How to determine the size of folding nuclei of protofibrils from the concentration dependence of the rate and lag-time of aggregation. II. Experimental application for insulin and LysPro insulin: aggregation morphology, kinetics, and sizes of nuclei.

Insulin is a commonly used protein for studies of amyloidogenesis. There are a few insulin analogues with different pharmacokinetic characteristics, in particular the onset and duration of action. One of them is LysPro insulin. The behavior of LysPro insulin in the process of amyloid formation has not been studied in detail yet. To quantitatively investigate the differences between insulin and LysPro insulin in the aggregation reaction, we used thioflavin T fluorescence assay, electron microscopy, X-ray diffraction methods, and theoretical modeling. Kinetic experimental data for both insulin samples demonstrated the increase of the lag-time for LysPro insulin at low concentrations of monomers, particularly at 2 and 4 mg/mL, which corresponds to the pharmaceutical concentration. However, the morphology of insulin and LysPro insulin fibrils and their X-ray diffraction patterns is identical. Mature fibrils reach 10-12 µm in length and about 3-4 nm in diameter. The obtained analytical solution allow us to determine the sizes of the primary and secondary nuclei from the experimentally obtained concentration dependences of the time of growth and the ratio of the lag-time duration to the time of growth of amyloid protofibrils. In the case of insulin and LysPro insulin, we have exponential growth of amyloid protofibrils following the "bifurcation + lateral growth" scenario. In accord with the developed theory and the experimental data, we obtained that the size of the primary nucleus is equal to one monomer and the size of the secondary nucleus is zero in both insulin and LysPro insulin.

How to determine the size of folding nuclei of protofibrils from the concentration dependence of the rate and lag-time of aggregation. I. Modeling the amyloid protofibril formation.

The question about the size of nuclei of formation of protofibrils (which constitute mature amyloid fibrils) formed by different proteins and peptides is yet open and debatable because of the absence of solid knowledge of underlying mechanisms of amyloid formation. In this work, a kinetic model of the process of formation of amyloid protofibrils is suggested, which allows calculation of the size of the nuclei using only kinetic data. In addition to the stage of primary nucleation, which is believed to be present in many protein aggregation processes, the given model includes both linear growth of protofibrils (proceeding only at the cost of attaching of monomers to the ends) and exponential growth of protofibrils at the cost of growth from the surface, branching, and fragmentation with the secondary nuclei. Theoretically, only the exponential growth is compatible with the existence of a pronounced lag-period (which can take much more time then the growth of aggregates themselves). The obtained analytical solution allows us to determine the size of the primary and secondary nuclei from the experimentally obtained concentration dependences of the time of growth and the new parameter-the ratio Lrel of the lag-time duration to the time of growth of amyloid protofibrils.

Development and testing of PFFSol1.1, a new polarizable atomic force field for calculation of molecular interactions in implicit water environment.

A detailed calculation of protein interactions with explicitly considered water molecules takes enormous time. If water is considered implicitly (as media rather than as molecules), calculations become faster. These calculations are less precise, unless one uses voluminous computations of solvent-accessible areas. Our goal is to obtain parameters for nonbonded atom-atom interactions in implicitly considered water without computation of solvent-accessible areas. Because the "in-vacuum" interactions of atoms are obtained from experimental structures of crystals and enthalpies of their sublimation, the "in-water" interactions must be corrected using solvation free energies obtained from Henry's constants. Thus, we obtained parameters for the in-water van der Waals, electrostatic, and polarized interactions for atoms typical of protein structures. Parameters of covalent interactions were taken from the ENCAD force field and partial charges of atoms from quantum-mechanical calculations. The sought parameters of the in-water nonbonded interactions were optimized to achieve the best description of crystal structures and their sublimation and solvation at the room temperature. With the optimized parameters, the correlation between the calculated and experimental cohesion of molecules in crystals is 98.3% in the in-vacuum case (the supplementary force field PFFSubl1.1) and 95.4% the in-water case (the sought force field PFFSol1.1).