Báo cáo khoa học: Dynamic model of Escherichia coli tryptophan operon shows an optimal structural design

doi:10.1046/j.1432-1033.2003.03641.x

Eur. J. Biochem. 270, 2644–2651 (2003) (cid:1) FEBS 2003

Dynamic model of Escherichiacolitryptophan operon shows an optimal structural design

Sharad Bhartiya1, Subodh Rawool1 and K. V. Venkatesh1,2 1Department of Chemical Engineering and 2School of Biosciences and Bioengineering, Indian Institute of Technology, Bombay, Mumbai, India

ranges from a rapid and underdamped to a sluggish and highly overdamped response. To test model fidelity, simu- lation results are compared with experimental data available in the literature. We further demonstrate the significance of the biological structure of the operon on the overall per- formance. Our analysis suggests that the tryptophan operon has evolved to a truly optimal design.

Keywords: trp operon; transcriptional control; dynamic modeling; tryptophan; feedback repression. A mathematical model has been developed to study the effect of external tryptophan on the trp operon. The model accounts for the effect of feedback repression by tryptophan through the Hill equation. We demonstrate that the trp operon maintains an intracellular steady-state concentration in a fivefold range irrespective of extracellular conditions. Dynamic behavior of the trp operon corresponding to varying levels of extracellular tryptophan illustrates the adaptive nature of regulation. Depending on the external tryptophan level in the medium, the transient response

and experimental data corresponding to the initial dynamic behavior. Further, their work does not discuss variation of intracellular tryptophan concentration.

Cells use genetic switches to activate or repress expression of genes to a given stimuli. Many genetic switches have been discovered to date. Examples include lac operon [1], trp operon [2], k-phage switch [3] and gal switch [4]. The trp operon has been the subject of numerous studies, both, at the molecular level [5–11] and through mathematical models [12–21]. Experiments in molecular biology have demonstra- ted the basic biological structure of the operon. Although quantification and analysis of these switches have been attempted, the specific effect of the genetic structure on the design of these switches is not yet clear. In this article, we focus on modeling and analysis of the trp operon with an emphasis on its structural properties.

All of the above mathematical models [12–21] do not provide insights into the design principle of the trp operon. Mathematical models can be used to answer questions relating to the evolutionary structure of the operon. For example, why has the repressor evolved to bind two tryptophan? Does the autoregulation of molecules of tryptophan make the switch sensitive? What are the limits on the intracellular tryptophan concentration due to regulation by the operon? In this paper, we develop a new model of the trp operon to answer the above questions. Our model is related to the model by Santillan & Mackey [17] in terms of inclusion of physical phenomena during regulation. However, unlike their model, which uses rate expressions to explain the behavior of the trp operon, the model developed herein employs the Hill equation to describe translation and transcription processes. Use of the Hill equation explicitly builds information about the operon structure in the model.

Initial models (e.g., see [13]) lacked details of pheno- menological interactions in the operon. Due to lack of experimental data, subsequent models (e.g., see [12]) only tested sensitivity of model parameters. However, no new biological insights were provided. Recently, Santillan & Mackey [17] reported a model incorporating repression, enzyme feedback inhibition, transcriptional attenuation and autoregulation. The model also incorporated the effect of tryptophan transport from the medium as well as the various time delays involved in the transcription and translation processes. They compared model prediction of dynamic enzyme synthesis in complete absence of trypto- phan in the medium with experimental data. Results showed that the model adequately captured the steady-state char- acteristics of enzyme synthesis. However, considerable discrepancy is observed between the model predictions

Correspondence to K. V. Venkatesh, Indian Institute of Technology, Bombay, Powai Mumbai, Maharastra 400076, India, Fax: + 91 22 25723480, Tel.: + 91 22 25767223, Email: venks@che.iitb.ac.in (Received 2 February 2003, revised 10 April 2003, accepted 28 April 2003)

For purposes of model development, the various cellular processes are categorized into two time scales. Processes with fast dynamics are assumed to be at quasi steady-state. These are the proteinÆDNA interaction, tryptophan–protein interaction, transcription and extracellular import of tryp- tophan. On the other hand, translation and tryptophan synthesis are characterized as slow processes. Thus, at a particular intracellular concentration of tryptophan, the status of the trp operon is obtained in the form of the Hill equation. The Hill coefficient in the Hill equation illustrates the sensitivity of the response to tryptophan. If the value of the Hill coefficient equals one, then it demonstrates a typical Michalis–Menten response. Important biological functions like adaptation and transcription require a more sensitive response [22]. For such sensitive responses, the Hill coeffi- cient is greater than one.

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Based on the above ideas, we have addressed some of the fundamental questions relating to the architecture of the trp operon. The model has been further used to study the effect of extracellular tryptophan on the performance of the switch.

Methods

Model description

Fig. 1. Schematic representation of autocatalytic regulation by the tryptophan operon. Tryptophan concentration is influenced by (a) enzyme synthesis (E) with kinetic constant k1, (b) enzyme catalyzed reaction for tryptophan synthesis from a nitrogenous substrate (NS) with kinetic constant kd, and (c) instantaneous uptake from the envi- ronment. Sufficient availability of tryptophan, T, leads to binding with the aporepressor molecule R, with a dissociation constant K1. The resulting holorepressor next binds with the free operon, O, with a dissociation constant K2, resulting in transcriptional and translational repression of enzyme synthesis.

Santillan & Mackey [17] considered four elements that contribute to the dynamic autoregulation observed in the trp operon: (a) availability of free operon for transcription of mRNA molecules; (b) description of free ribosome- binding sites of the structural genes; (c) synthesis of the catalytic enzyme (anthranilate synthase); (d) availability of the amino acid as determined by both, intracellular synthesis and uptake of extracellular tryptophan.

and complexes by equilibrium kinetics and stoichiometry. Dissociation constants are related to the concentrations as follows:

ð4Þ K1 ¼ ðR (cid:6) TÞ RT

In addition to repression and enzyme feedback inhibi- tion, they also modeled transcriptional attenuation and included various time delays to account for assembly and transport. In the current work, binding of the free operon by the holorepressor and the production of trp mRNA are assumed to be fast processes relative to the synthesis of the enzyme and tryptophan, and therefore, not rate limiting. The resulting quasi steady-state free operon concentration is represented by a Hill equation that is traditionally used for quantifying enzyme synthesis. Model simulations validate the steady-state assumptions made, thereby, focusing attention on important dynamic attri- butes of regulation in trp operon. The next section details the steady-state modeling of free operon concentration, which in turn, determines the dynamic rates of synthesis of the enzyme and tryptophan. The reduced model is also used to validate the interactions present in the structure of the operon. ð5Þ K1 ¼ ðRT (cid:6) TÞ RT2

Quasi steady state model for free operator concentration ð6Þ Kd ¼ ðO (cid:6) RT2Þ ORT2

An overall mass balance for the total operator and tryptophan concentrations yields the following:

ð7Þ Ot ¼ O þ ORT2

ð8Þ Tt ¼ T þ RT þ 2RT2 þ 2ORT2

ð9Þ Rt ¼ R þ RT þ RT2 þ ORT2

K1

Subscript t denotes total concentration of the species. Fraction of the total free operator concentration, p, is defined as follows: Trp-aporepressor is a dimer molecule [6]. As a first step, a single tryptophan molecule, T, binds with the aporepressor molecule, R, to form the intermediate, RT. Next, another tryptophan molecule binds with RT in a sequential and noncooperative manner [5,7] to yield the holorepressor, RT2. The holorepressor binds with the free operator, O, and forms operon–holorepressor complex, ORT2, thus, repress- ing tryptophan synthesis. Gene expression can ensue only if the state of operon is free of any binding by the holorepressor. A schematic representation of the trp operon system is provided in Fig. 1. Formation of the complexes can be described by the following equations:

K1

p ¼ ð10Þ RT R þ T (cid:2)(cid:2)*)(cid:2)(cid:2) ð1Þ O Ot

Kd

RT þ T (cid:2)(cid:2)*)(cid:2)(cid:2) ð2Þ RT2

t ¼

Solving Eqn (4)–(10), we obtain Tt* in terms of p, Ot* and Rt*, where the superscript * denotes normalization with respect to Kd. For example, Tt* ¼ Tt/Kd. ð3Þ O þ RT2 (cid:2)(cid:2)*)(cid:2)(cid:2) ORT2 (cid:2) (cid:1) (cid:2) (cid:1) T(cid:7) (cid:6) þ RT(cid:7) þ 2 þ 2ð1 (cid:2) pÞ (cid:6) O(cid:7) t 1 (cid:2) p p 1 (cid:2) p p K1 RT(cid:7) (cid:8) Kd ð11Þ where, K1 and Kd are the dissociation equilibrium constants for the respective interactions. Assumption of steady-state behavior constrains concentrations of the various molecules

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t (cid:2) R(cid:7) t

where RT* is given by, s ! (cid:1) þ ð1 (cid:2) pÞ (cid:6) O(cid:7) RT(cid:7) ¼ (cid:2)1 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (cid:2) 1 (cid:2) free operator concentration, that is, k1pOt and (b) concen- tration change owing to dilution as the cell grows. Thus the enzyme concentration neglecting degradation is represented as follows: 1 (cid:2) p p 4p 1 (cid:2) p

t

ð12Þ (cid:6) (cid:2) lE ð17Þ ¼ k1Ot 1 (cid:2) p 2p dE dt KgH i;1 i;1 þ TgH KgH

The fraction of free operator concentration, p, was evalu- ated for a given Ot*, and Rt* using Eqns (11) and (12) at various normalized tryptophan concentrations (Tt*). Further, a form of the Hill equation as given below was used to represent the fractional response:

t

p ¼ ð13Þ where k1 is the rate constant for protein expression, and l represents the specific growth rate. The explicit role of transcriptional attenuation is not considered as repression of transcription is the main mechanism in the operon regulation. It has been shown by Yanofsky et al. [11] that the regulation due to transcriptional attenuation is only 10% of the total feedback effect by tryptophan. T(cid:2)gH t i;1 þ T(cid:2)gH K(cid:2)gH

Tryptophan activity inside the cell is made up of two components, synthesized tryptophan, Ts, and exogenous tryptophan transported inside the cell, To. Thus, the total intracellular activity Tt can be represented as,

ð18Þ Tt ¼ Ts þ To

Santillan & Mackey [17] assumed fast transport of exogen- ous tryptophan. They modeled extracellular tryptophan To by the following static equation:

t

ð19Þ To ¼ To;max (cid:8) p ¼ ð14Þ þ e Text (cid:7) Text 1 þ Tt f where gH is the Hill coefficient and Ki,1 is the half saturation constant (that is Ki,1 ¼ Tt|P ¼0.5). Tt, in Eqn (13) represents the intracellular concentration of tryptophan due to synthesis by the trp operon and extracellular import. It is noted that in the above equation, the value of gH is positive. Thus, the explicit negative exponent ensures that Eqn (13) represents repression by tryptophan. The above equation can be further simplified as below: KgH i;1 i;1 þ TgH KgH

t jp¼0:9

The empirical parameters gH and Ki,1 indirectly capture the biophysical description by representing the number of binding sites on the repressor and the binding constants, respectively. The transcription is assumed linear to mRNA concentration, which is true for most prokaryotic systems and especially for E. coli [23]. The resulting production of enzyme due to free operator concentration (described by Eqn 14) is an overall manifestation of gene expression. Hill coefficient, gH, can be evaluated considering the values of Tt* at p ¼ 10% and p ¼ 90% as follows: where Text is the extracellular tryptophan concentration provided in the medium. To,max is the maximum intracel- lular tryptophan concentration which takes into account the consumption and dilution of the external tryptophan in the cell. Parameters f and e represent inhibition of tryptophan transport and Michalis–Menten saturation constant, respectively. Thus, the tryptophan uptake rate depends on the intracellular tryptophan concentration (which, as seen below, depends on synthesis and consumption of trypto- phan as well as on dilution due to cell growth). Enzyme catalyzed synthesis of tryptophan is given by, ð15Þ gH ¼ (cid:8) (cid:7) log T(cid:7) (cid:2) g ð20Þ log 81 t jp¼0:1=T(cid:7) ¼ k2E (cid:2) lTs dTs dt Ki;2 Ki;2 þ Tt Ts Kg þ Ts

Equation (15) is a general equation to obtain the value of the Hill coefficient for a regulatory system with a repressor. Thus, upon substituting Eqn (11) into Eqn (15), we obtain the expression of gH,

9 þ RT(cid:7)jp¼0:9 þ 2

K1 Kd(cid:6)RT(cid:7)jp¼0:1 K1 Kd(cid:6)RT(cid:7)jp¼0:9

log 81 gH ¼ ! log (cid:6) 9 þ RT(cid:7)jp¼0:1 þ 18 þ 1:8 (cid:6) O(cid:7) t (cid:6) 1 9 þ 0:2 (cid:6) O(cid:7) t

ð16Þ The first term in above equation represents enzymatic synthesis of tryptophan. It includes repression of the enzyme activity by tryptophan [17]. The second term on the right hand side represents consumption of tryptophan for protein synthesis in cell and the third term represents dilution due to cell growth. Initial conditions used to generate results presented in the next section are Ts(t ¼ 0) ¼ 0, and E(t ¼ 0) ¼ 0. The values of various parameters used are summarized in Table 1 and are referred from the work of Santillan & Mackey [17].

Results

The steady state analysis, thus, yields the fractional expres- sion of enzyme at various total intracellular tryptophan con- centration. Parameters Ki,1 and gH identified from the above analysis are used in the dynamic model presented below.

Dynamic model for tryptophan production

Equation (16) yields the value of gH at different operator concentration (Ot*). A plot of the same (Fig. 2) shows that at lower concentrations of Ot*, gH tends towards 1.92 and steeply drops towards zero at higher concentrations, where the in vivo aporepressor concentration becomes limiting. At in vivo operator concentration in E. coli, the gH and Ki,1 Enzyme synthesis (E) is governed by (a) rate of enzyme generation by expression of structural genes determined by

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Table 1. Tabulation of the model parameters used in simulation of in vivo enzyme and tryptophan synthesis in E. coli [17].

Parameter

Value

e f g Kd Ki,1 Ki,2 kg K1 k1 k2 Ot Rt To,max l

0.9 lM 380 lM 25 lM-min)1 0.0026 lM 3.53 lM 810 lM 0.2 lM 60.3 lM 65 min)1 25 min)1 0.0033 lM 0.8 lM 100 lM 0.01 min)1

gH ¼ 1.92. The results were obtained by solving Eqns (17)–(21) using the ODE15S SOLVER in MATLAB (version 6.0.0.88, Release 12; The Mathworks Inc.).

Fig. 2. Steady state variation of Hill Coefficient, gH, with normalized operator concentration, Ot* (Ot/Kd). The arrow shows in vivo concen- tration of Ot* in E. coli.

Figure 3 shows the dynamic response of intracellular tryptophan concentration at various extracellular trypto- phan concentrations. If there is no tryptophan available in the medium, all intracellular tryptophan is synthesized by the enzymes transcribed by the operon (Fig. 3A), that is, Tt ¼ Ts. The dynamic response obtained in this case is seen to be typical of a higher order underdamped response, observed commonly in feedback control systems. With an increase in extracellular tryptophan concentration, the total intracellular tryptophan, Tt, results from contributions by both, enzyme synthesis (Ts) and extracellular transport (To). The response is observed to be typical of an overdamped, second or higher order system (Fig. 3B). A tenfold increase in extracellular concentration leads to complete repression of trp enzymes resulting in the total intracellular tryptophan concentration to equal that trans- ported from the medium, that is, Tt ¼ To (Fig. 3C). Figure 3D shows dynamic enzyme levels for the three cases discussed above. In the absence of extracellular tryptophan, the enzyme profile exhibits a corresponding underdamped response as exhibited by the intracellular tryptophan (see Fig. 3A). Further, the profile shows a good match to the experimental data points obtained by Yanofsky and Horn [9]. Specifically, the model captures the initial oscillations suggested by the data points as well as the final steady-state behavior. However, further experimental work is needed to verify reproducibility of the transient oscillations. With an increase in extracellular tryptophan concentration, the enzyme levels decrease and exhibit overdamped response. In case of saturation by the extracellular tryptophan, Fig. 3C shows no tryptophan synthesis by enzyme. How- ever, as seen from the corresponding case (curve iii in Fig. 3D), about 10% of enzyme is transcribed by the operon. As tryptophan also inhibits enzyme activity, the 10% enzyme synthesis at high extracellular tryptophan levels is essentially inactive.

values obtained for the tryptophan repression regulatory system are 1.92 and 3.5 lM, respectively (as indicated by an arrow in Fig. 2). Thus, the probability of gene expression, p, in terms of tryptophan concentration, Tt, is given by,

t

ð21Þ p ¼ K1:92 K1:92 i;1 i;1 þ T1:92

Equation (21) is the key result of the steady-state analysis. It provides a simpler representation of gene expression, which was described by two dynamic state equations by Santillan & Mackey [17]. The gH value of greater than unity indicates that the trp switch is sensitive in response to tryptophan feedback. If only one tryptophan molecule binds to the aporepressor, the gH value would equal one (result not shown). Thus, the sensitivity is a result of the stoichiometry of the two molecules binding to the repressor (see Eqns 1–3). this section presents results using The remainder of The model is used to answer questions pertaining to the role of the sensitive regulation of tryptophan operator. To study this, the simulations were reworked at various Hill coefficient values. Figure 4 shows the steady-state concen- tration of intracellular tryptophan and the transcribed enzyme for different extracellular tryptophan concentra- tions for the cases where gH ¼ 1 (Fig. 4C,D) and gH ¼ 2 (Fig. 4A,B). For a wild-type strain (gH ¼ 1.9), the maxi- mum intracellular tryptophan concentration that can be achieved is 82 lM and corresponds to saturation of extra- cellular tryptophan [12]. The maximum tryptophan syn- thesized through transcribed enzyme, which corresponds to Text ¼ 0, is roughly 20% of the maximum intracellular tryptophan concentration of 82 lM. This indicates that the intracellular tryptophan concentration can vary only between 16 and 82 lM in E. coli for all conditions. The trp switch maintains intracellular tryptophan within a fivefold range. Such regulation is achieved by a switch-like response of enzyme synthesis as demonstrated by the Hill coefficient of 1.92 (Fig. 4B). It is noted that the steep regulation of enzyme synthesis and activity occurs beyond 0.2 lM of extracellular tryptophan. To study the effect of one binding site on the repressor molecule for tryptophan, Eqns (11) and

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Fig. 3. Dynamic behavior of tryptophan (T) and enzyme concentrations (E) for different extracellular tryptophan concentrations. Tryptophan and enzyme concentrations are normalized by 82 lM and 1 lM, respectively. (A) Text ¼ 0, (B) Text ¼ 0.14 lM, (C) Text ¼ 1.4 lM, (D) enzyme concentrations for (i) Text ¼ 0; (ii) Text ¼ 0.14 lM; (iii) Text ¼ 1.4 lM.

For gH ¼ 1, no overshoot is observed. Increasing gH further to 5 does not show any variation in the enzyme and tryptophan profiles as compared to gH ¼ 2.

In summary, we have quantified the regulatory behavior in trp operon using a reduced model. The model is able to capture the underdamped dynamic transient as observed in experimental data (Fig. 3D). Further, the model explicitly addresses the issue of correlating sensitivity of regulation to its structure. (12) were used to calculate the corresponding values of gH and Ki,1 and were found as 1 and 0.2 lM. Based on these new parameter values for one binding site, the steady-state response of tryptophan and enzyme concentrations are shown in Fig. 4C,D, respectively. It is clear that with one binding site, the operon can maintain a weak regulation within a twofold range of tryptophan concentration. It is noted that unlike the two binding site case, here the regulation is not switch like. Further, the switch responds at a much lower extracellular tryptophan concentration.

Discussion

Tryptophan, an amino acid, is essential for synthesis of proteins in the cell. However, its synthesis is expensive for the cell in terms of energy and nutrient requirements. The trp regulation has, thus, evolved to optimally regulate the intracellular tryptophan concentration. We have demon- strated that the trp switch maintains a steady-state trypto- phan concentration in the cell. To achieve this, a tight regulation of the trp operon is essential depending on the availability of extracellular concentration. Figure 5 shows the effect of gH on the dynamic response of enzyme and tryptophan concentrations. Although, the steady-state enzyme levels are similar for the cases gH ¼ 1 and gH ¼ 2 (see Fig. 5A), the initial transients are markedly different. In contrast, Fig. 5B reveals a significant difference in the corresponding steady-state intracellular tryptophan concentrations. The tryptophan concentration at steady- state for gH ¼ 1 is about one-fourth the steady-state concentration achieved in E. coli cells in absence of extracellular tryptophan (16 lM). The maximum concen- tration attained during the transient when gH ¼ 2 is 37 lM.

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Fig. 4. Steady-state concentrations of normalized tryptophan and enzyme at varying extracellular tryptophan concentrations for gH = 1 and gH = 1.9. Tryptophan and enzyme concentrations are normalized by 82 lM and 1 lM, respectively. (A) tryptophan concentrations at gH ¼ 1.9, (B) enzyme concentrations at gH ¼ 1.9, (C) tryptophan concentrations at gH ¼ 1, (D) enzyme concentrations at gH ¼ 1.

loops, feedback

seen from Fig. 4C,

The first structural requirement to achieve this control is the feedback loop. The binding of the tryptophan to the repressor molecule acts as a sensing mechanism to regulate synthesis. This feedback mechanism is captured by the Hill equation, implying that if gH ¼ 0, there is no feedback typically regulation. Conventional employed in industrial control systems, share the common attribute of feedback of measurement of the regulated variable. The second structural requirement is to achieve a sensitive response to extracellular tryptophan concentra- tion. The trp operon switches between two steady state levels: (a) synthesis in absence of extracellular tryptophan and (b) transport of extracellular tryptophan into cell (see Fig. 4A). To achieve homeostasis of tryptophan concen- tration in the cell, transition between the two states requires sensitive regulation. The near-perfect regulation of intracellular tryptophan for values of Text £ 0.2 lM (see Fig. 4A) is a result of enzyme regulation to balance the availability of tryptophan in the medium. Stoichiometric binding of two molecules of tryptophan to the repressor in the operon is the cause of this sensitivity. We have quantified this sensitivity by the Hill coefficient. The trp switch has a Hill coefficient of 1.9, which is due to the repressor having two binding sites for tryptophan. For gH ¼ 1, that is, the sensitivity equivalent to a Michaelis– Menten type response, the switching between the two states (discussed above) is no longer sensitive. Further, the steady-state values of tryptophan are lower than that obtained for gH ¼ 1.9. As the synthesis of tryptophan due to enzyme shuts off com- pletely beyond Text ¼ 0.02 lM. This value is an order of magnitude lower than that for gH ¼ 1.9. It is further observed, that when gH ¼ 1, the operon fails to provide the minimum level of tryptophan observed in in vivo conditions. The response of the operon to very small levels of extracellular tryptophan makes it susceptible to noise. The response to small values of extracellular tryptophan is evident from Fig. 4D, where regulation of enzyme synthe- sis begins at Text ¼ 0.02 lM. For any gH value >2, the switch performs similar to the case of gH ¼ 2. This clearly

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Fig. 5. Dynamic behavior of normalized enzyme (Fig. 5A) and tryptophan (Fig. 5B) concentrations in absence of extracellular tryptophan for gH = 1 (solid line) and gH = 2.5 (dotted line). Tryptophan and enzyme concentrations are normalized by 82 lM and 1 lM, respectively.

demonstrates that E. coli may have evolved evolutionarily to have this structural detail.

case of tryptophan operon in absence of extracellular tryptophan. Another notable feature is the adaptation of the regulatory mechanism to the level of extracellular tryptophan. The dynamic responses of Ts in Figs 3A,B,C indicate that the regulator gain (seen from the corres- ponding enzyme levels in Fig. 3D) takes on a different value for the three cases. This nonlinear behavior is due to the feedback inhibition and repression. It can be noted from Fig. 5B, that the design criterion of overshoot for quick availability of tryptophan is not observed when gH ¼ 1.

for Although we have not taken into account the effect of transcription attenuation, model simulations describe the experimental data reasonably well. One reason could be that the operator repression mechanism is more dominant in trp system. Repression and attenuation accounts for about 80 and sixfold variations, respectively [11]. It is known that the attenuation mechanism is active only during very severe tryptophan deficiency. Numerical simulations by Xieu et al. [21] confirm the same. Koh et al. [15] have shown that the attenuation mechanism alone operates over a narrow trp concentration range of trp repressor 1–5 lM compared with 1–100 lM, mechanism.

The model developed in the current work brings into focus the time-scale separation of the fast processes such from the slower tryptophan import as extracellular processes (such as translation). This characteristic is used to endow modularity to mathematical representation of the operon. The consequent model reduction based on time-scale separation is, therefore, a more parsimonious representation relative to the model by Sanillan & Mackey [17].

The analysis presented in this article demonstrates the significance of the biological structure to the overall performance of the genetic switch. Depending on sensitivity required for a specific biological function, a unique struc- tural design may have evolved. The methodology demon- strated in this article maybe extended to analyze such structural details to evaluate steady state and dynamic response of different genetic structures.

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