Generalized Net Model of the Lac Operonin Bacterium E. Coli

Kalin KosevInstitute of Biophysics and

Biomedical EngineeringBulgarian Academy of Sciences

105 Acad. G. Bonchev Str.,1113 Sofia, Bulgaria

Email: kalinkosev@yahoo.com

Pedro Melo-PintoCentre for the Research and

Technology of Agro-Envirinmentaland Biological Sciences – UTAD

5001-801 Vila Real, PortugalEmail: pmelo@utad.pt

Olympia Roeva, Member IEEEInstitute of Biophysics and

Biomedical EngineeringBulgarian Academy of Sciences

105 Acad. G. Bonchev Str.,1113 Sofia, Bulgaria

Email: olympia@biomed.bas.bg

Abstract—In this paper a Generalized Net (GN) model of theregulation of the lac operon in E. coli bacterium is presented.Prokaryotes, such the bacterium E. coli, have an efficient mecha-nism for metabolizing lactose. Three proteins that are importantin lactose metabolism are all encoded in a single expressible unitof DNA, called the lac operon. The GN model, presented in thispaper, describes the use of lactose as an energy source from E.coli. The net presents the turn on of several genes in the lacoperon, which are required for lactose metabolism.

Keywords—Generalized nets; Lactose; Glucose; Metabolism;lac Operon; E. coli

I. INTRODUCTION

The theory of the Generalized nets (GN) [8], [9], [11]proved to be quite successful when applied to the descriptionof the functioning of expert systems, machine learning anddifferent technological processes. Up to now GN are used asa tool for modelling of parallel processes in several areas [7],[8], [9], [10], [11] - economics, transport, medicine, computertechnologies etc. Already GN are used to model geneticnetworks [12], [13], [14], [15]. Genetic networks are anapproximation mathematical model that is consistent with thecontemporary knowledge on gene-gene interactions. As in allmodels various assumptions on the nature of such relations aremade (whether they will be expressed as differential equations,probabilistic distributions, topological criterion, etc. is a choicepredetermined by the structure of the data implemented in themodel).

Gene control in a cell is achieved through combined actionof genetic interaction and outside factors. Genetic activationand/or suppression is done by gene products activated bydifferent signal pathways. The first control system for enzymeproduction shown at molecular level described the control ofenzymes that are produced in response to the presence of thesugar lactose in E. coli cell. The work was performed by Jacoband Monod in 1962 for which they were awarded the NobelPrize. The following is a pathway that leads to the productionof glucose and galactose.

This paper is partially supported by the Bulgarian National Scientific Fundunder the Grants DID 02/29 “Modeling Processes with Fixed DevelopmentRules (ModProFix)” and DMU 03-38 “Management and modeling of biomed-ical processes and data with application of generalized nets and linked data”.

In this paper a Generalized Net (GN) model of the lacoperon in E. coli is presented. Operon is a functioning unitof genomic DNA containing a cluster of genes under thecontrol of a single regulatory signal or promoter. The genesare transcribed together into an mRNA strand and eithertranslated together in the cytoplasm, or undergo trans-splicingto create monocistronic mRNAs that are translated separately,i.e. several strands of mRNA that each encode a single geneproduct. The lac operon is an operon required for the transportand metabolism of lactose in Escherichia coli.

II. lac OPERON

The lac operon is an operon required for the transport andmetabolism of lactose in Escherichia coli and some otherenteric bacteria [6]. The lac operon is regulated by severalfactors including the availability of glucose and of lactose.Gene regulation of the lac operon was the first complex geneticregulatory mechanism to be elucidated and is one of theforemost examples of prokaryotic gene regulation.

Gene transcription can be switched on and off by generegulation proteins. The lac operon in E. coli is an exampleof that dual control. Glucose and lactose levels control theinitiation of transcription of the lac operon, i.e. whether thelac operon is switched ”ON” or ”OFF”.

• In an E. coli cell growing in the absence of lactose, arepressor protein binds to the operator, preventing RNApolymerase from transcribing the lac operon’s genes. Theoperon is OFF.

• When the inducer, lactose, is added, it binds to the repres-sor and changes the repressor’s shape so as to eliminatebinding to the operator. As long as the operator remainsfree of the repressor, RNA polymerase that recognizesthe promoter can transcribe the operon’s structural genesinto mRNA. The operon is ON.

The Fig. 1 presented the lac operon of E. coli [5].Near the lac operon is another gene, called lacI , or just

“I”. It codes for the lac repressor protein, which plays anessential role in lac operon control. The lac repressor gene isexpressed “constitutively”, meaning that it is always on (butat a low level). It is a completely separate gene, producing

84

The lac operon of E. coli.

Fig. 1. lac operon of E. coli

TABLE ICONTROL CIRCUIT FOR THE lac OPERON

lac Operon Gene Gene FunctionI Gene for repressor proteinP Promoter – aids in RNA polymerase bindingO Operator – “on/off” switch

binding site for the repressor proteinlacZ Gene for β-galactosidaselacY Gene for β-galactoside permeaselacA Gene for β-galactoside transacetylase

a different mRNA than the lac operon. Just upstream fromthe transcription start point in the lac operon are two regionscalled the operator (O) and the promoter (P ). Neither regioncodes for protein: they act as binding sites on the DNAfor important proteins. The promoter is the site where RNApolymerase binds to start transcription. Promoters are foundupstream from all protein-coding genes. The operator is wherethe actual control occurs. The control regions of the lac operonare presented in Table I.

The lac operon consists of three structural genes, and apromoter, a terminator, regulator, and an operator. The threestructural genes which code for polypeptides of about 1000,260, and 275 amino acids are [1], [2], [3], [4]: LacZ, LacY,and LacA

• lacZ encodes β-galactosidase (LacZ), an intracellularenzyme that cleaves the disaccharide lactose into glucoseand galactose.

• lacY encodes β-galactoside permease (LacY), amembrane-bound transport protein that pumps lactoseinto the cell.

• lacA encodes β-galactoside transacetylase (LacA), anenzyme that transfers an acetyl group from acetyl-CoAto β-galactosides.

Only lacZ and lacY appear to be necessary for lactosecatabolism.

III. GN-MODEL OF THE lac OPERON IN E. coli

Glucose is the sugar of choice of E. coli and if glucoseis in supply, then the bacteria will preferentially break down

glucose over lactose. If glucose is present, the lac operon willbe repressed (presented with transition Z1). RNA polymerasehas a low affinity for the promoter of the lac operon unlesshelped by a regulatory protein - cAMP receptor protein (CRP).CRP only becomes activated if the concentration of cyclicAMP (cAMP) is high. Glucose inhibits the formation ofcAMP:

• If the concentration of glucose is high, the concentrationof cAMP is low;

• If the concentration of glucose is low, the concentrationof cAMP is high.

Therefore, if the concentrations of glucose and lactose arehigh, the concentration of cAMP will be low, CRP will not beactivated, RNA polymerase will not be able to bind well to thepromoter, and the operon will be operating at a very low level(i.e. almost off). However, if the concentrations of glucose islow and lactose is high, the concentration of cAMP will behigh, CRP will be activated and bind to the DNA which willpromote RNA polymerase binding and initiate transcription(presented with transition Z6).

The cell can use lactose as an energy source by producingthe enzyme -galactosidase to digest that lactose into glucoseand galactose. However, it would be inefficient to produceenzymes when there is no lactose available, or if there isa more readily-available energy source available such asglucose (presented with transition Z1). The lac operon usesa control mechanism to ensure that the cell expends energyproducing three enzymes only when necessary. It achievesthis with the lac repressor, which halts the production inthe absence of lactose. This control mechanism causes thesequential utilization of glucose and lactose in two distinctgrowth phases, known as diauxic. Similar diauxic growthpatterns have been observed in bacterial growth on mixturesof other sugars as well, such as mixtures of glucose andarabinose, etc. When lactose is unavailable a repressor proteinis bound to the operator region located few bases upstreamof the operon (presented with transition Z2 and Z3). Thisrepressor protein prevents RNA polymerase from binding tothe promoter and stops the transcription of the operon. When

lactose is present in the growth media, a lactose moleculebinds to the allosteric center of the repressor protein (pre-sented with transition Z4 and Z5). This changes the nativeconformation of the protein and allows RNA polymerase toattach itself to the promoter. After transcription and translationthe enzymes - β-galactosidase, β-galactoside permease andthiogalactoside transacetylase are produced (presented withtransition Z7 and Z8). The first enzyme β-galactosidase isresponsible for lactose degradation to glucose and galactose.The second enzyme β-galactoside permease helps with lactosetransport over the cellular membrane. The role of the thirdenzyme is not yet clear. For the proper assimilation of lactosethe first two enzymes are sufficient (see Z9).

The GN-model GN-model of the lac operon in E. coli ispresented in Fig. 1.

Initially, in places l1, l2, l3, l10 and l11 stay tokens α withinitial and current characteristics:“Initial concentration of glucose”,“Initial concentration of lactose”,“Initial concentration of repressor protein”,“Initial concentration of RNA polymerase”,“Initial concentration of cAMP”.

The form of the first transition is:

Z1 = 〈{l1, l2, l17}, {l4, l5},

l4 l5l1 true falsel2 false W1

l22 false W1

,

∨(l1, l2)〉.

whereW1= ”no existence of token in place l1”.

In this transition glucose and lactose enter in the cell.Glucose is metabolized predominantly. Lactose is metabolizedonly if the levels of glucose are low or there is none presentin the medium.

The tokens have the following characteristics:in place l4 – “concentration of glucose”;in place l5 – “concentration of lactose ”.

The characteristic of the token in the place l22 is “concen-tration of lacY ”.

The form of the second transition is:

Z2 = 〈{l3, l7, l14}, {l6, l7},

l6 l7l3 false truel7 true truel14 false true

,

∧(l3)〉.

Here repressor synthesis occurs slow and steady in thebacterial cell. If lactose is not present the repressor proteinbinds to the operator region of the lac operon.

After the transition Z2 the tokens obtain the followingcharacteristics:in place l6 – “current concentration of repressor protein”;in place l7 – “total concentration of repressor protein”.

The characteristic of the token in the place l14 is “concen-tration of repressor protein”.

The form of the third transition is:

Z3 = 〈{l4, l5, l24}, {l8, l9}, r3,∨(l4, l5)〉.

l8 l9l4 true falsel5 false W1

l24 true false

,

The new characteristics of the tokens are:in place l8 – “concentration of glucose”;in place l9 – “lactose molecule bound with repressor protein”.

In the presented here GN model the place l8 is actually theinput place in the developed GN model in [15].

The form of the fourth transition is:

Z4 = 〈{l9, l15}, {l12, l13, l14, l15},

l12 l13 l14 l15l9 W2 true false truel15 false W3 true true

,

∧(l9, l15)〉.

whereW2= ”existence of token in place l13”; W3= ”existence oftoken in place l9”.

After the transition Z4 the tokens obtain the followingcharacteristics:in place l12 – “concentration of lactose ”;in place l13 – “signal for DNA transcription”;in place l14 – “repressor protein”;in place l15 – “repressor protein bound with lactose”.

If glucose is present it is metabolized predominantly [15].If there is lactose in the medium, one molecule binds to therepressor protein allosteric center. This changes the confor-mation of the protein and it releases the operator site of theoperon.

The form of the fifth transition is:

Z5 = 〈{l9, l15}, {l14, l15},

l16 l17l10 W2 truel17 W2 true

,

∨(l10, l17)〉.

Here tokens obtain the following characteristics:in place l16 – “RNA polymerase attached to the promotor”;in place l17 – “RNA polymerase”.

This transition represents responsibilities of RNA poly-merase for transcription of DNA to a RNA molecule.

The form of the sixth transition is:

Fig. 2. Generalized net model

Z6 = 〈{l11, l19}, {l18, l19},

l18 l19l11 W2 truel19 W2 true

,

∨(l11, l19)〉.

Here tokens obtain the following characteristics:in place l18 – “cAMP attached to the promotor”;in place l19 – “cAMP”.

The quantity of cAMP depends on the glucose availabilityand plays part for the operon activation.

The form of the seventh transition is:

Z7 = 〈{l13, l16, l18}, {l20},

l20l13 truel16 truel18 true

,

∧(l13, l16, l18)〉.

After the transition Z7 the tokens in place l20 obtain thecharacteristic “RNA”. RNA synthesis of the operon occurs inthe bacterial cell. This RNA molecule is translated into theprotein language by ribosomes in the cytoplasm.

The form of the eighth transition is:

Z8 = 〈{l20, l23}, {l21, l22, l23},

l21 l22 l23l20 false false truel23 true true true

,

∧(l20, l23)〉.

The two enzymes responsible for the lactose degradationreceive their native conformation and start metabolizing it. Thenew characteristics of the tokens are as follows:in place l21 – “lacZ ”;in place l22 – “lacY ”;in place l23 – “concentration of β-galactosidase andβ-galactoside”.

The form of the ninth transition is:

Z9 = 〈{l12, l21}, {l24, l25},

l24 l25l12 W4 W4

l21 true true,

∨(l12, l21)〉.

whereW4= ”existence of token in place l21”.

Here, lactose is metabolized to glucose and galactose.The tokens obtain the following characteristics:

in place l24 – “concentration of glucose”;in place l25 – “concentration of galactose”.

IV. CONCLUSION

A GN model of the regulation of the lac operon inEscherichia coli bacterium is presented in this paper. TheGN model illustrates very well the dependence of the generegulation on genetic and extra cellular factors. The cell doesnot transcribe the operon unless the predominant source ofenergy is not available. The GN model reports the influence ofthe repressor protein on the transcription of the lac operon. Italso shows that transcription occurs only in specific time whenoutside factor is present – in this case cAMP and lactose arethe factors. If glucose and lactose are available the quantityof cAMP is the factor preventing the transcription. Amongthe complexity in molecular interactions in the cell this modelpresents a simple view into the regulation of the operon. Italso shows the clever way in which the bacterium preservesenergy and resources until the need occurs.

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