Free-energy calculations in metalloproteins

Background.

The structure and function of complex biomolecules, like proteins, DNA or polysaccharides have been studied by molecular simulation and led to insight into the molecular mechanisms that underlie experimental observations (Karplus and Kuriyan, 2005; van Gunsteren et al., 2006). The quality of molecular simulations should be validated against experimental data (van Gunsteren et al., 2018) and will critically depend on two main aspects: sampling and scoring. Sampling involves the generation of a sufficiently large amount of molecular structures to describe the orientations and conformations that in reality lead to the observed (average) quantities that are measured. These may be obtained from a variety of search techniques, like systematic searches, molecular dynamics or Monte Carlo simulations. Scoring, on the other hand, is required to evaluate how likely the generated conformations are in terms of energy. The assignment of an energy value to a molecular conformation can again be done in a variety of ways, ranging from advanced quantum mechanical (QM) calculations, through force fields to empirical, or knowledge-based scoring functions.

In recent years, we have contributed significantly to the development of the GROMOS force fields for biomolecular simulations (Oostenbrink et al, 2004, Reif et al, 2012). However, for some molecular systems and processes, a classical force field is a rather poor representation of reality. Whenever the making or breaking of bonds is to be described (e.g. by an enzyme), a QM description of the process is needed. Similarly, the coordinative bond between metals and biomolecules is typically somewhat in between an ionic interaction and a covalent bond. The exact electronic state of the metal ion furthermore can take up variations in the coordination. Such systems are best described by QM. However, QM methods are typically too expensive to describe an entire protein in solution. Mixed resolution systems are available to describe parts of the system with QM and the rest with a classical (MM) force field. Novel approaches to include machine learning in the quantum description may furthermore be included in such hybrid models. However, for any of these cases, available methods to compute free-energy differences have not been well described. The calculation of alchemical free-energy differences, highly relevant in drug design, is an issue that has not been resolved in the context of quantum mechanical calculations.

Aims and methods.

Within BioToP, we have strong collaborations with experimental groups working on metalloproteins (OBINGER, LUDWIG) and in this project we will bring the description of such systems to a new level. We will evaluate and improve the possibilities of QM/MM calculations in the GROMOS simulation package, of which we are part of the development team. This will lead to an improved description of metal-ligand interactions. Furthermore, we plan to develop the methodology to be able to quantify the free energy of coordinative binding of small substrates to e.g. heme iron.

One particular challenge in this sense is the inclusion of alchemical perturbations to a QM description of interactions. In classical force fields, alchemical methods offer a powerful way to quantify e.g. the relative binding free energy of two related ligands via an unphysical (alchemical) modification of one ligand into the other. Intrinsically, this is done by defining a linear combination of two force-field descriptions. Applying the same techniques for a QM description is not at all straightforward, but crucial to bring modern drug-design methods further. Challenges arise due to highly unfavourable interactions at intermediate states and overlapping particles leading to singularities in the hybrid energy function. In the past, attempts have been described to perform such calculations via a transition from QM to MM (Hudson et al, 2018), or by empirical valence bond systems (Warshel et al, 1980). We will investigate the possibilities of a more direct pathway, possibly by modifying the functional form of the interpolation between the two states of the systems, using e.g. the enveloping distribution sampling (EDS) methodology (Christ et al, 2007; Perthold and Oostenbrink, 2018). In the latter approach, a unified Hamiltonian is constructed based on the sum of the Boltzmann probabilities of two end-states. A smooth energy surface is constructed, which basically follows the most likely molecular state. This can be expected to circumvent the highly unfavourable energies in one particular state and hence lead to an elegant solution of the problem. Possible implementations include both a direct EDS approach and a gradual interpolation based on the EDS functional form.

Another interesting recent development is the inclusion of machine learning techniques to approximate QM interactions. Efficient neural networks can be trained to reproduce the QM interaction energy from the atomic positions (Handley and Behler, 2014; Gastegger and Marquetand, 2015). While this is still beyond reach for complete protein systems, training a neural network for the metal ion in metalloproteins should be relatively straightforward. The trained network can subsequently be used almost as a classical interaction, although it represents the local quantum effects of a coordinative bond. At longer distances (e.g. beyond a cutoff of 0.8 nm), a classical description is likely to be sufficient. This would lead to a highly efficient way to accurately improve the interactions of metalloproteins and possibly avoid the issues that follow from alchemical modifications outlined above.

The methodology outlined above, will be developed on small model systems and subsequently applied to relevant protein systems in the BioToP consortium. Examples are the corproheme decarboxylases, chlorite dismutases, heme binding protein from S. solfataricus (in collaboration with OBINGER) and lytic polysaccharide monooxygenases (in collaboration with LUDWIG).

Collaborations in this thesis involve LUDWIG and OBINGER.

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