Response Delays and the Structure of
Transcription Networks
Nitzan Rosenfeld and Uri Alon*
Department of Molecular Cell
Biology and Department of
Physics of Complex Systems
Weizmann Institute of Science
Rehovot 76100, Israel
Sensory transcription networks generally control rapid and reversible
gene expression responses to external stimuli. Developmental transcription
networks carry out slow and irreversible temporal programs of gene
expression during development. It is important to understand the design
principles that underlie the structure of sensory and developmental transcription
networks. Cascades, which are chains of regulatory reactions,
are a basic structural element of transcription networks. When comparing
databases of sensory and developmental transcription networks, a striking
difference is found in the distribution of cascade lengths. Here, we
suggest that delay times in the responses of the network present a design
constraint that influences the network architecture. We experimentally
studied the response times in simple cascades constructed of well-characterized
repressors in Escherichia coli. Accurate kinetics at high temporal
resolution was measured using green fluorescent protein (GFP) reporters.
We find that transcription cascades can show long delays of about one
cell-cycle time per cascade step. Mathematical analysis suggests that such
a delay is characteristic of cascades that are designed to minimize the
response times for both turning-on and turning-off gene expression. The
need to achieve rapid reversible responses in sensory transcription networks
may help explain the finding that long cascades are very rare in
databases of E. coli and Saccharomyces cerevisiae sensory transcription networks.
In contrast, long cascades are common in developmental transcription
networks from sea urchin and from Drosophila melanogaster. Response
delay constraints are likely to be less important for developmental networks,
since they control irreversible processes on the timescale of cellcycles.
This study highlights a fundamental difference between the architecture
of sensory and developmental transcription networks.
q 2003 Elsevier Science Ltd. All rights reserved
Keywords: design principles; systems-biology; network motifs;
computational biology*Corresponding author
Introduction
Gene expression is governed by networks of
transcription interactions between transcription
factors and the genes they regulate. These networks
are complex, and unifying design principles
are needed to help make sense of their structure.1–14
One way to approach this is to attempt to understand
the structure of transcription networks
based on databases of experimentally verified
interactions. For Escherichia coli, this was pioneered
by the work of Collado-Vides and associates,6
based on an extensive transcription database.
Recently, an approach for identifying the basic
building blocks of networks, termed network
motifs, has been presented,15,16
and applied to transcription
networks. Network motifs were defined
as patterns that occur more often in the real network
than in randomized networks.17 – 20
Three
types of recurring network motif circuits
were found to describe most of the E. coli and
Saccharomyces cerevisiae transcription
networks.15,16,21
Each of these network motifs was
suggested to have a specific function in
information processing.
One important feature of the network architecture
is the distribution of transcription cascade
lengths. Transcription cascades are defined by a
set of transcription factors that regulate each other
0022-2836/03/$ - see front matter q 2003 Elsevier Science Ltd. All rights reserved
E-mail address of the corresponding author:
urialon@weizmann.ac.il
Abbreviations used: IPTG, isopropyl-b-Dthiogalactopyranoside;
GFP, green fluorescent protein.
doi:10.1016/S0022-2836(03)00506-0 J. Mol. Biol. (2003) 329, 645–654
sequentially.21,22
The first-step transcription factor
activates or represses the second-step transcription
factor, which in turn activates or represses the
third-step transcription factor, and so on. In a
study of the architecture of the transcription
network of E. coli, it was observed that it has a
strikingly shallow architecture, with most genes
regulated by overlapping cascades of length one
to two.15
Here, we find that two different databases
of transcription interactions in yeast (YPD,23
and a
genome-wide location analysis experiment21
) also
have a shallow architecture where long cascades
are much less common than short cascades. What
is the reason for a design that favors short
cascades?
To address this, we experimentally study the
response times of simple transcription cascades.
To study the properties of transcription cascades
in a setting that minimizes unknown coupling to
the rest of the cell, we constructed a two-step
cascade made of well-characterized bacterial
repressors.22
A similar approach has been used for
studying oscillators,24
toggle switches,25
negative
autoregulation circuits26,27
and logic-gates28
in
bacteria. We find that cascades can show long
delays. We suggest that response delays in transcription
cascades constitute one of the design constraints
that underlie the observed transcription
network architecture. We show that developmental
transcription networks, in which response delays
may not be an important constraint, have a
different architecture, with many long cascades.
Results
Long cascades are rare in databases of E. coli
and S. cerevisiae transcription interactions
We analyzed a database of experimentally
verified direct transcription interactions in E. coli,15
which includes 578 interactions and 423 operons.
The database consists of operons regulated by at
least one transcription factor. For each operon, we
traced the maximal number of transcription steps
back to a transcription factor that is not regulated
by any other transcription factor. We find that the
most common cascades in E. coli are one-step
cascades. Two-step cascades are less common, and
longer cascades are rare (Figure 1(a)).
A similar cascade distribution was found in a
database of transcription interactions for the yeast
S. cerevisiae, based on transcription interactions
from the YPD literature-based database,23
which
includes 1052 interactions and 678 genes. Again,
one-step cascades are the most common, and the
occurrence of longer cascades decreases sharply
with length (Figure 1(b)). We analyzed the network
of interactions found by a genome-wide location
analysis experiment in yeast,21
which includes
3969 interactions and 2341 genes. This dataset
does not have the methodological biases that may
occur in databases of interactions collected from
the literature, such as the possibility of unsaturated
genetics. In this database, there are more long
cascades than in the literature database, which
may reflect multi-step processes in yeast.29
On the
whole, short cascades are much more frequent
than long ones (Figure 1(c)).21
Long cascades are common in transcription
networks for sea urchin and Drosophila
early development
Developmental networks in metazoans are well
known to contain elaborate transcription cascades.4
In order to compare such networks to the sensory
networks of micro-organisms on an equal footing,
we enumerated the cascades in the rather limited
developmental databases that are currently available.
We analyzed the transcription network governing
endomesoderm specification during sea
urchin development.30
This network contains 44
genes and 82 interactions. We find that long cascades
are about as common as short cascades
(Figure 1(d)). We analyzed the network of the
established direct transcription interactions
involved in early development in Drosophila melanogaster,
based on the direct interactions listed in
the GeNet database. This network is comprised of
45 direct interactions and 25 genes, including gap
genes (hunchback, kruppel, knirps, giant, torso, tailless,
and caudal), pair-rule genes (hairy, odd-paired, oddskipped,
fushi tarazu, even-skipped, paired, runt, and
sloppy-paired), homeotic genes (Ultrabithorax,
empty-spiracles, Abdominal-A, and Antennapedia),
and segment polarity genes (decapentaplegic,
engrailed, wingless, and gooseberry-neuro), as well as
bicoid and achete. In this network, long cascades
are at least as common as short ones (Figure 1(e)).
Experiment on a two-step cascade shows a
delay of about one cell-cycle per step
To study the response kinetics of a cascade, we
used a synthetic repressor cascade in E. coli (Figure
2). In this cascade, the LacI repressor, made
chromosomally by the cell, represses a gene
encoding the Tet repressor. The Tet repressor, in
turn, represses the tet promoter that controls a
reporter gene (gfp) encoding the green fluorescent
protein (GFP) (Figure 2(a) and (c)).31
The promoter
activity of the first step in the cascade is monitored
in a separate reporter strain carrying the lac
promoter controlling gfp and a blank tet promoter
plasmid (Figure 2(b)). The system is induced by
adding external saturating isopropyl-b-D-thiogalactopyranoside
(IPTG) to inactivate the LacI
repressor. Fluorescence and cell density during
exponential growth were measured automatically
from both reporter strains in a multi-well fluorimeter
at about a five minute resolution.32,33
After
induction, the fluorescence from the first step
reporter rises, and fluorescence from the second
step is repressed after a delay (Figure 3).
We repeated this experiment at two different
646 Sensory Transcription Networks
Figure 1. Distribution of cascade lengths in databases of transcription interactions for (a) E. coli literature,15
(b) S. cerevisiae literature,23
(c) S. cerevisiae genome-wide location analysis,21
(d) endomesoderm development in sea
urchin,30
(e) Drosophila early development literature (GeNet database). The cascade length for each gene is defined as
the maximal number of transcription steps that link it to a transcription factor that is not regulated by any other
transcription factor.
Sensory Transcription Networks 647
Figure 2. (a) Plasmid system used
to implement the two-step transcription
cascade. A chromosomally
encoded LacI repressor represses
the tetR gene. The TetR repressor
represses the second step promoter,
which controls a gfp reporter gene.
The inducer IPTG binds and inactivates
lacI, causing the expression
of tetR and the repression of the
reporter gene. (b) The first-step
reporter strain bears a gfp gene
under control of the lac promoter
(step 1). As a control, this strain
bears a compatible plasmid with
the tet promoter upstream of a
multicloning site (MCS). (c) The
second-step reporter strain carries
tetR under the lac promoter, and
the second plasmid with tet promoter
controlling gfp. The two
strains allow the monitoring of the
promoter activity of both cascade
steps in parallel.
Figure 3. Normalized GFP fluorescence
from the two cascade steps
as a function of time. At time ¼ 0
the inducer IPTG was added. The
first step shows an increase in fluorescence,
and the second step shows
a delayed decrease in fluorescence.
Continuous lines, 36 8C; broken
lines, 27 8C.
648 Sensory Transcription Networks
temperatures, 27 8C where the cell-cycle is
155 minutes, and 36 8C where the cell-cycle is
80 minutes. The kinetics is slower at 27 8C than at
36 8C (Figure 3). When plotting the response versus
cell-cycles, determined by the number of optical
absorbance doublings, we find that the kinetics
overlap (Figure 4). The response-time of a system
is defined as the time to reach half of the change
between the pre-induced steady-state and the
post-induced steady-state.2,27
The response-time of
the first step in the cascade is about 1.1 cell-cycles,
and the response-time of the second step in the
cascade is about 1.9 cell-cycles.
Mathematical analysis suggests that
transcription cascades give rise to response
delays of about one cell-cycle per step
We extend a previous mathematical treatment of
transcription cascades27,34,35
(see Materials and
Methods), to ask what is the optimal responsetime
of a cascade given that it needs to respond
rapidly to reversible changes. How should the cascade
be designed, in order to minimize the sum of
the response times in the ON and OFF directions?
For simplicity, we consider long-lived proteins
with threshold-like transcription activation.36
Assume that protein X begins to be produced at a
constant rate upon induction, and is diluted out
by cells dividing with a cell-cycle time of t. When
X is induced, its concentration in the cells rises,
such that the distance to the steady-state level
decreases exponentially with time (Figure 5(a),
continuous line). The response-time of this first
step in the cascade (time to reach half of its maximal
level) is one cell-cycle (equation (4), in
Materials and Methods). When the production of
X is turned off, X is diluted. Its concentration
decays exponentially to zero, and the responsetime
is one cell-cycle as well (Figure 5(b), continuous
line, and see equation (7)).
Gene Y is activated when the concentration of X
exceeds a threshold value. If the threshold is low,
Y is turned on quickly (Figure 5(a)). However, it
takes a long time until Y is turned off when the
concentration of X decreases (Figure 5(b)). The
opposite is true if the threshold is high; it takes a
short time for Y to respond when the concentration
of X decreases (Figure 5(f)) but a long time to
respond when the concentration of X rises
(Figure 5(e)). In Materials and Methods, we show
that the optimal case, for which the sum of the
response-times in both directions is minimal, is
when the threshold is set at one-half of the steadystate
concentration of X (Figure 5(c) and (d)). In
this case, the response-time of Y in either direction
(the time for Y to reach half of its maximal change)
is two cell-cycles. Continuing in this fashion, it is
easy to show that the optimal response-time for
the nth step in a cascade is n cell-cycles.
These conclusions hold approximately also if the
transcription regulation is a graded function rather
than a threshold-like function. In the case of
rapidly degradable proteins, the cell-cycle time
should be replaced by the degradation half-life27,37
(see Materials and Methods)†, and each step in
the cascade contributes a delay of the order of the
degradation half-life of the protein produced in
that step.
Discussion
Synthetic networks built of well-characterized
components are useful for analyzing the general
Figure 4. GFP fluorescence from
the two cascade steps as a function
of cell-cycles. Data for Figure 3
were plotted against time in units
of cell-cycles. The response-time is
the time at which the response
reaches half of its maximal change.
This is the time at which the curves
intersect the horizontal line at
relative fluorescence ¼ 0.5. The
response-time of the first step in
the cascade is about 1.1 cell-cycles,
and the response-time of the second
step in the cascade is about 1.9 cellcycles.
Continuous lines, 36 8C;
broken lines, 27 8C.
† See also math primer, in downloadable data, at
http://www.weizmann.ac.il/mcb/UriAlon/
Sensory Transcription Networks 649
properties of network architecture.24 – 28,38
Here, we
analyzed the kinetics of a two-step cascade. The
response-time,2
which is the time taken to reach
half of the change upon induction, was found to
be about one cell-cycle per cascade step. Our
experiment further suggests that the cell-division
time is the basic time unit in the cascade response,
for long-lived proteins. A mathematical model
suggests that in order to minimize the response
times in both the ON and OFF directions, the cascade
should be designed so that the delays are of
about one cell-cycle per cascade step.
The slow responses of multi-step cascades may
explain the observation that the known E. coli transcription
network displays primarily short
cascades (Figure 1(a)). The exceptional long
cascades in the network typically correspond to
processes that take a few cell-cycles to complete.
For example, a three-step transcription cascade
controls flagella biosynthesis.32,39
The genes in this
cascade are termed class 1, class 2 and class 3,
according to their cascade step. In the flagella system,
the class 1 transcription factor FlhDC activates
the class 2 genes that make up the basal-body
motor and hook. One of these, FliA, activates class
3 genes that make up the flagellum filament and
the chemotaxis navigation system. To complete
the de -novo synthesis of a flagellum requires two
to three cell-cycles,32,40
which is roughly one cellcycle
per cascade step. Thus, a long transcription
cascade is well suited for controlling this system.
In contrast, the E. coli SOS DNA repair system33,41,42
and the arginine biosynthesis system43
both require
rapid responses and both show a one-step cascade
architecture.15
We note that even within a singlestep
transcription cascade, subtle timing differences
in expression can be achieved by using a
hierarchy of activation (or repression) thresholds
for the various genes controlled by a single
transcription factor.32,33,44
In contrast to transcription networks, proteinlevel
signaling networks are well known for
having long cascades such as map-kinase
cascades.45
Since protein signaling interactions are
typically much faster than transcription interactions,
response delays in long signaling protein
cascades need not be a serious design constraint.
The transcription network of the eukaryote
S. cerevisiae is found to show a cascade distribution
similar to that of the prokaryote E. coli, one-step
cascades being the most common. Though long
cascades exist, for example in the cell-cycle
system,29
the majority of genes are regulated by
short cascades. The arguments regarding slow
cascade responses should apply to all cell types.
One important point is that in eukaryotes, many
regulatory proteins have a life-time shorter than
the cell-cycle. In contrast, prokaryotic transcription
factors are often relatively long-lived (with important
exceptions as in the heat-shock system46
). For
rapidly degradable gene products, the cell-cycle in
all of the above arguments should be replaced by
the protein half-life (see footnote on previous
page).27,37
Thus, the incremental response time in
each cascade step is of the order of the lifetime of
the protein produced in that step. Increased protein
degradation can speed responses, though at a
cost of increased production. Other mechanisms
for speeding responses include negative auto-
regulation.2,27
Figure 5. Mathematical model of
the delay times between two steps
in a cascade of genes producing
long-lived proteins. The horizontal
axes are time in cell-cycles, the
vertical axes are relative protein
concentration. Continuous lines,
relative concentration of the first
step in the cascade, the upstream
protein X; heavy broken lines, relative
concentration of the second
step in the cascade, the downstream
protein Y. Protein X production is
turned on (a, c and e) or off (b, d
and f) at time ¼ 0. Y is expressed
only when the concentration of X is
above a threshold value (horizontal
broken line). The time of onset of Y
transcription after turn-on of X production,
T1, increases as the
threshold rises (a, c and e). On the
other hand, the time of shutoff of Y
production after turn-off of X, T2,
decreases with increasing threshold
(b, d and f). The optimal case in
which T1 þ T2 is smallest occurs at
intermediate values of the threshold
(c and d).
650 Sensory Transcription Networks
In the single-celled organisms considered above,
transcription cascades mainly orchestrate
responses to fluctuating external conditions (we
term these sensory networks). The decisions made
by sensory networks are mostly reversible. In
contrast, multi-cellular organisms have extensive
transcription networks that are responsible for
the cell-fate decisions made during
development.30,47– 51
Since developmental networks
often control processes on the order of many cellcycle
times, response-time considerations may be
less important than in sensory networks. Such
delays may even prove useful during development
by providing the cells with a measure of the number
of cell-cycles that have elapsed.† Furthermore,
since developmental networks often control processes
of deciding cell-fates, which are seldom
reversed, they need not optimize both turn-on and
turn-off times and can achieve fast timing in one
direction by having appropriate activation
thresholds (Figure 5). Therefore, developmental
transcription networks need not be biased against
long cascades. We find that developmental networks
from sea urchin and from Drosophila indeed
have a cascade structure that is very different
from that of sensory networks. Developmental networks
display many long cascades. Long cascades
are also found in developmental-like processes in
single-celled organisms, as in the sigma-factor cascade
regulating sporulation in the bacterium Bacillus
subtillis.52
The difference in the cascade
distributions suggests a fundamental difference
between the architecture of sensory and developmental
transcription factor networks. It would be
fascinating to discover additional constraints and
the design principles they induce in biological
networks.
Materials and Methods
Measuring cascade lengths in network databases
For each gene in the network, we traced back along
the network by an exhaustive search of all the possible
direct transcription cascades that affect this gene, and
found the longest cascade that affects each gene. When
a closed transcription loop was encountered, the cascade
was terminated. The E. coli network15
contains no closed
loops, though in rare cases two transcription factors are
encoded on the same operon, such as marA and marR,
and regulate their own promoter, thus effectively regulating
each other. The two yeast networks contain several
loops (the YPD23
network contains one two-gene and one
three-gene loop, and the genome-wide location
network21
contains two two-gene loops and one threegene
loop). The much smaller developmental networks
contain several closed loops (which may function as irreversible
switches25,30,36
): the sea urchin network30
has two
closed two-gene loops and the even smaller drosophila
network contains four two-gene loops and one threegene
loop. The drosophila network is from the GeNet
database (www.csa.ru/Inst/gorb_dep/inbios/genet/
s8prrl.htm).
Bacterial strains and plasmids
E. coli strain Dh5a which expresses lacI was used in all
experiments. Plasmid pairs pZS12-gfp þ pZE21-MCS,
pZS12-TetR þ pZE21-gfp were used for the cascade in
Figure 2(b) and (c).24
All plasmids were based on the
modular system described by Lutz & Bujard.53
For the
GFP variant used in this study, the biosynthetic delay
between transcription onset and production of the first
fluorescent GFP molecules is estimated at several
minutes.32,54
For pZS12-TetR and pZS12-gfp, the gene of
interest (tetR or gfp31
) was cloned into a vector containing
the low-copy SC101 origin of replication, ampicillinresistance
gene, and PLlacO1 promoter.53
pZE21-MCSI
contained a ColE1 origin, kanamycin-resistance gene, PLtetO1
promoter, and a multiple-cloning site, which was
replaced by gfp to create pZE21-gfp. This plasmid configuration
was chosen because it allows functional
modulation of step 2 by IPTG. Other plasmid choices,
such as placing tetR on a high-copy number plasmid,
were non-functional, in the sense that they failed to
cover much of the range of tet-promoter activity under
various levels of IPTG (data not shown).22
Growth conditions and measurements
Cultures (2 ml) inoculated from single colonies were
grown overnight in defined medium (M9 salts, 0.05%
(w/v) Casamino acids, 0.5% (v/v) glycerol, 2 mM
MgSO4, 0.1 mM CaCl2, 1.5 mM thiamine and antibiotics:
50 mg/ml of kanamycin, 100 mg/ml of ampicillin) at
37 8C with shaking at 300 rpm. The cultures were diluted
1:100 (v/v) into fresh medium in a flat-bottom 96-well
plate, at a final volume of 200 ml per well, and were
grown at 36 8C or 27 8C with shaking. The bacteria were
allowed to reach 1/10 stationary absorbance, and then
diluted 1:10 (v/v) into fresh medium at the same temperature,
with and without 1 mM IPTG to induce the
lac promoter. Cultures were grown in a Wallac Victor2
multiwell fluorimeter set at 36 8C or 27 8C, and assayed
with an automatically repeating protocol of shaking
(2 mm double-orbital, normal speed, 20 seconds) absorbance
(A) measurements (600 nm filter, 0.1 second,
absorbance through approximately 0.5 cm of fluid),
fluorescence readings (485 nm and 535 nm filters, 0.5
second, CW lamp energy 10,000 units), and a delay (100
seconds).32,33
The time between repeated measurements
was 5.75 minutes. Triplicates were averaged, and the
fluorescence of the induced cultures was divided by the
fluorescence of the uninduced culture. The relative fluorescence
was normalized for each strain to a maximum
level of one and a minimum of zero. The day-to-day
reproducibility error of the normalized fluorescence is
about 0.1.
Optimal response times for rapid turn-on and
turn-off
Assume that protein X is produced at a rate of Sx, is
degraded with a half-life of tdeg,x, and is diluted out by
cells dividing with a cell-cycle time of t. The basic
equation describing X, the concentration of protein X,
† Drosophila early development occurs in the absence
of cell divisions, and the time-scale may therefore be set
by the degradation times of the regulatory proteins.
Sensory Transcription Networks 651
is†:27,35
dX=dt ¼ Sx 2 axx ð1Þ
in which the first term describes production and the
second degradation plus dilution, with an effective
dilution rate of ax ¼ log(2)/tx, where the effective lifetime
is tx ¼ (1/t þ 1/tdeg,x)21
. For rapidly degradable
proteins with half-life tdeg,x much smaller than t, the
effective life-time is approximately equal to the degradation
half-life tx < tdeg,x. For long-lived proteins with
tdeg,x q t, the effective life-time is tx < t. The biosynthetic
delays between transcription onset and production
of the first active product can usually be
neglected because it is much shorter than typical protein
life-times.
The solution, assuming that X(t ¼ 0) ¼ 0, is
(Figure 5(a), continuous line):
XðtÞ ¼ Xmax½1 2 expð2axtފ ð2Þ
where the steady-state level is Xmax ¼ Sx/ax. The
response-time of this first step in the cascade, which is
the time needed to reach Xmax/2:
XðTON;1Þ ¼ Xmax½1 2 expð2axTON;1ފ ¼ Xmax=2 ð3Þ
is one effective life-time unit (one cell-cycle for longlived
proteins):
TON;1 ¼ tx ðfirst stepÞ ð4Þ
When the production of X is turned off, its concentration
decays exponentially (Figure 5(b), continuous line):
XðtÞ ¼ Xmaxexpð2axtÞ ð5Þ
and the response-time:
XðTOFF;1Þ ¼ Xmax expð2axTOFF;1Þ ¼ Xmax=2 ð6Þ
is one effective life-time unit as well (one cell-cycle for
long-lived proteins):
TOFF;1 ¼ tx ðfirst stepÞ ð7Þ
Suppose now that X activates the transcription of a
downstream gene Y, with a degradation half-life of tdeg,y,
an effective life-time of:
ty ¼ ð1=t þ 1=tdeg;yÞ21
and an effective dilution rate of ay ¼ log(2)/ty. Assume,
for simplicity, that X activates the transcription of Y
through a step-like activation function,34,36
so that Y is
produced at a rate Sy when X is at a greater than a
threshold concentration Xthreshold, and at rate zero otherwise.
More elaborate models, using Hill functions, produce
the same qualitative results. The concentration of
protein Y (Y) changes according to:
dY=dt ¼ FðXÞ 2 ayY; where FðX , XthresholdÞ
¼ 0 and FðX . XthresholdÞ ¼ Sy ð8Þ
The time when Y begins to be expressed is the time T1
when X crosses the threshold (Figure 5(a), (c) and (e)).
This occurs when:
XðT1Þ ¼ Xmax½1 2 expð2axT1ފ ¼ Xthreshold ð9Þ
The solution for T1 is:
T1 ¼ 2ð1=axÞlogð1 2 Xthreshold=XmaxÞ ð10Þ
After time T1, Y rises with a constant production rate Sy,
and has simple exponential kinetics (as in equation (2)):
YðtÞ ¼ Ymax½1 2 expð2ayðt 2 T1Þފ ð11Þ
with Ymax ¼ Sy/ay. The time to reach half-way to the
maximal level, Ymax, is shifted by T1 with respect to the
response-time of X (equation (2)), and is equal to T1 plus
one effective protein life-time unit (T1 þ t for long-lived
protein Y):
TON;2 ¼ ty þ T1 ðsecond stepÞ ð12Þ
Conversely, Y is turned off when X crosses the threshold
from above. Assume that at t ¼ 0, protein X is at its
steady-state level X ¼ Xmax, and then the production of
X is turned off. The concentration of X decays as
equation (4). It crosses the threshold Xthreshold at a time
T2 given by (Figure 5(b), (d) and (f)):
T2 ¼ 2ð1=axÞlogðXthreshold=XmaxÞ ð13Þ
At time T2, the production of Y ceases and it decays by
dilution. Since the kinetics of Y is like that of X (equation
(5)), only shifted by a time T2, the corresponding OFF
time is given by:
TOFF;2 ¼ ty þ T2 ðsecond stepÞ ð14Þ
The sum of the ON and OFF response-times is:
Tsum ¼ TON;2 þ TOFF;2 ¼ 2ty þ T1 þ T2 ð15Þ
The bigger Xthreshold is, the bigger is TON and the smaller is
TOFF (Figure 5). The activation threshold where Tsum is
minimal can be found by differentiating Tsum by Xthreshold,
and seeking d(Tsum)/d(Xthreshold) ¼ 0. The optimal
threshold that gives minimal Tsum is:
X
optimal
threshold ¼ Xmax=2 ð16Þ
At this optimal solution the response-times in the ON
and OFF directions are equal to each other, and both
equal to the sum of the effective life-times of the two
proteins:
TON;2;optimal ¼ TOFF;2;optimal ¼ tx þ ty ð17Þ
For long-lived proteins, tx ¼ ty ¼ t, and we obtain:
TON;2;optimal ¼ TOFF;2;optimal ¼ tx þ ty ¼ 2t ð18Þ
Continuing in this fashion, it is easy to show that the
response-time for the nth step in the cascade of longlived
proteins is n cell-cycles:
TON;n;optimal ¼ TOFF;n;optimal ¼ nt ð19Þ
Acknowledgements
We thank M. Elowitz and S. Leibler for plasmids
and discussions. We thank S. Shen-Orr and
P. Bashkin for their contributions. We thank
O. Hobart, M. Surette and all members of our
laboratory for discussions. This work was supported
by the Minerva fellowship, the Israel
Science Foundation and the Human Frontiers
Science Project. We dedicate this work to the
memory of Yasha (Yaakov) Rosenfeld.
References
1. Savageau, M.A. (1971). Parameter sensitivity as a
652 Sensory Transcription Networks
criterion for evaluating and comparing the performance
of biochemical systems. Nature, 229, 542–544.
2. Savageau, M. A. (1974). Comparison of classical and
autogenous systems of regulation in inducible
operons. Nature, 252, 546–549.
3. Savageau, M. A. (1977). Design of molecular control
mechanisms and the demand for gene expression.
Proc. Natl Acad. Sci. USA, 74, 5647–5651.
4. Arnone, M. I. & Davidson, E. H. (1997). The hardwiring
of development: organization and function
of genomic regulatory systems. Development, 124,
1851–1864.
5. Savageau, M. (2001). Design principles for elementary
gene circuits: elements, methods, and examples.
Chaos, 11, 142–159.
6. Thieffry, D., Huerta, A. M., Perez-Rueda, E. &
Collado-Vides, J. (1998). From specific gene regulation
to genomic networks: a global analysis of
transcriptional regulation in Escherichia coli. Bioessays,
20, 433–440.
7. Hartwell, L. H., Hopfield, J. J., Leibler, S. & Murray,
A. W. (1999). From molecular to modular cell biology.
Nature, 402, C47–C52.
8. McAdams, H. H. & Arkin, A. (2000). Towards a
circuit engineering discipline. Curr. Biol. 10,
R318–R320.
9. Barkai, N. & Leibler, S. (1997). Robustness in simple
biochemical networks. Nature, 387, 913–917.
10. Alon, U., Surette, M. G., Barkai, N. & Leibler, S.
(1999). Robustness in bacterial chemotaxis. Nature,
397, 168–171.
11. Yi, T. M., Huang, Y., Simon, M. I. & Doyle, J. (2000).
Robust perfect adaptation in bacterial chemotaxis
through integral feedback control. Proc. Natl Acad.
Sci. USA, 97, 4649–4653.
12. Barkai, N. & Leibler, S. (2000). Circadian clocks
limited by noise. Nature, 403, 267–268.
13. Vilar, J. M., Kueh, H. Y., Barkai, N. & Leibler, S.
(2002). Mechanisms of noise-resistance in genetic
oscillators. Proc. Natl Acad. Sci. USA, 99, 5988–5992.
14. Wyrick, J. J. & Young, R. A. (2002). Deciphering gene
expression regulatory networks. Curr. Opin. Genet.
Dev. 12, 130–136.
15. Shen-Orr, S. S., Milo, R., Mangan, S. & Alon, U.
(2002). Network motifs in the transcriptional regulation
network of Escherichia coli. Nature Genet. 31,
64–68.
16. Milo, R., Shen-Orr, S., Itzkovitz, S., Kashtan, N.,
Chklovskii, D. & Alon, U. (2002). Network motifs:
simple building blocks of complex networks. Science,
298, 824–827.
17. Wagner, A. & Fell, D. A. (2001). The small world
inside large metabolic networks. Proc. R. Soc. ser. B,
268, 1803–1810.
18. Guelzim, N., Bottani, S., Bourgine, P. & Kepes, F.
(2002). Topological and causal structure of the yeast
transcriptional regulatory network. Nature Genet. 31,
60–63.
19. Newman, M. E., Strogatz, S. H. & Watts, D. J. (2001).
Random graphs with arbitrary degree distributions
and their applications. Phys. Rev. ser. E, 64, 026118.
20. Maslov, S. & Sneppen, K. (2002). Specificity and
stability in topology of protein networks. Science,
296, 910–913.
21. Lee, T. I., Rinaldi, N. J., Robert, F., Odom, D. T., BarJoseph,
Z., Gerber, G. K. et al. (2002). Transcriptional
regulatory networks in Saccharomyces cerevisiae.
Science, 298, 799–804.
22. Yokobayashi, Y., Weiss, R. & Arnold, F. H. (2002).
Directed evolution of a genetic circuit. Proc. Natl
Acad. Sci. USA, 99, 16587–16591.
23. Costanzo, M. C., Crawford, M. E., Hirschman, J. E.,
Kranz, J. E., Olsen, P., Robertson, L. S. et al. (2001).
YPD, PombePD and WormPD: model organism
volumes of the BioKnowledge library, an integrated
resource for protein information. Nucl. Acids Res. 29,
75–79.
24. Elowitz, M. B. & Leibler, S. (2000). A synthetic oscillatory
network of transcriptional regulators. Nature,
403, 335–338.
25. Gardner, T. S., Cantor, C. R. & Collins, J. J. (2000).
Construction of a genetic toggle switch in Escherichia
coli. Nature, 403, 339–342.
26. Becskei, A. & Serrano, L. (2000). Engineering stability
in gene networks by autoregulation. Nature, 405,
590–593.
27. Rosenfeld, N., Elowitz, M. B. & Alon, U. (2002).
Negative autoregulation speeds the response times
of transcription networks. J. Mol. Biol. 323, 785–793.
28. Guet, C. C., Elowitz, M. B., Hsing, W. & Leibler, S.
(2002). Combinatorial synthesis of genetic networks.
Science, 296, 1466–1470.
29. Simon, I., Barnett, J., Hannett, N., Harbison, C. T.,
Rinaldi, N. J., Volkert, T. L. et al. (2001). Serial regulation
of transcriptional regulators in the yeast cell
cycle. Cell, 106, 697–708.
30. Davidson, E. H., Rast, J. P., Oliveri, P., Ransick, A.,
Calestani, C., Yuh, C. H. et al. (2002). A genomic
regulatory network for development. Science, 295,
1669–1678.
31. Cormack, B. P., Valdivia, R. H. & Falkow, S. (1996).
FACS-optimized mutants of the green fluorescent
protein (GFP). Gene, 173, 33–38.
32. Kalir, S., McClure, J., Pabbaraju, K., Southward, C.,
Ronen, M., Leibler, S. et al. (2001). Ordering genes in
a flagella pathway by analysis of expression kinetics
from living bacteria. Science, 292, 2080–2083.
33. Ronen, M., Rosenberg, R., Shraiman, B. I. & Alon, U.
(2002). Assigning numbers to the arrows: parameterizing
a gene regulation network by using accurate
expression kinetics. Proc. Natl Acad. Sci. USA, 99,
10555–10560.
34. McAdams, H. H. & Arkin, A. (1998). Simulation of
prokaryotic genetic circuits. Annu. Rev. Biophys.
Biomol. Struct. 27, 199–224.
35. Monod, J., Pappenheimer, A. & Cohen-Bazire, G.
(1953). La cinetique de la biosynthese de la
b-Galactosidase chez E. coli consideree comme fonction
de la croissance. Biochim. Biophys. Acta, 9.
36. Thomas, R. (1973). Boolean formalization of genetic
control circuits. J. Theor. Biol. 42, 563–585.
37. Schimke, R. (1969). On the roles of synthesis and
degradation in regulation of enzyme levels in
mammalian tissues. Curr. Top. Cell. Regul. 1, 77–124.
38. Judd, E. M., Laub, M. T. & McAdams, H. H. (2000).
Toggles and oscillators: new genetic circuit designs.
Bioessays, 22, 507–509.
39. Aldridge, P. & Hughes, K. T. (2002). Regulation of
flagellar assembly. Curr. Opin. Microbiol. 5, 160–165.
40. Aizawa, S. I. & Kubori, T. (1998). Bacterial flagellation
and cell division. Genes Cells, 3, 625–634.
41. Little, J. W. & Mount, D. W. (1982). The SOS regulatory
system of Escherichia coli. Cell, 29, 11–22.
42. Courcelle, J., Khodursky, A., Peter, B., Brown, P. O. &
Hanawalt, P. C. (2001). Comparative gene expression
profiles following UV exposure in wild-type and
SOS-deficient Escherichia coli. Genetics, 158, 41–64.
43. Makarova, K. S., Mironov, A. A. & Gelfand, M. S.
Sensory Transcription Networks 653
(2001). Conservation of the binding site for the
arginine repressor in all bacterial lineages. Genome
Biol. 2, 13.
44. Laub, M. T., McAdams, H. H., Feldblyum, T., Fraser,
C. M. & Shapiro, L. (2000). Global analysis of the
genetic network controlling a bacterial cell cycle.
Science, 290, 2144–2148.
45. Ferrell, J. E., Jr & Machleder, E. M. (1998). The biochemical
basis of an all-or-none cell fate switch in
Xenopus oocytes. Science, 280, 895–898.
46. Bourke Arnvig, K., Pedersen, S. & Sneppen, K.
(2000). Thermodynamics of heat-shock response.
Phys. Rev. Letters, 84, 3005–3008.
47. Lawrence, P. (1992). The Making of a Fly, Blackwell
Science Inc, Oxford.
48. Maduro, M. F. & Rothman, J. H. (2002). Making
worm guts: the gene regulatory network of the
Caenorhabditis elegans endoderm. Dev. Biol. 246,
68–85.
49. Kim, S. K., Lund, J., Kiraly, M., Duke, K., Jiang, M.,
Stuart, J. M. et al. (2001). A gene expression map for
Caenorhabditis elegans. Science, 293, 2087–2092.
50. Rothenberg, E. V. & Anderson, M. K. (2002).
Elements of transcription factor network design for
T-lineage specification. Dev. Biol. 246, 29–44.
51. Tomancak, P., Beaton, A., Weiszmann, R., Kwan, E.,
Shu, S., Lewis, S. E. et al. (2002). Systematic
determination of patterns of gene expression
during Drosophila embryogenesis. Genome Biol, 3,
research0088.1–research0088.14.
52. Stragier, P. & Losick, R. (1990). Cascades of sigma
factors revisited. Mol. Microbiol. 4, 1801–1806.
53. Lutz, R. & Bujard, H. (1997). Independent and tight
regulation of transcriptional units in Escherichia coli
via the LacR/O, the TetR/O and AraC/I1-I2 regulatory
elements. Nucl. Acids Res. 25, 1203–1210.
54. Cluzel, P., Surette, M. & Leibler, S. (2000). An ultrasensitive
bacterial motor revealed by monitoring
signaling proteins in single cells. Science, 287,
1652–1655.
Edited by M. Yaniv
(Received 30 January 2003; received in revised form 1 April 2003; accepted 9 April 2003)
654 Sensory Transcription Networks