Skip to main content
SpringerLink
Log in

Bistability and Asynchrony in a Boolean Model of the l-arabinose Operon in Escherichia coli

  • Original Article
  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

The lactose operon in Escherichia coli was the first known gene regulatory network, and it is frequently used as a prototype for new modeling paradigms. Historically, many of these modeling frameworks use differential equations. More recently, Stigler and Veliz-Cuba proposed a Boolean model that captures the bistability of the system and all of the biological steady states. In this paper, we model the well-known arabinose operon in E. coli with a Boolean network. This has several complex features not found in the lac operon, such as a protein that is both an activator and repressor, a DNA looping mechanism for gene repression, and the lack of inducer exclusion by glucose. For 11 out of 12 choices of initial conditions, we use computational algebra and Sage to verify that the state space contains a single fixed point that correctly matches the biology. The final initial condition, medium levels of arabinose and no glucose, successfully predicts the system’s bistability. Finally, we compare the state space under synchronous and asynchronous update and see that the former has several artificial cycles that go away under a general asynchronous update.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. The trp operon has been modeled in a discrete framework, but using Petri nets, see Simao et al. (2005).

  2. Between the writing and publication of this paper, the ADAM software was overhauled and replaced with a crowd-sourced version called TURING, currently in Beta by Honsy and Laubenbacher (2017).

References

  • Albert R (2004) Boolean modeling of genetic regulatory networks. In: Complex networks. Springer, Berlin, pp 459–481

  • Busenberg S, Mahaffy J (1985) Interaction of spatial diffusion and delays in models of genetic control by repression. J Math Biol 22(3):313–333

    Article  MathSciNet  MATH  Google Scholar 

  • Crick F et al (1970) Central dogma of molecular biology. Nature 227(5258):561–563

    Article  Google Scholar 

  • Chaouiya C, Naldi A, Thieffry D (2012) Logical modelling of gene regulatory networks with GINsim. In: van Helden J, Toussaint A, Thieffry D (eds) Bacterial molecular networks: methods and protocols. Springer, New York, pp 463–479

    Chapter  Google Scholar 

  • Cheng D, Qi H, Li Z (2011) Analysis and control of Boolean networks: a semi-tensor product approach. Springer Science & Business Media, Berlin

    Book  MATH  Google Scholar 

  • Doyle ME, Brown C, Hogg RW, Helling RB (1972) Induction of the ara operon of Escherichia coli B/r. J Bacteriol 110(1):56–65

    Google Scholar 

  • Dimitrova E, García-Puente LD, Hinkelmann F, Jarrah AS, Laubenbacher R, Stigler B, Stillman M, Vera-Licona P (2011) Parameter estimation for boolean models of biological networks. Theor Comput Sci 412(26):2816–2826

    Article  MathSciNet  MATH  Google Scholar 

  • De Jong H (2002) Modeling and simulation of genetic regulatory systems: a literature review. J Comp Biol 9(1):67–103

    Article  Google Scholar 

  • Dimitrova ES, Jarrah AS, Laubenbacher R, Stigler B (2007) A gröbner fan method for biochemical network modeling. In: International symposium on symbolic and algebraic computation, pp 122–126. ACM

  • Fauré A, Naldi A, Chaouiya C, Thieffry D (2006) Dynamical analysis of a generic boolean model for the control of the mammalian cell cycle. Bioinformatics 22(14):e124–e131

    Article  Google Scholar 

  • Greenblatt J, Schleif R (1971) Arabinose C protein: regulation of the arabinose operon in vitro. Nat New Biol 233(40):166–170

    Article  Google Scholar 

  • Hinkelmann F, Brandon M, Guang B, McNeill R, Blekherman G, Veliz-Cuba A, Laubenbacher R (2011) ADAM: analysis of discrete models of biological systems using computer algebra. BMC Bioinform 12(1):295

    Article  Google Scholar 

  • Hinkelmann F, Laubenbacher R (2011) Boolean models of bistable biological systems. Discrete Cont Dyn Sys Ser S 4(6):1443–1456

    Article  MathSciNet  MATH  Google Scholar 

  • Honsy A, Laubenbacher R (2017) TURING: algorithms for computation with finite dynamical systems. Published electronically at http://www.discretedynamics.org/

  • Jacob F, Perrin D, Sánchez C, Monod J (1960) L’opéron: groupe de gènes à expression coordonnée par un opérateur. C.R. Acad Sci 250:1727–1729

    Google Scholar 

  • Kauffman SA (1969) Metabolic stability and epigenesis in randomly constructed genetic nets. J Theor Biol 22(3):437–467

    Article  MathSciNet  Google Scholar 

  • Laubenbacher R, Stigler B (2004) A computational algebra approach to the reverse engineering of gene regulatory networks. J Theor Biol 229(4):523–537

    Article  MathSciNet  Google Scholar 

  • Laubenbacher R, Sturmfels B (2009) Computer algebra in systems biology. Am Math Monthly 116(10):882–891

    Article  MathSciNet  MATH  Google Scholar 

  • Murrugarra D, Veliz-Cuba A, Aguilar B, Arat S, Laubenbacher R (2012) Modeling stochasticity and variability in gene regulatory networks. EURASIP J Bioinform Sys Biol 2012(1):1–11

    Article  Google Scholar 

  • Ogden S, Haggerty D, Stoner CM, Kolodrubetz D, Schleif R (1980) The Escherichia coli L-arabinose operon: binding sites of the regulatory proteins and a mechanism of positive and negative regulation. Proc Natl Acad Sci 77(6):3346–3350

    Article  Google Scholar 

  • Raeymaekers L (2002) Dynamics of Boolean networks controlled by biologically meaningful functions. J Theor Biol 218(3):331–341

    Article  MathSciNet  Google Scholar 

  • Richard A (2010) Negative circuits and sustained oscillations in asynchronous automata networks. Adv Appl Math 44(4):378–392

    Article  MathSciNet  MATH  Google Scholar 

  • Richard A (2015) Fixed point theorems for Boolean networks expressed in terms of forbidden subnetworks. Theor Comput Sci 583:1–26

    Article  MathSciNet  MATH  Google Scholar 

  • Robert R (1980) Iterations sur des ensembles finis et automates cellulaires contractants. Linear Algebra Appl 29:393–412

    Article  MathSciNet  MATH  Google Scholar 

  • Remy É, Ruet P, Thieffry D (2008) Graphic requirements for multistability and attractive cycles in a Boolean dynamical framework. Adv Appl Math 41(3):335–350

    Article  MathSciNet  MATH  Google Scholar 

  • Saadatpour A, Albert I, Albert R (2010) Attractor analysis of asynchronous Boolean models of signal transduction networks. J Theor Biol 266(4):641–656

    Article  MathSciNet  Google Scholar 

  • Schleif R (2000) Regulation of the L-arabinose operon of Escherichia coli. Trends Genet 16(12):559–565

    Article  Google Scholar 

  • Shih M-H, Dong J-L (2005) A combinatorial analogue of the Jacobian problem in automata networks. Adv Appl Math 34(1):30–46

    Article  MathSciNet  MATH  Google Scholar 

  • Shmulevich I, Dougherty ER (2010) Probabilistic Boolean networks: the modeling and control of gene regulatory networks. SIAM

  • Snoussi EH (1998) Necessary conditions for multistationarity and stable periodicity. J Biol Syst 6(01):3–9

    Article  MATH  Google Scholar 

  • Saier MH, Roseman S (1976) Sugar transport. Inducer exclusion and regulation of the melibiose, maltose, glycerol, and lactose transport systems by the phosphoenolpyruvate: sugar phosphotransferase system. J Biol Chem 251(21):6606–6615

    Google Scholar 

  • Simao E, Remy E, Thieffry D, Chaouiya C (2005) Qualitative modelling of regulated metabolic pathways: application to the tryptophan biosynthesis in e. coli. Bioinformatics 21(suppl 2):ii190–ii196

    Article  Google Scholar 

  • Thomas R, d’Ari R (1990) Biological feedback. CRC Press, Boca Raton

    MATH  Google Scholar 

  • Thomas R (1973) Boolean formalization of genetic control circuits. J Theor Biol 42(3):563–585

    Article  Google Scholar 

  • Thomas R (1981) On the relation between the logical structure of systems and their ability to generate multiple steady states or sustained oscillations. In: Della Dora J, Demongeot J, Lacolle B (eds) Numerical methods in the study of critical phenomena. Springer, Berlin, pp 180–193

    Chapter  Google Scholar 

  • Veliz-Cuba A, Arthur J, Hochstetler L, Klomps V, Korpi E (2012) On the relationship of steady states of continuous and discrete models arising from biology. Bull Math Biol 74(12):2779–2792

    Article  MathSciNet  MATH  Google Scholar 

  • Veliz-Cuba A, Jarrah AS, Laubenbacher R (2010) Polynomial algebra of discrete models in systems biology. Bioinformatics 26(13):1637–1643

    Article  MATH  Google Scholar 

  • Veliz-Cuba A, Stigler B (2011) Boolean models can explain bistability in the lac operon. J Comp Biol 18(6):783–794

    Article  MathSciNet  Google Scholar 

  • Van Hoek MJA, Hogeweg P (2006) In silico evolved lac operons exhibit bistability for artificial inducers, but not for lactose. Biophys J 91(8):2833–2843

    Article  Google Scholar 

  • Yildirim N (2012) Mathematical modeling of the low and high affinity arabinose transport systems in Escherichia coli. Mol BioSyst 8(4):1319–1324

    Article  Google Scholar 

  • Yildirim N, Mackey MC (2003) Feedback regulation in the lactose operon: a mathematical modeling study and comparison with experimental data. Biophys J 84(5):2841–2851

    Article  Google Scholar 

  • Yildirim N, Santillan M, Horike D, Mackey MC (2004) Dynamics and bistability in a reduced model of the lac operon. Chaos 14(2):279–292

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Georgia, Athens, GA, USA

    Andy Jenkins

  2. Department of Mathematical Sciences, Clemson University, Clemson, SC, USA

    Matthew Macauley

Authors
  1. Andy Jenkins
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Matthew Macauley
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matthew Macauley.

Additional information

Partially supported by a National Science Foundation grant (DMS-1211691) and a Simons Foundation collaboration grant for mathematicians (Award #358242).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jenkins, A., Macauley, M. Bistability and Asynchrony in a Boolean Model of the l-arabinose Operon in Escherichia coli . Bull Math Biol 79, 1778–1795 (2017). https://doi.org/10.1007/s11538-017-0306-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11538-017-0306-1

Keywords

Mathematics Subject Classification

Search

Navigation