Genetically Structured Mathematical
Modeling of trp Attenuator Mechanism
Boon Tong Koh,1 Reginald B. H. Tan,1 Miranda G. S. Yap2
1
Department of Chemical Engineering, National University of Singapore,
10 Kent Ridge Crescent, Singapore 119260; telephone: +65-8746360; fax:
+65-7791936
2
Bioprocessing Technology Centre, National University of Singapore
Received 2 April 1997; accepted 10 October 1997
Abstract: A genetically structured mathematical model
of the trp attenuator in Escherichia coli based on known
coupling mechanisms of the transcription of the trp
leader region and translation of the trp leader peptide
region is proposed. The model simulates, both qualitatively and quantitatively, the effects of tryptophan on the
repression of cloned gene products. It shows that repression by attenuation mechanism alone operates over a
narrow trp concentration range of 1 to 5 µM compared
with 1 to 100 µM for trp repressor mechanism. This implies that attenuation by transcription termination is not
relaxed until tryptophan starvation is severe. Simulation
results show that the attenuator starts to derepress when
the repressor is about 40% repressed, and becomes significantly derepressed only when the repressor repression decreased to about 20%. Unlike the case of repressor-operator interaction, the operating range of tryptophan concentration in the attenuator mechanism is not
sensitive to plasmid copy number. © 1998 John Wiley &
Sons, Inc. Biotechnol Bioeng 58: 502–509, 1998.
Keywords: genetically structured mathematical model;
trp operon; cloned gene expression control
INTRODUCTION
The ability to regulate the expression of cloned genes is an
important factor in ensuring maximum productivity of valuable recombinant proteins in Escherichia coli. A typical
process involves effectively repressing cloned gene transcription during cell growth and, subsequently, inducing the
cells to maximize protein production (Lee et al., 1988; Seow
et al., 1989). Such intracellular control can be achieved by
expression systems built into the cloned genes. A common
example is the trp promoter which has been used to control
the expression of a number of useful proteins such as human
tumor necrosis factor (Seow et al., 1989) and human interferon-b (Mizukami et al., 1986).
The detailed mechanism of how the trp promoter controls
the expression of the cloned genes has been elucidated in
recent years. This involves understanding the mechanism of
the trp operon transcription, which is regulated by both
repression and attenuation. Repression, located at the promoter-operator site, regulates transcription initiation in reCorrespondence to: Reginald B. H. Tan
© 1998 John Wiley & Sons, Inc.
sponse to changes in the intracellular level of free tryptophan (Koh and Yap, 1993; Yanofsky and Crawford, 1987).
Attenuation, located downstream of the promoter-operator
site, regulates transcription termination in response to
changes in the levels of charged and uncharged tRNATrp
(Elsenberg et al., 1980; Lee and Yanofsky, 1977; Oxender
et al., 1979; Zurawski et al., 1978). The combined action of
these two regulatory mechanisms allows expression of the
operon to be varied over a 500-fold range; repression and
attenuation account for about 80- and 6-fold variations, respectively (Bertrand and Yanofsky, 1976; Yanofsky et al.,
1984).
In a previous article (Koh and Yap, 1993), a genetically
structured mathematical model of the trp repressor-operator
interactions was developed, which accounts for the operation of the trp operon under repression control. The model
showed that the important parameters governing expression
include tryptophan, indoleacrylic acid, aporepressor concentrations, and the plasmid copy number. The model could
be used to optimize these parameters for effective repression and subsequent derepression of the cloned gene. For
example, our model predicts that for high copy number
plasmids, full repression is not possible in spite of high
tryptophan levels in the medium, because of limiting aporepressor concentration in recombinant E. coli cells. This
accounts for ‘‘leaky’’ expressions often reported in literature (Kawai et al., 1986; Mizukami et al., 1986; Schroeckh
et al., 1992; Siegel and Ryu, 1985). However, the above
model is not complete because it does not account for the
effect of attenuator control on protein expression.
Attenuator control is known to influence expression levels. An increase in cloned gene product has been observed
when the attenuator is eliminated (Seow et al., 1989; Tacon
et al., 1983). In addition, compared with the repressor
mechanism, repression of the attenuator is achieved at much
lower concentration (Yanofsky et al. 1984). In this article, a
mathematical model, which complements the previous one,
is developed based on present understanding of trp attenuation mechanisms. The model seeks to predict the important
parameters and the range of tryptophan levels that govern
trp attenuation. It can be combined with the previous model
on trp repressor to fully describe the regulation of the trp
CCC 0006-3592/98/050502-08
repressor and attenuator mechanisms on cloned gene repression. It also enables us to compare the differences in the
regulation of trp repressor and attenuator mechanisms and
increase our understanding of their biological functions.
MODEL DESCRIPTION
Attenuation in the trp operon is a dynamic process involving both transcription of the leader region and translation of
the trp leader peptide (#27–71) located within this leader
region (#1–140). This leader transcript is capable of folding
to form mutually exclusive secondary structures as shown in
Figure 1. Formation of the {3:4} terminator structure causes
transcription termination, while the {2:3} antiterminator
structure allows transcription to proceed into the structural
genes of the operon. Attenuation control, which involves the
formation of one of these two alternative structures, is governed by the coupling effects of the translation of the peptide coding region (#27–71), specifically the two tandem trp
codons (#54–59), and the transcription of the leader region.
A brief description of the complex mechanisms, which have
been reported in detail (Landrick and Yanofsky, 1984; Lan-
drick et al., 1985; Landrick and Yanofsky, 1987; Roesser
and Yanofsky, 1988; Roesser et al., 1989; Yanofsky and
Crawford, 1987), is presented.
At the onset of transcription from the trp promoter, the
RNA polymerase transcribes the trp leader region and
pauses after leader segment 2 (up to #92) is synthesized.
The transcription pause has been found to be due to the
formation of the {1:2} pause structure (Fig. 2a) (Landrick
and Yanofsky, 1984; Winkler and Yanofsky, 1981). Translation of the short leader peptide coding region begins with
the ribosome binding to the start codon. The translating
ribosome approaches and disrupts the {1:2} pause structure,
releasing the paused transcription complex. This ensures
that the translating ribosome would be placed on the transcript close behind the transcribing polymerase (Landrick et
al., 1985; Winkler and Yanofsky, 1981). Polymerase and
ribosome then move in unison over the leader region and
leader transcript, respectively. It has been reported that the
70S translating ribosome is estimated to mask 13 base pairs
of RNA both upstream and downstream of the codon being
translated (Roesser and Yanofsky, 1988; Yager and von
Hippel, 1987). Hence, in the model we assume that the
Figure 1. Alternate secondary structures of Escherichia coli trp leader transcript.
KOH ET AL.: MODEL OF Trp ATTENUATOR
503
Figure 3. Schematic diagram showing mechanism of trp attenuator: alternative scenarios for (a) trp excess and (b) trp limiting.
Figure 2. Schematic diagram showing mechanism of trp attenuator: (a)
ribosome loading, and (b) pause disruption.
ing ribosome) to the time when the ribosome is released
from the stop (#71) codon. This period comprises the average translation times from val to gly (#42 to #53), two trp
codons and from arg to ‘‘stop’’ (#60 to #71) codons, and the
ribosome release time, tr. Hence,
4
4
+ Trp Stalling Time ~TST! +
ATL
ATL
+ RB release time ~tr!
TL =
{1:2} pause structure is disrupted when the ribosome starts
translating the val codon (at #42), four codons before the
first trp codon (Fig. 2b).
The position of the two tandem trp codons at the beginning of structure {1:2} plays an important role in attenuation control. After disruption of the pause structure, two
basic scenarios are possible. In the presence of sufficient
available tryptophan, the translating ribosome continues uninterrupted to the ‘‘stop’’ codon where it is subsequently
released from the transcript (Fig. 3a). The position of the
ribosome at segment ‘‘2’’ interferes with the formation of
the {2:3} structure. This sequence of events favors the formation of the {3:4} terminator structure and results in transcription termination of the trp mRNA. Under tryptophan
starvation, the translating ribosome stalls at either of the two
trp codons at segment ‘‘1,’’ preventing the formation of the
{1:2} structure. This condition facilitates the formation of
the antiterminator structure {2:3}. This allows transcription
to proceed into the cloned genes (Fig. 3b). Hence, attenuation control involves coupling of the translation and transcription processes (Roesser et al., 1989).
The mathematical equations are developed as follows.
The total translating time (TL) is defined as the period
from the start of translation of the val (#42) codon (ie., the
moment the {1:2} pause structure is disrupted by the mov-
504
(1)
where ATL is the average rate of translation in codons/s.
The trp stalling time (TST), defined as the time it takes to
translate the two tandem trp codons, may be derived from
the mechanism of translation and amino-acryl tRNA synthetase as outlined below.
The general rate propagation equation for translation of
amino acids 1 to n is derived in the Appendix to be
d
!=
~RB − mRNA − tRNA1→n
n
dt
kI
~RB! ~mRNA!~tRNAn!
kP1
)S
n
i=1
kPi
~tRNAii!
~tRNAi!
D
(2)
When n, the last codon to be translated is trp, Equation (2)
becomes
kI
d
!=
~RB − mRNA − tRNA1→n
~RB!~mRNA!~tRNAn!
n
dt
kP1
n−1
~tRNAii!
~tRNATrp
Trp !
kPi
(3)
i
Trp
~tRNA !
~tRNA !
i=1
S )S
DD
Assuming that only tryptophan is limiting, that is, all other
amino acids are available in excess, then translation of the
BIOTECHNOLOGY AND BIOENGINEERING, VOL. 58, NO. 5, JUNE 5, 1998
trp leader peptide is limited by charged tRNATrpTrp at the
two trp codons. By collecting constant terms together as
ATL, and assuming constant intracellular volume, Equation
(3) as applied to translation of trp codons only, can be
simplified to:
~Cdn!
@tRNATrp
Trp#
= ATL
Trp
Dt
@tRNA #
(5)
The relationship of tRNATrp/tRNATrp
Trp and tryptophan concentration can be derived from the mechanism of tryptophanyl tRNA synthetase (TrpRS). A simplified two-step
mechanism of tryptophanyl tRNA synthetase (TrpRS) in
charging tRNAtrp is as follows (Hershey, 1987):
TrpRS + T
←→
~B!
TrpRS-T + tRNATrp ←→
TrpRS-T
Km2
T + tRNATrpTrp
tRNATrpTrp + TrpRS
Km
←→
tRNATrpTrp
Km2 =
@T# @TrpRS#
@TrpRS − T#
(6)
@TrpRS − T# @tRNATrp#
@tRNATrp
Trp#@TrpRS#
(7)
Assuming constant intracellular volume, we combine Equations (6) and (7) to yield
@tRNATrp
Trp#
@tRNA
Trp
#
=
@T#
Km
(8)
where Km 4 Km1 × Km2
Values of Km1 and Km2 have been reported to be 20 mM
and 1, respectively (Hershey, 1987).
From a material balance of tryptophan, the total trp concentration can be expressed as
@Tt# = @T# + @TrpRS − T# + @tRNATrp
Trp# + 2@PCN#
TST =
2 Km
1.08
ATL @T1# − 0.004PCN
(9)
The term 2 [PCN] arises from the Trp molecules involved in
induction. The equation does not account for Trp molecules
which are complexed with aporepressor that are not yet
bound with DNA, as the amount in this transient state is
considered insignificant.
Assuming constant cell volume, [PCN] is equivalent to
approximately 0.002 mM per plasmid. Further, because
there are typically about 800 molecules of aminoacryl-
(10)
(11)
Therefore, the total time taken for translation, TL, derived
by accounting for: (a) the average translation rate, ATL; (b)
trp stalling time at either of the two trp codons, which is
assumed to depend on an equilibrium process between
available tryptophan and charged tRNATrp; and (c) average
ribosome release time at the stop codon (#69–71), tr, is
4
1.08
2 Km
4
+
`
+t
ATL
ATL @T1# − 0.004PCN ATL r
(12)
Transcription of mRNA, which had paused at the {1:2}
structure (#92), resumes when the translating ribosome disrupts this pause structure. The position of the leader transcript (#TC) with respect to translation is obtained as
#TC 4 92 + BTC TL
The equilibrium relationship can be written as:
Km1 =
@T1# − 0.004PCN
1.08
Substituting Equations (8) and (10) into Equation (5), we
have
TL =
Km1
~A!
@T# =
(4)
where Cdn is the number of trp codons being translated.
Hence, the total trp stalling time at two tandem trp
codons is
2 @tRNATrp#
TST =
ATL @tRNATrp
Trp#
tRNA synthetase per cell (Hershey, 1987; Neidhardt et al.,
1977), [TrpRS] is approximately 1.6 mM. Substituting
Equations (6) and (8) into (9), rearranging, and noting that
[tRNATrp]/20 << 1, we obtain the following expression for
[T]:
(13)
where BTC is the average transcription rate of trp mRNA.
We have defined the repression coefficient under attenuator control, RA, as the probability of forming {3:4} terminator structure. Because {2:3} and {3:4} structures are mutually exclusive, RA can be written as
RA 4 P{3:4} 4 1 − P{2:3}
(14)
where P{3:4} and P{2:3} are the probabilities of forming
{3:4} terminator and {2:3} antiterminator structures, respectively.
A simulation of all the possible positions of the leader
transcript, #TC in Equation (13), determines the relative
probabilities of {2:3} or {3:4} formation, which are mutually exclusive.
The average transcription (BTC) and translation (ATL)
rates for E. coli at 37°C are 50 nucleotides/s and 17 codons/
s, respectively (Gausing, 1972; Kassavetis and Chamberlin,
1981). The average ribosome release time (tr) at the UGA
codon in the translational reading frame of the prfB gene has
been estimated to be 0.6 seconds (Curran and Yarus, 1988).
These values used for the model simulation are summarized
in Table I.
It is known that the {3:4} termination structure followed
by the series of seven uracil nucleotides causes transcription
termination of the trp mRNA (Landick and Yanofsky, 1987;
Yanofsky and Crawford, 1987). Hence, we assume that
once transcription has passed the seven uracil nucleotides
(#141) without the formation of {3:4} terminator structure,
KOH ET AL.: MODEL OF Trp ATTENUATOR
505
Table I. Values of ATL, BTC, Km and tr used in our model.
Values Used
ATL
17 codons/s
BTC
51 nucleotides/s
Km
tr
20 mM
0.6 s (average value)
References
Kassavetis and Chamberlin (1981);
Roesser et al. (1989)
Kassavetis and Chamberlin (1981);
Roesser et al. (1989)
Hershey (1987)
Curran and Yarus (1988); Roesser
et al. (1989)
transcription termination would not be possible. Attenuation
repression, RA, would then be at its minimum (given an
arbitrary value of 0).
SIMULATION RESULTS
Figure 4, generated from Equations (11) and (12) for PCN
of 10, shows the position of the mRNA being transcribed
when translation of the peptide coding region is at (a) second trp codon, and (b) ‘‘stop’’ codon before ribosome is
released. Points (i) and (ii) on the graph indicate the position
of the ribosome at the time nucleotide #141 is being transcribed. To the left of point (i) in Figure 4, RA would be
minimum, because ribosome is binding on segment 1 which
prevents structure {1:2} formation. This results in formation
of antiterminator {2:3}. Hence, P{2:3} is maximum and RA
is minimum (given an arbitrary value of 0).
Similarly, to the right of point (ii) in Figure 4, RA would
be maximum, because ribosome binds segment 2 which
prevents structure {2:3} formation. This results in terminator {3:4} formation. Hence, P{2:3} is minimum resulting in
maximum RA (given an arbitrary value of 1). For tryptophan
levels between those represented by points (i) and (ii), both
{2:3} and {3:4} structures compete for formation, and we
have assumed that the probabilities would then depend on
Figure 4. Position of mRNA being transcribed (#TC) when translation of
the peptide coding region is at second trp codon (a), and at ‘‘stop’’ codon
(b); and repression coefficient (c) as predicted by our model. Generated for
PCN of 10. X-axis represents total tryptophan, [Tt].
506
the relative stability of each structure corresponding to their
Gibbs free energy (Roesser and Yanofsky, 1988; Roesser et
al., 1989). Because the Gibbs free energy for both structure
are approximately equal (Oxender et al., 1979; Zuker and
Stieger, 1981), a linear relationship of RA is assumed between the two extreme points as shown in Figure 4.
To study the effects of the level of tryptophan and plasmid copy number (PCN) on attenuator mechanism, simulation was carried out for E. coli with PCN of 10 and 100.
Results of the attenuator repression coefficients (RA) calculated from the mathematical model are shown in Figure 5.
The attenuator mechanism appears to be ‘‘on/off’’ control
over a narrow range of tryptophan concentration for both
low and high copy number plasmids. This shows that attenuator is not deactivated until very low tryptophan concentration and that the rate of derepression is extremely
sensitive to tryptophan concentration at low levels. In addition, increasing the PCN from 10 to 100 has no significant
effect on the regulating range of tryptophan.
The repressor repression coefficient calculated from our
previous model (Koh and Yap, 1993) is also plotted for
comparison. As shown in Figure 5, the regulating range of
tryptophan for the trp repressor mechanism is much wider.
This suggests that the response to changes in tryptophan
levels are milder. For high PCN, such as RR(100), effective
repression was not achieved because of limiting aporepressor levels (Koh and Yap, 1993).
DISCUSSION
From our modeling, it appears that the tryptophan requirement for attenuation action is very low compared with the
repressor mechanism. The total tryptophan level regulated
by attenuator falls within a narrow range of 1 to 5 mM, as
compared with 1 to 100 mM for repressor mechanism. This
shows that attenuation by transcription termination is not
Figure 5. Repressor (RR) and Attenuator (RA) repression coefficients
calculated from our models for PCN of 10 (---) and 100 (—). Repression
coefficients 0 and 1 are arbitrary values representing minimum and maximum repression respectively.
BIOTECHNOLOGY AND BIOENGINEERING, VOL. 58, NO. 5, JUNE 5, 1998
relaxed until tryptophan starvation is severe. In contrast,
repressor action regulated transcription initiation over the
range from excess to moderate tryptophan levels. These
findings are consistent with observations reported in literature. One reason suggested by Yanofsky et al. (1984) is that
tRNATrp charging did not become growth-limiting until the
intracellular tryptophan concentration was so low that it was
inadequate to activate the trp aporepressor. This suggests
that repressor and attenuator come into play independently
in regulating trp operon transcription.
The different requirements of tryptophan in controling
repression and attenuation can be explained by the low concentration of aporepressor and high level of tRNATrp present
in E. coli. From their mechanisms, both the repressor and
attenuator require only two tryptophan molecules per active
site for repression. However, the tryptophan level required
to effectively repress repressor mechanism is relatively
high. This is because of the low aporepressor concentration
(30 copy/cell) present in the cell. Hence, from equilibrium
analysis (KI 4 [ApoR-T][T]/[TrpR] 4 30 mM), a low aporepressor level will result in a high tryptophan concentration
to generate sufficient trp repressor, [TrpR], for effective
repression. For the attenuator, because of the relative abundance of tRNATrp (about 6000 copy/cell) and a reasonable
equilibrium constant (Km 4 20 mM), the requirements of
tryptophan to form charged tRNATrp are lower. Hence, a
significantly smaller amount of tryptophan is needed to activate the attenuator mechanism.
The overall repression coefficient for the combined action of trp repressor and attenuator, ROV, can be written as
Figure 7. Plot of RR, RA and ROV for PCN of 10, assuming minimum and
maximum RA of 0.05 and 0.85, respectively.
ROV, calculated by Equation (15), are shown in Figure 6
(PCN 10) and Figure 7 (PCN 100) by assuming the minimum and maximum RA of 0.05 and 0.85, respectively. The
last value corresponds to basal level expression of about
15% as reported in literature (Roesser et al., 1989). For both
cases of high and low PCN, simulation results show that the
attenuator mechanism starts to deactivate only when the
repressor is about 40% repressed, and becomes significantly
deactivated when the repressor is about 20% repressed.
In conclusion, Figure 8 compares the repressor with attenuator mechanisms (ROV) and repressor mechanism without attenuator (RR) for plasmid copy number of 10 and 100.
The small difference between ROV(10) and ROV(100) compared with RR(10) and RR(100) implied that unlike the repressor mechanism, the attenuator mechanism is not significantly affected by changes in the plasmid copy number. The
operating range of tryptophan remains the same from 1 to 5
mm. Control of cloned gene expression appeared to be ‘‘onoff’’ over this narrow range for the attenuator mechanism.
The simulation results also showed that presence of attenuator downstream of the repressor significantly reduce the
expression of cloned genes. This is evidenced by reports
that removal of the attenuator improved the expressed effi-
Figure 6. Plot of RR, RA and ROV for PCN of 10, assuming minimum and
maximum RA of 0.05 and 0.85, respectively.
Figure 8. Plot of RR and ROV for PCN of 10 (---) and 100 (—), assuming
minimum and maximum RA of 0.05 and 0.85, respectively.
ROV (repressor and attenuator) 4 1 − (1 − RR)(1 − RA) (15)
KOH ET AL.: MODEL OF Trp ATTENUATOR
507
ciency of the plasmid (Seow et al., 1989; Tacon et al.,
1983).
RB − mRNa
K
tRNAii
tRNAij
(b) releasing uncharged tRNAi
NOMENCLATURE
ATL
BTC
Km1, Km2
mRNA
PCN
RA
RR
ROV
RB
TST
TL
tr
tRNAi
tRNAii
TrpRS
#TC
P{2:3}
P{3:4}
[ApoR]
[T]
[Tt]
average rate of translation (codons/s)
average rate of transcription (nucleotides/s)
equilibrium constants as defined by Equations (6) and (7)
messenger RNA
plasmid copy number
attenuator repression coefficient as defined by Equation (12)
repressor repression coefficient as defined in (Koh and Yap,
1993)
overall repression coefficient as defined by Equation (13)
translating ribosome
total trp stalling time at both trp codons (s)
total translation time (s)
ribosome release time (s)
uncharged amino acid i transfer RNA
charged amino acid i transfer RNA
tryptophanyl tRNA synthetase
position of leader transcript wrt to translation
probability of forming {2:3} antiterminator structure
probability of forming {3:4} terminator structure
total aporepressor concentration
free tryptophan concentration
total tryptophan concentration
By assuming steady state and equating rate of formation
(2a) to rate of depletion (2b), we can derive:
d
~RB − mRNA − tRNAi−j
j !=
dt
k1 kpj~RB!~mRNA!
where kpj =
k3j
k4j
(tRNAii)
(tRNAi!
~tRNAij!
(A2)
(dimensionless)
(c) on further propagation with another charged tRNAk
k3k
k
RB − mRNA − tRNAi−j
i + tRNAk ←→ RB
tRNAi−j
j
− mRNA
tRNAkk
K
By similar derivations, we have:
APPENDIX
The derivation of translation rate equations from mechanism of translation:
1. Initiation: RB and charged tRNA bind to mRNA in two
elementary steps.
K1
~a!
RB + mRNA ←→ RB − mRNA
~b!
RB − mRNA +
tRNAii
k2
→ RB − mRNA −
tRNAii
(RB-mRNA) 4 K1(RB)(mRNA)
The rate of reaction (b) is written as
d~RB − mRNA − tRNAii!
= k2~RB − mRNA!~tRNAii!
dt
d~RB − mRNA − tRNAii!
= k1~RB!~mRNA!~tRNAii!
dt
(A1)
where kI 4 K1 ? k2 (4(moles2-s)−1)
2. Propagation: Addition of more amino acid(s) to mRNA:
(a) adding charged TRNAi
k3j
RB − mRNA − tRNAii + tRNAjj ←→
d
!=
~RB − mRNA − tRNAi−j−k
k
dt
kI kpj kpk ~RB!~mRNA!
(tRNAii) (tRNAij)
(tRNAi) (tRNAj)
(tRNAkk) (A3)
By extending the expression for amino acids 1 to n,
We propose the hypothesis that step (a) is significantly
faster than step (b), on the grounds that charged species
are not involved in the reaction. This allows us to apply
the steady-state assumption, giving
508
K
tRNAii k4j
←→ RB − mRNA
tRNAjj
i
− tRNAi−j
j + tRNA
RB − mRNA
d
!=
~RB − mRNA − tRNA1→n
n
dt
kI
~RB!~mRNA!~tRNAn!
kp1
n
)
i=1
S
kpi
~tRNAii!
~tRNAi!
D
(A4)
References
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the leader region of the tryptophan operon of E. coli involves tryptophan or its metabolic product. J. Mol. Biol. 103: 339–349.
Curran, J. F., Yarus, M. 1988. Use of tRNA suppressors to probe regulation
of E. coli release factor 2. J. Mol. Biol. 203: 75–83.
Elsenberg, S. P., Soll, L., Yarus, M. 1980. Role of tRNATrp and leader
RNA secondary structure in attenuation of the trp operon, p. 469–479.
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Kassavetis, G. A., Chamberlin, M. J. 1981. Pausing and termination of
transcription within the early region of bacteriophage T7. J. Biol.
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Kawai, S., Mizutani, S., Iijima, S., Kobayashi, T. 1986. On-off regulation
of the tryptophan promoter in fed-batch culture. J. Ferment. Technol.
64: 503–510.
Koh, B. T., Yap, M. G. S. 1993. A simple genetically structured model of
trp repressor-operator interactions. Biotechnol. Bioeng. 41: 707–714.
Landick, R., Yanofsky, C. 1984. Stability of an RNA secondary structure
affects in-vitro transcription pausing in the trp operon leader region. J.
Biol. Chem. 259: 11550–11555.
Landick, R., Carey, J., Yanofsky, C. 1985. Translation activates the paused
transcription complex and restores transcription of the trp operon
leader region. Proc. Natl. Acad. Sci. USA 82: 4663–4667.
Landick, R., Yanofsky, C. 1987. In: F. C. Neidhardt (ed.), Escherichia coli
and Salmonella typhimurium: Cellular and molecular biology, Vol. 2,
pp. 1276–1301. American Society for Microbiology, Washington, DC.
Lee, F., Yanofsky, C. 1977. Transcription termination at the trp operon
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