EDITORIAL
Towards a theory of biological robustness
Molecular Systems Biology 18 September 2007; doi:10.1038/msb4100179
Robustness is one of the fundamental characteristics of
biological systems. Numerous reports have been published
on how robustness is involved in various biological processes
and on mechanisms that give rise to robustness in living
systems (Savageau, 1985a, b, 1998; Barkai and Leibler, 1997;
Alon et al, 1999; von Dassow et al, 2000; Bhalla and Iyengar,
2001; Csete and Doyle, 2002, 2004; Kitano et al, 2004, 2004a, b;
Stelling et al, 2004; Kitano and Oda, 2006; Kitano, 2007a). With
increasing interest in systems biology, properties at the system
level such as robustness have attracted serious scientific
research. Nevertheless, a mathematical foundation that
provides a unified perspective on robustness is yet to be
established. For systems biology to mature into a solid
scientific discipline, there must be a solid theoretical and
methodological foundation. Often, systems biology is equated
with computer simulation of cells and organs. Although
computer simulation is a powerful technique for clarifying
the complex dynamics of biological systems, it is also a useful
tool for exploring the foundation of biological systems. While
investigation on the dynamic properties of specific aspects of
organisms is scientifically significant and can be widely
applied, it is a study on specific instances of design within a
design space that is shaped by fundamental principles,
structural, environmental, and evolutionary constraints. The
scientific goal of systems biology is not merely to create
precision models of cells and organs, but also to discover
fundamental and structural principles behind biological
systems that define the possible design space of life (Figure 1).
The value of understanding fundamental and structural
theories is that they provide deeper insights into the governing
principles that complex evolvable systems including biological
systems follow. Building a solid theoretical foundation of
biological robustness, and in particular defining a mathema-
tical formulation of robustness, represents a key challenge in
Systems Biology. Such a framework would be enormously
useful, as it would provide general constraints on possible
architectural features of living organisms.
The concept of robustness
First of all, there must be a common understanding on what
‘robustness’ means. Defining any scientific term is a nontrivial
issue, but in this paper, the following definition will be used:
‘robustness is a property that allows a system to maintain its
functions against internal and external perturbations.’ (Kitano,
2004a). A similar definition with a slightly different phrasing
was used by others, such as ‘robustness, the ability to maintain
performance in the face of perturbations and uncertainty, is a
long-recognized key property of living systems’ (Stelling et al,
2004) and is thus considered to be the most appropriate
definition. It is important to choose the most reasonable and
appropriate definition, rather than creating yet another
definition of robustness. To discuss robustness, one must
identify system, function, and perturbations.
It important to realize that robustness is concerned with
maintaining functions of a system rather than system states,
which distinguishes robustness from stability or homeostasis.
Homeostasis is described as follows: ‘The coordinated
physiological processes which maintain most of the steady
states in the organism are so complex and so peculiar to living
beings—involving, as they may, the brain and nerves, the
heart, lungs, kidneys, and spleen, all working cooperatively—
that I have suggested a special designation for these states,
homeostasis. The word does not imply something set and
immobile, a stagnation. It means a condition—a condition
which may vary, but which is relatively constant (Cannon,
1932)’. According to this definition, homeostasis is clearly
a property that maintains the state of the system rather than
its functions. Homeostasis, stability, and robustness will be
identical if the function to be preserved is the one that
maintains the state of the system. In addition, the robustness
of a subsystem often contributes to homeostasis of the system
at the higher level. Such examples can be seen in yeast diauxic
shift (DeRisi et al, 1997) and glycolytic shift in tumor
metabolism (Mazurek and Eigenbrodt, 2003) in which the
state of the system changes at the level of metabolic functions
that maintain ATP production despite environmental pertur-
bations. This illustrates that robustness—not stability or
homeostasis—of subsystems may contribute to homeostasis
of the whole system when the function maintained, ATP
production in our example, is related to the stability of the
system state at the higher level. Whereas homeostasis and
stability are somewhat related concepts, robustness is a more
general concept according to which a system is robust as long
as it maintains functionality, even if it transits through a new
steady state or if instability actually helps the system to cope
with perturbations (Figure 2). Such transition between states
is often observed in biological systems when facing stress
conditions. An extreme example can be seen in the anhy-
drobiosis of tardigrade that suspends metabolism almost
completely, if not entirely, under extreme dehydration and
enters the dormant state, surviving for years (Crowe and
Crowe, 2000). This dormant state is attained by extensive
production of trehalose and tardigrade become active again
upon rehydration. Such dramatic shifts can be observed in
other organisms as well (Singer and Lindquist, 1998), and
some have argued that this is a third form of life called
‘cryptobiosis’ (Clegg, 2001). These examples of extreme
robustness under harsh stress conditions show that organisms
can attain an impressive degree of robustness by switching
from one steady state to the other, rather than trying to
maintain a given state. Such a phenotypic switch is also
& 2007 EMBO and Nature Publishing Group Molecular Systems Biology 2007 1
Molecular Systems Biology 3; Article number 137; doi:10.1038/msb4100179
Citation: Molecular Systems Biology 3:137
& 2007 EMBO and Nature Publishing Group All rights reserved 1744-4292/07
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observed in bacteria and can be considered to be important for
drug-resistance (Balaban et al, 2004). Robustness is also not
identical to stability. Some species gain robustness by
increasing instability in a part of its system. The HIV-1 virus
is robust against numerous therapeutic interventions due to a
high mutation domain (Larder and Kemp, 1989; Tisdale et al,
1993), which is one of the general mechanisms for viral
survivability (Eigen, 1993), and tumors are robust against
various chemotherapies, because chromosome instability
enhances heterogeneity within a tumor cell population (Baisse
et al, 2001; Rasnick, 2002). In summary, whereas robustness is
a general concept, homeostasis or stability can be considered
as particular instances of robustness.
Under modern control theory, a set of sophisticated methods
generally called ‘robust control’ has been developed. Robust
control assumes uncertainties in a model and defines a method
of applying stable control over the system such that proper
control is guaranteed even if the model deviates from the real
system due to modeling errors (Zhou and Doyle, 1997). Note
that robust control assumes a control system that stabilizes the
target system so as to be robust against model errors; this
mechanism for robustness is consistent with the definition of
robustness given above. Nevertheless, control theory assumes
a system that is designed to meet given criteria, and so it
cannot be directly applied to biological systems that have
evolved and for which the desirable state of the system is not
explicit. In addition, most of the mathematics used to describe
robustness are mostly based on control theory, which tend to
focus on stability and performance of monostable systems.
A theory that take into account multistability and evolution of
instable systems needs to be developed and new theoretical
avenues need to be explored to provide a broad and unified
account of robustness of biological systems.
A particularly interesting topic in the context of robustnesss
is its trade-offs. What kind of trade-off exists in biological
systems? Is robustness conserved? Does a trade-off between
robustness and fragility indicate some kind of conservation
principle as claimed by Csete and Doyle (2002)? Highly
optimized tolerance (HOT) theory demonstrates, taking the
example of a forest fire, that a system that is optimized for a
specific perturbation inevitably entails extreme fragility for
unexpected perturbations (Carlson and Doyle, 1999, 2002)
(see Box 1). Commercial jet airliners with fly-by-wire control
are highly robust against most component failures and
atmospheric perturbations, but become extremely fragile
against highly improbable events such as a total power failure
as they depend entirely on electric control. The Wright Flyer,
on the other hand, is a non-robust system but free from power
failure problems, because it does not use any electric system.
Biological examples of such trade-offs are abundant. Some
diseases can be considered as manifestations of such trade-offs
(Kitano et al, 2004; Kitano, 2004b; Kitano and Oda, 2006), and
the efficacy and side effects of drugs may be related to
robustness trade-offs (Kitano, 2007b).
In addition, biological trade-offs may actually not only
involve robustness and fragility, but also resource demands
and performance of the system. For example, having an entire
backup copy of the system enhances robustness against
component failure due to redundancy, but it doubles the
resources required and may therefore degrade the perfor-
mance of the system. Thus, when robustness of the system
against certain perturbations is increased, it may result in
increased fragility against unexpected perturbation, increased
resource demands, and degradation of performance. A
simultaneous increase of robustness and reduction of fragility
Living organisms as
instances of design
St
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ur
al
p
rin
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le
s
Th
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rie
s
on
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es
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tra
in
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Fundamental principles.
Theories on elementary matters and interactions
Environmental constraints
Ev
ol
ut
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n
A
possible
design
space
Figure 1 Fundamental principles, structural principles, and design. Living
organisms are designed through evolution and perturbed under environmental
constraints. Each instance of design is an actual life form that exists in the past,
present, and future. Viable design is only possible within the constraints of
fundamental principles and structural principles. Fundamental principles include
basic laws such as quantum theory, Maxwell’s equations, basic chemistry, and
physics that apply to almost everything universally. Structural principles govern
properties of systems and have a specific architecture such as control theory,
communication theory, and various principles applied to specific configurations of
components that are generally architecture-specific and context-dependent. For
systems biology to be truly successful, not only studies on specific instances of
life, but also studies on principles governing the entire design space are required.
Steady state 3
Steady state 2
Steady state 1
Figure 2 Stability, homeostasis, and robustness. Assume that the initial state
of the system is at the center of steady state 1. A perturbation may drive the state
of the system toward the boundary of the basin of attractor of steady state 1.
When the state of the system returns to its original state, it is called ‘stability’ and
‘homeostasis’. When it transits to steady state 2, stability is once lost and the
system regains its stability in the new steady state. If the system’s functions are
still intact, such transition of state is considered a part of robust response. The
system is considered to be robust if it maintains functions regardless of whether it
is in steady state 1 or 2. On extreme case, the system may continue to transit
between multiple steady state points to cope with perturbations.
Editorial
H Kitano
2 Molecular Systems Biology 2007 & 2007 EMBO and Nature Publishing Group

may be achieved when additional resources are integrated
properly into the system or if system performance is reduced.
Alternatively, a system’s performance can be maximized by
giving up robustness of the system against various perturba-
tions. We should also note that these features are not
independent. Performance, in terms of maneuverability of
some animals in a hostile environment, may translate into
robustness against predator attacks. Increased resource
demands may translate into fragility against severe resource
competition as well as perturbation on available resources.
The key issue is whether it is possible to find a formalism in
which robustness and its trade-offs could be defined so that
robustness is a conserved quantity or whether the trade-offs
discussed above are bound to remain at the level of useful
but empirical observations. Understanding such trade-offs
would be critically important to understand the basic design
principles of life at the level of individual organisms and cells.
It may also explain the origin of the diversity of life through
evolutionary selection of design space under competitive
environments. Mammals have evolved to be highly robust
against a broad range of perturbations, but require important
resources for their development and maintenance of their daily
life. Bacteria, on the other hand, have adopted a set of rather
simple mechanisms at the individual level, but can reproduce
very quickly and sustain huge populations due to smaller
resource demands than other species. How can we map
different evolutionary niches within a map based on robust-
ness and its trade-offs?
Mathematical formulation of biological
robustness
The effort towards formalizing a theory of robustness and its
trade-offs is still in its infancy and much remains to be
completed to build a mature theory. For a theory to be useful, it
must be able to predict characteristics and behaviors of the
system. This means that the theory has to be framed to
explicitly describe constraints that bind the system.
First of all, mathematical definitions of terms are
given. Robustness can be defined as a system’s characteristics
that maintain one or more of its functions under external
and internal perturbations. Under this definition, robust-
ness (R) of the system (s) with regard to function (a) against
a set of perturbations (P) can be mathematically described
as:
Rsa;P ¼
Z
P
cðpÞDsa pð Þdp ð1Þ
The function c(p)is the probability for perturbation ‘p’ to take
place, and this should be 1 when all perturbation to take place
at equal probability. D (p) is an evaluation function under
perturbation (p), and P is the entire perturbation space. The
evaluation function determines if the system still maintains
function under a perturbation and to what degree, and is
defined as:
DsaðpÞ ¼
0; p 2 A  P
faðpÞ=fað0Þ; p 2 PnA

ð2Þ
A is a set of perturbations where the system failed to maintain
its function. This means that D(p) is zero when a function does
not meet a defined criteria under perturbation (p) and D(p)
returns a relative viability of a function under perturbation
compared against non-perturbed condition otherwise. For
example, ATP production drop 20% under a certain perturba-
tion compared with ATP production under unperturbed state,
then 0.8 shall be returned. Note that p in this equation
represents a specific instance of a perturbation. Figure 3
illustrates definition of robustness.
A system ‘S1’ can be said to be more robust than a system
‘S2’ with regard to a function ‘a’ against a certain set of
perturbations ‘Y’, when
RS1a;Y4RS2a;Y
The HOT model argues that systems that have evolved to have a higher level of complexity are optimized for specific perturbations but, at the same time,
are also inevitably extremely fragile against unexpected perturbations (Carlson and Doyle, 1999, 2002). Carson and Doyle use the example of a forest
fire to illustrate the intrinsic nature of the system. Forest buffer zone patterns and tree planting patterns that are optimized for specific types of fire can be very
fragile against unexpected types of fire, and may ignite from unexpected directions (A). They also argued that a design that can utilize a large degree of
freedom can be more optimal for anticipated perturbations, and therefore can be extremely fragile against unexpected perturbations (Reynolds et al, 2002).
Of course, such trade-offs are not limited to the simple design of buffer zone locations. If one decides to plant all trees around a city in a circular manner to be
able to cope with fire from any direction, such design may actually allow a fire to spread and encircle the city, causing major damage to the city as well (B).
However, if one cuts all the trees in the field, the city will be very fragile against the rainy season as flooding will be more likely due to the resulting loss of water-
absorbing capacity of the trees (C). The point here is that such trade-off is inherent and cannot be avoided. This figure is inspired by the HOT theory (Carlson and
Doyle, 1999, 2002).
Box 1 Robustness-fragility trade-offs in forest fire countermeasures
Editorial
H Kitano
& 2007 EMBO and Nature Publishing Group Molecular Systems Biology 2007 3

However, considering an entire perturbation space (P), or
sufficiently broad perturbation space, robustness-fragility
trade-off should hold, thus difference of robustness (DR)
between two systems shall be:
DRS1;S2a;P ¼
Z
P
cðpÞðDS1a ðpÞ  DS2a ðpÞÞdp ¼ RS1a;P  RS2a;P ¼ 0 ð3Þ
which is reminiscent of the Bode integral formula. If
robustness is conserved, then above equation should be zero
with equiprobability over the perturbation space (see also
Figure 4A). Assuming that S1 and S2 are the same system but
with parameters optimized for different subset of perturba-
tions, this equation implies that any increase in robustness
against a subset of perturbation will be off-set by decrease of
robustness against other perturbations. In fact, the notion that
trade-offs between robustness and fragility represents a
conservation of robustness (fragility) in biological systems
was initially inspired by the so-called Bode Integral formula
(Csete and Doyle, 2002):
Z1
0
log SðoÞj jdoX0 ð4Þ
Where S(o) is the sensitivity of a system at a frequency o. The
Bode integral formula represents conservation of sensitivity of
a negative feedback (NFB) system along the frequency axis
(Bode, 1945) (the relevance of this theorem to biological
systems is best described in (Csete and Doyle, 2002); see also
Box 2). The Bode theorem indicates that an improvement of
sensitivity gained by NFB in the low-frequency range is traded-
off by increased instability in the high-frequency range. In
addition, within the theoretical framework of Metabolic
Control Analysis, the summation and connectivity theorems
represent constraints that are imposed on parametric changes
in metabolic pathways (Fell, 1992), implying that the
sensitivity of the network is conserved for changes in rate
constants.
As noted above, trade-off between robustness and perfor-
mance also need to be considered. It is often the case that
systems that are particularly well tuned for a specific task
under a given environment are fragile against change in the
environment. In contrast, systems with moderate performance
tend to be more robust and thus can remain functional under
a broader range of conditions (Figure 4B). Such trade-offs
need to be formulated as well. In electric engineering,
amplifier design is known to involve constraints on Gain-
Band Width, which represent similar trade-offs. How such
trade-off can be generalized to biological systems remains to
be explored. Similar argument apply to trade-offs between
robustness and resource use, where robustness against
component failure can be improved by having a greater level
of redundancy, hence increased resource demand. An example
of this is provided by reliability engineering, which offers a
mathematical basis for reduced fault rate (Figure 4C). Never-
theless, it is unclear whether formulations for each trade-off
can be integrated into a single unified system of equations.
However, efforts to further elaborate such relationships shall
provide us deeper mathematical insights into biological
systems.
Future challenges
This article briefly discussed a primitive concept of how
biological robustness may be formulated mathematically and
raised some of the key issues that remain to be resolved.
Although there are numbers of challenges ahead, it is clearly
understood that much of the basic mathematics are already in
place provided we deal with a well-chosen set. Further
theoretical studies should be able to utilize such formalization
as a starting point. Bode integral theorem and a set of theorems
from metabolic analysis have already illustrated the conserva-
tion of robustness to some extent, and reliability engineering is
a solid basis for component failure analysis. Mathematical and
experimental studies are still required to characterize the
trade-off relationship between robustness and performance.
It will be a major challenge to find out under which
conditions trade-offs exist and how to calculate system-level
properties such as robustness or performance, when additional
D (p)=0
Features that are perturbedFeatures that are perturbed
D
eg
re
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o
f p
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tu
rb
at
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D
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f p
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tu
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p16 p26 p36 p46 p56
p10 p20 p30 p40 p50 p60
p11 p21 p31 p41 p51 p61
p12 p22 p32 p42 p52 p62
p13 p23 p33 p43 p53 p63
p14 p24 p34 p44 p54 p64
p15 p25 p35 p45 p55 p65
p66
D
eg
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e
o
f p
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tu
rb
at
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p16 p26 p36 p46 p56
p10 p20 p30 p40 p50 p60
p11 p21 p31 p41 p51 p61
p12 p22 p32 p42 p52 p62
p13 p23 p33 p43 p53 p63
p14 p24 p34 p44 p54 p64
p15 p25 p35 p45 p55 p65
p66
D
eg
re
e
o
f p
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tu
rb
at
io
n
g1 g2 g3 g4 g5 g6 g1 g2 g3 g4 g5 g6
Figure 3 Robustness. Perturbations are imposed on each feature and at different degree if applicable. The figure illustrate coarse grain view of perturbation space
where there are six features to be perturbed each of which is perturbed at six different degree. Colors on box for each perturbation indicate how system responded to
each perturbation. Red box indicate that system fail to maintain its function. Different blue colors show the level of degradation of the function. Although the area the
function is maintained is same in (A and B), (A) is considered more robust as the function is better preserved than (B).
Editorial
H Kitano
4 Molecular Systems Biology 2007 & 2007 EMBO and Nature Publishing Group

resource use is accompanied with changes in system configu-
ration. In the long run, the theory should be extended to deal
with major structural changes. This will require the elaboration
of definitions based on biological network properties, and the
development of a comprehensive set of innovative computa-
tional methods to derive such characteristic quantities for
Robustness–fragility trade-off
Robustness–performance trade-off
A
–
R
Robustness–resource trade-off
Pe
rfo
rm
a
n
ce
(f
(p)
)
A high-performance low-robustness case
Degree of perturbation
g10– +
Y b = f (0)b
R b
A low-performance high-robustness case
Y a = f (0)a
Pe
rfo
rm
a
n
ce
(f
(p)
)
Degree of perturbation
g10– +
R a
R
ob
u
st
ne
ss
Resource used
A
–
2R
2R
Features that are perturbed
g1 g2 g3 g4 g5 g6 g1 g2 g3 g4 g5 g6
D
eg
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o
f p
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tu
rb
at
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p16 p26 p36 p46 p56
p10 p20 p30 p40 p50 p60
p11 p21 p31 p41 p51 p61
p12 p22 p32 p42 p52 p62
p13 p23 p33 p43 p53 p63
p14 p24 p34 p44 p54 p64
p15 p25 p35 p45 p55 p65
p66
Features that are perturbed
D
eg
re
e
o
f p
er
tu
rb
at
io
n
p16 p26 p36 p46 p56
p10 p20 p30 p40 p50 p60
p11 p21 p31 p41 p51 p61
p12 p22 p32 p42 p52 p62
p13 p23 p33 p43 p53 p63
p14 p24 p34 p44 p54 p64
p15 p25 p35 p45 p55 p65
p66
Figure 4 Robustness trade-offs. (A) If robustness is strictly conserved, then any increase in robustness for specific perturbation shall be compensated by increase in
fragility elsewhere. Left panel is a profile of robustness of a hypothetical system that responds equally to perturbations of each feature (from g1 to g6). Now, if the system
is tuned to cope better with perturbations of a subset of features (g1, g2, and g3), then robustness against other subset of perturbations are significantly reduced (right
panel). Total robustness of both systems over this perturbation space remains equal. (B) If the robustness-performance trade-off holds, a system that is tuned to attain
high performance might be less robust than a system with moderate performance but a higher level of robustness. Let’s assume Y a¼f a(0) for system A where f (0) is the
performance of the function of the system under perturbation ‘0’ (no perturbation) and Ra is the robustness of the system over some defined perturbations. Although
the figure simply refers to the colored areas for ease of understanding, the exact Ra needs to be calibrated based on Equation (1). The horizontal dashed lines indicate
the threshold under which the system fails to perform the function considered. A robustness-performance trade-off would then imply that Y aR a¼Y bR b. (C) Identical
circuits with slightly difference resource use are shown. Both use NFB loop, but one uses only one resistor in the loop, whereas the other one uses two resistors in
parallel. Parallel use of components significantly improves robustness of the system against component failure, but requires more resources. Here, the probability of
degradation of system function can be computed using basic equations from reliability engineering so that the difference of robustness can be derived for simple example
like this one. It is however challenging to derive an expression for more complex systems under various perturbations. The question is how can we compute
DRS1;S2a;P ¼ UðmS1  mS2Þ
where mS1 and mS2 are resource used for system S1 and S2, respectively. The function U would relate the difference of resource use to the difference in robustness as a
function of some design principles according to which resources are used. Whether it is at all possible to define such a function and, more fundamentally, whether such
conservation actually exists, in either relative or strict manner, remains open.
Editorial
H Kitano
& 2007 EMBO and Nature Publishing Group Molecular Systems Biology 2007 5

large systems. This is an important undertaking as it may bring
abstract theory to practical utility by providing specific
constraints underlying the organization of biological organ-
isms and subsystems. As progresses in theoretical research
will derive more concrete constraints, we should be able to
better predict and reverse-engineer the structures of biological
networks. Combined with various high-throughput experi-
mental data, we should be able to derive the structures and
dynamics of networks with higher accuracy.
The current mathematical formulations are mostly con-
cerned with the stability and maintenance of the system’s
functions against perturbations. As discussed at the outset,
robustness is a broader concept than stability. A theory
that would account for phase transition and instability as
means to achieve robustness would need to be formulated
and integrated with theories on stability. Although instability-
based robustness involves survival of the fittest under
selective pressure, it needs to be integrated with mathematical
framework on evolution, genetics, and game theory (Maynard-
Smith, 1982).
Ultimately, the theory will have to be interfaced with
thermodynamics. Studies on nonequilibrium dissipative
systems are mostly focused on chemical reactions and some
are trying to extend theories on nonequilibrium dissipative
systems to the principles of life (Prigogine et al, 1974).
However, the theories still do not take into account the
G
ai
n
Frequency
BWa
BWb
G
a
G
b
Frequency
N
or
m
a
liz
e
d
fra
gi
lity
-
Frequency
N
or
m
a
liz
e
d
fra
gi
lity
A
B
A = B
-
There have been various studies on the trade-offs between robustness, fragility, and performance in engineering systems as well as in physics. In amplifier design,
the trade-off between stability in specific frequency range provided by a NFB loop is compensated by increased instability in higher frequency region and less
overall gain of the amplifier. This is a central issue in electric circuit design, and has been intensively investigated in control theory. Assuming a simple feedback
circuit as seen in amplifiers, the steady-state sensitivity (S) against a perturbation (d) of the system having feedback gain (G) is defined by S¼1/(1þG).
Therefore, a larger gain reduces the sensitivity and hence increases the robustness against perturbations. However, frequency domain analysis shows that such
increase in robustness increases fragility in a specific frequency domain (A and B). Sensitivity of the system against perturbations (fragility) is conserved. An
increase in feedback level reduces the sensitivity in a specific frequency range (A), but creates a region of instability elsewhere as shown by the peak of normalized
fragility in the middle. With a larger feedback strength, sensitivity in a specific frequency range may be significantly reduced, but fragility would be larger as a result
(B; adapted from Yi et al, 2002). The mathematics behind this trade-off is well known, but is particularly well documented by Yi et al (2002) related to biological
examples which they describe as follows: given the output of the system (Y(o)) and disturbance (D(o)), sensitivity function (S(o)) can be defined as S(o)¼Y(o)/
D(o). Let S0(o) be a sensitivity function of the open loop system, then we can define a base line sensitivity as
log S0 oð Þj j
Normalized sensitivity, hence fragility of the system, can be obtained by subtracting the sensitivity of the feedback system and the base line sensitivity of the system
without feedback. This normalized sensitivity can be described by the equation:
Z1
0
log SðoÞj jdo  0
This inequality is essential as it implies that feedback control cannot improve overall sensitivity; it only improves sensitivity in one place in a trade-off for fragility
elsewhere. In addition, theories that integrate trade-offs between robustness and fragility in a feedback system with a feedback channel of limited capacity have
been developed recently, thus expanding the horizon of intrinsic trade-offs involved (Martins et al, 2004, 2007). It has been argued that the same trade-off may
apply to biological systems, and so increased robustness against certain perturbations inevitably results in extreme fragility elsewhere (Csete and Doyle, 2002). At
the same time, using NFB reduces the overall gain of the amplifier. Trade-offs between robustness and performance have also been thoroughly investigated. In
amplifier design, it is well known that the gain-bandwidth product (GBWP) is conserved (C). In this case, the gain of the amplifier is considered as performance of
the system, and the bandwidth corresponds to how broadly the system can ensure a certain level of insensitivity to disturbances on this circuit within the frequency
region where the sensitivity is reduced by the feedback loop. For an amplifier with a gain of 1000 at 1 kHz (GBWP¼1000 1¼1000), the bandwidth can be
extended using NFB to 100 kHz by reducing the gain to 10 (GBWP¼100 10¼1000). This high-frequency cut-off is extended due to the feedback loop.
Box 2 Robustness trade-offs in engineering and physics
Editorial
H Kitano
6 Molecular Systems Biology 2007 & 2007 EMBO and Nature Publishing Group

heterogeneity and structured nature of biological systems as
well as selection through evolution and it is a major challenge
to attempt bridging this gap. The situation is similar for the
fields of nonlinear dynamics and chaos, for which theories that
embrace the characteristics of biological systems are yet to
emerge.
Formulation of a fundamental theory of biological systems is
one of the grand challenges in biology. In very general terms,
this will involve resolving the gap between the level of
description used in thermodynamics and other basic physical
sciences—for example, the properties of ensemble of mole-
cules in a medium—and the abstraction level used to define
the concepts elaborated in this article, which involve networks
of biological interactions. Hopefully, the ideas and concepts
discussed in this article will stimulate discussions and provide
some stepping stones for research directed towards these
ambitious objectives.
Acknowledgements
This research is supported, in part, by ERATO/SORST program (Japan
Science and Technology Agency: JST), Sweden-Japan Strategic
Collaboration Program (JST), Genome Network Project (Ministry of
Education, Sports, Culture, Science, and Technology).
References
Alon U, Surette MG, Barkai N, Leibler S (1999) Robustness in bacterial
chemotaxis. Nature 397: 168–171
Baisse B, Bouzourene H, Saraga EP, Bosman FT, Benhattar J (2001)
Intratumor genetic heterogeneity in advanced human colorectal
adenocarcinoma. Int J Cancer 93: 346–352
Balaban NQ, Merrin J, Chait R, Kowalik L, Leibler S (2004) Bacterial
persistence as a phenotypic switch. Science 305: 1622–1625
Barkai N, Leibler S (1997) Robustness in simple biochemical networks.
Nature 387: 913–917
Bhalla US, Iyengar R (2001) Robustness of the bistable behavior of a
biological signaling feedback loop. Chaos 11: 221–226
Bode HW (1945) Network Analysis and Feedback Amplifier Design.
Melbourne, FL: Krieger
Cannon W (1932) The Wisdom of the Body. New York: WW Norton &
Company Inc
Carlson JM, Doyle J (1999) Highly optimized tolerance: a mechanism
for power laws in designed systems. Phys Rev E Stat Phys Plasmas
Fluids Relat Interdiscip Topics 60: 1412–1427
Carlson JM, Doyle J (2002) Complexity and robustness. Proc Natl Acad
Sci USA 99 (Suppl 1): 2538–2545
Clegg JS (2001) Cryptobiosis—a peculiar state of biological
organization. Comp Biochem Physiol B Biochem Mol Biol 128:
613–624
Crowe JH, Crowe LM (2000) Preservation of mammalian cells-learning
nature’s tricks. Nat Biotechnol 18: 145–146
Csete ME, Doyle J (2004) Bow ties, metabolism and disease. Trends
Biotechnol 22: 446–450
Csete ME, Doyle JC (2002) Reverse engineering of biological
complexity. Science 295: 1664–1669
DeRisi JL, Iyer VR, Brown PO (1997) Exploring the metabolic and
genetic control of gene expression on a genomic scale. Science 278:
680–686
Eigen M (1993) Viral quasispecies. Sci Am 269: 42–49
Fell DA (1992) Metabolic control analysis: a survey of its theoretical
and experimental development. Biochem J 286 (Part 2): 313–330
Kitano H (2004a) Biological robustness. Nat Rev Genet 5: 826–837
Kitano H (2004b) Cancer as a robust system: implications for
anticancer therapy. Nat Rev Cancer 4: 227–235
Kitano H (2007a) Biological robustness in complex host-pathogen
systems. Prog Drug Res 64: 239, 241–263
Kitano H (2007b) A robustness-based approach to system-oriented
drug design. Nat Rev Drug Disc 6: 202–210
Kitano H, Oda K (2006) Robustness trade-offs and host-microbial
symbiosis in the immune system. Mol Syst Biol 2: 0022
Kitano H, Oda K, Kimura T, Matsuoka Y, Csete M, Doyle J, Muramatsu
M (2004) Metabolic syndrome and robustness tradeoffs. Diabetes
53 (Suppl 3): S6–S15
Larder BA, Kemp SD (1989) Multiple mutations in HIV-1 reverse
transcriptase confer high-level resistance to zidovudine (AZT).
Science 246: 1155–1158
Martins NC, Dahleh MA, Doyle JC (2007) Fundamental limitations of
disturbance attenuation in the presence of side information. IEEE
Transactions on Automatic Control 52: 56–66
Martins NC, Dahleh MA, Elia N (2004) Stabilization of uncertain
systems in the presence of a stochastic digital link. IEEE Conference
on Decision and Control. Nassau: IEEE
Maynard-Smith J (1982) Evolution and the Theory of Games.
Cambridge, UK: Cambridge University Press
Mazurek S, Eigenbrodt E (2003) The tumor metabolome. Anticancer
Res 23: 1149–1154
Prigogine I, Nicolis G, Babloyantz A (1974) Nonequilibrium problems
in biological phenomena. Ann NYAcad Sci 231: 99–105
Rasnick D (2002) Aneuploidy theory explains tumor formation, the
absence of immune surveillance, and the failure of chemotherapy.
Cancer Genet Cytogenet 136: 66–72
Reynolds D, Carlson JM, Doyle J (2002) Design degree of freedom and
mechanisms for complexity. Phys Rev E: 016108
Savageau MA (1985a) Mathematics of organizationally complex
systems. Biomed Biochim Acta 44: 839–844
Savageau MA (1985b) A theory of alternative designs for biochemical
control systems. Biomed Biochim Acta 44: 875–880
Savageau MA (1998) Demand theory of gene regulation. I.
Quantitative development of the theory. Genetics 149: 1665–1676
Singer MA, Lindquist S (1998) Thermotolerance in Saccharomyces
cerevisiae: the Yin and Yang of trehalose. Trends Biotechnol 16:
460–468
Stelling J, Sauer U, Szallasi Z, Doyle III FJ, Doyle J (2004) Robustness of
cellular functions. Cell 118: 675–685
Tisdale M, Kemp SD, Parry NR, Larder BA (1993) Rapid in vitro
selection of human immunodeficiency virus type 1 resistant to
30-thiacytidine inhibitors due to a mutation in the YMDD region
of reverse transcriptase. Proc Natl Acad Sci USA 90: 5653–5656
von Dassow G, Meir E, Munro EM, Odell GM (2000) The segment
polarity network is a robust developmental module. Nature 406:
188–192
Yi TM, Sauro HM, Ingalls B (2002) Limits of control and Bode’s integral
formula. Basic Control Theory for Biologists 2002
Zhou K, Doyle J (1997) Essentials Of Robust Control. New Jersey, USA:
Prentice Hall
Hiroaki Kitano1,2,3
1Sony Computer Science Laboratories Inc., Shinagawa, Tokyo, Japan,
2The Systems Biology Institute, Shibuya, Tokyo, Japan and
3Department of Cancer Systems Biology, Cancer Institute of Japanese
Foundation of Cancer Research, Koutou-ku, Tokyo, Japan
Editorial
H Kitano
& 2007 EMBO and Nature Publishing Group Molecular Systems Biology 2007 7