For modern proteomics, research and prediction of protein interactions are very important tasks, since they determine the function of proteins at levels from the cell to the whole organism. For proteins whose structure is known, the search for intermolecular interactions according to known data on the conformation of their tertiary structure reduces to the problem of searching for geometric complementarity of the sections of two interacting molecular surfaces and modelling their contacts, the so-called molecular docking [System Computer Biology . Monograph. Novosibirsk: Publishing House of the SB RAS.2008.769 p]. The task of molecular docking is the task of a conformational search algorithm, which reduces to a search for the conformational space of the formed biological complex due to the variation of the torsion angles of protein molecules.
Modern conformational search algorithms in most cases find conformations that are generally
close to the experimentally found structures in a relatively short time. However, there are factors that also have a significant impact on the success of the docking, which are often not taken into account in standard algorithms. One such factor is the conformational mobility of the target protein. The mobility range can be different beginning with a small «adjustment» of the side chains and ending with scale domain movements[Betts M.J., Sternberg M.J. An analysis of conformational changes on protein-protein association: implications for predictive docking.// Protein Eng. 1999. V. 12. Pp. 271-283] These movements play an important role.
At the same time, for each rotation configuration, estimates are made for the evaluation function.
The evaluation function is based on surface complementarity (the mutual correspondence
of complementary structures (macromolecules, radicals), determined by their chemical properties),
electrostatic interactions, van der Waals repulsion, and so on. The problem with this approach is that calculations throughout the configuration space require a lot of time, rarely leading to a single solution, which in turn does not allow us to speak of the uniqueness of the target protein and ligand interaction variant. So, while modelling by the methods of molecular dynamics, from 200 to 10 000 possible combinations of the formation of a protein
complex with a ligand were found. Such a large number of modifications, along with the lack of a criterion for selecting the most probable variants of the bound structures of biological complexes (which would allow a radical reduction in their number) makes it very difficult to interpret the
theoretical results obtained for practical use, namely, the finding of catalytic centers and a qualitative assessment of the dissociation constant of interacting substances.
In contrast to the above computer simulation algorithms, mathematical algorithms have been developed in this soft that allow determining the detection of proteins active regions and detecting the stability of different regions of protein complexes (linear docking) by analyzing the potential energy matrix of pairwise electrostatic interaction between different sites of the biological complex.
At first glance, the most logical solution to this problem is to take into account the mobility of the protein in a docking program. Unfortunately,modern computational tools do not allow such modelling to be performed in an acceptable time frame since a protein molecule is very large, and allowing for mobility over all degrees of freedom can lead to a so-called «combinatorial explosion» (an astronomical increase in the number of possible variants). Only in some programs is there a limited mobility of protein binding sites (usually at the level of a small adaptation of conformations of the side chains of the active center residues). Another approach to this problem consists in docking the same protein in several different conformations and then selecting the best solutions from each docking run. The third approach is to find a universal structure of the target protein in which docking would produce fairly good results for different classes of ligands. In this case, the number of «missed» (but correct) solutions decreases, but the number of incorrect options [Pyrkov T.V., Ozerov I.V., Balitskaya E.D., Efremov R.G. Molecular docking: the role of non-valence interactions in the formation of protein complexes with nucleotides and peptides// Bioorganic Chemistry 2010. V.36. N4. Pp.482-492] also increases significantly. It should also be noted that most programs for the theoretical docking of proteins work according to the following principle: one protein is fixed in space, and the second is rotated around it in a variety of ways.
The proposed theoretical method allows one to take into account the change in the stability of the biological complex with point mutations in proteins, as well as to obtain information about the distribution of the potential energy of electrostatic interaction, indicating the location of key amino acid residues during the formation of the biological complex. Based on a visual map of the interaction of two proteins, which is obtained on the basis of our
method, we can determine whether the complex is stable when replacing key amino acid residues, which in turn will coincide qualitatively with the experimental Kd measured value. Thus, the analysis of the results allows us to identify amino acid
residues that play a significant role in the formation of a biological complex.
Also the developed and physically grounded mathematical approach, in addition to work on molecular dynamics, will theoretically predict the passage of the biochemical reaction in the selected
direction with the given amino acid sequences, identify the stability of different areas of protein complexes by analyzing the potential energy matrix of electrostatic interaction between different sites of the biological complex, and also examine the effect of point mutations in BH3 peptides on the stability of the biological complex formed by them with the proapoptic proteins of the Bcl-2 family and qualitatively determine the dissociation constants when different BH3 peptides bind to the Bcl-2 proteins.
The developed software package will allow:
— to determine the key amino acid residues in the protein complex, which account for the maximum values of the potential energy of electrostatic interaction;
— to determine the effect of point mutations in peptides on the stability of the resulting biological complexes with protein. To qualitatively determine the range of variation of Kd during point mutations, when peptides bind to the active protein site;
— to determine the active sites of the interactions between proteins, when it is formed in succession of a.a. residues of two proteins with an unknown three-dimensional structure of proteins.
In the future, it will allow to solve fundamental and applied problems of medicine, for example, to develop new drugs, to study the processes occurring in the development of diseases.
Scanning Amino Acid Residues including three-dimensional structure. 3D static algorithm
An algorithm and software package was developed for analyzing protein interactions, taking into account their three-dimensional structure from the PDB database.
Numerical of results including three-dimensional structure
In this section, numerical calculations of a map of protein interactions are given, the location of key amino acid residues in peptides are described, and the procedure for finding the range of Kd values for point mutations in the polypeptide chain of a peptide, when it is bound to a protein, are described.
Scanning Amino Acid Residues including three-dimensional structure. 3D algorithm rotation
This algorithm, taking into account the rotation of one amino acid residue allows you to find the stability of protein-protein complexes, peptide-peptide complex, peptide-protein complex with a known amino acid sequence and with the known three-dimensional structure from x-ray analysis.
Calculation of Entropy Change
For obtain a numerical estimate of this interaction we estimate the change differential entropy for one-dimensional and multidimensional case.
Results of numerical calculations Bcl-2-BH3-Bax with using the change one-dimensional differential entropy
In this section, we present the numerical results of the change in entropy by the formula for multidimensional differential entropy in the interaction of two amyloid wild-type peptides from 11th amino acid residue to 42th amino acid residue.
Scanning amino acid residues excluding three-dimensional structure. Description of the physical model.
In this part that allow determining the detection of proteins active regions and detecting the stability of different regions of protein complexes (linear docking) by analyzing the potential energy matrix of pairwise electrostatic interaction between different sites of the biological complex, such as the homodimer of the histone chaperone Nap1-Nap1, which are responsible for the entry of a whole protein molecule into biochemical reactions.
Model linear docking. Algorithm 1
Model linear docking. Algorithm 2
Model linear docking. Algorithm 3
Examples of calculations using Algorithm 1 and Algorithm 2
Examples of calculations using Algorithm 3
Mathematical modelling of the effect of phosphorylation on the stability of the formation of biological complexes
Results of a numerical calculation of the formation of biological complexes by different sites of the p53, Mdm2 and p300 proteins, taking into account the effect of phosphorylation of the flexible N-terminus of the p53 protein
Behaviour of hydrophobic molecules
Let us consider this process in more detail. We assume that the positive values of the potential energy of pairwise electrostatic interactions between hydrophobic amino acid residues tell us about the repulsive nature at small distances (in our case). However, when placed in an aqueous environment, which is characterized by a high degree of ordered interaction between water molecules, they will lead to the formation of hydrophobic clusters. Note that water molecules tend to interact strongly due to the uneven distribution of electron density between the oxygen atom and the two hydrogen atoms. The nature of the formation of a water molecule is such that the electron density around the oxygen atom is much greater than in the region of hydrogen atoms, which in turn leads to the formation of a dipole, the ends of which carry a charge of different value. The presence of such a dipole of water molecule leads to a strong interaction with another dipole of a water molecule: the oxygen atom region carries a negative charge, and the hydrogen atom regions carry a positive charge. As a result, the oxygen atom of one molecule attracts a hydrogen atom of another water molecule, forming a highly ordered structure of dipoles of water molecules.
We assume that if hydrophobic molecules are placed in such an ordered structure of water dipoles, between which there are no such strong dipole-dipole interactions, but dispersion interactions, then they will break it.
Note that dispersion interactions occur between atoms as a result of fluctuations of the electron density around the nucleus, which leads to the appearance of an instantaneous dipole moment of atoms and the interaction of the corresponding dipoles. Dispersion interaction takes place between any atomic groups, but it is greater in those cases when the atomic polarizability is higher. Also in proteins, dispersion forces play an important role in the interaction of amino acid residues having hydrophobic (non-polar) side radicals, such as, for example, valine, leucine, tryptophan, phenylalanine, tyrosine. The energy of dispersion interactions is inversely proportional to the sixth power of the distance between the interacting atoms. This means that the dispersion forces play a significant role only at very short distances.
The criterion for the stability of protein molecules structure
Theorem. The criterion for the stability of protein molecules structure is the condition number of the matrix, the elements of which are determined through the potential energies of the electrostatic interactions between pairwise taken amino acid residues biological complexes for various structures.