Conserved Moiety Determination

Introduction to Moiety Conservation

Conserved moieties represent quantities that are conserved in a system, regardless of the individual reaction rates.

Consider this simple network:

reaction 1: A -> B
reaction 2: B -> C
reaction 3: C -> A

Regardless of the rates of reactions 1, 2, and 3, the quantity A + B + C is conserved throughout the dynamic evolution of the system. This conservation is termed structural because it depends only on the structure of the network, rather than on details such as the kinetics of the reactions involved. In the context of systems biology, such a conserved quantity is sometimes referred to as a conserved moiety. A typical, real-world example of a conserved moiety is adenine in its various forms ATP, ADP, AMP, etc. Finding and analyzing conserved moieties can yield insights into the structure and function of a biological network. In addition, for the quantitative modeler, conserved moieties represent dependencies that can be removed to reduce a system’s dimensionality, or number of dynamic variables. In the previous simple network, in principle, it is only necessary to calculate the time courses for A and B. After this is done, C is fixed by the conservation relation.

Algorithms for Conserved Cycle Calculations

The sbioconsmoiety function analyzes conservation relationships in a model by calculating a complete set of linear conservation relations for the species in the model object.

sbioconsmoiety lets you specify one of three algorithms based on the nature of the model and the required results:

'qr' — sbioconsmoiety uses an algorithm based on QR factorization. From a numerical standpoint, this is the most efficient and reliable approach.
'rreduce' — sbioconsmoiety uses an algorithm based on row reduction, which yields better numbers for smaller models. This is the default.
'semipos' — sbioconsmoiety returns conservation relations in which all the coefficients are greater than or equal to zero, permitting a more transparent interpretation in terms of physical quantities.

For larger models, the QR-based method is recommended. For smaller models, row reduction or the semipositive algorithm may be preferable. For row reduction and QR factorization, the number of conservation relations returned equals the row rank degeneracy of the model object's stoichiometry matrix. The semipositive algorithm can return a different number of relations. Mathematically speaking, this algorithm returns a generating set of vectors for the space of semipositive conservation relations.

In some situations, you may be interested in the dimensional reduction of your model via conservation relations. Recall the simple model, presented in Introduction to Moiety Conservation, that contained the conserved cycle A + B + C. Given A and B, C is determined by the conservation relation; the system can be thought of as having only two dynamic variables rather than three. The 'link' algorithm specification caters to this situation. In this case, sbioconsmoiety partitions the species in the model into independent and dependent sets and calculates the dependence of the dependent species on the independent species.

Consider a general system with an n-by-m stoichiometry matrix N of rank k, and suppose that the rows of N are permuted (which is equivalent to permuting the species ordering) so that the first k rows are linearly independent. The last n – k rows are then necessarily dependent on the first k rows.

The matrix N can be split into the following independent and dependent parts,

$N = (\begin{array}{l} N R \\ N D \end{array})$

where R in the independent submatrix N_R denotes 'reduced'; the (n – k)-by-k link matrix L0 is defined so that N_D = L0*N_R. In other words, the link matrix gives the dependent rows N_D of the stoichiometry matrix, in terms of the independent rows N_R. Because each row in the stoichiometry matrix corresponds to a species in the model, each row of the link matrix encodes how one dependent species is determined by the k independent species.

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For examples of determining conserved moieties, see: