Molecular dynamics simulations on the effect of size and shape on the interactions between negative Au18(SR)14, Au102(SR)44 and Au144(SR)60 nanoparticles in physiological saline

Abstract

Molecular dynamics simulations employing all-atom force fields have become a reliable way to study binding interactions quantitatively for a wide range of systems. In this work, we employ two recently developed methods for the calculation of dissociation constants K_D between gold nanoparticles (AuNPs) of different sizes in a near-physiological environment through the potential of mean force (PMF) formalism: the method of geometrical restraints developed by Woo et al. and formalized by Gumbart et al. and the method of hybrid Steered Molecular Dynamics (hSMD). Obtaining identical results (within the margin of error) from both approaches on the negatively charged Au₁₈(SR)₁₄ NP, functionalized by the negatively charged 4-mercapto-benzoate (pMBA) ligand, we draw parallels between their energetic and entropic interactions. By applying the hSMD method on Au₁₀₂(SR)₄₄ and Au₁₄₄(SR)₆₀, both of them near-spherical in shape and functionalized by pMBA, we study the effects of size and shape on the binding interactions. Au₁₈ binds weakly with K_D = 13mM as a result of two opposing effects: its large surface curvature hindering the formation of salt bridges, and its large ligand density on preferential orientations favoring their formation. On the other hand, Au₁₀₂ binds more strongly with K_D = 30μM and Au₁₄₄ binds the strongest with K_D = 3.2nM.

Keywords: Aggregation; Gold Nanoparticles; Molecular Dynamics.

Figure 1

Systems under study: (a) the Au₁₈(SR)₁₄ (diameter ~ 1nm), (b) Au₁₀₂(SR)₄₄ (diameter ~ 1.5nm) and (c) Au₁₄₄(SR)₆₀ (diameter ~ 2nm) NPs represented as spheres (with van der Waals radius). The coloring scheme throughout this paper is the following: sulfur, yellow; oxygen, red; carbon, cyan; hydrogen, white; gold, golden; nitrogen, blue. All graphics were rendered with VMD.[41]

Figure 2

Building procedure: (a) The Au₁₈ NP structure undergoing energy minimization after attaching 14 ligands to its surface. (b) The Au₁₈ NP pair undergoing free rotation in the presence of a layer of Na⁺ located at the middle, in order to find the orientation of minimum energy.

Figure 3

Methodology for the PMF calculations of Au₁₈ (a) The hSMD method: The displacements of the 3 pulling gold atoms are shown as magenta cones. (b) The method of geometrical restraints: The six groups employed to define the six collective variables are represented as purple spheres. The groups involved in the definition of each collective variable are connected by purple lines.

Figure 4

hSMD results : (a) Histogram of the coordinates of the three Au₁₈ pulling atoms during the bound-state equilibration employed to compute Z₀. (b) The distances among the three Au₁₈ pulling atoms and the angle formed by them vs. time during the unbound-state equilibration employed to compute Z_∞. (c) The PMFs of Au₁₈, Au₁₀₂ and Au₁₄₄ as functions of distance between the cores’ centers of mass. The minima are connected by a dashed pink line. The snapshot shows the bound state of Au₁₈ and all ions within 20 Å of the NPs colored by name: sodium, blue; chloride, magenta. (d) The energetic contributions to the PMF from the van der Waals interactions.

Figure 5

The radial density distributions of the carboxyl groups and the sodium counterions around (a) Au₁₈(SR)₆₀, (b) Au₁₀₂(SR)₄₄ and (c) Au₁₄₄(SR)₆₀. The snapshots on the top show the density maps of sodium around the NP (the coloring scheme in order of increasing density is: blue, green, red). The snapshots at the bottom show the sodium lying within 3.5 Å of both NPs simultaneously at the bound state.

Figure 6

Geometrical restraint results for Au₁₈: The (a) conformational (i.e. RMSD), (b) angular (i.e. Euler angles for orientation and spherical angles for location) and (c) radial collective variables as functions of time for the 20 ns of simulation under equilibrium conditions. The PMF profiles as functions of the (d) conformational, (e) angular and (f) radial collective variables. The values employed as minima for each of the constraint potentials are indicated by arrows.

Figure 7

(a) The center-to-center separation between two free Au₁₈ NPs vs. time. Starting with an initial value of 3 nm, their aggregation is spontaneous. (b) The PMF of Au₁₈ as a function of the center-to-center separation, employing the opposite orientation as in Fig. 4. The net interaction becomes repulsive.