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
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 onQR
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,
where R
in the independent submatrix NR
denotes
'reduced'; the (n
– k
)-by-k
link
matrix L0
is defined so that ND =
L0*NR
. In other words, the link
matrix gives the dependent rows ND
of
the stoichiometry matrix, in terms of the independent rows NR
.
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: