Role of metallic core for the stability of virus-like particles in strongly coupled electrostatics

Electrostatic interactions play important roles in the formation and stability of viruses and virus-like particles (VLPs) through processes that often involve added, or naturally occurring, multivalent ions. Here, we investigate the electrostatic or osmotic pressure acting on the proteinaceous shell of a generic model of VLPs, comprising a charged outer shell and a metallic nanoparticle core, coated by a charged layer and bathed in an aqueous electrolyte solution. Motivated by the recent studies accentuating the role of multivalent ions for the stability of VLPs, we focus on the effects of multivalent cations and anions in an otherwise monovalent ionic solution. We perform extensive Monte-Carlo simulations based on appropriate Coulombic interactions that consistently take into account the effects of salt screening, the dielectric polarization of the metallic core, and the strong-coupling electrostatics due to multivalent ions. We specifically study the intricate roles these factors play in the electrostatic stability of the model VLPs. It is shown that while the insertion of a metallic nanoparticle by itself can produce negative, inward-directed, pressure on the outer shell, addition of only a small amount of multivalent counterions can robustly engender negative pressures, enhancing the VLP stability across a wide range of values for the system parameters.

(1) where C is a constant. The solution to the above set of equations in the region outside the spherical NP can be expressed as the sum of a special solution (rst term below), representing the bulk solution G 0 (r, r ) = e −κ|r−r | /(4πεε 0 |r − r |), and a homogenous solution (second term below) due to the presence of the NP [1], where k l (·) are modied spherical Bessel functions of the second kind, P l (·) are Legendre polynomials, and we have dened r = |r|, r = |r |, and ϑ as the angle between r and r . The coecients B l are in general functions of r . The rst term above can be expanded as [1] exp(−κ|r − r |) |r − r | = κ ∞ l=0 (2l + 1)i l (κr < )k l (κr > )P l (cos ϑ), in which i l (·) are modied spherical Bessel functions of the rst kind and r < and r > denote the smaller and larger values of r and r . Since the potential on the metallic sphere is constant and does not depend on ϑ, and using r < = r = R 0 and r > = r , we nd C = B 0 k 0 (κR 0 ) + κ 4πεε0 i 0 (κR 0 )k 0 (κr ), for l = 0, 0 = B l k l (κR 0 ) + κ 4πεε0 (2l + 1)i l (κR 0 )k l (κr ), for l > 0.
(4) and, hence, i0 (κR0) k0(κR0) k 0 (κr ), which give the solution in the outside region, r, r ≥ R 0 , as The constant C can be xed by using the fact that the metallic NP is assumed to be electroneutral; hence, using Gauss's law and after straightforward manipulations, we nd where we have used the explicit expressions The nal expression for the Green's function can thus be obtained as G(r, r ) = G 0 (r, r ) + G im (r, r ), where G im (r, r ) is the contribution representing salt/dielectric image eects, In the VLP model used in the main text, the charge distribution of the inner and outer spherical shells (of radii R 1 and R 2 ) can formally be expressed as Other explicit charges in the system include multivalent ions each of charge valency q, located at positions {r i }, giving the local charge distribution function The Hamiltonian associated with electrostatic interactions in the system can in general be written as Let us rst focus on the case of only one multivalent ion in the system positioned at b (note that multivalent ion positions are restricted to remain outside the inner shell, i.e., b = |b| > R 1 ). We will thus have The rst term in Eq. (14) is the self-energy of the multivalent ion and its image interaction. We subtract the redundant (innite) vacuum self-energy of the multivalent ion, and the ion-image interaction term is found as The second term in Eq. (14) is the interaction between the ion and the surface charge, including both the direct DH and the image interactions. For the α-th shell, it yields The direct interaction is where ϑ is the angle between r and b. The direct interaction term can be evaluated as The image interaction part, on the other hand, is obtained as The net contribution from the second term in Eq. (14) is thus obtained as For the last part of Eq. (14), which gives the contribution from surface-surface interaction (including the relevant image eects), we can write The direct interaction part here is given by or, giving The rst angular integration above can be done straightforwardly, and since the result is independent of the angle between the two vectors, the second angular integration only yields a constant. Thus, The image interaction part, on the other hand, is obtained as or, similarly as before, Hence, we have Putting the three terms contributing to the Hamiltonian together, i.e., H = H im + H σ + H σσ , we have Now, when we have N multivalent ions in the system, the Hamiltonian can straightforwardly be expressed as This completes the derivation of the expressions given in Eqs.

III. NET PRESSURE ON THE OUTER SHELL
In the absence of a metallic core within the VLP, the net electrostatic potential of the two charged shells with radii R 1 and R 2 is obtained as ϕ 2 (R 1 < r ≤ R 2 ) = e 0 εε 0 σ 2 e −κR2 R 2 sinh κr κr + (σ 1 R 1 sinh κR 1 ) e −κr κr , ϕ 3 (r > R 2 ) = e 0 εε 0 (σ 1 R 1 sinh κR 1 + σ 2 R 2 sinh κR 2 ) e −κr κr . (34) The corresponding net (osmotic) pressure acting on the outer shell follows from where V 2 = 4πR 3 2 /3 is the volume of the outer shell and the partial derivative is taken at xed value of the total surface charge of this shell, i.e., Q 2 = 4πR 2 2 σ 2 . We thus nd The contribution of multivalent ions to the osmotic pressure follows as (see Refs. [2,3]) where the potential ϕ is dened in piece-wise fashion throughout the space according to expressions (32)-(34). Since multivalent ions are restricted to remain outside the inner shell |r i | > R 1 , we shall only require and In the presence of a metallic core within the VLP, the potential derivative can be obtained from the second term in Eq. (30) as This expression can be used to construct the contribution of multivalent ions to the osmotic pressure, as expressed in Eqs. (12) and (13)  The algorithm can be summarized in practical terms as follows.
Set up an initial simulation with the number of multivalent ions being set equal to N ini = N 0 + C N , where C N and N 0 are chosen for practical convenience as C N = 10 and N 0 = max{1, c 0 V b − (Q 1 + Q 2 )/q} (these choices are of no physical signicance for the outcomes). Here, c 0 V b is the number of multivalent ions in the simulation box of volume V b , if they were to be distributed evenly within the box, and −(Q 1 + Q 2 )/q is the excess number of multivalent ions required to compensate the sum of the xed charges on the two shells Q α = 4πσ α R 2 α , where α = 1 and 2 for the inner and outer shells, respectively. Run a full simulation using N ini multivalent ions and obtain the resulting bulk concentration c ini .
If c ini < c 0 , re-run the initial simulation by setting N ini → N ini + C N and repeat as necessary until a situation with c ini > c 0 is reached.
The initial bracketing of the prescribed c 0 is thus achieved by storing the last values of N ini and c ini obtained through the preceding steps as the upper-bound values N max = N ini and c max = c ini , and by setting the lower-bound values as N min = 1 and c min = 0; the latter are reasonable lower-bound choices, as with just one multivalent ion, our simulations give a nearly vanishing bulk concentration. Repeat the above four steps until |c * − c 0 | < c 0 is satised, where the convergence relative error is conventionally taken as = 10 −3 .
• The data from the nal simulation achieved through the above iterative steps produce the desirable outcomes.