The zero deficiency theorem
*
Patrick De Leenheer
†
September 18, 2009
Abstract
We prove the zero deficiency theorem in Theorems 4 and 8 below.
1
Chemical reaction networks (CRN’s)
Notation
When
x, y
∈
R
n
, we abuse notation and define the vector
xy
by taking entrywise
products of the vectors
x
and
y
. The natural inner product of two vectors
x
and
y
in the Euclidean
space
R
n
is denoted as
< x, y >
.
When
x
∈
int(
R
n
+
), we let ln(
x
) denote the
n
vector obtained from
x
by taking entrywise
natural logarithms. Similarly, for
y
∈
R
n
, we let e
y
denote the
n
vector obtained from
y
by taking
entrywise exponentials. If
x
∈
R
n
+
and
a
∈
Z
n
+
, then we define
x
a
:=
n
Y
i
=1
x
a
i
i
,
where we define 0
a
i
:= 1 if
a
i
>
0 and
x
0
i
:= 1 for all
x
i
≥
0.
A CRN is given by a set of
r
reactions between
p
complexes involving
n
species. Each complex
is a linear combination of the species with nonnegative integer coefficients. The concentrations
obey:
˙
x
=
SER
(
x
)
,
(1)
where
S
is a
n
×
p
matrix in which the
j
th column contains the stoichiometric coefficients of the
species in the
j
th complex, and
E
describes the reactions taking place between the complexes.
This matrix is a
p
×
r
matrix for which each column corresponds to a unique reaction, and it has
exactly one entry equal to +1, one entry equal to

1 and the other entries equal to 0. The
i
th
entry of the
k
th column of
E
is

1 (+1) if the
k
th reaction has the
i
th complex as its reaction
(product) complex. Notice in particular that all columns of
E
add to 0:
1
T
E
= 0
.
Finally, the vector
R
(
x
) is the
r
vector containing the reaction rates of the various reactions. We
assume that this vector has nonnegative entries that depend in some sufficiently smooth way on
the state vector
x
, and that if
x
∈
int(
R
n
+
), then
R
(
x
)
>
0. In addition we assume that
R
i
(
x
) = 0
if
x
j
= 0 for some species
j
appearing in the reaction complex of reaction
i
.
This is a natural
assumption which simply expresses that the reaction does not take place if one of its reactants is
missing. As a consequence it is not hard to prove that
Fact 1.
R
n
+
is forward invariant for (1).
The proof is omitted but relies on the easily established fact that if
x
i
= 0, then the
i
th component
of the vector field of (1) is nonnegative. The geometrical interpretation of this condition is that
the vector field does not point away from
R
n
+
on its boundary. The result should now be intuitively
clear.
*
These notes were written for the graduate students of UF taking MAP 6487 Biomath Seminar I in the fall of
2009.
†
Email: [email protected] Department of Mathematics, University of Florida.
1
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By setting
Γ =
SE,
we obtain a second description for the evolution of the species concentrations:
˙
x
= Γ
R
(
x
)
(2)
The matrix Γ is usually called the stoichiometric matrix.
Perhaps this name should have been
reserved for the matrix
S
. Notice that every column of Γ is the differences of two columns of the
matrix
S
. For some purposes, description (1) is useful, but for others (2) is. Later we will see a
third way to represent the ODE for the concentrations.
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 Summer '06
 DeLeenheer
 Linear Algebra, Graph Theory, CRN, mass action kinetics

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