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bilinear map Let X, Y, Z be vector spaces. biochemical motif A motif describing a bio-
A map B : X × Y → Z is called bilinear if it is chemical relationship between two compounds
linear in each factor, i.e., in the donor-acceptor formalism.
Comment: The constraint for bimolecular
B(αx + βy, z) = αB(x, z) + βB(y, z), relationship permits use of the common donor-
acceptor language. A reaction may have more
B(z, αx + βy) = αB(z, x) + βB(z, y). than one such relationship. Note that the bio-
chemical donor-acceptor relationship is often
binary A binary number system is based on opposite to that of the chemical one: thus a
the number 2 instead of 10. Only the digits 0 and phosphoryl donor is a nucleophile acceptor. See
1 are needed. For example, the binary number also chemical, dynamical, functional, kinetic,
5 4 3 2 mechanistic, phylogenetic, regulatory, thermo-
101110 = 1 · 2 + 0 · 2 + 1 · 2 + 1 · 2 +
0
0 · 2 = 46 in decimal notation. dynamic, and topological motifs.
A binary operation is an operation that
depends on two objects. Addition and, multi- biochemical network A mathematical net-
work N(V, E, P, L) representing a system R
plication are binary operations.
of biochemical reactions, their participating
molecular species; descriptive, transformational,
binomial The formal sum of two terms,
thermodynamic, kinetic, and dynamic param-
e.g., x + y.
eters describing the reactions singly and com-
posed together; and labels giving the names of
binomialcoefficients Thecoefficientsinthe reactions, molecules, and subnetworks. V is the
n
expansion of (x + y) . The (k + 1)st binomial bipartite set of vertices: V representing molecu-
m
coefficient of order n is the coefficient of x n−k k lar species; V representing reactive conjunctions
y ,
r
and it is given by of molecules, V = V ∪ V . E = E ∪ E ∪ E is
d
c
s
m
r
thesetofrelationsbetweenmoleculeandreactive
n n!

= . conjunction vertices, e(λ, v m,i ,v ) ∈ E, where
r,j
k k!(n − k)!
for each pair (v m,i ,v ), λ is one and only one
r,j
of {s, d, c}= L: a molecule is a member of
This is also the number of combinations of n
the set of coreacting species that appear sinis-
things k at a time.
tralaterally, dextralaterally, or catalytically in the
reaction equation. Members of the parameter set
bioassay A test to determine whether a
P apply to vertices, edges, and connected graphs
chemical has any biological function (sometimes
of vertices and edges as biochemically appro-
also called activity). This is usually accom-
priate and as such information is available. If
plished by a set of chemical reactions leading
there are no parameters (P =∅), the network
to an observable change in biological systems or
N(V, E, P, L) reduces to its graph N (V, E, L).

in test tubes.
Labels apply to vertices, edges, and subnetworks
and take the form of one of the elements of
biochemical graph A set of biochemical {l m,i ,l ,l ((m,i),(r,j)) ,l {V m ,V r ,E} }.
r,j
reactions, their participating molecules, and Comment: The network is a biochemical
labels for reactions, molecules, and subgraphs, graph whose nodes, edges, and subgraphs have
represented as a graph. qualitative and quantitative parameters. Thus
Comment: The considered biochemical concentration is a property of a compound node;

0
graphs are sometimes hypergraphs, mathemat- G is a property of a set of compound and reac-
ically. However, the key results and algorithms tive conjunction nodes, and their incident edges;
of the two objects are equally applicable; the k cat is a property of the edge joining an enzyme to
common usage in computer science is to use its reaction; molecular structure is a property of a
the word “graph.” Notice this is simply the compound node; etc. Not all nodes or edges need
biochemical network with an empty parameter be so marked; and in fact much known informa-
set. See also biochemical network. tion is at present unavailable electronically.
© 2003 by CRC Press LLC