The higher the number of independent reaction vectors, the smaller the orthogonal left null space. The higher the number of independent reactions, the fewer conservation quantities exist.
FUNDAMENTAL SUBSPACES OF S
The dimensions of both the column and row space is r (rank; number of linearly independent rows and columns that the matrix contains).
dim(Col(S)) = dim(Row(S)) = r
Since the dimension of the concentration vector is m, we have
dim(Left Null(S)) = m− r
Similarly, the flux vector is n-dimensional; thus,
dim(Null(S)) = n – r
![*]() Null space. The null space of S contains all the steady-state flux distributions allowable in the network. The steady state is of much interest since most homeostatic states are close to being steady states.
![*]() Row space. The row space of S contains all the dynamic flux distributions of a network and thus the thermodynamic driving forces that change the rate of reaction activity.
![*]() Left null space. The left null space of S contains all the conservation relationships, or time invariants, that a network contains. The sum of conserved metabolites or conserved metabolic pools do not change with time and are combinations of concentration variables.
![*]() Column space. The column space of S contains all the possible time derivatives of the concentration vector and thus shows how the thermodynamic driving forces move the concentration state of the network.
Singular Value Decomposition
SVD states that for a matrix S of dimension m× n and of rank r, there are orthonormal matrices U (of dimension m ×m) and V (of dimension n × n) and a matrix with diagonal elements ∑ = diag(σ1, σ2, ... , σr ) with σ1 ≥ σ2 ≥ ··· ≥ σr > 0 such that S = U∑VT
A non-negative real number σ is a singular value for M if and only if there exist unit-length vectors u in Km and v in Kn such that
Mv=σu and M*u=σv
The vectors u and v are called left-singular and right-singular vectors for σ, respectively.
In any singular value decomposition
M=UΣV*
the diagonal entries of Σ are necessarily equal to the singular values of M. The columns of U and V are, respectively, left- and right-singular vectors for the corresponding singular values.
The columns of U are called the left singular vectors and the columns of V are the right singular vectors. The columns of U and V give orthonormal bases for all the four fundamental subspaces of S (see Figure 8.3). The first r columns of U and V give orthonormal bases for the column and row spaces, respectively. The lastm− r columns of U give an orthonormal basis for the left null space, and the last n − r columns or V give an orthonormal basis for the null space.
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