Comparison of deterministic and stochastic models of the lac operon genetic network.

The lac operon has been a paradigm for genetic regulation with positive feedback, and several modeling studies have described its dynamics at various levels of detail. However, it has not yet been analyzed how stochasticity can enrich the system's behavior, creating effects that are not observed in the deterministic case. To address this problem we use a comparative approach. We develop a reaction network for the dynamics of the lac operon genetic switch and derive corresponding deterministic and stochastic models that incorporate biological details. We then analyze the effects of key biomolecular mechanisms, such as promoter strength and binding affinities, on the behavior of the models. No assumptions or approximations are made when building the models other than those utilized in the reaction network. Thus, we are able to carry out a meaningful comparison between the predictions of the two models to demonstrate genuine effects of stochasticity. Such a comparison reveals that in the presence of stochasticity, certain biomolecular mechanisms can profoundly influence the region where the system exhibits bistability, a key characteristic of the lac operon dynamics. For these cases, the temporal asymptotic behavior of the deterministic model remains unchanged, indicating a role of stochasticity in modulating the behavior of the system.

[1]  T. Steitz,et al.  Crystal structure of lac repressor core tetramer and its implications for DNA looping. , 1995, Science.

[2]  J. Chen,et al.  Wild-type operator binding and altered cooperativity for inducer binding of lac repressor dimer mutant R3. , 1994, The Journal of biological chemistry.

[3]  G. Chang,et al.  Crystal Structure of the Lactose Operon Repressor and Its Complexes with DNA and Inducer , 1996, Science.

[4]  M. Calos,et al.  DNA sequence for a low-level promoter of the lac repressor gene and an ‘up’ promoter mutation , 1978, Nature.

[5]  B. Müller-Hill,et al.  Quality and position of the three lac operators of E. coli define efficiency of repression. , 1994, The EMBO journal.

[6]  J. Griffith.  Mathematics of cellular control processes. II. Positive feedback to one gene. , 1968, Journal of theoretical biology.

[7]  W. Reznikoff,et al.  The lac Promoter , 1980 .

[8]  J M Rosenberg,et al.  Kinetic studies of inducer binding to lac repressor.operator complex. , 1980, The Journal of biological chemistry.

[9]  Tianhai Tian,et al.  A Mathematical Model for Genetic Regulation of the Lactose Operon , 2005, ICCSA.

[10]  W. Vieth,et al.  Role of Active Transport of Inducer in Enzyme Biosynthesis and Process Scale-Up , 1982 .

[11]  J. Ross,et al.  Stability and relative stability in reactive systems far from equilibrium. II. Kinetic analysis of relative stability of multiple stationary states , 1977 .

[12]  B. Müller-Hill.  The lac Operon: A Short History of a Genetic Paradigm , 1996 .

[13]  C Cruz,et al.  Genetic studies of the lac repressor. XIV. Analysis of 4000 altered Escherichia coli lac repressors reveals essential and non-essential residues, as well as "spacers" which do not require a specific sequence. , 1994, Journal of molecular biology.

[14]  T. Kepler,et al.  Stochasticity in transcriptional regulation: origins, consequences, and mathematical representations. , 2001, Biophysical journal.

[15]  H. Okano,et al.  Mathematical description of gene regulatory units. , 2006, Biophysical journal.

[16]  J E Bailey,et al.  Genetically structured models forlac promoter–operator function in the Escherichia coli chromosome and in multicopy plasmids: Lac operator function , 1984, Biotechnology and bioengineering.

[17]  Ertugrul M. Ozbudak,et al.  Multistability in the lactose utilization network of Escherichia coli , 2004, Nature.

[18]  W. Gilbert,et al.  Mutants that make more lac repressor. , 1968, Proceedings of the National Academy of Sciences of the United States of America.

[19]  A. Riggs,et al.  lac repressor--operator interaction. II. Effect of galactosides and other ligands. , 1970, Journal of molecular biology.

[20]  M. Moo-young,et al.  Cell growth and extracellular enzyme synthesis in fermentations , 1973, Biotechnology and bioengineering.

[21]  W. Vieth,et al.  Regulation of lac operon expression in mixed sugar chemostat cultures , 1987, Biotechnology and bioengineering.

[22]  J. Chen,et al.  Deletion of lactose repressor carboxyl-terminal domain affects tetramer formation. , 1992, The Journal of biological chemistry.

[23]  D. Ramkrishna,et al.  Complex growth dynamics in batch cultures: Experiments and cybernetic models , 1991, Biotechnology and bioengineering.

[24]  S. Bell,et al.  Charting a course through RNA polymerase , 2000, Nature Structural Biology.

[25]  D. Sherrington.  Stochastic Processes in Physics and Chemistry , 1983 .

[26]  J. Miller,et al.  Mutations affecting the quaternary structure of the lac repressor. , 1976, The Journal of biological chemistry.

[27]  T. Elston,et al.  Stochasticity in gene expression: from theories to phenotypes , 2005, Nature Reviews Genetics.

[28]  B. Müller-Hill,et al.  The three operators of the lac operon cooperate in repression. , 1990, The EMBO journal.

[29]  T. Imanaka,et al.  A kinetic model of catabolite repression in the dual control mechanism in microorganisms. , 1977, Biotechnology and bioengineering.

[30]  W. Gilbert,et al.  ISOLATION OF THE LAC REPRESSOR , 1966, Proceedings of the National Academy of Sciences of the United States of America.

[31]  Leonor Saiz,et al.  DNA looping in gene regulation: from the assembly of macromolecular complexes to the control of transcriptional noise. , 2005, Current opinion in genetics & development.

[32]  J. Hasty,et al.  Noise-based switches and amplifiers for gene expression. , 2000, Proceedings of the National Academy of Sciences of the United States of America.

[33]  P. Swain,et al.  Gene Regulation at the Single-Cell Level , 2005, Science.

[34]  Daniel T. Gillespie.  On the calculation of mean first passage times for simple random walks , 1981 .

[35]  M. Cohn,et al.  ANALYSIS OF THE DIFFERENTIATION AND OF THE HETEROGENEITY WITHIN A POPULATION OF ESCHERICHIA COLI UNDERGOING INDUCED β-GALACTOSIDASE SYNTHESIS , 1959, Journal of bacteriology.

[36]  Jeffrey H. Miller,et al.  DNA sequence alteration resulting from a mutation impairing promoter function in the lac repressor gene , 1980, Molecular and General Genetics MGG.

[37]  F. Jacob,et al.  L'opéron : groupe de gènes à expression coordonnée par un opérateur [C. R. Acad. Sci. Paris 250 (1960) 1727–1729] , 2005 .

[38]  A Constantinides,et al.  Modeling the role of cyclic AMP in catabolite repression of inducible enzyme biosynthesis in mircobial cells , 1978, Biotechnology and bioengineering.

[39]  K. Toda,et al.  Dual control of invertase Biosynthesis in chemostat culture , 1976, Biotechnology and bioengineering.

[40]  A. Kepes.  Études cinétiques sur la galactoside-perméase d'Escherichia coli , 1960 .

[41]  J. Lee,et al.  Construction of a dimeric repressor: dissection of subunit interfaces in Lac repressor. , 1994, Biochemistry.

[42]  D. Gillespie.  A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions , 1976 .

[43]  M. Caruthers,et al.  Binding of synthetic lactose operator DNAs to lactose represessors. , 1977, Proceedings of the National Academy of Sciences of the United States of America.

[44]  J. Mahaffy,et al.  Stability analysis for a mathematical model of the lac operon , 1999 .

[45]  J. Keasling,et al.  Mechanistic modeling of prokaryotic mRNA decay. , 1997, Journal of theoretical biology.

[46]  A. Arkin,et al.  Simulation of prokaryotic genetic circuits. , 1998, Annual review of biophysics and biomolecular structure.

[47]  J. Monod,et al.  [Operon: a group of genes with the expression coordinated by an operator]. , 1960, Comptes rendus hebdomadaires des seances de l'Academie des sciences.

[48]  Liskin Swint-Kruse,et al.  Perturbation from a distance: mutations that alter LacI function through long-range effects. , 2003, Biochemistry.

[49]  K. Matthews,et al.  Glycine Insertion in the Hinge Region of Lactose Repressor Protein Alters DNA Binding* , 1999, The Journal of Biological Chemistry.

[50]  Elizabeth A. Burroughs,et al.  LOCA 1.0 Library of Continuation Algorithms: Theory and Implementation Manual , 2002 .

[51]  W. McClure,et al.  Mechanism of CRP-cAMP activation of lac operon transcription initiation activation of the P1 promoter. , 1984, Journal of molecular biology.

[52]  Daniel T. Gillespie,et al.  A pedestrian approach to transitions and fluctuations in simple nonequilibrium chemical systems , 1979 .

[53]  J E Bailey,et al.  Genetically structured models for lac promoter–operator function in the chromosome and in multicopy plasmids: Lac promoter function , 1984, Biotechnology and bioengineering.

[54]  R Ramakrishna,et al.  Cybernetic modeling of growth in mixed, substitutable substrate environments: Preferential and simultaneous utilization. , 1996, Biotechnology and bioengineering.

[55]  M. Lewis,et al.  A closer view of the conformation of the Lac repressor bound to operator , 2000, Nature Structural Biology.

[56]  D. Gillespie.  Exact Stochastic Simulation of Coupled Chemical Reactions , 1977 .

[57]  D. A. Mcquarrie.  Stochastic approach to chemical kinetics , 1967, Journal of Applied Probability.

[58]  M. Shuler,et al.  Structured model of genetic control via the lac promoter in Escherichia coli , 1994, Biotechnology and bioengineering.

[59]  D. Gillespie.  Transition time statistics in simple bi-stable chemical systems , 1980 .

[60]  Gregory Stephanopoulos,et al.  On physiological multiplicity and population heterogeneity of biological systems , 1996 .

[61]  A. Arkin,et al.  Stochastic mechanisms in gene expression. , 1997, Proceedings of the National Academy of Sciences of the United States of America.

[62]  A. Novick,et al.  ENZYME INDUCTION AS AN ALL-OR-NONE PHENOMENON. , 1957, Proceedings of the National Academy of Sciences of the United States of America.

[63]  Y. Ohshima,et al.  Binding of an inducer to the lac repressor. , 1974, Journal of molecular biology.

[64]  A. Arkin,et al.  Stochastic kinetic analysis of developmental pathway bifurcation in phage lambda-infected Escherichia coli cells. , 1998, Genetics.

[65]  Jose M. G. Vilar,et al.  Modeling network dynamics: the lac operon, a case study , 2004 .

[66]  E. Padan,et al.  Site-specific mutagenesis of histidine residues in the lac permease of Escherichia coli. , 1985, Proceedings of the National Academy of Sciences of the United States of America.

[67]  A. Riggs,et al.  Lac repressor-operator interaction. I. Equilibrium studies. , 1970, Journal of molecular biology.

[68]  G. Yagil,et al.  On the relation between effector concentration and the rate of induced enzyme synthesis. , 1971, Biophysical journal.

[69]  J. Keasling,et al.  Investigating autocatalytic gene expression systems through mechanistic modeling. , 1999, Journal of theoretical biology.

[70]  A. Narang.  Effect of DNA looping on the induction kinetics of the lac operon. , 2007, Journal of theoretical biology.

[71]  Jeffrey Miller,et al.  Genetic Studies of Lac Repressor: 4000 Single Amino Acid Substitutions and Analysis of the Resulting Phenotypes on the Basis of the Protein Structure , 1996, German Conference on Bioinformatics.

[72]  Ertugrul M. Ozbudak,et al.  Predicting stochastic gene expression dynamics in single cells. , 2006, Proceedings of the National Academy of Sciences of the United States of America.

[73]  J. Keasling,et al.  Mathematical Model of the lac Operon: Inducer Exclusion, Catabolite Repression, and Diauxic Growth on Glucose and Lactose , 1997, Biotechnology progress.

[74]  A. Arkin,et al.  Stochastic amplification and signaling in enzymatic futile cycles through noise-induced bistability with oscillations. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[75]  M L Shuler,et al.  Ribosomal protein limitations in Escherichia coli under conditions of high translational activity , 1994, Biotechnology and bioengineering.

[76]  R. Toral.  Noise‐induced transitions vs. noise‐induced phase transitions , 2011 .

[77]  M. Mackey,et al.  Feedback regulation in the lactose operon: a mathematical modeling study and comparison with experimental data. , 2003, Biophysical journal.

[78]  Michael C Mackey,et al.  Influence of catabolite repression and inducer exclusion on the bistable behavior of the lac operon. , 2004, Biophysical journal.

[79]  R. Dickerson,et al.  Equilibrium binding of inducer to lac repressor.operator DNA complex. , 1980, The Journal of biological chemistry.