BIOINFORMATICS
ORIGINAL PAPER
Vol. 21 no. 11 2005, pages 2698–2705
doi:10.1093/bioinformatics/bti348
Systems biology
A new measure of the robustness of biochemical networks
Bor-Sen Chen1,∗ , Yu-Chao Wang1, Wei-Sheng Wu1 and Wen-Hsiung Li2,3,4
1 Lab
of Control and Systems Biology, Department of Electrical Engineering, National Tsing Hua University,
Hsinchu, 300, Taiwan, ROC, 2 Department of Evolution and Ecology, University of Chicago, 5801 South Ellis,
Chicago, IL 60637, USA, 3 Academia Sinica, Taipei, Taiwan, ROC and 4 Genomics Research Center, Academia
Sinica, Taipei, Taiwan, ROC
Received on November 11, 2004; revised on January 25, 2005; accepted on February 20, 2005
Advance Access publication February 24, 2005
ABSTRACT
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. Robustness also plays an important role
in the fail-safe 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, we
consider only the robustness measurement of a biochemical network
at the steady state.
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 (robustness) of a biochemical network without testing many
parametric perturbations. We find that a biochemical network with
a large tolerance can also better attenuate the effects of variations
in rate parameters and environments. Compensatory parameter variations and network redundancy are found to be important mechanisms
for the robustness of biochemical networks. Finally, four biochemical
networks, such as a cascaded biochemical network, the glycolytic–
glycogenolytic pathway in a perfused rat liver, the tricarboxylic acid
cycle in Dictyostelium discoideum and the cAMP oscillation network
in bacterial chemotaxis, are used to illustrate the usefulness of the
proposed robustness measure.
Supplementary information: http://www.ee.nthu.edu.tw/∼bschen/
robustness_bio-networks/
Contact: [email protected]
INTRODUCTION
In recent years, the robustness of biochemical systems has attracted much attention (Barkai and Leibler, 1997; Alon et al., 1999;
Von Dassow et al., 2000; Wagner, 2000; Yi et al., 2000; Meir et al.,
2002; Morohashi et al., 2002). Robustness in metabolism, cell cycle
and intercellular signaling pathways has been widely investigated.
These biochemical networks must operate reliably under vastly different environmental conditions that can cause changes in the internal
‘parameters’ of a network.
∗ To
whom correspondence should be addressed.
2698
A change in an internal parameter can be caused by a mutation or a
disease and might result in an altered enzyme activity. In vivo, small
changes in parameter values of a biochemical system are the rule.
They occur continually and propagate throughout the network. The
organism may respond to these changes by returning to the original
physiological steady state or by assuming a slightly changed steady
state (Voit, 2000).
We say a biochemical network is robust if its steady state is preserved despite the changes of its parameter values. Robustness is
defined as a measure of the tolerance for parameter perturbations
with the steady state of the biochemical network preserved. Sensitivity analyses are conventionally employed to assess the robustness
of biochemical networks (Savageau, 1971). The sensitivity of the
steady-state concentration of a metabolite with respect to a change in
a parameter is considered as the inverse of robustness of biochemical
networks (Voit, 2000). For example, Heinrich and Schuster (1996,
1998) and Fell (1997) discussed a number of types of sensitivities in
the analysis of the regulation and control of metabolic and cellular
systems. However, because sensitivity says little about the loss of
stability of a steady state, it is difficult to use parameter sensitivity
to assess the robustness of a biochemical network. Therefore, we
propose a new measure of robustness for assessing the tolerance of
parameter perturbations in a biochemical network with respect to the
maintenance of steady state.
Robustness has been widely investigated in linear control systems (Weinmann, 1991) but it is still very difficult to measure the
robustness of complex non-linear systems (Qu, 1998; Carlson and
Doyle, 2000). In general, biochemical systems are complex nonlinear systems, hence it is difficult to evaluate their robustness by the
conventional control system methods (Weinmann, 1991; Qu, 1998;
Carlson and Doyle, 2000). However, many important characteristics of a biochemical system at or close to a steady state can be
analyzed using simple means that only require linear algebra and
calculus (Weinmann, 1991; Voit, 2000). This is of great importance
because most biochemical systems in nature operate close to a steady
state.
In this study, we develop a robustness measure of biochemical networks subject to kinetic parameter variations at the steady state. We
use the S-system representation, which is a well-studied approach
in modeling biochemical systems (Savageau, 1976; Voit, 2000). It
is a type of power-law formalism that uses non-linear differential
equations in which the component processes are characterized by
power-law functions. The structure of the S-system is rich enough
© The Author 2005. Published by Oxford University Press. All rights reserved. For Permissions, please email: [email protected]
A new measure of robustness
to capture many relevant biological dynamics and the S-system
allows customizing analytical and computational methods (Voit,
1991, 2000), especially with respect to steady-state evaluation, control analysis and sensitivity analysis. Considering logarithm on the
state variables makes the steady state of the S-system equivalent to
an algebraic linear system (Voit, 2000), facilitating the robustness
analysis of a biochemical network.
Using this approach, we show below that the tolerance of a biochemical network can be measured from the system matrix at the
steady state without testing many parametric variations. We also
discuss the relationship between the robustness of a biochemical
network and the sensitivity of variations of rate constants and environments. We point out two mechanisms that can improve the robustness of a biochemical network. One is to increase the connectivity
of the system matrix in order to tolerate large parameter perturbations. The other is to prevent violent parameter perturbations,
which may make the steady state unstable. Network redundancy
and compensatory parameter variation may belong to this kind of
mechanism. Finally, we use a cascaded biochemical network, the
glycolytic–glycogenetic pathway, the tricarboxylic acid (TCA) cycle
of Dictyostelium discoideum and the cAMP oscillation network as
four examples to illustrate the usefulness of our robustness measure.
METHODS AND RESULTS
Mathematical notations
For convenience, some mathematical notations are given below. For a
vector
X = [X1 , X2 , . . . , Xn ]T , the l2 norm for X is defined by X2 =
X12 + · · · + Xn2 . We say X ∈ l2 , if X2 < ∞. For y = AX, the l2 induced
matrix norm is defined as A2 = supX∈l2 y2 /X2 , i.e. the gain from X
to y. It has been shown that A2 = σmax (A) = maxi , λi (AT A), where
T
σmax (A) denotes the largest singular value of A and λi (A A) denotes the i-th
eigenvalue of AT A (Gill et al., 1991; Weinmann, 1991). A2 < 1 if and
only if AAT < I , i.e. A is contractive, where I is the identity matrix (Gill
et al., 1991; Weinmann, 1991).
In general, a biochemical network is a collection of enzymatic reactions
that serve to process metabolites within the cell and to convert intercellular
metabolites into intracellular metabolites and vice versa. For a systematic
description of a biochemical network, dynamic mass balances are achieved
for each metabolite in the network. In biochemistry, one often measures rates
of reactions or fluxes, and the rates correspond directly to changes in concentrations. When we express such changes in terms of concentrations of
substrates, enzymes, factors or products, we can write the relationship in
terms of differential equations. Suppose that the biochemical network has
both dependent and independent variables. The following S-system representation has been an efficient model for describing the dynamic system of a
biochemical network in the last three decades (Savageau, 1976; Irvine and
Savageau, 1985a,b; Heinrich and Schuster, 1996, 1998; Voit, 2000)
Ẋ1 = α1
g
Xj 1j
− β1
j =1
n+m
n+m
j =1
g
Xj ij −βi
j =1
n+m
h
Xj ij
j =1
..
.
Ẋn = αn
n+m
j =1
g
Xj nj −βn
n+m
j =1
Consider the steady state of the biochemical network in (1), i.e. inputs and
outputs are balanced (Voit, 2000).
αi
n+m
g
Xj ij = βi
j =1
n+m
h
Xj ij
i = 1, . . . , n.
(2)
j =1
h
Xj nj
i = 1, . . . , n,
ln αi +
n+m
gij ln Xj = ln βi +
j =1
n+m
hij ln Xj
i = 1, . . . , n.
(3)
j =1
Then, after some rearrangements,
n
(gij − hij ) ln Xj = ln βi − ln αi −
j =1
n+m
(gij − hij ) ln Xj
j =n+1
i = 1, . . . , n.
Introduce new variables and coefficients as follows:
yj = ln Xj ,
aij = gij − hij ,
bj = ln(βi /αi ).
(4)
The steady state of a biochemical system consists of n linear equations in
n + m variables as follows:
h
Xj 1j
..
.
Ẋi = αi
Robustness measure of a biochemical system
Assume that none of the rate constants and variables in (2) is zero. Taking
the logarithm on both sides of (2), we obtain
Model of a biochemical network
n+m
where X1 , . . . , Xn+m are the metabolites, such as substrates, enzymes, factors
or products of a biochemical network, in which X1 , X2 , . . . , Xn denote
n-dependent variables and Xn+1 , . . . , Xn+m denote the independent variables. In biochemical networks, intermediate metabolites and products are
dependent variables, whereas substrates and enzymes are independent variables. Ẋi , the rate of change in Xi , is equal to the difference between two
terms, one for production or accumulation and the other for degradation or
clearance. Each term is the product of the rate constant, αi or βi , which is
positive or zero, and all dependent and independent variables that affect directly the production or degradation, respectively. Each variable Xj is raised
to the power of a kinetic parameter, gij or hij , which represent an activating effect of Xj to Xi when their values are positive and an inhibitive
effect when their values are negative. The non-linear Equation (1) describes
the dynamic evolution among dependent variables. How to construct the
S-system representation of a biochemical network and how to estimate its
parameters from experimental data can be found in Voit (2000) and references
therein.
The system of equations in (1) is called an S-system, where S refers to
synergism and saturation of the investigated biochemical system (Savageau,
1969a,b, 1970; Heinrich and Schuster, 1996; Voit, 2000). Synergism and
saturation are two fundamental properties of biochemical and biological
systems. Note that all the S-system equations have the same mathematical
form but differ in their parameters (Voit, 2000).
In general, it is difficult to directly study the robustness of the nonlinear system in (1). Fortunately, many important characteristics of a
system at or close to the steady state can be analyzed using simple
algebraic methods. Since most biochemical systems in nature operate
close to the steady state in which inputs and outputs are almost balanced, we shall focus on the robustness of biochemical systems at steady
state.
(1)
a11 y1 + a12 y2 + · · · + a1n yn = b1 − a1,n+1 yn+1 − · · · − a1,n+m yn+m
a21 y1 + a22 y2 + · · · + a2n 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
..
.
an1 y1 + an2 y2 + · · · + ann yn = bn − an,n+1 yn+1 − · · · − an,n+m yn+m .
(5)
In the above set of equations, the dependent (unknown) variables are separated
from the independent (known) variables.
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B.-S.Chen et al.
Let us denote
tolerance of a biochemical network to parameter perturbations with respect
to the maintenance of steady state.
Suppose that the parameter perturbation owing to mutation or disease can
alter the kinetic properties of the steady state of a biochemical system in (6)
as follows:
 

 

y1
yn+1
b1
.
 . 
.


 

YD = 
 ..  , b =  ..  , YI =  .. 
yn
bn
yn+m




a11 · · · a1n
a1,n+1 · · · a1,n+m
 .
 .
.. 
.. 
..
..



AD = 
.
.
 ..
.  , AI =  ..
. ,
an1 · · · ann
an,n+1 · · · an,n+m
(AD + AD )(YD + YD ) = (b + b) − (AI + AI )YI ,
where the parameter perturbations of the biochemical network are defined by

where AD denotes the system matrix of the interactions between dependent
variables and AI indicates the interactions between the dependent variables
YD and independent variables YI . 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 obtain the steady state of a biochemical system as follows:
YD = A−1
D (b − AI YI )
(7)
Remark 1. Let us decompose AD by the following singular value decomposition (Press et al., 1992; Gill et al., 1991)
AD =
n
σi uTi vi
i=1
 
v1
  v2 
 
 . 
 . 
 . 
 
  vi 
 
 . 
  .. 
vn
σn

σ1




T T
T
T 
= u1 , u2 , . . . , ui , . . . , un 




0
σ2
..
.
σi
..
0
.
= U T V ,
(8)
where σi denotes the i-th singular value and ui , vi ∈ R 1 × n denote the corresponding left and right singular vectors, respectively. Then with U U T =
I , V V T = I and σ1 ≥ σ2 ≥ · · · ≥ σi≥ · · · ≥ σn ≥ 0, we obtain (Gill et al.,
1991; Press et al., 1992)
A−1
D
n
1 T
=
v ui
σi i
i=1
1
 σ1







T T
= v1 , v2 , . . . , viT , . . . , vnT 








= V T −1 U .


1
σ2
0
..
.
1
σi
0
..
.
u1

 
 u 
  2
 
 
 . 
  .. 
 
 
 
  ui 
 
 
 . 
 . 
 . 
1  
un
σn
(9)
If all singular values are non-zero, the inverse of AD exists. However, if
at least one singular value of AD is perturbed to zero, the inverse of these
singular values will become infinite and the inverse of AD will cease to exist.
In this study, the robustness of a biochemical network lies in how to avoid
any parameter perturbation that will lead to a zero singular value(s).
Conventionally, parameter sensitivity is introduced to measure the change
of the steady state in response to parameter variations. The drawback of
parameter sensitivity analyses lies in that it is not simple to discuss directly the
2700
(10)

a11
···
a1n
 .

.
AD = 
. 
 ..
aij
. 
···
ann
an1


···
(g1n − h1n )
(g11 − h11 )


.
.

=
.
.


.
.
(gij − hij )
···
(gnn − hnn )
(gn1 − hn1 )


b1
 . 

b = 
 ..  ,
bn


(g1,n+1 − h1,n+1 )
···
(g1,n+m − h1,n+m )


.
.
,
AI = 
.
.


.
.
(gi,n+j − hi,n+j )
···
(gn,n+m − hn,n+m )
(gn,n+1 − hn,n+1 )
where AD denotes the parameter perturbations owing to the kinetic parameter variations gij and hij of dependent variables, b denotes the
parameter perturbations owing to rate constant variations and AI denotes
the parameter perturbations owing to the kinetic parameter variations of
independent variables.
In general, the effect of b and AI on the magnitude of YD + YD can
−1
be discussed by sensitivity matrices A−1
D and AD AI , respectively (Savageau,
1969a,b, 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. The relationship between sensitivity and robustness
will be discussed in the sequel.
From (10), we derive
AD (I + A−1
D AD ) (YD + YD ) = (b + b) − (AI + AI )YI .
(11)
If the following robustness condition holds (Noble and Daniel, 1988; Gill
et al., 1991; Weinmann, 1991)
−1
(12)
AD AD < 1,
2
then the singular values of I + A−1
D AD are free of zero and the inverse (I +
−1 exists. Therefore, the steady state of the biochemical network
A−1
A
)
D
D
in (11) is uniquely solved as
−1 −1
(YD + YD ) = (I + A−1
D AD ) AD ((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 the condition (12)
does not hold, some singular values of I + A−1
D AD may be zero and the
−1 may not exist, and the steady state Y + Y may
inverse (I + A−1
D
D
D AD )
cease to exist under parameter perturbation AD .
The physical meaning of (12) is that if the l2 norm of the normalized
perturbation of kinetic parameters is less than one or A−1
D AD is contractive,
the effect of the kinetic parameter perturbation AD can be tolerated by the
biochemical network. Therefore, the inequality in (12) can be used to test the
robustness of the biochemical system under the kinetic parameter perturbation
AD owing 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):
T −T
A−1
D AD AD AD < I
or
AD ATD < AD ATD ,
(14)
A new measure of robustness
i.e. AD ATD is the upper bound of AD ATD . 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:
R = AD ATD − AD ATD > 0.
(15)
If the robustness matrix R is a symmetric positive definite matrix, the steady
state of a biochemical network is still preserved, under the 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).
Let us denote λi (R) as the i-th eigenvalue of R. Then, the following
inequalities
λi (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
AD ATD = U T 2 U .
(17)
Therefore, if a parameter variation is specified as follows
AD = −σi uTi vi
i = 1, 2, . . . , n,
(18)
then






T
AD + AD = U 






σ1
..
.






V.





0
σi−1
0
σi+1
..
0
.
(19)
σn .
Obviously, the inverse (AD + AD )−1 does not exist under the parameter
perturbations in Equation (18).
Moreover,
R = AD ATD − AD ATD
 2
σ1

..

.


2
σi−1


= UT 





0







 U,





0
0
2
σi+1
..
.
(20)
σn2
which is not positive definite (Gill et al., 1991) and violates the robustness
condition in Equation (15).
Obviously, a perturbation with a magnitude σi in the uTi vi direction will
destroy the steady state of a biochemical network. Since the biochemical
network is most weak in the uTn vn direction, a perturbation in this direction
with a magnitude larger than the smallest singular value σn will destroy the
steady state of the biochemical network.
Remark 2. In the classic system control theory, robustness has been linked
to the distances between the eigenvalues λi (AD ) and the j ω-axis in the complex plane, i.e. the tolerance of parameter perturbations (or robustness) is
equal to the smallest real part of eigenvalues λi (AD ). Actually, the tolerance
of parameter perturbations could also be determined by checking the size
of the last term in the Routh–Hurwitz array of the characteristic polynomial
det(λI − AD − AD ), i.e. the robustness means the tolerance of parameter
perturbation AD that should not lead to any sign change in the last term of
the Routh–Hurwitz array (Weinmann, 1991; Voit, 2000). However, the above
classical control analysis of robustness can only be checked case by case,
hence it is difficult to measure robustness systematically. From Equations
(15) and (18)–(20), it is seen that the eigenvalues of R are related to the absolute values of the real parts of the eigenvalues of AD . The eigenvalues of R
indicate the tolerance of parameter perturbations by which the corresponding
eigenvalues λi (AD ) are not to be perturbed so as to cross the j ω-axis.
Relation between robustness and sensitivity
From (6), it is easy to understand 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) and Voit (2000):
YD /b = A−1
D
or
(YD /b)T (YD /b) = (AD ATD )−1 .
(21)
Obviously, from (14), the sensitivity from b to YD is inverse to the robustness of a biochemical network, i.e. the more robust a biochemical network
is, the less sensitive it is to the variation of rate constant, and vice versa.
Similarly, the sensitivity from environment variation YI to output
variation YD is given by Savageau (1970) and Voit (2000):
YD /YI = −A−1
D AI
or
(YD /YI )T (YD /YI ) = ATI (AD ATD )−1 AI .
(22)
Obviously, from (14), the sensitivity is also inversely related to the robustness
and a biochemical network with strong robustness will be more resistant to
the effect of environmental variation YI . A trade off between robustness
and sensitivity (a measure of fragility) of biochemical networks denotes and
constrains evolution and biology, i.e. a strong robust biochemical network
will constrain evolution and phenotype will be conserved in evolution, but a
more sensitive biochemical network is more possible to undergo a revolution.
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
A−1
D AI 2 in order to achieve an optimal robustness design has been an
important topic in modern control theory during this period.
Remark 3. From (14), it is seen that there are two mechanisms for increasing the robustness of a biochemical networks. One is to increase AD ATD in
order to tolerate large parameter perturbations in AD . The other is to prevent
the occurrence of large parameter perturbations in AD so that the robustness
condition in (14) cannot be easily violated. The redundancy and compensatory parameter variation [i.e. gij = hij so that AD = 0 in (10)] may be
two major sources for this kind of robustness. Compensatory parameter variations make AD small. Some examples of redundancy and compensatory
parameter variation are given below. A negative self-regulation of a metabolite in a pathway will lead to a parameter variation compensation. Thus,
a negative self-regulation is thought to provide the advantage of increasing
robustness of gene expression. About 10% of yeast genes encoding regulators are negative self-regulated so the mechanism seems to be important to
maintain robustness in yeast (Lee et al., 2002). If the kinetic activity of the
metabolite is increased or decreased, owing to negative self-feedback, the
increase or decrease in the kinetic activity will decrease or increase the influx
to balance the concentration of the metabolite. If a redundant pathway is produced by duplicate genes, the scale of parameter variation will be reduced
by the sharing of a pathway. Interestingly, a study of yeast and Escherichia
coli revealed that metabolic proteins tend to have more duplicate genes than
non-metabolic proteins (Marland et al., 2004), which apparently can increase
the robustness of metabolic pathways (Gu et al., 2003). Network redundancy
would be a buffer to prevent violent kinetic perturbations in AD , which may
violate the robustness condition in (14). This will be discussed in detail by the
simulation examples later. Therefore, as discussed in the following examples,
network redundancy and compensatory parameter variation are two efficient
mechanisms to attenuate parameter perturbations to prevent the violation of
the robustness condition in (14).
EXPERIMENTAL SIMULATIONS
To confirm the validity of our robustness measure, we conducted
four experimental simulations. The first example is the cascaded
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B.-S.Chen et al.
The dynamic response of this cascaded network is shown in
Figure 2a, i.e. the solution of the dynamic system in (24). The upper
bound of the tolerance is given by

0.2625 −0.2 −0.025
0.5
−0.25  .
=  −0.2
−0.025 −0.25
0.5

AD ATD
(25)
If the robustness condition (14) holds, then the parameter perturbation AD will be tolerated with respect to the maintenance of
steady state. Otherwise, the steady state of the cascaded network may
cease to exist under AD . Suppose the network suffers parameter
perturbations owing to a gene mutation as follows:
Fig. 1. A cascaded pathway with three steps and two feedbacks.
biochemical network in Figure 1. The computational experiments
suggested that if the robustness condition is violated, the steady
state of the biochemical network ceases to exist. The second
example is the glycolytic–glycogenolytic pathway in a perfused
rat liver (Torres, 1994; Voit, 2000). The third example is the
TCA in D.discoideum (Newsholme and Start, 1973; Kelly et al.,
1979; Voit, 2000). The fourth example is the cAMP oscillation
network in bacterial chemotaxis (Laub and Loomis, 1998). The
parameter perturbations in the four examples were all generated by
the random number generator in Matlab. To save space, the last
three examples are given in the supplementary information (http://
www.ee.nthu.edu.tw/∼bschen/robustness_bio-networks/).
Ẋ1 = 10X2−0.086038 X3−0.023759 X4 − 5X10.50495
X1 (0) = 0.2
X2 (0) = 0.5
X3 (0) = 0.1
Ẋ2 =
2X10.35939
−
1.44X20.43778
Ẋ3 =
3X20.22108
−
7.2X30.49843
X4 = 0.75.
In this kinetic parameter perturbation case,

0.026241

0
0.00157

−0.00495 0.013962
AD = −0.14061 0.06222
0
−0.27892
(26)
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). In
this biochemical network, some inputs lead to an increased production of metabolites at the first level, which activates the process at the
second level. Savageau (1976) reviewed a number of cascaded mechanisms and pointed out their significance for amplification, speed of
response, control and efficiency. Cascaded mechanisms are found in
diverse areas of biochemistry and physiology, including hormonal
control, gene regulation, immunology, blood clotting and visual
excitation.
The cascaded network involves three steps and two feedbacks.
Suppose the S-system is employed to describe the cascaded regulation mechanism. A precursor X4 is converted into metabolite X1
and the metabolites synthesized early in the cascade affect synthesis
at the next step of the cascade. The amplification process is slowed
down when the products X2 and X3 are available in sufficient quantity. The feedback regulation is presented by the kinetic parameters
g12 and g13 . The cascaded network can be represented as follows
(Voit, 2000):
Ẋ1 = 10X2−0.1 X3−0.05 X4 − 5X10.5
X1 (0) = 0.2
Ẋ2 = 2X10.5 − 1.44X20.5
X2 (0) = 0.5
Ẋ3 = 3X20.5 − 7.2X30.5
X3 (0) = 0.1

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 2b,
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.
Suppose that the cascaded biochemical network suffers the following kinetic parameter variations:
Ẋ1 = 10X20.01756 X30.1155 X4 − 5X10.45325
X1 (0) = 0.2
Ẋ2 = 2X10.54667 − 1.44X20.2174
X2 (0) = 0.5
Ẋ3 =
X3 (0) = 0.1
3X20.56494
−
7.2X30.4086
X4 = 0.75.
In this case, we have

0.04675 0.11756
AD = 0.04667 0.2826
0
0.06494
(23)
X4 = 0.75.
In this case, the system matrix AD of the cascaded network is as
follows:


−0.5 −0.1 −0.05
0 .
(24)
AD =  0.5 −0.5
0
0.5
−0.5
2702
and
and

0.2191
R = −0.2354
−0.0478
−0.2354
0.418
−0.2684

0.1655
0 
0.0914
(27)

−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
A new measure of robustness
Fig. 2. The dynamic response of the cascaded network in simulation experiment 1 under different perturbative cases. (a) The case without perturbation;
(b) The case with parameter perturbation in Equation (27); (c) The case with parameter perturbation in Equation (28); and (d) The case with perturbation from
negative feedback to positive feedback.
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 2c, 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.
From the control system point of view (Weinmann, 1991; Qu,
1998), feedback inhibition plays an important role in the robustness of a biochemical network. In the cascaded biochemical network
in (23), 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
Ẋ1 = 10X20.3 X30.15 X4 − 5X10.5
X1 (0) = 0.2
Ẋ2 =
X2 (0) = 0.5
2X10.5
−
1.44X20.5
Ẋ3 = 3X20.5 − 7.2X30.5
X4 = 0.75.
X3 (0) = 0.1
In this case,

AD ATD
0.2
= 0
0
0
0
0

0
0 ,
0
with the eigenvalues of R as follows (−0.0432, 0.3325, 0.7731),
thereby violating robustness condition (15). The dynamic response of
this cascaded network is shown in Figure 2d, in which the steady state
is not preserved. Therefore, adequate negative feedback inhibition
has the robustness property as in control theory.
In the supplementary information the changes in the two parameters g12 and g13 that push the system over the stability edge in
the parametric space are shown to illustrate where the robustness is
maintained and where it is lost. These results can be confirmed by
the smallest eigenvalue of the robustness matrix R in Supplementary
Figure S1. Therefore, the smallest eigenvalue of R is an indicator of
the robustness of a biochemical network.
Furthermore, suppose the negative feedback inhibition from X2 to
X1 in the cascaded network of Figure 1 consists of duplicated pathways but with the same flux, then the first equation in (24) is modified
as Ẋ1 = 10X2−0.05 X2−0.05 X3−0.05 X4 −5X10.5 . In this situation, a failure
of the second pathway will lead to Ẋ1 = 10X2−0.05 X3−0.05 X4 −5X10.5 ,
which will cause a smaller AD ATD than that in the failure of
2703
B.-S.Chen et al.
the feedback loop from X2 to X1 without redundancy. Therefore,
redundancy is a source of robustness via the mechanism of decreasing
AD ATD .
Other simulation experiments
From simulation experiment 2 of the glycolytic–glycogenolytic pathway (Voit, 2000) in Supplementary Figures S2 and S3, simulation
experiment 3 of the TCA cycle (Voit, 2000) in Supplementary Figures
S4 and S5 and simulation experiment 4 of bacterial chemotaxis (Alon
et al., 1999; Yi et al., 2000) in Supplementary Figures S6 and S7
(see Supplementary information), we also find that if the parameter
perturbations AD have not violated the robustness condition (14) or
(15), then the steady states of these biochemical networks will be preserved; otherwise, the steady states may cease to exist. Furthermore,
in simulation experiment 2 (see Supplementary information), we see
that the steady state of the biochemical network will be preserved
even with large compensatory perturbations with g22 = h22 = 5
in a negative self-regulatory pathway of X2 , i.e. if a large kinetic parameter perturbation h22 in the forward path can be compensated
by a large kinetic parameter perturbation g22 in a negative selfregulation path, then the biochemical network is robust at the steady
state with only a small change in the transient state.
These experimental simulations show that the proposed robustness measure is a good indicator of the robustness of biochemical
networks under parameter perturbations.
DISCUSSION
Mutations and diseases are unavoidable and can permanently alter
the kinetic properties of a biochemical network. Such alterations are
reflected in dynamic models as numerical changes in one or more
of the system parameters. The effects of permanent changes in a
biochemical network have been examined in this study using robustness analysis. We found that a biochemical network can tolerate
parameter variations with respect to the existence of steady state if
the robustness condition in (14) or (15) is not violated.
Our robustness analysis using the S-system model showed that if
the parameter variation AD ATD is less than the upper bound AD ATD
or the robustness matrix R is positive definite, the network is robust at
the steady state. As seen in the simulation examples, adequate negative feedback and redundancy may contribute significantly to the
robustness of a biochemical network. A redundant pathway owing
to duplicate genes may reduce the scale of parameter variation to
attenuate AD ATD to improve the robustness of a biochemical network. For example, knocking out one or two of the Per1, Per2 and
Per3 genes in the circadian network produces no visible phenotype
(Kitano, 2002) because the three genes were derived from two gene
duplication events.
We also found that in the compensatory parameter variation case,
the biochemical network could tolerate very large parameter perturbations with only a small change in the transient period. Therefore,
it is also an effective mechanism to preserve the steady state of a
biochemical network. A self-regulatory metabolite (or gene) with
negative feedback can lead to compensatory parameter variation.
Real biochemical networks must be sufficiently robust to tolerate
parameter and environmental variations or else they cannot respond
efficiently to small but persistent parameter and environmental perturbations. Therefore, if a model of a biochemical network has a small
robustness measure, it is often a sign of structural inadequacies of
2704
the model, which provides a means for identifying inconsistencies in
data. Thus, the proposed robustness measure scheme is also a good
way to validate models of biochemical networks.
Our robustness measure has been confirmed by four metabolic networks with numerical simulations. From the simulations, we found
that if the parameter variation measure AD ATD violates the upper
bound of the tolerance AD ATD , the steady state of the biochemical network may cease to exist. Therefore, in the evolutionary process of a
biochemical network, parameter perturbations owing to DNA mutation should be less than AD ATD or the steady state of the biochemical
network may cease to exist and the mutation will be eliminated by
natural selection.
It has been claimed (Nijhout, 2002) that non-linearity is the nature
of robustness. However, non-linearity can simply be the nature of a
biochemical network because almost all biochemical networks are
non-linear. If the system matrix AD is singular, the robustness of
a non-linear biochemical network is lost. The above examples of
unstable biochemical networks when some parameters were perturbed to the extent of having negative eigenvalues in R are of these
cases, i.e. their steady states become unstable even when their biochemical networks are still non-linear. Obviously, non-linearity is
not a guarantee of robustness for a biochemical network. The robustness of a biochemical network can be increased by increasing the
upper bound of the tolerance (i.e. AD ATD ) or attenuating the perturbation (i.e. AD ATD ). From the above experimental simulations,
we also found that an adequate negative feedback pathway may
increase AD ATD and redundancy, as well as compensatory parameter
perturbation can attenuate the parameter variation AD ATD , i.e.
adequate negative feedback, compensatory parameter perturbation
and redundancy may be three mechanisms for increasing the robustness of a biochemical network, which is consistent with recent system
theory (Weinmann, 1991; Qu, 1998) and system biology (Yi et al.,
2000; Hood et al., 2004). In this situation, we claim that these three
may also be the nature of biochemical network robustness. According to the robustness analysis, understanding diseases by studying
the biochemical pathways or networks involved is helpful for drug
design from the viewpoint of improving the robustness of biochemical pathways or networks (Hood et al., 2004). For example, in order
to improve the robustness of biochemical network, some pathways
could be added or some drug could be designated to change kinetic
parameters of the network to increase the eigenvalues of to improve
the robustness.
ACKNOWLEDGEMENT
Bor-Sen Chen would like to thank Professor Chang-I Wu for his
valuable discussions and comments. This study was supported by an
NSC grant and by Academia Sinica, Taiwan.
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