A new measure of
biochemical network
robustness
Prof. Bor-Sen Chen 陳博現
Lab of Control and System Biology
Department of Electrical Engineering
National Tsing Hua University
1
Motivation



The robustness of a biochemical network is defined as
the tolerance of variations in kinetic parameters with
respect to the maintenance of steady state. (phenotype)
Robustness also plays an important role in the failsafe mechanism in the evolutionary process of
biochemical networks. The purposes of this paper are
to use the synergism and saturation system (S-system)
representation to describe a biochemical network and
to develop a robustness measure of a biochemical
network subject to variations in kinetic parameters.
Since most biochemical networks in nature operate
close to the steady state, only the robustness
measurement of a biochemical network at the steady
state is considered.
2
Results



We show that the upper bound of the tolerated
parameter variations is related to the system matrix
of a biochemical network at the steady state.
Using this upper bound, we can calculate the
tolerance of a biochemical network without testing
all possible parametric perturbations and gain much
insight into the robustness of a biochemical network.
We find that a biochemical network with a large
tolerance can also better attenuate the effects of
variations in rate parameters and environments.
3


Compensatory parameter variations and network
redundancy are found to play important roles in the
robustness of biochemical networks.
Finally, four biochemical networks, i.e., a cascaded
biochemical network, the glycolytic-glycogenolytic
pathway in a perfused rat liver, the tricarboxylic acid
(TCA) cycle in Dictyostellium discoideum and the
cAMP oscillation network in bacterial chemotaxis, are
used to illustrate the usefulness of the proposed
robustness measure.
4
METHODS and Results
Model of a biochemical network
The following S-system model has been an efficient model for
describing the dynamic system of a biochemical network in the last
three decades (Savageau1976; Irvine and Savageau, 1985a,b;
Heinrich and Schuster,1996, 1998; Voit, 2000)

n m
X 1  1  X
j 1

n m
X i  i  X
j 1

n m
g1 j
j
j 1
 1  X j 1 j
gi j
j
X n  n  X
nm
gn j
j
h
j 1
nm
 i  X j i j
h
i  1,
, n.
(1)
j 1
nm
 n  X j n j
h
j 1
where X 1 X nm are the metabolites such as substrates, enzymes,
factors and products of a biochemical network, in which X 1 , X 2 X n
denote n dependent variables and X n1 X nm denote the independent
variables.
5
Robustness measure
of a biochemical system
Consider the steady state of a biochemical network in
(1), i.e., inputs and outputs are in balance (Voit, 2000).
n m
i  X
j 1
gi j
j
nm
 i  X j i j
i  1,
h
, n.
(2)
j 1
Assume none of rate constants and variables in (2) is
zero. Taking the logarithm on both sides of (2), we get
n+ m
n+ m
ln  i   g i j ln X j  ln  i   h i j ln X j
j 1
i  1,
, n. (3)
j 1
6
Then, after some rearrangements,
n
 (g
j 1
ij
 hi j )ln X j  ln  i  ln  i 
n+m
 (g
j n1
ij
 hi j )ln X j
i  1,
, n.
Introduce new variables and coefficients as follows
yi  ln X j ,ai j  g i j  hi j
, bi  ln(  i / i )
(4)
The steady state of a biochemical system consists of
n linear equations in n+m variables
7
a11 y1  a12 y2   a1n yn  b1  a1,n1 yn1   a1,nm ynm
a21 y1  a22 y2   a2 n yn  b2  a2,n1 yn1   a2,nm ynm
a31 y1  a32 y2   a3n yn  b3  a3,n1 yn1   a3,nm ynm
(5)
an1 y1  an 2 y2   ann yn  bn  a1,n1 yn1   an,nm ynm
In the above equations, the dependent (unknown)
variables are separated from the independent
(known) variables.
8
Let us denote
 
YD   
 yn 
 
,b   
 bn 

AD  
 an 1

ann 

y1
a11
b1
a1 n
 
, YI   
 yn  m 
yn  1

, AI  
 an,n+1
a1, n +1

an , n+ m 

a1 , n+ m
where AD denotes the system matrix among the
interactions of dependent variables and AI indicates
the interactions between the dependent variables
and independent variables YI.
9
We obtain the steady state equation
AD YD  b  AI YI
(6)
In the nominal parameter case, we assume that the
inverse of AD exists, and then get the steady state of a
biochemical system as follows
YD  AD1 (b  AI YI )
(7)
10
n
AD    i uiT vi
(8)
i 1
 u1T u2T
uiT
 1
 
2


T

un  
i

 0


v1 

0 
v
2
 
 
 
 vi 
 
 
 n 
 vn 
U T V
where  i denotes the ith singular value and
ui , vi  R1n denote the corresponding left and right
singular vectors, respectively.
11
Then with UUT =I, VVT =I and  1   2 
 i 
 n  0
we obtain (Gill et al., 1991; Press et al., 1992)
n
A 
1
D
i 1
1
i
viT ui
 v1T v2T
 V T  1 U
(9)
viT
1

 1
1

 
2


vnT  



0




0
1
i



 u1 
 u 
 2 
 
 
 ui 
 
 
 un 
1 

n 
12
Suppose the parameter perturbation due to mutation
or disease can alter the kinetic properties of the
steady state of a biochemical system in (6) as follows
( A D A D )(YD  YD )  (b  b)  ( AI  AI )YI
(10)
where the parameter perturbations of biochemical
network are defined by
 a11
A  

 an1
ai j
a1n   (g11  h11 )
(g1n  h1n ) 


(g ij  hij )
 

ann  ( g n1  hn1 )
(g nn  hnn ) 
 (g1, n+1  h1,n+1 )
(g 1,n+1  h1,n+1 ) 
 


b    , C  
(g i,n+ j  hi,n+ j )

 bn 
( g n, n+1  hn, n+1 )
(g n, n+ m  hn, n+ m ) 
b1
13
△ AD
denotes the parameter perturbations due to the
kinetic parameter variations △gij and △hij of dependent
variables, △b denotes the parameter perturbations due
to rate constant variations and △AI denotes the
parameter perturbations due to the kinetic parameter
variations of independent variables.
In general, the effect of △b and △AI can not influence
the existence of steady state. Their influences on the
magnitude of YD +△YD can be discussed by sensitivity
1
matrices AD
and AD1 AI , respectively (Savageau,
1969a, 1969b, 1970; Voit, 2000). The robustness is
mainly to check the tolerance for △AD with respect to
the maintenance of steady state of the perturbed
biochemical network.
14
From (10), we obtain
A D ( I  AD1A D )(YD  YD )  (b  b)  ( AI  AI )YI
(11)
If the following robustness condition holds (Noble and
Daniel, 1988; Gill et al., 1991; Weinmann, 1991)
AD1  AD  1
2
(12)
then the singular values of I  AD1 AD are free of zero
and the  I  AD1 AD 1 inverse exists.
15
Therefore, the steady state of the biochemical network
in (11) is uniquely solved as
YD  YD    I  A
1
D
AD

1
AD1   b  b    AI  AI  YI 
(13)
The above analysis says that if the robustness
condition in (12) holds, then the steady state of a
biochemical system is preserved under parameter
variations AD , i.e., YD  YD in (11) has a small
difference YD from the nominal YD in (7) under small
parameter perturbations. However, if condition (12)
does not hold, some singular values I  AD1AD of may
1
1
be zero and the inverse  I  AD AD  may not exist, and
the steady state YD  YD
may cease to exist under
parameter perturbation .
16
The physical meaning of (12) is that if the l2 norm of
the normalized perturbation of kinetic parameters is
less than one or AD1AD is contractive, the effect of the
kinetic parameter perturbation AD can be tolerated
by the biochemical network and the steady state of the
biochemical network is preserved. Therefore, the
inequality in (12) can be used to test the robustness of
the biochemical system under kinetic parameter
perturbation AD
due to mutation or disease.
Equivalently, (12) can be rewritten as a more intuitive
robustness condition as follows (Noble and Daniel, 1988;
Gill et al., 1991; Weinmann, 1991)
AD1AD ADT ADT  I
T
D
i.e., AD A
or
AD ADT  AD ADT
is the upper bound of AD A
T
D
(14)
.
17
If the robustness condition (14) holds, the steady state of the
perturbed biochemical network still exists.
Let us denote the robustness condition in (14) as follows
T
T
R  AD AD
 AD AD
0
If the robustness matrix R is a symmetric positive
definite matrix, the steady state of biochemical network
is still preserved, under parameter perturbation  AD .
This is a simple criterion to check whether the
parameter variation  AD is tolerated or not. It has been
shown that the robustness matrix R is positive definite
if and only if all its eigenvalues are real and strictly
positive (Gill et al., 1991; Noble and Daniel, 1988).
18
Let us denote i  R as the ith eigenvalue of R. Then the
following inequalities
i  R   0
i  1, 2,
,n
(16)
are the robustness conditions for the biochemical
network to tolerate  AD .
From the singular value decomposition in (8), we have
T
AD AD
 U T 2 U
(17)
Therefore, if a parameter variation is specified as
follows
 AD   i uiT vi
i  1, 2,
,n
(18)
19
Then
 1



0




 i 1


AD   AD  U T 
0
V


 i 1


 0


 n 

(19)
1
A


A


Obviously, the inverse D
does not exist
D
under the parameter perturbations in (18).
20
Moreover,
 12



0




2

i

1


T
T
T 
U
R  AD AD  AD AD  U
0


2
 i 1




0


2


n


(20)
which is not positive definite (Gill et al., 1991) and
violates the robustness condition in (15).
21
Obviously, the perturbation with a magnitude  i at
the uiT vi direction will destroy the steady state of a
biochemical network. That is, at the uiT vi direction,
the biochemical network has a weak structure, and a
T
kinetic parameter perturbation  i at the ui vi
direction will disrupt the steady state. If the
T
perturbation un vn is at the direction, then the
biochemical system will tolerate the parameter
perturbation to the value less than  n , i.e., the
biochemical network is most weak at the unT vn
direction, in which a smallest perturbation  n will
destroy the steady state of the biochemical network.
22
Relation between robustness and
sensitivity
From (6), it is easy to know the effects of variations △b and
△YI of rate constant and the environment on the output
variation △YD.
The sensitivity from △b to △YD is given by (Savageau, 1970;
Voit, 2000)
YD / b 
AD1
or
 YD / b   YD / b   
T

T 1
AD AD
(22)
Obviously, from (14), the sensitivity from △b to △YD is inverse
to the robustness of biochemical network, i.e., if the
biochemical network is robust, it is less sensitive to the
variation of rate constant, and vice versa.
23
Similarly, the sensitivity from environment variation △YI to
△YD output variation is given by (Savageau, 1970; Voit,
2000)
YD / YI 
 AD1 AI
or
 YD / YI   YD / YI  
T
AIT


T 1
AD AD
AI
(23)
Obviously, from (14), the sensitivity is also inverse to
robustness and a biochemical network with strong
robustness will be more resistant to the effect of
environmental variation △YI . In the last two decades,
system control theory has addressed the robustness
problem about the effect of the environment on the system
output at some operation points. How to design a feedback
control to minimize the sensitivity AD1 AI 2 to achieve an
optimal robustness design is an important topic in modern
control theory for the last two decades.
24
T
T
From (14), i.e.,  AD AD  AD AD it is seen that there are two
mechanisms for the robustness improvement of biochemical
networks. One mechanism is to increase AD ADT in order to
tolerate large parameter perturbations in  AD . Another
mechanism is to prevent the occurrence of large parameter
perturbations in △AD so that the robustness condition in (14)
can not be easily violated. The redundancy and
compensatory parameter variation (i.e., △gij = △hij so that
△AD=0 in (10)) may be two major sources of this kind of
robustness. Compensatory parameter variations make △AD
small. Network redundancy will be a buffer to prevent
possible violent kinetic perturbations in △AD , which may
violate the robustness condition in (14). Therefore, network
redundancy and compensatory parameter variation may be
two efficient mechanisms to attenuate the parameter
perturbation to prevent violating the robustness condition
in (14), which will be discussed in the following examples.
25
Experimental Simulations

Simulation experiment 1
The role of the cascaded system in Figure 1 has been investigated as an
amplifier for biochemical signals (Savageau, 1976; Voit, 2000).
Cascaded mechanisms are found in diverse areas of biochemistry and
physiology, including hormonal control, gene regulation, immunology,
blood clotting, and visual excitation.
26
The cascaded network can be represented as follows (Voit,
2000)
X 1  10 X 20.1 X 30.05 X 4  5 X10.5
X1  0   0.2
X 2  2 X10.5  1.44 X 20.5
X 2  0   0.5
X 3  3 X 20.5  7.2 X 30.5
X 3  0   0.1
(24)
X 4  0.75
In this case, the system matrix AD of the cascaded
network is as follows
 0.5 0.1 0.05
AD   0.5 0.5
0 
 0
0.5 0.5 
(25)
27
Suppose the network suffers parameter perturbations
due to gene mutation as follows
X 1  10 X 20.086038 X 30.023759 X 4  5 X10.50495
X 1  0   0.2
X 2  2 X10.35939  1.44 X 20.43778
X 2  0   0.5
X 3  3 X 20.22108  7.2 X 30.49843
X 3  0   0.1
X 4  0.75
In this kinetic parameter perturbation case,
 0.00495 0.013962 0.026241

 AD   0.14061 0.06222
0


0
0.27892 0.00157 
(27)
28
and ,
 0.2616  0.2016  0.0211
R   0.2016 0.4764  0.2326  0


  0.0211  0.2326 0.4222 
which can be checked
by the following strictly positive eigenvalues of
R (0.073, 0.3595, 0.7276). From the simulation in Figure
2(b), the steady state is preserved. However, since the
parameters of the cascaded network are perturbed, the
steady state YD+△YD has some changes even when the
characteristics of the steady state are preserved after
some parameter perturbations.
29
Suppose the cascaded biochemical network suffers the
following kinetic parameter variations
X 1  10 X 20.01756 X 30.1155 X 4  5 X 10.45325
X 2  2 X 10.54667  1.44 X 20.2174
X  3 X 0.56494  7.2 X 0.4086
3
2
3
X 4  0.75
X 1 (0)  0.2
X 2 (0)  0.5
X 3 (0)  0.1
In this case, we have
0.04675 0.11756 0.1655
AD  0.04667 0.2826
0 


0.06494 0.0914
 0
(28)
30
and
 0.2191  0.2354  0.0478
R   0.2354
0.418
 0.2684


  0.0478  0.2684 0.4874 
, which is not a
positive definite matrix because its eigenvalues are not
all strictly positive (i.e., -0.0176, 0.3877, 0.7544).
Obviously, the robustness condition in (14) or (15) is
violated and the existence of the steady state of the
cascaded biochemical network is not guaranteed.
From the simulation result in Figure 2(c), the steady
state of the cascaded biochemical network ceases to
exist under this parameter perturbation. These
computational results confirm the claim of our
robustness condition.
31
From the control system point of view (Weinmann,
1991; Qu, 1998), feedback inhibition plays an
important role in the robustness of biochemical
networks. In the cascaded biochemical network in (24),
the kinetic parameters g12 and g13 model the feedback
inhibition. Even with small changes, they have much
influence on the robustness of the cascaded
biochemical network, especially with a sign change.
Suppose the negative feedback of the cascaded
network is perturbed into positive feedback; for
example, g12 changes from –0.1 to 0.3 and g13 from –
0.05 to 0.15,respectively. In this situation, the cascaded
network becomes
32
X 1  10 X 20.3 X 30.15 X 4  5 X 10.5
X 2  2 X 10.5  1.44 X 20.5
X  3 X 0.5  7.2 X 0.5
3
2
X 4  0.75
In this case,
3
 0 .2 0 0 
AD ADT   0 0 0


 0 0 0
X 1 (0)  0.2
X 2 (0)  0.5
X 3 (0)  0.1
, with the
eigenvalues of R as follows (-0.0432, 0.3325, 0.7731),
and the robustness condition (15) is violated. The
dynamic response of this cascaded network is shown
in Figure 2(d), in which the steady state is not
preserved. So we find that adequate negative feedback
inhibition has the robustness property as in control
theory, which contributes significantly to the
robustness of the biochemical system.
33
Furthermore, suppose the negative feedback
inhibition from X2 to X1 in the cascaded network of
Figure 1 consists of duplicated pathways but with
same flux, then the first equation in (24) is modified
as X  10 X X X X  5 X . In this situation, a failure of
one redundancy will lead to X 1  10 X 20.05 X 30.05 X 4  5 X 10.5
,
which will cause a smaller AD ADT than that in the
failure of the feedback loop from X2 to X1 without
redundancy. Therefore, redundancy is a source of
robustness via the mechanism of decreasing AD ADT .
1
0.05
2
0.05
2
0.05
3
4
0.5
1
34
35

Simulation experiment 2
Consider the glycolytic-glycogenolytic pathway in
perfused rat liver (Scrutton and Utter, 1968; Torres,
1994a, b, c).
36
The kinetic properties of the pathway are obtained as
follows (Voit, 2000), which are also shown in Figure 3
X 1  0.077884314 X 40.66 X 6  1.062708258 X 11.53 X 20.59 X 7
X 2  0.585012402 X 10.95 X 20.41 X 50.32 X 70.62 X 100.38
X 1 (0)  0.07
X 2 (0)  0.5
 7.93456  10 4 X 23.97 X 33.06 X 8
X 3  7.93456  10 4 X 23.97 X 33.06 X 8  1.05880847 X 30.3 X 9
X 4  10, X 5  5, X 6  3, X 7  40, X 8  136, X 9  2.86, X 10  4
X 3 (0)  0.16
(29)
where
X 1  glucose - 1 - P (G1P)
X 7  phosphoglu comutase
X 2  glucose - 6 - P (G6P)
X 8  phosphoglu cose isomerase
X 3  fructose - 6 - P (F1P)
X 9  phosphofru ctokinase
X 4  Pi
X 10  glucokinas e
X 5  glucose
X 11  glycogen
X 6  phosphoryl ase 
37
In this case, the system matrix AD is obtained as
0 
 1.53 0.59
AD   0.95  4.38 3.06 


3.97  3.36
 0
The dynamic response of the nominal glycolyticglycogenolytic pathway in (29) is given in Figure 4(a)
and the upper bound of the tolerance is given by
 4.0377
2.3423 
 2.689
AD ADT   4.0377 29.4505  27.6702


 2.3423  27.6702 27.0505 
That is, if the parameter perturbation measure AD ADT
is less than AD ADT , i.e. the robustness matrix R > 0, the
characteristics of the steady state will be preserved.
38
Suppose the glycolytic-glycogenolytic pathway suffers
from a kinetic perturbation as follows
X 1  0.077884314 X 40.66 X 6  1.062708258 X 11.5007 X 20.64825X 7
X 2  0.585012402 X 11.0113 X 20.6152 X 50.32 X 70.62 X 100.38
X 1 (0)  0.07
X 2 (0)  0.5
 7.93456  104 X 23.8804 X 33.0351 X 8
X 3  7.93456  10 4 X 23.8804 X 33.0351 X 8  1.05880847 X 30.3 X 9
X 4  10, X 5  5, X 6  3, X 7  40, X 8  136, X 9  2.86, X 10  4
X 3 (0)  0.16 (30)
in which the parameter perturbation is of the following
form
0 
0.0293 0.05825
AD   0.613  0.1156  0.0249


 0.00896 0.0249 
 0
39
In this perturbed case,
 0.0043  0.0049  0.0005
AD ADT   0.0049 0.0177
0.0004  and


0.0007 
  0.0005 0.0004
 4.0328
2.3428 
 2.6847
R   4.0328 29.4328  27.6706 ,


 2.3428  27.6706 27.0498 
which is a positive definite
matrix with its eigenvalues all positive (i.e., 0.0204,
2.8261, 56.3208). From our computational result and
the perturbed dynamic response shown in Figure 4(b),
the robustness of steady state is preserved.
40
Suppose another parameter perturbation occurs such
that the glycolytic-glycogenolytic pathway is
perturbed as
X 1  0.077884314 X 40.66 X 6  1.062708258 X 11.3581X 20.69463X 7
X 2  0.585012402 X 10.80419 X 20.18704 X 50.56977 X 70.59214 X 100.48904
 7.93456  104 X 23.7771 X 33.0793 X 8
X 3  7.93456  10 X
4
4.0071
2
X
3.0793
3
X 8  1.05880847 X
X 1 (0)  0.07
X 2 (0)  0.5
(31)
0.3
3
X9
X 4  10, X 5  5, X 6  3, X 7  40, X 8  136, X 9  2.86, X 10  4
X 3 (0)  0.16
In this case, we have
0.10463
0 
0.0405 0.0191 0.0039
 0.1719
AD   0.14581 0.42225 0.0193  AD ADT  0.0191 0.1999 0.0153


0.0039 0.0153 0.0017 .
0
0.0371  0.0193 ,

41
 4.0568
2.3384 
 2.6845
R   4.0568 29.2506  27.6855


 2.3384  27.6855 27.0488 
is not positive definite,
because its eigenvlaues are not all positive (i.e., -0.0915,
2.7967, 56.2427). Therefore, the robustness condition is
violated. From the dynamic response shown in Figure 4(c),
we can see that the steady state of the biochemical network
ceases to exist.
In order to confirm the compensatory parameter variations
gall
hsuch
(i.e.,
for
that
) in
2,
ij  i,j
ij
ARemark
D 0
we let
△g22 = △h22 =5
42
43

Simulation experiment 3
The tricarboxylic acid (TCA) cycle in Dictyostelium, a soil-living
amoeba, produces ATP very efficiently while decomposing pyruvate to
water and CO2 via acetyl-CoA. Under a nutrient-rich condition, the
cycle is fed by ingested proteins that are broken down into amino acids.
44
X 1  Oxalacetat a 1 (OAA 1)
X 16  Succinate dehydrogen ase
X 2  Oxalacetat e 2(OAA 2)
X 17  Fumarase
X 3  Acetyl - CoA (ACO)
X 18  Malate dehydrogen ase
X 4  Isocitrate (ISOC)
X 19  Malic enzyme
X 5  Pyruvate (PYR)
X 20  Ala  Pyr
X 6  Glutamate( GLU)
X 21  Pyruvate dehydrogen ase complex
X 7  Aspartat (ASP)
X 22  Oaa 2  Asp
X 8  Alaninee (ALA)
X 23  Asp  Oaa 2
X 9  Citrate 1 (CIT 1)
X 24  Citrate synthetase
X 10    Ketoglutar ate (2KG1)
X 25  Aconitase
X 11  Succinate (SUC)
X 26  Isocitrate dehydrogen ase
X 12  Fumarate (FUM)
X 27  Glu  Suc
X 13  Malate (MAL 1)
X 28  Aspartate transamin ase
X 14  Glutamate dehydrogen ase
X 29  Alanine transamin ase
X 15    Ketoglutar ate dehydrogen ase complex
45
The S-system model is shown as follows (Voit, 2000)
0.108
0.0848 0.599
0.181
0.915 0.0847
X 1  0.8282 X 10.038 X 60.0204 X 70.106 X 100.114 X 130.7 X 180.807 X 28
X 31
X 46 X 48
 1.3423 X 1 X 30
X 33
0.0848 0.915
0.0848 0.915
0.0341
X 2  1.3401X 10.915 X 70.0848 X 23
X 30  17.166 X 20.706 X 30.0716 X 22
X 24 X 47
X  0.3231X 0.405 X 0.156 X 0.427 X 0.573 X 0.422 X 0.405 X 0.418  9.6952 X 0.376 X 0.489 X 0.554 X 0.446 X 0.00206
3
3
5
21
35
46
47
48
2
3
24
41
47
X 4  X 9 X 25  0.152 X
X 26 X
X
0.465
0.756
0.244
0.748
X 5  1.875 X 70.0274 X 80.465 X 130.336 X 190.535 X 20
 0.01923 X 30.717 X 50.413 X 60.306 X 80.29 X 100.0883 X 21
X 29
X 46
0.958
4
0.0348
46
0.862
48
0.718
0.741
 X 47
X 48
0.813 0.276
0.6413
X 6  2.459 X 10.00921X 60.0154 X 70.0162 X 100.086 X 110.276 X 28
X 32 X 39
 1.1528 X 50.0.0963 X 61.01 X 80.204 X 100.062 X 140.0518
0.277
0.171 0.5
0.0222
0.0191
 X 27
X 29
X 45 X 46
X 48
0.129
0.129
0.741
0.129
0.165 0.129
0.577
X 7  2.1167 X 10.129 X 20.129 X 22
X 33
X 34
 3.4893 X 10.0187 X 60.0311X 70.868 X 100.174 X 23
X 28
X 31 X 40
0.197
0.803
0.375
0.625
X 8  0.5724 X 50.111 X 60.247 X 80.234 X 100.0713 X 29
X 38
 1.9369 X 8 X 20
X 44
X  16.242 X 0.679 X 0.0782 X X 0.0372  X X
9
2
X 10  0.156 X
0.724
4
3
X
0.106
5
24
X
0.259
6
47
X
9
0.223
8
X
0.0679
10
25
0.756
0.188
0.0506
0.672
X 140.0568 X 26
X 29
X 46
X 48
 0.8063 X 10.0101X 60.0168
0.0891 0.882
0.879
0.881
 X 70.0177 X 100.99 X 110.879 X 150.911 X 28
X 46 X 47
X 48
0.166
0.335 0.483 0.481 0.483
0.166
0.261
X 11  2.0031X 60.166 X 100.491 X 110.481 X 150.499 X 27
X 36
X 46 X 47 X 48  2.4373 X 110.495 X 120.00542 X 160.574 X 32
X 42
X  1.271X 0.106 X 0.00836 X 0.885 X 0.115  9.1694 X 1.89 X 1.24 X 0.911 X 0.0893
12
11
12
16
37
12
13
17
43
1.98
0.139
X 13  8.289 X 12
X 131.36 X 17  0.9387 X 10.0197 X 70.0196 X 130.775 X 180.618 X 190.382 X 46X 0.458 X 48
X 1 (0)  0.003, X 2 (0)  0.003, X 3 (0)  0.065, X 4 (0)  0.01, X 5 (0)  0.32, X 6 (0)  6.63, X 7 (0)  2.035
X 8 (0)  5.313, X 9 (0)  0.0275, X 10 (0)  0.01, X 11 (0)  0.9, X 12 (0)  0.04, X 13 (0)  0.24
X 14  0.977, X 15  7610, X 16  3.15, X 17  25.7, X 18  77.8, X 19  3.08, X 20  0.196
X 21  258, X 22  74, X 23  0.1, X 24  8.24, X 25  80, X 26  271, X 27  0.133, X 28  9.95
X 29  2.67, X 30  800, X 31  0.1, X 32  1, X 33  74, X 34  1.06, X 35  2.07, X 36  1.62
X 37  0.36, X 38  2.03, X 39  1.86, X 40  0.446, X 41  27.2, X 42  1.57, X 43  7, X 44  0.326
X 45  0.24, X 46  0.072, X 47  0.1, X 48  0.18
46
The TCA network in the above equation is shown in Figure
5, and the dynamic response is shown in Figure 6(a). The
system matrix AD is obtained as
0
0
0
0
  1.04
 0.915  0.706  0.0716
0
0

 0.376  0.894
0
0.156
 0

0
0
 0.958
0
 0
 0
0
0.717
0
 0.41

0
0
0
 0.10
  0.01
AD   0.148 0.129
0
0
0

0
0
0
0.111
 0
 0
0.379
0.0782
0
0

0
0
0.724 0.106
 0.01
 0
0
0
0
0

0
0
0
0
 0
 0.02
0
0
0
0

 0.02
0.7 
0
0.085
0
0
0
0
0
0 

0
0
0
0
0
0
0
0 

0
0
0
1
0
0
0
0 
 0.31 0.027 0.755
0 0.089
0
0
0.336 

 1.03 0.016 0.204
0 0.148 0.276
0
0 
0.031  0.87
0
0  0.17
0
0
0 

0.247
0
 1.234 0  0.07
0
0
0 
0
0
0
1
0
0
0
0 

0.276  0.02  0.223 0  1.06 0.879
0
0 
0.166
0
0
0 0.491  0.976 0.005
0 

0
0
0
0
0
0.106  1.9 1.24 
0
 0.02
0
0
0
0
1.98  2.14
0.106
0
0
0.114
0
0
47
From AD , we can calculate the upper bound of the
T
perturbation tolerance AD AD . That means when the
network is perturbed by AD such that the robustness
condition (14) is violated, and the steady state of the
biochemical network may not exist. Suppose there is a
perturbation AD as follows
M
AD  102  
P
N
Q 
(34)
48
where
0
0 0 0 2.67 0.0292
2.09
M  2.19 4.12 0 0 0
0
0.0305


 2.78 0 0 5.8
0
0 
 0
,
0
1.57 1.57
0
0
0 
 0
 0
0
0
0
6.11  1.98  0.02 


0
0
0
 3.44 1.11
0.01 
 0.87


0
0
0
0.492 0.0054
0.039 0.726
 0
0
0
0
4.13  1.33
0 
P
,
2.16
0
0
0
0
0 
 0
 1.19
0
1.79 1.79  4.72 1.53
0.017 


0
0
0
0
2.07
0 
 0
 0
0
0
0
0
0
0 


0
0
0
0
0
0.017 
 1.24
1.35
0
 0
 0.571
0
 4.18

0
2.35
  0.32

0
1.04
 0
 0.39
0
 2.82
Q
1.49
0
 0
  0.44
0
3.23

0
4.38
 0
 0
0
0

0
0
 0
0 
0
0
 1.05

1.57
0
0 

0
0
0 
0
0
0 
.
0
0
0 
2.15
0
0 

2.93 1.03
0 
2.39 0.843 0.9 

0
0.79
0.84 
0
0 0 5.65 0 0 1.42
N  0 0 0 0 0 0 


0 0 0 0 0 0 
0
49
In the perturbed case, the robustness matrix R is
not a positive definite matrix with its eigenvalues
(-0.006, 0.0218, 0.0955, 0.3666, 0.5360, 0.8629,
0.9511, 1.3141, 2.2116, 2.4675, 2.9605, 3.9098,
13.7447) and the dynamic response of the
perturbed TCA cycle is shown in Figure 6(b).
Obviously, the steady state of the perturbed TCA
cycle ceases to exist.
50
51

Simulation experiment 4
Periodic responses are often encountered in organisms ranging from
bacteria to mammals. A periodic oscillation can be considered as one
kind of steady state phenomenon from the system point of view.
52
A periodic network in Figure 7 is modeled as follows to
produce the spontaneous oscillations in cAMP observed
during the early development of D. discoideum (Laub and
Loomis, 1998) and account for the synchronization of the
cells necessary for chemotaxis (Yi et al., 2000; Ma and
Iglesia, 2002).
X  ACA
X  internal cAMP
1
5
X 2  PKA
X 6  external cAMP
X 3  ERK2
X 7  CAR1
X 4  REG A
X 1  2 X 7  0.9 X 1 X 2
X 2  2.5 X 5  1.5 X 2
X  0.6 X  0.8 X X
3
7
2
X 1 ( 0 )  1 .5
X 2 ( 0)  1
3
X 3 ( 0 )  1 .5
X 4  1  1.3 X 3 X 4
X 5  0.3 X 1  0.8 X 4 X 5
X  0.7 X  4.9 X
X 4 ( 0 )  2.5
X 7  23 X 6  4.5 X 7
X 7 ( 0 )  1 .5
6
1
6
X 5 ( 0)  1
X 6 ( 0)  0
53
From (14) or (15), we can find that if the perturbation
measure AD ADT
is large such that the robustness
condition R  A A  A A  0 is violated, the robustness of
the steady state may not be preserved. Suppose the
parameter perturbation as follows
D
T
D
0
0

0

AD  0
0

0
0
D
T
D
0 0
0
0
0 0
0
0
0 0
0
0
0 0
0
0
0 0  0.1 0
0 0
0
0
0 0
0
0
0
0
0

0 0.1

0
0
0
0

0
0
0.1 0 
0
In this case, all eigenvalues of R are greater than 0, and
the oscillation still exists as shown in Figure 8(b).
54
0
0
0
0
0
0.1 
 0
 0
0.2
0
0
0.1 0
0 


0.2  0.2
0
0
0
0.1 
 0


AD   0
0
0.2  0.2 0
0
0 
  0.6 0
0
0.1 0.1 0
0 


0
.
1
0
0
0
0
0
.
02
0


 0
0
0
0
0
0.1 0.02
the robustness condition R > 0 is violated, i.e., the
smallest eigenvalue of R is less than zero. The
oscillation disappears and the steady state of the
biochemical network ceases to exist as shown in Figure
8(c). Obviously, the proposed robustness measure is an
important indicator for the robustness of biochemical
networks under parameter perturbations.
55
56
Discussion

Mutations and diseases are unavoidable in biosystems,
and they can permanently alter the kinetic properties
and capabilities of a biochemical network. Such
alterations are reflected in dynamic models as
numerical changes in one or some of the system
parameters. The effects of permanent changes in
parameters of a biochemical network have been
examined in this paper using robustness analysis. We
found that the biochemical network can tolerate the
parameter variations with respect to existence of
steady state if the robustness condition in (14) or (15)
is not violated. Obviously, the robustness analysis is an
important research topic for biochemical networks.
57

This study proposed an efficient method for
measuring the robustness of biochemical networks
using the S-system model. The proposed robustness
measure scheme can tell how much parameter
variation a biochemical network can tolerate in order
to preserve the steady state of the network. We found
that if the parameter variation AD ADT is less than the
upper bound AD ADT or the robustness matrix R is
positive definite, the network is robust at the steady
state. The proposed robustness measure scheme gives
an efficient method for computing the upper bound to
tolerate the parameter variations of biochemical
networks.
58
• As seen in the simulation examples, adequate negative
feedback and redundancy may contribute significantly to
the robustness of biochemical networks. We also found
that in the compensatory parameter variation case, the
biochemical network could tolerate very large parameter
perturbation only with a small change in the transient
period. It is also a good way to preserve the steady state for
biochemical networks.
• In real biochemical networks, they must have enough
robustness to tolerate the parameter and environmental
variations or else they can not respond immediately to
small but persistent parameter and environmental
perturbations. Therefore, if a model of a biochemical
network has a small robustness measure, i.e., a model
lacking robustness, it is often a sign of structural
inadequacies of the model and provides a tool for
identification of inconsistencies in data. Thus, the proposed
robustness measure scheme is also a good way to validate
the models of biochemical networks.
59

Conventionally, sensitivity analyses are used to assess
the robustness of biochemical networks. That is, the
sensitivity of the steady state concentration of a
metabolite with respect to a change in a parameter is
used to indicate the robustness of biochemical networks.
Because conventional sensitivity analyses in biochemical
systems pay more attention to the effects of b and YI
on YD but not AD on YD , it is not easy to use parameter
sensitivity to assess quantitatively whether the steady
state of a biochemical network is preserved or not.
Furthermore, from (22) and (23) we found that if the
biochemical network is more robust, it is also less
sensitive to the variations of rate constant and
environment, i.e., b and YI .
60


The robustness measure is confirmed by four metabolite
networks with several numerical simulations. From the
simulation examples, we found that if the parameter
variation measure AD ADT violates the upper bound of the
tolerance AD ADT , then the steady state of the biochemical
network may cease to exist. Therefore, in the evolutionary
process of the biochemical network, parameter
perturbations due to DNA mutation in genes should be
less than AD ADT
or the steady state of the biochemical
network may cease to exist and may be eliminated by
natural selection.
It has been claimed (Nijhout, 2002) that the nonlinearity is
the nature of robustness. In general, it is not true. The
nonlinearity is only the nature of biochemical network.
Robustness of biochemical networks should be more
related to increasing the upper bound of the tolerance
(i.e., AD ADT ) or attenuating the perturbation (i.e., A D ADT ).
61

From the above experimental simulation examples, we also
found that an adequate negative feedback pathway may
increase AD ADT and redundancy as well as compensatory
parameter perturbation can attenuate the parameter
variation AD ADT
, i.e., adequate negative feedback,
compensatory parameter perturbation and redundancy
may be three mechanisms providing the robustness of a
biochemical network, which is consistent with the recent
robustness results of system theory (Weinmann, 1991; Qu,
1998) and system biology (Yi et al., 2000; Hood et al., 2004).
In this situation, we claim that adequate negative feedback,
compensatory parameter perturbation and redundancy
may be the nature of biochemical network robustness.
According to the robustness analysis, understanding the
diseases by studying the biochemical pathways or networks
is helpful for the design of potential drugs from the
viewpoint of improving the robustness of biochemical
pathways or networks (Hood et al., 2004).
62