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Origin of Bistability in the Lac Operon

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Origin of Bistability in the Lac Operon 3830 Biophysical Journal Volume 92 June 2007 3830–3842 Origin of Bistability in the lac Operon M. Santilla´n,*y M. C. Mackey,yz and E. S. Zeron§ *Unidad Monterrey, Centro de Investigacio´n y Estudios Avanzados del Instituto Polite´cnico Nacional, Monterrey, Me´xico; Escuela Superior de Fı´sica y Matema´ticas, Instituto Polite´cnico Nacional, Me´xico DF, Me´xico; yCentre for Nonlinear Dynamics in Physiology and Medicine, zDepartments of Physiology, Physics, and Mathematics, McGill University, Montreal, Canada; and §Departamento de Matema´ticas, Centro de Investigacio´n y Estudios Avanzados del Instituto Polite´cnico Nacional, Me´xico DF, Me´xico ABSTRACT Multistability is an emergent dynamic property that has been invoked to explain multiple coexisting biological states. In this work, we investigate the origin of bistability in the lac operon. To do this, we develop a mathematical model for the regulatory pathway in this system and compare the model predictions with other experimental results in which a nonmetabolizable inducer was employed. We investigate the effect of lactose metabolism using this model, and show that it greatly modifies the bistable region in the external lactose (Le) versus external glucose (Ge) parameter space. The model also predicts that lactose metabolism can cause bistability to disappear for very low Ge. We have also carried out stochastic numerical simulations of the model for several values of Ge and Le. Our results indicate that bistability can help guarantee that Escherichia coli consumes glucose and lactose in the most efficient possible way. Namely, the lac operon is induced only when there is almost no glucose in the growing medium, but if Le is high, the operon induction level increases abruptly when the levels of glucose in the environment decrease to very low values. We demonstrate that this behavior could not be obtained without bistability if the stability of the induced and uninduced states is to be preserved. Finally, we point out that the present methods and results may be useful to study the emergence of multistability in biological systems other than the lac operon. INTRODUCTION At the molecular level, biological systems function using two stability corresponds to a true switch between alternate and types of information: genes, which encode the molecular coexisting steady states, and so allows a graded signal to be machines that execute the functions of life, and networks of turned into a discontinuous evolution of the system along regulatory interactions, specifying how genes are expressed. several different possible pathways. Multistability has cer- Substantial progress over the past few decades in biochem- tain unique properties not shared by other mechanisms of istry, molecular biology, and cell physiology has ushered in a integrative control. These properties may play an essential new era of regulatory interaction research. Recent analysis role in the dynamics of living cells and organisms. More- has revealed that cell signals do not necessarily propagate over, multistability has been invoked to explain catastrophic linearly. Instead, cellular signaling networks can be used to events in ecology (1), mitogen-activated protein kinase cas- regulate multiple functions in a context-dependent fashion. cades in animal cells (2–4), cell cycle regulatory circuits in Because of the magnitude and complexity of the interactions Xenopus and Saccharomyces cerevisiae (5,6), the generation in the cell, it is often not possible to understand intuitively of switchlike biochemical responses (2,3,7), and the estab- the systems behavior of these networks. Rather, it has lishment of cell cycle oscillations and mutually exclusive cell become necessary to develop mathematical models and analyze cycle phases (6,8), among other biological phenomena. On the behavior of these models, both to develop a systems-level the other hand, there are also serious doubts that multistability understanding and to obtain experimentally testable predic- is the dynamic origin of some biological switches (9). Never- tions. This, together with the fact that DNA micro-arrays, theless, it is generally accepted that two paradigmatic gene- sequencers, and other technologies have begun to generate regulatory networks in bacteria, the phage l switch and the lac vast amounts of quantitative biological data, has accelerated operon (at least when induced by nonmetabolizable inducers), the shift away from a purely descriptive biology and toward a do show bistability (10–12). In the former, bistability arises predictive one. through a mutually inhibitory double-negative-feedback loop, Recent computer simulations of partial or whole genetic while in the latter, a positive-feedback loop is responsible for networks have demonstrated collective behaviors (commonly the bistability. called systems, or emergent, properties) that were not apparent The lac operon, the phage-l switch, and the trp operon, from examination of only a few isolated interactions alone. are three of the best known and most widely studied systems Among the various patterns of complex behavior associated in molecular biology. The inducible lac operon in E. coli is the with nonlinear kinetics, multistability is noteworthy. Multi- classic example of bistability. It was first noted by Monod and co-workers more than 50 years ago, although it was not fully recognized at the time. The bistable behavior of the lac operon has been the subject of a number of studies. It was Submitted November 27, 2006, and accepted for publication February 5, 2007. first examined in detail by Novick and Weiner (13) and Cohn Address reprint requests to M. Santilla´n, E-mail: [email protected]. and Horibata (14). Later experimental studies include those Ó 2007 by the Biophysical Society 0006-3495/07/06/3830/13 $2.00 doi: 10.1529/biophysj.106.101717 Origin of Bistability in the Lac Operon 3831 of Maloney and Rotman (15) and Chung and Stephanopou- TABLE 1 Full set of equations for the model of lactose operon los (10). Over the last few years, the lac operon dynamics regulatory pathway depicted in Fig. 1 have been analyzed mathematically by Wong et al. (16), Eq. No. Vilar et al. (17), Yildirim and Mackey (18), Santilla´n and _ M ¼ DkMPDðGeÞPRðAÞgMM (1) Mackey (19), Yildirim et al. (20), Tian and Burrage (21), and _ E ¼ kEM À gEE (2) Hoek and Hogeweg (22). This was possible because this _ L ¼ kLb ðLeÞb ðGeÞQ À 2f MðLÞB À g L (3) system has been experimentally studied for nearly 50 years L G M L A ¼ L (4) and there is a wealth of biochemical and molecular infor- Q ¼ E (5) mation and data on which to draw. Recently, Ozbudak et al. B E=4 ¼ ÀÁÀÁ (6) (23) performed a set of ingenious experiments that not only 1 pp 1 pcðGeÞÀÁkpc À 1 (7) confirm bistability in the lac operon when induced with the PDðGeÞ¼ 1 1 pppcðGeÞ kpc À 1 nonmetabolizable inducer thiomethylgalactoside (TMG), but nh K (8) p ðGeÞ¼ G also provide new and novel quantitative data that raise ques- c nh 1 nh KG Ge tions that may be answered via a modeling approach. 1 P ðAÞ¼ (9) Previous studies have analyzed bistability and its dynamic R j rðAÞ 1 1 rðAÞ 1 123 ðÞ1 1 j rðAÞ ðÞ1 1 j rðAÞ properties as a systems phenomenon. However, to our knowl- 2 3 4 edge, none of them have dealt with questions like: ‘‘How did KA (10) r A r ð Þ¼ max 1 multistability arise within the context of gene regulatory KA A Le networks?’’ or ‘‘What evolutionary advantages do bistable b Le (11) Lð Þ¼ 1 regulatory networks have when compared with monostable kL Le Ge ones?’’ In this article, we address these issues from a mathe- b ðGeÞ¼1 À f (12) G Gk 1 Ge matical modeling approach, basing our examination on the G L (13) dynamics of the lac operon. MðLÞ¼ kM 1 L Differential Eqs. 1–3 govern the time evolution of the intracellular THEORY concentration of mRNA (M), polypeptide (E), and lactose (L) molecules. Ge and Le, respectively, stand for the extracellular glucose and lactose Model development concentrations. A, Q, and B represent the intracellular allolactose, permease, and b-galactosidase molecule concentrations. The functions PD, PR, bL, A mathematical model for the lac operon is developed in and bG, respectively, account for the negative effect of external glucose on Appendices A–C. All of the model equations are tabulated in the initiation rate of transcription (via catabolite repression), the probability Table 1 and they are briefly explained below. The reader may that the lactose promoter is not repressed, the positive effect of external lac- tose on its uptake rate, and the negative effect of external glucose on lactose find it useful to refer to Fig. 1, where the lac operon regu- uptake (inducer exclusion). The expression 2fMM is the rate of lactose me- latory pathway is schematically represented. tabolism per b-galactosidase. Finally, pc and r represent internal auxiliary The model consists of three differential equations (Eqs. variables. 1–3), that respectively account for the temporal evolution of mRNA (M), lacZ polypeptide (E), and internal lactose (L) concentrations. ceptor protein (CRP) to form the CAP complex. Finally, Messenger RNA (mRNA) is produced via transcription of CAP binds a specific site near the lac promoter and enhances the lac operon genes, and its concentration decreases because transcription initiation. The probability of finding a CAP mole- of active degradation and dilution due to cell growth. The cule bound to its corresponding site is given by pc (Eq. 8) and value gM represents the degradation plus dilution rate, kM is is a decreasing function of Ge. Moreover, PD is an increas- the maximum transcription rate per promoter, and D is the ing function pc and, therefore, a decreasing function of Ge. average number of promoter copies per bacterium.
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