Dynamic regulation of the tryptophan operon: A modeling study and comparison with experimental data Moise´ s Santilla´ n* and Michael C. Mackey† Department of Physiology and Centre for Nonlinear Dynamics, McGill University, McIntyre Medical Sciences Building, 3655 Drummond Street, Montreal, QC, Canada H3G 1Y6 Edited by John Ross, Stanford University, Stanford, CA, and approved November 3, 2000 (received for review June 30, 2000) A mathematical model for regulation of the tryptophan operon is the operon, and the operon end product (tryptophan). Never- presented. This model takes into account repression, feedback theless, they consider neither feedback inhibition nor transcrip- enzyme inhibition, and transcriptional attenuation. Special atten- tional attenuation and neglect inherent time delays. tion is given to model parameter estimation based on experimental We present a mathematical model of the tryptophan operon data. The model’s system of delay differential equations is numer- regulatory system. This model considers repression, enzyme ically solved, and the results are compared with experimental data feedback inhibition, and transcriptional attenuation, as well as on the temporal evolution of enzyme activity in cultures of Esch- the system’s inherent time delays. In Section 2, an outline of the erichia coli after a nutritional shift (minimal ؉ tryptophan medium mathematical model is presented. A list of the model variables to minimal medium). Good agreement is obtained between the and symbols is given in Table 1. The model equations are shown numeric simulations and the experimental results for wild-type E. in Table 2. A list of all the parameters and their estimated values coli, as well as for two different mutant strains. is given in Table 3. The variables’ steady-state values are presented in Table 4. In Section 3, the numerical method used 1. Introduction to solve the model equations is described. The procedure to In recent decades, we have witnessed spectacular advances in numerically simulate a given set of experiments and the com- molecular biology, in particular the explosive growth in knowl- parison of the theory with the experiment are given. Some edge concerning gene control systems. However, the mathemat- concluding remarks are given in Section 4, along with a discus- ical modeling of these molecular regulatory systems lags far sion of the feasibility of the model and possible future directions. behind the experimental work. In other areas of biology (neu- Supplementary material (which is published on the PNAS web robiology, for instance), the consideration of experimental data site, www.pnas.org) is given in two sections. The equation for the within the context of biologically accurate and realistic mathe- dynamics of repression is derived in supplemental section A. This matical models has helped to sharpen experimental questions is a partial result of the development of the model. The estima- asked as well as interpretations of new data and to suggest new tion of all the model parameters is described in supplemental experiments. The present mathematical modeling study of the section B. tryptophan operon is an attempt to help close the gap between 2. The Model experimental and theoretical studies of regulation at the molec- ular level. The choice of the tryptophan operon was made In this section, we introduce a mathematical model of the trp deliberately because this operon and the lactose operon are the operon regulatory system. A schematic representation of this molecular systems most extensively studied as prototypic gene regulatory system is given in Fig. 1. As any other model, the control systems. Thus there is a large body of experimental data present one is oversimplified in some sense. Many simplifying on which to draw. assumptions (discussed below) are made during its development. The term ‘‘operon’’ was first proposed in a short paper in the However, it is our premise that the model still considers enough proceedings of the French Academy of Sciences in 1960 (1). of the system essential characteristics to reproduce some exper- From this paper, the so-called general theory of the operon was imental dynamic observations. In Table 1, a list of the model developed. This theory suggested that all genes are controlled by independent variables is presented. The exact meaning of each means of operons through a single feedback regulatory mech- one is discussed in the forthcoming paragraphs. anism: repression. Later, it was discovered that the regulation of For the purpose of the present model, the tryptophan operon genes is a much more complicated process. Indeed, it is not is considered to be constituted by the major structural genes, possible to talk of a general regulatory mechanism, as there are preceded by a controlling section, where both repression and many, and they vary from operon to operon. Despite modifica- transcription initiation take place. Consider all the trp operons tions (2), the development of the operon concept is considered in a bacterial (Escherichia coli) culture. The controlling sections one of the landmark events in the history of molecular biology. can be in one of three states: free (OF), repressed (OR), or bound Shortly after the operon concept was presented, a mathemat- by a mRNA polymerase (OP) (here the same symbols are used ical model for it was proposed (3). Bliss et al. (4) proposed a more to represent the chemical species and their concentration, unless detailed model for the tryptophan operon that considered otherwise stated). We assume that there is a single type of repression and feedback inhibition. The system’s inherent time repressor molecule, produced by the trpR operon, whose active delays, caused by transcription and translation, were also taken form competes with mRNA polymerase (mRNAP) to bind free into account. More recent experimental results reveal that the dynamics of the interaction between repressor and tryptophan This paper was submitted directly (Track II) to the PNAS office. molecules are different than considered in Bliss’ model. Fur- *Permanent address: Escuela Superior de Fı´sicay Matema´ticas, Instituto Polite´cnico Nacio- thermore, the Bliss model did not take into account another nal, 07738, Me´xico D.F., Me´xico. regulatory mechanism at the DNA level, which was discovered †To whom reprint requests should be addressed. E-mail: [email protected]. later and is called transcriptional attenuation. More recently, The publication costs of this article were defrayed in part by page charge payment. This other models have been proposed (5–7). They take into account article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C. (with more detail) interactions among the repressor molecules, §1734 solely to indicate this fact. 1364–1369 ͉ PNAS ͉ February 13, 2001 ͉ vol. 98 ͉ no. 4 Downloaded by guest on September 30, 2021 so, we point out that the estimated values of kr, kϪr, and kp reveal that the binding rate of repressor molecules to free operons is two orders of magnitude larger than the corresponding binding rate of mRNAPs. This fact justifies a quasisteady-state assump- tion for the repression process. From this assumption, the resulting equation for the dynamics of OF is given by Eq. 1 of Table 2. The details of its derivation are given in supplemental section A. The mRNA molecules synthesized by transcription encode five different polypeptides. These polypeptides are used to build up the enzymes that participate in the catalytic pathway that synthesizes tryptophan from chorismic acid. The first enzyme in this pathway (anthranilate synthase) is a complex of two TrpE and two TrpD polypeptides, which are, respectively, the first and second proteins encoded by the trp mRNA. From the regulatory point of view, anthranilate synthase is the most important of the enzymes in the catalytic pathway. This is because it catalyzes the first reaction in the tryptophan synthesis pathway and because it is subject to feedback inhibition by tryptophan. Because there is evidence supporting the assump- tion that the production rates of all five polypeptides encoded by the trp mRNA are very similar under normal conditions (8), we focus on the production of TrpE polypeptide and assume that the anthranilate synthase production rate is just one-half that of TrpE. Let MF represent the concentration of free TrpE-related ribo- some-binding sites. MF increases because of transcription. Never- theless, not all of the mRNAPs that initiate transcription produce BIOCHEMISTRY functional mRNAs. Many of them terminate transcription prema- turely depending on the availability of charged tRNATrp. The higher Fig. 1. Schematic representation of the tryptophan operon regulatory sys- Trp tem. See text for details. the concentration of charged tRNA , the more probable that a transcribing mRNAP aborts transcription at a premature stage. This regulatory mechanism is known as transcriptional attenuation. Trp controlling sections. Let RA denote the concentration of active Because the amount of charged tRNA depends on the trypto- repressor molecules. The repression process is assumed in this phan concentration, we assume that the probability of premature model to be a first-order reversible chemical reaction with transcription termination [A(T)] is just a function of tryptophan A T 2 forward and backward rate constants kr and kϪr, respectively. concentration. The functional form of ( ) is given by Eq. of These constants are estimated in supplemental section B and Table