Cooperativity and Frustration Effects (or Lack Thereof) in Polarizable and Non-polarizable Force Fields

Abstract

Understanding cooperativity and frustration is crucial for studying biological processes such as molecular recognition and protein aggregation. Force fields have been extensively utilized to explore cooperativity in the formation of protein secondary structures and self-assembled systems. Multiple studies have demonstrated that polarizable force fields provide more accurate descriptions of this phenomenon compared to fixed-charge pairwise nonpolarizable force fields, thanks to the incorporation of polarization effects. In this study, we assess the performance of the AMOEBA polarizable force field and the AMBER and OPLS nonpolarizable pairwise force fields in capturing positive and negative cooperativity recently explored in neutral and charged molecular clusters using density functional theory. Our findings show that polarizable and nonpolarizable force fields qualitatively reproduce the relative cooperativity observed in electron structure calculations. However, AMBER and OPLS fail to describe absolute cooperativity. In contrast, AMOEBA accounts for the absolute cooperativity by considering interactions beyond pairwise interactions. According to the energy decomposition analysis, it is observed that the electrostatic interactions calculated with the AMBER and OPLS force fields seem to play an important and counterintuitive role in reproducing the adiabatic interaction energies calculated with density functional theory. However, it is important to note that these force fields, due to their nature, do not explicitly incorporate many-body effects, which limits their ability to accurately describe cooperativity. On the other hand, frustration in polarizable and nonpolarizable force fields is caused by changes in bond stretching and angle bending terms of the building blocks when they are forming a complex.

Figure 1.

Relative cooperativity (left) and frustrativity (right) for (a) NH₃(H₂O)_n, (b) Li⁺(H₂O)_n, and (c) F⁻(H₂O)_n systems. Solid lines are single-point values calculated from optimized DFT structures, while dashed lines are calculated in optimized FF structures.

Figure 2.

Average root-mean-square deviation (RMSD) between DFT and force field coordinates for the for (a) NH₃(^H₂^O)_n, (b) Li⁺(H₂O)_n, and (c) F⁻(H₂O)_n systems shown in blue, red, and green bars, respectively. Error bars indicate the standard deviation.

Figure 3.

Energy decomposition analysis for the NH₃(H₂O)_n system. (a) OPLS, (b) AMOEBA. Solid lines represent single point values calculated from optimized DFT structures, while the dotted line values were calculated using optimized FF structures.

Figure 4.

Energy decomposition analysis for the Li⁺(H₂O)_n system. (a) AMBER, (b) AMOEBA. Solid lines represent single point values calculated from optimized DFT structures, while the dotted line values were calculated using optimized FF structures.

Figure 5.

Energy decomposition analysis for the F⁻(H₂O)_n system. (a) OPLS, (b) AMOEBA. Solid lines represent single point values calculated from optimized DFT structures, while the dotted line values were calculated using optimized FF structures.

Figure 6.

Corrected interaction energies for: (a) NH₃(H₂O)_n system. y = 1.044x – 0.743 and y = 1.181x + 1.202 were used to correct the adiabatic interaction energies for the AMOEBA and OPLS force fields, respectively. (b) Li⁺(H₂O)_n system. y = 2.725x + 0.448 and y = 3.600x + 1.535 have been used to correct the adiabatic interaction energies for the AMOEBA and AMBER force fields, respectively. (c) F⁻(H₂O)_n system. y = −0.724x + 6.217 and y = −0.564x + 9.661 have been used to correct the adiabatic interaction energies for the AMOEBA and OPLS force fields, respectively.