurement chambers with a thickness of ~200 µm. The erythrocyte elongation index (EI) is defined as (a-b)/(a+b), where a and b are the lengths of major and minor axes of the fitting ellipse on the laser diffraction pattern (Figure 7 B and C). The elongation index was determined at pressure differences of 023 Pa.

4.7. Receptor-mediated NP adhesion to membranes: a kinetic model We perform Brownian dynamics simulations to investigate the irreversible binding of NPs to receptors on the membrane (for details see Supporting Information). Lengths in simulation and experiment are connected via the NP diameters, and times via the NP diffusion coefficients. NPs and receptors are modeled by a fixed number of spheres with radii rNP and rrec that diffuse in bulk and on a planar membrane, respectively. (For wrapping calculations membrane curvature becomes important when the curvatures of membrane and NP become comparable.21,31 In our case, the ratio of the largest NP radius I = 100 and the inverse average mean curvature 4#J = 2,500 of the erythrocyte membrane,59 is small.) Periodic boundary conditions are used parallel to the membrane and a repulsive hard wall borders the simulation box at the side opposite to the membrane. We use a box with dimensions 250x250x1000 simulation units (su) cubed, where 1 su corresponds to 26.47 nm. Therefore, for NPs with diameter 27 nm, the lowest NP concentration in bulk, = 3.7 10J5 K #J, corresponds to N = 625 NPs and the highest NP concentration, = 9.24J? K #J , to N = 107,100 NPs. Analogously, for 45 nm NPs the numbers of nanoparticles in the simulations that correspond to the concentrations used in the experiments varies between 11 ≤ ≤ 58,000 , and for 100 nm NPs between 2 ≤ ≤ 2,120 . The repulsive interaction force between two NPs i and j is given by M'N = O2%=> − %'N P%SQR for %'N ≤ 2 %=> , and M'N = 0 for 31 ACS Paragon Plus Environment


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%'N > 2 %=> . When an NP enters a thin slab with a thickness of NP radius R above the membrane, the number of receptors within a circular patch of radius R around the projected NP position is calculated. An NP is considered to be bound and its position on the membrane is fixed if the required minimal number rmin of receptors bind a particle simultaneously, otherwise the NP does not bind and can diffuse back to the bulk solution. Further receptors can bind to an already attached NP until the maximal number rmax of receptors is reached.

4.8. Receptor-mediated NP adhesion to membranes: free-energy calculations We use a lattice model to predict wrapping fractions of NPs for given receptor-ligand bond energy

and receptor density

on the membrane. Assuming that the carboxyl groups on the

NPs act as ligands, we calculate areas per ligand of for the

and the

lipid head group area,92 we use

for the

NPs and of

NPs. Because the areas per ligand a smaller than a typical (which corresponds to a circular patch with radius

) as area per site for our model. If every lipid that is in contact with an NP binds to a ligand, the membrane bending energy per receptor is energy can be neglected for calculating the receptor free energy if

. The bending , i.e.,

. Using U = 15 kBT 93,94 and κ = 50 kBT,95,96 we find and

for the

NPs. We, therefore, do not take bending energy into account

for calculating the receptor free energy (further details of the calculations can be found in the Supporting Information). The number of receptor-ligand bonds for given NP size and wrapping fraction obtained from the free-energy calculations is

(6) .


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We minimize the sum of the adhesion energy

, the deformation energy

, and the receptor free energy for a given UV , with respect to

to predict the

wrapping fraction of the NP. 4.9. NP-induced erythrocyte shape changes via area-difference elasticity Erythrocyte shapes have been found to depend on the area difference between the two monolayers that form the bilayer.

42,43,69

The energy of a vesicle determined by the area

difference elasticity (ADE) energy functional is

(7) where D ≈ 3 nm is the membrane thickness, A is the membrane area, difference that corresponds to the discocyte shape, between the areas of the monolayers, and

is the area

is the actual area difference induced

is the Gaussian saddle-splay modulus. The

integral is taken over the entire membrane area of the closed vesicle. The total area difference between the outer and the inner layer induced by

NPs with radius

and wrapping fraction

is . Using

(8)

, we obtain an effective spontaneous curvature of the NP-decorated

membrane,

(9)

4.10. NP-induced erythrocyte shape changes via attached spherical caps Partial-wrapped NPs can be considered as spherical-cap inclusions that induce catenoidal membrane deformations in their surroundings.59–61 We predict an effective spontaneous curvature c0 as function of the NP radius R , the wrapping fraction fw,, and the NP density using geometrical calculations. Within a cylindrically-symmetric approximation, we calculate 33 ACS Paragon Plus Environment


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radius and slope of the outer boundary of the membrane patch. Fitting this to a sphere allows us to determine an effective spontaneous curvature of the NP-decorated membrane. Further details can be found in the Supporting Information.

4.11. NP-induced erythrocyte shape changes via reduced volume with V=100µm3 and

Healthy erythrocytes have reduced volumes

A=137 µm2.69 Upon complete uptake of NPs by wrapping, the volume of the erythrocyte increases while the area of its plasma membrane decreases. For partial uptake, we find an effective reduced-volume increase ,

(10)

with the effective volume increase ,

(11)

and the effective plasma membrane-area decrease .

(12)

Supporting Information The Supporting Information contains measurements for the calibration of the FACS measurements, dynamic light scattering data to characterise the NPs, experimental and simulation data for the binding kinetics of NPs, a study of the desorption kinetics, systematic parameters studies for the Brownian dynamics simulations, and details for the analytical calculations for receptor model and erythrocyte shape predictions. This material is available free of charge via the Internet at http://pubs.acs.org. Acknowledgments 34 ACS Paragon Plus Environment

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Support from the EU FP7 NMP collaborative project PreNanoTox (309666) is gratefully acknowledged. KS and SD acknowledge support from the International Helmholtz Research School of Biophysics and Soft Matter (IHRS BioSoft). We thank Pierre Sens (Paris), Emilio Benfenati (Milan), Thomas Eisenstecken (Jülich), and Arvind Ravichandran (Jülich) for helpful discussions. Author Contributions TA, RK, and GG designed the study. RK designed the experiments. AB and LH-A performed the experiments. TA, SD, and GG designed the simulations. KS performed the simulations. AB and KS analysed the data. All authors discussed the results. AB, KS, TA, RK, and GG wrote the manuscript. Notes The authors declare no competing financial interest.

Alexander Barbul and Karandeep Singh contributed equally to this work.

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