Definition of Labeling Patterns

A positional isotopomer has an exactly determined labeling pattern, described by a specific number of atoms in specific positions of the molecule. Generally, 2 isotopomers are possible for a compound with n carbon atoms. The 8 different positional isotopomers of a compound with 3 carbon atoms are shown in Fig. (1). 

Schmidt et al. suggested isotopomer distribution vectors (IDV) to quantitatively describe distributions of positional isotopomers, whereby the elements of an IDV contain molar fractions of single positional isotopomers [14]. The indexing of positional isotopomers as elements of an IDV is based on the binary code with ones for labeled and zeros for non-labeled carbons following the standard rules of numbering the carbons within a molecule. The positional isotopomers in Fig. (1) are ordered in this way. The sum of all elements in the IDV equals 1. The IDV of pyruvate can serve as an example (Eq. 1). 

A mass isotopomer x +i is specified by the number i of atoms in the molecule, but not by their positions. A compound with n carbons has -i-1 different mass isotopomers, ranging from the non-labeled U- C mass isotopomer (x ) 

In their approach of applying transition matrices for the description of the carbon transfer in metabolic networks, Wiechert et al. [16] defined the labeling pattern of a molecule via cumulated isotopomer (cumomer) fractions, whereby the isotopomers belonging to the same cumomer fraction share labelings at specific carbon positions. For a Cj compound, three 1-cumomer fractions 

The labehng state of a molecule with n carbons can be also expressed as molar enrichment (ME) [15] or summed fractional labeling [17], describing the weighted sum of mass isotopomer fractions (Eq. 3). 

---