The Importance of Mathematics in the Development of Science and Technology. BİLİM ve TEKNOLOJİNİN GELİŞİMİNDE MATEMATİĞİN ÖNEMİ

Transkript

1 The Importance of Mathematics in the Development of Science and Technology BİLİM ve TEKNOLOJİNİN GELİŞİMİNDE MATEMATİĞİN ÖNEMİ

2 BİLİM ve TEKNOLOJİNİN GELİŞİMİNDE MATEMATİĞİN ÖNEMİ Prof.Dr.Mustafa BAYRAM Yıldız Teknik Üniversitesi Kimya-Metalurji Fakültesi Matematik Mühendisliği Bölümü İstanbul

3 Essence and Role of Mathematics Mathematics is an autonomous intellectual discipline, one of the clearest exponents of the creative power of the human mind. On the other hand, it plays a fundamental role in modern Science, has a strong influence on it and it has been influenced by it in an essential way.

4 Essence and Role of Mathematics There are two different Matehmatics: A first dimension of Mathematics is in fact the PURE aspect, Mathematics as an art in its own right, a game that is played in our minds. A second dimension of Mathematics is in fact the APPLICATION aspect.

5 Essence and Role of Mathematics Mathematicsc is an art that expresses beauty in the form of axioms, theorems and logical or numerical relations; Applied role of Mathematics is more essential. In fact: Mathematics has played a fundamental role in the formulation of modern Science since the very beginning; a scientific theory is a theory that has an adequate mathematical model;

6 Essence and Role of Mathematics The Mathematics that can be applied today covers all the fields of the mathematical science and not only some special topics; it concerns Mathematics of all levels of difficulty and not only simple results and arguments; The capabilities of scientific computation have made numerical simulation an indispensable tool in the design and control of industrial processes.

7 Essence and Role of Mathematics In this presentation we will deal with this aspect whereby Mathematics is language in which the pages of Science are written. Thanks to it there has been a development of the combination Science-Technology. In deed, the daily practice of the physical sciences and engineering hides huge amounts of higher mathematics.

8 Essence and Role of Mathematics Moreover, the very concepts on which their theories are based are essentially mathematical concepts. In the last decades we have seen the trend towards mathematization reach other disciplines, like Economics, particularly the financial market, branches of Chemistry, Biology and Medicine, and even the social sciences.

9 Essence and Role of Mathematics Mathematics should permit to assimilate the data and to understand the phenomena. In the hands of the engineer, it is the tool that makes possible to buid a numerical or qualitative model whose analysis allows to make decisions and design artifacts in an efficients and reliable way.

10 Essence and Role of Mathematics Applied mathematics cover the classical areas like mathematical physics and mathematical methods for engineering, but it has today broader contours with the advent of scientific computation and numerical simulation. Modeling, computational simulation and data analysis are essential tools in modern science and industry.

11 Galile s and Newton s Heirs (Galile ve Newton un Mirascıları) Two great historical figures fixed the key role of Mathematics ın the moments in which modern Science was being born. Galileo formulated it, Newton demonstrated it.

12 Galile s and Newton s Heirs (Galile ve Newton un Mirascıları) Leonardo da Vinci guessed the role of Mathematics in Science. As is well known, modern Science appeared in Europe at the end of the Renaissance. It is not based upon Mathematics alone.

13 Galile s and Newton s Heirs (Galile ve Newton un Mirascıları) The fundamental pillar of the building in germ was aptly formulated by the English philosopher and politician Francis Bacon circa 1620 and consists of the experimental method. Nature becomes the preferential object of philosophical investigation, we should learn to read and to understand it, and eventually to control it; observation is the means for comprehension and expertiment is the test of our predictions.

14 Galile s and Newton s Heirs (Galile ve Newton un Mirascıları) The sciences were formed around this method, first Physics, then Biology, Geology, Chemistry and so on. Galileo was of course a committed defender of the experimental method, to which he contributed his famous astronomical and mechanical observations.

15 Galile s and Newton s Heirs (Galile ve Newton un Mirascıları) Isaac NEWTON ( ), who shows the incontestable success of Galileo s proposal as applied to mechanics. He attacks the basic problems debated during the century. F ma

16 Galile s and Newton s Heirs (Galile ve Newton un Mirascıları) Universal attraction formula F Gmm / r 2

17 The Century of Reason and Lights During the following three centuries, a part of that ocean has been filled with truth, science and mathematics, Science and Technology, the basis of the Industrial Revolution, have advanced with theories, reasoning and experiments.

18 The Century of Reason and Lights As a consequence, the society of the XXth century has changed more radically with respect to the XVIIth century than anything that had happened in several thousand years before, since the onset of the great agricultural civilizations. The comfort of house, transportation and communications, and the healty of the present-day citizen rest upon technical bases completely unknown to the people of the XVIIth century.

19 The Century of Reason and Lights Starting with G.W.Leibniz, a great philosopher and Newton s rival in the famous and a bit sad dispute of the Calculus, a series of brilliant mathematicians, like Bernoulli family, Euler, D Alembert,.exploited the potential of the new Calculus and formulated mathematically all types of mechanical problems: shooting problems, problems concerning the fall of bodies, the motion of fluids, mechanical vibrations, minimization,..

20 The Century of Reason and Lights Toward the year 1738 Johann and Daniel Bernoulli establish the theoretical science of Hydrodynamics on the idealized basis of the so-callled perfect fluids. The study is continued by Euler, who writes the famous equations(1755)

21 The Century of Reason and Lights This equation is u u u p 0, u 0 t

22 The Century of Reason and Lights Whose analytical solution turns out to be intractable at the time.

23 The XIXth Century, the Great Century of Sciene The contribution of the XIXth century to Mathematics,both pure and applied,is surprising by its novelty, by its richness and multiplicity of topis, and by its very unexpetedness. Let us begin our review with Physis.

24 The XIXth Century, the Great Century of Sciene ELETRICITY and MAGNETISM: From Mihael Faraday toj.c.maxwell,ex periments and partial laws over a road that counts the names of Gauss, Ampeere, Biot, Savart,Lenz,...till we arrive at the system of partial diferential equa tions that relates the electric and magnetic felds(1863), the work of James Clerk MAXWELL. Maxwell's equations are one of the major achievements of Mathematics in the 19 th century.

25 The XIXth Century, the Great Century of Sciene THE REAL FLUIDS: The Navier-Stokes equations desribe real fluids and they govern the behavior of atmospheric phenomena (climate, Meteorology, Hydrology, the future Aeronautis).

26 The XIXth Century, the Great Century of Sciene THERMODYNAMCIS: which studies the exhange of heat, aquires solid mathematical foundations with James Joule, Saadi Carnot, J.R.Mayer,...It has strong infuene on the calculus with partial derivatives and the concept of exact diferential. This theory inludes the famous Second Law of Thermodynamics (law of entropy growth in the universe), a fundamental law in sciene.

27 The XIXth Century, the Great Century of Sciene Finally, let us mention STATISTICAL MECHANISC, assoiated to the names of L.Boltzmann and Gibbs, who carved a branch of Mathematial Physis on the basis of the calculus of probabilities, a disipline that had remained very much at the margin of this scientific adventure.

28 The XXth Century, a Century of Wonders A mainfeature that stands out is the progressive mathematization of other scienes,which makes them appear as new horizons for Applied Mathematics. THE THEORY OF RELATIVITY. Albert EINSTEIN,

29 The XXth Century, a Century of Wonders The fundamental formula E=mc 2 about the mathematical equivalence of mass and energy

30 The XXth Century, a Century of Wonders QUANTUM MECHANICS:Quantum mechanics often describes and predicts the movement and behavior of particles at the atomic and subatomic levels. These include particles such as atoms, electrons, protons, and photons. According to quantum mechanics, the behavior and movement of particles at these microscopic levels are counter-intuitive, radically differing from anything observed in everyday life

31 The XXth Century, a Century of Wonders AERONAUTIS: Aeronautics is the study of the science of flight. Aeronautics is the method of designing an airplane or other flying machine. THE CALCULUS of PROBABILITIES: As a mathematical foundation for statistics, probability theory is essential to many human activities that involve quantitative analysis of large sets of data.

32 The XXth Century, a Century of Wonders DETERMINISTIC CHAOS: Chaos theory is a field of study in applied mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions.

33 The XXth Century, a Century of Wonders NEW CONEPTS of SOLUTION in DIFFERENTIAL EQUATIONS(1930) Engineering and Mathematics in the last revolution of the Century. Computers and Computational Mathematics

34 The XXth Century, a Century of Wonders The Computational Word, a new word for Mathematics. The computer word is changing little by little the daily life of the citizen: bankig transactions, electronic mail, ticket reservations,...

35 The XXth Century, a Century of Wonders Mathematical Modeling, Mathematical and Numerical Analysis, Simulation, Vizualization, control. Compuational Biology, Computaional Fluid Dynamic. Weather prediction, Astrophysics, mining, indusrial engineering,.

36 The XXth Century, a Century of Wonders A view has emerged where Computational Science is now the third leg of the scientific method together with Theory and Experiment, and this view is nowadays strongly practiced in Physics, Chemistry and Engineering.

37 Trends at the beginning of the XXIth century Mathematisc in the Sciences, Industry, Management and Business

38 Trends at the beginning of the XXIth century Celestial Mechanics. Problems of aerospatial sciene. Stability and chaos in dynamical systems. Strange attrators. Mechanics of solids and fuids in zero gravity. Theory of fuids.appliation to meteorology and climatology

39 Trends at the beginning of the XXIth century Aeronautis. Hydrodynamical problems, supersonic and transonic fight. Shock waves and hyperbolic equations. Modern Physis.The Mathematics of the atomic world and of elementary partiles. Group theory, renormalization and Galule theories, supersymmetry, Yang Mills equations, instantons, dilatons, branes,...

40 Trends at the beginning of the XXIth century Astrophysis. General relativity, stella rmodels Mathematics of plasma physis, magnetohydrodynamis. Kinetic equations (Boltzmann,Landau,Fokker Plank, Vlasov,...). Geosienes. Problems of resources and mining.environmental Problems(PD equations)

41 Trends at the beginning of the XXIth century Materials Science. Modeling and simulation of composite materials, magnetic material, polymers, glass, and paper. (Linear and nonlinear elasticity, calculus of variations. Homogenization theory.)

42 Trends at the beginning of the XXIth century Nanotechnology. Integrated optics, optical networks.(wigner equation) Indusrial Engineering. Steel industry, blast furnaces, car industry. Communication. Telecommunication and optical networks. (Maxwell equations and Fourier heat theory)

43 Trends at the beginning of the XXIth century Management. (Discrete mathematics, graph theory, combinatorics.) Computer Sciences. Mathematical logic, algorithmics, computational complexity, parallezization..

44 Trends at the beginning of the XXIth century Control, Optimal control, robust control, nonlinear control, Predictive control,. Automation and Robotics. Algebraic geometry and computation. Information theory. Coding of messages, error-correcting codes.

45 Trends at the beginning of the XXIth century Statistics in Science. Industry, Government and Business. Estimation and hypothesis testing, design of experiments. Reliabilitiy, survival analysis. Chemistry. Quantum Chemistry: simulation of atomic and molecular structures through fundamental equations.

46 Trends at the beginning of the XXIth century Optimization Theory and Mathematical Programming. Integer Programming: Facets, Subadditivity, and Duality.Nonlinear programming, convex programming. Iterative Methods.Industrial Design Optimization. Numerical methods, partial differential equations, calculus of variations, combinatorics, linear algebra.

47 Trends at the beginning of the XXIth century Problems of optimal transportation. Problems of traffic (with continuous and discrete modeling). Network planning. Traffic in the Web. Economy. Financial mathematics (option pricing,derivative trading, risk management, ) unites stochastic differential equations partial differential equations and free boundary problems. Models for the global economy.

48 Trends at the beginning of the XXIth century Biology: Mathematical Ecology, Epidemiology, Biometrics, Bio-informatics. Mathematics of Genetics, Computational phylogenetics. Nucleic Acid structure and function. Molecular evolution. Proteomics. Regulatory and developmental pathway inference. DNA computation. Sequence alignment, fuzzy reasoning. Mathematical modeling in biopolymerization.

49 Trends at the beginning of the XXIth century Medicine: interaction fluid-structure as a model for the blood flow. Modeling and simulation of the function of other organs: brain, lungs and liver. Self-organization and fractal geometries. Computational asistence of surgery. Pharmacokinetics, tumor growth modeling. Computational neuroscience. The Mathematics of infectious diseases and epidemic spreading. Artificial organs, immune system modeling.

50 Trends at the beginning of the XXIth century Medical imaging methods. Tomography: computerized tomography, 3D image reconstruction. Fourier and Radon Transforms, inverse problems.

51 Trends at the beginning of the XXIth century Though Computational Mathematics (as different from Computer Science) permeates all fields of application, it deserves a mention in itself: numerical methods and codes; efficient algorithms; approximation, (a priori and a posteriori) error estimates, adaptive methods and adaptive models, multigrid and domain decomposition, multiscale analysis, numerics of random processes,

52 Trends at the beginning of the XXIth century On the other hand, Mathematical Modeling in its different variants (deterministic, continuous, discrete, ) leads to the problems of Model Validation and the techniques of obtaining and elaborating data on which validation is based

53 Trends at the beginning of the XXIth century (see Statistics above), as well as the quite important (and debated) concept of hierarchy of models, a progressive way of approaching reality that is nowadays recognized and embedded into the toolkit of the applied scientist (the old idealists with their eternal truth will revolve in their graves; or will they not?).

54

55 METABOLIC CONTROL ANALYsis Prof.Dr.Mustafa BAYRAM Yıldız Teknik University Istanbu

56 Abstract it is important to the establishment of mathematical models for kinetic analysis of a biochemical system. Metabolic control analysis allows one to quantify the behaviour of a metabolic pathway in steady state in terms of dimensionless coefficients. From the definition of metabolite and flux control coefficients and elasticities we are able to derive symbolic forms of these parameters, in terms of conventional kinetic parameters. At the simplest level we are able to substitute values of these kinetic parameters, to yield values for the metabolic control coefficients. Since we are substituting into symbolic equations we can always quarantee the conservation relationships hold. The basic relationships are the summation and connectivity theorems. The ability to define the control coefficient equations in matrix form not only allows easy solution by numerical inversion but also opens up the possibility of obtaining the algebraic solutions by symbolic manipulation of the matrix. However for matrixes longer than rank 4 or 5, this latter possibility, if done by hand, becomes very tedious and is prone to error. The solution to this problem is to develop computer software to automatically carry out this procedure.

57 Metabolism The chemical changes that take place in a cell or an organism that produce energy and basic materials needed for important life processes. Anabolism-Synthesis of complex molecules from simpler ones (e.g amino-acid synthesis) Catabolism-Breaking down of complex molecules to simpler, smaller molecules (e.g. glycolysis)

58 Metabolic Networks Reactions: A B Pathways: A B C D Networks: A B C E D

59 Metabolic Network Definitions (I) Reaction Intermediate A B C E Substrate D Product Active reaction Inactive reaction

60 Metabolic Network Definitions (II) Exchange flux A B C E D System Boundary Internal flux Flux The production or consumption of mass per unit area per unit time.

61 Metabolic Network Analysis: The Stoichiometric Matrix Let us consider following enzymes kinetic systems v 1. 1 X 1 2. * v 2 X 1 + X 6 X X 2 + 2X 5 X 3 X 3 X 4 + X 5 X 4 v 3 v 4 v 5 X 5 + X 6 + *

62 Metabolic Network Analysis: The Stoichiometric Matrix Where, (i=1,2,,6) are concentration metabolites. We define the metabolite concentration vector as: x x x X x x x x i X

63 Metabolic Network Analysis: The Stoichiometric Matrix In order to construct the model, we first write the stoichiometric reaction scheme that describes how the mrtabolites combine [reder, 1988]. It will be convenient to associate to the reaction scheme the matrix of rows and columns N m r constructed as follows: the column j of represents the reaction j, and we write in this column at row. For example, for metabolite and reaction 1, N( i, j) N(1,1). N i x1

64 Metabolic Network Analysis: The Stoichiometric Matrix N ij, if the reaction j produces molecules of x1., if the reaction j consumes molecules of x1. 0, if the reaction j neither produces nor consumes of x1.

65 Metabolic Network Analysis: The Stoichiometric Matrix For the system v 1. 1 X 1 2. * v 2 X 1 + X 6 X X 2 + 2X 5 X 3 X 3 X 4 + X 5 X 4 v 3 v 4 v 5 X 5 + X 6 + *

66 Metabolic Network Analysis: The Stoichiometric Matrix The stoichiometric is: N

67 Metabolic Network Analysis: Rate Vector We assume that the rate of change of the concentration of metabolite is the sum of the r reaction rates, each weighted by the corresponding stoichiometric coefficient of metabolites. Let v j denotes the rate of the reaction J, The rate vector of the above system can be written as: v v v v v v T

68 Definition 1. For each metabolite X i : dx [ ] dt dx dt i Sv ij j j Nv where S ij is the stoichiometric coefficient of the reactant X i in the reaction j with the flux v j. Negative if X i is a substrate, positive if X i is a product.

69 Metabolic Network Analysis: Rate Vector For the system v 1. 1 X 1 * v 2 X 1 + X 6 X 2 v 3 X + 2 2X 5 X 3 v 4 X + 3 X 4 X 5 5. X 4 + v 5 X + 5 X 6 *

70 Metabolic Network Analysis: Rate Vector We write v v v v v x x x x x x dt d

71 Definition 2.(Mathematical Model) A mathematical model of the biochemical kinetics is written as following differential equations: dx ( z) : Nv( x; z) dt where v j is a function of concentransion of metabolite x i (the number of x i is m) and external parameters z (the number of z is p).

72 Definition 3. At Steady-state Nv 0

73 Example: Stoichiometric Matrix b 2 b 1 v 1 v 2 v 6 A B C E b 4 v 3 v 4 v 5 v 7 D Balance Equations A : v b B : v v v v Matrix Notation N C : v v v b 0 D : v v v v b E : v v b b 3 Stoichiometric Matrix = Nv= Internal fluxes Exchange Fluxes

74 Metabolic Control Analysis Çok enzimli bir metabolik sistemde yer alan enzimlerin kinetik özellikleri bu sistemin özelliklerinin belirlenmesinde etkilidir. Ancak sistemi oluşturan enzimlerden herhangi birinin konsantrasyonunda yapılacak küçük bir değişimin sistemin hızını nasıl etkileyeceği, enzimlere ait kinetik çalışmalar ile açıklanamaz.

75 Metabolik Kontrol Analizi-Tanım Metabolik kontrol analizi, çok enzimli bir metabolik sistemde, sistem hızının ve metabolik konsantrasyonlarının bu sistemde yer alan enzimler tarafından nasıl kontrol edildiğini inceleme metodudur.

76 Metabolik Kontrol Analizi- Biraz Tarihçe Metabolik kontrol analizine ait ilk teori Kacser ile Burn [8] ve Heinrich ile Rapoport [6, 7] tarafından ileri sürülmüştür. Bunların dışında gerek biyokimyacılar gerekse matematikçiler tarafından bu konuda yapılmış bir çok çalışma vardır

77 Metabolik Kontrol Katsayıları Metabolik kontrol katsayıları; Sistem hızı(flux) kontrol katsayıları, Konsantrasyon kontrol katsayıları ve Esneklik katsayıları olmak üzere üç kısımdan oluşur.

78 Sistem Hızı Kontrol Katsayıları Çok enzimli bir sistemde, herhangi bir enzimin aktivitesindeki değişiklikler, o enzime ait kinetik parametrelerden herhangi birinde yapılacak bir değişikle veya enzim konsantrasyonunda yapılacak bir değişiklikle gerçekleştirilir. Çok enzimli bir sistemde, enzimlerden birinin konsantrasyonlarında yapılacak küçük değişiklik, bu enzimin katalizlediği reaksiyonun hızında da değişikliğe neden olacaktır. Bu ise, sistemin hızında bir değişiklik meydana getirir. Bu değişiklikler, sistem hızı kontrol kantsayıları ile tanımlanır.

79 Sistem Hızı Kontrol Katsayıları Bir metabolik sistem için E i sistem hızı de i. enzimin konsantrasyonu olmak üzere, sistem hızı kontrol katsayısı C v E i olarak tanımlanır. v v / v ln v E / E ln E i i i

80 Sistem Hızı Kontrol Katsayıları Çok enzimli lineer bir sistemde, sistem hızı kontrol katsayısı 0 ile 1 arasında değişen değerler alır. Lineer olmayan sistemlerde bu değerler daha küçük veya daha büyük olabilirler. Sistemdeki herhangi bir enzim için SHKK nın 0 olması bu enzimin sistem hızı üzerinde etkisi olmadığı, SHKK nın 1 olması ise bu enzimin sistem hızını tek başına tümden kontrol ettiği anlamına gelir.

81 Sistem Hızı Kontrol Katsayıları Lineer olmayan bir sistemde SHKK nın sıfırdan küçük olması enzim aktivitesindeki artışın sistem hızında bir düşüşe neden olacağını gösterir.

82 Konsantrasyon Kontrol Katsayıları -Tanım Konsantrasyon kontrol katsayısı (KKK), çok enzimli bir sistemdeki herhangi bir enzimin aktivitesinde yapılacak değişimin, sistemdeki ara metabolit konsantrasyonlarına etkisini gösterir.

83 Konsantrasyon Kontrol Katsayıları KKK nın pozitif değeri enzimin aktivitesindeki artışla birlikte metabolit seviyesinde de bir artış olduğunu gösterirken, negatif KKK değeri, enzimin aktivitesindeki artışla metabolit seviyesinde bir düşüş olduğu anlamına gelir.

84 Konsantrasyon Kontrol Katsayıları - Tanım bir metabolik sistem için S bir ara metabolit ve E i de i. enzimin konsantrasyonu olmak üzere konsantrasyon kontrol katsayısı, C S E i S / S ln S E / E ln E olarak tanımlanır. i i i

85 Esneklik Katsayıları Çok enzimli bir sistemde, her bir enzimin katalizlediği reaksiyon, diğer kısmından ayrı olarak incelenebilir. Bu şekilde elde edilen bilgiler sadece o enzimin katalizlediği reaksiyona ait olduğu için lokal özellikler olarak adlandırılır. Bir enzim için esneklik katsayısı (EK), sistemdeki substratlardan veya ürünlerden herhangi birindeki küçük değişimin o enzimin katalizlediği reaksiyonun hızına etkisinin bir ölçüsüdür.

86 Esneklik Katsayıları Bir metabolik sistemde i. reaksiyonun hızı ve S de bu reaksiyonun ürün veya substratlarından biri olmak üzere esneklik katsayısı, v / v ln v S / S ln S vi i i i S v i şeklinde tanımlanır.

87 Esneklik Katsayıları SHKK ve KKK bütün bir sistem ile ilgili olduğu için sistem özelliği olarak isimlendirilir. EK ise tek bir reaksiyonu ilgilendirdiği için lokal bir özelliktir.

88 Toplam Teoremleri Çok enzimli bir metabolik sisteme ait SHKK ları toplam teoremlerini sağlamak zorundadır. Teorem 1: n-tane enzim içeren bir sistem için v sistem hızı, E de i. enzimin i konsantrasyonu olmak üzere dir [8]. n i1 C v E i 1

89 Toplam Teoremleri Teorem 2: n-tane enzim içeren bir sistem için S j herhangi bir ara metabolite ve E de i i.enzimin konsantrasyonu olmak üzere n i1 C S E 0 eşitliği sağlanır [4, 6, 7]. i j

90 Konnektivite Bağıntıları SHKK ve KKK bir sistem özelliği olup sistemi oluşturan tüm enzimlere ve parametrelere bağlıdır. Esneklik katsayısı ise sistemdeki tek bir enzim ile ilgili lokal bir özelliktir. Konnektivite bağıntıları, esneklik katsayıları ile kontrol katsayıları arasında ilişki kuran bağıntılardır

91 Konnektivite Bağıntıları n-tane enzim içeren dallanmasız çok enzimli bir sistemde n 2 tane bilinmeyen vardır. Toplam teoremleri ile bir tanesi, SHKK ile ilgili diğer n-1 tanesi de KKK ile ilgili olmak üzere n-tane bağıntı tanımlamaktadır. O 2 halde böyle bir sistem için n n tane konnektivite bağıntısı vardır.

92 Konnektivite Bağıntıları SHKK ile EK lar arasındaki ilişkiyi gösteren bağıntı ilk kez Kacser ve Burns [3, 4, 8] tarafından bulunmuş olup S j bir iç metabolit olmak üzere v v v v v v E S E S E S C C C L 1 j 2 j 3 j şeklinde tanımlanır. 0

93 Konnektivite Bağıntıları KKK ile EK lar arasındaki ilişkiyi gösteren bağıntı ilk kez Wasterhoff ve Chen [4, 18] tarafından ileri sürülmüş olup S ve S j k birer ara metabolit olmak üzere C C C k v1 k v2 k 3 L 1 j 2 j 3 j S S S v E S E S E S şeklinde tanımlanır. 1 eğer k j ise 0 eğer k j ise

94 Applications Let us consider the following two enzymes system: aspartate aminotransferase (AAT) and malate dehydrogenase(mdh), that is, v AAT 1 aspartate ketoglutarate glutamate oxaloacetate v MDH 2 oxaloacetate NADH malate NAD

95 Applications The system given above contain two substrates and two producs and the system at steady state, That is, v v v v Where v is overall flux.

96 Applications Flux control coefficients for the given system: and C C v AAT v MDH AAT v v AAT MDH v v MDH To calculate the above flux control coefficients we have taken the begining value of the experimental date.

97 Applications Sum of all the flux control coefficients of a reaction system is C v AAT v MDH C Summation theorem is provided.

98 Applications Metabolic control coefficients (Concentration control coefficients): and C oaa AAT AAT oaa oaa AAT C oaa MDH MDH oaa oaa MDH

99 Applications Sum of metabolic control coefficients C oaa AAT oaa MDH C

100 Applications Finally, Elasticity control coefficients, v oaa v 1 1 oaa v oaa and v oaa oaa v v oaa

101 Application Connectivity theorem: v v C C AAT oaa MDH oaa v v The connectivity theorem is provided.