Extracellular noise-induced stochastic synchronization in heterogeneous quorum sensing network

Mathematical Modelling of Bacterial Quorum Sensing: A Review

Bacterial quorum sensing (QS) refers to the process of cell-to-cell bacterial communication enabled through the production and sensing of the local concentration of small molecules called autoinducers to regulate the production of gene products (e.g. enzymes or virulence factors). Through autoinducers, bacteria interact with individuals of the same species, other bacterial species, and with their host. Among QS-regulated processes mediated through autoinducers are aggregation, biofilm formation, bioluminescence, and sporulation. Autoinducers are therefore "master" regulators of bacterial lifestyles. For over 10 years, mathematical modelling of QS has sought, in parallel to experimental discoveries, to elucidate the mechanisms regulating this process. In this review, we present the progress in mathematical modelling of QS, highlighting the various theoretical approaches that have been used and discussing some of the insights that have emerged. Modelling of QS has benefited almost from the onset of the involvement of experimentalists, with many of the papers which we review, published in non-mathematical journals. This review therefore attempts to give a broad overview of the topic to the mathematical biology community, as well as the current modelling efforts and future challenges.

Dynamics of the quorum sensing switch: stochastic and non-stationary effects

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 Vibrio fischeri based on the synthetic strains lux01 and lux02. Our approach takes into account the key regulatory interactions between LuxR and LuxI, the autoinducer transport, the cellular growth and the division dynamics. By using both deterministic and stochastic models, we analyze the response and dynamics at the single-cell level and compare them to the global response at the population level. Our results show how fluctuations interfere with the synchronization of the cell activation and lead to a bimodal phenotypic distribution. In this context, we introduce the concept of precision in order to characterize the reliability of the QS communication process in the colony. We show that increasing the noise in the expression of LuxR helps cells to get activated at lower autoinducer concentrations but, at the same time, slows down the global response. The precision of the QS switch under non-stationary conditions decreases with noise, while at steady-state it is independent of the noise value.

Conclusions

Our in silico experiments show that the response of the LuxR/LuxI system depends on the interplay between non-stationary and stochastic effects and that the burst size of the transcription/translation noise at the level of LuxR controls the phenotypic variability of the population. These results, together with recent experimental evidences on LuxR regulation in wild-type species, suggest that bacteria have evolved mechanisms to regulate the intensity of those fluctuations.

Robust synchronization analysis in nonlinear stochastic cellular networks with time-varying delays, intracellular perturbations and intercellular noise

Naturally, a cellular network consisted of a large amount of interacting cells is complex. These cells have to be synchronized in order to emerge their phenomena for some biological purposes. However, the inherently stochastic intra and intercellular interactions are noisy and delayed from biochemical processes. In this study, a robust synchronization scheme is proposed for a nonlinear stochastic time-delay coupled cellular network (TdCCN) in spite of the time-varying process delay and intracellular parameter perturbations. Furthermore, a nonlinear stochastic noise filtering ability is also investigated for this synchronized TdCCN against stochastic intercellular and environmental disturbances. Since it is very difficult to solve a robust synchronization problem with the Hamilton-Jacobi inequality (HJI) matrix, a linear matrix inequality (LMI) is employed to solve this problem via the help of a global linearization method. Through this robust synchronization analysis, we can gain a more systemic insight into not only the robust synchronizability but also the noise filtering ability of TdCCN under time-varying process delays, intracellular perturbations and intercellular disturbances. The measures of robustness and noise filtering ability of a synchronized TdCCN have potential application to the designs of neuron transmitters, on-time mass production of biochemical molecules, and synthetic biology. Finally, a benchmark of robust synchronization design in Escherichia coli repressilators is given to confirm the effectiveness of the proposed methods.

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

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.

Signal propagation in small-world biological networks with weak noise

An emerging notion in systems biology is that biological networks have evolved to function well while their components behave stochastically. Thus, the dynamics in a biological network consist of two parts, deterministic and stochastic. A fundamental question is to find a quantitative relation between the two parts. We term such a relation as a deterministic-stochastic principle (DSP) and propose a model for a DSP with regard to signal propagation in biological networks. In this model, (i) the dynamics in a biological network is supposed to be captured by a stochastic differential equation which has been a standard approach in modeling systems with internal noise; (ii) the internal noise of a biological network is weak as is apparent in experimental observations; and (iii) a biological network is organized as small-world as suggested by recent studies. We introduce the concept of a signaling sample path. Using this concept we relate the structure of a biological network to its dynamics. The network structure characterizes the deterministic part of the dynamics, which in turn ensures a probability for a signal to propagate. The weakness of the internal noise characterizes the stochastic part of the dynamics. Analysis of the proposed model yields a quantitative description as follows: In a small-world biological network with weak internal noise, the signaling pathways (induced by the network structure) for a signal may ensure a probability near 0 for the signal propagation. Despite such a small probability, a correct response to the signal will still occur with a probability close to 1 provided that this signal propagation can take a certain amount of time. Computer simulations are performed to illustrate this result. We also discuss how a recent study on the reconstruction of a transcription network in Saccharomyces cerevisiae has tested the proposed model against real data.

Class-II neurons display a higher degree of stochastic synchronization than class-I neurons

We describe the relationship between the shape of the phase-resetting curve (PRC) and the degree of stochastic synchronization observed between a pair of uncoupled general oscillators receiving partially correlated Poisson inputs in addition to inputs from independent sources. We use perturbation methods to derive an expression relating the shape of the PRC to the probability density function (PDF) of the phase difference between the oscillators. We compute various measures of the degree of synchrony and cross correlation from the PDF's and use the same to compare and contrast differently shaped PRCs, with respect to their ability to undergo stochastic synchronization. Since the shape of the PRC depends on underlying dynamical details of the oscillator system, we utilize the results obtained from the analysis of general oscillator systems to study specific models of neuronal oscillators. It is shown that the degree of stochastic synchronization is controlled both by the firing rate of the neuron and the membership of the PRC (type I or type II). It is also shown that the circular variance for the integrate and fire neuron and the generalized order parameter for a hippocampal interneuron model have a nonlinear relationship to the input correlation.