Development of a Simple Electron Transfer and Polarization Model

Electron Transfer and Polarization Model and Its Application to Biological Systems ... Publication Date (Web): November 29, 2016 ... The intent of...
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Development of a Simple Electron Transfer and Polarization Model and Its Application to Biological Systems David J. Diller*,† Data2Discovery Consulting, East Windsor, New Jersey 08520, United States ABSTRACT: Here we present a new method for point charge calculation which we call QET (charges by electron transfer). The intent of this work is to develop a method that can be useful for studying charge transfer in large biological systems. It is based on the intuitive framework of the QEQ method with the key difference being that the QET method tracks all pairwise electron transfers by augmenting the QEQ pseudoenergy function with a distance dependent cost function for each electron transfer. This approach solves the key limitation of the QEQ method which is its handling of formally charged groups. First, we parametrize the QET method by fitting to electrostatic potentials calculated using ab initio quantum mechanics on over 11,000 small molecules. On an external test set of over 2500 small molecules the QET method achieves a mean absolute error of 1.37 kcal/mol/electron when compared to the ab initio electrostatic potentials. Second, we examine the conformational dependence of the charges on over 2700 tripeptides. With the tripeptide data set, we show that the conformational effects account for approximately 0.4 kcal/mol/electron on the electrostatic potentials. Third, we test the QET method for its ability to reproduce the effects of polarization and electron transfer on 1000 water clusters. For the water clusters, we show that the QET method captures about 50% of the polarization and electron transfer effects. Finally, we examine the effects of electron transfer and polarizability on the electrostatic interaction between p38 and 94 small molecule ligands. When used in conjunction with the Generalized-Born continuum solvent model, polarization and electron transfer with the QET model lead to an average change of 17 kcal/mol on the calculated electrostatic component of ΔG.

1. INTRODUCTION Electrostatics play a central role in all aspects of molecular interactions, and as a result they play a central role in many molecular modeling applications including 3D-QSAR, conformational analysis, docking, and molecular dynamics. Though not a physical quantity,1 the majority of molecular modeling applications approximate electrostatic interactions by assigning atomic point charges. Even with the vast amount of work done to derive optimal point charges for use in molecular mechanics calculations, there remains a significant amount of debate as to how to optimally assign atomic point charges.2 Indeed, the selection of method is often context dependent. Accordingly, there are a large number of approaches for calculating atomic point charges. For an excellent discussion of the strengths and weaknesses of the various approaches see the discussion by Gilson and co-workers.3 The Gold standard methods are often viewed as deriving point charges from ab initio quantum mechanically calculated electrostatic potential values calculated at discrete points surrounding the molecule. These include methods such as CHELP,4 CHELPG,5 and RESP.6,7 Typically, in this approach the electrostatic potential is calculated at hundreds of points either on the Connolly surface or in the volume surrounding the molecule. The point charges are then chosen via a least-squares fit to the quantum mechanical electrostatic potential at the selected points. To minimize conformational dependence and minimize the effect of overfitting, the best practice is to constrain the point charges © XXXX American Chemical Society

of topologically equivalent atoms to be identical and to average the point charges over as many conformations as is practical. For an overview of the strengths and weaknesses of these approaches to charge calculation see the comparison by Sigfridsson and Ryde.8 From a practitioner’s point of view, the main shortcoming of ab initio quantum mechanical methods is the amount of computer time needed: tens of minutes to several hours for small molecules. The ab initio methods are not applicable to even modestly sized biological systems due to their poor scalability. At the opposite extreme are those methods that rely solely on molecular topology to assign atomic point charges. The advantage of these approaches is that they are not dependent on conformation and are therefore extremely fast. Perhaps the most commonly used method is based on the seminal work of Gasteiger.9 Recently, Gilson and co-workers described another such method based on electronegativity equalization.3 Particularly, when dealing with large numbers of small molecules, such as building large databases for virtual screening, these methods are essentially the only methods that can be used. The challenge with these methods relates to balancing parametrization versus quality of point charges. Because they depend only on topology to derive sufficiently accurate charges an atom’s type depends on its topological neighbors. Given the Received: August 28, 2016 Published: November 29, 2016 A

DOI: 10.1021/acs.jctc.6b00852 J. Chem. Theory Comput. XXXX, XXX, XXX−XXX

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Journal of Chemical Theory and Computation diversity of functional groups it can be daunting to develop a family of atom types that is sufficiently general to ensure accurate results across large sets of molecules. A third approach to point charge calculation, which is growing in popularity, is the AM1-BCC10,11 model which is intermediate to the two families of methods described above. Initial charges are calculated via a semiempirical AM1 population analysis. These charges are then adjusted by bond charge corrections. This method relies on the AM1 calculation to capture the gross effects, while the bond charge corrections allow refinement to better match HF/6-31G* electrostatic potentials. By relying on a semiempirical calculation, the model is much more efficient than the ab initio methods described above. As with the topology based methods, the AM1-BCC model relies on an atom typing scheme to maximize the quality of the final charges. The goal of this work is to develop a model that can be used to investigate polarization and electron transfer in large biologically relevant systems. Either because they do not depend on conformation or because they are too computationally expensive the methods described above are not suitable for studying electron transfer in large systems. We chose to build our approach on the QEQ model proposed by Rappe and Goddard.12 The QEQ model is remarkable in that with no parametrization it gives reasonable point charges for a large array of neutral organic molecules. The QEQ model has been further refined by Yang and Sharp13 to better reproduce solvation free energies using the Poisson−Boltzmann equation.14 Despite it being physically intuitive, the QEQ model can exhibit nonintuitive behavior. As pointed out by Gilson and coworkers,3 the shortcoming of this model is that it significantly underestimates the charges on formally charged groups. The poor handling of charges on formally charged groups stems from a physically unreasonable behavior. A simple system that exhibits this behavior, shown in Figure 1A, consists of a negatively charged acetic acid and a protonated ammonium. One would expect that as the distance between the two molecules gets very large they would essentially decouple, i.e., the ideal charge distribution for the system would decompose into the ideal charge distribution for each molecule as if it were in isolation. In particular, we expect that the net charge on the acetic acid would approach −1 and the net charge on the ammonium would approach +1. Figure 1B shows that with the QEQ model molecules do not decouple as their separation becomes large: the net charge on the ammonium approaches +0.2 rather than +1.0 as the distance between the two molecules gets large. While the system in Figure 1 might seem like a trivial example, this effect occurs many times in a typical biological system. A large biological system consists of many charged groups, e.g., charged amino acids such as arginine, aspartic acid, etc. and charged ions, such as Na+ and Cl−, many of which are separated at relatively large distances. Here we describe a new approach to charge assignment which we call charges by electron transfer or QET for short. The QET method for charge assignment maintains the intuitive framework of the QEQ method but eliminates the decoupling problem leading to a much better handling of formally charged groups. Briefly, to assign point charges the QET method tracks all electron transfers between pairs of atoms. It uses a pseudoenergy function similar to that used in the QEQ model but includes an additional cost term that accounts for the cost of each pairwise electron transfer. Because this cost function becomes infinite as the distance between atoms becomes large,

Figure 1. Decoupling of formally charged molecules at large distances. A. The system under study. The system consists of a negatively charged acetic acid and a positively charged ammonium. We are interested in the behavior of the system as the two molecules move away from one another. B. The net charge on the ammonium as a function of the distance between the center carbon of the acetic acid and the nitrogen of the ammonium. The solid curve is from the QEQ model, and the dashed curve is from the QET model. C. The electrostatic interaction between the ammonium and acetic acid as a function of their distance. D. The difference between the electrostatic interaction energy as calculated with fixed charges and as calculated with the full QET model.

molecules decouple as the distance between them becomes large thus preventing electron transfer between atoms far from one another and thereby solving the decoupling problem seen with the QEQ model in Figure 1. Here we describe the parametrization and testing of the QET model using quantum mechanically calculated electrostatic potentials on a data set of B

DOI: 10.1021/acs.jctc.6b00852 J. Chem. Theory Comput. XXXX, XXX, XXX−XXX

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Journal of Chemical Theory and Computation over 11,000 small molecules. Further, we examine the conformational dependence of the QET charges using small peptides and the nonlinearity of the electrostatic potential using water clusters. Finally, we examine the impact of allowing dynamic point charges when compared to static point charges on protein−ligand interactions using a data set of map kinase p38/ligand cocrystal structures.

Ek = Pk − Fk +

The Tkj are found by minimizing the equation Energy =

k

∑ bk(Ek − ak)2 + ∑ ∑ Ikj(Ek , Ej , dkj) + ∑ ∑ C(dkj)Tkj2 k

k

j2700 tripeptides. With this data set we show that there are significant conformational effects and that the QET model captures a significant portion of this effect. Third, we examine the extent to which electrostatics contributes to the nonadditivity of molecular interactions using a series of water clusters. In particular, with the water clusters we examine the nonlinearity, with respect to the molecules, of the electrostatic potential. Here we show that while the assumption of linearity captures the majority, ∼95%, of the variance of the electrostatic potential there is a significant component missed and that the QET model captures a significant portion of this nonlinearity. Thus, the QET model could be useful in understanding the nonadditivity of molecular interactions. The most telling is the fourth test case: p38/small molecule complexes. With this data set we show that the point charges on the ligands exhibit seemingly small changes in going from

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

David J. Diller: 0000-0003-4998-8374 Present Address †

CMDBioscience Inc., 5 Science Park, New Haven, CT 06511.

Notes

The author declares no competing financial interest.

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REFERENCES

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DOI: 10.1021/acs.jctc.6b00852 J. Chem. Theory Comput. XXXX, XXX, XXX−XXX