The measure of the relaxation of dendrimers was determined from the autocorrelation function of the squared radius-of-gyration [6], C R g 2 ( t ) which is evaluated from the expression: C R g 2 ( t ) = < R g 2 ( 0 ) R g 2 ( t ) > − < R g 2 > 2 < R g 4 > − < R g 2 > 2

The resulting graph for dendrimers is presented in Figure 6 and in all examined models C R g 2 ( t ) decays to 0 at timescales shorter than 1 ns. We can conclude that the simulation times are sufficient to sample enough independent configurations for averaging of the static properties. The details of the five systems studied are listed in Table 6.

The dendrimer size can be quantified by considering the radius-of-gyration R_g of the molecular structure. This is especially useful to characterize polymer solutions to have a rough measure for the compactness of a structure. It is computed as follows: R g = ( ∑ i ‖ r i ‖ 2 m i ∑ i m i ) 1 2where m_i is the mass of atom i and r_i the position of atom i with respect to the center-of-mass of the molecule. The R_g values estimated by MD simulations of the GnBPO dendrimers are listed in Table 7, while Figure 7(a) shows the behavior of R_g as a function of the generation number. The calculated values are in good agreement with those obtained from the Small Angle X-ray Scattering (SAXS) for G_n+1 generation of the unfunctionalized PAMAM dendrimer [7].

Figure 7(b) shows the behavior of the R_g as a function of the dendrimer number of atoms N in logarithmic scale. The linear relation between log R_g and log N is an indication for a space-filling structure and the related fractal dimensionality d_c takes account of the compactness of the dendrimer series. In this case, the size scale N ∼ R g 2.68,and the d_c is 2.68, indicating that these dendrimers have a more polymeric-like structure than the PAMAM dendrimers, where d_c is 3, showing an homogeneous and densely packed structure [8]. This compact structure requires a space filling geometry that implies back-folding of these dendrimers. This back-folding is evident from several lines of evidence, as will be discussed below.

A structure-persistent characteristic of these dendrimers can be obtained by considering the average values of the three principal moments of inertia I_z, I_y, I_x, where I_z, I_y, I_x represent the eigenvalues of the moment of inertia tensor in descending order. The ratio of these three principal moments is a measure of the eccentricity (minor-major axes ratio) of the ellipsoid shape of the dendrimer. Figure 8(a) shows the average aspect ratios I_z/I_x and I_z/I_y for the different generation dendrimers (Table 7). The values of the I_z/I_y are in the range 1.1–1.3 and I_z/I_x are in the range 1.2–2.4. This means that these dendrimers are compact ellipsoid-spheroid in shape.

The change in shape is better reflected by the relative shape anisotropy, which is defined as [9]: δ = 1 − 3 < I 2 > < I 1 2 >where I₁ and I₂ are the first and second invariants of the radius-of-gyration tensor: I₁ = I_x + I_y + I_z, I₂ = I_xI_y + I_yI_z + I_xI_z. This quantity assumes values between 1 (for a linear array of atoms) and 0 (for shapes of high 3D similarity). Figure 8(b) reveals that the relative shape anisotropy of the simulated dendrimers of the G₀BPO₄ to G3BPO32 generations is still far from adopting compact, spherical structures although the relative shape anisotropy does not decrease monotonically, and only the higher generations assume a more globular structure.

We checked the position of the ethylendiamine core (EDA) to see if it stays in the center of the dendrimers. We measured the distance between the centers of mass of the whole dendrimer and of the EDA core. The average distances from G₀BPO₄ to G₄BPO₆4 were 4.04, 2.56, 4.70, 2.77, 3.57 Å respectively. So it is obvious that the core deviates from the center of the dendrimers, especially in G0BPO4 and G2BPO16 generations.

Typical equilibrated conformations of the dendrimer generations are shown in Figure 9. To simplify the figure, the hydrogen atoms have been omitted, the carbon atoms are depicted cyan, nitrogen atoms blue, oxygen atoms red and sulfur atoms yellow. These snapshots are generated using VMD software [10].

The conformation of the dendrimer can be conveniently expressed through the average radial monomer density ρ(r), that can be defined by counting the number N(r) of atoms whose centers of mass are located within the spherical shell of radius r and thickness δr. Hence, the integration over r yields the total number of atoms as: N ( r ) = 4 π ∫ 0 ∞ r 2 ρ ( r ) d r, where r is the distance from the center-of-mass of the dendrimers. This number was divided by the volume of the shell to produce ρ(r).

The distribution of atoms within the dendrimers is shown using radial density profiles. The curves corresponding to the five generations of dendrimers, from G₀BPO₄ to G₄BPO₆4, are shown in Figure 10.

For all dendrimers the maximum density is close to the center-of-mass and decays toward the edge of the molecule. The density measured in the region of the center-of-mass is not for the core part. In our model the core deviates from the center-of-mass and the dendrimers here have not perfect symmetrical structures. The smaller dendrimers G₀BPO₄ and G1BPO8 show a quick decrease in atom density after about 4–6 Å of the center-of-mass, while for the upper generations G₂BPO1₆ to G₄BPO₆4 the inner parts of the dendrimer exhibit a high atom density and at longer distances a steady decrease in atom density takes place, and it is the upper generation G₄ that shows the higher density profile, suggesting that the dendrimer's atoms are densely packed. Finally, in the tail zone, the density decays monotonically toward the exterior of the molecules.

As can be readily verified in Figure 10, in all cases the water density reaches the bulk value at distances beyond the dendrimer boundary, after the dendrimer density distribution drops to zero. Voids inside the dendrimer are occupied by water molecules, as shown by the increase in the water concentration accompanying any decrease in dendrimer atom density.

Radial monomer density profiles are displayed for all the dendrimers and for each type of monomer, shown in Figure 11. These profiles allow us to determine whether or not the different residue types are present within the interior of the dendrimer. The EDA and PAA density distribution of these monomers show, for all dendrimers, a displacement from the center-of-mass of the molecule. The value of the radial density is very noisy near the center-of-mass because the error is big when the number is divided by the very small volume within the shell near the center of the mass.

To elucidate the NOE effect found between the hydrogen atoms of the BPO-Ph subunit and the hydrogens of the methylenes (l, m) corresponding to the PAH monomer in the G₂BPO₁₆, and with the methylene (t) in the G₄BPO₆₄, we evaluated the density distribution functions between these atoms (Figure 12(a,c)). Both distributions show sharp peaks in the range between 2.5 and 5 Å. Therefore, these atoms are close enough to have a NOE effect between them.

In a similar way, Figure 12(b,d) show the density distribution functions between alpha and beta methyl protons of different BPO subunits in G₂BPO₁₆ and G₄BPO₆₄ respectively. These distributions show a higher density for G₄BPO₆₄ in the range between 2.0 and 3.0 Å.

As the dendrimer generation increases, the probability of finding a monomer, PAH or BPO, throughout the molecule also increases. The number of terminal monomers doubles with each generation, so the BPO groups are spread over an increasingly large volume. These results suggest a back-folding of the terminal monomers. The density of these groups moves generally to larger distances as the generation increases, indicating considerable flexibility in the dendritic structure.

The benzylpenicilloyl residues could be considered as negative charged thiazolidine rings (BPO-CO₂⁻) plus the rest of the benzylpenicilloyl fragments (BPO-Ph). Figure 13 shows the radial density profiles of these subunits for all the dendrimers. In all of the dendrimers, the apolar subunit (BPO-Ph) is always oriented toward the core of the dendrimers whereas the polar BPO-CO₂⁻ is oriented toward the exterior. In higher generations this subunit shows a plateau at the periphery, suggesting a more uniform distribution on the outside of the molecule.

It is observed that sodium counterion distribution accompanies the BPO moieties. In addition, the BPO distribution increases with the generation of the dendrimers, and always on the same side of the BPO-CO₂⁻ subunit. Notice that back-folding in dendrimers is only entropically favorable, but electrostatically unfavorable when there are no counterions present.

Poly(amidoamine) dendrimers (ethylenediamine core) with amino surface groups (PAMAM generations 0–4) were purchased from Dendritic Nanotechnologies, Inc. (dnt). Benzylpenicillin sodium salt was supplied by CEPA, and S.L. Standard chemicals were obtained from Aldrich or Merck.

The conjugates Bu-BPO and G_nBPO were prepared using procedures previously described and adapted for each generation (Table 2) [2].

¹H NMR, ¹³C NMR and ¹H,¹H-NOESY experiments were performed on a Bruker Avance III 400 MHz NMR instrument and chemical shifts (δ) are given in ppm relative to the residual solvent peak. All NMR spectra were recorded using D₂O as solvent (with Na₂CO₃ to ensure a basic pD = 11) at room temperature. ¹H NMR spectra were acquired with a spectral window of 14 ppm, an acquisition time of 3 s and a relaxation delay of 1 s. ¹³C NMR spectra were acquired with a spectral window of 240 ppm, an acquisition time of 0.7 s and a relaxation delay of 2 s. NOESY 2D-NMR spectra of conjugates were collected with 3,605 Hz spectral windows in f1 and f2, a 0.5 s mixing time and 9.28 μs ¹H 90° pulse width. The experiments were done with a 1 s relaxation delay and a 0.284 s acquisition time. Eight transients were averaged for each 256 × 2,048 complex t1 increments.

Dendrimers were built using the program Starmaker which is part of Silico [11]. This is a flexible program which can be used to construct a wide variety of dendrimers, including heterogeneously functionalized (variegated) dendrimers. Starmaker builds dendrimers from a core by repeated addition of generations of monomers. As each monomer unit is added, a stochastic search is used to attain an extended conformation for the monomer, while leaving the remainder of the dendrimer intact. This extended conformation is reached through application of a penalty varying inversely with distance, coupled with a significantly larger penalty for each close contact.

Each dendron structure was built using four different residue types: a central ethylendiamine (EDA), a repetitive polyamidoamine (PAA), a terminal polyamidoamine (PAH), and a terminal benzylpenicilloyl group (BPO). The missing bonds, angle torsions, or van der Waals parameters not included in the parm99 force field were transferred from the general AMBER force field (GAFF) [12]. These residue types were optimized and subjected to a conformational analysis using TINKER-Software Tools for Molecular Design [13]. The conformation of minimum energy was submitted to MP2/6-31G* calculation to obtain the electrostatic potentials using G09 [14]. Next, the restrained potential (RESP) method was used for charge fitting [15]. Finally, the residue was created using the Antechamber module of AmberTools (version 1.5) [16].

Molecular dynamics simulations were performed with the program NAMD2 (version 2.6b1) [17,18], using the AMBER force field which has previously been successfully applied in past computational studies for the description of PAMAM dendrimers [19-21]. All simulations were run using a timestep of 2 fs, a cutoff of 10 Å with a switching function 8 Å, and the Langevin thermostat. Full periodic boundary conditions were used and to treat the long-range electrostatic interactions the particle-mesh Ewald (PME) algorithm was employed [22]. The PME algorithm used a tolerance of 1 × 10⁻⁶ Å and an interpolation order of 4. Unit cell sizes were chosen to ensure that the number of points along any dimension in the PME grid was composed of factors of 2, 3 and 5. Bond lengths involving bonds to hydrogen atoms were constrained using the SHAKE algorithm [23]. Water molecules were modeled using the TIP3P model [24].

At the first step, the dendrimers were subjected to a conjugate gradient energy minimization (100,000 steps) in vacuum using a relative dielectric constant of 1. In the second stage, to preserve the overall charge neutrality and to represent more realistic conditions, an appropriate number of Na⁺ counterions were added, and the molecules were then hydrated in cubic cells the dimension of which was chosen to provide at least a 10 Å solvation shell around the dendrimer structure. These systems were minimized (100,000 steps) and followed by four cycles of constant volume, constant temperature ensemble (NVT) simulated annealing, each approx. 2 ns long (100 ps of heating to 600 K, followed by 1.9 ns of cooling to 300 K). The third stage of the simulation protocol involved an equilibration period of at least 2 ns using a constant temperature and constant pressure ensemble (NPT, p = 1 atm, T = 300 K) [25]. Finally, starting from the configurations produced by the above procedure, production runs of 14–15 ns trajectories were performed under an NPT ensemble. All of the molecular dynamics simulations were carried out working in parallel on 32 processors of the IBM calculation cluster of the Supercomputing and Bioinformatic Center of the University of Malaga.