Mathematical modeling of living cell metabolism using the method of steady-state stoichiometric flux balance.

This approach uses a set of algebraic linear equations for reaction rates (the method of steady-state stoichiometric flux balance) to model the purposeful metabolism of the living self-reproducing biochemical system (i.e. cell), which persists in steady-state growth. Linear programming (SIMPLEX method) is used to derive the solution for the model equations set (determining reaction rates which provide flux balance at given conditions). Here, we demonstrate the approach through the mathematical modeling of steady-state metabolism in Saccharomyces cerevisiae mitochondria.

Role of the electrostatic interactions in pre-orientation of subunits in the formation of protein-protein complexes.

Theoretical estimation of contribution of the electrostatic interactions to pre-orientation of ribonuclease subunits in process of complex formation was carried out. The subunit was considered as a multipole consisting of partial charges of all atoms of the molecule. The object of investigation was a system of two subunits with their centers of gravity fixed at some distance in vacuum. It was proposed that each subunit independently could rotate freely around its fixed center of gravity. The relative orientation states of the subunits in such system were searched at which the system has electrostatic energy minima (equilibrium states). In first approximation the equilibrium states were found using especially designed approximate method for electrostatic interaction energy calculation, which permitted to calculate and compare the energies of the system in 24(5) (approximately 8 10(6)) states with different mutual orientation of subunits. The angular coordinates of the found equilibrium states were further specified by calculation with gradient sliding method. Angular coordinates of the equilibrium states and the shapes of energy surface cuts along each coordinate angle were calculated also for the intersubunits distances diminished down to 50 angstroms. The dispersions of the angular coordinates of equilibrium states caused by heat movement (at T=300 degrees) and their changes with shortening the distance between centers of gravity of subunits were estimated. Mutual orientation of subunits in the equilibrium states of the system under consideration was found to be similar to their mutual orientations in complex. Also it was found that relaxation time of the system, caused by electrostatic interaction of subunits, after removing the system from an equilibrium state, is much less in vacuum than the mean time between their Brownian collisions at room temperature. It follows from these results that in the case of ribonuclease in vacuum the electrostatic interactions of its subunits must be strong enough to realize the effective pre-orientation of subunits during their Brownian approach from distances of the order 100 angstroms. Preliminary consideration taking into account the effect of surrounding water molecules on the electrostatic interactions of ribonuclease subunits showed that weakening of the interaction must be much less than in the case when one uses in its calculation the macroscopic dielectric permeability value equal to 80. So the results obtained for vacuum seem to be true for water solution also. More strict theoretical analysis of this problem will be carried out in the following publication.

About factors providing the fast protein-protein recognition in processes of complex formation.

A package of programs for the examination of areas of subunit contacts (interface) in protein-protein (PP) complexes has been created and used for a detailed study of amino acid (AA) composition and interface structure in a large number of PP complexes from Brookhaven database (PBD). It appeared that in about 75% of the complexes, the AA composition of the subunit surface is not important. This suggests that, along with the surface AA composition, interactions between AA from the inner parts of protein globules may play a significant role in PP recognition. Such interactions between relatively distant AA residues can only be of electrostatic nature and contribute to the total electric field of the protein molecule. The configuration of the electric field itself appears to determine the PP recognition. The total electric field created by protein molecules can be calculated as a result of superimposition of the fields created by the protein multipole (i.e. by the totality of partial electric charges assigned to each atom of the molecule). We performed preliminary calculations for the distant electrostatic interaction of ribonuclease subunits in a vacuum. The results reveal that the effect of the electric fields of the protein multipole is strong enough to orient protein molecules prior to their Brown collision.

[What forces can determine the formation of highly specific protein-protein complexes?]. / Kakie sily opredeliaiut obrazovanie vysokospetsificheskikh belok-belkovykh kompleksov.

A software package was designed and used in a detailed study of the contact regions (interfaces) of a large number of protein-protein complexes using the PDB data. It appeared that for about 75% of the complexes the amino acid composition of the subunit surface in the contact region is not essential. Thus one may suggest that, along with the amino acid residues at the interface, the residues in the interior of the globules substantially contribute to protein-protein recognition. Such interactions between quite remote residues are most probably of electrical nature, and are involved in recognition by contributing to the overall electric field created by the protein molecule; the configuration of this field is perhaps the definitive factor of recognition. The overall field of the protein molecule is additively built of the fields created by each constituent residue, and it can be calculated as a sum of the fields created by the protein multipole (aggregate of 'partial' electric charges assigned to every atom of the protein molecule). Preliminary calculations of the remote electrostatic interaction have been performed for ribonuclease subunits in vacuum. The results are indicative of a real possibility that the electric field created by the protein multipole can strongly influence the mutual orientation of molecules before Brownian collisions.

Molecular mechanisms of protein-protein recognition: whether the surface placed charged residues determine the recognition process?

We studied the structure and composition of contact areas in 812 different kind dimeric protein-protein complexes from Brookhaven data base (PDB ) in order to reveal their pecularities with regard to protein-protein recognition. We have found, that the large portion of complexes (approximately 70%) have oppositely charged residues in the contact areas (interfaces) on the subunits surfaces, which form electrostatic contacts - R:E, R:D, K:E, K:D, H:E, H:D. These results are consistent with the current view that high rate complex formation may be driven by the long-range electrostatic interaction between charged AA residues of subunits surfaces. However, there are many complexes among the studied ones (approximately 30%), which have no electrostatic contacts at all in their contact area. Thus a question arises: what forces account for high complex formation rates (i.e. for the distant orienting of subunits before encounter) by forming complexes where the surface contact areas lack electrostatic contacts? We believe that the long-range orienting electrostatic interaction of subunits may account for all cases of efficient complex formation if one drops the traditional view that protein subunits interact mainly through their surfaces. We suggest that the distant orienting being due to the electrostatic interaction between the whole aggregates of partial electric charges of atoms of each complex subunits. Our preliminary model calculations (unpublished) made for ribonuclease dimer (does not have electrostatic contacts) conform this suggestion.

The enzyme activity allosteric regulation model based on the composite nature of catalytic and regulatory sites concept.

A new kinetic model of enzymatic catalysis is proposed, which postulates that enzyme solutions are equilibrium systems of oligomers differing in the number of subunits and in the mode of their assembly. It is suggested that the catalytic and regulatory sites of allosteric enzymes are of composite nature and appear as a result of subunits joining. Two possible joining modes are postulated at each oligomerization step. Catalytic site may arise on oligomer formed only by one of these modes. Effector acts by fastening together components of certain oligomeric form and increases the life time of this form. It leads to a shift of oligomer equilibrium and increases a proportion of effector-binding oligomers. Effectors-activators bind the oligomers carrying composite catalytic sites and effectors-inhibitors bind the oligomers, which do not carry active catalytic sites. Thus, catalytic activity control in such system is explained by effector-induced changes of a catalytic sites number, but not of a catalytic site activity caused by changes of subunit's tertiary structure. The postulates of the model do not contradict available experimental data and lead to a new type of general rate equation, which allows to describe and understand the specific kinetic behavior of allosteric enzymes as well as Michaelis type enzymes. All known rate equations of allosteric The equation was tested by modeling the kinetics of human erythrocyte phosphofructokinase. It enabled to reproduce quantitatively the 66 kinetic curves experimentally obtained for this enzyme under different reaction conditions.

[Protein-protein recognition: testing of some hypotheses of binding site's structure using Brookhaven data bank]. / Belok-belkovoe uznavanie: proverka nekotorykh gipotez o strukture kontaktnykh oblastei v kompleksakh belkov na materiale Brukkheivenskogo banka dannykh.

Four amino acid's complementarity hypotheses have been checked using the data on the structure of 122 protein complexes taken from protein Brookhaven Data Bank. No one of hypotheses was conformed at the analysis of the contact region structures of the protein complexes examined.

[Optimal structure of the multienzyme system of the tricarboxylic acid cycle of E. coli during growth on various carbon sources]. / Optimal'naia struktura polifermentnoi sistemy tsikla trikarbonovykh kislot E. coli pri roste na razlichnykh istochnikakh ugelroda.

The optimal structure (stoichiometry) of the tricarboxylic acid cycle multienzyme system of E. coli has been computed using experimental data for the E. coli biomass monomeric content and the results of the steady-state metabolic fluxes calculation. The calculation was performed for the cases of E. coli growth on acetate, glycolate, glycerate, glutamate, glucose, gluconate and pyruvate as carbon sources. The possible self-control mechanisms make it possible to optimize the cellular multienzyme system structure for each of the carbon sources under study. The relationship between the optimal structure and the maximal specific rate of the cell growth is discussed.

[Theoretical analysis of H+/O stoichiometry during aerobic oxidation in E. coli in the stationary state]. / Teoreticheskii analiz stekhiometrii H+/O pri aérobnom okislenii v statsionarnom sostoianii u E. coli.

A general scheme of E. coli respiratory chain under aerobic oxidation with NAD.H2 is considered. The ratio H+/O is calculated by the currents method for the respiratory chain in a stationary state. The maximal possible stoichiometry is shown to equal 8. The origin of different H+/O values in the respiratory chains is discussed.

[Mathematical model of transitional processes in the chemostat culture of microorganisms]. / Matematicheskaia model' perekhodnykh protsessov v khemostatnoi kul'ture mikroorganizmov.

A mathematical model of the growth of the cell culture was developed. The model takes into account changes of the levels of the enzymes which define the metabolism rate, transport of the substrate into the cell, regeneration of the donors of energy. The model is based on the proposition that the rate of overall protein synthesis in the cell is defined by the concentration of a few aminoacids limiting the growth. The chemostat culture of the methanol-assimilating yeast was used as the object of modelling. The model allows to explain the experimental kinetics of alterations in cell number (biomass) and other measurable characteristics of the culture during the transient process when the dilution rate was changed.

[Relation of DNA replication and cell division frequency in a population of microorganisms to the cell growth rate]. / Sviaz' replikatsii DNK i chastoty deleniia kletok v populiatsii mikroorganizmov so skorost'iu rosta kletki.

DNA replication and the frequency of cell division were studied in a microbial population in relation to the rate of cell growth. The relationship is based on the law of cell biomass linear increase during the cell cycle and on the exponential law of mean cell mass increase, and depends on the specific rate of population growth. The cell mass in the initiation of DNA replication is correlated with the number of initiation points basing on the Cooper-Helmstetter theory of DNA replication and taking account of the linear growth of mass in one cell. Possible applications of these relationships are discussed.

[Calculation of the minimal substrate expenditures for energy metabolism in microorganisms]. / Raschet minimal'nykh zatrat substratov na énergeticheskii obmen u mikroorganizmov.

The paper presents a method for calculating the minimal expenses of substrates from the medium necessary to compensate energy donor disbalances in an intracellular process under study (for instance, in the production of a unit biomass or in the synthesis of one mole of monomer) between the number of regeneration events and the number of events in which each energy donor is utilized. The method employs the performance tables of energetic processes and is based on linear programming method. The paper includes the performance tables RE, RU and RS describing the stoichiometry of energetic processes in Escherichia coli, and the results of calculation for the cases of E. coli growth on glucose and acetate as sources of carbon.

[Characteristics tables for monomer biosynthesis by microorganisms]. / Kharakteristicheskie tablitsy biosinteza monmerov mikroorganizma.

The pathways of biosynthesis of amino acids, nucleotides as well as membraneous and cell wall components (target monomers) were analysed in bacteria growing on different carbon sources. The analysis has made it possible to single out "junction" metabolites in the "branching points" of metabolic pathways of monomer synthesis whatever growth substrates are. The authors have compiled stoichiometry tables for the synthesis of all cellular target monomers from "junction" metabolites and for the synthesis of "junction" metabolites from different carbon sources (ethanol, galactose, glycerol, mannitol, glucose, acetate, glutamate and succinate). The tables are referred to as characteristic ones. The paper presents characteristic tables for Escherichia coli.

[Mathematical model of tryptophan synthesis and excretion into the environment by E. coli cells]. / Matematicheskaia model' sinteza triptofana i vydeleniia ego v sredu kletkoi E. coli.

A quantitative kinetic model is suggested for the auto-regulated tryptophan synthesis by E. coli trp-operon system and for tryptophan excretion from the cell mediated by transport systems. Applications of the model for the calculation of several parameters characterizing tryptophan metabolism are considered. In order to explain experimental data it is suggested that a system of tryptophane excretion from the cell exists which is induced by high tryptophan concentrations. The rate of tryptophan excretion was studied as a function of various genetic effects (derepression or reduction of retroinhibition) as well as of changes in intracellular concentrations of substrates of tryptophan synthesis (chorismic acid and serine). Possible ways of making the cell to excrete markedly higher quantities of tryptophan without changing the genotype are discussed.