Citation: The Journal of Chemical Physics 143, 144902 (2015); doi: 10.1063/1.4932372 View online: http://dx.doi.org/10.1063/1.4932372 View Table of Contents: http://aip.scitation.org/toc/jcp/143/14 Published by the American Institute of Physics

THE JOURNAL OF CHEMICAL PHYSICS 143, 144902 (2015)

Influence of the shell thickness and charge distribution on the effective interaction between two like-charged hollow spheres Daniel G. Angelescu1,a) and Dan Caragheorgheopol1,2 1

Romanian Academy, “Ilie Murgulescu” Institute of Physical Chemistry, Splaiul Independentei 202, 060021 Bucharest, Romania 2 Technical University of Civil Engineering Bucharest, Lacul Tei Blvd., 122-124, 020396 Bucharest, Romania

(Received 2 July 2015; accepted 23 September 2015; published online 8 October 2015) The mean-force and the potential of the mean force between two like-charged spherical shells were investigated in the salt-free limit using the primitive model and Monte Carlo simulations. Apart from an angular homogeneous distribution, a discrete charge distribution where point charges localized on the shell outer surface followed an icosahedral arrangement was considered. The electrostatic coupling of the model system was altered by the presence of mono-, trivalent counterions or small dendrimers, each one bearing a net charge of 9 e. We analyzed in detail how the shell thickness and the radial and angular distribution of the shell charges influenced the effective interaction between the shells. We found a sequence of the potential of the mean force similar to the like-charged filled spheres, ranging from long-range purely repulsive to short-range purely attractive as the electrostatic coupling increased. Both types of potentials were attenuated and an attractive-to-repulsive transition occurred in the presence of trivalent counterions as a result of (i) thinning the shell or (ii) shifting the shell charge from the outer towards the inner surface. The potential of the mean force became more attractive with the icosahedrally symmetric charge model, and additionally, at least one shell tended to line up with 5-fold symmetry axis along the longest axis of the simulation box at the maximum attraction. The results provided a basic framework of understanding the non-specific electrostatic origin of the agglomeration and long-range assembly of the viral nanoparticles. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4932372]

I. INTRODUCTION

The electrostatic interactions among highly charged colloids play a dominant role in governing the phase stability and the physicochemical properties in the aqueous solutions. Theoretical developments and simulation studies revealed that the phase stability of highly charged colloidal systems could not be described qualitatively by mean-field theories averaging over small ions such as the Poisson-Boltzmann equation.1,2 Nonetheless, by an explicit treatment of the small ions, elaborated theories3,4 predicted and computational models5,6 confirmed that the short-range attractive forces may appear between like-charged colloids, and that they originated from spatial correlations between counterions residing near different colloids.3,5–8 Such non-specific electrostatically driven attraction was affected by the shape of the colloids9 and documented for like-charge planes,10 cylinders,7,11 and spherical7,12,13 macroions. The control of the phase stability of the nanoparticles solutions and the possibility to reach high-order nanoparticle structures are the key factors in the synthesis of the advanced hybrid biomaterials. Self-organization of hollow nanoparticles, such as virus capsids, virus-like nanoparticles, and inorganic shells, has attracted the interest of the researches over the past decade because of the potential impact in a)Author to whom correspondence should be addressed. Electronic mail:

[email protected] 0021-9606/2015/143(14)/144902/14/$30.00

advanced applications such as fabrication of electrical devices, solar cells and catalysts, magnetic or bioactive materials.14,15 Hollow inorganic nanoparticles have recently gained more prominence than the conventional mesoporous materials in applications such as drug delivery,16 anti-reflection coating,17 and electrochemistry18 because of their enhanced loading capacity and controlled release and the particular optical and chemical properties.19 Hollow nanoparticles of variety of materials such as SiO2,20 ZnO,21 TiO2,22 Al2O3,23 and with various size and shell thickness were achieved using micelles, biopolymers, or etchable materials as soft templates. Viruses are another type of hollow nanoparticles with rod-like or nearly spherical appearance. In the latter case, a well-established number of one or a few slightly different protein self-assemblies into a closed icosahedral shaped shell whose inner void can host the viral genome, either dsDNA or ssRNA/DNA,24 synthetic polyions,25 or inorganic nanoparticles.26 Thus, the multi-protein shells, referred usually as viral capsids, can be used as scaffolds in nanomaterials research, such as the synthesis of bioorganic/inorganic hybrid materials and the enhanced gene delivery to cells or transduction of retroviral particles.14 For these practical applications, it is noteworthy to control not only the steadiness of the loaded capsid but also the phase stability of the virus-like nanoparticles under different solution conditions. Notably, the self-assembly of virus-like nanoparticles often required the addition of polyions or highly asymmetric salts, hinting thus at an important role of the nonspecific electrostatic interactions.27 The presence

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of multivalent species can either modify significantly the interaction between the loaded material and the surrounding capsid or yield an effective attraction between the capsid constituents.27,28 For sufficiently highly charged closed shell or permeable droplets, the multivalent counterions could strongly accumulate inside and resulted in a negative osmotic pressure, stabilizing the shell against breakdown.29 Another important feature reported upon addition of multivalent species was given by the accumulation or longrange organization of the virus-like nanoparticles. Thus, by alternating the electrostatic assembly through the layerby-layer techniques, high-order assemblies consisting of Autographa Californica Nuclear Polyhedrosis Virus (AcNPV) and oppositely charged polyelectrolytes could be produced.30 Ordered hexagonal organization of Turnip Yellow Mosaic Virus (TYMV) was reported at the planar oil-water interface in the presence of divalent cations Cd2+ and the positively charged dehydroabeityl amine (DHAA).31 Complexes containing viruses with 3D architectures could be also obtained. The self-assembly of Cowpea Chlorotic Mottle Virus (CCMV) could be induced by the presence of spermine functionalized dendrons and the experimental investigation combined with the molecular dynamics simulations revealed that a certain degree of multivalency is required for efficient virus selfassembly.32 Moreover, the assembly was not related to the presence of any material held inside the capsid,33 and the ionic strength delivered control over the reversibility of the assembly formation.34 Several recent investigations had underlined the role of electrostatic interactions, in particular the valency of added cations, in the specific cases of the virus aggregation or adsorption onto solid interfaces. Wong et al. stated that the presence of divalent counterions enhanced the aggregation of Human Adenovirus (HAdV) as well as its adsorption to the surface.35 Additionally, Gutierrez et al.36 showed that the Rotaviruses could aggregate in the presence of divalent cations, whereas no aggregation was detected in the monovalent cations containing solution. By measuring the aggregation kinetics of the bacteriophage MS2, Mylon et al.37 indicated that the virus aggregation could be induced in divalent electrolytes whereas no aggregation could be found within a reasonable kinetic time frame in monovalent electrolytes. Phase agglomeration was also reported for the bacteriophage Qbeta. In contrast with the previous mentioned studies, higher multivalency was required to flocculate the virus solution as AlO4Al12(OH)247+ cation, resulting from the solubilisation of polyaluminum chloride (PACl), led to

an enhanced flocculation as compared with the Al3+ cations provided by AlCl3 or Al2(SO4)3 electrolytes.38 Despite the fact that these investigations revealed the importance of the electrostatic interactions in the selfassembly of the viral particles, the exact mechanism driving the complexation remained largely elusive, being rather speculated that the ion-ion correlations could rise to the viruses aggregation in solutions.37 As a consequence, the present work attempts to fill in the gap concerning the electrostatic-driven instability of the virus-like nanoparticles, by investigating how the capsid properties influence the effective interaction between such highly charged hollow objects in the salt-free limit. To accomplish this goal, we employed Monte Carlo simulations within a simple model system containing two charged spherical shells, possessing either a homogeneous or a discrete, icosahedrally symmetric charge distribution, and mono- or multivalent counterions. The nature and amplitude of the mean force and of the potential of the mean force acting between the two shells were evaluated by varying the shell thickness as well as the radial and angular location of the shell interacting sites. The results provided evidences for an effective attraction between the shells in the presence of the multivalent counterions, with quite significant effects induced by the shell characteristics. The paper is organized as follows. The model system, the simulation aspects, as well as the method used to calculate the mean force and the corresponding potential acting between the charged hollow spheres are described in Section II. Section III presents the results, with an initial focus in Section III A on the nature of the mean force in the presence of mono- and multivalent counterions. Then, Sections III B and III C deal with the mean force in the increasing electrostatically coupled systems at different shell thicknesses and radial position of the homogeneous shell charge, respectively. Section III D is devoted to the influence of the discrete shell charge model with icosahedral symmetry on the effective interaction mediated by the trivalent counterions. Finally, Section IV summarizes the conclusions.

II. METHOD A. Model system

Our model is based on the structural characteristics of the viruses that exhibited self-aggregation upon addition of asymmetric salts or small polyions. Table I summarizes the size

TABLE I. Characteristics of viruses experiencing phase instability in the presence of multivalent cations: inner R int and outer R ext radius, net surface charge of the bare capsid Z c, experimental zeta potential ζ at low ionic strength, and calculated charge density σ provided that the charge is located at either outer or inner surface. σ (C/m2) Virus MS2 bacteriophage Qbeta bacteriophage Cowpea Chlorotic Mottle virus Human adenovirus Rotavirus

R int (Å)39 105 107 94 326 303

R ext (Å)39

Z c (e)39

ζ (mV)

Outer surface

Inner surface

144 147 144 474 498

+180 +300 −120 +5760 −11 760

−64.737

−0.032 −0.013 −0.012 −0.006 −0.007

−0.55 −0.07 −0.093 −0.05 −0.15

−32.456 −32.440 −35.035 −35.636

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and the net charge of the viral capsid (as the sum of all surface exposed charged residues) as given by the VIPER database.39 The zeta potential, derived from the reported electrophoretic mobility experiments carried out at ionic strength lower than 1 mM, is also given in Table I. It is seen that the external radius of the viral capsid Rext ranges from 144 to 498 Å, and the normalised thickness d S = (RS,ext − RS,int)/RS,ext from 0.27 to 0.39, yet not directly related to the capsid size. Noteworthy, the zeta potential values are all negative despite the fact that the capsid bears in some cases a significant positive net charge. This apparent disagreement indicates that the simplified picture of the surface exposed charged residues obtained from the crystallographic determination cannot be simply related to the net capsid charge.40 It is also suggested that the charge patches at the capsid surfaces, which vary to a great extent among the viral capsids, and the dissociation constants of amino acids depending on the detailed molecular environments should be taken into consideration.41 Other aspect raised by the negative zeta potentials is the presence of the viral genome which can influence the net charge balance of the viruses, in particular when overcharging is expected upon capsid loading.42 This statement is supported by the reported electrophoretic mobilities of the emptied and loaded viral capsid. The charge density at the outer surface of the capsid seemed to control the electrostatics as the electrophoretic mobilities of the MS2 and CCMV capsids were not significantly influenced by the presence of the encapsidated ssRNA.43 By contrast, the bacteriophages with different fractions of packaged dsDNA, mostly found at the capsid inner surface, experienced different electrophoretic mobilities.44 Thus, one could consider the extreme case when the net charge location is either at the outer or at the inner capsid surface. This oversimplified approach enables us to assess the surface charge density of the viruses in Table I by approximating the icosahedral shape of the capsid with a spherical shell and identifying the potential at the outer capsid surface as zeta potential. The non-linear Poisson-Boltzmann equation was numerically solved using the cell model,45 and the results shown also in Table I indicate a charge density ranging from −0.006 C/m2 to −0.032 C/m2 when the outer surface was the location preferred and from −0.05 C/m2 to −0.55 C/m2 otherwise.

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Given the rather wide range of the viral properties displayed in Table I, our model system did not intend to emulate any particular virus, instead aimed to capture the salient properties required for understanding the electrostatic interaction between two virus-like nanoparticles in the presence of the multivalent species. The option chosen was to use a scale-down model that proved previously to be applicable for modelling the viral genome conformation inside the capsid as well as the thermodynamics of the encapsidation process.46 Thus, the capsid shape is reduced to a hollow sphere, whereas the capsid symmetry is transferred to the capsid charge distribution in order to discriminate between the radial and angular influences. The outer radius of the shell is RS,int = 62 Å, and the inner radius RS,int varies from 0 to 50 Å. The corresponding normalised shell thickness d S ≡ DS/RS,ext = (RS,ext − RS,int)/RS,ext is in the range 0.19–1.0, including thus a filled sphere and shells whose thicknesses are similar to those of the viruses in Table I. The shell carries a total charge ZS = −252 located at the normalized radial distance d σ = dint/DS, with d σ = 0 and 1 representing the outer and inner capsid surface, respectively. At given d σ, the charge could be distributed following (i) a homogeneous or (ii) a discrete, icosahedrally symmetric charge model, with the charge distribution parameter λ giving the fraction of the capsid charge ZS ascribed to the latter model. Thus, λ = 0 corresponds to a fully homogeneous surface charge density of −0.083 C/m2 at d σ = 0 and −0.128 C/m2 at d σ = 1. The discrete charge model is characterized by 252 interacting sites located at each of the vertices of an icosahedron and at eight evenly distributed positions along each edge of the icosahedron. At λ = 1, each interacting site bears an elementary charge unit. To separate out the radial variation of the discrete charge model, the edges connecting the vertices are projected onto the sphere circumventing the isosahedron. Note that the resulting shell charge distribution still retains the icosahedral symmetry, and the corresponding electrostatic potential obtained at the outer surface resembles the one generated by the CCMV capsid, as can be seen from Figure 1. These coarse approximations and simplifying assumptions of the virus capsid are considered as reasonable for capturing the salient aspects of the mean force behavior.

FIG. 1. Electrostatic potential (kT/e units) generated by (a) the capsid charges of Cowpea Chlorotic Mottle Virus at R ext (cf. Table I) and (b) the discrete charge model (λ = 1.0) at r = R S,ext + R I = 64 Å. The coordinates of the viral capsid charges were obtained from Virus Particle Explorer (VIPER) database.39

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FIG. 2. Schematic illustration of the two shells and of the typical counterions of Systems I-III within the cylindrical cell of L cyl length and R cyl radius. The shells are separated by the distance r and characterized by the internal radius R S,int, external radius R S,ext, thickness DS, and radial charge position d int.

Nonetheless, refined models would be directed towards an elaborate mapping of the capsid charge reflecting the patchy interacting sites of opposite sign found mainly at capsid surfaces,41,47 the ionizable residue’s dependence on the local pH and ionic strength,48 an explicit treatment of the monovalent electrolyte solution, and a detailed quantification of the dielectric discontinuities induced by the proteinaceous capsid. The model used to evaluate the mean force in the saltfree limit is composed of two such charged hollow spheres and the counterions constituting altogether an electroneutral system which is enclosed in a cylindrical cell possessing hard walls (see Figure 2 for an illustration of the system). The two shells are positioned along C∞ axis, the z symmetry axis of the cylinder cell, and symmetrically with respect to the plane z = 0 at z = ±zSS. The electrostatic coupling into the cell box is altered by taking into account three types of counterions. In System I and II, the counterions are hard spheres of RI = 2 Å radius and having valence of ZI = 1 and 3, respectively. System

III involves a dendritic construct composed of a neutral hard sphere on which are grafted three individual chains, each one composed of three hard spheres of 2 Å radius and valence 1. Thus, the charge of the multivalent structure was 9. B. Interacting potentials

The properties of the model system are described within the primitive model. Thus, the total energy of the system is assumed to be pairwise additive and given by U = Uel + Ubond + Uangle + Ucell.

(1)

The first term represents the hard-sphere and Coulomb interactions between the charged species and it can be regarded as a sum of two contributions, Uel =

2 ui j r i j + uS,i (r S,i ) .

i< j

S=1

(2)

i

The first contribution is ui j (r i j ) = ui j ( r i − r j ) ∞, r i − r j < Ri + R j Z i Z j e2 = , , r i − r j ≥ Ri + R j 4πε ε r − r 0 r i j

(3)

where i and j indices represent the counterions or the sites of the discrete charge model of the capsid, and r i − r j the distance between the center of the interacting species; ε 0 is the dielectric permittivity of vacuum and ε r the relative permittivity of the system. The second term in Eq. (2) outlines the interaction of the hollow sphere with the counterions and is given by ∞, RS,int − Ri ≤ |ri ± z S S k| ≤ RS,ext + Ri 252 2 Z i Z j e2 Zi Z S e , |ri ± z S S k| < RS,int − Ri (1 − λ) +λ 4πε 0ε r(RS,ext − d int) 4πε 0ε r ri − r j uS,i (r S,i ) = , j=1 252 2 2 Zi Z j e Zi Z S e , |ri ± z S S k| > RS,ext + Ri (1 − λ) 4πε 0ε r |ri ± z S S k| + λ 4πε 0ε r ri − r j j=1

(4)

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with i and j indices standing for the counterions and site of the discrete charge model, respectively, and zSS the z-coordinates of the centers of the hollow spheres. Note that Eqs. (3) and (4) involve the same dielectric permittivity of the shell and the solvent, whereas the relative permittivity of the proteic capsids is usually smaller, between 2 and 5. A lower permittivity of the shell leads theoretically to the presence of a polarization surface charge density at the dielectric discontinuities. This surface polarization was generally treated as an image charge49 or as a dynamic variable within the free energy functional.50 Applying the variation of the permittivity across the shell would increase the effort of solving the present model system substantially as the potential energy induced by the image charge is not trivial. However, when the image charge model was addressed for calculating the potential of the mean force between two like-charged macroions with a low dielectric permittivity in the salt-free limit using the cylindrical cell model, the effect of the surface polarization was short ranged and led to an attenuation of the counterion accumulations at the macroion surfaces.49 The second and third terms in Eq. (1) imply the counterions with dendritic architecture only and are given by Ubond =

N k bond (r i j − r 0)2 Ωi j , 2 i< j

(5)

where Ωi j = 1 when bead i and j are bonded, otherwise 0; r 0 = 5 Å is the equilibrium bond distance and kbond = 0.4 Nm−1 the bond force constant, and Uangle =

N −1 i=1

kangle (α i − α0)2 Ωi−1,i,i+1, 2

Cell radius, R cyl Cell length, L cyl Temperature, T Relative permittivity, εr

and |z| ≤ L cyl/2 , (7)

where x i, yi, zi are the Cartesian coordinates of the counterion i. The values characterizing the analysed systems are summarized in Table II. C. Theory and simulation details

It was shown in Ref. 5 that there were different ways to express the mean force within the cylindrical cell model. Here, we consider an approach that is independent of the shape of the charged objects, and according to which the mean force acting on one shell is divided as a sum of four contributions calculated across the midplane of the cylindrical box (z = 0),

248 Å 744 Å 298 K 78.4

Hollow sphere Number of hollow sphere, NS Outer radius, R S,int Inner radius, R S,int Charge, Z S Radial location of the charge, d int Charge distribution parameter, λ

2 62 Åa 0–50 Å −252 0–10 Å 0, 1

Counterions System Radius of beads, R I (Å) Valence, Z I Number of beads, Nb Number of counterions, NI a Volume

I 2 1 1 504

II 2 3 1 168

III 2 9 9 56

fraction of the hollow objects was 0.0139.

according to F(r) = Felec(r) + Fideal(r) + Fhs(r) + Fbond(r).

(8)

The first term is given by

(6)

i

1/2 ≤ Rcyl x 2i + yi2 otherwise

General

Felec(r) =

where α i is the angle between the position vectors ri+1 − ri and ri−1 − ri with Ωi−1,i,i+1 = 1 if bead i − 1, i, and i + 1 refer to the same side chain or neutral node, otherwise 0; α0 = 180◦ is the equilibrium angle and kangle = 0.51 × 10−24 J deg−2 the angular force constant. The last term entering Eq. (1) represents the confining cylindrical box and is given by Ucell = ucell(r i ) 0, = ∞, i

TABLE II. Model systems investigated.

N ′ −∇r i j uelec ri j , ij

(9)

i< j

r i j standing for the electrostatic interactions across with uelec ij the mid-plane, and i and j indices of interacting charged particles (hollow spheres or counterions) located on different sides of the plane z = 0. Here and in the following, ⟨. . .⟩ denotes a configuration average. The second term is Fideal(r) = −kT ρi (z = 0) − ρi (z = L cyl/2) A, (10) where ρi represents counterion densities at the midplane and at the end of the cylindrical simulation box (z = L cyl/2), and A the cross-section area of the cylinder. The third term is given by Fhs(r) =

N ′

−∇r i j uhs i j ri j ,

(11)

i< j

where uihsj r i j is the hard sphere repulsion between two small ions i and j at the mid-plane. The fourth term intervenes in case of the counterions with dendritic structure and is evaluated according to ′ Fbond(r) = −∇r i j ubond (r i j ) , (12) k k

where k is the bond connecting bead i and j found on each side of the midplane and ubond is given by Eq. (5). k The corresponding potential of the mean force U pmf (r) is given by

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r U pmf (r) = −

F(r ′)dr ′

(13)

∞

and, conventionally, U pmf (r) is considered zero at the largest shell-shell separation investigated. The mean force was sampled from individual simulations carried out at fixed separation between the two charged shells distributed symmetric along the C∞ symmetry axis of the box. For the discrete charge model, i.e., λ = 1, the shells were allowed to rotate freely or they had fixed orientations with respect to C∞ axis. The relative orientation of the two distributions was quantified by d x y, i = ((x 1,i − x 2,i )2 + ( y1,i − y2,i )2)1/2/Rext,

(14)

where i index stands for the most departed (i) vertices (5fold axis) and (ii) centers of faces (3-fold axis) of the shells indexed as 1 and 2, and x, y are their Cartesian coordinates in the box frame. For consistency with the symmetry axis notation, i ≡ 5-5 or 3-3. When the shell orientation was fixed, one of the symmetry axes was parallel to C∞ axis, and the particular cases depicted in Figure 3 were considered. Thus, the discrete charge distribution of each shell had the 2-fold, 3-fold, or 5-fold symmetry axis along C∞, and additionally, the two charge distributions were symmetric with respect to z = 0 plane. Figure 3 displays also the d xy,5-5 and d xy,3-3 parameters for these fixed orientations. The equilibrium of the model system was obtained by employing Monte Carlo simulations. The counterions were free to move in the available cell, i.e., the inner and the outer space of the hollow objects. The counterions of Systems I and II were subjected to single particle trial move with a translational displacement parameter larger than the shell thickness, which rendered the hollow spheres permeable to the counterions. As to counterions of System III exhibiting internal degrees of freedom, the individual beads were subjected to single particle moves and two collective trial moves were additionally considered: (i) translation of the whole dendritic structure with a trial displacement larger than the shell thickness and (ii) pivot rotation of the short chains.

The contributions of the mean force given by Eq. (8) were evaluated from the production run involving 106 trial moves per particle that followed to an equilibration run of 104 trial moves. The uncertainties were obtained by dividing the production run in ten sub-batches and assessed as one standard deviation of the mean. It should be mentioned that the exact axial ratio of cylindrical box was not important as long as the box cross section was not sufficiently narrow to interfere with the counterion distribution around the two shells. It turned out that the mean force was not altered within the numerical precision for Rcyl > 200 Å at the constant hollow sphere fraction of 0.0139 and therefore the box dimensions were fixed at Rcyl = 4RS,ext = 248 Å and L cyl = 12RS,ext = 744 Å. The simulations were all performed by using the integrated Monte Carlo/molecular dynamics/Brownian dynamics simulation package MOLSIM.51

III. RESULTS AND DISCUSSION A. Electrostatic coupling in Systems I-III at constant shell parameters

Figure 4 shows the cylindrical box containing Systems I-III where the two hollow spheres have the relative thickness d S = 0.193, and the charge fully smeared out at the outer surface, i.e., d σ = 0 and λ = 0. The two shells are separated by the normalized distance r/2RS,ext = 1.113. The main features observed are given by (i) the counterion partitioning between the inner and the outer space of the shells, and (ii) the increasing inhomogeneity of the counterion distribution in the outer space with increasing electrostatic coupling, which leads eventually to an apparent full adsorption in the case of the dendritic counterions (System III). The mean forces of Systems I-III are shown in Figure 5 as a function of the normalized distance between the shells r/2RS,ext. From the plot we observe that the mean force is highly dependent on the counterion valence, in line with the mean force acting between filled like-charged objects.5 Starting with System I, the mean force is repulsive

FIG. 3. Picture of the two hollow spheres (red shells) illustrating three particular orientations of the interacting sites (green dots) with respect to C∞ axis (yellow arrow): 5-fold, 3-fold, and 2-fold symmetry axis parallels C∞, and the shell orientation ensured additionally the symmetry of the two distributions with respect to z = 0 plane. The relative radial shell-shell separation was r /2R S,ext = 1.13 and dσ = 0. See text for the explanation of d xy,5-5 and d xy,3-3 parameters.

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FIG. 5. Mean force F /kT as a function of the normalized distance between the two shells r /2R S,ext for System I (•), II (■), and III () at d S = 0.193, d σ = 0, and λ = 0.

FIG. 4. Typical configurations of System (a) I, (b) II, and (c) III containing two charged hollow spheres (red shells) with d S = 0.193, d σ = 0, λ = 0 and separated by the distance r /2R S,ext = 1.113 and counterions with valence 1 (green), 3 (yellow), and small dendrimers (blue).

and increases steadily with decreasing r/2RS,ext so that it amounts to ∼6.4 Å−1 near the shell-shell contact. When the monovalent counterions were replaced by the trivalent ones, the nature of mean force changes and depends on the shell-shell separation. It remains repulsive at very short separations, yet turns attractive with a shallow minimum at r/2RS,ext ≈ 1.06 and essentially zero at r/2RS,ext > 1.14. These features describing the appearance of a short-ranged mean force are also encountered for System III. In more detail, the separation range exhibiting an attractive mean force extends up to r/2RS,ext ≈ 1.25, the attractive force deepens at constant shell-shell separation, and the maximum amplitude is shifted to r/2RS,ext ≈ 1.1. Note that the mean force remained still repulsive at very short separations in all systems, denoting the lack of available space between the two shells for accommodating the adsorbed species. If one views the increase of the counterion valency as a process, the features mentioned above demonstrate that, similar to the filled charged-like spheres, the electrostatic coupling is determinant for the nature of mean force acting between the shells. B. Hollow spheres with variable inner radius

We have shown in Sec. III A that the shells behaved electrostatically similar to the filled spherical colloids as

the nature of the mean force depended on the valence of the counterions. To investigate how the shell characteristics influence the mean force, we first examine the hollow spheres having constant outer shell and radial location of the charge and variable inner radius. Thus, the electrostatic potential originating from the shell charge is maintained constant inside and outside the shell, and the counterion presence inside the hollow sphere (cf. Figure 4) is directly related to the inner space volume. Figure 6 shows the mean force between the two shells F/kT and the corresponding potential of the mean force U pmf/kT as a function of the shell-shell separation for Systems I-III. The relative thickness of the shell d S ranged from 1, i.e., filled sphere, to 0.193, whereas the charge was smeared out at d int = 10 Å from the outer surface. F/kT of System I is repulsive independent of the shell thickness and decreases for d S < 0.838 at constant shell-shell separation. Alternatively, U pmf/kT is strong and long-ranged, with thinner shells weakening the repulsion. When the electrostatic coupling increased at System II, the short-range attractive mean force is observed for the filled sphere. As the shell became increasingly thinner, the mean force is again constant at d S > 0.838, and thereafter the force magnitude decreases steadily in spite of the fact that the range of the attractive domain is the same. Notably, the weak attractive interaction turns eventually to a repulsive one at d S = 0.193. As regards the potential of the mean force, U pmf of the filled sphere exhibits a minimum of ≈ − 5kT at r/2RS,ext ≈ 1.06, followed by a local weak barrier of 0.3 kT at r/2RS,ext ≈ 1.2. The amplitude of the attractive minimum reduces rapidly at d S < 0.838 and a long-range repulsive potential is found instead at d S = 0.193. In the case of System III, the reduced effective shell-shell attraction upon decreasing d S is observed, and, by contrast to System II, a

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FIG. 6. (i) Mean force F /kT and (ii) potential of the mean force U pmf /kT as a function of the normalized distance between the two shells r /2R S,ext for System (a) I, (b) II, and (c) III at d int = 10 Å, λ = 0, and d S = 1.0 (•), 0.838 (■), 0.516 (), 0.193 (N); the inset represents the charge of the counterions Z i n i,int in the inner space of the shell for System I (•), II (■), and III () as a function of d S at r /2R S,ext = 1.3.

transition to a pure-repulsive regime does not occur for the thinnest shell. Because the inner radius was the varying shell parameter, the features mentioned above originated from the increasing available space for the counterions inside the hollow sphere. This statement is supported by the inset of Figure 6 displaying, for all systems, the mean charge of the encapsulated counterions Zini,int as a function of thickness d S at the shell-shell separation r/2RS,ext = 1.3. We should mention first that the functional form of the curves does not depend critically on the particular shell-shell separation. Second, no counterions are encapsulated at d S ≥ 0.838, that is, an internal radius RS,int ≤ 10 Å, irrespective of the counterion type. Thus, although the void of 10 Å could accommodate monovalent and trivalent counterions with a radius of 2 Å and dendrimers whose radius of gyration was 4.2 Å, the small ions were not confined because the entropy loss associated with their entrapment is still larger than the energetic gain at the counterions transfer to the inner space. One may also infer that a less permeable shell would further deplete the inner cavity at d S < 0.838. We remind that the small ions exchange across the shells, which could occur

through pores located around the symmetry axes52 or may be absent at least for poliovirus,53 was not considered explicitly by our model. Third, with further decrease of d S, Zini,int starts increasing steadily, and the higher the counterion valence, the larger encapsulated charge. The assessment of the small ions encapsulated at varying shell inner radius cannot explain completely the rich behavior of F/kT, in particular the attraction-to-repulsion transition encountered in System II. To obtain further insights into the role of counterion partitioning, we investigated the components of the mean force. They are displayed in Figure 7 as a function of d S for Systems I-III found at r/2RS,ext = 1.048. Again, the functional form of the curves does not depend critically on the distance between the two shells. As a first remark, the components are all constant at d S ≥ 0.838 due to the lack of encapsulated counterions. Starting with System I, Felec increases at d S ≤ 0.838, in particular due to the net transfer of small ions to the inner space, while Fhs decreases because of the counterion loss at the midplane. Fideal possessed similar dependence as Fhs, a feature also ascribed to the counterion depletion at the midplane as the shell becomes thin.

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dendritic structure. On the other hand, the high amplitude of Fbond indicates the presence of bonds across the midplane that belongs to dendrimers adsorbed onto both shells. C. Shells with variable radial position of the charge

FIG. 7. The contributions Felec (•), Fideal (■), Fhs (), and Fbond (N) to the mean force as a function relative shell thickness d S for System (a) I, (b) II, and (c) III at d int = 10 Å and r /2R S,ext = 1.048.

As to System II, the salient feature is given by the negative values of Felec, which originates from the shell-counterion correlation effects expected at large electrostatic coupling. In more detail, Fideal and Felec display similar trends as in System I at decreasing shell thickness, whereas Fhs vanishes practically because of the narrowed double layer at the outer surface (cf. Figure 4(b)). The presence of the connected beads in System III leads to high values of Fideal and Fbond mean force components. However, they exhibit at the same time a large degree of cancellation such that they supersede several times their sum. Fideal, Fhs, and Felec exhibit the same trends as in System II upon decreasing d S, with the additional observation that the Felec amplitude is lowered because of the smaller bead charge. Nevertheless, the overall mean force was more attractive in System III owing to the additional strongly attractive bond term Fbond. The negative bond contribution originated from the fact that the bond length was larger than the beadbead equilibrium separation, which in turn was due to the electrostatic repulsion between the connected beads of the

The mean force between shells with the same thickness and various relative radial position of the charge within the shell is examined. Figure 8 shows the mean force F/kT and corresponding potential U pmf as a function of shellshell separation for the hollow spheres with d S = 0.193 and a homogeneous charge distribution located in the shell at d σ ranging from 0 to 0.833. Thus, the charge is at the outer surface in the former case, and embedded and near the inner surface in the latter situation. The long-ranged repulsive interaction in System I is independent of the location of the capsid charge, and F/kT and U pmf decrease by shifting the shell charge towards the inner surface at given shell-shell separation (Figure 8(a)). Turning to System II (Figure 8(b)), the shortrange effective attraction found at d σ = 0 weakened and turned repulsive by moving the charge close to the inner surface. In case of the more electrostatically coupled System III (Figure 8(c)), the nature of the mean force does not change irrespective of the shell charge locations. Nevertheless, in line with System II, the larger d σ parameter, the weaker the magnitude, and the larger domain of the attractive U pmf , features that were also observed by decreasing the shell thickness (cf. Figure 6(c)). The mean charge of the counterions found inside the shell Zini,int at the shell-shell separation r/2RS,ext = 1.3 is illustrated in the inset of Figure 8 as a function of the radial position of the shell charge. As expected to occur by increasing the electrostatic potential inside a cavity, the net charge of the encapsulated counterions increases steadily for all investigated systems when the radial location of the shell charge approaches the inner surface. The inset also shows a complex dependence of the encapsulated charge on the counterion valence at given d σ. Thus, the net charge of encapsidated small ions decreases with increasing valence at low d σ, whereas the trend is reversed at d σ higher than 0.2. To get further insights into the effect of the radial location of the shell charge, we examine the components of the mean force. They are shown in Figure 9 as a function of d σ at the shell-shell separation r/2RS,ext = 1.048, where attractive forces manifested in System II and III. All components of System I were positive with a steady increase of Felec and decrease of Fideal and Fhs at the radial move of the shell charge towards the inner surface. At the elevated electrostatic coupling of Systems II-III, the component analysis results in similar trends of Felec and Fideal at increasing d σ and the main attractive component given by Felec, in System II, and Fbond in System III. Notably, if one compares the behaviors of mean force components by varying either d S or d σ (cf. Figures 7 and 9), one can conclude that the net transfer of the charged species regulates the counterion density at the outer shell surface and mid plane. The former provides the shellcounterion correlation effects responsible for the appearance of an effective attraction and the latter is bound to the overlapping counterion layers.

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FIG. 8. (i) Mean force F /kT and (ii) potential of the mean force U pmf /kT as a function of the normalized distance between the two shells r /2R S,ext for System (a) I, (b) II, and (c) III at d σ = 0 (•), 0.5 (■), and 0.833 (), d S = 0.193, and λ = 0; the inset represents the charge of the counterions Z i n i,int in the inner space of the hollow spheres for System I (•), II (■), and III () as a function of d σ at r /2R S,ext = 1.3.

Next, the mean force component analysis is complemented with the examination of the counterions near the shell surfaces. This is carried out by determining the number density of the counterions ρn found in a slab with a radial cutoff of 4 Å as a function of z coordinate. Figure 10 displays ρn(z) for Systems I-III at the shell-shell separation r/2RS,ext = 1.048 and three d σ values. We mention that ρn(z) is normalized to the overall number of counterions in the slab. Figure 10 indicates that non-homogeneous counterion distributions take place at both surfaces. In more detail, the monovalent counterions accumulate between the two shells (low absolute values of z) at both surfaces. When d σ increased, the relative accretion enhances at the outer surface and remains the same at the inner surface. By contrast, System II exhibits an accumulation only at the outer surface, whereas a weak depletion is observed at the inner surface at low absolute z values. The former feature is affected to a larger extent as compared to System I, whereas the depletion at the inner surface remained rather constant by the variation of d σ. Notably, ρn(z) of the dendritic counterions (System III) reveals damped oscillation behavior, pointing

likely at the presence of long-range spatial correlations among the dendrimer beads. Note also that the higher amplitudes are obtained at the outer surface and the oscillation period is dependent on d σ. D. Shell with icosahedral discrete charge model

In this section, the homogeneous charge distribution is replaced by the icosahedral distribution described in detail in Section II A. Thus, we examined the shells characterized by the relative thickness d s = 0.193 and the charge located at the outer surface d σ = 0 and the charge distribution parameter λ = 1. Apart from the case where the two shells were allowed to rotate freely, additional simulations at fixed shell orientation having one of the distribution symmetry axis along C∞ axis of the box were carried out. The resulting mean force and the potential of the mean force of System II as a function of shellshell separation are displayed in Figure 11. One sees that the short-range attractive character is not altered upon changing λ parameter, despite the strong dependence of the maximum

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FIG. 9. The contributions Felec (•), Fideal (■), Fhs (), and Fbond (N) to the mean force as a function of radial location of the charge inside the shell d σ for System (a) I, (b) II, and (c) III at d S = 0.193 and r /2R S,ext = 1.048.

amplitude and of the range of the attractive domain on the shell angular orientation. In more detail, (i) r/2RS,ext domain experiencing attraction shifts towards shorter separation by making the capsid charge discrete and (ii) the global minimum of F(r) amounting to −0.5 kT/e at λ = 0, decreases to −5.9, −2.7, and −1.2 kT/e for 5-5, 2-2, and 3-3 fixed orientations. Note also the particular case of 3-3 orientation, where the attractive domain shifts pronouncedly such that the repulsion at very close shell-shell separations is absent. These observations could be also drawn from the potential of the mean force in Figure 11(b). It can be additionally pointed out that the amplitude of the local repulsive minimum found at the intermediate distance r/2RS,ext ≈ 1.2 for λ = 0 is not affected by the capsid orientation. The effect of the shell charge discreteness is particularly illustrated by the freely oriented model. The functional form of the mean force is similar to those found at fixed orientations (cf. Figures 11(a) and 11(b)), and the angular averaging results in intermediate values at given shell-shell separation. As the

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global minimum of U pmf decreased from −3.5 kT at λ = 0 to −5.2 kT at λ = 1, it can be concluded that the discrete model leads to a more effective attraction. An enhanced attractive U pmf (r) was reported for like-charged filled objects when the discrete charges on the surface were either mobile13 or fixed at random or symmetric angular orientation.9,13 We should also mention that the discrete charge distribution applied to System I was found to reduce the repulsion between the two shells (data not shown), a feature that had been reported recently by a theoretical investigation of the interaction of two permeable spherical particles carrying surface charge distributions with icosahedral symmetry.54 Notably, the reduced repulsion in System I could not be turned into an effective attraction as a result of increasing the local variation of the shell charge. One can infer that a discrete charge model improvement by considering patches of opposite sign may result in a repulsiveto-attractive transition for System I, as recently suggested by Božiˇc et al.54 We will further on continue to investigate the freely oriented shells by identifying the relative orientation of the two charge distributions at the separations that are representative of the profile of the potential of the mean force: the largest separation r/2Rs,ext = 1.47 having U pmf (r) conventionally zero and at 1.04 < r/2RS,ext < 1.1, where U pmf (r) was attractive (cf. Figure 11(b)). We recall that the relative orientation is quantified by d xy,5-5 and d xy,3-3 parameters defined in Section II C. Representative distribution functions P(d xy,5-5, d xy,3-3) at the shell-shell separations mentioned above are shown in Figure 12. At r/2Rs,ext = 1.47, a wide range of orientations is obtained, and high values are noted at 0.5 0.8 (cf. Figure 12(d)). At r/2Rs,ext < 1.04, the distribution was centered at d xy,5-5 = 0 and d xy,3-3 = 0.917 (data not shown), indicating the 5-5 orientation becomes additionally symmetric with respect to z = 0 plane. These results demonstrate that the high electrostatic potential developed at the vertices (cf. Figure 1(b)) determines the relative orientation of the two shells at very short separations only. Finally, we relate the effective attraction between the shells bearing icosahedral charge distribution to the shellcounterion correlations. We found that the net charge of the encapsulated counterions, calculated at the separation where U pmf reached its minimum, decreased from 25.5 e to 9.5 e by making the shell charge discrete. Thus, a net transfer of the small ions between the inner and outer shell space occurs

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FIG. 10. Number density of counterions ρ n(z) found in a spherical layer of 4 Å at (i) the inner and (ii) the outer surface as a function of z coordinate for System (a) I, (b) II, and (c) III at d σ = 0.0 (•), 0.5 (■), and () 0.833 and d S = 0.193 and r /2R S,ext = 1.048; ρn(z) was normalized to the total number of counterions in the layer.

by varying either the radial or the angular location of the shell charges and can explain the enlarged correlation effects of the discrete charge model. On the other hand, it turned out that the number of encapsulated counterions was virtually unaffected by the shell orientations, and implicitly the number of trivalent counterions adsorbed onto the shell outer surface was rather the same. Nonetheless, the large variation of the potential depth (cf. Figure 11(b)) suggests that the relative

orientation of the two shells has also a strong impact on the shell-charge-counterion spatial correlation. To describe this effect qualitatively, we assessed the number density of the counterions ρn(z) near the outer surface of the shell in a similar manner to the examination carried out at λ = 0 (cf. Figure 10). As the region between the two outer surfaces is expected to have the largest impact on U pmf ,13 we found that ρn(±zmin), with zmin the lowest absolute coordinate of the outer shells,

FIG. 11. (a) Mean force F /kT and (b) potential of the mean force U pmf /kT for System II as a function of r /2R s,ext for the charge distribution parameter λ = 0 (dashed curve) and 1 (continuous curves) at fixed (•) 5-5, () 3-3, and (■) 2-2, and (N) free orientations; d σ = 0, d s = 0.193.

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FIG. 12. Contour plot of the 2D distribution function P(d xy,5-5, d xy,3-3) for freely oriented shells at the normalized separation r /2R s,ext = (a) 1.47, (b) 1.08, (c) 1.06, and (d) 1.05; d σ = 0, d s = 0.193, and λ = 1.

reached the largest value of 1.2 × 10−3 at 5-5 orientation, then amounted to near 0.8 × 10−3 at 2-2 orientation and for the freely oriented shells and to virtually zero at 3-3 orientation. This counterion accumulation or depletion at ±zmin obtained by freezing the mobile shell at 5-5 or 3-3 orientation can be understood in terms of the counterion adhesion to the icosahedral contour, explaining thus the variation of the U pmf amplitude with the shell orientations in Figure 11(b). IV. CONCLUSION

The mean force acting between two charged hollow spheres and mediated by the mono-, trivalent, or short dendritic counterions has been examined in the salt-free limit using a cylindrical cell model and canonical Monte Carlo simulations. Besides the conventional construction of a homogeneous charged surface, a discrete model including interacting sites localized on the shell outer surface has been considered. The charged shell represents a coarse and scaleddown description of the viruses with icosahedral symmetry exhibiting experimental superstructures and agglomerations in the presence of multivalent electrolytes. Similar to the cases of like-charged macroions, the counterion multivalency tailored the spatial correlations among the

shells and the counterions adsorbed at the shell outer surface and determined a transition from a purely repulsive to a purely attractive behavior. Both types of forces were attenuated, and furthermore, a repulsion-to-attraction transition could take place in the presence of trivalent counterions, as a result of a net transfer of the counterions from the outer space to the inner void. The counterion balance between these two regions was influenced by the inner radius and the radial location of the charge of the shell at constant shell size. With icosahedrally distributed shell charge, the counterion encapsulation was reduced and rendered an enhanced shellshell attraction in the presence of the trivalent counterions. Orientations to line up the vertices (5-fold symmetry axis) of both icosahedral distribution along the longest axis of the simulation box were encountered only at very short separations and were not required to increase the effective attraction. Other preferred orientations at the shell-shell separations exhibiting an effective attraction involved near alignment of 5fold symmetry axis for one shell and 2-fold or 3-fold symmetry axis for the other shell. Albeit our model system did not emulate any particular real virus, the results confirm the virus agglomeration experimentally observed in the presence of multivalent species originated from the correlation effects of electrostatic origin,

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and we expect that our results would be altered quantitatively in a way, but not qualitatively, by more elaborated viral capsid constructions. This work can be thus seen as an important step towards a deeper understanding of the non-specific electrostatic interactions of the virus-like nanoparticles. ACKNOWLEDGMENTS

This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS–UEFISCDI, Project No. PN-II-RU-TE-2012-3-0036. We also thank Professor Per Linse for helpful and constructive comments. 1V.

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