Elementary modes

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Elementary modes^[1] may be considered minimal realizable flow patterns through a biochemical network that can sustain a steady state. This means that elementary modes cannot be decomposed further into simpler pathways. All possible flows through a network can be constructed from linear combinations of the elementary modes.

The set of elementary modes for a given network is unique (up to an arbitrary scaling factor). Given the fundamental nature of elementary modes in relation to uniqueness and non-decomposability, the term `pathway' can be defined as an elementary mode. Note that the set of elementary modes will change as the set of expressed enzymes change during transitions from one cell state to another. Mathematically, the set of elementary modes is defined as the set of flux vectors, ${\displaystyle \mathbf {v} }$, that satisfy the steady state condition,

${\displaystyle \mathbf {N} \ \mathbf {v} (\mathbf {x} ,\mathbf {p} )=0}$

where ${\displaystyle \mathbf {N} }$ is the stoichiometry matrix, ${\displaystyle \mathbf {v} }$ is the vector of rates, ${\displaystyle \mathbf {x} }$ the vector of steady state floating (or internal) species and ${\displaystyle \mathbf {p} }$, the vector of system parameters.

An important condition is that the rate of each irreversible reaction must be non-negative, ${\displaystyle \mathbf {v} _{irr}\geq 0}$.

A more formal definition is given by:^[2]

An elementary mode, ${\displaystyle \mathbf {v} _{i}}$, is defined as a vector of fluxes, ${\displaystyle v_{1},v_{2},\ldots }$, such that the three conditions listed in the following criteria are satisfied.

1. The ${\displaystyle \mathbf {v} _{i}}$ vector must satisfy: ${\displaystyle \mathbf {N} \mathbf {v} _{i}=0}$, that is: the steady state condition.
2. For all irreversible reactions: ${\displaystyle v_{i}\geq 0}$. This means that all flow patterns must use reactions that proceed in their most natural direction. This makes the pathway described by the elementary mode a thermodynamically feasible pathway.
3. The vector ${\displaystyle \mathbf {v} _{i}}$ must be elementary. That is, it should not be possible to generate ${\displaystyle \mathbf {v} _{i}}$ by combining two other vectors that satisfy the first and second requirements using the same set of enzymes that appear as non-zero entries in ${\displaystyle \mathbf {v} _{i}}$. In other words, it should not be possible to decompose ${\displaystyle \mathbf {v} _{i}}$ into two other pathways that can themselves sustain a steady state. This is called elementarity. A more formal test is that the null space of the submatrix of ${\displaystyle \mathbf {N} }$ that only involves the reactions of ${\displaystyle \mathbf {v} _{i}}$ is of dimension one and has no zero entries.^[2]