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ABG and LHZ conceived and designed the study. HBZ performed the experiments. DJT, KBW, and TL developed software and performed the numerical computations. ABG designed and analyzed the models and wrote the paper.

The authors have declared that no competing interests exist.

Understanding of the intracellular molecular machinery that is responsible for the complex collective behavior of multicellular populations is an exigent problem of modern biology. Quorum sensing, which allows bacteria to activate genetic programs cooperatively, provides an instructive and tractable example illuminating the causal relationships between the molecular organization of gene networks and the complex phenotypes they control. In this work we—to our knowledge for the first time—present a detailed model of the population-wide transition to quorum sensing using the example of

Molecular networks, which integrate signal transduction and gene expression into the unified decision circuitry, are ultimately responsible for the realization of all life activities of biological cells including internal developmental programs and responses to environmental factors. One of the main challenges of systems biology is to uncover and understand the relationships between the properties of these molecular circuits and the macroscopic cellular phenotypes that are controlled by them [

QS refers to the ability of bacterial populations to collectively activate certain gene expression programs, e.g., toxin release or antibiotic production, once some critical population density has been reached. QS is found in a vast variety of bacterial species and has been extensively studied experimentally [

This transition on both intracellular and population-wide scales is the focus of our study. We investigate the phenomenon of QS in the soil-dwelling plant pathogen

Functional significance of QS for the control of Ti plasmid conjugation remains an ecological and evolutionary puzzle. It is widely believed [

An experimental approach to this problem is often complicated by the technical difficulty of work in real ecosystems. On the other hand, mathematical modeling can significantly aid and complement experimental methods in answering biological questions that involve spatial and temporal scales of the QS phenomenon. Some aspects of either intracellular [

All genes that are thought to constitute the QS network are located on the Ti plasmid itself [

Blue ellipses represent proteins, red rectangles indicate mRNA species, and green flattened circles denote metabolites. Open circle arrowheads represent enzymatic activation of a reaction or transport, and open triangle arrowheads denote translation. Essential bimolecular reactions are shown explicitly as open squares. S represents two substrates of the autoinducer synthetase TraI, and Ø denotes protein degradation.

Several lines of evidence suggest that active transporters facilitate traffic of the QS signaling molecules through the cell wall in a number of bacterial species including

We first asked whether the intracellular model constructed by us purely from the individual molecular interactions indeed describes known biological properties of the QS network. Although numerical simulation of the full model can answer this question in principle, it does not bring qualitative insight into the system's behavior. Thus, we reduced the full model to only two nonlinear differential equations describing TraRd and intracellular AAI that are readily amenable to qualitative analysis (see _{e},_{i}_{e}

Nullclines represent lines on which the respective variable does not change with time (e.g., _{i}_{e}

Sensitivity of the QS network to the changes in the extracellular concentration of AAI and consequently to the population density is defined by passive permeability of the cell wall to AAI and any active transport, if existent. Passive diffusion largely depends on the physical properties of the AAI molecule (e.g., length of the hydrocarbon chain) and cannot be controlled by the QS network. On the other hand, our model shows that an active importer, such as putative transporter Imp, can exert significant control over the transition threshold. Indeed, we found that the critical extracellular AAI concentration

The solid line is fitted to the computed data points indicated by the filled squares. The value of

Our model clearly demonstrates that the negative feedback provided by

Additional information about the contribution of various components of the QS network to the control of the TraRd copy number can be gained from the analysis of sensitivity of the stationary concentration of TraRd to variation in the network parameters. One of the popular methods to assess sensitivity of molecular networks [_{e}_{e} =_{e}_{e}

We next set out to investigate whether the sharp switch-like behavior of the QS network predicted by the deterministic model is preserved when fluctuations in the number of molecules are considered. To answer this question, we simulated the full intracellular model stochastically (see

TraRd concentration averaged over 6 × 10^{6} s (filled squares) is plotted against extracellular concentration of AAI. The prediction of the deterministic model is shown as a solid line for comparison.

Undetectable in the average value, fluctuations in the copy number of TraRd deserve special attention as they can dramatically affect the network behavior. In particular, the off state predicted by our model is biologically meaningful only if the fluctuations of TraRd are controlled as tightly as its average concentration. Rare but significant departures from the off state in the absence of the extracellular AAI signal (_{e}_{e}^{−5}. This demonstrates that in the absence of external autoinducer the QS network maintains robust control of the fluctuations in the TraRd copy number and effectively prevents spontaneous transitions to the on state. At the same time, other molecular species whose copy number is not controlled by the QS network, e.g., TraI, TraM, and intracellular AAI, exhibit large-amplitude fluctuations around their average levels, in agreement with earlier reports for other molecular networks [_{e}_{e}

We first investigated the transition to QS in the simplest case of a population growing exponentially in a homogeneous liquid medium. The stochastic population model was simulated to imitate actual population dynamics of motile bacteria in a small volume element (_{e}^{−5} ml) of a liquid medium bulk. During approximately 7 h of simulation time, the population grew more than 100 times to reach the maximal density ^{9} cells/ml.

Intracellular concentrations of TraRd (thick red line) and TraM (thin blue line, filled circles) are plotted against the population density that exponentially grows with time.

(A) Dynamics of an individual cell in the stochastic population model.

(B) Dynamics of the stochastic population model averaged over ten bacteria.

(C) Behavior of the deterministic population model.

The value of the critical AAI concentration _{e}_{c}

The difference between the actual critical concentration _{e}^{c}_{e}^{c,∞}_{c}_{e}^{c}_{e}^{c,∞}_{e}^{c} − A_{e}^{c,∞}∝T^{ − 0.61}_{c}

The threshold population density depends on ^{c}^{9} cells/ml by the stochastic model and ^{9} cells/ml by the deterministic approach). In response to depletion of the nutrients in the liquid medium, a typical ^{ c}

We interpreted the previous finding as an indication that the simplest experimental scenario of growth in a spatially homogeneous liquid medium is not ecologically relevant and the QS network is not “tuned on” to support the transition to QS in such conditions. Indeed, in nature, the transition to QS and subsequent bacterial conjugation take place in the thin but dense biofilm on the surface of a Ti-plasmid-induced plant tumor. In laboratory conditions, the experiments on detection and quantification of conjugation are normally performed in the quasi two-dimensional environment of polymer filters that provide bacteria with firm support for attachment and conjugation [^{8} cells/cm^{2} for the faster and 1.33 × 10^{8} cells/cm^{2} for the slower diffusion, are considerably lower than the homogeneous bulk value. From analysis of electron microscopy images of ^{2} (0.5–5.0 × 10^{8} cells/cm^{2}). The values predicted by our model are thus well within the natural range and can be readily reached by a biofilm growing in the optimal nutrition conditions, e.g., on the feeding surface of a plant tumor.

(A) Spatial gradient of autoinducer created by the biofilm after 7 h of growth at two different values of AAI diffusion coefficient: (a) 10^{−5} cm^{2}/s and (b) 10^{−6} cm^{2}/s. The

(B) Intracellular copy numbers of TraRd and TraM averaged over 20 randomly selected bacteria versus the population density for the slower diffusion of AAI.

Using the phenomenon of bacterial QS as an enlightening example, we investigated the relationship between the dynamics of an intracellular molecular network and the population-wide phenotype that is controlled by this network. We first reconstructed the QS network of

One of the long-standing questions in this field is whether all cells in a population produce autoinducer at the same rate and experience transition to QS simultaneously [

Despite large variability between individual cells, a single population-wide value can be meaningfully defined for both critical concentration of autoinducer and the threshold population density for the transition to QS if the intracellular dynamics of individual cells is averaged over the population. These values are robust to stochasticity of individual bacteria and can be predicted with sufficient accuracy by a deterministic population model provided that spatial heterogeneity is insignificant. Our model demonstrates that when driven by exponential growth, the population transition to QS does not occur under steady state conditions and therefore the critical values depend on the parameters of population growth. Thus, the critical concentration of autoinducer depends on the growth rate.

Importantly, bacteria grown in different experimental conditions require different population densities to reach QS. Transition to QS in the bulk of the liquid medium appears to be the least favorable and requires much higher population density than transition in a biofilm. This result suggests potential ecological and evolutionary significance of the QS phenomenon for Ti plasmid propagation. In natural conditions a bacterial population dwells in a heterogeneous habitat with both bulk (e.g., soil) and the attracting surface (the plant–soil interface). Given the difference in cell density thresholds, it is likely that the transition to QS occurs in the surface-attached biofilm but not in the bulk. Therefore, it is tempting to speculate that the Ti plasmid utilizes QS to detect whether its bacterial host is firmly attached to the biofilm in close proximity to the source of nutrition (octopine) and, therefore, is in favorable conditions to initiate the conjugal transfer. Contribution of QS to the maturation of biofilms has been suggested for a number of species (for review see [

Thus, our systems-level model provides experimentally testable quantitative predictions regarding both the dynamics of the intracellular control network and the population-wide characteristics of the transition to QS. Experimental verification of these predictions can be achieved, for example, by using a combination of classical genetic techniques and modern fluorescent confocal microscopy. One of the less obvious model predictions amenable to this approach is the existence of an appreciable intracellular pool of TraM in the off state. Insertion of a fluorescent reporter, like the green fluorescent protein or one of its derivatives, into some or all operons controlled by TraRd would result in the development of a sensitive gage to directly observe the dynamics of the transition to QS in vivo. With such a reporter it should be possible to observe bistability in the transition region by simultaneously monitoring populations of cells in the off and on states, e.g., as has recently been done in a study of the lactose utilization network [

In addition to producing these predictions, our study suggests an answer to the long-standing question regarding the ecological and evolutionary role of the QS phenomenon in the genetic conjugation of Ti plasmids. Finally, our analysis demonstrates how computational modeling connects multiple scales of biological phenomena, from the level of molecular networks to that of multicellular populations.

We formulated the chemical kinetics of the

with three and one effective reaction constants, respectively. The action of the putative AAI importer is modeled according to the standard enzymatic mechanism:

where _{e}_{i}

Model Variables and Equations

We extracted a number of crucial model parameters, e.g., lifetime of TraRd and reaction constant of TraRd and TraM, directly from the literature. Many parameters of a more general nature, such as velocities of transcription and translation, are not reported for our system and were estimated based on values obtained for other prokaryotic systems. We estimated the average volume of a bacterial cell _{b}^{−13} cm^{3} (a cylinder with diameter 0.3 μm and length 2 μm) using high-quality electronic microscopy images of

The model (solid line) is fitted to the experimental data (filled squares). All parameters are as in _{11}, which is set to zero to model the inability of mutated TraM to sequester TraRd.

We performed a local analysis of sensitivity of the stationary states of the full deterministic model to variation of the model parameters using the formalism of the metabolic control analysis [_{i}_{j}

These coefficients were computed using Jarnac, which was integrated into Systems Biology Workbench, a freely available software platform [

We used standard methods of chemical kinetics to reduce the dimensionality of the full QS network model to only two equations describing the dynamics of TraRd and intracellular AAI. We first eliminated variables describing vacant and occupied TraRd binding sites (see

where

Finally, we obtained the reduced model with two equations:

Since the full model for the intracellular dynamics is expressed entirely in mass-action rate law equations, it can be simulated with both deterministic and stochastic methods. To model the transition to QS in the exponentially growing bacterial population deterministically, we complemented the intracellular model with two equations describing the dynamics of cell density _{e}

where _{b}_{e}

In the stochastic formulation, each bacterium is represented by a separate agent endowed with an independent stochastic realization of the QS network model. We assumed that bacteria can either randomly move in the medium (planktonic form) with effective diffusion coefficient _{b}^{−6} cm^{2}/s [_{A}^{−5} cm^{2}/s. Cells periodically divide, resulting in two replicas that are exactly identical at the moment of division and diverge thereafter. In both deterministic and stochastic representations average cell cycle is 1 h (α = 1.9 × 10^{−4} s^{−1}) according to the estimate based on our experimental data.

All deterministic simulations were performed with Matlab (MathWorks, Natick, Massachusetts, United States). The stochastic model of the intracellular network was simulated with the exact Gillespie algorithm [

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The NCBI Entrez Protein (

We acknowledge many people from the Bioinformatics Institute who contributed their help and advice. In particular we are grateful to the present and former members of the Cellware team: Pawan Dhar, Li Ye, Tan Chee Meng, Wu Song, and Sandeep Somani for their indispensable help with incorporation of their software into our computation platform. We thank Arun Krishnan, Francis Tang, and Stephen Wong for help with parallel programming and cluster computing. We also acknowledge inspirational and fruitful discussions with Guna Rajagopal and Santosh Mishra. We thank Vinay Sarathy and Jeffrey Pang for their contribution during the period of their SMA student internships. Many thanks go to Herbert Sauro and Brian Ingalls, who provided invaluable advice and software for the sensitivity analysis. ABG gratefully acknowledges the help of Baltazar Aguda, who read the manuscript and provided useful suggestions. This work was supported by the Agency for Science, Technology and Research of Singapore.

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