Computer Simulation and Modelling (Co.S.Mo.) Lab, Parc Científic de Barcelona, C/ Baldiri Reixac 4, 08028 Barcelona, Spain

Abstract

Background

A wide range of bacteria species are known to communicate through the so called quorum sensing (QS) mechanism by means of which they produce a small molecule that can freely diffuse in the environment and in the cells. Upon reaching a threshold concentration, the signalling molecule activates the QS-controlled genes that promote phenotypic changes. This mechanism, for its simplicity, has become the model system for studying the emergence of a global response in prokaryotic cells. Yet, how cells precisely measure the signal concentration and act coordinately, despite the presence of fluctuations that unavoidably affects cell regulation and signalling, remains unclear.

Results

We propose a model for the QS signalling mechanism in

Conclusions

Our

Background

Bacteria, long thought having a solitary existence, were found to communicate with one another by sending and receiving chemical messages

The QS systems in gram-negative bacteria share a core network architecture. In this regard, a characteristic model system is the LuxR/LuxI regulatory network in

A number of studies have shown that noise plays an important role in bistable systems

In the context of QS modelling, most research has focused on the understanding of the intracellular circuit

Methods

Modelling of the LuxI/LuxR gene regulatory network

The regulatory interactions that control the wild-type lux operon are more complex than first thought _{2}). The latter binds to the promoter region activating both the transcription of _{
ext
}) that can be eventually modified by an external influx of molecules (^{∗}) and a dilution protocol (see below). In our model we consider that signalling molecules degrade at the same rate whether they are cytoplasmic or not. Finally, we consider a

Scheme of the LuxI/LuxR regulatory network

**Scheme of the LuxI/LuxR regulatory network.** The LuxR (

As revealed by the set of reactions (1), we assume that the regulatory complex (_{2} activates the transcription of _{
A
}=0, and an exogenous supply of the signalling molecule is required to induce the system. The expression rates of _{
R
}
_{
R
} and _{
I
}
_{
I
}, even though the regulatory complex (_{2} is not bound to the promoter region of the _{
R
},_{
I
}≪1 (see parameter values below), the maximum transcriptional rates take place when the activator complex is bound.

Deterministic and stochastic approaches: cell growth and division

The equations (1) lead to a Master equation description that can be sampled exactly by means of the Gillespie algorithm _{
c,tot
}(Figure _{
X
}, in _{
c,tot
}. Therefore, this model can only be used to study the dynamics of species averaged over all the cells in the population. From an experimental point of view, the population average can be measured determining the average bulk fluorescence of the

**Text S1.** Chemical equations for the deterministic model.

Click here for file

**Video S1.** Movie of the stochastic simulation. Movie of the stochastic simulation for the _{
R
}=_{
I
}=4. Cells are modelled as individual compartments containing a copy of the LuxR/LuxI regulatory network. The Gillespie algorithm (see text for details) is used to integrate the stochastic dynamics of the whole system of cells. Cell growth and division is explicitly taken into account as well as a certain degree of stochasticity in the cell cycle duration. Cells movement is purely aesthetic since we do not include any spatial effects in our model and consider a well-mixed environment. The number of cells (

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Scheme of the deterministic and stochastic modelling approaches

**Scheme of the deterministic and stochastic modelling approaches. A**: In the deterministic model, the population of cells is described by a unique volume with average and continuous concentrations of all species, including the DNA carrying the QS network (small circles). Cellular growth is also taken into account in this approach. **B**: In the stochastic model, cells are modelled as individual compartments that can grow and divide and all molecular species are represented as discrete entities. In both cases, **A** and **B**, we assume that all species are well-stirred inside the cells and in the medium. In order to maintain a constant cell density, as in the experiments we aim to model, we implement a dilution protocol. In the deterministic model the dilution removes continuously cytoplasmic material in order to compensate the cell growth. In the stochastic model individual cells are removed every time a new cell is born (see Additional file

We notice that our _{
c,tot
}(_{0,tot
}2^{
t/τ
}. Where _{0,tot
}=_{0}, _{0}the volume of a single cell at the beginning of the cell cycle. As a consequence, the cellular growth introduces dilution terms, _{
ext
}. On the other hand, cell division events lead to the duplication of the genetic material. The latter is taken into account by adding the term

In our simulations, as in the experiments we aim to reproduce, the cell density is kept constant. This can be achieved by means of an external dilution protocol (see below) that compensates for cell proliferation. We then keep the volume _{
c,tot
} constant and define the external volume, _{
ext
}, such that the total volume of the cell culture reads _{
tot
}=_{
ext
} + _{
c,tot
}. Accordingly, the parameter _{
c,tot
}/_{
ext
}. We assume that molecules are homogeneously distributed inside both the cytoplasm and the external volume (i.e. spatial effects are disregarded). Finally, the resulting ODEs are numerically integrated.

In order to study the role of noise in a population of cells communicating by QS, we build also a stochastic model of a population of bacteria. In this case, each bacterium is described as a single cell carrying a copy of the regulatory network. The ensemble of all the chemical reactions in all cells, including the diffusion reaction, are treated as one global system. We apply the Gillespie algorithm _{
ext
}changes the probabilities of an autoinducer molecule to diffuse into any other cell. Thus, all the cells are coupled through the diffusion reaction. We note that while a possible optimization of the algorithm relies on parallelizing the code such that each cell evolves independently

As mentioned above, cell growth introduces a dilution of the molecules in a cell. We implement cell growth in our stochastic model by allowing the volume of cell

where _{0}is the volume of a cell at the beginning of the cell cycle (same for all cells), _{
i
}is the duration of the cell cycle of cell _{
i
} the cell _{0}respectively. Moreover, when a cell divides, proteins, mRNAs and signalling molecules are binomially distributed

The duration of the cell cycle, _{
i
}is different for each cell and is set independently after a division according to the following stochastic rule

where

In this way, we allow variability from cell to cell in regards of the duration of the cell cycle, yet setting a minimum cell cycle duration,

Finally, we notice that in principle the Gillespie algorithm needs to be adapted in order to take into account the time-dependent cell volume. The propensity of a second-order reaction at cell _{
i
}(_{0}
_{0}/_{
c,i
}(_{0} stands for propensity of the reaction at division time when _{
c,i
}(0)=_{0}. The propensity _{0}are derived from the corresponding reaction rate, _{0}=_{0}. In addition to the change in the propensities of the reaction channels, the algorithm would also need to be adapted to compute the time till next reaction

Gene expression noise: burst size

During translation mRNA molecules are translated into proteins following a bursting dynamics _{
X
}, is defined as the ratio between the protein _{
X
} is directly related to the intensity of gene expression noise _{
X
} is, the more fluctuating expression dynamics is displayed by protein _{
X
}as a parameter to tune the noise intensity at the level of _{
R
}=_{
I
}=20.

External dilution protocol

In controlled experimental setups it is advantageous to keep the cell density constant. This is carried out by means of an external dilution protocol that compensates for cell growth. Experimentally, this is usually achieved by periodic dilutions of the cell culture

In the deterministic model, as shown in Figure _{
c,tot
}. Cell density is controlled by a continuous efflux that removes cytoplasm and culture medium at a rate that compensates exactly for the cell growth, such that the volume _{
c,tot
}remains constant. Concurrently, a continuous influx of equal and opposite rate brings fresh medium to the cell culture. In our _{
ext
}, from the medium and washing away cells by “deleting” a cell picked at random in the population each time a new cell is born.

In our simulations, as in the experiments we aim to reproduce, the exogenous autoinducer concentration

where _{
ext
}:_{
tot
}/_{
ext
}≃1. In the absence of synthesis (e.g.

**Figure S1.** Intra and extracellular autoinducer as a function of exogeneous autoinducer concentration. Response curves to autoinducer induction for **A**, **C** and **E**) and **B**, **D** and **E**) operons. Total autoinducer concentration **A** and **B**), intracellular concentration _{
A
}(**C** and **D**), and extracellular concentration **E** and **F**), as a function of the exogenous autoinducer concentration,

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Parameters

The parameters used in our model are listed in Table _{
N
}=5·10^{8}cells/mL. Moreover, in order to keep the computational time within reasonable limits, we choose a system size of _{
N
}=_{
tot
} and _{
ext
}=_{
tot
}−_{0}, where _{
tot
}=2·10^{−4}
_{
A
}. The latter is adjusted such that the lower bound of the hysteresis region extends up to

**Parameter**

**Description**

**Value**

**Reference**

_{
d1}

dissociation constant of LuxR to A

100

unbinding rate of LuxR to A

10 ^{−1}

estimated

_{
d2}

dissociation constant of

20

fitted

dissociation rate of dimer (_{2}

1 ^{−1}

estimated

_{
A
}

synthesis rate of A by LuxI

0.04 ^{−1}

fitted

_{
dlux
}

dissociation constant of (_{2} to the lux promoter

200

fitted

dissociation rate of (_{2} to the lux promoter

10 ^{−1}

estimated

burst size

20

_{
R
}

transcription rate of

200/^{−1}

fitted

_{
I
}

transcription rate of

50/^{−1}

fitted

_{
R
}

translation rate of

_{
mR
}
^{−1}

_{
I
}

translation rate of

_{
mI
}
^{−1}

_{
R
}

ratio between unactivated and activated rate of expression of

0.001

fitted

_{
I
}

ratio between unactivated and activated rate of expression of

0.01

fitted

_{
A
}

degradation rate of

0.001 ^{−1}

degradation rate of (_{2}

0.002 ^{−1}

estimated

_{
C
}

degradation rate of

0.002 ^{−1}

estimated

_{
R
}

degradation rate of LuxR

0.002 ^{−1}

estimated

_{
I
}

degradation rate of LuxI

0.01 ^{−1}

estimated

_{
mR
}

degradation rate of

0.347 ^{−1}

_{
mI
}

degradation rate of

0.347 ^{−1}

effective diffusion rate of

10 ^{−1}

cell cycle duration (doubling time) in RM/succinate at 30 C

45

relative weight between the det./sto. components of the cell cycle

0.8

_{0}

cell volume at the beginning of cell cycle

1.5 ^{3}

_{
tot
}

total cell culture volume

2·10^{−4}

First passage time analysis

The mean first passage time at a given autoinducer concentration quantifies the average time that a cell takes to get activated or deactivated. For computing the first passage time in transitions, from low (high) to high (low) state, we take a single cell at the low (high) state and follow its dynamics until the GFP expression level reaches the high (low) state. We point out that the maximum GFP concentration refers to that of the deterministic simulations. In order to get enough statistics, we repeat this procedure, departing from the same initial condition, 10^{3} times for each concentration of autoinducer.

Results

The deterministic model reproduces the experimental observations at the population level

The chemical kinetics formalism leads to a set of ODEs that describes the population average dynamics in terms of the concentration of the different species considered in our model (see Additional file

We use the deterministic simulations as a benchmark of the regulatory interactions included in our model and also to fit/estimate some parameters such that the experimental data are reproduced (see

Response curves to autoinducer induction in the population-average model

**Response curves to autoinducer induction in the population-average model.****A**) and **B**) operons. The normalized GFP concentration is plotted as a function of the exogenous autoinducer concentration

Further simulations to check if the dynamics of our model is compatible with the experimental data refer to the behaviour of the system under non-stationary induction conditions and to the serial dilution protocol of the external medium

The stochastic simulations reveal the interplay between non-stationary effects and noise

Cells are subjected to intrinsic noise at the level of the mRNAs, regulatory proteins, i.e. LuxR and LuxI, and at the level of signalling molecules. In order to analyze the behaviour of individual cells and reveal how noise affects the QS switch, we perform stochastic simulations of a population of growing and dividing cells as described in the Methods section (see Additional file

Figure

**Figure S2.** Cell response distribution during decreasing-concentration trajectories. Cell response distribution for decreasing-concentration trajectories for

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Cell response distribution to autoinducer induction in the stochastic model

**Cell response distribution to autoinducer induction in the stochastic model.** Cell response probability after 10 hours (top: **A**, **B**) and 100 hours (middle: **C**, **D**) of induction at different autoinducer concentrations for the **A**, **C**) and **B**, **D**) operons in the stochastic model. The distribution reveals the coexistence of two subpopulations with low and high GFP expression when the cells are induced at intermediate autoinducer concentrations. The region of bistability (precision) is defined by the range of **A**, **B**) most cells are still in a transient state if **C**, **D**), the bimodality region shrinks and the precision increases. The population average curves of the induction and dilution experiments in the stochastic model (bottom: **E**, **F**, dashed lines) show that the intrinsic noise allows cells to jump to the high state inside the deterministic bistable region. On the other hand, the transition from high to low follows the deterministic path thus indicating that the switching rate in this case is close to zero.

Individual cell trajectories for autoinducer induction in the stochastic model

**Individual cell trajectories for autoinducer induction in the stochastic model.** Individual cell trajectories (blue lines), cell population average (orange line) and deterministic solution (red dashed line) for an induction experiment at

The heterogeneity in terms of the jumping statistics is revealed in Figure

Lineage tree of an induced population of cells in the stochastic model

**Lineage tree of an induced population of cells in the stochastic model.** Linage tree of a population of cells induced at a fixed autoinducer concentration

As expected the intrinsic noise decreases the precision of the QS switch with respect to the deterministic case. Still, noise helps cells to become activated before the critical concentration of a fluctuations-free system under all induction conditions. Moreover, in steady-state conditions the high state is globally achieved before the critical deterministic concentration. This phenomenon is recapitulated in Figure

The features of the QS switch depends on the transcriptional noise of LuxR

For the same concentration of the external autoinducer, the stochastic dynamics of the regulatory network arises from the noise at the level of LuxI and LuxR. We now analyze the individual contribution of those by modulating the burst size of LuxR and LuxI, _{
R
}and _{
I
}respectively. We notice that the burst size modulates the stochasticity levels while maintaining the average protein copy numbers. Additional file

**Figure S3.** Trajectory of chemical species in individual cells. Trajectory of chemical species LuxR mRNA (mR), LuxR, LuxI, intracellular autoinducer (AI), regulatory complex (_{2}(AL2) and promoter bound to complex (P10), in an individual cell for the following control parameter and burst size values: (A)

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Mean first passage time of cell activation for different burst size values

**Mean first passage time of cell activation for different burst size values.** Mean first passage time of cell activation as a function of autoinducer concentration for different values of the burst size for LuxR (_{R}) and LuxI (_{I}) and for the deterministic solution: (**A**) low to high transition MFPT in the **B**) low to high transition MFPT in the _{R}=_{I}=20 (blue shaded region) and _{R}=_{I}=0.01 (green shaded region). The MFPT reveals a non-trivial behaviour: for low autoinducer concentration noise helps cells to jump quicker to the high state, while for high autoinducer concentration noise slows down the cells activation (see text). Intersections of the quantile 10% and quantile 90% curves with a horizontal line at _{R}since GFP does not contribute to the activation process.

We observe these effects both for the _{
low
} and _{
high
} for which, at a given _{
low
}and a FPT>_{
high
}are 10%, i.e. the 10% and 90% quantiles respectively. The shadings in Figure _{
R
}=_{
I
}=20 and _{
R
}=_{
I
}=0.01. The precision of the switch after _{
R
} and _{
I
}. Notice that the region of bimodality does not vary when changing the burst size for LuxI. However, decreasing the burst size in LuxR reduces the region of bimodality thus increasing the precision of the switch. Furthermore, the noise at the level of LuxR helps some cells to become activated at lower concentration levels of the autoinducer. Once more, this phenomenon does not depend on the levels of transcriptional noise of LuxI. That is, while the global coordination increases as the transcriptional noise of LuxR decreases, more concentration of the autoinducer is required to start activating cells. Figure

Cell response distribution in the transient regime for different burst size values

**Cell response distribution in the transient regime for different burst size values.** Cell response distribution (jumping probability) after 10 hours of induction (transient state) at different autoinducer concentrations for the **A**) _{R}=_{I}=20 (**B**) _{R}=4,b_{I}=20 (**C**) _{R}=20,b_{I}=4 (**D**) _{R}=_{I}=4 (**E**) _{R}=_{I}=0.01. Width of bistable region: (**A**) = 60 nM (**B**) 25 nM (**C**) 70 nM (**D**) 27.5 nM (**E**) 25 nM. The black line stands for the concentration of GFP (normalized) as a function of

Cell response distribution at the steady-state for different burst size values

**Cell response distribution at the steady-state for different burst size values.** Cell response distribution at the steady-state (100 h induction), at different autoinducer concentrations for the **A**) _{R}=_{I}=20 (**B**) _{R}=_{I}=4 (**C**) _{R}=_{I}=0.01. The probability density of getting a particular GFP expression level is indicated by means of a density plot. The width of bistable region barely depends on the stochasticity levels, ≈7 nM. The black line stands for the concentration of GFP (normalized) as a function of

Discussion

The response of bacterial colonies driven by the QS signalling mechanism under noisy conditions has been addressed, in a broad sense, by different authors. In particular, the characterization of the collective response as a synchronization phenomenon where the phenotypic variations can be generically predicted has been proposed

Herein, we have characterized how the precision of the QS switch depends on the stochasticity levels and, importantly, elucidated which noisy component of the LuxI/LuxR regulatory network drives the observed phenomenology. Thus, we have found that under non-stationary conditions, LuxR controls the phenotypic variability and that changing the noise intensity at the level of LuxI has no effect on the precision of the switch. A plausible explanation for this reads as follows. The fluctuations at the level of LuxI are transmitted to the autoinducer. However, the diffusion mechanism rapidly averages out the stochasticity levels of the latter. This is not possible for LuxR which is kept within the cell. As a consequence the amount of activation complex, that is ultimately the responsible for the activation, is driven by the fluctuations of LuxR but not by those of LuxI.

Recent experimental work has measured the bioluminescence levels of individual

Finally, our simulations indicate that non-stationary effects are essential during the activation of the QS response. While speculative, these results can be extrapolated to growing colonies where the cell density is not kept constant. A good supply of nutrients implies short induction times since the concentration of autoinducer will quickly grow (exponentially) as the population size does. According to our results, this fast growing condition decreases the precision of the switch and, consequently, promotes variability at the population level (see Figure

The growth rate conditions the phenotypic variability

**The growth rate conditions the phenotypic variability.** In the context of a growing colony, the autoinducer concentration increases as the colony does: purple lines show schematically two exponential growth conditions for the autoinducer concentration as a function of time. Our results on the MFPT, valid at fixed autoinducer concentrations, can be extrapolated, qualitatively, to the case of increasing autoinducer levels. Fast growth results in a large cell variability and large critical colony size for achieving a global response, while slow growth produces reduced cell variability and a smaller critical population size. Increasing fluctuations in LuxR have two opposite effects: in the slow growth case, increasing the noise (blue curves: _{R}=20; green curves: _{R}=0.01;) decreases the critical population size while hardly changing the variability, in the fast growth case, increasing noise increases the critical population size and increases greatly the variability.

Conclusions

Herein we have introduced deterministic and stochastic modelling approaches for describing the core functionality of the LuxI/LuxR regulatory network in quorum sensing systems. We have focused on synthetic constructs,

By computing the statistics of the activation dynamics of cells, we have shown that the QS precision depends on the gene expression noise at the level of LuxR and is independent from that of LuxI. Our results, together with the experimental evidences on LuxR regulation in wild-type species, suggest that the noise at the level of LuxR controls the phenotypic variability of the LuxR/LuxI QS systems and that bacteria have evolved to control its intensity. In addition, the robust stabilization of the phenotype once is fully induced indicates that, albeit synthetic strains as

Most insight in regards of the effect of LuxR noise on the dynamics of cell activation is given by the study of the mean first passage time (MFPT). In terms of the timing of activation, we have observed two opposite effects depending on the control parameter

In summary, our results indicate that in bacterial colonies driven by the QS mechanism there is a trade-off between the activation onset and a global response due to non-stationary and stochastic effects. On one hand, large levels of noise at the level of LuxR imply that cells require smaller autoinducer levels for achieving an activation onset but, at the same time, a global response requires a substantial autoinducer concentration. On the other hand, if the LuxR noise levels are small, the activation onset is shifted toward larger values of the autoinducer concentration but the global response is achieved for smaller concentration values. Our study could be useful for Synthetic Biology approaches that exploit the QS mechanism. The fact that some important features of the QS mechanism, e.g. precision, rely on the burst size of one component, opens the door to modifications of the LuxI/LuxR operon for regulating the response depending on the problem under consideration. Finally, further research is needed about the general validity and applicability on the noise-induced stabilization phenomenon of particular phenotypic states in other gene regulatory systems beyond the QS mechanism. Work in that direction is in progress.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

MW and JB designed the experiments. MW carried out the simulations. MW and JB analyzed the data. All authors read and approved the final manuscript.

Acknowledgements

We thank Oriol Canela Xandri and Nico Geisel for fruitful comments. Financial support was provided by MICINN under grant BFU2010-21847-C02-01/BMC, and by DURSI through project 2009-SGR/01055. We also acknowledge support from the European Science Foundation through the FuncDyn programme. M.W. acknowledges the support of the Spanish MICINN through a doctoral fellowship (FPU AP2008-03272).