Why Does the Lac Enzyme Continues to Be Produced Even in the Absence of Lactose

lac Operon

L. Swint-Kruse , K.S. Matthews , in Encyclopedia of Biological Chemistry (Second Edition), 2013

Multiple Sites/Multiple Targets – DNA Looping

In addition to the primary operator, LacO, shown in Figure 1 , two 'pseudo-operator' sequences are present within the lac operon sequence and contribute to repression. The DNA sequences of the pseudo-operators are very similar, but not identical, to LacO and are bound by LacI more weakly. The presence of two DNA-binding sites in LacI protein tetramer suggested a mechanism by which pseudo-operators could enhance repression – one LacI tetramer could bind two separate operators and generate a looped DNA structure. Experimental evidence for DNA looping has been obtained from a variety of laboratories. Recent evidence indicates that the angle between the two dimers must open for loop formation to occur, as illustrated in Figure 4 . These looped structures are highly stabilized, accounting for the significant repression of lacZYA expression observed in bacterial cells. Indeed, DNA containing multiple operator sequences and with the supercoiling density characteristic of E. coli exhibits a half-life for the complex that exceeds 2 days. However, even these looped complexes respond rapidly (in less than 30 s) to the presence of inducer sugars, allowing quick adaptation to an external lactose source that may be transient.

Figure 4. Looped DNA structure. The teal blue curved line depicts the lac operon DNA (with shading to indicate nearness to observer), which contains three possible LacI-binding sites (two of which, O₁ and O₂, are shown bound to LacI). The pseudo-operator sequence O₂ is located within the lacZ gene, and the primary operator sequence O₁ overlaps the promoter sequence for the lacZYA metabolic genes ( Figure 1 ). Tetrameric LacI is shown at the bottom of the figure as simultaneously interacting with O₁ and O₂. This structure loops the DNA and generates a complex with very high stability. Note that the dimers within a LacI tetramer separate and adopt a larger angle between them when looping between O₁ and O₂ than in the absence of looping (i.e., the structure shown in Figure 3(a) ). The need for flexibility between the dimers for looping to occur is supported by experimental evidence.

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lac Operon

Liskin Swint-Kruse , Kathleen S. Matthews , in Encyclopedia of Biological Chemistry, 2004

LacI Function

The role of LacI is to inhibit mRNA production for proteins encoded by the lac operon. Transcription is not completely eliminated, but lacZYA mRNA is transcribed only at very low levels. This function is accomplished by specific binding of LacI protein to the lac operator DNA sequence to inhibit transcription via a variety of mechanisms. Since the lac operator (LacO) overlaps the promoter, LacI binding directly competes with RNA polymerase for binding this region. LacI can also impede transcription initiation and/or block elongation of mRNA. The LacI·LacO association and consequent transcription repression occur when no lactose is available to serve as the substrate of the lac metabolic proteins.

When lactose is available as a carbon source, the low levels of metabolic enzymes allow a small amount of this sugar to be transported into the bacterium by LacY. Next, residual LacZ metabolizes lactose to glucose and galactose, which produces energy for the bacterium. Notably, this catalytic process also generates low levels of allolactose (a rearrangement of the β-1,4 linkage between glucose and galactose to a β-1,6 linkage; Figure 2) . The by-product allolactose binds to LacI and elicits a conformational change in the protein that results in release of the operator DNA sequence (induction). Consequently, RNA polymerase is freed to generate numerous copies of mRNA encoding the lac enzymes. When translated into proteins, these enzymes allow the bacterium to transport and metabolize large quantities of lactose as its carbon energy source, taking advantage of environmental opportunity. One result of the studies by Jacob and Monod was the discovery that a variety of non-natural galactoside sugars (e.g., IPTG; Figure 2) can induce LacI and relieve transcription repression of lacZYA.

Figure 2. Chemical structures for lactose, allolactose (the natural inducer), the sugar components of lactose (glucose and galactose), the gratuitous inducer IPTG, and cAMP.

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RNA Transcription and Control of Gene Expression

John W. Pelley , in Elsevier's Integrated Review Biochemistry (Second Edition), 2012

Regulation of the Lac Operon

The lack of a nuclear membrane in prokaryotes gives ribosomes direct access to mRNA transcripts, allowing their immediate translation into polypeptides. This makes transcription the rate-limiting step in prokaryotic gene expression and, therefore, a major point of regulation. The classic example of prokaryotic gene regulation is that of the lac operon. This operon is a genetic unit that produces the enzymes necessary for the digestion of lactose ( Fig. 16-13).

The lac operon consists of three contiguous structural genes that are transcribed as continuous mRNA by RNA polymerase. An operator sequence located at the 5′ end serves as a binding site for a repressor protein that blocks RNA polymerase. The repressor protein is produced constitutively (continuously) by the i gene, which is not under regulatory control. The repressor itself is formed from subunits that self-assemble to form the active tetramer. When present, the inducer, allolactose, binds to the repressor subunits, preventing their assembly into an active tetramer. Allolactose is produced from lactose by β-galactosidase at a steady low rate and thus serves as a lactose signal. Another regulatory component is the catabolite activator protein (CAP). CAP forms an active complex with intracellular cyclic adenosine monophosphate (cAMP), which accumulates in the absence of glucose (cAMP is a starvation signal). RNA polymerase binds to the lac promoter effectively only when the CAP-cAMP complex is also bound. This ensures that the lac operon will be expressed only when glucose is absent.

The lac operon exhibits both negative control and positive control. Under negative control, a regulatory factor is needed to prevent expression of the lac operon, whereas under positive control, a regulatory factor is needed to permit expression of the lac operon.

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Negative control (conditions: glucose only; prevent expression of lac operon). If lactose is absent and glucose is present (see Fig. 16-13A), the gene products from the lac operon are not needed. Thus a regulatory factor, the repressor protein, prevents lac operon expression. Since the repressor is produced constitutively and spontaneously assembles as its active tetrameric form, it is available to bind to the operon and prevent transcription.

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Positive control (conditions: lactose only; permit expression of lac operon). If no glucose is present and lactose is present (see Fig. 16-13B), the gene products from the lac operon are needed to use the lactose for energy. Thus a regulatory factor, the CAP-cAMP complex is needed to permit expression of the operon. Because cAMP is a starvation signal indicating an absence of glucose, it is available to form the CAP-cAMP complex and permit transcription.

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Positive control (conditions: lactose and glucose; do not permit expression of the lac operon even if not prevented by repressor). If both lactose and glucose are present (see Fig. 16-13C), the regulatory mechanisms act to avoid wasteful expression of the lac operon. Even though the repressor is inactivated by the presence of lactose, RNA polymerase cannot bind to the promoter, since the CAP-cAMP complex is absent owing to the presence of glucose.

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Operon

J.L. Ramos , ... Z. Udaondo , in Brenner's Encyclopedia of Genetics (Second Edition), 2013

The lac Operon

The lactose operon (also known as the lac operon) is a set of genes that are specific for uptake and metabolism of lactose and is found in E. coli and other bacteria. The lac operon consists of three structural genes: lacZ, which codes for β-galactosidase, which acts to cleave lactose into galactose and glucose; lacY, which codes for lac permease, which is a transmembrane protein necessary for lactose uptake; and lacA, which codes for a transacetylase that transfers an acetyl group from coenzyme A (CoA) to the hydroxyl group of galactosides. In the 5′ end with respect to lacZ is the lacI gene, which encodes a repressor of the lac operon, which is transcribed independently from the structural genes ( Figure 1 ).

Figure 1. Scheme of the lac operon structure.

In front of the lacZ gene is the promoter whose expression is modulated by the LacI repressor and the catabolite activator protein (CAP, also known as cAMP receptor protein (CRP)). Expression of this operon is activated only when lactose levels outside the cell are high and glucose levels are low. E. coli utilizes preferentially glucose and, as a result, will not activate the genes to metabolize lactose until there is a sufficiently high level of external lactose, that acts as an effector and a sufficiently low level of glucose.

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Embryonic

Klaus I. Matthaei , in Handbook of Stem Cells, 2004

Bacterial lac Operon

The lac operon in the bacterium Escherichia coli functions by a repression mechanism in which an inhibitor protein (lacI) binds to regulatory sites (lacO) in the promoter and turns off transcription (Fig. 59-2). On the addition of lactose, the lacI protein undergoes a conformational change, which changes its binding affinity for the lacO sequences. The lacI protein thereby comes off the lacO sites, and transcription can occur. E. coli uses this system to tightly control the genes required for the use of lactose, and it is completely reversible.

Figure 59-2. Bacterial lac operon. The lac operon functions by a repression mechanism. (A) An inhibitor protein, lacI, binds to regulatory sites lacO in the promoter (P) and turns off transcription of the genes required for lactose metabolism. (B) On the addition of lactose, the lacI protein undergoes a conformational change, which changes its binding affinity for the lacO sequences. The lacI protein thereby comes off the lacO sites and transcription of the lac genes can occur. (A, transacetylase; Y, permease; and Z, β-galactosidase.)

(adapted from Mills ⁵² )

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Regulation of Transcription in Prokaryotes

David P. Clark , Nanette J. Pazdernik , in Molecular Biology (Second Edition), 2013

Activators, Repressors and Operons

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In prokaryotes, many genes are clustered into operons and transcribed and regulated as a group.

Prokaryotic gene arrangement often includes operons, which are multiple coding regions under the control of the same promoter. Often, operons include genes involved in the same metabolic pathway, and therefore, are needed at the same time. One advantage of operons is that the genes within the operon are regulated simultaneously.

The lactose operon of E. coli is the classic example of an operon and is often used when discussing prokaryotic regulation. The lac operon consists of three coding regions in tandem, lacZ, lacY, and lacA. The lacZ gene encodes β-galactosidase, which degrades lactose. The lacY gene product, lactose permease, transports lactose into the cell, and the lacA gene product, lactose acetylase, has an unknown and not usually necessary function. The lac operon is repressed by LacI, encoded by lacI. The lacI gene is upstream of lacZYA and faces in the opposite direction. The repressor, LacI, binds to the operator sequence upstream of lacZYA and prevents transcription of those genes unless the inducer molecule is present. The inducer molecule, allo-lactose, signifies the presence of lactose. Allo-lactose binds to LacI and prevents it from repressing transcription of lacZYA, thus the operon is derepressed. In addition to allo-lactose, LacI may bind other chemicals such as IPTG (see Figures 16.11 and 16.12 Figure 16.11 Figure 16.12 ).

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Activator and repressor proteins take part in positive and negative transcriptional regulation.

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Some regulatory proteins may function as either a repressor or an activator depending on the situation.

Activators typically bind upstream of a promoter sequence on the DNA and help RNA polymerase bind to the promoter to initiation transcription of the genes. Repressors bind to operator sequences downstream of the promoters to prevent RNA polymerase from either binding or moving forward, which prevents transcription.

Some regulatory proteins behave as both activators and repressors, depending upon the conditions. The AraC regulatory protein is involved in regulation of the utilization of arabinose, a sugar source for cells. When arabinose binds to AraC, the regulatory protein is converted from a repressor to an activator. The araBAD and araFG operons are repressed in the absence of arabinose and activated in the presence of arabinose.

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Regulatory proteins are often controlled in turn by binding small signal molecules.

Small inducer molecules often bind to repressors and change the shape of the protein such that it is no longer able to bind DNA and repress transcription of its target genes. An example includes allo-lactose for the lac operon of E. coli. Additionally, some repressors are not able to bind DNA unless they have a small signal molecule present, called a co-repressor. This type of regulation is often seen for biosynthetic pathways where if the product of the pathway is present in the cell, then the pathway is repressed.

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Transcription | lac Operon Regulation☆

Liskin Swint-Kruse , ... Kathleen S. Matthews , in Encyclopedia of Biological Chemistry (Third Edition), 2021

Catabolite Repressor Protein

CRP Function

The lac operon includes two regulatory switches – one for lactose and one for glucose. This arrangement allows E. coli to leverage the energetic balance between glucose and lactose utilization. By metabolizing lactose only when glucose is unavailable, bacteria receive an energetic benefit: When glucose is available, cells can avoid the additional energetic cost of transcribing and translating the proteins required to use lactose. Unlike the lactose signal, glucose signaling is accomplished indirectly and inversely through the presence of the signaling molecule cAMP (Fig. 2). Glucose exerts an inhibitory influence on adenylate cyclase, the enzyme that produces cAMP (Peterkofsky and Gazdar, 1974; reviewed in Kolb et al., 1993; Busby and Ebright, 1999; Harman, 2001; Won et al., 2009). Thus, levels of glucose and cAMP are inversely related. As glucose levels decrease, cAMP levels rise, and cAMP binds to CRP. The CRP-cAMP complex binds in turn to the lac promoter (Harman, 2001).

The CRP-cAMP complex has high affinity for a wide range of DNA sequences within the bacterial genome (Figs. 1 and 3(c)) and helps to regulate many different genes and operons (Harman, 2001). One of the DNA target sequences for cAMP-CRP is found within the promoter for the lac metabolic proteins, upstream of LacO1 (Fig. 1). The two regulatory proteins have opposite functional mechanisms: cAMP binding to CRP enhances its binding to DNA (Harman, 2001; Lawson et al., 2004), whereas inducer binding to LacI decreases its binding to DNA. Likewise, DNA binding by the cAMP-CRP complex enhances RNA polymerase transcription of lacZYA mRNA, whereas LacI binding represses transcription. The end result of this co-regulation is that the proteins for lactose metabolism are expressed at high levels only when environmental glucose is low and lactose levels are high.

Similar to LacI, CRP has been used to regulate transcription in other host systems. Recent studies utilized CRP fused to the VP16 transactivation domain of Herpes simplex virus with gene expression mediated by cAMP in mammalian cells (Durai et al., 2019). This synthetic transcription factor is able to activate expression of a reporter gene in mammalian cells, including tumorigenic cells with altered cAMP signaling. This result further illustrates the varied ways that prokaryotic proteins can be used in mammalian systems. Finally, modern techniques have provided important new insights into CRP regulation; for example, deep sequencing of bacterial populations under different selective pressures for mutations in the crp gene identified more than 100 novel CRP variants that will provide even greater insights into the structure and function of this E. coli master regulator (Frendorf et al., 2019).

CRP Structural Characteristics

Like many regulatory proteins, CRP is multimeric (in this case dimeric) and has multiple domains (Fig. 3(c)). Note that the symmetry seen in the two types of lac operon regulatory sites is a common feature of regulatory DNA binding sites and is associated with multimeric states common to regulatory proteins. Several different structures have been solved for CRP (Parkinson et al., 1996a,b; Popovych et al., 2009; Sharma et al., 2009). The N-terminal domain facilitates dimer formation and binds at least one molecule of cAMP (two/dimer) (Harman, 2001). In the absence of cAMP, the two DNA binding domains of the dimer are mis-aligned for sequence-specific DNA recognition (Sharma et al., 2009; Popovych et al., 2009). In the presence of cAMP, the structure rearranges to bring the two DNA-binding helix-turn-helix structures into the proper orientation for effective DNA binding (Parkinson et al., 1996a,b; Sharma et al., 2009). Similar to LacI, CRP-cAMP binding also elicits a bend in its target DNA-binding site (Parkinson et al., 1996b). Likewise, each monomer only binds to one-half of the DNA-binding site, which is symmetric through the center of the sequence (see Fig. 3(c)). Recent cryoelectron microscopy of the CRP-cAMP-DNA complex demonstrates CRP wrapping of upstream DNA sequences and interactions that recruit RNA polymerase (Liu et al., 2017).

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Regulation of Transcription in Prokaryotes

David P. Clark , ... Michelle R. McGehee , in Molecular Biology (Third Edition), 2019

Although the lactose operon genes were the first whose regulation was characterized in detail, and although they are often cited as a typical example, they are aberrant in several ways. Curiously, lactose itself is not the inducer. Lactose, which consists of glucose linked to galactose, is converted to allo-lactose, an isomer in which the same two sugars are linked differently. This transformation is carried out by β-galactosidase, which normally splits lactose, but makes a small amount of allo-lactose as a side reaction. It is allo-lactose that actually binds to the LacI protein and acts as an inducer.

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Regulation of Transcription in Prokaryotes

David P. Clark , Nanette J. Pazdernik , in Molecular Biology (Second Edition), 2013

Conceptual Questions

1.

The lactose operon was the first operon studied, and many mutations have been created to understand how the operon functions. In genetic nomenclature, a small superscript "+" denotes that the gene is normal, whereas a superscript "-" denotes the gene is defective or mutated. When a gene is deleted, the "Δ" symbol precedes the gene designation. In the following chart, fill in the expected phenotype for each of the genetic mutants using your knowledge of lac operon control. The++++designates enzyme activity.

2.

In E. coli, partial diploids can be created by adding an extrachromosomal ring of DNA called a plasmid that has an origin and terminator regions so it is replicated and maintained throughout each cell division. The plasmid can be modified in the lab to contain different genes such as all the genes in the lac operon. When the plasmid is introduced into the bacteria, there are two copies of each of the genes in the lac operon, and this is denoted as I⁺P⁺O⁺Z⁺Y⁺/I⁺P⁺O⁺Y⁺Z⁺ where the genes before the "/" are chromosomal and those genes after the "/" are on the plasmid. The following partial diploid, I⁻O⁺Z⁺Y⁻/I⁻O⁺Z⁺Y⁺, has β-galactosidase activity and lactose permease activity with or without an inducer. Why?

3.

Two different types of E. coli bacteria were grown in rich broth. The first type of E. coli is wild-type for H-NS and the second type has a deletion that removes the gene for H-NS. The mRNA from each culture was isolated and analyzed for the total amount of mRNA for each of the following genes using quantitative PCR. Each mRNA is compared to an internal control mRNA, and the fold induction is presented. Based upon this data, what genes are regulated by H-NS? What type of DNA structure is most likely present in the promoter for these genes?

4.

The trp operon is under control of an aporepressor called TrpR that binds to the operator site upstream from the biosynthetic genes of tryptophan biosynthesis. How would you expect the presence or absence of tryptophan to affect transcription of the operon? Unlike the inducible LacI repressor, aporepressors do not bind the operator site unless bound by a co-repressor. If tryptophan is the co-repressor, describe the binding of TrpR to the operator with and without tryptophan.

5.

Look at the following promoter sequence. Underline the sequences necessary for σ⁵⁴ factor binding based on the information in Table 16.01.

−40 GATCGCAGCCGGATTGGCAATATCCTTGCAATACTTAAATC +1

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Computer Methods, Part C

Necmettin Yildirim , Caner Kazanci , in Methods in Enzymology, 2011

4 An Example: Lactose Operon in E. coli

We use the lactose operon (the lac operon) of E. coli and a modified version of the Yildirim–Mackey model (Mackey et al., 2004; Yildirim and Mackey, 2003; Yildirim et al., 2004) developed for this bacterial regulatory circuit to demonstrate the methods and analysis described in previous sections. The lac operon is the classical example of an inducible circuit which encodes the genes for the transport of external lactose into the cell and its conversion to glucose and galactose. A cartoon that depicts the major components of this circuit is shown in Fig. 12.9. The molecular mechanism of the lac operon works as follows: The lac operon has a small promoter/operator region (P and O) and three larger structural genes lacZ, lacY, and lacA. There is a regulatory gene lacI preceding the lac operon. lacI is responsible for producing a repressor (R) protein. In the presence of allolactose, a binary complex is formed between allolactose and the repressor that makes binding of the repressor to the operator region impossible. In that case, the RNA polymerase bound to the promoter is able to initiate transcription of the structural genes to produce mRNA(M). However, in the absence of allolactose (A) the repressor protein R binds to the operator region O and prevents the RNA polymerase from transcribing the structural genes. Once the mRNA has been produced, the process of translation starts. The lacZ gene encodes the portion of the mRNA that is responsible for the production of β-galactosidase (B) and translation of the lacY gene produces the section of mRNA that is ultimately responsible for the production of an enzyme permease (P). The final portion of mRNA produced by transcription of the lacA gene encodes for the production of thiogalactoside transacetylase which is thought not to play a role in the regulation of the lac operon (Beckwith, 1987). This positive control system works as follows: When there is no glucose available for cellular metabolism but if lactose (L) is available in a media, the lactose is transported into the cell by the permease. This intracellular lactose is then broken down into glucose, galactose, and allolactose by β-galactosidase. The allolactose is also converted to glucose and galactose by the same enzyme β-galactosidase. The allolactose feeds back to bind with the lactose repressor and enable the transcription process which completes the positive feedback loop.

Figure 12.9. Schematic representation of the lactose operon regulatory system. See the text for details.

Yildirim et. al. (Mackey et al., 2004; Yildirim and Mackey, 2003) devised a mathematical model which takes into account the dynamics of the permease, internal lactose, β-galactosidase, the allolactose interactions with the lac repressor, and mRNA. The final model consists of five nonlinear differential delay equations with delays due to the transcription and translation process. We modified this model in this study and eliminated the delay terms. This change reduced the original model to a five-dimensional system of ODEs. The equation of this model are given in Eqs. (12.25)–(12.29). The estimated values for the model parameters from the published data are listed in Table 12.1. The details on the development of this model and estimation of the parameters can be found in Mackey et al. (2004), Yildirim and Mackey (2003), Yildirim et al. (2004) (Table 12.2).

Table 12.1. The model parameters estimated from experimental data (from Yildirim and Mackey, 2003)

n	2	μ max	3.47   ×   10−   2  min−   1
γ M	0.411   min−   1	γ B	8.33   ×   10−   4  min−   1
γ A	0.52   min−   1	Γ0	7.25   ×   10−   7  mM/min
K	7200	α M	9.97   ×   10−   4  mM/min
K L 1	1.81   mM	α A	1.76   ×   104  min−   1
K A	1.95   mM	α B	1.66   ×   10−   2  min−   1
γ L	0.0   min−   1	β A	2.15   ×   104  min−   1
α L	2880   min−   1	K L	9.7   ×   10−   4M
K L e	0.26   mM	γ P	0.65   min−   1
β L 2	1.76   ×   104  min−   1	α P	10.0   min−   1
K 1	2.52   ×   10−   2 (μM)−   2	β L 1	2.65   ×   103  min−   1
K L 2	9.7   ×   10−   4M

Table 12.2. The equations describing the evolution of the variables M, B, L, A, and P in the Yildirim–Mackey model for the lac operon

(12.25) $\frac{\text{d}[M]}{\text{d}t}={\alpha }_M\frac{1+K_1{[A]}^n}{K+K_1{[A]}^n}+{\mathrm{\Gamma }}_0-{\tilde{\gamma }}_M[M]\text{.}$

(12.26) $\frac{\text{d}[B]}{\text{d}t}={\alpha }_B[M]-{\tilde{\gamma }}_B[B]\text{.}$

(12.27) $\frac{\text{d[}L\text{]}}{\text{d}t}={\alpha }_L\frac{[P][L_e]}{K_{L_e}+[L_e]}-{\beta }_{L_1}\frac{[P][L]}{K_{L_1}+[L]}-{\beta }_{L_2}\frac{[B][L]}{K_{L_2}+[L]}-{\tilde{\gamma }}_L[L]\text{.}$

(12.28) $\frac{\text{d}\left[A\right]}{\text{d}t}={\alpha }_A\frac{[B][L]}{K_L+[L]}-{\beta }_A\frac{[B][A]}{K_A+[A]}-{\tilde{\gamma }}_A[A]\text{.}$

(12.29) $\frac{\text{d[}P\text{]}}{\text{d}t}={\alpha }_P[M]-{\tilde{\gamma }}_P[P]\text{.}$

In this model ${\tilde{\gamma }}_i={\gamma }_i+\mu \text{,}i\in \left\{M,B,L,A,P\right\}$ _.

We studied this model using both deterministic and stochastic approaches. To see if the modified model captures the experimentally observed bistable behavior (Cohn and Horibata, 1959; Novick and Wiener, 1957; Ozbudak et al., 2004), we set the left-hand side of each equation in the system Eqs. (12.25)–(12.29) to zero and solve the resultant system of five nonlinear equation for a range of L _e concentration after keeping all the other parameters as in Table 12.1 for μ  =   2.26   ×   10^{−   2}  min^{−   1}. The result is shown in Fig. 12.10. Our modified model predicts that there is a physiologically range for the external lactose concentration that corresponds to the S-shaped curve in this figure. When the external lactose concentration falls in this range, the lac operon can have three coexisting steady states.

Figure 12.10. Bistability arises in the lac operon model as the external lactose (L _e ) concentration changes when μ  =   2.26   ×   10^{−   2}  min^{−   1}. Notice that the parameter values, there exists a range of L _e concentration for which there are three coexisting steady states for β-galactosidase concentration. Our calculations estimate this range as [0.026, 0.057] mM of [L _e].

Figure 12.11 shows how the bistability arises in evolution of β-galactosidase concentration in the deterministic simulation of the model. In this simulation, all the parameters are kept constant as in Table 12.1 when μ  =   2.26   ×   10^{−   2}  min^{−   1} and we chose [L _e] as [L _e]   =   53   ×   10^{−   3}  mM. As shown in Fig. 12.10, there are three steady states for this particular concentration of L _e. We calculate these steady state values numerically as in Table 12.3. To produce this figure, the initial values for the concentrations of all the proteins were kept constant at their steady state values on the middle branch of the S-shaped curve when [L _e]   =   53   ×   10^{−   3}  mM except mRNA concentration. Then three initial values of the mRNA concentration were chosen slightly below its steady value on the middle branch and another three initials were chosen slightly above its steady state concentration on the same branch and the model equations in Eqs. (12.25)–(12.29) was solved numerically for [L _e]   =   80   ×   10^{−   3}  mM, which corresponds to the external lactose concentration on a steady on the upper branch of the S-shaped curve. When the simulation is started from an initial starting point that is exactly on the middle branch of S-shape curve, the β-galactosidase concentration stays constant over time (the horizontal line in Fig. 12.11) as it is a steady state for this system. Since the middle branch is unstable, small perturbations around the middle branch can kick the simulation either to the lower or the upper stable branches of S-shape curve. All the other runs converge to the stable steady states either on the lower branch or on the upper branch as seen in this simulation. We observe that the ones started initially above the steady state concentration of mRNA on the middle branch converged to the steady state on the upper branch, the ones started initially below the steady state concentration of mRNA on the middle branch converged to the steady state on the lower branch.

Figure 12.11. Semilog plot of β-galactosidase concentration over time showing effects of the initial values of concentration mRNA around the middle branch of S-shape curve in Fig. 12.10 in the numerical simulation.

Table 12.3. The steady state values calculated from Eqs. (12.25)–(12.29) by setting the time derivatives zero

	[M*]	[B*]	[A*]	[L*]	[P*]
Lower branch	2.80   ×   10−   6	1.98   ×   10−   6	1.00   ×   10−   2	1.88   ×   10−   1	4.17   ×   10−   5
Middle branch	5.33   ×   10−   6	3.78   ×   10−   6	2.04   ×   10−   2	2.11   ×   10−   1	7.93   ×   10−   5
Upper branch	6.56   ×   10−   4	4.65   ×   10−   4	3.37   ×   10−   1	2.46   ×   10−   1	9.75   ×   10−   3

All the parameters are kept constant as in Table 12.1, when μ  =   2.26   ×   10^{−   2}  min^{−   1} and [L _e ]   =   53   ×   10^{−   3}  mM for which there exist three steady states (see Fig. 12.10).

In Fig. 12.12, the deterministic and stochastic simulation of the Yildirim–Mackey lac operon model is shown. To produce this plot, we run six simulations by choosing the steady state value on the lower branch of the S-shaped curve as the initial starting point when [L _e]   =   53   ×   10^{−   3}  mM and μ  =   2.26   ×   10^{−   2}  min^{−   1} while all other parameters are kept constant as in Table 12.1. As seen in this simulation, the average of the stochastic simulations are about the same as the solution of differential equations. Since we pick the initial concentrations from the bistable region, there is a slow transition before reaching to the steady state in both simulations. The deterministic model estimates this transition period about 120 min. The stochastic simulations predicts a significant variance in this transition period and estimate that this period may take up to 500 min for individual cells.

Figure 12.12. Deterministic and stochastic simulation of the Yildirim–Mackey lac operon model given by Eqs. (12.25)–(12.29). In this simulation, the solid lines show the ODE solutions and the broken lines represent the results of the stochastic simulations. To produce this plot, we chose the steady state value on the lower branch of the S-shaped curve as the initial value when [L _e ]   =   53   ×   10^{−   3}  mM and μ  =   2.26   ×   10^{−   2}  min^{−   1} while all the other parameters are kept constant as in Table 12.1 and run six stochastic simulations for the external lactose concentration [L _e ]   =   80   ×   10^{−   3}  mM which corresponds to a steady state value on the upper branch of the S-shaped curve in Fig. 12.10.

We investigate the effects of stochasticity in the bistable region. To this end, we run the stochastic simulation eight times starting from the stable steady state on the lower branch of the S-shaped curve and another eight runs starting from the stable steady state on the upper branch of the S-shaped curve for [L _e]   =   53   ×   10^{−   3}  mM. The results are shown in Fig. 12.13. In a bistable system, the random fluctuations can push the system from one stable steady state to the other one. The frequency of this transition is higher for systems with higher noise levels. We observe that all simulations starting from the lower branch of the S-shaped curve ended up converging to the stable steady state on the upper branch. However, simulations initialized at the upper branch never switch to the lower steady state and stay on the upper branch. This indicates that the steady state on the upper branch is more robust, and is resistive against fluctuations in the protein concentrations compared to the steady state on the lower branch. As seen in this simulation, the time required to shift from the lower steady state to the upper steady state can change significantly from one run to another. This transition can happen as early as in 60 min and as late as 600 min. Another surprising result is the variance at steady levels of β-galactosidase and lactose. When [L _e]   =   53   ×   10^{−   3}  mM, the steady state concentrations of β-galactosidase and lactose concentration are around 50 and 23,000   mM, respectively. In general, we expected to see less variation when concentration of a molecular species is high. In other words, relative noise is less for high concentrations. However, our stochastic simulation results indicate that relative noise appears to be about the same for both of these proteins (results are not shown). One conclusion we can derive from this simulation result is about the sensitivity of concentration of β-galactosidase, that significant changes in the concentration of β-galactosidase is not likely to have an impact on the entire system, because it will most likely be dominated by noise anyway.

Figure 12.13. Sixteen stochastic simulations of the Yildirim–Mackey lac operon model is shown, with the same parameters used for Fig. 12.12. Eight simulations use the lower steady state value as the initial condition, while the others use the upper steady state as the initial condition. We observe that all the simulations starting from the lower branch converge to the upper steady state and the simulations initialized from the upper branched stay on that steady state (only one of the simulations is plotted here).

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