Elementary modes

In <a href="/facts/Biochemistry/fa2pWEBp">biochemistry</a>, elementary modes may be considered minimal realizable flow patterns through a <a href="/facts/Metabolic_network/BYx7DDMH">biochemical network</a> that can sustain a <a href="/facts/Steady_state_(chemistry)/w7mgo1pD">steady state</a>. 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 <a href="/facts/Enzyme/DSI1Fh7y">enzymes</a> change during transitions from one cell state to another. Mathematically, the <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> of elementary modes is defined as the set of <a href="/facts/Flux_(metabolism)/xABkvHfh">flux</a> vectors 
 
 
 
 
 v
 
 
 
 {\displaystyle \mathbf {v} }
 
 that satisfy the steady state condition

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

where 
 
 
 
 
 N
 
 
 
 {\displaystyle \mathbf {N} }
 
 is the <a href="/facts/Stoichiometry/I5p9cJzV">stoichiometry</a> matrix, 
 
 
 
 
 v
 
 
 
 {\displaystyle \mathbf {v} }
 
 is the vector of <a href="/facts/Reaction_rate/Prwvxq0y">rates</a>, 
 
 
 
 
 x
 
 
 
 {\displaystyle \mathbf {x} }
 
 the vector of steady state floating (or internal) <a href="/facts/Chemical_species/A8BxM3oW">species</a> and 
 
 
 
 
 p
 
 
 
 {\displaystyle \mathbf {p} }
 
, the vector of system <a href="/facts/Parameter/zFvVFMv9">parameters</a>.
An important condition is that the rate of each <a href="/facts/Committed_step/v2Y7dwsL">irreversible reaction</a> must be non-negative, 
 
 
 
 
 
 v
 
 
 i
 r
 r
 
 
 ≥
 0.
 
 
 {\displaystyle \mathbf {v} _{irr}\geq 0.}

A more formal definition is given by:
An elementary mode, 
 
 
 
 
 
 v
 
 
 i
 
 
 
 
 {\displaystyle \mathbf {v} _{i}}
 
, is defined as a vector of fluxes, 
 
 
 
 
 v
 
 1
 
 
 ,
 
 v
 
 2
 
 
 ,
 …
 
 
 {\displaystyle v_{1},v_{2},\ldots }
 
, such that the three conditions listed in the following criteria are satisfied.

<ol><li>The 
 
 
 
 
 
 v
 
 
 i
 
 
 
 
 {\displaystyle \mathbf {v} _{i}}
 
 vector must satisfy: 
 
 
 
 
 N
 
 
 
 v
 
 
 i
 
 
 =
 0
 
 
 {\displaystyle \mathbf {N} \mathbf {v} _{i}=0}
 
, that is: the steady state condition.</li>
<li>For all irreversible reactions: 
 
 
 
 
 v
 
 i
 
 
 ≥
 0
 
 
 {\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.</li>
<li>The vector 
 
 
 
 
 
 v
 
 
 i
 
 
 
 
 {\displaystyle \mathbf {v} _{i}}
 
 must be elementary. That is, it should not be possible to generate 
 
 
 
 
 
 v
 
 
 i
 
 
 
 
 {\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 
 
 
 
 
 
 v
 
 
 i
 
 
 
 
 {\displaystyle \mathbf {v} _{i}}
 
. In other words, it should not be possible to decompose 
 
 
 
 
 
 v
 
 
 i
 
 
 
 
 {\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 
 
 
 
 
 N
 
 
 
 {\displaystyle \mathbf {N} }
 
 that only involves the reactions of 
 
 
 
 
 
 v
 
 
 i
 
 
 
 
 {\displaystyle \mathbf {v} _{i}}
 
 is of dimension one and has no zero entries.</li></ol>

Elementary modes open-in-new

Elementary modes