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Molecular dynamics simulations of PAMAM and PPI dendrimers using the GROMOS-compatible 2016H66 forcefield. Mayk Caldas Ramos, Vitor A. C. Horta, and Bruno A.C. Horta J. Chem. Inf. Model., Just Accepted Manuscript • DOI: 10.1021/acs.jcim.8b00911 • Publication Date (Web): 15 Mar 2019 Downloaded from http://pubs.acs.org on March 17, 2019

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Journal of Chemical Information and Modeling

Molecular Dynamics Simulations of PAMAM and PPI Dendrimers Using the GROMOS-compatible 2016H66 Forcefield. Mayk C. Ramos,† Vitor A. C. Horta,‡ and Bruno A. C. Horta∗,† †Instituto de Qu´ımica, Universidade Federal do Rio de Janeiro, Rio de Janeiro, 21941-909, Brazil ‡Departamento de Ciência da Computa¸cão, Universidade Federal de Juiz de Fora, Juiz de Fora, 36036-900, Brazil E-mail: [email protected] Abstract A systematic evaluation of the accuracy of the GROMOS-compatible 2016H66 forcefield in the simulation of dendrimers is performed. More specifically, the poly(amido amine) (PAMAM) and the poly(propylene imine) (PPI) are considered due to the availability of experimental data and simulation results in the literature. A total of 36 molecular systems are simulated and the radius of gyration, asphericity, density profiles and the self-diffusion coefficient are monitored in terms of the generation number and pH (low, medium and high) condition. Overall, the results support the recommendation of the 2016H66 forcefield for the simulation of dendrimer systems. The natural building-block based strategy adopted in the definition of 2016H66, together with a careful parametrization of the chemical functional groups to reproduce thermodynamic properties in environments of different polarity, and also the ability to accurately reproduce the expected structural and dynamic features of dendrimers, as

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shown in the present work, make this forcefield an attractive option for the simulation of such systems and alike.

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Introduction

Dendrimers are highly-branched synthetic polymers 1 usually organized according to three structural building blocks: (i) the core block as the central piece and the origin of the branching points; (ii) the intermediate block as the repeating branched unit (also referred to as monomer); and (iii) the terminal block as the end group. The dendrimer structure forms layers of branched units that grow from the core block to the outermost shell of terminal groups. Each layer corresponds to the so-called “generation of growth” and a dendrimer with N generations is usually referred to as a GN dendrimer (e.g. PAMAM G4). Dendrimer synthesis is usually highly controlled in terms of the generation. 1,2 Since dendrimers are highly customizable synthetic molecules 3 - usually created to a specific-purpose application, and subjected to the creativity of synthetic chemists - the above definition can be and has been extended. There are reports on non-symmetrical (e.g. Janus 4 ) dendrimers, hybrid dendrimers, 5 surface-modified dendrimers, 6 and many others. Due to the high degree of customization, dendrimers have a broad spectrum of applications. One notable example concerns drug-delivery systems (DDS), 7–12 which exploit the cavities within the dendrimers for drug-loading purpose, and exploit the tunable surface properties for targeting and solubility control. Dendrimers have also found applications in many other technological fields, such as catalysis, 13,14 biosensors, 15,16 separation process, 17,18 light-harvesting 19,20 and material science. 21,22 The study of structural properties of dendrimers is crucial for understanding their physical behavior and their interactions with the environment and with other compounds. These studies are also of fundamental importance in the design of novel structures and in the discovery of new applications. Dendrimers can be studied by means of several experimen-

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tal techniques, 23,24 such as NMR, 25 fluorescence, 26 UV-Vis, 27 SAXS, 28–30 SANS 31–35 and AFM. 36 However, these techniques are usually limited to characterize the chemical structure and to probe particle size. The experimental access to structural and dynamical properties at the molecular level is a challenging task and suffers from several limitations. Simulation techniques, e.g. molecular dynamics (MD) 37–39 and Monte Carlo (MC), 40,41 provide an alternative route in the study of dendrimer properties and have been considered as the main tool for accessing information at the molecular level. 42,43 Among all possible dendrimer structures, two classes deserve special attention: poly(amidoamine) (PAMAM) (Figure 1) and poly(propylene imine) (PPI) (Figure 2). They are commercially available, well studied and have a broad spectrum of applications. 44 For instance, they have been considered as promising DDS due to their drug-loading capacity and the ability to incorporate into biomembranes and cells thus being capable of delivering small active compounds in the intracellular region. 45–49

Figure 1: Representation of the chemical structure of a amine-terminated EDA-cored PAMAM G3 dendrimer. Each monomer layer is displayed with different colors. Terminal amine groups are displayed in pink and the dendrimer core is displayed in red.

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Figure 2: Representation of the chemical structure of a amine-terminated DAB-cored PPI G4 dendrimer. Each monomer layer is displayed with different colors. Terminal amine groups are displayed in pink and the dendrimer core is displayed in red.

Over the last decades, several atomistic MD simulations of PAMAM and PPI dendrimers have been carried out and provided substantial structural information at the molecular level, mainly in terms of size, shape and density distributions. In addition, in some studies, the dependence of these properties with the generation number and the pH condition have been examined. A variety of forcefields such as Dreiding, 50 CVFF, 51 AMBER, 52 GAFF, 53 CHARMM 54 and, in same cases, a combination of different forcefields, have been employed in such simulation studies. So far, only a few simulation studies have employed the GROMOS forcefield for the simulation of such systems. The philosophy underlying the GROMOS forcefield involves a careful parametrization of the non-bonded interactions aiming at reproducing thermodynamic properties of small molecules representative of the most common chemical functions, and use them as building blocks for modeling more complex systems. The first dendrimer simulation using a GROMOS forcefield, by Kim et al., 55 aimed at calculating the adsorption free energy of a PAMAM G3 dendrimer on the surface of biological membranes. Although this is a relatively recent study (2014), they employed an old version of the GROMOS forcefield, the 37C4 (also known as GROMOS87), released on 1987. 56 Still, according to a recent study, this forcefield was able to predict a negative free energy of interaction between the dendrimer and the 4


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membrane (as expected, according to experimental data 57 ), whereas CHARMM and GAFF resulted in repulsive interactions (∼20 kcal/mol). 58 Moreover, also recently, constant-pH MD simulations using the GROMOS 54A7 forcefield 59 were able to predict in good agreement with experiment the titration curve of a PAMAM G2 dendrimer. 60 These studies evidence the predictive power of a forcefield that is calibrated using thermodynamic properties as target. In 2011, the parameter set 61 labeled GROMOS 53A6OXY was introduced with the aim at reproducing, simultaneously, pure-liquid properties and solvation free energies of small organic molecules, as a hope to unify the 53A5 (liquid forcefield) 62 and 53A6 (biomolecular forcefield). 62 This set has been extended and validated in several contexts, 63–76 culminating in the recently introduced GROMOS-compatible 2016H66 forcefield, 77 which includes optimized parameters for a vast number of organic chemical functions, including the ones commonly present in dendrimers, such as amides, amines, alcohols, ethers and esters. In principle, this parametrization strategy, focused on thermodynamics and not on structure, may confer predictive power, specially considering processes such as partition in environments of different polarity, complexation/aggregation phenomena, folding and binding. These are all fundamental processes in dendrimer science and the use of such a thermodynamically consistent forcefield might be highly beneficial. Other advantages of using the 2016H66 for dendrimer simulations are: (i) the compatibility with the biomolecular (aminoacids, lipids, carbohydrates, nucleobases) building blocks, 77 small organic molecules, 77 polymers (e.g. such as PEO and PEG) 78 and other building blocks included in the forcefield distribution, avoiding the bad practice of constructing chimera forcefields combining parameter sets that are a priori not compatible; (ii) the united-atom representation of CHn groups, 79 reducing the computational cost of aliphatic chains without any clear evidence of accuracy loss (e.g. for PAMAM, this representation has about half the number of atoms in an all-atom forcefield); (iii) the a priori compatibility with other recent versions of GROMOS, such as 54A7, 59 and also with the ATB; 80 (iv) the compatibility with the newly developed dendrimer

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topology builder pyPolyBuilder. 81 Due to the above mentioned advantages of this forcefield, the main idea is that it will enable more accurate predictions of free energies of dendrimer-drug complex formation, a better description of dendrimer-biomolecule interactions, and other relevant processes. Therefore, the goal of the present work is to evaluate the performance of the GROMOS-compatible 2016H66 forcefield 77 in the context of dendrimer simulations, assessing the extent to which this forcefield is able to reproduce the expected structural and dynamic features of such systems. More specifically, PAMAM and PPI dendrimers were chosen as benchmarks due to the availability of a vast number of experimental data and simulation results. A total of 36 molecular systems were simulated and the radius of gyration, asphericity, density profiles and the self-diffusion coefficient were monitored in terms of the generation number and pH condition.

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Metodology

2.1

Simulated Systems

Ethylenediamine(EDA)-cored PAMAM and 1,4-diaminobutane(DAB)-cored PPI dendrimers from generation 0 to 5 and 1 to 6, respectively, were simulated at low (pH< 4), neutral (pH' 7) and high (pH> 10) pH conditions. Table 1 reports the 36 simulated systems along with the number of dendrimer atoms, counterions and solvent molecules. In order to mimic the pH conditions above, the protonation states for the PAMAM dendrimers were assigned according to the experimental work by Cakara et al. 82 At high pH, all amine groups were considered deprotonated. At neutral pH, the primary amines were considered protonated. At low pH, both primary and tertiary amines were considered protonated. Figure S.1 shows the initial strucures used in the simulations, highlighting the protonation scheme. The protonation state of the PPI dendrimer was assigned considering the Ising model. 83,84 6


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Table 1: Molecular systems considered in the present work. PAMAM and PPI systems are characterized in terms of the generation number GN , pH environment, number of dendrimer atoms, number of Cl− counterions and number of water molecules.

GN G0 G1 G2 G3 G4 G5 G0 G1 G2 G3 G4 G5 G0 G1 G2 G3 G4 G5

PAMAM #Atoms 54 142 348 Low 670 1374 2782 52 136 304 Neutral 640 1312 2656 48 128 288 High 608 1248 2528 pH

#Cl− 6 14 30 62 126 254 4 8 16 32 64 128 0 0 0 0 0 0

#Water 1674 4292 11803 10394 19248 36214 1666 4033 8550 13241 17982 34096 1348 3653 6670 15943 14713 23565

GN G1 G2 G3 G4 G5 G6 G1 G2 G3 G4 G5 G6 G1 G2 G3 G4 G5 G6

PPI #Atoms #Cl− 36 6 84 14 180 30 Low 372 62 756 126 1524 254 34 4 78 10 172 22 Neutral 356 46 724 94 1460 190 30 0 70 0 150 0 High 310 0 630 0 1270 0 pH

#Water 1348 2921 5257 8486 13075 23546 1140 2416 3593 5014 9723 14865 1143 1482 2258 3213 4136 6351

In this scheme, the dendrimer at neutral pH is considered to be 2/3 protonated. The primary amines are all protonated and the tertiary amines correspond to alternating shells of protonated and deprotonated sites (see Figure 2 in Ref. 83,84 ). The initial structures and topologies were created using our in-house software pyPolyBuilder. 81 A manuscript describing the functionalities of this program is in preparation. Briefly, pyPolyBuilder has two modules: general assembler and dendrimer assembler. With the general module, it is possible to combine any number of predefined building blocks as long as a connectivity file is provided. The individual building blocks are written according to the definition of a molecular topology in either the Gromacs (itp) or GROMOS (mtb) format. With the dendrimer module, one needs to specify which building block is the core, intermediate and terminal one, and also provide the respective anchoring and branching points. Once the required input files are ready, with a single command line, the program generates

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the molecular topology and a starting set of coordinates. The atom lists for the molecular building blocks used in this work are reported in the Supplementary Tables S.5-S.16. The associated bonded parameters are listed in Table S.17.

2.2

Simulation Protocol

Molecular dynamics simulations were performed for all 36 systems using the GROMACS 5.1.4 85 simulation package and the 2016H66 forcefield. 77 Newton’s equations of motion were integrated using the leap-frog scheme with a time step of 2 fs. Periodic boundary conditions based on cubic boxes were adopted in all cases. The center of mass translational motion was removed every 100 steps. All bond lengths were constrained to their reference values and the SPC water molecules kept fully rigid by application of the LINCS algorithm. 86 Electrostatic interactions were calculated with the PME method 87 using a real-space cutoff of 1.2 nm and a grid spacing of 0.12 nm. The Lennard-Jones interactions were truncated at 1.2 nm using a straight cutoff. Atomic-based pairlist generation was used as implemented in the Verlet algorithm. 88 Although this procedure differs from the original scheme used in the forcefield parametrization, a systematic study has been recently carried out 89 showing that the present scheme is valid and preferable when using recent versions of the Gromacs package. In order to evaluate the effect of using a lattice-sum treatment for the Lennard-Jones interactions, the simulations were also carried out with LJ-PME, using a 1.0 nm real-space cutoff for both electrostatics and Lennard-Jones interactions and a grid spacing of 0.12 nm. Temperature and pressure were controlled by weak-coupling 90 of the entire system to external heat and volume baths. In the case of thermal coupling, an extra stochastic term was included in order to enhance kinetic-energy equipartition (velocity-rescale thermostat 91 ). For thermal coupling, the time constant was set to 0.1 ps and for pressure coupling, the time constant and isothermal compressibility were set to 2.0 ps and 4.5 × 10−5 bar−1 , respectively. Each system was firstly energy-minimized in vacuum for 5000 steps and then placed in a cubic simulation box in such way that all dendrimer atoms were, at least, 1 nm distant 8


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from the box edges. The box was then filled with a sufficiently large number of SPC water molecules. 92 In systems with a non-zero charge (see Table 1), the simulation box was neutralized by adding the appropriate number of Cl− counterions. Systems were subsequently energy minimized under periodic boundary conditions. The equilibration of each system was firstly performed under the NVT ensemble in a stepwise manner. Initial velocities were generated according to a Maxuell-Boltzmann distribution corresponding to a temperature of 50 K. Six successive 200 ps simulations were employed in order to gradually raise the temperature from 50 K to 300 K in intervals of 50 K. Pressure coupling was then switched on and the system was equilibrated under the NPT ensemble for 200 ps with a reference pressure of 1 bar. In order to completely remove any bias from the initial structure, an additional NPT simulation was carried out at 400 K for 200 ps and cooled down to 300 K for more 200 ps. For each system, the production run was carried out at 298.15 K and 1 bar for 50 ns. The coordinates were written to file every 1 ps for analysis.

2.3

Trajectory analysis

The simulation trajectories were analyzed in terms of the radius of gyration Rg , asphericity δ, radial distribution function gαβ (r) and self-diffusion coefficient D. 2.3.1

Radius of gyration

Dendrimer size can be quantitatively estimated by the mean-square radius of gyration (Rg ), computed as: 93,94

Rg =

1 hRg2 i = M

*

q

N X

hRg2 i ,

(1)

+ [mi (|ri − R|2 )]

i=1

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,

(2)


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where N is the number of dendrimer atoms, M is the total dendrimer mass, mi and ri are, respectively, the mass and the coordinate vector of the ith atom, R is the coordinate vector of the center of mass and the angle brackets stand for time averaging. The last 10 ns of the simulation trajectories were considered for averaging. Average Rg values, were compared to experimental values derived by SAXS and SANS, as well as values from other simulation studies and served as the main validation property.

2.3.2

Shape Tensor

In way to study the shape of the dendrimer at different generations and protonation states, the moments of inertia were calculated in order to compute the asphericity parameter δ that was compared with other simulations studies. The shape tensor that describes the mass distribution shape, is defined by: 93,94

Gmn =

N 1 X [mi (rmi − Rm )(rni − Rn )] , M i=1

(3)

where the subscripts m, n = x, y, z. The calculation of δ was performed as recommended by Rudnick and Gaspari: 95

δ =1−3

hI2 i hI1 2 i

,

(4)

where I1 = Ix + Iy + Iz and I2 = Ix Iy + Ix Iz + Iy Iz . In this formulation, the closer to zero the value of δ is, the more spherical is the molecule.

2.3.3

Radial Distribution Function

The radial distribution function gαβ (r) was calculated from a distance histogram constructed by counting the number of atoms located in spherical shells of radius r and thickness ∆r:

gαβ (r) = (4πr2 ρ∆r)−1 hNαβ (r; ∆r)i ,

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(5)

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where ρ is the average number density, ∆r is the bin width of the distance histogram, N is the number of β sites around α sites at a distance between r − ∆r/2 and r + ∆r/2, and the angle brackets denotes trajectory averaging.

2.3.4

Self-Diffusion Coefficient

The self-difussion coefficient D was evaluated based on the long-time limit of the mean-square displacement (MSD) of the dendrimer center of mass using the Einstein relation. 40,96 h|rcm (τ + t) − rcm (τ )|2 iτ t→∞ 6t

D = lim

(6)

where rcm is the coordinate vector of the dendrimer center of mass and h...iτ stands for averaging over time origins τ . The last 40 ns of each trajectory were used for calculating the MSD. A least-square fitting over the 3-5 ns time interval was performed to estimate D.

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Results and discussion

The 36 molecular systems considered in this study (Table 1) were simulated using two different approaches for the LJ interactions: straight truncation with cutoff value of 1.2 nm and and PME, as explained in Section 2.2. The systems were analyzed in terms of size, shape, density distributions and self-diffusion coefficient, as detailed in Section 2.3. The calculated values for these properties were essentially the same with both simulation setups. This is in agreement with the results reported in Ref. 89 for simulations in solvents of high polarity and, especially, in water. For sake of conciseness, only the results obtained with straight cutoff are shown in the main document and the LJ-PME results are reported graphically and numerically in the Supplementary Material.

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3.1 3.1.1

PAMAM PAMAM Size

The radius of gyration Rg was calculated (see Section 2.3.1) for each simulated PAMAM system (Table 1) and plotted as a function of the dendrimer generation in Figure 3. The available experimental and simulation values considering the same systems under similar thermodynamic conditions were extracted from the original references and are also displayed for a comparative analysis. The corresponding results using PME for the Lennard-Jones interactions are displayed in Figure S.5. The numerical values are reported in Table S.2.

Figure 3: Radius of gyration Rg plotted as a function of generation and considering low (left panel), medium (middle panel) and high (right panel) pH environments. The results obtained with the 2016H66 forcefield are compared to those extracted from previous experimental and simulation studies: Prosa1997, 28 Lee2002, 97 Rathgeber2002, 29 Maiti2004, 93 Maiti2005, 98 Opitz2006, 99 Porcar2008, 35 Maiti2009, 94 Maingi2012, 100 Caballero2013, 49 Barraza2018, 101 Kanchi2018. 58

Considering the results for low pH, SANS experimental data are available for generations 3, 4 and 5 showing an almost linear increase of Rg from ∼1.7 nm (G3) to ∼2.7 nm (G5). A previous simulation study by Lee et al. 97 using the CVFF forcefield reports values that systematically and significantly overestimate the experimental ones. The radial growth is slightly non-linear with the largest deviation from the experimental value occurring for G5, which shows Rg = 3.8 nm. The study by Opitz and Wagner 99 reports Rg values obtained using the Dreiding forcefield with an implicit solvent model. The results are in very good agreement with the SANS data in spite of the continuous solvation. The Dreiding results by Maiti et al., 98 obtained with explicit solvent, systematically underestimate the SANS

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data by ∼0.2 nm. The results obtained in the present work using the 2016H66 forcefield show a linear behavior with the same angular coefficient compared to the SANS data, but systematically shifted to larger values by ∼0.2-0.3 nm. For instance, the simulated values for G3, G4 and G5 are 1.9, 2.5 and 3.0 nm, respectively, compared to SANS values of 1.7, 2.2 and 2.7 nm. Considering neutral pH, there are three experimental data sets available. 28,29,35 The SAXS data set by Prosa et al. 28 significantly deviates from linearity, showing almost no difference from G3 to G4 (1.6 → 1.7 nm) and a remarkable difference from G4 to G5 (1.7 → 2.4 nm). The SAXS data set by Rathgeber et al. 29 is the most complete, including all generations considered in the present work, and shows a close to linear behavior. The SANS data set by Porcar et al. 35 includes G3 to G5, shows a linear behavior and systematically overestimates the two SAXS data sets by ∼0.1-0.3 nm. The simulation results by Lee et al. 97 with the CVFF forcefield again significantly overestimate the Rg values. As opposed to the results for low pH, the two Dreiding data sets now exhibit an inverted trend. The implicit solvent model 99 systematically underestimate the Rg values compared to the explicit solvent one. 98 The results obtained with the 2016H66 forcefield are in very good agreement with the two SAXS data sets and only slightly underestimate the SANS data. In addition, the 2016H66 results agree well with the GAFF results by Barraza et al. 101 for G0 to G3 and also with the CHARMM results by Caballero et al. 49 for G3 to G5 dendrimers. Considering high pH, all simulation results significantly underestimate the SANS data set, 35 which is the only available experimental data set in this case. Interestingly, all reported simulation results lie within the same region, regardless of the employed forcefield. The 2016H66 results show a close to linear behavior and stand well within the range of results obtained with the other forcefields. Figure 4 shows Rg as a function of pH for generations 3, 4 and 5. The general trend observed with molecular dynamics simulations is that, regardless of the considered forcefield, Rg depends upon pH. More precisely, simulations show an increase in Rg upon acidification.

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This can be explained by an increase in solvation of protonated amines, which tend to result in more expanded dendrimer conformations. However, the magnitude of this swelling behavior can vary depending on the considered forcefield. CVFF shows a more pronounced swelling behavior compared to the Dreiding and 2016H66 forcefields. The results obtained with the 2016H66 forcefield show a close to linear behavior for all three generations considered, whereas CVFF and Dreiding exhibit a noticeable deviation from linearity. It is also important to highlight the close agreement between 2016H66 and GAFF. The latter is nowadays often employed in dendrimer simulations especially after the development of the Dendrimer Building Toolkit. 100 A modified version of the Dreiding forcefield shows an almost perfect agreement with the SANS data, but it was parametrized with the aim at reproducing this behavior. 102 Unfortunately, SAXS experimental results are only available for neutral pH. Still, the deviations between SAXS and SANS results, and also between the two SAXS data sets for neutral pH indicate that the underlying model assumptions for extracting Rg from scattering experiments play a significant role. The complete lack of swelling indicated by the SANS measurement is rather suspicious. A more recent measurement for PAMAM G6, 103 using SAXS with a more sophisticated modeling approach corroborates the swelling upon acidification, with a Rg value of 3.55 nm for pH=5.5 compared to 3.04 for pH=8.5. Taking this into account, it seems that the simulation results correctly describe the swelling phenomenon.

Figure 4: Radius of gyration Rg plotted as a function of pH considering PAMAM dendrimers of generation 3 (left panel), 4 (middle panel) and 5 (right panel). The results obtained with the 2016H66 forcefield are compared to those extracted from previous experimental and simulation studies: Prosa1997, 28 Lee2002, 97 Rathgeber2002, 29 Maiti2005, 98 Opitz2006, 99 Porcar2008, 35 Liu2009, 102 Maingi2012, 100 Barraza2018. 101

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In summary, based on the results described above, the 2016H66 forcefield is at least as appropriate as the GAFF and CHARMM forcefields for the description of PAMAM size. In this context, these three forcefields can be considered more accurate compared to Dreiding and CVFF.

3.1.2

PAMAM Shape

Figure 5 shows the results for the asphericity δ as a function of the dendrimer generation and considering low, medium and high pH conditions. The corresponding results using PME for the Lennard-Jones interactions are displayed in Figure S.7.

Figure 5: Asphericity δ of PAMAM dendrimer plotted as a function of generation. Left, middle and right panels show, respectively, the results for low, medium and high pH conditions. The results obtained with the 2016H66 forcefield are compared to those extracted from previous simulation studies: Maiti2004, 93 Tanis2009, 47 Maingi2012, 100 Barraza2018, 101 Kanchi2018. 58

At low pH, the 2016H66 results indicate a monotonic decrease of δ, tending to zero (perfect sphere) upon increasing the generation, slightly overestimating those obtained with the AMBER forcefield. At neutral pH, the 2016H66 results decrease with generation, but in a less pronounced fashion compared to low pH. Also, a slightly larger value of δ is obtained for G4 compared to G3. Similar values are obtained with the GAFF forcefield. The AMBER forcefield results in a much more spherical dendrimer. At high pH, the δ values obtained with 2016H66 decrease slightly from G0 to G2, but a pronounced decrease is observed from G2 to G3. Again, G4 is less spherical than G3.

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This is also evidenced by the results obtained with GAFF and Dreiding, which, at least qualitatively, agree with 2016H66. The AMBER forcefield results indicate again a more spherical dendrimer. The increase in δ observed from G3 to G4 at neutral and high pH conditions is probably a consequence of an asymmetry induced by a high degree of terminal backfolding (see below).

3.1.3

PAMAM Atom Densities

The atom densities for the PAMAM dendrimers considered (G0 to G5) were calculated in terms of radial distribution function g(r) as explained in Section 2.3.3, and the results are shown in Figure 6 and supplementary Figures S.2-S.3. The corresponding plots using PME for the Lennard-Jones interactions are displayed in Figures S.8-S.10. For sake of improving the visualization, some of these distributions were normalized by density and volume while others were not, as explained in the figure captions. The three left panels on Figure 6 show the non-normalized density distributions of the monomers with respect to the dendrimer center of mass for low (top), neutral (middle) and high (bottom) pH. The most immediate observation is that, regardless of the pH condition, the maximum of the distribution tends to larger distance values and the distribution becomes broader as the generation increases. The effect of pH is such that dendrimers become more compact from low to high pH (top → bottom). The level of structuration also increases from low to high pH, as evidenced by the increasingly pronounced “shoulders” on the distributions at neutral and high pH (especially for G4 and G5). The middle panels show the normalized density distributions of the terminal groups with respect to the dendrimer center of mass for low (top), neutral (middle) and high (bottom) pH. In general, and in special for G4 and G5, terminal groups exhibit a broad distribution indicating a certain degree of backfolding. It is easy to note that the degree of backfolding increases from low to high pH, as shown by the intensity of the density peaks at short distances. This is in agreement with theoretical models that predicts that the PAMAM core

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Figure 6: Radial distribution function g(r) plots for the PAMAM dendrimer considering generations G0 to G5. The top, middle and bottom rows indicate the pH condition, respectively, low, neutral and high pH. The 3 left panels consider the non-normalized g(r) of all dendrimer monomers with respect to the dendrimer center of mass. The 3 middle panels consider the normalized g(r) of the dendrimer terminal groups with respect to the dendrimer center of mass. The 3 right panels consider the normalized g(r) of the water molecules with respect to the dendrimer center of mass.

becomes denser upon decreasing the acidity of the medium. 42,104–106 At neutral and high pH conditions, G4 and G5 exhibit a qualitatively different behavior compared to dendrimers up to G3. This is consistent with a more compact dendrimer, highly backfolded and, probably for this reason, less spherical (as indicated by Figure 5). The right panels show the corresponding analysis in terms of water radial distribution function with respect to the center of mass, providing information on the level of water penetration and hydration. Overall, water penetration is more pronounced at low pH as a 17



consequence of the protonation of the more internal tertiary amine groups. This swelling of the dendrimer at low pH explains the larger Rg and the less compact structure. At neutral and high pH, water permeation is less pronounced due to intradendrimer interactions. Overall, the above results are in agreement with previous studies, 100,102,107,108 indicating that the 2016H66 forcefield is able to reproduce the expected structural features of the PAMAM dendrimer. 3.1.4

Self-Diffusion Coefficient

The self-diffusion coefficient D for the PAMAM dendrimers considered (G0 to G5) were calculated as explained in Section 2.3.4, and the results are shown in Figure 7 and also reported numerically in Table S.4. The supplementary Figure S.4 shows the corresponding MSD curves used for the least-square fitting.

Figure 7: Self-diffusion coefficient of the PAMAM dendrimer as a function of the generation and considering low (left panel), medium (middle panel) and high (right panel) pH environments. The results obtained with the 2016H66 forcefield are compared to those extracted from previous experimental and simulation studies: Fritzinger2005, 109 Maiti2009, 110 Rathgeber, 29 Garcia-Fernandez2014. 111

A look at the data corresponding to the simulations with 2016H66 in Figure 7 reveals that there is a clear dependence of D with the dendrimer generation, indicating that larger dendrimers tend to have a slower diffusion. The 2016H66 results are compared with the ones obtained by Maiti and Badchi 110 that reported a simulation study using the Dreiding force field and the TIP3P water model in which they investigated the self-diffusion coefficient of PAMAM in solution, suggesting that D decreases with size in a non-Stokesian fashion and slightly increases with pH (see Figure 4 therein). They compared their results to two NMR 18


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studies, one by Fritzinger and Scheler 109 and another by Rathgeber et al., 29 both carried out in deuterated methanol. These curves are all displayed in Figure 7 for a clear comparison. It is important to mention that none of these methanol studies reported the pH condition used in the experiments. They were regarded by Maiti and Badchi 110 as if they were carried out at high pH (fully deprotonated). The 2016H66 results are in good agreemente with the Dreiding results and with two experimental sets, the one by Rathgeber et al. 29 and a more recent study by Garcia-Fernandez and Paulo. 111 The latter provides a more direct comparison since it was measured in water. The NMR data by Fritzinger and Scheler 109 systematically overestimate the values of D in comparison to all other measurements and calculations displayed for high pH condition. The experiments by Garcia-Fernandez and Paulo 111 considered also the effect of pH on D. Unfortunately, for this part, only PAMAM G4 was considered. There is an excellent agreement between these results and the ones calculated with 2016H66, suggesting that this forcefield is able to quantitatively predict D and its pH dependence.

3.2 3.2.1

PPI PPI Size

The radius of gyration Rg was calculated (see Section 2.3.1) for each simulated PPI system (Table 1) and plotted as a function of the dendrimer generation (G1 to G6) in Figure 8. The corresponding plot using PME for the Lennard-Jones interactions is shown in Figure S.15. The available experimental and simulation values considering the same systems under similar thermodynamic conditions were extracted from the original references and are also displayed for a comparative analysis. The corresponding numerical values are reported in Table S.3. Considering low pH, no experimental data was found and only two simulation results, both for PPI G5, have been reported. One of these results was obtained by Wu 112 using a chimera COMPASS/OPLS forcefield, and the other by Jain et al. 113 using GAFF. The two 19



Figure 8: Radius of gyration Rg as a function of generation considering low (left panel), neutral (middle panel) and high (right panel) pH environments. The results obtained with the 2016H66 forcefield are compared to those extracted from previous experimental and simulation studies: Prosa1997, 28 Scherrenberg1998, 31 Topp1999, 32 Wu2010, 112 Maingi2012, 100 Jain2013. 113

values reported, respectively, 1.59 and 1.63 nm, are in agreement with the corresponding value obtained using the 2016H66 forcefield, 1.65 nm. For the other generations, the 2016H66 forcefield leads to a linear behavior, ranging from 0.5 nm for G1 to 1.99 nm for G6. For neutral pH, three experimental data sets are available. 28,31,32 One is a SAXS data set by Prosa et al., 28 including Rg values for G3 to G5. The others are two SANS data sets, one by Sherrenberg et al. 31 including Rg values for G1 to G5, and one by Topp et al. 32 for G4 and G5. The deviations between the three experimental data sets are on the order of ∼0.1-0.3 nm. Overall, the results obtained with the 2016H66 forcefield are slightly above the experimental data, except if compared to the SAXS results for G3 and G4. In broad terms, the agreement between the 2016H66 results and the experimental data can be considered satisfactory. Simulation studies of PPI under similar conditions are scarce and only values for G5 have been reported. PPI G5 was simulated by Wu, 112 Maingi et al. 100 and Jain et al. 113 The first work was carried out using COMPASS/OPLS and the others using GAFF. Similarly to the situation at low pH, the agreement between their results and the results using 2016H66 for G5 is excellent. The values for other generations simulated with 2016H66 are very similar to the low pH values, ranging linearly from 0.49 nm for G1 to 1.99 nm for G6. For high pH, no experimental data was found and simulation results are also scarce. The

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results obtained with 2016H66 now deviate slightly from a linear behavior and in terms of magnitude are different from the situation at low and neutral pH, indicating a significantly more compact structure for PPI G4 to G6 at high pH. Comparison to previous studies is again made in terms of the simulation results for PPI G5 by Wu, 112 Maingi et al. 100 and Jain et al., 113 that reported Rg values of 1.23, 1.31 and 1.27 nm, respectively. The value of 1.32 nm obtained with 2016H66 is in very good agreement. Figure 9 shows Rg as a function of pH for generations 3, 4 and 5. The 2016H66 results indicate a modest decrease in Rg upon increasing the pH for G3. For G4 and G5, this decrease is more accentuated. The neutral pH values lie within the spread of the experimental data. For G5, it is also possible to compare with COMPASS/OPLS and GAFF. All simulation results are in good agreement, reflecting a small change in Rg from low to neutral pH and a more pronounced change from neutral to high pH.

Figure 9: Radius of gyration Rg plotted as a function of pH considering PPI dendrimers of generation 3 (left panel), 4 (middle panel) and 5 (right panel). The results obtained with the 2016H66 forcefield are compared to those extracted from previous experimental and simulation studies: Prosa1997, 28 Scherrenberg1998, 31 Topp1999, 32 Wu2010, 112 Maingi2012, 100 Jain2013. 113

Overall, the Rg results obtained for PPI using the 2016H66 forcefield can be considered good. In spite of the lack of experimental data, the quantitative agreement with COMPASS/OPLS and GAFF for G5 indicates a very consistent behavior by the three models that were derived on the basis of different philosophies.

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3.2.2

PPI Shape

Figure 10 shows the results for the asphericity δ as a function of the dendrimer generation and considering low, medium and high pH conditions. The corresponding plot using PME for the Lennard-Jones interactions is shown in Figure S.17.

Figure 10: Asphericity δ of PPI dendrimer plotted as a function of generation. Left, middle and right panels show, respectively, the results for low, medium and high pH conditions. The results obtained with the 2016H66 forcefield are compared to those extracted from previous simulation studies: Wu2010, 112 Maingi2012, 100 Jain2013. 113

At low pH and neutral pH, the 2016H66 results indicate a monotonic decrease of δ upon increasing the generation number, reaching values close to zero for G5 and G6. Previous simulations using COMPASS/OPLS and GAFF only report values for G5 and are in very good agreement with the 2016H66 results. 100,112,113 At high pH, the 2016H66 results show a decrease in δ from G1 to G3, reaching a plateau at δ ∼1.4 from G3 to G5 and, only then, decaying to a close-to-zero value for G6. The GAFF result by Maingi et al. 100 for PPI G5 is in good agreement with the corresponding value for 2016H66. However, Jain et al. 113 reported a smaller value also using GAFF and Wu 112 obtained a much more spherical PPI G5 using COMPASS/OPLS.

3.2.3

PPI Atom Density

The atom densities for the PPI dendrimers considered (G1 to G6) were calculated in terms of radial distribution function g(r) as explained in Section 2.3.3, and the results are shown in Figure 11 and supplementary Figures S.12 and S.13. The corresponding plots using PME 22


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for the Lennard-Jones interactions are displayed in Figures S.18-S.20. For sake of improving the visualization, some of these distributions were normalized by density and volume and others were not, as explained in the figure captions.

Figure 11: Radial distribution function g(r) plots for the PPI dendrimer considering generations G1 to G6. The top, middle and bottom rows indicate the pH condition, respectively, low, neutral and high pH. The 3 left panels consider the non-normalized g(r) of all dendrimer monomers with respect to the dendrimer center of mass. The 3 middle panels consider the normalized g(r) of the dendrimer terminal groups with respect to the dendrimer center of mass. The 3 right panels consider the normalized g(r) of the water molecules with respect to the dendrimer center of mass.

The three left panels on Figure 11 show the non-normalized density distributions of the monomers with respect to the dendrimer center of mass for low (top), neutral (middle) and high (bottom) pH. As expected, and also similarly to what is observed for the PAMAM dendrimer, regardless of the pH condition, the maximum of the distribution tends to larger 23



distance values and the distribution becomes broader as the generation increases. Considering the pH effect, PPI dendrimers become more compact from low to high pH (top → bottom), but as explained in Section 3.2.1, the difference is more pronounced from neutral to high pH. The presence of “shoulders” is not as obvious as in the case of PAMAM, but becomes more apparent for G5 and G6, specially at high pH. The middle panels show the normalized density distributions of the terminal groups with respect to the dendrimer center of mass for low (top), neutral (middle) and high (bottom) pH. For low and neutral pH, and except for generations G1 and G2, PPI dendrimers show a considerable degree of backfolding, as evidenced by the large density of terminal groups close to the center of mass. At high pH, the level of backfolding is further increased for G3 to G6 and even G2 evidences a small but noticeable probability of backfolding. The right panels show the water radial distribution function with respect to the center of mass. At low and neutral pH conditions, and for all generations considered (G1 to G6) it is possible to note a clear first density peak at a distance of ∼0.4 nm, corresponding to the hydration near the core moiety. The height of the first peak becomes much less pronounced at high pH, indicating that the hydration level is severely hampered from neutral to high pH. This is in good agreement with the results reported by Wu 112 (see Figure 5c therein).

3.2.4

Self-diffusion Coefficient

The self-diffusion coefficient D for the PPI dendrimers considered (G1 to G6) were calculated as explained in Section 2.3.4, and the results are shown in Figure 12 and also reported numerically in Table S.4. The supplementary Figure S.14 shows the corresponding MSD curves used for the least-square fitting. The 2016H66 results show a monotonically decreasing behavior of D as a function of the generation for all pH conditions considered. For high pH, the values systematically underestimate the two experimental sets displayed, 114,115 but the shape of the curve approaches that of Ref. 114 Note that these experiments only serve for a qualitative comparison since

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Figure 12: Self-diffusion coefficient of the PPI dendrimer as a function of the generation. The The results obtained with the 2016H66 forcefield are compared to those exatracted from previous experimental and simulation studies: Rietveld2000 114 (PPI in methanol), Pavlor2002 115 (nitrile terminated PPI in chloroform).

the data by Rietveld and Bedeaux 114 were measured in methanol and the data by Pavlov et al. 115 consider nitrile terminated PPI dendrimers in chloroform.

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4

Conclusion

The GROMOS-compatible 2016H66 forcefield was parametrized so as to reproduce thermodynamic properties of molecular liquids and solvation free energies in polar and apolar environments. The objective of the present study was to investigate the transferability of such parameters in the description of structural and dynamic properties of dendrimers. In special, PAMAM and PPI dendrimers were selected due to their importance and availability of experimental and simulation data. Considering the results obtained for the PAMAM dendrimer, the 2016H66 forcefield leads to a good agreement in terms of Rg values derived from SAXS experiments that have been measured at neutral pH. Moreover, 2016H66 leads to Rg and δ values very close to those calculated using the best regarded forcefields in the area, CHARMM and GAFF. In terms of the swelling behavior upon acidification, suggested by several simulation studies, 2016H66 provides a close-to-linear dependence of Rg with pH, in a very good agreement with the results provided by the highly used GAFF forcefield. The self-diffusion coefficients are in good agreement with the experimental values measured in water and also with the simulation results using the Dreiding forcefield. Considering PPI, the Rg results at neutral pH condition obtained using 2016H66 agree very well with the three sets of the available experimental data and also with simulation studies using GAFF and COMPASS/OPLS. At low and high pH conditions, for which no experimental data are available, the results obtained using 2016H66 were compared to simulation results using GAFF and COMPASS/OPLS, showing a very good agreement, although only data for G5 were available. The self-diffusion coefficients show a monotonically decreasing behavior upon increasing Rg and are in qualitative agreement with experiments that considered chloroform and methanol as solvents. In summary, the results presented in this work support the recommendation of the 2016H66 forcefield for the simulation of dendrimer systems. The natural building-block based strategy adopted in the GROMOS philosophy, together with a careful parametrization 26


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of the chemical functional groups to reproduce thermodynamic properties in environments of different polarity, and also the ability to accurately reproduce the expected structural features of dendrimers, as shown in the present work, make this forcefield an attractive option for the simulation of such systems and alike. Moreover, in terms of practical aspects, this forcefield can be considered a convenient alternative for the dendrimer simulation community due to its compatibility with all sorts of molecular topology building blocks included in the 2016H66 distribution that can be downloaded from the CSMS Website. 116 This distribution includes parameters for a broad spectrum of biological building blocks (e.g. aminoacids, nucleotides, lipids, carbohydrates), organic solvents, small molecules and polymers such as PEO and PEG. Finally, in order to facilitate the preparation of MD input files (topologies and initial coordinates), a software under the name pyPolyBuilder has been developed and will be released soon.

Acknowledgements This research was also carried out with the support of N´ ucleo Avan¸cado de Computa¸caõ de Alto Desempenho (NACAD/COPPE/UFRJ). This study was financed in part by the Coordena¸caõ de Aperfei¸coamento de Pessoal de N´ıvel Superior - Brasil (CAPES) - Finance Code 001. FAPERJ (grant numbers: E-26/203.198/2016 and E-26/010.002420/2016) and CNPq are also acknowledged for financial support.

Supporting Information Available Figures showing the illustration of the protonation states, normalized and non-normalized R(g) plots using the setup with cutoff, as well as all results obtained with the PME scheme for both PAMAM (Figures S1-S10) and PPI (Figures S11-S20). Tables reporting the numerical results (Tables S1-S4). The topological definition of the molecular building blocks used to construct PAMAM and PPI dendrimers (Tables S5-S16) along with the corresponding 27



bonded parameters (Table S17). This material is available free of charge via the Internet at http://pubs.acs.org.

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