This book is devoted to the physical and mathematical modeling of the formation of complexes of protein molecules. The models developed show remarkable sensitivity to the amino acid sequences of proteins, which facilitates experimental studies and allows one to reduce the associated costs by reducing the number of measurements required according to the developed criteria. These models make it possible to reach a conclusion about the interactions between different amino acid chains and to identify more stable sites on proteins. The models also take the phosphorylation of amino acid residues into account. At the end of the book, the authors present possible directions of application of their physical and mathematical models in clinical medicine. Tatiana Koshlan graduated from St. Petersburg State University, the department of Molecular Biophysics and Physics of Polymers. She is Master of Science in the field of biophysics. Now she is a post-graduate student at the department of Photonics, St. Petersburg State University. Her interdisciplinary research is in the field of biological and physical sciences. Her research is devoted to studying the interaction of biological molecules by physical methods, using mathematical tools to develop new technology and software with the ability to perform systematic measurements of various data sets of biological interactions. Kirill Kulikov has been a full professor since 2014 at Peter the Great St. Petersburg Polytechnical University, Institute of Applied Mathematics and Mechanics, Department of Higher Mathematics. He received his Ph.D. in Physics and Mathematics with Mathematical Modeling of the Optical Properties of Multilayer Biological Systems and Structures in their Heterogeneous Conjugation (2004). He has habilitation at the State Polytechnical University (Great St. Petersburg Polytechnical University) of St. Petersburg, Russia (Doctor Science in Physics and Mathematics). His Doctor of Science thesis title was Analytical models of interaction of laser radiation with complex heterogeneous biological tissues (2014). His research interests are theory diffraction theory, electrodynamics, physics of lasers, tissue optical methods of mathematical modeling in biological tissue optics and numerical method, biophysics. 1. Introduction 2. Physical methods for studying proteins Describe the various experimental approaches for determine the structure of molecules, the relative location of domains in space, the identification of active protein centers. Their advantages and disadvantages are shown: 2. Physical methods of protein studies 2.1. Electrophoresis method. 2.2. Chromatographic method 2.3. Spectral method. 2.4. X-ray analysis of protein crystals 2.5. Spectroscopy in the ultraviolet and visible range. 2.6. Spectrofluorimetric method 2.7. Circular dichroism. 2.8. Conclusions 3 Physical properties of amino acids and proteins. tions 3.1. Structural organization of proteins. 3.2. The solubility of proteins. 3.3. Conclusions 4. Selection of a biological objects. Presents the rationale for choosing a biological objects will be presented. A description will be given of DNA in the nucleosome core, the structure and functions of the histone proteins H2A, H2B, H3, H4, the histone chaperone Nap1, the functions of the p53, Mdm2, P300 proteins, the Bcl-2 protein family functions. 4.1. The formation of DNA in the nucleosome core. 4.2. The structure and function of histone proteins H2A, H2B, H3, H4. 4.3. Structure and function of the histone chaperone Nap1. 4.4. The structure and function of the p53 protein. 4.5. The structure and function of the Mdm2 protein. 4.6. The structure and function of the P300 protein. 4.7. Structure and function of the Bcl-2 protein family. 4.8. Conclusions 5. Mathematical Simulation of Complex Formation of Protein Molecules Allowing for Their Domain Structure A physical model of the interactions between protein molecules is presented, and an analysis of their propensity to form complex biological complexes is performed. The reactivities of proteins are studied using electrostatics methods based on the example of the histone chaperone Nap1 and histones H2A and H2B. The capability of proteins to form stable biological complexes that allow for different segments of amino acid sequences is analyzed. The ability of protein molecules to form compounds is considered by calculating matrices of electrostatic potential energy of amino acid residues constituting the polypeptide chain. The method of block matrices is used in the analysis of the ability of protein molecules to form complex biological compound 5.1. Description of the physical model 5.2. Numerical simulation of biological systems 5.3. Heterodimers formation (H2A-H2B) and (H3-H4). 5.4. Analysis of the histone chaperone homodimer formation of protein Nap1. 5.5. The algorithm for finding active sites taking into account different protein sites. 5.6. Conclusions 5.7 Description of the program complex with the listing of programs. 6. Mathematical modeling of histone dimers formation in vitro with solutions of different ionic strength in the presence of monovalent salts The chapter develops a physical model for the interaction of protein molecules and their ability to form complex biological complexes for the case in vitro in a solution of a monovalent salt. Introduction. 6.1. Accounting for the screening effect in a salt solution. 6.2. Simulation of complex dimers histones (H3-H4) and (H2A-H2B) formation in the monovalent salts solution. 6.3. Effect of salt solution on the domain binding. 6.4. Description of the program complex with the listing of programs. 6.5. Conclusion 7. Mathematical modeling of the temperature effect on binding of monomeric proteins in aqueous solutions by example on histones H2A, H2B, H3 and H4. Introduction The chapter is devoted to mathematical modeling of temperature influence on the stability of histone dimers H2A-H2B and H3-H4 by studying their behavior in different temperature regimes from C. Introduction. 7.1. Description of the physical model. 7.2. Numerical simulation of the temperature effect on the binding of the monomeric proteins dimers H2A-H2B, H3- H4 in aqueous solutions. 7.3. Description of the program complex with the listing of programs 7.4. Conclusion 8. Mathematical modeling of the temperature effect on binding of different sites of protein BCL-XL in aqueous solutions Presents a mathematical model of the temperature effect on the formation and stability of homodimers formed by the Bcl-Xl proteins and its truncated amino acid sequences by studying their behavior in different temperature regimes( C) . A numerical calculation allows to analyze the contribution of different segments of the polypeptide chain into formation and stabilization Biological complex, as well as allow to determine sites of the protein molecules responsible for protein binding in aqueous solutions under different temperature regimes C 8.1. The nature and structure of homodimer formation by the BCL-XL. 8.2. Numerical modeling of the temperature influence on the nature homodimer formation by the BCL-XL according to different sites of the proteins in aqueous solutions. 8.3.Description of the program complex with the listing of programs. 8.4. Conclusion. 9. Mathematical modeling of the phosphorylation effect on the nature formation of biological complexes P53-MDM2 and P53-P300. Develops a physical model of the effect of the phosphorylation of the amino acid residues of the polypeptide chain on the formation of biological complexes, for example, phosphorylation on two amino acid residues of the flexible N-terminus of the p53 protein and analysis of the stability of the biological complexes P53-Mdm2 and P53-P300 formed before and after phosphorylation. 9.1. The structure of the biological complex formations of P53-MDM2 and P53-P300. 9.2. Competing P300 and MDM2 proteins for binding with the N-terminus of the protein P53. 9.3. Numerical simulation of the phosphorylation effect on the formation of biological complexes P53-MDM2 and P53-P300 . 9.4. Description of the program complex with the listing of programs. 9.5. Conclusions. References
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