Computational Study of the Forces Driving Aggregation of Ultrasmall

Mar 17, 2017 - Nanoparticle (NP) aggregation can lead to prolonged retention in tissues or embolism, among other adverse effects. Successful use in ...
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Computational Study of the Forces Driving Aggregation of Ultrasmall Nanoparticles in Biological Fluids Sergio A. Hassan* Center for Molecular Modeling, OIR/CIT, National Institutes of Health, U.S. DHHS, Bethesda, Maryland 20892, United States ABSTRACT: Nanoparticle (NP) aggregation can lead to prolonged retention in tissues or embolism, among other adverse effects. Successful use in biomedicine thus requires the capability to make NPs with limited aggregative potential. Rational design is presently a challenge due to incomplete knowledge of their interactions in biofluids. Recently, ultrasmall gold NPs passivated with endogenous antioxidant glutathione have shown promise for use in vivo. Computer simulations are here conducted to identify the forces underlying aggregation (or lack thereof) of these NPs in a cell culture. Electrostatic interactions are insufficient to induce association, but the van der Waals forces exerted by cations, anions, and net-neutral polar species can promote the formation of stable dimers. The entropic effects of depletion are negligible, but the combined effect of depletion and macromolecular crowding at physiological concentrations can stabilize aggregates containing just a few NPs. Interparticle interactions are controlled by modest changes in both the structure and dynamic of the interfacial liquid. The molecular origin of these effects and their dependence on NP size are described. The liquid is shown to be highly structured, with large and long-lived hydrogen-bonded water clusters developing often in the interparticle space; their potential role as transient, long-range proton wires connecting and enveloping neighboring NPs is discussed. The basis for a parsimonious theory of ultrasmall NPs in complex fluids is established. KEYWORDS: ultrasmall nanoparticles, nanoparticle aggregation, depletion forces, crowding effects, simulations in biological fluids

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NPs can still aggregate under certain conditions. These observations suggest that colloidal stability could be obtained by using ultrasmall NPs of well-defined coating-dependent size below a certain threshold. Such NPs would constitute an optimal platform for further functionalization in a broad range of biomedical applications,14,15 including targeted and controlled drug delivery. In the ultrasmall size scale NPs are expected to display characteristics of both colloidal and macromolecular systems. Ultrasmall NPs are comparable in size to an average ∼50 kDa protein and are smaller than albumin and other serum proteins. Because of their relatively large sizes, most commonly studied NPs can develop a protein corona when exposed to a biological medium;5,16 this is a layer of adsorbed proteins that forms on the NP surface and plays a central role in their interaction with cells and tissues.17 This notion is no longer applicable to ultrasmall NPs, which have the potential to associate with proteins as other proteins do, in a variety of modes and with a

anoparticles (NPs) with core diameters (⦶) below ∼100 nm have emerged as promising agents for diagnostics and therapeutics,1−3 e.g., as imaging probes, biomarkers, harvesters, and drug carriers. Despite their versatility and potentialities, their physiological fate and the effects on the environment upon excretion are not yet well established, and concerns exist regarding their role in inflammation and toxicity.4,5 The origin of these problems lies, in part, in the tendency of NPs to aggregate or bind to macromolecules, such as plasma proteins, which could lead to prolonged retention in tissues or embolism, among other conditions.6 The propensity of NPs to aggregate can be mitigated or even controlled by changing two features of the NP design:7−11 core size and surface chemistry. In particular, ultrasmall (⦶ < ∼3 nm) AuNPs passivated with the endogenous antioxidant glutathione (GSH) have recently been shown to present some advantages for use in vivo,8,12 including reduced toxicity and faster renal clearance, as well as tumor accumulation. However, even within this size subclass a change in behavior has recently been reported13 around a critical core diameter ⦶c ≈ 2 nm: smaller NPs remain colloidally stable in a variety of biological fluids, whereas larger This article not subject to U.S. Copyright. Published 2017 by the American Chemical Society

Received: February 12, 2017 Accepted: March 17, 2017 Published: March 17, 2017 4145

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25 pair is broad, with a local minimum at an interparticle separation rcc ≈ 4.5 nm, similar to the shallow metastable state of the AuNP-14 pair at rcc ≈ 3.3 nm. The broad, shallow minima at close contact are observed in all the simulations and are typically located within a ∼0.5 nm range around rcc (indicated by dashed lines). These equilibrium configurations arise from a balance of forces exerted by the different components of the solution, none of which, individually, is responsible for the local stabilization of the pairs. This is apparent from the decompositions in Figure 2, which shows the potentials generated by cations, anions, neutral species, and water. (Because the direct NP−NP forces and the forces induced by the cations are large and act in opposite directions, the latter is given as the sum of both.) All the behaviors discussed in this section are based on general trends and correlations of the potentials consistent across simulations and common in varying degrees to both NP pairs. As the interparticle separation decreases, the average force induced by each component of the solution changes in magnitude and direction. In particular, as the AuNP-14 pair approaches the close contact configuration, the direct interparticle repulsion is balanced by the attractive forces induced by water, anions, and cations. For this pair, the forces induced by the neutral species are negligible within the errors. For AuNP-25 the situation is different: around the close contact configuration, anions and neutral species induce repulsion and water attraction, whereas the attractive forces induced by the cations are significantly strengthened so as to overcome the direct electrostatic repulsion; the reversal of forces occurs around r ≈ 5 nm for both types of ions. Despite their strength, the attractive forces generated by the cations are insufficient to stabilize the potentials at close contact, and anions are required to offset the direct repulsion in both pairs. The potentials generated by cations and anions tend to anticorrelate, as do those generated by water and the rest of the CC species despite their quite different interaction mechanisms with the NPs. The molecular origin of these correlations is not always immediately apparent. The observation that no single component of the system is responsible for the local equilibrium at close contact is in line with a previous study13 of the same NPs in phosphate buffer solution (PBS) and may be a general feature of large, complex interfaces in biological fluids. This contrasts with the effect of salts on simpler molecular solutes18 in which localization of individual ions at the solute/solute interface can induce strong stabilizing forces; this also appears to be the case for ultrasmall NPs in simpler aqueous solutions.19,20

range of affinities and association/dissociation rates. A theoretical description based on common mean-field approximations would thus be incomplete. Ultimately, the rational design of NPs with desired structural, kinetic, and dynamic features in vivo calls for an atomic-level understanding of their interactions in and with a biological medium (biointeractions), mainly in the blood plasma and in the intracellular and interstitial fluids. The present study is undertaken with this objective in mind. The behavior of ultrasmall GSH-coated AuNPs with core diameters at both sides of ⦶c is studied in a cell culture at ambient conditions. Several questions are investigated: (i) the effects of the individual components of the solution on the interparticle interactions, including water, cations, anions, and net-neutral polar species; (ii) the entropic effects of depletion and macromolecular crowding and the effects of ion sequestration by proteins; (iii) the role of water in mediating and inducing interparticle forces; (iv) the structural and dynamic changes of the interfacial liquid with the interparticle separation and their dependence on NP size. The implications for the development of a phenomenological theory of ultrasmall NPs in complex fluids are discussed.

RESULTS AND DISCUSSION Figure 1 shows the interparticle potentials of mean force, V(r), of the two NP pairs in a cell culture (CC; cf. Methods). The

Figure 1. Consensus means of the interparticle potentials of mean force and standard errors for the two pairs of glutathione-coated nanoparticles in cell culture at ambient conditions.

larger NPs favor association and the smaller dissociation, with an average energy difference of ∼30 kcal/mol at their respective close-contact configurations. This result is statistically significant and in agreement with recent experimental observations of the same NPs in CC.13 The basin of the potential of the AuNP-

Figure 2. Decompositions of the potentials generated by the components of the solution: cations: Na+, K+, Ca2+, Mg2+, L-Arg+, L-Lys+; anions: Cl−, SO42−, PO4H2−, PO4H2−, CO3H−, C3O3−; neutral: L-Gln, L-Ile, D-glucose, HEPES. Error bars shown at specific distances above rcc. 4146

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In general, a spatially homogeneous distribution of cations would induce repulsion between two negatively charged NPs due to the imbalance of forces resulting from the exclusion of cations from the space occupied by the NPs. Anions would induce attraction for similar reasons; likewise, for a homogeneous distribution of polar, net-neutral species, including water, due to the long-range charge−dipole interactions (dielectric response). Similarly, the long-range dispersive component of the vdW forces would induce repulsion, whereas the short-range repulsive component (hydrostatic pressure) would have no effects unless the NPs overlap, in which case attraction would ensue. Departures from these trends are due to the spatial heterogeneities and reconfiguration of the liquid as the distance between the NPs changes and occur in the vicinity of the NPs. There are several sources of heterogeneity; the most obvious is the counterion distribution, which reverses the direction of induced forces from repulsion (bulk) to strong attraction. For the NPs studied here the situation is rather complex, as neutral species and even anions can approach and bind to the NPs, thereby exerting significant electrostatic and vdW forces. Figure 5 shows the spatial distributions (cf. Methods) of cations, anions, and neutral species in the regions where restructuring and dynamic changes occurs. These regions are qualitatively divided in the outer (Ψo) and inner (Ψi) NP/liquid interfaces and the outer (Ωo) and inner (Ωi) NP/NP interfaces (cf. inset). The corresponding distributions in the bulk (not shown) are indeed quite homogeneous, so the corresponding background forces will not be discussed. Idiosyncratic behaviors of the solution species are consistent across simulations, including preferential binding and specific binding modes. Analysis of the individual trajectories show that divalent cations, here Ca2+ and Mg2+, tend to bind strongly to the GSH molecules and become highly coordinated; for AuNP25 these ions can be adsorbed into the coating, as observed previously in CaCl2 solutions.13 Their electrostatic effects on the interparticle interactions are significant, especially when they are attracted to the NP/NP interface. Other divalent and trivalent cations not considered in this study (cf. Methods) may play a significant role as well despite their low concentrations and warrant future studies. L-Arginine+ and L-lysine+ are frequently found on the NP surfaces, forming hydrogen bonds with GSH either through their side chains or through the CO2− or NH3+ termini. Strong, persistent binding occurs when the three groups are H-bonded simultaneously to one or more GSH molecules. These multiple H-bond interactions and the high coordination of divalent cations are possible only because of the close packing of GSH molecules and befitting surface curvature. This suggests a specific dependence of behavior on coating density and NP size. Na+ ions are quite mobile and highly concentrated on the NP surfaces and induce the largest interparticle forces; K+ ions do not seem to play a specific role. Anions are also found in the four Ψ and Ω regions, but tend to occupy the second hydration layer. Anions are generally bridged to the NP through one or more cations, usually Na+, which bind to the carboxyl groups of GSH. (Na+mediated associations are also observed with other cations, specifically between the carboxyl terminal groups of Arg+ or Lys+ and GSH.) Divalent anions such as PO4H2− and SO42− are highly stabilized in this way, as is CO3H−, which is found in disproportionate amounts on the NP surfaces, especially on AuNP-25. Nonetheless, some of these species, bicarbonate in particular, also bind directly to the NP surface through H-bonds

The forms of the potentials in Figure 2 are the result of specific molecular processes that unfold as the NPs approach one another from their initial separation. In general, the behavior of the bulk solution provides a nonspecific background modulation of the interactions, e.g., through its dielectric response. The static dielectric permittivity of water itself is dependent on the concentration and chemical nature of the solution components, which modify the structure and dynamics of water, hence the screening of electrostatic forces. In general, these bulk effects can be described through a mean field approximation. However, as discussed below, the nuances of the potentials, including the reversal of forces suggested by the means, result from specific structural and dynamic changes that take place at the NP/liquid interface, especially in the interparticle space, henceforth referred to as the NP/NP interface regardless of the NP−NP separation. The analysis here is restricted to electrostatics and van der Waals (vdW) interactions as defined by the classical, nonpolarizable force field used (cf. Methods); hydrogen bond contributions are not treated separately as they are accounted for implicitly through the parametrization of the nonbonded terms. The corresponding contributions are shown in Figure 3.

Figure 3. Electrostatic (solid line) and van der Waals (dashed) contributions to the total potentials of mean force of the AuNP-14 (thin line) and AuNP-25 (thick) pairs. Errors are calculated independently for each contribution.

Although the electrostatic forces generated by the solution always induce attraction and are much stronger than the corresponding vdW forces, they are nonetheless insufficient to stabilize either pair. At close contact the electrostatic potential of mean force is similar for both NPs within the standard errors, and it is only through the vdW forces that pairs can be stabilized. However, the direct NP−NP vdW interactions (not shown) are negligible except at very short interparticle separations when the coating molecules begin to overlap; this occurs below the minima of the vdW potentials in Figure 3 and is ∼0.5 nm smaller than rcc in both pairs (Figure 1). The fact that the direct vdW interactions play a negligible role in solution suggests that the conclusions drawn in this study are applicable to other core materials as well. Figure 4 shows the mean electrostatic and vdW contributions of each component of the solution. A comparison with Figure 2 shows that the directions of forces and overall shape of the potentials correlate strongly with electrostatics (except for distances with substantial coating overlap), including the reversal of forces at ∼5 nm in AuNP-25. The vdW forces are generally attractive, except for water. The rest of the section provides the molecular basis for the forms of the potentials in Figures 2 and 4. 4147

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Figure 4. Electrostatic and van der Waals contributions to the potentials generated by the components of the solution. The local forces are determined by changes in the structure (e.g., in A and B) and dynamics (e.g., D and C) of the interfacial liquid. Error bars are shown only at specific distances above rcc.

with the amine group of GSH. Cl− ions are also found in direct contact with these groups, but their association appears to be weak. These direct anion−NP interactions are transient, albeit frequent, and occur once the electric field created by the nearby carboxyl groups of GSH is diminished by the presence of a nearby cation, suggesting a correlation between the spatial localization of both types of ions. PO4H2− does not seem to play a specific role and neither does pyruvate. The forces exerted by anions close to the NPs are strong, which highlight the importance of ion pairing and higher orders of ion clustering in the NP biointeractions. Cluster formation in CC is quite rare in the bulk, but quite common near the NPs due to the localization of cations. The rich variety of interactions involving anionic species on the NP surface also suggests a dependence on the coating density. Finally, polar, net-neutral species can also form ion-mediated or H-bond associations with the NPs. Their direct effects on NP−NP interactions are, as expected, much smaller than in the case of ions, but not negligible in AuNP-25 (Figure 2). Their indirect effect, however, might be more important, as most of the neutral species are large and exclude ions and water from the vicinity of the NPs. This is apparent in Figure 5, where regions of high concentrations of neutral molecules tend to overlap with regions of low concentrations of ions. HEPES is found in relatively large amounts on the surface of AuNP-25 and much less on AuNP-14. It is often H-bonded to GSH either through its hydroxyl or sulfonyl hydroxide groups and tends to adopt a perpendicular conformation relative to the NP surface. In many cases, however, it can lie parallel to the surface, forming multiple H-bonds and become highly stabilized. HEPES is a buffering agent commonly used to maintain physiological pH; the present simulations, however, suggest a more active role. LIsoleucine and L-glutamine also form H-bonds with GSH through their amine or carboxylic termini and can form multiple H-bonds as well; their binding to GSH can also be mediated by ions. D-Glucose is an exception among all the species in that it is seldom found on the NP surface despite its

relatively high concentration. Hydrophobic associations are common in the bulk and are sometimes observed on the NP surfaces. The local details in the average potentials of Figure 4 appear to have a molecular origin related to changes in the structural and dynamic behavior of the solution species and water in the NP/NP interface. Cation-induced forces change sharply in magnitude at two interparticle separations (A and B in Figure 4). At position A, the mean forces become sufficiently strong to offset the direct repulsion in AuNP-25, but fail to do so in AuNP-14; at B, the opposite is true. Similar changes are observed in the anion-induced forces and essentially at the same positions. These correlations in the cation and the anion subsystems and, to a lesser extent, in water as well suggest a physical underlying mechanism rather than a statistical anomaly. Indeed, analysis of the independent trajectories indicates that the changes in magnitude and directions of the forces are directly related to the ion reconfiguration at the NP/ NP interface. Figure 6 shows the charges (open circles) at the interface and the corresponding forces (−dV/dr, closed circles; in arbitrary units) as a function of the interparticle separation averaged over all the simulations. The changes in magnitude and direction of forces correlate with the inflow and outflow of interfacial ions and explain the changes at A and B. The correlation is much stronger in the AuNP-25 pair, where even the positions of the maxima and minima of the forces and the charges coincide. For r > rcc these variations in charge are small, 4e at most in AuNP-25, but their effects are quite strong, especially as the interfacial space is reduced and the charge density augments accordingly. The variations in positive and negative charges are mutually correlated. As the AuNP-25 pair approaches the close-contact configuration, both positive and negative charges are pulled into the interface, which explains the reversal of forces in A. For AuNP-14 the effect is much smaller but similar in nature. For r < rcc the interfacial ions are pushed away by the direct action of the coating, although a few anions remain trapped at the AuNP-25 interface. This residual 4148

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Figure 5. Spatial distributions of cations, anions, and neutral species on a plane (in practice a 5−Å thick slab) across the nanoparticle centers at four representative interparticle separations (close-contact distances labeled in red). The plots are contour maps, so that the darker the color, the higher the value; values below a certain threshold are shown in white; for the neutral species, areas above certain threshold are shown in green. The nanoparticles are drawn to scale as gray discs with effective radii of 1.44 and 1.96 nm (cf. Methods); the Au cores are represented to scale as yellow discs. The insets show the glutathione molecule as obtained from a snapshot of the simulations (sulfur atom in yellow; attached gold atom not shown) and the inner and outer NP/liquid (Ψ’s) and NP/NP (Ω’s) interfaces.

charge in Ωi (noticeable in Figure 5) and the accumulation of anions in Ωo contribute to the persistent anion-induced repulsion below rcc seen in Figure 4. The changes in spatial configurations just discussed involve mostly ions of small molecular weight, which can diffuse more easily and rapidly than the larger species within the time frame of the present simulations. Therefore, these observations are less likely to be limited by the statistics of the calculations. The analysis of the independent trajectories also indicates that neutral molecules tend to induce electrostatic repulsion because they typically populate the outer shell Ψo instead of Ψi. The corresponding forces are stronger than the attractive forces induced by neutral molecules in the bulk, which act as a dielectric medium. These species do not undergo major restructuring at the NP/NP interface, and this is reflected in the lack of local details in the potentials. The form of the van der Waals potentials is also the result of changes in the behavior of the interfacial liquid, albeit of a different nature. Cation, anions, and neutral species all exert

attraction on both pairs, although the main contributions come from cations and water. For cations, the short-range component of the vdW forces dominates, whereas dispersion plays a more modest role. This is qualitatively similar in both pairs and is illustrated here for AuNP-14 (red lines in Figure 4). The contribution from short-range repulsion is relatively small at large interparticle distances because the hydrostatic pressure effected by cations is homogeneously distributed over the NP surfaces. As the distance between the NPs decreases, there is a rapid imbalance of forces where attraction sets in (C in Figure 4). The calculations show that the average number of atoms from all the cationic species in direct contact with the NPs is very similar in Ψo and Ψi for all interparticle separations. In addition, there is no discernible redistribution of cations within Ψo and Ψi that could explain the imbalance of repulsive forces. The onset of attraction is thus attributed to a change in the dynamic behavior of cations in Ψi relative to Ψo that directly affects the strength and/or frequency of collisions with the NPs. These dynamic changes appear to be correlated with the 4149

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Figure 6. Interfacial charges (open circles) in units of the elementary charge e as a function of the interparticle distance, calculated as the total charge in Ψi, Ωi, and Ωo (cf. inset in Figure 5; for these calculations h was set as the diameter of the NPs, so Ωo ≈ 0). The interparticle forces are also shown (solid circles; in arbitrary units); attractive forces are above the horizontal lines for cations and below for anions. The strong attractive forces generated by cations include the strong repulsive forces of the direct NP−NP interactions, hence the changes in directions of cations-induced forces.

structural changes discussed above in connection with electrostatics: the onset of vdW attraction in C coincides with the restructuring at B, and the weakening of the vdW attraction apparent in D is seemingly coincidental with A. It is unclear why this should be the case, but the behavior of water may be the underlying common cause. Water is highly structured in Ψ and becomes increasingly structured and less dynamic in Ω as the NPs approach one another. Because water has a strong preference for fully hydrating ions, changes in its structure and dynamics would naturally affect the behavior of ions as well. Figure 7 shows the spatial distribution of water on a plane containing the AuNP-14 centers. At large interparticle separations the NPs are surrounded by structured hydration shells ∼5.6 Å thick, similar to AuNP-25 (not shown). The changes in structure and dynamics of water start when these shells enter in mutual contact. These changes are subtle, and one way to quantify them is through the behavior of water clusters. Within the extended hydration shells, long-lived networks of hydrogen-bonded water develop often (cf. Figure 7). The general behavior described here is observed in all the simulations and in both NP pairs, thus statistically meaningful. At long interparticle distances the shells are effectively separated, and clusters containing up to ∼40 water molecules with survival times of up to ∼30 ps are observed; as a rule, the larger the cluster, the shorter their lifespans (for a comparison with bulk water, see Methods). As the separation between the NPs decreases and the hydration shells begin to merge, the number, size, and lifetimes of the clusters increase. At the same time, clusters that form at the NP/NP interface begin to connect the NPs (Figure 7). Around the close contact configuration clusters containing up to ∼60 water molecules with survival times as long as ∼40 ps are formed. At close enough separations, stable water networks can no longer form in Ωi but still can bridge the NPs across Ωo (Figure 7). For AuNP-25 the clusters tend to be larger, although the survival times do not differ much. These changes in water behavior explain the forms of the water potentials in Figure 2 and the indirect effects on the ion dynamics. As the NPs are brought

Figure 7. Spatial distributions of water (left panel) and watercluster distributions (right) for representative interparticle separations (only AuNP-14 shown; qualitatively similar to AuNP25; close-contact distance shown in red). The NPs are represented as discs on the left (cf. caption Figure 5) and as solvent-accessible surfaces on the right (arbitrary snapshot used to illustrate). The red dots represent the positions of the oxygen atoms of all the water clusters formed during the 1 ns dynamics at the corresponding separation (shown only for one of the simulations). Only clusters larger than decamers that survive more than 10 ps are shown. The cluster sizes and lifespans increase with decreasing separation. As the NPs approach one another, water clusters begin to form at the interface and start bridging the NPs; two such clusters formed at different times are shown for each interparticle distance (vdW representations in blue). For AuNP-25 the clusters tend to be larger, but their survival times are comparable to those in AuNP-14. The left and right panel yield different, complementary information on the structure and dynamics of water.

together, mechanical work is needed to remove the structured water from the increasingly constrained NP/NP interface. This explains the broad desolvation barrier observed in both pairs in Figure 2. The situation is qualitatively similar to that found in 4150

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the hydration of small polar/charged molecules:18,21 water− charge attraction is stronger than water−water attraction, so there is an energy cost for removing the hydrating water between the solutes as they are forced into close contact; the removed water then H-bonds to other water molecules, thereby lowering the system’s energy. For the NPs the situation is qualitatively similar albeit at a grander scale, as much of the water at the NP/NP interface is indeed water of hydration. The difference between the pairs appears to be mostly quantitative, with the larger NPs containing more structured water and thus requiring more work of desolvation. This also explains the destabilizing effect of water on the AuNP-25 pair at close contact. The electrostatic/vdW decomposition of the water potential in Figure 4 is more difficult to explain and remains to be studied. Figure 8 shows the entropic effects of depletion and crowding on the NP interactions. For both pairs, the effects

CONCLUSIONS Computer simulations were carried out to gain insight into the molecular origin of the forces driving association of ultrasmall NPs in a cell culture medium. With statistical certainty it was shown that, even within the ultrasmall size subclass, the smaller NPs remain colloidally stable, whereas the larger NPs can aggregate under the same conditions. This is in agreement with recent experimental results.13 An effort was made to draw conclusions that are expected to hold beyond the limitations of the force field and the statistics of the present simulations. The molecular mechanisms identified are thus based on general trends and correlations of the potentials, which are generally consistent across simulations and common to both NP pairs. Although the electrostatic forces induced by the solution are strong and help stabilize both pairs, they alone are insufficient to promote association. For this to occur, attraction induced by the van der Waals interactions with cations, anions, and netneutral, polar molecules is needed. Electrostatic forces are controlled by modest but consequential changes in the structure of the interfacial liquid, which is manifested in the inflow and outflow of charge at the interface as the NPs approach one another; by contrast, the vdW forces are controlled by changes in the liquid dynamics, which leads to an imbalance in the frequency/strength of collisions of the solution components with the NPs. These observations are qualitatively similar in both pairs, but favor association of the larger NPs. A difference in behavior with NP size was also observed in the entropic forces of depletion and macromolecular crowding. Depletion forces of the solution components alone are rather small, but the combined effects of depletion and macromolecular crowding can contribute to the stabilization of small aggregates. These effects also tend to favor association of the larger NPs. The microscopic picture that emerges in the ultrasmall regime is different from that in larger, most commonly studied NPs. For the latter, even average-sized proteins would act as depletants, and the entropic effects of the solution components could be neglected. Large NPs tend to develop a protein corona, a concept that loses its meaning in the ultrasmall regime. Also, dispersive interactions between large NP cores could play a more prominent role and could be modeled by a Hamaker-like potential. Changes in hydrostatic pressure would be negligible in comparison, and a mean field approximation for the combined effects of electrostatic and van der Waals interactions would suffice, e.g., the DLVO theory commonly used to model colloidal suspensions in electrolyte solutions. These approximations are known to break down as the particles decrease in size, and ad hoc corrections have been proposed. Still, for ultrasmall NPs in biological media these notions should be revisited systematically as the forces that control aggregation (or lack thereof) appear to change their relative importance. The direct association of the NPs with the solution species affects the interactions, as do the subnanometer lengthscale changes in the structure and dynamics of the interfacial liquid. These effects are difficult to generalize and capture in a continuum model, and atomistic descriptions, such as those employed in the study of protein−protein interactions, may be needed. The decompositions of the potentials in Figures 2 and 4 suggest that lowering the concentrations of ions should destabilize both NPs, albeit for reasons different from the commonly invoked electrostatic “screening” by ions, as

Figure 8. (Left) Potentials of mean force generated by the entropic forces of depletion and/or crowding in pure cell culture (CC), in an aqueous solution of human serum albumin (HSA), and in a solution of HSA in CC. The restraining potential used in the configurational biased Monte Carlo simulations is shown schematically in the inset. The maximum r for which V(r) = 0 is given by r = R − dM, where R is the radius of the container, and dM is the sum of the radius of the NP and the diameter of the largest of the CC species; this ensures a full CC layer between the NPs and the border of the container. (Right) Snapshot of the simulation of the AuNP-14 pair (shown as yellow spheres of radius 1.44 nm; cf. Methods); all the CC species are shown in blue; reverse coarsening (fine graining) to atomic resolution was carried for albumin to illustrate.

of depletion in pure CC are small ( t but reforms before t′ + δt, the cluster is assumed to have experienced only a structural fluctuation and remained intact during δt; if separate clusters merge at t′ but split again before t′ + δt, the clusters are assumed to have remained separate during δt. Both rc and δt are estimated from a 1 ns MD simulation of pure bulk water under the same conditions used in the simulations of the NPs. The cutoff rc was thus set at 3.3 Å, which covers the ∼2.8 Å average O−O distance of Hbonded water. The longest time a water molecule was observed to remain spatially localized was 10 ps, and this is the value chosen for ΔtL. By convention, ΔrL = rc and δt = ΔtL. With these values, dimers and trimers were common in pure bulk water; a few tetramers appeared rarely and a pentamer only once.

of the NIH HPC Biowulf cluster (http://hpc.nih.gov) and was supported by the Intramural Research Programs of the Center for Information Technology, National Institutes of Health, Bethesda, MD, USA.

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AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. ORCID

Sergio A. Hassan: 0000-0003-3319-078X Notes

The author declares no competing financial interest.

ACKNOWLEDGMENTS The author thanks Alioscka Sousa for the cell culture formulation and discussions and Alan Heckert for help with DATAPLOT. This work utilized the computational resources 4153

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