PHYSICAL REVIEW E 86, 067101 (2012)
Conservation-dissipation structure of chemical reaction systems
Wen-An Yong
*
Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing 100084, China
(Received 31 August 2012; published 5 December 2012)
In this Brief Report, we show that balanced chemical reaction systems governed by the law of mass action
have an elegant conservation-dissipation structure. From this structure a number of important conclusions can
be easily deduced. In particular, with the help of this structure we can rigorously justify the classical partial
equilibrium approximation in chemical kinetics.
DOI: 10.1103/PhysRevE.86.067101 PACS number(s): 82.20.−w, 05.70.Ln, 82.40.Qt
Chemical reactions are fundamental for many natural
phenomena, ranging from the rusting of iron to cell cycling and
from photosynthesis to apoptosis. Since J. Wei [1] proposed a
set of axioms to characterize general chemical systems in 1962,
there have been huge efforts in seeking a unified mathematical
formulation of chemical reactions. Oster and Perelson [2], for
example, presented a geometric structure by casting much of
the classical theory of kinetics into the language of differential
geometry in 1974. Very recently in Ref. [3] van der Schaft
et al. derived a general graph-theoretic formulation, which
is basically already contained in the innovative paper by
Sontag [4]. Other general studies of the mathematical structure
of chemical systems include those of Aris [5–7], Bowen [8,9],
Coleman and Gurtin [10], Feinberg [11–13], Horn [14], Horn
and Jackson [15], Krambeck [16], Sellers [17], Shapiro and
Shapley [18], Wei [19], Wei and Prater [20], and so on. See
Refs. [21,22] for more references.
In this report, we present a conservation-dissipation struc-
ture of the chemical reaction equations. It will be seen that
this new structure is different from all those mentioned above.
Consider a reaction system with n
s
species participating in n
r
reversible reactions
ν
′
i 1
S
1
+ ν
′
i 2
S
2
+···+ ν
′
in
s
S
n
s
k
if
⇀
↽
k
ir
ν
′′
i 1
S
1
+ ν
′′
i 2
S
2
+···+ ν
′′
in
s
S
n
s
(1)
for i = 1,2,...,n
r
. Here S
k
is the chemical symbol for the
kth species, the nonnegative integers ν
′
ik
and ν
′′
ik
are the
stoichiometric coefficients of the kth species in the i th reaction,
and k
if
and k
ir
are the respective direct and reverse constants
of the i th reaction. The reversibility means that both k
if
and
k
ir
are positive.
Denote by u
k
the concentration of the kth species S
k
.
According to the law of mass action, the evolution of u
k
=
u
k
(t ) obeys the ordinary differential equations (see, e.g.,
Ref. [22]):
du
k
dt
=
n
r
i =1
(ν
′′
ik
− ν
′
ik
)
k
if
n
s
j =1
u
ν
′
ij
j
− k
ir
n
s
j =1
u
ν
′′
ij
j
. (2)
*
wayong@tsinghua.edu.cn
Set u = (u
1
,u
2
,...,u
n
s
)
T
with the superscript T for the
transpose. The free energy of the reaction system is defined as
F (u) =
n
s
k=1
(u
k
ln u
k
− u
k
ln u
∗
k
− u
k
)
with u
∗
= (u
∗
1
,u
∗
2
,...,u
∗
n
s
)
T
a constant state to be specified
below. Clearly, the gradient of F = F (u) is
∂F
∂u
=
(
ln u
1
− ln u
∗
1
, ln u
2
− ln u
∗
2
,..., ln u
n
s
− ln u
∗
n
s
)
T
(3)
and F (u) is convex with respect to u.
The aforesaid conservation-dissipation structure is given in
the following theorem, which will be proved at the end of this
report.
Theorem. Assume that the system described by Eq. (2)
satisfies the principle of detailed balance: there are n
s
positive
numbers, u
∗
i
> 0, such that
k
if
n
s
j =1
(u
∗
j
)
ν
′
ij
= k
ir
n
s
j =1
(u
∗
j
)
ν
′′
ij
(4)
for i = 1,2,...,n
r
. Then there is a symmetric and nonpositive
definite matrix, S = S (u), defined for u with u
i
> 0(i =
1,2,...,n
s
), such that the kinetic Eq. (2) can be rewritten
as
du
dt
= S (u)
∂F
∂u
(5)
and the null space of S (u) is
span
(
ν
′′
i 1
− ν
′
i 1
,ν
′′
i 2
− ν
′
i 2
,...,ν
′′
in
s
− ν
′
in
s
)
T
,i = 1,2,...,n
r
⊥
,
which is independent of u with u
i
> 0(i = 1,2,...,n
s
).
Remark that the constant state u
∗
= (u
∗
1
,u
∗
2
,...,u
∗
n
s
)
T
needs not be unique and is often not unique. The symmetry
of S (u) is reminiscent of the celebrated Onsager reciprocal
relation [23], but it is different from that due to the dependence
of S (u) on u. The null space of S (u) is just the right null space
of the n
r
× n
s
-stoichiometric matrix
⎛
⎜
⎜
⎜
⎝
ν
′′
11
− ν
′
11
ν
′′
12
− ν
′
12
··· ν
′′
1n
s
− ν
′
1n
s
ν
′′
21
− ν
′
21
ν
′′
22
− ν
′
22
··· ν
′′
2n
s
− ν
′
2n
s
.
.
.
.
.
.
.
.
.
.
.
.
ν
′′
n
r
1
− ν
′
n
r
1
ν
′′
n
r
2
− ν
′
n
r
2
··· ν
′′
n
r
n
s
− ν
′
n
r
n
s
⎞
⎟
⎟
⎟
⎠
and is just the orthogonal complement of the stoichiometric
subspace [13]. Its independence on u has a clear physical
067101-1 1539-3755/2012/86(6)/067101(3) ©2012 American Physical Society