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THE MATHEMATICAL MODELING AND COMPUTER ANALYSIS OF GENE NETWORKS. molecular-genetic systems – the main object s for our studies. *peroxisome. *nucleus. *ER lumen. *cytoplasm. *chloroplast. *mitochondria. *cell membrane.

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## THE MATHEMATICAL MODELING AND COMPUTER ANALYSIS OF GENE NETWORKS

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**THE MATHEMATICAL MODELING**AND COMPUTER ANALYSIS OF GENE NETWORKS**molecular-genetic systems– the main objectsforour studies***peroxisome *nucleus *ER lumen *cytoplasm *chloroplast *mitochondria *cell membrane**The generalized chemical kinetic approach (biochemical**reactions)**The generalized chemical kinetic approach(Generalisations of**biochemical reactions) Generalized Michaelis-Menten reaction**Prm-PR locus of lambda bacteriophage**Prm up1 mutation Biochemical model of Prm-PR locus activity regulation by TF CI and Cro**Activity of Prm, PR and Prm up1 promoters**PR Prm Promoter activity, arbitrary units CI,nM Prm up1 CI,nM Points - experimental data, curves - in silico simulation**Generalised Hill Functions**Generalized Hill Functions are the nonnegative rational polynoms with nonnegative parameters which also are Generalized Hill Functions. Subset GHF which is deduced from structural models of promoters and enzymatic reactions.**The enzymatic reaction kinetic, catalyzed by tryptophan**sentitive2-dehydro-3-deoxyphosphoheptonate aldolase(DDPHA) 1 / V, arbitrary units 1 / V, arbitrary units Influence of various concentration Trp on DDPHA-reaction rate. Points – experimental data from (Akowski, Bauerle, 1997); curves – experiment in silico; values of parametres: kf = 20.6 s-1; Km,S1 = 35 M; Km,S2 = 5.3 M; hS1 = 2.6; hS2 = 2.2; Ki,P1 = 1 mM; klR,Vmax = 1.7; kR,Vmax = 5 M; klR,KmS1 = 0.85; kR,KmS1 = 25 M; hR,KmS1 = 0.6; klR,hS1 = 1.1; kR,hS1 = 1 M; hR,hS1 = 1; klR,hS2 = 0.47; kR,hS2 = 1 M; hR,hS2 =2.**GHF model of DDPHA-reaction**V - specific rate of reaction; S1 - concentration of E4P, S2 - concentration of PEP, P1 - concentration of PI, P2 - concentration of 3DDAH7P, R - concentrations of Trp.**GHF- modellingof expression: cyоABCDEoperon efficacy is**dependentfrom oxygen concentration Promoter cyoAp structure: Dependence of promoter activity on the oxygen concentration : parameters value Points- experimental data: Tsenget al., J. Bacteriol., 1996**With application of a hierarchical method of modeling we**have developed the following mathematical models: The phage lambda ontogenesis Artificial gene Networks The cholesterol homeostasis The Auxin synthesis and transport in a plant root and shoot The evolution of the elementary self-reproduced system The endocellular stage in development of an influenza virus Hepatitis virus C replicon replication The frog life water stage cycle in development Trematoda parasite life cycle And some others**The Model of the phage lambdaontogenesis**• Phage lambda genetic elements described as elementary models • PromotersPi, PL, Prm, PR, Pre, PR. • Transcription terminationsti, tL3, tL2, tL1, tR1, tR2,tQ • Protein-coding phage lambda genes • Antitermination sitesNUTL, NUTR • Origin replication site ori • Spacers.**Adaptation: the model of the bacteriophage lambdaontogenesis**а б 80 60 ДНК (штуки/ клетку) 40 20 мин мин Dynamics of synthesis of total DNAin phage and its mutants having defects in genes exo and gamma, at their reproduction inE.coli polA¯. a – experiment in silico, б – the data of experimentsin vitro(Skalka, Enquist, 1974). % % б a Percentage of replicated in the teta -form ring DNAs from total number of replicated rings.(а) – experiment in silico, (б)– the data of experiments in vitro (Better, Freifelder,1983). мин мин**In silico calculation of DNA, mRNA and protein dinamics**Unit/cell Min after infection Protein CI synthesis (experimentin silico). 1- N¯ inE. сoliwith low concentration of the HflB (right scale), 2 – cro¯ inE. сoliwithhighconcentrationof the HflB (left scale), 3 - cro¯CI857 inE. сoli, withhighconcentrationof the HflB, temperate is 37ºС (left scale). Dynamics of changes in concentration of various forms of phage ДНК (experiment in silico, simulating the lytic phage wtdevelopment inE.coli). 1 – supercoil ring DNAs, 2 – relaxed ring DNAs, 3 – ring DNAs, entering into -forms4 –autonomous ring DNAsof phage mRNA synthesis(experiment in silico), simulating lytic phasein phagedevelopments 1 - mRNAPL- tL1, 2 - mRNAPR-tR1, 3 – mRNA PL–xis-…, 4 –mRNA PR–Q-…, 5 –mRNA PR–R-…, 6 – mRNA Pre-…, 7 – mRNA Prm-…**Artificial genetic system**(Elowitz, Leibler, Nature, 2000)**Simulation of Elowitz-Leibler repressilator**Experimental data [Elowitz, Leibler, 2000] Experimentin silico**The hierarchical organisation of gene networks structure, as**dynamic systems Structural level functional level parameter level HGN**HGN Elements**Multimer Genetic element (g) – HGN elementary subsystem protein mRNA gene g Regulation connection () – GHN structural unit g2 g1**HGN Structure presented**by the structural graphs g g g g g g g g g g g g g g g g 1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4 HGN diagrams HGN structural graphs**HGN functional level:regulation the expression efficiencyin**genetic elements as elementary subsystem of gene network Тип 1 • участок инициации • экспрессии - сайт регуляции - синтезируемый продукт -гетеромультимер -гомультимеры**Models for regulation of expression in genetic elements :**type 2 and type 3 Type 2 Type 3**HGN behaviour with different types of regulation**x2 Graph structure x3 x1 models chaos circle**1-base of oriented graph (definition)**Let we have oriented graphG(V,W). The subset UV called 1-base then and only then: 1. v, u U (v, u) and (u, v) W, 2. u V\U v U (v,u) W Criterion of stable steady state points (SSP) for special gene Networkclasses Let we haveG(V,W). Denotes Let Let U1,…,Um – all 1-bases of G. Thenexistsuch0 и 0, thatfor all >0 и >0 and for each Uj, i=1,…m, Mj(G), j=1,2,3,4have unique stable stationary point.Other stable stationary points are not present. Examples: no 1-bases -> no SSP’s 3 1-bases -> 3 SSP’s**(n,k)-Criterion for symmetric HGNs with structural graphGn,k**definition of Gn,k Тype1 • graph nodes are located on a circle • From every node there are arches to the next right(k-1) nodes • Other arches at graph are not present G6,3 Let Gn,k is structural graph of HGN, which is described by the model Then if k is a divider n, for enough big and , HGN has k stable stationary points, if the greatest general divider d=GGD (n, k) =k, GGNhas d stable periodic trajectories.**The theory of the gene Network modelling**Models of matrix processes were investigated. The mathematical formulation of the first limiting theorem. Theorem http://www.bio.miami.edu/dana/250/prokaryotetranscrip.jpg**ИМ**ИЦиГ Демиденко Г.В. Голубятников В.П. Евдокимов А.А. Когай В.В. Фадеев С.И. Акбердин И.Р. Безматерных К.Д. Казанцев Ф.В. Лашин С.А. Матушкин Ю.Г. Миронова В.В. Мищенко Е.Л. Подколодная О.А. Ратушный А.В. Омельянчук Н.А. Хлебодарова Т.М. ИТиПМ Латыпов А.Ф. Медведев А.Е. Никуличев Ю.В. IG&BS, USA Mjelsness E.M.

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