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notes-zero-def

# notes-zero-def - The zero deciency theorem Patrick De...

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The zero deficiency theorem * Patrick De Leenheer September 18, 2009 Abstract We prove the zero deficiency theorem in Theorems 4 and 8 below. 1 Chemical reaction networks (CRN’s) Notation When x, y R n , we abuse notation and define the vector xy by taking entrywise products of the vectors x and y . The natural inner product of two vectors x and y in the Euclidean space R n is denoted as < x, y > . When x int( R n + ), we let ln( x ) denote the n vector obtained from x by taking entrywise natural logarithms. Similarly, for y R n , we let e y denote the n vector obtained from y by taking entrywise exponentials. If x R n + and a Z n + , then we define x a := n Y i =1 x a i i , where we define 0 a i := 1 if a i > 0 and x 0 i := 1 for all x i 0. A CRN is given by a set of r reactions between p complexes involving n species. Each complex is a linear combination of the species with non-negative integer coefficients. The concentrations obey: ˙ x = SER ( x ) , (1) where S is a n × p matrix in which the j th column contains the stoichiometric coefficients of the species in the j th complex, and E describes the reactions taking place between the complexes. This matrix is a p × r matrix for which each column corresponds to a unique reaction, and it has exactly one entry equal to +1, one entry equal to - 1 and the other entries equal to 0. The i th entry of the k th column of E is - 1 (+1) if the k th reaction has the i th complex as its reaction (product) complex. Notice in particular that all columns of E add to 0: 1 T E = 0 . Finally, the vector R ( x ) is the r vector containing the reaction rates of the various reactions. We assume that this vector has non-negative entries that depend in some sufficiently smooth way on the state vector x , and that if x int( R n + ), then R ( x ) > 0. In addition we assume that R i ( x ) = 0 if x j = 0 for some species j appearing in the reaction complex of reaction i . This is a natural assumption which simply expresses that the reaction does not take place if one of its reactants is missing. As a consequence it is not hard to prove that Fact 1. R n + is forward invariant for (1). The proof is omitted but relies on the easily established fact that if x i = 0, then the i th component of the vector field of (1) is non-negative. The geometrical interpretation of this condition is that the vector field does not point away from R n + on its boundary. The result should now be intuitively clear. * These notes were written for the graduate students of UF taking MAP 6487 Biomath Seminar I in the fall of 2009. Email: [email protected] Department of Mathematics, University of Florida. 1

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By setting Γ = SE, we obtain a second description for the evolution of the species concentrations: ˙ x = Γ R ( x ) (2) The matrix Γ is usually called the stoichiometric matrix. Perhaps this name should have been reserved for the matrix S . Notice that every column of Γ is the differences of two columns of the matrix S . For some purposes, description (1) is useful, but for others (2) is. Later we will see a third way to represent the ODE for the concentrations.
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notes-zero-def - The zero deciency theorem Patrick De...

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