Physical Foundation of Biomolecular Interactions

National Library Of Medicine

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Abstract

Biomolecular interactions define the internal mechanics of functioning of a biological system: how transcription factors recognize their DNA binding sites, how proteins interact and form complexes, how molecular receptors bind to only certain ligands, and so on. Since many biological molecules bear considerable electric charge, electrostatic interactions are among the most important when studying biomolecular interactions. However, electrostatic interactions in biological systems are difficult to calculate accurately in practice. Aside from the significant charges carried by biomolecules such as DNA and proteins, dielectric properties of the solvent itself, namely, water, result in nontrivial electrostatic effects. Furthermore, hydrogen bonds, known to be involved in numerous important interactions, such as helix formation in both DNA and proteins, are essentially electrostatic in origin. Indeed, it seems that electrostatic effects often drive the physical-chemical processes in biological systems and, thereby, determine biological functions. Therefore, any attempt to perform molecular dynamics (MD) simulations of biological systems requires an adequate description of these electrostatic forces. Previously, we have developed a rigorous surface charge method (SCM) to calculate the crucial electrostatic forces in a biomolecular system. Exact analytical results have been obtained for systems with sufficient symmetry. For example, we developed methods to compute the electrostatic forces for a biomolecular system in which the atoms are represented by spheres. The method is rigorous in the context of the model and the accuracy can be tuned to any desired level. Additionally, we have significantly improved the speed of computation by successfully reformulating the SCM to sidestep the need to compute Wigner rotation matrix elements for every pair of spheres without loss of rigor. Our efforts in the past years have been to extend this exact formalism to a more realistic and flexible model in which a biomolecule is represented by an arbitrary surface. One way to do so is to incorporate intrinsic multipole moments other than just point charges. This should provide a good description of the system when the separation between biomolecules is large enough. It is known that under physiological conditions, the biomolecules are mostly found in electrolyte solution. One choice to deal with this is to explicitly introduce ions as small dielectric spheres, but one will end up spending too much in computational resources on billions of objects whose individual dynamics do not matter. Having an implicit ion approach is desirable but challenging. Even Lev Landau, possibly the most famous Russian Nobel Laureate, worked on this problem but obtained only an approximate pairwise force between two macro dielectric spheres. Because of its importance, we turned our attention to the implicit ion problem. Finally, the last piece of the puzzle for a rigorous implicit ion formalism was found in the later part of 2020. We now have a full implicit ion formalism, covering Landau's approximation as a special case, that also does not require Wigner rotation for its application. Recently we focused on investigating the outcome from the rigorous ionic screening formalism by examining a few dielectric spheres immersed in electrolyte as the testing ground. Surprising results were obtained and published in the European Journal of Physics E. We examined the full implicit ion formalism in the limit of strong ionic screening, developing a rigorous expansion that we expect to provide insight into the qualitative behavior of biomolecular interaction under these conditions. We also applied the rigorous ionic screening formalism to a planar geometry that represents electrostatic interaction of biomolecules at close approach but is exact in the context of the model and discovered a much more complex and nuanced behavior of electrostatic forces than one would expect in such a symmetric system. This year we have pursued the question of the limits of a classical formulation. When atoms or molecules come close enough together, one might expect quantum mechanics to become necessary to describe the biological system, as is certainly true when chemical bonding occurs. However, previous work that we published in 2016 demonstrated that the classical picture remains applicable at surprisingly close approach, at least in the case of atoms with sufficiently simple electronic structure (closed shell atoms). This year we extended this investigation to atoms with more general electronic structure, namely open p-shells. Interaction energies between atoms (dielectric spheres) with non-uniform charge distributions were calculated utilizing the full rigorous electrostatics formalism developed earlier in our group. The charge distributions were chosen to reproduce electronic charge configurations in open-shell atoms. The calculated classical energies were compared to those obtained with a “gold standard” quantum method (the coupled cluster method, which is systematically derived from the general quantum mechanical framework and accurately reproduces experimental binding energies for pairs of atoms) and also with density functional theory (DFT). The results of the relatively fast and simple classical method were in reasonable agreement with the coupled cluster method. Quite remarkably, the classical interaction energies were once again more accurate than those from ad hoc DFT methods. This finding (accepted for publication in the Journal of Chemical Physics) overturns a common assumption that DFT results, by virtue of being obtained with a quantum method, are sufficiently reliable to serve as a gold standard, whereas, in fact, DFT methods are not as reliable as our classical calculations that accurately capture collective electron motion resulting in polarization of atoms. Therefore, the framework of classical electrostatics remains a priority for modeling pre-binding interactions between biological molecules, both small and large. Longer-term efforts in this direction will involve extending the classical approach to larger and more general biomolecules by building up the charge distribution of such molecules from a library of carefully calculated local charge distributions for the various valence states of the atomic building blocks.