notes-zero-def - The zero deficiency theorem * Patrick De...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 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 (CRNs) 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 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: Department of Mathematics, University of Florida. 1 By setting = SE, we obtain a second description for the evolution of the species concentrations: x = R ( x ) (2) The matrix is usually called the stoichiometric matrix. Perhaps this name should have been reserved for the matrix S . Notice that every column of is the differences of two columns of the....
View Full Document

This note was uploaded on 09/12/2011 for the course MAP 4305 taught by Professor Deleenheer during the Summer '06 term at University of Florida.

Page1 / 12

notes-zero-def - The zero deficiency theorem * Patrick De...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online