1.1 Introduction

Understanding the mechanisms of gene expression is critically important for understanding the regulation of cellular behavior. Transcription of genes (messenger RNA (mRNA) synthesis), translation of mRNA (protein synthesis), degradation of mRNA and proteins, and protein–protein interactions are all involved in the control of gene expression where proteins may bind with DNA, with mRNA, and with other proteins, leading to complex networks of interactions. Cells have many more genes than they need to express under any given set of environmental conditions. Transcription and translation are energetically expensive processes, so cells should only express the genes required for the environmental circumstances in which they find themselves.

Certain genes, often termed housekeeping genes, are required to support basic life processes and are expressed continuously. The expression of many other genes, though, is contingent upon environmental or physiological factors. The expression of these regulated genes is controlled by the cell to ensure efficient use of its energy and materials. In this chapter we will focus on a set of regulated genes in Escherichia coli (E. coli), which are expressed only when lactose is the sole sugar available.

The most efficient point for controlling gene expression is at the level of transcription where the cell can control whether or not the gene is transcribed, at what rate it is transcribed, and under what conditions transcription occurs. Bacteria control transcription through the binding of specific proteins to their DNA. Some DNA binding proteins block transcription, while others cause the DNA to bend in a manner that facilitates the action of RNA polymerase. Still other proteins, the polymerase-associated sigma factors, confer sequence-specific binding ability on the RNA polymerase, allowing it to transcribe genes accurately. In the more complex eukaryotic cells of higher organisms, multiple proteins may bind to multiple sites in the DNA, and protein–protein interactions are also involved in the control of transcription. DNA sequences controlling transcription may be found at considerable distances from the gene to be controlled and may be brought to the vicinity of the RNA polymerase binding site (the promoter) by the bending of the DNA. This highly complex structure presents significant experimental challenges in the process of understanding and describing cellular behavior.

Mathematics provides a formal framework for organizing the overwhelming amounts of disparate experimental data and for developing models that reflect the dependencies between the system’s components. Different types of mathematical models have been developed in an attempt to capture gene regulatory mechanisms and dynamics.

Various broad classifications are used in reference to such models. Deterministic models generate the exact same outcomes under a given set of initial conditions while in stochastic models the outcomes will differ due to inherent randomness. Dynamic models focus on the time-evolution of a system while static models do not consider time as part of the modeling framework. Among the dynamic models, time-continuous models utilize time as a continuous variable, while in time-discrete models time can only assume integer values. Space-continuous models refer to situations where the model variables can assume a continuum of values while in space-discrete models those variables can only assume values from a finite set. Space-continuous models of gene regulation are often constructed in the form of differential equations (in the case of continuous time) or difference equations (in the case of discrete time) and focus on the fine kinetics of biochemical reactions. We will refer to such models as DE models. Discrete-time models built from functions of finite-state variables are referred to as algebraic models.

In a DE model, all variables assume values from within biologically feasible ranges. Modelers usually need comprehensive knowledge of the interactions between variables, which may include detailed information of recognized control mechanisms, rates of production and degradation, minimal and maximal biologically relevant concentrations, and so on. In an algebraic model only values from a finite set are allowed. The special case of a Boolean network allows only two states, e.g., 0 and 1, generally representing the absence or presence of gene products in a model of gene regulation. In contrast to DE models, the information necessary to construct a Boolean model requires only a conceptual understanding of the causal links of dependency. Thus, in general, DE models are quantitative while Boolean models are qualitative in nature.

Historically, DE models have been the preferred type of mathematical models used in biology. This type of dynamical modeling has proved to be essential for problems in ecology, epidemiology, physiology, and endocrinology, among many others. Boolean models were first introduced to biology in 1969 to study the dynamic properties of gene regulatory networks [1]. They are appropriate in cases where network dynamics are determined by the logic of interactions rather than finely tuned kinetics, which may often be unknown.

In this chapter we present some of the fundamentals of creating Boolean network models for one of the simplest and best understood mechanisms of gene regulation: the lactose (lac) operon that controls the transport and metabolism of lactose in E. coli. Since the seminal work by Jacob and Monod [2,3], the lac operon has become one of the most widely studied and best understood mechanisms of gene regulation. It has also been used as a test system for virtually every mathematical method of modeling gene regulation (see, e.g., [4–10]).

The rest of the chapter is organized as follows: In Section 1.2 we outline the basic structure of the lac operon. This section is meant only as a quick introduction and is not comprehensive in any way (see [11] for a more thorough introduction). Section 1.3 focuses on the construction and initial testing of a mathematical model with an emphasis on Boolean networks. A primer on Boolean algebra is included to make the chapter self-contained. We consider several Boolean models of the lac operon, then introduce and utilize the web-based suite of applications Discrete Visualizer of Dynamics[12] to perform initial testing and validation of the models. In Section 1.4 we turn to the question of determining the steady states (fixed points) of Boolean networks, casting the question in the broader context of polynomial dynamical systems and the use of Groebner bases for solving systems of polynomial equations. In Section 1.5 we point out directions for extending and generalizing the models and provide some concluding remarks regarding the possible use of this material in the undergraduate mathematics and biology curricula.

1.2 E. Coli and the LAC Operon

E. coli is a short rod-shaped bacterium which is a common intestinal resident of mammals and birds. It has been the object of extensive study for decades. DNA replication, transcription, and translation were all elucidated in E. coli before they were studied in eukaryotic cells. Its physiology is well-understood and its entire gene sequence is known. For an overview of its importance to the study of genetics, see, for example, [13].

Since it lives in the intestines, any given E. coli bacterium’s nutrition depends upon the diet of the animal whose digestive tract it inhabits. Digestion of the complex biomolecules in the foods consumed by the animal generally provides the bacterium with all of the simple biomolecules it needs. Digestion of starches provides the monosaccharide (simple sugar) glucose, digestion of proteins provides all of the amino acids, and, whenever milk is consumed by the host, E. coli is also exposed to lactose (milk sugar). Lactose is a disaccharide consisting of one glucose sugar linked to one galactose sugar. Galactose is a six carbon simple sugar which is an isomer of glucose. It has the same chemical formula as glucose but differs in the position of one hydroxyl group. Like glucose, galactose can be used as an energy source, although some additional enzymatic manipulation will be required.

In order to import sugars across their plasma membranes, cells must produce specific sugar transport proteins (e.g., glucose requires a glucose transporter; lactose requires a lactose transporter, and so on). Once the sugar has been imported into the cell, specific enzymes will act on the sugar, either to use it to make a required cellular molecule (in the constructive processes collectively called anabolism) or to break it down in order to harvest its chemical energy in the form of ATP (in the destructive processes collectively called catabolism). Glucose is the preferred energy source for all cells, as it is used in glycolysis, the first and most fundamental energy-providing pathway of both anaerobic and aerobic cellular respiration.

The interactions between sugars and their transport proteins and the enzymes involved in their utilization within the cell are very specialized and cells must have the capacity to make specific proteins for every sugar they take up and use. However, it would not make sense for the E. coli to make the lactose transport protein if no lactose is present in its environment. It would also be inefficient to make the enzyme necessary to break lactose into glucose and galactose if there were no lactose inside the cell. If a cell had to make all of its proteins all of the time, it would be expending a lot of cellular energy in the making of proteins for which it has no use. Instead, cells have the ability to make certain of their proteins only when the environmental conditions warrant. Such proteins are called inducible proteins, because their synthesis is induced as a consequence of a certain cellular condition. The proteins needed for lactose uptake and utilization are inducible proteins, as they are only made if lactose is present and glucose, the preferred energy source, is not. Their concentrations increase 1000-fold under these conditions. Inducible genes belong to the group of regulated genes, in that they are only transcribed under certain specific conditions.

In E. coli, when lactose is the only sugar present, the transporter protein lactose (lac) permease and the enzyme -galactosidase are both required in order to utilize it. Lac permease is a transmembrane protein which binds the disaccharide and brings it across the plasma membrane into the cytoplasm of the cell. The lac permease mediated transport of lactose through the cellular membrane is reversible, meaning that high levels of lactose inside the cell result in reverse transport of lactose to the outside of the cell. -galactosidase catalyzes the hydrolysis of lactose into glucose and galactose. It also catalyzes the rearrangement of lactose into allolactose. These two proteins, lactose permease and -galactosidase are produced by a tightly coordinated mechanism described by Francois Jacob and Jacques Monod and termed the lac operon. In the lac operon, due to the arrangement of its genes LacZ, LacY, and LacA, a single messenger RNA encodes both -galactosidase and lactose permease, as well as a third enzyme, transacetylase, not involved in the metabolism of lactose (see Figure 1.1). Thus, when the lac promoter is active, all three proteins are produced.

Adjacent to the lac genes lacZ, lacY, and lacA, is the gene LacI that encodes a regulatory protein, the lac repressor (see Figure 1.2A). When there is no lactose present in the cell’s environment, the lac repressor protein will bind to the operator, preventing the RNA polymerase from producing the lac mRNA. When lactose is present in the medium outside of the cell, a small amount of it will be transported into the cell by the few lactose permease molecules found in the plasma membrane. Once the lactose is inside the cell, it is converted to allolactose by the few -galactosidase molecules present. Allolactose binds the lac repressor and causes it to undergo a conformational change, such that it can no longer bind to the operator (see Figure 1.2B). As a result, the RNA polymerase is able to read right through, produce the mRNA, and the three proteins are produced. This causes more lactose to be brought into the cell by lactose permease and then broken down by -galactosidase. We say that the operon is on. When all of the lactose is used up, there will be no allolactose, so the lac repressor protein will bind the operator and the synthesis of mRNA will stop. Levels of lactose permease and -galactosidase will fall. The operon has turned itself off.

If both glucose and lactose are available to E. coli, glucose, the preferred energy source, is always used up before the bacterium begins to utilize the lactose. This is controlled by the mechanism of catabolite repression. Without it, the presence of lactose would cause allolactose to be made, preventing the lac repressor protein from binding the operator and allowing the transcription of the lac operon. In reality, the availability of glucose represses this process, and when both glucose and lactose are available, RNA polymerase is not able to initiate transcription sufficiently to ensure the levels of production of mRNA, -galactosidase, and lactose permease reached under the condition of exclusive lactose availability. This failure of transcription initiation is called catabolite repression.

The mechanism of catabolite repression requires a DNA-binding protein, called CAP (for catabolite activator protein), which is the product of the crp (for cAMP receptor protein) gene. The CAP protein is capable of binding DNA only in the presence of cAMP. This small molecule is a common cellular signal or messenger and is produced by the enzyme adenylate cyclase. When glucose is present, adenylate cyclase does not produce cAMP. When there is no glucose, cAMP is produced. When cAMP binds CAP, the CAP-cAMP complex attaches to the DNA at the lac promoter and facilitates the attachment of RNA polymerase at the lac promoter. If there is no cAMP present, the CAP protein can’t bind to the DNA, the RNA polymerase won’t bind at the lacpromoter, and the lac mRNA is not made. No mRNA means no -galactosidase or lactose permease, and the operon will be off.¹ The key to the entire system is cAMP. If glucose is present, E. coli uses the glucose and the catabolism of glucose keeps cAMP levels low. After the glucose is used up, cAMP levels rise. The CAP protein binds cAMP, and the CAP-cAMP complex binds the DNA and facilitates the attachment of RNA polymerase. Figure 1.3 presents a schematic of the whole lac operon regulatory mechanism.

1.3 Boolean Network Models of the LAC Operon

In this section we design some basic dynamical models of the lac operon mechanism, capable of reflecting its biological behavior. At the very minimum, a model should imply that when lactose is absent from the medium, the operon is off, and that when lactose is present but glucose is not, the operon is on. We consider Boolean models with various levels of complexity and compare the results. The material can be used as a gentle entry into mathematical modeling, especially for students without Calculus background (see Section 1.5 for more details).

1.4 Determining the Fixed Points of Boolean Networks

The dynamical behavior of a Boolean network model with nodes is described by the transition functions , namely

(1.9)

A point is a fixed point for the set of Eqs. (1.9), when plugging the values for into the functions , from Eqs. (1.9) returns the same values for . In other words, is a fixed point for the set of Eqs. (1.9) when

(1.10)

We seek a method for determining all points that satisfies the condition in Eqs. (1.10).

The brute force approach in this context would be to test all possible states for the fixed-point property defined by Eqs. (1.10). However, since the size of the state space grows exponentially with the number of nodes in the network, we already decided that going over the entire state space to test all of its elements is not practical (see Exercise 1.2). The approach we describe next makes it possible to rephrase the problem of finding all fixed points of a Boolean network as a problem of solving systems of polynomial equations and determining a computationally efficient way of finding the solutions. This method uses a computational algebra approach and is based on some fundamental results in abstract algebra and algebraic geometry and on the use of Groebner bases for solving systems of polynomial equations [25].

Equations (1.10) state that the fixed points of the model are the solutions of the system of equations

(1.11)

where, the functions , are Boolean functions.

The general theory we will be applying here describes a method for solving such systems of equations in the special case where the functions , are polynomials of the variables . This may appear to be a serious restriction since in the case of Boolean networks the transition functions do not seem to satisfy this condition. However, as we will see next, each Boolean function can be rewritten as a Boolean polynomial, where a Boolean polynomial of the (Boolean) variables is a sum of terms of the form with .

To convert any Boolean function into a Boolean polynomial, the Boolean operations AND, OR, and NOT need to be converted to polynomials as follows:

To verify that the Boolean and polynomial functions (1)–(3) are identical, we need to compare their values and verify that the two functions return the same values for the same inputs. Table 1.3 contains the proof that .

Since addition and multiplication are performed over the field {0,1}, the following addition and multiplication rules apply for any value

Once the system of Boolean Eqs. (1.10) is translated into a system of polynomial equations, finding the fixed points becomes a problem of solving a polynomial system of equations. This can be done by employing a technique from computational algebra involving Groebner bases. With this technique, it is generally possible to rewrite a system of polynomial equations in a much simpler form while preserving the set of solutions. Under some additional conditions, the reduced form of the system of equations is guaranteed to have a form that makes finding the solutions possible by back-substitution. Readers familiar with the method of Gaussian elimination for systems of linear equations will notice that finding a Groebner basis that diagonalizes the system of equations can be viewed as a generalization of the Gaussian elimination method for polynomial functions.

For readers with appropriate mathematical background in abstract algebra, including the basic theory of rings, fields, ideals, and varieties, an outline of the theory of Groebner bases, as it relates to solving systems of polynomial equations, is presented in the online Appendix 1. For what follows, however, we do not assume familiarity with this theory. Instead, we illustrate how knowing the diagonalized forms of the polynomial systems (obtained by determining the Groebner basis with the use of computational software) can help to solve the system of polynomial equations and determine the fixed points.

Various computer systems including Macaulay 2, MAGMA, CoCoA, SINGULAR, and others, have the capability of computing Groebner bases. For our needs, using the web-based SAGE interface for Macaulay 2 for determining the Groebner basis for a diagonalized reduction is easy and convenient (http://www.sagemath.org/). We suggest that you use the online option first. Select “Try SAGE Online” and register for a SAGE notebook account. We consider a few examples below. The first two illustrate how the methods can be used to solve systems of polynomial equations over the real numbers, a problem that the reader is likely to be more familiar with. We then use the method to determine the fixed points of the Boolean model of the lac operon defined by Eqs. (1.8).

(1.12)

Next, we need to find the Groebner basis for the functions that form the left-hand sides of the equations: . To compute the Groebner basis for these functions in SAGE, we use the following commands (click on “evaluate” after entering each line)²:

In the first command, the variables in the system are listed on the left. On the right, RR denotes the real numbers (meaning we want to work over the reals) and three is the number of variables. For our purposes, we will always need to use order=‘lex’.

After evaluating the last command, SAGE returns the Groebner basis:

It follows from the general theory outlined in the online Appendix 1 that the systems of Eqs. (1.12) has a solution set equivalent to the solution set of the system

Notice that this system is “diagonal” in the sense that if we solve the first equation for z, we can use the values in the second equation to solve for y and then in the third equation for x. After doing this, we obtain the solutions for the system, which leads to the solutions and .

Evaluating, SAGE returns the following Groebner basis for the set of functions in the left-hand sides of Eqs. (1.13):[x1, x2, x3, x4 + 1, x5 + 1, x6, x7, x8, x9].

Thus, the system of equations from Eqs.(1.13) has the same solution set as the system of equations

This gives the steady state for .

The results from Example 1.10 and Exercise 1.20 show that the Boolean model of the lac operon from Eqs. (1.8) has the right qualitative behavior, predicting that the operon is On only when external lactose is available and external glucose is not. In this case all variables of the model, except for the repressor protein, are present. When glucose is available, the operon is Off. All fixed points are biologically feasible.

Now that we understand how fixed points may be found as solutions of systems of polynomial equations, we can use DVD again to analyze the model and compare the results to those from Example 1.10 and from Exercise 1.20. In DVD you can either enter and run the model four times for the four different combinations of the parameter values or use the approach we took in Example 1.5 (2) and include an additional “variable” for each of the parameters with an update rule that keeps that variable constant for all time steps. The analysis in DVD confirms the fixed points from Example 1.10 and from Exercise 1.20 and affirms that the state space does not contain limit cycles.

1.5 Conclusions and Discussion

Utilization of lactose by the bacterium E.coli requires the production of two proteins, the transporter lactose permease and the enzyme -galactosidase. Efficient use of cellular energy and materials requires that the bacterium only make these two proteins when needed. The regulatory mechanism of the lac operon ensures the repression of these two proteins when glucose, the preferred energy source, is present and the coordinated production of these two proteins when lactose alone is present. In this chapter we show how Boolean networks can be used to model the control mechanisms of the lac operon system. The chapter can be used as an introduction to mathematical modeling without calculus. A brief primer of Boolean algebra is included, so virtually no mathematical prerequisites are required. We introduce specialized web-based software for testing and analyzing the models, which is a convenient way to separate theory from applications. For readers interested in the theoretical underpinnings, the chapter provides an online appendix (Appendix 1) that outlines the use of Groebner bases as it pertains to solving systems of polynomial equations.

The models considered in this chapter are relatively simple and capture only the most significant qualitative behavior of the lac operon – its ability to turn on its lactose utilization mechanism when glucose is absent from the external medium and lactose is present. It is clear even from these simple models, that threshold values separating concentrations with Boolean value 0 from those with Boolean value 1 need to be selected carefully based on the biology. We generally assume that a value of 0 and 1 indicate, respectively, “low” and “high” concentrations, where “low” does not always mean a concentration of zero. For concentrations of mRNA, lac permease, and -galactosidase, as well as for other proteins and enzymes involved in the lac operon control, which increase by a factor of thousands when the operon is turned on in comparison with their baseline concentrations, such threshold values can easily be selected. The concentrations of lactose, allolactose, and glucose, however, depend on the external environment and change more gradually. For those, “low” could truly mean “absent” and medium-level concentrations are biologically completely feasible. In the model from Eqs. (1.8), the authors take care to introduce designated Boolean variables for low lactose and allolactose to ensure that “low” means “some but not zero” [22]. Other models (see, e.g., [26,27] and Chapter 2 of this volume) consider Boolean frameworks within which it is possible to distinguish between low, medium, and high lactose concentrations.

Focusing on the medium range of lactose concentration is of particular interest since the lactose operon has been shown to exhibit bistability for medium levels of lactose. Bistability is the ability of a system to settle in one of two different fixed points under the same set of external conditions. Which of these fixed points the system will reach depends on its history. It has been known since the 1950s [28] that for medium lactose concentrations, both induced and uninduced cells can be observed in a population of E.coli. Cells grown in an environment poor in extracellular lactose will likely remain uninduced for medium lactose levels while cells grown in a lactose-rich environment will likely retain their induced state. Chapter 2 of this volume examines Boolean network models of the lac operon that can capture the bistability property of the system.

Boolean models are important from an educational perspective since they require only a minimal mathematics background. At the introductory level, the construction of a simple model may essentially amount to translating the system’s interactions represented by a biology “cartoon” into a directed graph (wiring diagram), followed by a subsequent translation into logical expressions (the update rules). This makes Boolean models ideal for an early (below-calculus level) introduction to mathematical models, removing the need for calculus or other mathematical prerequisites. For mathematics students, such models can be introduced in low-level finite mathematics or discrete mathematics courses and used to provide an early demonstration of the important link between mathematics and biology.

At the more advanced mathematics level, Boolean models can be generalized to finite dynamical systems (FDS) and used to provide an introduction to some serious theoretical mathematical questions or as a path to questions appropriate for student research projects. In this chapter we examined the questions of determining the fixed points of FDS. This leads to a question of solving systems of polynomial equations over a finite field, for which the theory of Groebner bases provides a practical solution. The algorithm is essentially a generalization of the well-known process of Gaussian elimination for solving systems of linear equations.

The actual implementation of the method requires the use of specialized software (as even the verification that a given set of polynomials is a Groebner basis for an ideal is labor intensive and virtually impossible to do by hand). Although there are several open-source computational algebra systems that compute Groebner bases (e.g., Macaulay 2, MAGMA, CoCoA, SINGULAR, and others), most such systems require download and installation. For the purposes of this chapter the web-based SAGE interface to Macaulay 2 is appropriate, as it requires only a few straightforward commands. The students can then focus on the output and its interpretation with regard to the question of solving polynomial systems of equations.

For mathematics students, we see the use of the chapter material to be threefold. On one hand, it introduces them to a new modeling approach that is currently not taught in any of the mainstream undergraduate mathematics courses. On the other hand, FDS models provide links to important mathematical theory and results in abstract algebra and algebraic geometry that can be further pursued in advanced-level courses or as independent student research projects. Finally, the topic provides evidence for the important connections between modern biology and modern mathematics, and can be used to highlight mathematical and systems biology as career paths for mathematics majors.

For biology students, the chapter can be used as an introduction to mathematical modeling without calculus prerequisites. Our experience indicates that the “just-in-time” approach for developing the necessary mathematical concepts as a way to formalize specific aspects of the biology works well for Boolean models. It allows students to focus on the logical links that determine the variable interactions instead of on the detailed kinetics needed for calculus-based models. Concurrent or subsequent introduction to such models in calculus or differential equations courses will allow students to reinforce the conceptual framework, further improve their mathematical sophistication, and solidify the retention of basic ideas.

Acknowledgments

The authors gratefully acknowledge the support of the National Science Foundation under the Division of Undergraduate Education award 0737467.

1.6 Supplementary Materials

The online appendix, additional files, and computer code associated with this article can be found, in the online version, at <http://dx.doi.org/10.1016/B978-0-12-415780-4.00021-1> and from the volume’s website http://booksite.elsevier.com/9780124157804

References

1. Kauffman S. Metabolic stability and epigenetics in randomly constructed gene nets. J Theor Biol. 1969;22:437–467.

2. Jacob F, Perrin D, Sanchez C, Monod J. L’Operon: groupe de gène à expression par un operatour. C.R Seances Acad Sci. 1960;250:1727–1729.

3. Jacob F, Monod J. Genetic regulatory mechanisms in the synthesis of proteins. J Mol Biol. 1961;3:318–356.

4. Cheng B, Fournier RL, Relue PA. The inhibition of Escherichia coli lac operon gene expression by antigene oligonucleotidesmathematical modeling. Biotechnol Bioeng. 2000;70:467–472.

5. Doi A, Fujita S, Matsuno H, Nagasaki M, Miyano S. Constructing biological pathway models with hybrid functional Petri nets. In Silico Biology. 2004;4:271–291.

6. Farina M, Prandini M. A mathematical model for genetic regulation of the lactose operon. In: Lecture Notes in Computer Science 2007;693–697. Bemorad A, ed. Hybrid systems: computation and control. vol. 4461.

7. Romero-Campero FJ, Pérez-Jiménez MJ. Modelling gene expression control using P systems: the lac operon, a case study. Biosystems. 2008;91:438–457.

8. Setty Y, Mayo AE, Surette MG, Alon U. Detailed map of a cis-regulatory input function. Proc Natl Acad Sci USA. 2003;100:7702–7707.

9. Tian T, Burrage K. A mathematical model for genetic regulation of the lactose operon. In: Lecture Notes in Computer Science 2005;1245–1253. Gervasi O, ed. Computational science and its applications—ICCSA 2005. vol. 3481.

10. van Hoek M, Hogeweg P. The effect of stochasticity on the lac operon: an evolutionary perspective. PLoS Comput Biol. 2007;3:e111.

11. Muller-Hill B. The lac operon: a short history of a genetic paradigm. Berlin: Walter De Gruyter, Inc.; 1996.

12. Vastani H, Jarrah AS, Laubenbacher R, Visualization of dynamics for biological networks. http://dvd.vbi.vt.edu/dvd.pdf.

13. Russell PJ. iGenetics: A Molecular Approach. San Francisco: Benjamin Cummings; 2006.

14. Santillán M, Mackey MC, Zeron E. Origin of bistability in the lac operon. Biophys J. 2007;92:3830–3842.

15. Yildrim N, Mackey MC. Feedback regulation in the lactose operon: a mathematical modeling study and Comparison with Experimental Data. Biophys J. 2003;84:2841–2851.

16. Yildirim N, Santillán M, Horike D, Mackey MC. Dynamics and bistability in a reduced model of the lac operon. Chaos. 2004;14:279–292.

17. Laubenbacher R. Algebraic models in systems biology. In: Anai H, Horimoto K, eds. Algebraic Biology. Tokyo: Universal Academy Press; 2005;33–40.

18. Allen E, Fetrow J, Daniel L, Thomas S, John D. Algebraic dependency models of protein signal transduction networks from time-series data. J Theor Biol. 2006;238:317–330.

19. Delgado-Eckert E. An algebraic and graph theoretic framework to study monomial dynamical systems over a finite field. Complex Systems. 2009;19:307–328.

20. Jarrah AS, Laubenbacher R, Veliz-Cuba A. The dynamics of conjunctive and disjunctive Boolean network models. Bull Math Biol. 2010;72:1425–1447.

21. Veliz-Cuba A, Jarrah AS, Laubenbacher R. Polynomial algebra of discrete models in systems biology. Bioinformatics. 2010;26:1637–1643.

22. Stigler B, Veliz-Cuba A, Network topology as a driver of bistability in the lac operon. 2008, arXiv:0807.3995. http://arxiv.org/abs/0807.3995.

23. Saez-Rodriguez J, Simeoni L, Lindquist JA, et al. A logical model provides insights into T cell receptor signaling. PLOS Comp Biol. 2007;3:e163.

24. Astronomers count the stars. BBC News, July 22, 2003. http://news.bbc.co.uk/2/hi/science/nature/3085885.stm. Retrieved June 25, 2012.

25. Laubenbacher R, Sturmfels B. Computer algebra in systems biology. Am Math Monthly. 2009;116:882–891.

26. Hinkelmann F, Laubenbacher R. Boolean models of bistable biological systems. Discrete and continuous dynamical systems. 2011;4:1442–1456.

27. Veliz-Cuba A, Stigler B. Boolean models can explain bistability in the lac operon. J Comput Biol. 2011;18:783–794.

28. Novick A, Weiner M. Enzyme induction as an all-or-none phenomenon. Proc Natl Acad Sci U S A. 1957;43:553–566.