5. Biochemical Reaction Networks

The true yield coefficient for citrate is not reported by Christiansen and Nielsen (2002) since they only have data for a single chemostat experiment. Their approach is therefore somewhat more complicated, and to reduce the complexity we have here use a true yield coefficient that has been calculated from their data.

Calculation of c ₂ as described above follows from the theory of similarity transformation of matrices. If, as is the case here T ₁ has fewer columns than rows, then the product \( \user2{T}_1^{\rm{T}}{\user2{T}_1} \) is always a square matrix of dimension equal to the number of columns in T ₁. The matrix A in (2) therefore contains as eigenvalues the singular values σ _i of T ₁, and c ₂ is the square root of the ratio between the largest and smallest of |σ _i |. \( \user2{T}_1^{\rm{T}}{\user2{T}_1} \) is a square matrix of dimension equal to the number of rows in T ₁. It also has the singular values σ _i of T ₁ as eigenvalues, but in addition a number of eigenvalues σ _i = 0 that is equal to the difference between the number of rows and columns in T ₁. These extra eigenvalues are ignored, and therefore it does not matter whether we calculate \( \user2{T}_1^{\rm{T}}{\user2{T}_1} \) or T ₁ T ^T to obtain A. But, since the dimension of A is smaller in the first case we shall always as in (2) use \( \user2{T}_1^{\rm{T}}{\user2{T}_1} \) when the condition number for a rectangular matrix with fewer columns than rows is calculated by means of the eigenvalues of A.

The theory of similarity transformation is used in statistics (Principal Component Analysis, or PCA) to extract “the most important” correlations from large sets of data. In PCA the calculations of the present example are exactly followed, and the “extraction” of results mimics the treatment given to matrix T ₁.

Notice that it is normally necessary to consider natural labeling, which is approximately 1.1% ¹³C, but for simplicity we will neglect natural labeling here.