Noise regulation by quorum sensing in low mRNA copy number systems

Noise regulation by quorum sensing in lowmRNA copy number systemsWeber and Buceta

Weber and Buceta BMC Systems Biology 2011, 5:11http://www.biomedcentral.com/1752-0509/5/11 (20 January 2011)

RESEARCH ARTICLE Open Access

Noise regulation by quorum sensing in lowmRNA copy number systemsMarc Weber, Javier Buceta*

Abstract

Background: Cells must face the ubiquitous presence of noise at the level of signaling molecules. The latterconstitutes a major challenge for the regulation of cellular functions including communication processes. In thecontext of prokaryotic communication, the so-called quorum sensing (QS) mechanism relies on small diffusivemolecules that are produced and detected by cells. This poses the intriguing question of how bacteria cope withthe fluctuations for setting up a reliable information exchange.

Results: We present a stochastic model of gene expression that accounts for the main biochemical processes thatdescribe the QS mechanism close to its activation threshold. Within that framework we study, both numericallyand analytically, the role that diffusion plays in the regulation of the dynamics and the fluctuations of signalingmolecules. In addition, we unveil the contribution of different sources of noise, intrinsic and transcriptional, in theQS mechanism.

Conclusions: The interplay between noisy sources and the communication process produces a repertoire ofdynamics that depends on the diffusion rate. Importantly, the total noise shows a non-monotonic behavior as afunction of the diffusion rate. QS systems seems to avoid values of the diffusion that maximize the total noise.These results point towards the direction that bacteria have adapted their communication mechanisms in order toimprove the signal-to-noise ratio.

BackgroundGene regulation at the transcriptional level is one of thecorner stones of molecular and cellular biology [1].Recent studies in prokaryotes have revealed the exis-tence of antisense and alternative transcripts and multi-ple regulators per gene that imply a highly dynamictranscriptome more similar to that of eukaryotes thanfirst thought [2]. Still, prokaryotic gene regulationmainly relies on the binding of regulatory proteins thatattach to DNA for either stimulating or repressing tran-scription. These binding/unbinding events are intrinsi-cally probabilistic because of the significance of thermalfluctuations at that scale and the low number of mole-cules involved in the process. In this regard, over thepast years a growing number of experiments haveindeed characterized not only the levels of randomnessin cellular biochemical processes but also their function-ality [3-8].

Technical advances such as the use of fluorescent tagsin single-cell experiments have allowed for quantitativemeasurements of the noise in protein concentration andhave shed light on the mechanisms of gene expressionthat lead to cell-to-cell variability [9-11]. Moreover, theadvent of experimental approaches that permit to countindividual mRNA and protein molecules in single cellshas further evidenced the role played by fluctuationsand their characteristics [12,13]. Thus, in E. coli, thedirect measurement of integer-valued numbers ofmRNA as a function of time has revealed transcriptionalbursts with Poissonian statistics [14]. The latter is inagreement with the two-state gene expression modelwhere switching between the active and inactive tran-scriptional regimes occurs with constant probability[15]. It is worth noticing that these noisy sources, far forbeing a nuisance, have been recognized to play a con-structive role in many gene regulatory processes. Exam-ples in this direction include the efficiency of the phagelambda switch [16] or the differentiation into the com-petence state in bacteria [17]. All in all, it is now

* Correspondence: [email protected] Simulation and Modelling (Co.S.Mo.) Lab, Parc Científic deBarcelona, C/Baldiri Reixac 10-12, Barcelona 08028, Spain

Weber and Buceta BMC Systems Biology 2011, 5:11http://www.biomedcentral.com/1752-0509/5/11

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accepted that stochastic and non-linear approaches arerequired for understanding the randomness in thedynamics of biochemical reactions and the effects offluctuations in gene regulatory networks [7,8].While a lot of modeling studies have focused on the

single cell level [18,19], few of them have addressed therole played by noise at the colony level [20-23]. Ourrecent contributions within this topic, that illustrate theconstructive role of stochasticity, include the noise-induced coherence resonance phenomenon in multicel-lular circadian clocks [24] and the interplay between thestochasticity of the cell cycle duration and the proteinexpression noise in bacterial colonies [25]. One relevantquestion within this context is how cellular populationsdeal with stochasticity in communication processes. Inparticular, we focus on the simplest cell-to-cell commu-nication mechanism in prokaryotes: the so-calledquorum sensing (QS). The term QS generically refers tothe mechanism that allow bacteria to count their num-ber, i.e. the colony size, by producing, exporting/import-ing into/from the environment, and detecting a diffusivesignaling molecule, namely, the autoinducer [26,27]. Asthe population of bacteria grows, the autoinducer accu-mulates both in the extracellular medium and inside thecells. When the autoinducer concentration surpasses athreshold the expression of QS-controlled genes starts.This mechanism ultimately results in a response of thecolony in a cell density dependent manner. Importantly,QS has opened the door to the design of gene circuitsusing synthetic biology approaches that control cellpopulations at the collective level [28-31].Recent studies have shown that diffusion in QS

reduces the noise at the level of the autoinducer [23].However, to the best of our knowledge, the role playedby different sources of stochasticity and their contribu-tion to the dynamics of the signaling molecule has notbeen characterized yet. Moreover, while in eukaryotesthe diffusion seems to contribute for enhancing the pre-cision of regulatory processes, similar effects have notbeen reported in the context of QS [32]. We point outthat a deep understanding of these issues is key in orderto design robust synthetic circuits based on such bacter-ial communication system. Herein, we address theseproblems by studying the interplay between the QScommunication and the transcriptional noise in bacterialpopulations. First, we aim at understanding how thatinteraction determines the dynamics of the autoinducer.Second, we aim at shedding light on the mechanismsthat confer robustness to noise in QS communication.Transcriptional noise is expected to be particularly rele-vant when transcription events are short and rare.Under these conditions two main sources of stochasti-city naturally arise: the dichotomous fluctuatingdynamics of mRNA and the intrinsic noise due to low

copy number of species. Thus, we restrict ourselves tothe study of the aforementioned problems near the QSactivation threshold where we can assume that the tran-scription events produce basal constitutive levels ofmRNA as low as one molecule per cell at a time. Wenote that such mRNA production processes have beenexperimentally validated in prokaryotes revealing thatthe statistics of proteins bursts originates from thetranslation of a single mRNA molecule [33].Our main findings are threefold. First, we show how

the diffusion process leads to a repertoire of dynamicsin regards of the signaling molecule. Second, we demon-strate that, for a large range of diffusion rate values, themain contribution to the total noise of the autoinducerconcentration is the mRNA fluctuations. Finally, weshow that the total noise exhibits a non-monotomicbehavior as a function of the diffusion rate in contrastto previous results [23].The paper is organized as follows. In the Methods sec-

tion we introduce our modeling approach, the analyticalcalculations, and the parameter values used in ourin silico experiments. In the next section, Results, wepresent our findings and compare the results from sto-chastic simulations with those from analytical calcula-tions. Finally, we discuss further implications of ourresults in the Discussion, and summarize our findings inthe Conclusion.

MethodsModeling ApproachA large class of gram-negative bacteria use acyl homo-serine lactones (AHLs) as signaling molecules [27,34].These autoinducer molecules are typically synthesizedby enzymes of the LuxI family and can freely diffuseacross the cell membrane, i.e. by means of passive diffu-sion. When the concentration of autoinducer surpassesa critical threshold, it binds to its receptor (a cytoplas-mic protein of the LuxR family) which then activatesthe expression of target genes, e.g. in Vibrio fisheri spe-cies the luxICDABE operon, that is responsible for theexpression of luxI and luciferase. In contrast, when theautoinducer concentration is below the activationthreshold, the transcription of the luxI gene occurs at alow basal rate, thus producing low levels of the enzyme.In this regime, the feedback regulation of the luxI geneleading to autoinduction can be disregarded. As a mat-ter of fact, a number of QS systems lack autoinduction[35]. Consequently, to neglect the feedback in thosecases is a valid approximation even above the activationthreshold. Herein, for the sake of simplicity, we focus onthis situation and describe the dynamics of the autoin-ducer near, but below, the activation threshold whenfeedback loops can be neglected and the downstreamQS genes are not activated. We stress that, generally


Page 3 of 13

speaking, our results are not applicable above the activa-tion threshold since the dynamics of the autoinducer isobviously conditioned by feedback effects.Following previous approaches we consider a two-

stage model for gene expression/regulation [15,36-38].Thus, we assume that during the transcription events asingle mRNA molecule, e.g. a luxI transcript, is pro-duced and its dynamics can be then described by meansof a Markovian dichotomous process [39],

M Mi i0 1

⎯ →⎯⎯← ⎯⎯⎯ (1)

where Mi0 1 0 1, ,= stands for the number of mRNA

molecules at cell i and a and b for the transition ratesbetween these states; i.e. a and b account for the prob-abilities per unit of time of mRNA degradation andtranscription frequency respectively. Notice that the sto-chastic alternation of the mRNA between the values 0(no mRNA) and 1 (a mRNA molecule) is not memory-less, i.e. white. Once a mRNA molecule is produced,and until it becomes degraded, the cell keeps noticingits presence and keep producing the autoinducer. Thatis, the transcriptional noise is a colored noise, and itsautocorrelation decays exponentially with a characteris-tic time scale τc = (a + b)-1 [39].Once a mRNA molecule is produced the translational,

and post-translational processes (if any), leads to theappearance of functional LuxI synthetases. Yet, ourinterest here focuses on the dynamics of the signalingmolecule. It has been shown that the amount of thesynthetase substrate is not a limiting factor for the pro-duction of the autoinducer [40,41]. As a consequence,the levels of the signaling molecule depends directly onthe expression levels of the synthetase. Ignoring inter-mediate biochemical steps in the autoinducer synthesisreduces the number of noise sources and may evenchange, under some circumstances, the observeddynamics [42]. Still, it is a valid approximation in manysituations, and here we assume that the translation ofthe synthetase and the subsequent synthesis of the auto-inducer, A, can be effectively described by a single che-mical step with rate k+. In addition, we consider thatthe autoinducer becomes degraded at a rate k-, that is,

M M Ai k i i1 1

+⎯ →⎯⎯ + (2)

Ai k−⎯ →⎯⎯ ∅ (3)

Passive diffusion of the autoinducer can be implemen-ted by considering a new species, Aext, that accounts forthe number of signaling molecules in the extracellularmedium such that,

A Ai D

rD⎯ →⎯⎯← ⎯⎯⎯ ext (4)

where D stands for the diffusion rate and r = V/Vext

represents the ratio of the volume of a cell to the totalextracellular volume. We consider all cells to have thesame value of the diffusion rate. In addition, we assume awell-stirred system where spatial effects can be neglected.As the bacterial population grows the autoinducer

accumulates in the media. In experiments, in order tokeep the concentration of the autoinducer below the acti-vation threshold, such growth is compensated by meansof a dilution protocol. As detailed below (see Parameters)the latter constitutes the main source of effective degra-dation of the signaling molecule. Thus, hereinafter weassume the degradation rate of the signaling molecule tobe the same inside and outside the cell,

A kext

−⎯ →⎯⎯ ∅ (5)

Figure 1 schematically represents the biochemical pro-cesses considered in our approach. The set of reactions(1)-(5) characterizes the stochastic dynamics of the auto-inducer and that of the mRNA. Their probabilisticdescription is given by the corresponding master equa-tion that is exactly sampled by means of the Gillespiealgorithm in a N-cells system [43].

Analytical Calculations: Null Intrinsic Noise ApproximationFurther insight into the dynamics of the signalingmolecule can be obtained by analytical means as

Figure 1 Scheme of a simplified biochemical network of QSsystems near the activation threshold. Schematic representationof the biochemical processes considered in our approach fordescribing the dynamics of the signaling molecule, A, in cell i. ThemRNA dynamics satisfies a dichotomous process characterized bythe states M0,1 corresponding to zero and one moleculesrespectively. Once the autoinducer has been produced, it candiffuse in and out the cell leading to cell communication (see text).


Page 4 of 13

follows. Two stochastic contributions drive thedynamics of A: the mRNA fluctuations due to the ran-dom switching (mRNA present or not) and the mole-cular, i.e. intrinsic, noise due to low copy number ofthe autoinducer. As for the latter, it can be neglectedif over the course of time Ai/(Ai + 1) ≃ 1 (large num-ber of autoinducer molecules). While in our systemthis approximation is not totally justified (see para-meters values below), it is useful to implement it inorder to discriminate between the effects caused bydifferent stochastic contributions and to obtain analyti-cal expressions. In this case, it is straightforward todemonstrate that the dynamics of the autoinducer,Eqs. (1)-(5), can be described by the following coupledstochastic equations,

c k c t k c D c cA M A A Ai i i i= − + −+ −

1( ) ( )

ext (6)

c k c rD c c

k c rDN c c

A A

i

N

A A

A A A

iext ext ext

ext ext

= − + −

= − + ⟨ ⟩ −

−=

−

∑( )

(1

)),

(7)

where c A VA

ii = / , c t M V

Mi

i1

1( ) = , and

c A VAext ext ext= / stand for the concentration of speciesA and Mi

1 at cell i and for species Aext at the extracellu-lar medium respectively, N is the colony size (number ofcells), and ⟨·⟩ represents the population average. InEq. (6) the term c t

Mi1

( ) accounts for a dichotomousstochastic process characterized by the rates and states(a, b) and (0, V −1) respectively, and describes the fluc-tuating mRNA dynamics. We point out that in case thatD = 0, Eq. (6) has been proposed to study graded andbinary responses in stochastic gene expression. Interest-ingly, it has been shown that despite its simplicity it canactually reproduce some gene expression phenomena[37,44].We can further proceed with the analytical calcula-

tions by implementing, as in previous studies e.g. [45], aquasisteady approximation for the dynamics of the

external autoinducer, i.e. c Aext= 0, so that,

c ck

NDr

A Aext=

+ −

1

1. (8)

By substituting Eq. (8) into Eq. (6) we obtain an equa-tion for the concentration of the signaling moleculeinside a given cell that depends on the average ⟨cA⟩ (theindex i has been dropped),

c k c t Dk

Dc

cD

kN Dr

A M A

A

= ( ) − +⎛⎝⎜

⎞⎠⎟

++

+−

−

11

1

(9)

In the absence of diffusion, Eq. (9) reveals that theconcentration of the signaling molecule reaches a maxi-mum value of c k k VA

++= ( )− when c t VM1

1( ) = − . Interms of c A

+ and the time scale tc = 1/k-, the typical life-time of a signaling molecule, the dimensionless versionof Eq. (9) reads

c c t k c k cA M A A= ( ) + ( ) −+ −ˆ ,1

eff eff (10)

where D D k k D= = +−, ,_eff 1 and

k c c D NDr c tA A M+ ⟨ ⟩ = ⟨ ⟩ + ( )( )eff ( ) / ; ( ) 1 11

being aMarkovian dichotomous noise characterized by thestates ˆ ,c M1

0 1{ } = and the rates = −k and = −k Equation (10) can be formally closed byinvoking the following self-consistency condition:

⟨ ⟩ = ⟨ ⟩∫

c c c c dcA A A A AΩ

( ; ) , (11)

( ; ) c cA A⟨ ⟩ being the probability density solving

Eq. (10) and Ω its support [39]:

( ; ) ( ( ))

.( (

c c k c k c

k

A A A Ak⟨ ⟩ = ⟨ ⟩

+ ⟨

+

+

−

−1− −eff eff

eff

eff

1 cc k cA Ak⟩) )− −

−1−eff eff

(12)

with

=+ +

+⎡

⎣⎢

⎤

⎦⎥

+⎡⎣⎢

⎤⎦⎥ +

⎡

⎣⎢

⎤

⎦⎥

( )11

1 1

DD

D D

Γ

Γ Γ

(13)

being the normalization constant. The condition (11)can be exactly solved and leads to the following valuefor the average concentration:

⟨ ⟩ = ++ + +

= ++ +

⟨ ⟩ =

cDNr

DNr D

DNr

DNr Dc

A

A D

11

11 0

|

(14)


Page 5 of 13

where ⟨ ⟩ = +( )= c A D| /0 is the average concentra-

tion of signaling molecules in the absence of diffusion.Thus, as expected, ⟨ ⟩ < ⟨ ⟩ = c cA A D| 0 . For the sake of conci-sion, on what follows we drop the argument term ⟨ ⟩c Afrom the notation of ( ; ) c cA A⟨ ⟩ . Note that ( )c A hastwo states (barriers) that define its support. That is, theminimum and maximum values that the concentration ofthe autoinducer can reach as a function of the diffusion are:

cD Nr

D D DNrA− =

+ + + +

2

1 1( )( )

(15)

c cDA A

+ −= ++1

1(16)

It is easy to prove that the probability density ( )c A

shows a single extremum if,

, , k−eff (17)

Where the extremum is a maximum if , > −k eff

and a minimum if , < −k eff . In the other cases theprobability density does not display any extrema. There-fore, as a function of and , the probability density( )c A may show four different behaviours dependingon the value of the diffusion coefficient as schematicallyrepresented in Figure 2A. However, a constraint in our

modeling restricts the regions, i.e. behaviors, accessibleto the autoinducer dynamics. We have assumed a lowconstitutive expression such that only a single mRNAmolecule can be transcribed at a time. The latter impliesthat < (the degradation rate of the mRNA is largerthan the transcription rate) in order to assure that amaximum of one mRNA molecule is present in a cell ata given time. As a consequence, and independently ofthe diffusion value, the dynamics leading to the prob-ability density shown at the top-left region of Figure 2Acannot be considered as physical in the context of ourmodeling approach. We stress that this constraint is nota fundamental ingredient for obtaining our results (seeDiscussion).Finally, the noise of the autoinducer concentration

reads,

cc

AA

A

c

D DNr

D DNr D

22

2

2

2

1

1 1 1

=⟨ ⟩

=+ +( )

+( ) +( ) + + +(( )

(18)

where c A AAc c2 2 2= ⟨ ⟩ − ⟨ ⟩ .

Nonetheless we have disregarded the intrinsic noise inthe analytical calculations, Eq. (18) will allow us to eluci-date the contributions of different sources of noise asfollows. By means of the numerical simulations (Gille-spie) of the set of reactions (1)-(5) in a N-cell system,we can evaluate the total (intrinsic+transcriptional)noise. Hence, by substracting from that quantity thecontribution of the transcriptional noise, i.e. Eq. (18), weobtain the levels of intrinsic noise (see Results).

ParametersWe are interested in the role played by the fluctuationsof the signaling molecule, A, when its concentration isclose to the activation of the QS switch. Therefore, wefix the mean concentration of the autoinducer and mod-ulate the rest of the parameters in order to keep con-stant this value. Pai and You [46] have recently studiedthe core architecture of the QS mechanism for a com-prehensive set of systems. These authors have estimatedthat the critical concentration of autoinducer needed forthe activation of the QS genes ranges from 10 to 50 nMfor most of the bacterial species. In our model, weset the average concentration of A to a typical value ofC A

0 = 25 nM. Yet, our results do not depend on theparticular value we choose within that range. As shownbelow, see Results, this value fixes the level of intrinsicnoise of the system. However, the interplay between dif-fusion and transcriptional noise does not depend on

A B

Α�

Β�

1�D�

1�D�

�

�

� �

Γ1Γ2

Γ3 Γ40 5 10 15 200

5

10

15

20

Α�

Β�

Figure 2 Probability densities of the signaling molecule andparameter space. Panel A: Schematic representation of the differentprobability densities of the autoinducer concentration depending onthe value of and with respect to that of D . Given a set ofvalues ( , ) the dynamics of the autoinducer shows differentbehaviors depending on the value of the diffusion parameter sincethe transitions lines are located at , = 1 + D . The constraintsof our modelling in terms of the parameters values make the regionon the top-left corner non-accessible (see text). Panel B: Parameterspace diagram ( , ) indicating the sets of parameters used inthe simulations (solid squares): g1 = (8, 2), g2 = (15, 5), g3 = (8, 0.5), g4= (15, 0.5). The experimental values reported for the degradation rateof the mRNA leads to a biological meaningful range for (blueregion). The low constitutive expression assumption is prescribed bythe constraint > 2 (red colored region).


Page 6 of 13

that. Moreover, by defining the so-called sensing poten-tial, ν = (rN)-1 Pai and You estimated the range of criti-cal cell densities for the QS activation. They concludedthat its characteristic value is ν ~ 103 - 104. In our simu-lations we set this parameter to ν - 103. In the experi-mental setups the cells are typically present in a volumeof a few milliliters and the total number of cells is ofthe order of 108 - 109. Therefore, the concentration ofautoinducer in the medium is determined by theexchange of signaling molecules coming from manycells. In contrast, the behavior of QS systems with avery low number of cells can be significantly different,as shown by microfluidic confinement of cells in picoli-ter droplets [47-49]. In our study, in order to discardsmall system size effects, we choose a sufficiently largenumber of cells in the numerical simulations, N = 102.Since the typical volume of an E. coli cell is V = 1.5μm3 then Vext = 105V (i.e. r = 10−5). We point out thatkeeping ν to a constant value necessarily requires anexternal dilution protocol for maintaining constant thecell density. In experiments, the control of the dilutionrate is usually achieved by the use of chemostats ormicrofluidic devices [50]. The rate of dilution shouldcompensate for the cell growth, ~ 2 · 10−2 min−1 (i.e.cell cycle duration ~50 min). In our modeling, by keep-ing constant the number of cells and the average con-centration of the autoinducer, we tacitly assume adilution protocol too. Importantly, the dilution rateeffectively modifies the degradation rate of the signalingmolecule. In this regard, while some bacteria specieshave hydrolytic enzymes that degrade AHLs, generallyspeaking, bacteria that synthetize AHLs do not degradethem enzymatically. In fact, AHLs are chemically stablespecies in aqueous solutions [51]. The degradation rateof the homoserine lactone 3-Oxo-C6-AHL has beenmeasured in vitro revealing that this autoinducer israther stable: ~ 3 · 10−4 min−1 [51]. Measurements ofthe degradation rate of other AHL autoinducers showsimilar results. Based on experimental data and mathe-matical modeling, the degradation rate of the signalingmolecule in vivo has been also estimated [46]. Depend-ing on the pH of the medium, the latter ranges from ~5 · 10−3 min−1 to ~ 2 · 10−2 min−1. Consequently, thedilution process constitutes the main source of effectivedegradation of A, both inside and outside the cell, andhere we set k− = 2 · 10−2 min−1.As for the value of the diffusion rate, the coefficient of

passive diffusion has been estimated for the 3-Oxo-C6-AHL autoinducer based on the measure of the diffusionof glucose and lactose through the outer membrane ofE. coli [23]. For a typical cell volume of 1.5 μm3 the esti-mated coefficient of diffusion is ~ 103 min−1. Underthese conditions the typical value for the dimensionless

parameter D is of the order of 104. Yet, active transportmechanisms for the autoinducer leads to much smallereffective diffusion values (see Discussion) and weexplore the role played by this parameter.In regards to the mRNA dynamics, a, the degradation

rate, depends on the cell degradative machinery. To thisrespect, the half-lives of all mRNAs of Staphylococcusaureus have been recently measured during the mid-exponential phase. Most of the transcripts, 90%, havehalf-lifes shorter than 5 minutes [52,53].According to these studies we restrict the mRNA

degradation rate to the range ln(2)/5 min−1 < a < ln(2)/2 min−1. Consequently, > 1. As for the frequency ofthe transcription events, b is determined by particularcharacteristics of the gene regulatory process under con-sideration, e.g. the affinity of the regulatory proteins tothe operator site and the initiation rate of transcription.Due to the assumption of a low constitutive expression,we choose values of parameter b satisfying the relationa >b. In particular, we implement the more restrictivecondition a > 2b. Figure 2B recapitulate these con-straints and show the different sets of and valuesthat we have used in our simulations and analyticalcalculations.Summarizing, N, r, and k− are kept fixed in our simu-

lations and analytical calculations and we explore theparameter space a, b, and D within the ranges, andsatisfying the constraints, mentioned above. In everyparticular situation, once a set of those parameters isprescribed, we set the value of k+ by using Eq. (14) inorder to keep the average value of cA near its criticalconcentration value (25 nM).

ResultsComprehensive Study of the Autoinducer Dynamics as aFunction of the Diffusion RateThe distribution of cA at the steady-state is computedfor the different parameter sets according to the rangesand constraints described above (section Methods). Inorder to explore the role of the diffusion in thedynamics of the signaling molecule we first study thecase D = 0 . According to the analytical calculations,see Eq. (17), in this case two possible distributions forthe concentration of cA can be observed depending onthe value of . Since > 1 we can expect a maximumonly if > 1 (note that k− =eff 1 if D = 0 ), otherwiseextrema are not expected. The results of the numericalsimulations (Gillespie), Figure 3A, reveal that scenario.Note that in all cases the histogram obtained from thesimulations fits fairly well to the expression (12) exceptfor deviations due to the intrinsic noise that arenot taken into account by the analytical approach. Thedifferences among dynamics are evidenced by the


Page 7 of 13

trajectories, Figure 3B. Thus, for < 1 the dynamics ofthe autoinducer shows a burst-like behavior. If > 1the frequency of bursts is high enough to maintain theconcentration of signaling molecules near the averageand a single-peak distribution develops.If D > 0 we expect a more fruitful phenomenology

since the transition lines between behaviors in the para-meter space ( , ) shift as a function of the diffusion(see Figure 2A). According to the analytical calculationswe can anticipate that, for a given parameter set and asD increases, the system explores different dynamicalregimes. By taking as a reference the case g2, that is ( , ) = (15, 5), Figure 4 shows the effect of the diffusionon the distribution (left column) and dynamics (centercolumn) of cA in a given cell. The system initially dis-plays a single-peak distribution for D = 1. By increasingthe diffusion coefficient we observe transitions to theother phases (monotonically decreasing and double-peakdistributions). The corresponding dynamics of cA (rightpanels) show how the diffusion, acting as an additionaleffective degradation on A, first increases the sharpnessof the bursts of production. For D = 10, the diffusionis large enough to remove signaling molecules betweenconsecutive burst events, thus leading to a monotoni-cally decreasing distribution. Increasing the diffusionrate to D = 100 leads to the situation where both

and becomes smaller than 1 + D and a bistabledynamics develops. Under these circumstances the con-centration of autoinducer alternates between two statesthat correspond to a low concentration, when there isno mRNA production, and a high concentration, follow-ing the mRNA synthesis. As the diffusion furtherincreases, e.g. D = 2 · 103, the autoinducer moleculesdiffusing from the external medium into the cell set aconstitutive level of this species. The latter explains thepresence of A molecules in the cell even if no mRNA isproduced. Finally, at very large values of D , e.g. D = 5· 104, the low constitutive concentration of the autoin-ducer increases due to the influx of molecules when nomRNA is present whereas the concentration of A that isinternally produced decreases due to the efflux of mole-cules. In this case, the whole N-cells system can be con-sidered as a single volume with no diffusive barriersbetween cells. Thus, the burst events average out and,as a consequence, a single effective peak appears.

Diffusion and Intrinsic NoiseFigure 4 shows that the theoretical distribution capturesthe essential features of the dynamics obtained innumerical simulations (Gillespie). The noticeable devia-tions are due to the intrinsic noise that are not consid-ered in the theoretical analysis. Notice that as the

A B

100

Γ1

cA�nM� 100

Γ2

cA�nM� 1000

40Γ2

t �min�

c A�nM�

100

Γ3

cA�nM� 100

Γ4

cA�nM� 1000

130Γ4

t �min�

c A�nM�

Figure 3 Distributions and dynamics of the signaling molecule in a diffusionless system. Panel A: Distributions of cA at steady-state fordifferent sets of parameters ( , ) as indicated in Figure 2B. In all cases D = 0. The histogram obtained in simulations (blue bars) compareswell with the distribution from the analytical calculations (blue line). Yet, deviations are observed due to intrinsic noise (see text). Panel B: Thedynamics of the autoinducer show different behaviors depending on the region of the parameters phase space (see Figure 2). Two typicaltrajectories are shown with a grey-shaded background indicating the presence of a mRNA molecule in the cell.


Page 8 of 13

D��1

125cA�nM� 300

45

t �min�

c A�nM�

D��10

125cA�nM� 300

105

t �min�c A�nM�

D��100

125cA�nM� 300

120

t �min�

c A�nM�

D��2000

125cA�nM� 300

70

t �min�

c A�nM�

D��50000

125cA�nM� 300

40

t �min�

c A�nM�

Figure 4 Distributions and dynamics of the signaling molecule in a system with diffusion. Distributions (left column) and dynamics(center column) of cA at steady-state for different values of D . The right most column stands for a density plot of the distribution of c A1

as afunction of c A2

for discerning a putative increase in the molecular noise (see text). In all cases the parameters set ( , ) is g2 (see Figure2B). The production rate k + is modulated as a function of ( , , D ) in order to maintain constant the average ⟨cA⟩ = 25 nM. Thehistograms obtained in the stochastic simulations (blue bars, left column) are in qualitative agreement with the probability densities from theanalytical calculations (blue line, left column). When increasing the diffusion coefficient, the system explores different dynamics as revealed bythe trajectories shown in the center column. The grey-shaded background shown in the trajectories of cA indicates the presence of a mRNAmolecule in the cell. The density plots (right column) reveals that the diffusion does not contribute to an increase of the intrinsic noise since thespreading of the distributions in a direction perpendicular to the diagonal does not grow.


Page 9 of 13

diffusion increases those deviations seem to be larger.We stress that in our simulations we keep constant theaverage concentration of the autoinducer by modulatingthe effective production rate (see Eq. (14)). That is, asD becomes larger, we increase the production rate k+so that the average number of autoinducer moleculesper cell remains the same. Consequently, the deviationsbetween simulations and the theoretical analysis cannotbe ascribed to a putative intracellular decrease of thenumber of A molecules (i.e. to an increase of intrinsicnoise). Moreover, the deviations cannot be attributedeither to a failure of the quasisteady approximationintroduced in Eq. (7) because the larger the diffusion,the more accurate that approximation is. As indicatedby equation (18), the transcriptional noise behaves asD−1 . Thus, for large values of the diffusion rate, thetranscriptional noise level decreases. Therefore, we mustconclude that the deviations between the theoretical andthe numerical approaches become accentuated as thediffusion increases because there is a drop of the fluc-tuations related to the mRNA dynamics. This also indi-cates that for large enough diffusion rate, the intrinsicnoise constitutes the main source of stochasticity.In order to ensure that the intrinsic fluctuations are

not actually increasing due to diffusion we perform thefollowing in silico experiment. We consider a modifica-tion of our system such that a single mRNA moleculetranscript leads to two autoinducer molecules that areconsidered to be distinguishable. The latter can beexperimentally achieved by placing two consecutivecopies of the encoding sequence of the autoinducersynthetase labeled with different fluorescent tags in theoperon. Thus, we double the set of equations (2)-(5) inorder to account for Ai

1 and Ai2 molecules synthesis at

cell i due to a single mRNA transcript Mi1 . Following

[9], by plotting the distribution of c A1as a function of

c A2a putative increase of the intrinsic fluctuations can

be discerned by these means. Right column of Figure 4displays the results in this regard. The width of the dis-tribution in a direction perpendicular to the diagonal isa measure of the intrinsic fluctuations (see [9] fordetails). As shown, as the diffusion increases there is noamplification of this quantity.

Diffusion and Total NoiseIt is interesting to place the previous result in the con-text of the total noise of the autoinducer concentration.Figure 5 reveals that c A

2 shows a non-monotonic beha-vior. As a function of D the total noise first increases,reaches a maximum, and then decreases as the diffusionbecomes larger. While Figure 5 represents data for theg2 parameter set, this behavior applies for all the valuesof and D explored in our simulations (datanot shown). We point out that if c A

2 > 1 then the

dispersion is larger than the mean. Under this circum-stance the fluctuations can lead to catastrophic events.E.g., fluctuations are able to remove all the signalingmolecules within the cell. Note that the analytical calcu-lations, that just account for the transcriptional noise,are in agreement with the numerical simulations, thataccount for both the transcriptional and the intrinsicnoise, for a large range of D values. This indicates thatthe main contribution to the total fluctuations within alarge range of diffusion values is the transcriptionalnoise. Yet, as mentioned above, the latter diminishes asthe diffusion increases while the intrinsic fluctuationsremain constant. Consequently, the contribution of theintrinsic noise must become more relevant than themRNA stochasticity beyond some value of D .We address this point quantitatively by calculating

the relative importance of these noisy sources.To this end, we make use of the decomposition c c cA A A

2 2 2= +,int ,tran where c A ,int2 and c A ,tran

2 , standrespectively for the intrinsic and the transcriptional con-tributions to the total noise [54]. Thus, by subtractingthe analytical expression given by Eq. (18) to the totalnoise obtained in the numerical simulations, we are ableto compute the intrinsic noise as a function of the diffu-sion. By performing a linear regression of the pointsthat corresponds to the intrinsic noise we obtain thatthe slope of the curve is indeed zero in practical terms,~ 2 · 10−7. Therefore, in agreement with the qualitative

10�1 101 103

0.1

1

10

D�

ΗcA2

Figure 5 Noise of the signaling molecule as a function of thediffusion coecient. Noise c A

2 as a function of diffusioncoefficient D for the set of parameters g2 (see Figure 2B):stochastic simulations (circles) and analytical expression. Eq. (18),(solid line). By using the decomposition c c cA A A

2 2 2= +,int ,tran the differences between thecomputational and the theoretical distributions quantifies theamount of intrinsic noise (squares). As evidenced by the linearregression (blue short-dashed line) the later remains constant and isthe main contribution to the total noise only for large diffusionvalues, D > 104 (see text).


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results obtained in Figure 4 (right column), the intrinsicnoise remains constant as the diffusion increases,c A ,int

2 = 0.054 ± 0.003, and is the main stochastic com-ponent if D > 104.

DiscussionAs a matter of discussion, our modeling consider passivediffusion as the mechanism for the transport of the sig-naling molecule. This is indeed true in many QS sys-tems. However, in other cases the autoinducer isactively transported in and out of the cell. For example,in the bacterial species Pseudomonas aeruginosa, C4-HSL can freely diffuse but C12-HSL, a larger signalingmolecule, is subjected to active influx and efflux at ratesof ~ 10−2 min−1 and ~ 10−1 min−1 respectively [55].Other example corresponds to the AI-2 signaling mole-cule. The latter is present in many Gram-positive andGram-negative species and it is believed to allow forinterspecies communication [56]. In E. coli and Salmo-nella enterica extracellular AI-2 accumulates during theexponential phase, but then decreases drastically uponentry into the stationary phase. This reduction is due tothe import and processing of AI-2 by the Lsr transpor-ter [56,57]. Moreover, excretion from the cell of thisautoinducer also appears to be an active process invol-ving the putative transport protein YdgG (or alterna-tively named TqsA) [58]. In the case of E. coli theserates have been estimated by computational and experi-mental means: Dout ≃ 10-1 min−1 and Din ≃ 10-3 - 10-2

min−1 [59]. All in all, the diffusion rates when driven byactive processes are four orders of magnitude smallerthan the diffusion rates of small molecules through themembrane. Hence, the diffusion rates in QS systems canbe categorized into two main, well separated, classes:small diffusion rates due to active transport, and largediffusion rates due to passive mechanisms.In principle our model does not account for active dif-

fusion processes, but transport driven by concentrationdifferences. Still, it can be demonstrated that if the fol-lowing condition holds, r ≃ Din/Dout, then our simplemodel fairly describes the active diffusion with D = Dout.However, that regime is not accessible in our simula-tions near the critical sensing potential since that wouldimply a non-physical situation (N < 1). If in any case weconsider the rates of active transport of these QS sys-tems, they would fit in our model with a normalizedeffective diffusion coefficient in the range D Î [10-1,10]. Note that, either driven by passive diffusion or byactive transport, the region that maximizes the totalnoise, c A

2 , is not accessible: D ~ 5 · 101 − 102. Noticealso that this range of diffusion values corresponds tothe dynamics that produce separated peaks in the distri-bution of the autoinducer (see Figure 4). While ourmodeling is certainly very simple and the derived

consequences should be carefully taken, this observationsuggest that bacteria have developed mechanisms forcoping with the noise and keep their functional QSregime away from the region where c A

2 > 1. In thisregard, let us point out that in every form of informa-tion exchange the precision is key. If the precision ofthe information is fuzzy then the related biological func-tion lacks robustness. Thus, our results point towardsthe direction that bacteria have adapted their communi-cation mechanisms in order to improve the signal tonoise ratio.Diffusion can lessen the effects of fluctuations, both in

eukaryotes [32] and prokaryotes [23,60]. This behavioris certainly obtained in our model: if the diffusion rate islarger than a given amount, then the noise decreases asthe diffusion increases. However, to the best of ourknowledge, the reverse effect, i.e. the noise increases asthe diffusion increases, had not been reported. Theseopposed behaviors are responsible of the non-mono-tonic comportment of the noise described herein. Onemay wonder the reason underlying this phenomenology.By manipulating Eq. (18), it is easy to demonstrate thata) the slope of the noise at D = 0 is always positive andb) in the white noise limit, τc ® 0, the slope becomesnull and the transcriptional noise decreases monoto-nously as the diffusion increases. Consequently, theobserved behavior is due to the colored character of thetranscriptional noise, i.e. due to a competition of tem-poral scales: those of the diffusion and the noise correla-tion time. This result opens the possibility of finding asimilar phenomenology in other systems subjected torelevant levels of transcriptional noise and diffusive sig-nals. In addition, it points up the relevance of non-memoryless noisy sources in biological systems [11].Moreover, it shows that our results do not depend ondetails as the precise number of transcripts (one in ourcase). As long as there are distinct transcriptional phasesinside cells with characteristic time scales, that is, pre-sence versus absence of transcripts, the observed phe-nomenology is, qualitatively, the same. Yet, those phasesare prone to appear when the autoinducer is producedat constitutive levels and the number of transcripts issmall. Herein we restrict ourselves to the case of onetranscript simply because we are able to obtain analyti-cal results in that situation.

ConclusionsHerein we have explored the role played by cell-cellcommunication and transcriptional noise in QS systemsnear the activation threshold. Within this context, wehave shown that the dynamics of the signaling moleculeexhibits different behaviors depending on the diffusioncoefficient. When increasing the rate of diffusion, theprobability distribution of the autoinducer changes from


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single-peak distribution (sustained bursts dynamics), tomonotonically decreasing distribution (bursts dynamics),to double-peak distribution (bistable dynamics), andfinally to narrow single-peak distribution (diffusion-aver-aged dynamics).In addition, we have shown that the mRNA dynamics

plays a crucial role for regulating the total amount ofmolecular noise of the signaling molecules. Transcrip-tional noise is the main contribution to the total noisefor a large range of diffusion values, D < 104. Only forvery large values of the diffusion the intrinsic noise isthe major source of stochasticity. Due to a competitionof temporal scales, the total noise shows a non-mono-tonic behavior as a function of the diffusion rate. Forlarge values of the diffusion coefficient, the total noisedecreases as the diffusion rate increases. In this regard,our results are to be compared to previously reportednoise reduction mechanisms, as for example in the caseof bistable genetic switches coupled by QS communica-tion [60]. On the other hand, when the diffusion rate issmall enough compared to the characteristic rate oftranscription events, the total noise increases as the dif-fusion becomes larger. The values of the diffusion ratesin QS systems fall into two distinctive categories: eitherlarge values corresponding to passive transport mechan-ism or small values when an active transport mechanismapplies. Surprisingly, these two QS classes avoid diffu-sion rates that maximize the total noise. According tothese results, we conjecture that bacteria have engi-neered the communication mechanism for reducing thesignal-to-noise ratio and produce a more reliable infor-mation exchange.Our final comment refers to the possibility of consider-

ing other sources of stochasticity. Cell-to-cell variabilityand extrinsic noise have been proved to act as an impor-tant contribution in many cell processes [16,25,61,62]. Inthe context of the problem studied herein, we can envi-sion that variability, either at the level of the mRNAdynamics or at the level of the diffusion rate, can effec-tively lead to significant changes in the reported phenom-enology. In addition, by considering additional steps inthe synthesis of the autoinducer, the levels of intrinsicnoise would increase. Whether or not this extra level offluctuations, coupled with feedback regulation, may gen-erate new effects in the framework of QS is not known.Work in those directions is in progress.

AcknowledgementsWe thank Oriol Canela Xandri and Pieter Rein ten Wolde for fruitfulcomments. Financial support was provided by the Spanish MICINN undergrants FIS2009-11104 and BFU2010-21847-C02-01/BMC and by DURSIthrough project 2009-SGR/01055. M.W. acknowledges the support of theSpanish MICINN through a doctoral fellowship (FPU AP2008-03272) and thepartial support of the Catalan government by means of a FI doctoralfellowship (2009FI_B 00433).

Authors’ contributionsMW and JB contributed equally to construct the model, perform theanalytical calculations and write the paper. Numerical simulations werecarried out by MW. All authors read and approved the final manuscript.

Received: 28 July 2010 Accepted: 20 January 2011Published: 20 January 2011

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doi:10.1186/1752-0509-5-11Cite this article as: Weber and Buceta: Noise regulation by quorumsensing in low mRNA copy number systems. BMC Systems Biology 20115:11.

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Noise regulation by quorum sensing in low mRNA copy number systems