Elsevier

Biosystems

Volume 101, Issue 3, September 2010, Pages 200-212
Biosystems

A mathematical model of the Gac/Rsm quorum sensing network in Pseudomonas fluorescens

https://doi.org/10.1016/j.biosystems.2010.07.004Get rights and content

Abstract

I present a deterministic model of the dynamics of signal transduction and gene expression in the Gac/Rsm network of the soil-dwelling bacterium Pseudomonas fluorescens. The network is involved in quorum sensing and governs antifungal production in this important biocontrol agent. A central role is played by small untranslated RNAs, which sequester regulatory mRNA-binding proteins. The model provides a reasonable match to the available data, which consists primarily of time series from reporter gene fusions. I use the model to investigate the information-processing properties of the Gac/Rsm network, in part by comparing it to a simplified model capable of quorum sensing. The results suggest that the complexity and redundancy of the Gac/Rsm network have evolved to meet the conflicting requirements of high sensitivity to environmental conditions and a conservative, robust response to variability in parameter values. Similar systems exist in a wide variety of bacteria, where they control a diverse set of population-dependent behaviors. This makes them important subjects for mathematical models that can help link empirical understanding of network structure to theoretical insights into how these networks have evolved to function under natural conditions.

Introduction

Many species of bacteria modify their behaviors in response to the concentration of diffusible molecules that they themselves produce, or are produced by other species. The behaviors under the control of such diffusible signals cover a wide range, including antibiotic production, motility, bioluminescence, competence, virulence, biofilm production, and metabolism (Bejerano-Sagie and Xavier, 2007, Fuqua et al., 2001). In many cases, it appears that the diffusible molecules act as indicators of population density, allowing the bacteria to coordinate their behaviors in a population-dependent manner; this phenomenon is known as quorum sensing (QS) (Bassler, 1999). Alternative explanations for the role of the signals have been proposed, including the detection of local diffusibility in the environment (Redfield, 2002), and detection of a combination of population density, cell clustering, and diffusibility (Hense et al., 2007).

In most well-understood examples of QS (e.g. the LuxI/R system in Vibrio fischeri and the LasI/R system in Pseudomonas aeruginosa), the regulatory network includes a strong positive feedback loop, making the system autoinductive (Ng and Bassler, 2009). The signal is produced at a low constitutive level, but its production or detection is positively regulated by genes that are themselves upregulated in the presence of the signal. As the population grows, the signal concentration passes a threshold, and the positive feedback loop causes a sudden change in cell behavior. The overall structure of the positive feedback loop is often augmented with secondary positive or negative feedback loops (Goryachev et al., 2006, Ng and Bassler, 2009, Williams et al., 2008). An important problem in the theory of QS is to understand what role these auxiliary feedback loops play in fine-tuning the system's dynamics and processing information about the cell's environment.

In many γ-proteobacteria (including pseudomonads, Escherichia coli, Vibrio spp., Salmonella enterica, and Legionella pneumophila), QS is controlled at least partly by the Gac/Rsm signal transduction pathway (Bejerano-Sagie and Xavier, 2007, Heeb and Haas, 2001, Lapouge et al., 2008). Regulation of target genes is carried out post-transcriptionally by RNA-binding proteins of the RsmA/CsrA family. These proteins bind to target mRNA molecules, preventing them from being translated. The RsmA/CsrA proteins are themselves regulated by a family of small untranslated RNAs (sRNAs) such as RsmY and CsrB. These sRNAs bind to the RNA binding proteins, sequestering them and freeing the target mRNAs to be translated. Thus, an increase in the concentration of sRNAs causes the target genes to turn from “off” to “on”. The transcription of the sRNAs is under the control of the activator GacA (or one of its homologues, such as UvrY). GacA is part of the GacS/GacA two-component system; GacS is a sensor kinase that undergoes autophosphorylation in the presence of the diffusible signal. GacS passes the phosphoryl group to the response regulator GacA; phosphorylated GacA is then able to bind DNA and act as a transcriptional activator. Since the production of the diffusible signal is downregulated by RsmA/CsrA, the entire pathway is autoinductive. The Gac/Rsm network often coexists with other QS circuits in the same bacteria, but it is unclear to what extent these networks interact.

The basic structure and some of the detailed interactions in Gac/Rsm regulatory networks have been studied experimentally in a number of bacteria. However, it does not appear that the system is fully understood in any species. An important limitation to our understanding of the Gac/Rsm system is that the chemical structure of the diffusible signal has not been determined, although it appears not to be related to well known QS signals like N-acyl-homoserine lactones (Lapouge et al., 2008). Thus, details of the signal molecule's synthesis and transport are not known. Nevertheless, it is clear that the Gac/Rsm system as a whole is autoinductive and regulates social behaviors; this has led researchers to categorize it as an example of QS (Bejerano-Sagie and Xavier, 2007, Lapouge et al., 2008).

At this stage, a mathematical model can help to bridge the gap between what is currently known and what remains to be discovered. First, a model can help determine whether the network as currently understood is capable of producing the observed data. This helps to highlight gaps in our understanding, and prioritize areas for further study. Second, a model can be used to study the information-processing properties of the network. In particular, how does the network respond to fluctuations in the concentration of the signal molecule or to variability in transcription, translation, and binding rates? This can shed light on why such systems have evolved to their current forms, with a high level of complexity and redundancy. Finally, the Gac/Rsm system is a useful test case for incorporating sRNAs into models of gene expression. Because of the gaps in our empirical understanding of Gac/Rsm networks, any such model must be a somewhat speculative. However, enough information is available to construct a reasonable first-generation model that may lead to a virtuous cycle of experimentation and model refinement.

As a specific case study, I present a model of the Gac/Rsm system in the soil-dwelling bacterium Pseudomonas fluorescens. There are two reasons for this choice. First, the system is relatively well understood in P. fluorescens, with time series data available for several key components. Second, P. fluorescens is an important biocontrol agent; it synthesizes several antifungal compounds that protect host plants from pathogenic fungi (Haas et al., 2002, Heeb and Haas, 2001). The synthesis of these compounds is partially under the control of the Gac/Rsm system, so that QS in P. fluorescens has important implications for plant ecology and agriculture.

Mathematical models have been developed for a number of QS bacteria, particularly Vibrio fischeri and Pseudomonas aeruginosa (although the focus there has been on other pathways for QS, rather than Gac/Rsm). Most studies have represented the signal transduction and gene expression networks in terms of differential equations, and analyzed the resulting behavior (Dockery and Keener, 2001, Fagerlind et al., 2003, Gustafsson et al., 2004, James et al., 2000, Karlsson et al., 2007, Kuttler and Hense, 2008). In general, these systems are dominated by an autoinductive positive feedback loop, which gives rise to hysteresis; it appears that hysteretic switching underlies the rapid transition from “off” to “on” of QS systems. Because many QS bacteria live in biofilms, several theoretical studies have investigated the role of spatial structure on QS (Chopp et al., 2002, Chopp et al., 2003, Dockery and Keener, 2001, Goryachev et al., 2005, Hong et al., 2007). These studies suggest that the dynamics of the colony can be understood as a wave of hysteretic switching by individual cells, although the local diffusion of the signal molecule has important implications for the degree of synchronization among different cells and the threshold signal concentration required for switching “on”.

Other theoretical studies have focused on the information-processing properties of QS systems, often by incorporating some form of stochasticity into the model. The networks have likely evolved to perform optimally in the face of two apparently contradictory constraints (Goryachev et al., 2006). On the one hand, they need to be extremely sensitive, able to detect small changes in the local density of a signal molecule and amplify those changes into a cellular response. On the other hand, they need to be robust against small random fluctuations in the signal concentration and to the noise inherent in regulatory networks that may involve only a few copies of each molecule per cell. On these issues there is not yet a consensus about general results, nor indeed about the precise questions to ask or methods to use. Two studies (Muller et al., 2008, Williams et al., 2008) showed that a high concentration of sensor proteins can effectively compute a time-averaged signal concentration, buffering the system against high frequency random fluctuations in the signal. However, Hong et al. (2007) showed that random signal fluctuations can in principle act to synchronize cells in a biofilm. Tanouchi et al. (2008) showed that rapid turnover of transcriptional regulators and intracellular signal can buffer a QS system against low frequency fluctuations in protein concentrations and cellular machinery. Cox et al. (2003) showed that QS networks can shape the noise spectrum as information propagates through the system, although the biological significance of this is not yet clear. They also argued that stochasticity plays an important role in the hysteretic switching of QS networks, allowing the population to sample both sides of the threshold signal concentration. On the other hand, Goryachev et al. (2006) demonstrated that a fully stochastic model of a common QS motif behaved essentially the same as a deterministic (ODE) model, while for simplified networks the stochastic and deterministic models exhibited qualitatively different behavior.

In perhaps the most detailed and realistic such study to date, Goryachev et al. (2005) developed a spatially explicit, stochastic model for QS in Agrobacterium tumefaciens. Critically, they found that the behavior of the stochastic model could be captured at the population level by a simpler deterministic model. In addition, the primary effect of incorporating spatial structure was to lower the effective threshold concentration of the diffusible signal. Taken together, these results suggest that one possible function of the complex network structure of many QS systems is to yield deterministic, predictable behavior at the population level in spite of stochasticity and spatial heterogeneity. This implies that a deterministic, non-spatial model is a reasonable first approach for investigating the function of novel QS systems.

Section snippets

Biological system

The model that I present reflects the current understanding of the interactions in the Gac/Rsm network in P. fluorescens (Kay et al., 2005, Lapouge et al., 2008). The general schematics are outlined in Fig. 1A. In the presence of the signal, GacS undergoes autophosphorylation and subsequently phosphorylates GacA. Once phosphorylated, GacA activates transcription of the sRNAs RsmX/Y/Z. Phosphorylated GacA also increases the transcription of GacS. The sRNAs bind with the regulatory proteins

Mathematical model

The model consists of a set of coupled nonlinear differential equations for the cell population and concentrations of gene products in the Gac/Rsm network. This assumes that the system is completely deterministic, and that all cells are identical. In fact, regulatory events involving small numbers of molecules can be highly stochastic, and QS may involve a mixed population of cells in different states. However, there is not sufficient quantitative data to justify including these complications

Experimental growth conditions

Using the parameter values in Table 1, the model agrees reasonably well with the experimental data. Fig. 2 shows the model's prediction of β-galactosidase accumulation for RsmX and RsmE gene fusions, for various knockout mutations, along with corresponding experimental data. The β-galactosidase levels are scaled to represent Miller units (×1000). The qualitative and quantitative dynamics compare favorably with the data (cf. Kay et al., 2005, Reimmann et al., 2005, Valverde et al., 2003). For

Discussion

I have presented a deterministic model of the Gac/Rsm quorum sensing system in P. fluorescens, a network that shares features with analogous systems in many other bacteria. The model can be parameterized to provide a reasonable match to the available data, which consists primarily of semiquantitative time series from reporter gene fusions. The central role played by small untranslated RNAs, as well as the variety of cellular behaviors that are controlled by these networks, makes them important

Acknowledgements

I am indebted to Dieter Haas for sharing with me his thoughts on the structure of the Gac/Rsm network, and to Phoebe Lostroh for many useful insights about bacterial genetics. I thank my former students Chris Morin and Young Yi for their work on early versions of the model, and two anonymous reviewers for their thoughtful suggestions. This work was supported by NSF grant DEB 0120169.

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