_{L}LA) and Poly(glycolic acid)(PGA) Oligomers

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Structural characterization of poly-_{L}LA) and poly(glycolic acid) (PGA) oligomers containing three units was carried out with an atomistic approach. Oligomer structures were first optimized through quantum chemical calculations, using density functional theory (DFT); rotational barriers concerning dihedral angles along the chain were then investigated. Diffusion coefficients of

Nowadays, overcoming diseases and providing worldwide medical care are high priorities. Materials science, in conjunction with biotechnology, can meet this challenge by developing safe drug delivery systems and organ implants [

Among the several materials suitable for biomedical applications, polyester-based polymers hold great promises because of their feasible properties and biological affinity [_{L}LA and _{D}LA and the raceme _{L,D}LA). Primarily, they are well tolerated by the human body, as their degradation products are incorporated in the tricarboxylic acid cycle, or the Krebs’ one. Their degradation process is mainly due to hydrolysis mechanism: water diffuses into the material and breaks long chains in small oligomers which are able to diffuse within and out the polymeric matrix [_{g}) value with COMPASS force field. Furthermore, Entrialgo-Castaño

In this framework, the present work is focused on poly(_{L}LA and PGA. In the first part, torsional barriers through polymeric chains were obtained both through molecular mechanics and quantum chemistry computations. Molecular mechanics computations were performed with AMBER8^{®} [

Structural characterization was carried out by analyzing torsional potentials involving C-O and C-C rotations: the analysis was focused on computing energies with respect to the dihedral angles considered, C_{1}-O_{2}-C_{3}-C_{4} and O_{2}-C_{3}-C_{4}-O_{5}, as also shown in

PGA structure was first optimized _{1}-O_{2}-C_{3}-C_{4} and O_{2}-C_{3}-C_{4}-O_{5} angle values of this minimum energy geometry were estimated equal to −87° and −164°, respectively.

Then, energy scan was done through quantum chemistry computations, changing dihedral angles with a step of 10° and optimizing the structure at every step.

Analyzing torsional potentials involving C_{1}-O_{2}-C_{3}-C_{4} angle, energy profile shows one global minimum at −80°, which corresponds to the global optimized geometry (

The energy peak corresponds to a geometry in which the repulsion between oxygen atoms is the strongest one, since they are close and almost aligned; in the local minimum, repulsions between oxygen atoms are minimized except for the oxygen of the esters bond, which are aligned.

Referring to O_{2}-C_{3}-C_{4}-O_{5} angle, energy profile shows a global minimum at −180° (

The energy barrier between global minimum and maximum is equal to 4.27 Kcal/mol, while the barrier between the global minimum and the local maximum is equal to 4.05 Kcal/mol. The two maxima have a similar energy value; they present a difference in energy of 0.22 Kcal/mol. The energy barrier between the local minimum and the global and local maxima is equal to 2.42 Kcal/mol and to 2.20 Kcal/mol, respectively.

In the global minimum structure, oxygen repulsions are minimized, while in the global maximum, oxygen atoms are closer and in an unfavorable structure. In contrast, the local minimum structure presents a favorable conformation for oxygen atoms, except for the ones involved in esteric bonds, which are close each other. The local maximum exhibits a structure similar to the global maximum one, where oxygen interactions are not preferred.

The same torsional barriers were computed by means of molecular mechanics, using GAFF force field as implemented in AMBER8^{®} suite of programs. The comparison between the energy profiles computed through quantum chemistry and those obtained via molecular mechanics (

P_{L}LA structure was first optimized _{1}-O_{2}-C_{3}-C_{4} and O_{2}-C_{3}-C_{4}-O_{5} angle of values of the minimum energy structure are equal to 70° and −164°, respectively.

Energy profile of C_{1}-O_{2}-C_{3}-C_{4} angle shows a global minimum at 70°, which corresponds to the optimized geometry shown in

The global minimum corresponds to the overall optimized geometry, where a helix conformation minimizes the interactions between oxygen atoms: in the maximum energy structure, oxygen atom positions are less favorable, increasing repulsion interactions. The same situation can be seen in the local maximum structure.

The local minimum energy geometry exhibits a helix structure similar to the overall minimum structure, but the energy is higher because of the repulsion between carbonyl oxygen atoms.

O_{2}-C_{3}-C_{4}-O_{5} angle, energy profile shows a global minimum at −160°, which corresponds to the overall optimized geometry, and a global maximum at 110° (

In the global maximum structure, oxygen atoms of oligomer backbone are close each other, and thus are subjected to unfavorable repulsion interaction. The global minimum energy geometry is the same as before, where the helix conformation optimizes distances between oxygen atoms.

The local minimum structure has an optimized helix structure except for two close oxygen atoms of the ester bond, while the local maximum structure does not optimize oxygen atom positions.

Also, in this case, GAFF force field is not able to reproduce the torsional behavior of PLLA, as shown in _{1}-O_{2}-C_{3}-C_{4} angle, the global minimum is well characterized, but the other configurations are not in agreement with quantum chemistry computations. Energy profile concerning O_{2}-C_{3}-C_{4}-O_{5} angle shows a bad agreement with quantum chemistry torsional barriers. In order to understand the poor agreement of energy profiles, structures obtained through molecular mechanics can be compared with the corresponding ones characterized by means of quantum chemistry. It can be seen, indeed, that molecular mechanics geometries are very different from quantum chemistry ones. Even if a great number of minimization steps are performed, in order to obtain minimum energy structures using atoms coordinates as variables, the system is not able to relax properly. This can be related to a parameterization that is not suitable for such systems; GAFF force field, indeed, is intended to be used only with small organic molecules, and not with polymers. For the sake of completeness, an example referring to the PGA global maximum with respect to O_{2}-C_{3}-C_{4}-O_{5} angle is here reported (

Diffusion coefficients were computed by means of Einstein equation [

Where ^{−5} cm^{2}/s, and it is in good agreement with the experimental value of 0.993 × 10^{−5} cm^{2}/s reported by Ribeiro ^{−5} cm^{2}/s.

Up to our best knowledge, an experimental self diffusion coefficient for glycolic acid is not available in literature, but the computed result can be considered reasonable, since it is very similar to the one of lactic acid. The slope of log(

Starting structures were P_{L}LA and PGA oligomers composed by three monomeric units. First, structure geometries were optimized ^{®} suite of programs [_{L}LA and PGA structures were placed ^{6} cycles was carried out, applying a restrain to the investigated dihedral angles. As in the DFT approach, while the dihedral angle of interest was restrained, the molecule geometry was optimized through energy minimization in order to obtain the most probable structure with respect to that dihedral angle value. In particular, 100,000 minimization cycles were run with the steepest descent algorithm, while the remaining ones were accomplished through conjugate gradient algorithm, which is more efficient when the system is close to an energy minimum. Atomic charges were computed starting from electrostatic potentials (ESP) calculated through quantum chemistry at B3LYP/6–31G(d,p) level; then, charges were fitted following RESP (Restrained Electrostatic Potentials) [

In order to provide further proof as to the suitability of the adopted method, ^{®} suite of programs: in particular, the following simulation protocol was adopted. First of all, a 2000-cycle minimization, in which the solute is restrained with harmonic potential k(Δx)^{2} (where Δx is the displacement and k is the force constant, equal to 500 Kcal mol^{−1} Å^{−2}), was performed in order to cut out bad contacts which derive by the random placing of solvent molecules. 1000 minimization steps were realized through the steepest descent algorithm, while the remaining 1000 were accomplished through the conjugate gradient algorithm, which is more efficient when the system is close to an energy minimum. Then a 3500-cycle minimization was performed in order to minimize the energy of the whole system, without restraints. In this case, 2000 minimization steps were done through the steepest descent algorithm, and the remaining ones were realized with conjugate gradient algorithm. The temperature was raised from 0 K to 300 K by a simulated annealing of 20 ps at constant volume, imposing a weak restraint (k = 10 Kcal mol^{−1} Å^{−2}) on the solute with the purpose of avoiding wild fluctuations, and afterwards the system was relaxed with a 100 ps run at constant pressure in order to reach the correct density of the solution. Finally, molecular dynamics simulations were carried out, investigating a time span of 5 ns. SHAKE algorithm was used for all covalent bonds involving hydrogen atoms, thus allowing a time step of 2 fs. Simulations were run under steady conditions (300 K and 1 atm). Temperature and pressure were controlled using Langevin dynamics (with a collision frequency equal to 1.0 ps^{−1}) and isotropic position scaling, respectively.

Torsional barriers involving C-O and C-C bonds were investigated for PGA and PLLA oligomers, by means of quantum chemistry and molecular mechanics. The analysis shows that oligomers were allowed to conform in different structures, since there are also local minima in the energy profiles. Comparing torsional barriers computed through quantum chemistry with the ones obtained by means of molecular mechanics, it could be stated that the GAFF force field chosen is not able to describe the structural behavior. Such disagreement with quantum chemistry profile is related to parameters of dihedral term of the force field, which are not suitable for such analysis. Indeed, even if a large number of minimization steps were performed through molecular mechanics, structures were not able to relax properly. However, GAFF force field is surely suitable to characterize transport phenomena in water environment, since the computed diffusion coefficient for lactic acid satisfactorily matches the experimental one. This confirms the suitability of the method used to compute atomic charges, and that long range interactions are determined in a proper way.

Analyzed dihedral angles; R = H for poly(glycolic acid), and R = CH_{3} for L-lactic acid.

Optimized PGA structure

Comparison between energy barriers of C_{1}-O_{2}-C_{3}-C_{4} angle _{2}-C_{3}-C_{4}-O_{5} angle

Optimized P_{L}LA structure

_{L}LA structure, with respect to C-O bond; _{L}LA structure, with respect to C-C bond; _{L}LA structure, with respect to C-O bond; _{L}LA structure, with respect to C-O bond; _{L}LA structure, with respect to C-C bond; _{L}LA structure, with respect to C-C bond.

Comparison between energy barriers of C_{1}-O_{2}-C_{3}-C_{4} angle _{2}-C_{3}-C_{4}-O_{5} angle

Comparison between corresponding PGA structures obtained through quantum chemistry

(