Stochastic simulation of quorum sensing in Vibrio fischeri based on P System

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Abstract

Inspiration from nature and especially from biology is up today a multidisciplinary challenge where biologists aim to exploit ‘competences’ of a computer to understand phenomena of life through simulations and where artificial intelligence aims to develop new computational approaches inherent from natural processes. Situated in the border of evolutionary systems and bioinformatics, the computational model of P system is an evolved algorithm based upon the structure and evolution rules of biological living cells to outperform some conventional systems. Thus, this paper presents a version of multi-compartmental Gillespie algorithm used to simulate the parallelism of chemical reactions in the quorum sensing phenomena of a bacteria colony. This model is used to distribute the computation over the membranes. The aim is to achieve scalability and get mapping the communication between bacteria and their environment. The obtained results prove that this process adds an important value to the multi-compartmental Gillespie algorithm, which uses the organizational aspect of the membrane structure. They also prove that the independent parallel evolution of the membranes is basic to improve legitimacy of the strategy which leads to a reliable simulation and gives a more realistic representation of the system’s evolution.

Keywords

Cell communication Quorum sensing P systems Gillespie algorithm 

1 Introduction

Inspiration from nature and especially from biology is up today a multidisciplinary challenge where biologists aim to exploit ‘competences’ of a computer to understand phenomena of life through simulations and where artificial intelligence aims to develop new computational approaches inherent from natural processes. Situated in the border of evolutionary systems and bioinformatics, the computational model of P system is an evolved algorithm based upon the structure and evolution rules of biological living cells to outperform some conventional systems. In the perspective to simulate a powerful machine to solve NP-hard problems, we mainly find the self-organizational aspect of a cell and the emergent social behavior of cells that ensure the realization of an aimed goal. Recently, different stochastic strategies for modeling biological systems based on Stochastic Simulation Algorithm (Cao et al. 2006) are proposed and discussed in (Cazzaniga et al. 2006; Pérez-Jiménez and Romero-Campero 2006); and they are deterministic approach, multi-compartmental approach and dynamical probabilistic P system approaches. Where, membrane computing was used as a modeling framework in order to address structural and dynamical aspects of cellular systems (Păun and Romero-Campero 2008). The objective of these methods is to exploit the P system topology namely the hierarchical structure of its compartments.

The researchers confirmed in their work that these new computational modeling tools take into account: the discrete character of the quantity of components of biological systems, the inherently randomness in biological phenomena and the key role played by membranes in the function of living cells. To illustrate the importance and the applicability of these strategies, two famous biological systems are modeled in (Pérez-Jiménez and Romero-Campero 2006): the EGFR (Epidermal Growth Factor Receptor) Signaling Cascade and the Quorum Sensing System in the bacterium Vibrio fischeri. These stochastic approaches are reviewed in (Cazzaniga et al. 2006) to compare their respective results and to discuss their advantages as well as their limits and possible improvements. Another interesting work that exploited the quorum sensing as an expression gene means has concerned the opportunistic Pseudomonas aeruginosa bacterium (Bianco et al. 2006) were the authors in their approach derived a mechanistic P systems model to describe the behavior of a single bacterium.

A new research has also been developed in this area. Because of the high dimensionality of the problem as well as lack of biological laboratory data; Sarpe et al. (2010) used a generic particle swarm optimization algorithm to discover optimal solutions for the rule stochasticity constants. Romero-Campero et al. (2009) used an optimized version of Multi-compartmental approach to define modularity based on chemical specificity and spatial localization that can be easily specified and analyzed in P systems using sets of rewriting rules and membrane structure. This paper is decomposed in four sections: the first section describes the used biological concepts. The second section illustrates the stochastic P systems from the conceptually view. The third section exposes the quorum sensing of V. fischeri bacterium using a P system and shows the different cases study undertaken in order to highlight the contribution of parallelism in the multi-compartmental Gillespie algorithm. Finally, we conclude the paper by some interesting points.

2 Related background

2.1 Nature inspired P systems

Inspired from nature, Gheorghe Păun introduced a new computational model based upon chemical reactions across cell membranes (Păun 2000) which represents an asset for solving several types of problems in nature. Herein, the structure of membrane is the key concept in modeling any evolution system. As illustrated in Fig. 1, the membrane structure consists in a set of hierarchical compartments arranged inside a main compartment where the membrane that delimited such compartment is called skin membrane. Hence, at each region in the membrane structure a set of objects (multiset) is associated representing symbols or strings of symbols in addition to a set of reaction rules of a biochemical inspiration. Basically, a rule has the strength of transforming objects from state to another, and also enables the communication between adjacent compartments through the sending of objects. With respect of the variety of existing P systems, a basic P system of degree \(~n \geq 1\) (number of membranes) is formally a tuple \(\pi ~\):

Fig. 1

The P System structure composed of hierarchical membranes

$$\pi =\left( {V,\mu ,{M_1},{M_2}, \ldots ,{\text{~}}{M_n},{\text{~}}{R_1},{R_2}{\text{~}} \ldots ,{\text{~}}{R_n}} \right)$$
where, \(V\) is a finite alphabet of the system; \(\mu\) is the membrane structure, composed of \(n\) libeled membranes; \({M_1}, \ldots ,{M_n}\) are multisets over \(V\), where each multiset \({M_i}\), \(1 \leq i \leq n\), represents initial objects in the region \(i\) of the system; \({R_1}, \ldots ,{R_n}\) a set of evolution rules associated relatively to the \(n\) regions of \(\mu\).

Although a variety of P systems models were enumerated introducing each different specifications at the evolution rule level, these models are in most cases applied in a non-deterministic maximal parallel manner. It is important to note that three main P systems are available: Cell-like P Systems, Tissue-like P Systems, and Neural-like P Systems. The first type is none other than the classic model inspired from cell membrane identified by an arrangement of membranes delimiting compartments where multiset of objects are placed (Păun and Păun 2002). The most frequent type of rules are (1) multiset rewriting rules of the form \(u \to v\) where \(u\) and \(v\) are multisets of objects, and (2) transport rules, e.g. symport or antiport rules. A symport rule is the form \(\left( {u,in} \right)\) or \(\left( {u,out} \right)\) moving objects of multiset \(u\) through a membrane, and antiport rule is the form \(\left( {u,out;v,~in} \right)\) moving objects of multiset \(u\) outside a membrane and in the same time moving objects of \(v\) inside the membrane. In tissue-like P systems (Martin-Vide et al. 2003), many one-membrane cells are considered as evolving in a common environment enabling the communication either between adjacent cells or between cells and the environment. Finally, the third class of Neural-like P systems can be a simply extended model of Tissue-like P System or introduced as spiking neural P systems (Ionescu et al. 2006).

2.2 Quorum sensing

Among several natural phenomena, bioluminescence represents a wonderful classic that attracted scientists and prompted them to observe more closely the chemical process of production and emission of light by a living organism. In marine animals, bioluminescence is produced by symbiotic organisms as it is the case for V. fischeri bacterium indentified in 1983 (Engebrecht et al. 1983) and which lives in symbiosis with the squid as host ensuring it camouflage against predators. Bioluminescence puts the fact on the bacteria communication and reveals the natural means through which this communication enables bacteria to acquire social life. In fact, bacteria communicate and even better “talk” to each other using a chemical language to synchronize their behavior and act in union as higher multi-cellular organism. Such cell-to-cell communication among bacteria is also called quorum sensing.

Although a bacterium as a single entity is inoffensive, its impact, positive (beneficial) or negative (pathogenic), on the environment (host) is defined as a collegiate timely decision-making happened as response to pheromones activation. Indeed, at lower bacteria densities a signal molecule also called autoinducer (pheromone) is synthesized and accumulated in each bacterium in small amount. However, as well as the population bacteria grows in a surrounding medium, the concentration of released autoinducer increases until reaching a critical threshold concentration of signal molecules (Kaplan and Greenberg 1985; Weber and Buceta 2013). At this time, each bacterium in the surrounding, able to perceive this extracellular event, activates its receptor molecule that leads to a signal transduction cascade. This coordination mechanism in such bacteria colony called quorum sensing is achieved when the quorum, higher concentration density is induced by more and more bacterium.

Figure 2 illustrates the biochemical gene regulation mechanism (Stevens et al. 1995) triggered by quorum sensing. For a large class of bacteria, more known as Gram-negative species, the most commonly used signal molecule appears to be N-acyl-homoserine lactone AHL and most specifically for V. fischeri the AHL is the N-(3-oxo-hexanoyl)-l-homoserine lactone autoinducer (OHHL). In the transcription machinery of bioluminescence, the luxI gene codes for the LuxI (autoinducer synthase), luxR codes for the receptor protein, and lux operon (lux CDABE) codes for the production of luciferase.

Fig. 2

Schematic representation of the quorum sensing system of the V. fischeri bacterium illustrating the expression gene steps from down-regulated state to up-regulated state that yields to bioluminescence

Indeed, at down-regulated state, i.e. under low bacteria density, bacteria produce basal level of OHHL and as well the cell density increases on reaching a critical threshold concentration, the OHHL binds to its cognate LuxR receptor protein. Hence, the LuxR–OHHL complex as a transcriptional activator binds to the luxBox part of the quorum sensing gene region in order to activate the transcription of genes part of the luxCDABE operon. Besides the fact that LuxR–OHHL is responsible for the species specific response in our case the synthesis of the enzymes of bioluminescence, it also induces in high amount the expression of luxI because it is encoded in the luciferase operon (Kempner and Hanson 1968).

Indeed, the regulation of gene expression by quorum sensing results in phenotypic changes which include bioluminescence, virulence, symbiosis, conjugation, biofilm formation, synthesis of enzymes, antibiotic substances, etc. Quorum sensing regulation is a powerful computational mechanism used by both Gram-negative and Gram-positive bacteria like Vibrio harveyi, Agrobacterium tumefaciens, Erwinia carotovora, Bacillus subtilis, Staphylococcus aureus, etc. involving each a specific autoinducer. Currently, quorum sensing regulation has been found in more than 50 species of bacteria (Ahmer 2004; Fuqua et al. 1996; Kalia 2013; Khmel and Metlitskaya 2006; Papenfort and Bassler 2016; Zhu et al. 2002).

3 Stochastic P system

Stochastic P system is the modeling framework tool based on the stochastic simulation algorithm. The stochastic simulation algorithm introduced by Gillespie (1977), and lately developed by the same author (2001) and Cao et al. (2006), is an effective algorithm used to model chemical or biochemical system contained inside a single fixed volume.

3.1 Stochastic simulation algorithm

Obviously the chemical or biochemical system contains N molecular species \({S_1},~{S_2},~ \ldots ,~{S_N}\) interacting through M reactions \({R_1},~{R_2},~ \ldots .~,~{R_M}\) having reactions constants \({c_1},~{c_2},~ \ldots .~,~{c_M}\) which only depend on the physical condition and the properties of the molecules. Based on the fundamental hypothesis \({c_p}t\)—the average probability that a particular reactants combination will react in the next time interval \(t\) according to the reaction \({R_P}\)—the stochastic simulation algorithm computes both: the next reaction that will be applied in the system and its execution time (Gillespie 1977, 2001). For this it uses the following derivations of the probability density function:where, \({a_p}\) is the propensity function: \({a_p}={h_p}.{c_p}\), and \({a_0}=\sum\nolimits_{{p=1}}^{M} {{a_p}}\). \({h_p}~\) is the number of distinct reactant molecules combination. \({c_p}~\) is the stochastic rate constant associated to reaction \(~{R_p}\). \({r_1}\) and \({r_2}\) are two uniform random numbers.

Equations (3) and (4) determine respectively the next reaction \({R_p}\) which will be launched and its execution time \({\tau _p}\). In what follows, we shall study the multi-compartmental Gillespie algorithm which is one of the three stochastic P system’s strategies mentioned in the previous section.

3.2 Multi-compartmental Gillespie algorithm

Multi-compartmental Gillespie algorithm (Pérez-Jiménez and Romero-Campero 2006; Romero-Campero and Pérez-Jiménez 2008) is an extended version of Gillespie algorithm. It uses the membrane structure as the modeling framework for biological systems as well as the strategy defined in the Gillespie algorithm rather than the maximal parallel way to evolve the system. This strategy allows to retrieving the information for when and which one will be the next applied reaction at each step of the simulation in the whole system.

Contrary to the original Gillespie algorithm where only one volume is taken into account, Multi-compartmental Gillespie algorithm simulates the evolution of the chemical reactions in multiple regions so that the application of rules within a given region can also affect the content of another one. Multi-compartmental Gillespie algorithm uses the organizational aspect of the membrane structure, and evolves the system in a pure sequential way such that at each step of the simulation, only one reaction is selected to be applied in the whole system. In what follows, the main steps of the used Multi-compartmental Gillespie algorithm are given:

3.3 Parallel multi-compartmental Gillespie algorithm

It is clear that, in Multi-compartmental Gillespie algorithm, the environment send one signal, at each step, to one bacterium selected randomly and attributes the rule which will diffuse the signal OHHL to bacteria in the whole system. Compared to this existing version, the main modification done on this algorithm concerns the fact that through the Gillespie algorithm a rule is drawn to be applied not to whole system but only to the compartment that delimits a pointed region. This notion of locality implies that each membrane in the system will evolve independently of others keeping a local execution environment surrounding a region, a proper rule and a proper multiset of objects. By this manner, a bacterium ensures itself the propagation of diffusion signal instead of the environment. Thus, this modified version of the original algorithm simulates the parallel evolution of the chemical reactions in multiple regions. The steps in each region can be summarized in three main stages: the computation of the probability distribution for determining the next fired rule, the assignment of the objects to the rules and the updating of the multisets.

4 Case study

In the context of our study, the multi-compartmental Gillespie algorithm is used in the context of modeling the evolution bio-process called quorum sensing. In this paper, the quorum sensing process of the specified bacterium V. fischeri is formalized by membrane computing where each bacterium and the environment are represented by a membrane. This framework allows examining the individual behavior of each bacterium as well as of the whole colony.

The quorum sensing system relies on the synthesis, accumulation and subsequent sensing of the signal molecule OHHL. At high cell density the signal accumulates in the environment and can also diffuse to the inside of the bacterial cells. It is able to interact with the LuxR protein to form the complex LuxR–OHHL which is linked to a region of DNA called LuxBox causing the transcription of the luminescence gene. The bacteria lights up only when exploiting the light organs (when the LuxBox is occupied) and do not emit light when they are in the free living state, therefore the quorum sensing can explain why the bacteria are dark when in the free living state at low density and light at high cell density when colonizing the light organs. In what follows a model of the quorum sensing system in V. fischeri is presented using a P System (\({\pi _{({\text{N}})}}\)):
$${\pi _{\left( {\text{N}} \right)}}=\left( {O,\left\{ {e,b} \right\},\mu ,\left\{ {{w_1},{w_i}} \right\},\left\{ {{R_e},{R_b}} \right\}} \right),$$
where:
  1. 1.

    \(O\), the alphabet consisted of the signal OHHL, the protein LuxR, the complex protein-signal, regulatory region Lux-Box and the regulatory region occupied by the complex:

     
$$O=\left\{ {OHHL,LuxR,LuxR - OHHL,LuxBox,LuxBox - LuxR - OHHL} \right\}$$
  1. 2.

    \(\mu\), is the membrane structure containing two main regions named the environment and the bacteria. The environment will be represented by the membrane labeled \(e\) and each bacteria by a membrane labeled \(b\).

     
  2. 3.

    \({w_i}\), are the initial multisets that represent the initial conditions of the system. \({w_1}=\emptyset\) and \({w_i}=\left\{ {LuxBox} \right\}\), where \(i=2, \ldots ,N+1\).

     
  3. 4.
    \({R_b}\) and \({R_e}\) are the rules that correspond the chemical reactions of the quorum sensing system where \({R_b}\) is a program associated with the bacteria and \({R_e}\) is the program associated with the environment. and they rules are:
     

The protein LuxR and the signal OHHL together can form the complex LuxR–OHHL which in turn can dissociate into OHHL and LuxR (see rules \({r_3}\) and \({r_4}\)). The complex LuxR–OHHL acts as a transcription factor linking to the regulatory region DNA of the bacterium called LuxBox (see rules \({r_5}\)and \({r_6}\)), and allows to produce a massive increase in the transcription of the signal and of the protein (see rules \({r_7}\) and \({r_8}\)). The signal OHHL, the protein LuxR and the complex LuxR–OHHL undergo a process of degradation in the bacterium (see rules \({r_{10}}\), \({r_{11}}\) and \({r_{12}}\)). The signal can diffuse outside the bacterium and accumulate in the environment (see rules \({r_9}\)), or diffuse from the environment inside the bacteria and goes through a process of degradation (see rules \({r_{13}}\) and \({r_{14}}\)), rule \({r_{14}}\) represents the program associated with \({R_e}\).

5 Results and discussion

Here we will interested in examining how bacteria communicate to coordinate their behaviors, and how the system moves from a down regulated state where the protein and the signal are produced at basal rates to up regulated state where it produce light. Therefore, in the initial multisets we will suppose that there is nothing in the environment, and in the bacteria we will only have the genome LuxBox to start the transcription of the signal OHHL and protein LuxR. The stochastic rate constants parameters [19] that have been chosen to simulating the dynamic behavior of system are: c1 = 2, c2 = 2, c3 = 9, c4 = 1, c5 = 10, c6 = 2, c7 = 250, c8 = 200, c9 = 50, c10 = 30, c11 = 20, c12 = 20, c13 = 1, c14 = 20.

In order to study the behavior of the system, we have developed the algorithm presented in this paper using the platform java. To examine the evolution over time of the number of signal OHHL in the environment and in the bacterium we have considered populations of different size. For each population we have take the results of 5000 iterations, the time in all the figures that follow is expressed in hours.

In Figs. 3 and 4 the number of the signal OHHL in the environment and in a bacterium for a population of single bacterium are shown.

Fig. 3

The evolution over time of the number of signal in the environment for a population of single bacterium

Fig. 4

The number of signal inside the single bacterium over time

It is possible to see that, there exist a correlation between the number of signal in the bacterium and the occupation of LuxBox by the complex. When the complex occupied the LuxBox the bacterium can produce more signals using the rule \({r_7}\) which increase the probability to send a signal outside the bacterium using the rule \({r_9}\). From the previous figures, we can see that the bacterium can send the signal to the environment when the signal reaches a threshold of six molecules. Such that, when the number of signal in the bacterium drops so does the number of signal in the environment. In Figs. 5 and 6, the number of the signal OHHL in the environment and a in bacterium for a population of 10 bacteria are shown.

Fig. 5

The occupation of the LuxBox by the complex over time for the single bacterium

Fig. 6

The evolution over time of the number of signal in the environment for a population of 10 bacteria

The results shown in Figs. 6 and 7 can explain how the bacteria can sense their own population density. In Figs. 7 and 8, we can see that at the most times of the evolution of a specified bacterium, the LuxBox is not occupied by the complex and in despite of this reason the number of signal inside the bacterium is increased some times. That is to say that, this bacterium is at the stage to communicating with the environment to detect the signals produced by the surrounding bacteria using \({r_{13}}\). When the signal accumulates in the environment and reached some thresholds the number of signal in the bacterium increase. The bacteria are able to detect and respond to signals produced by surrounding bacteria when the signal reached some thresholds that mean it can sense the amount of signal present.

Fig. 7

The evolution over time of the number of signal inside a bacterium in a population of 10 bacteria

Fig. 8

The occupation of the LuxBox by the complex over time in a bacterium

In Figs. 9 and 10 the number of the signal OHHL in the environment and the number of quorated bacteria for a population of 300 bacteria are shown.

Fig. 9

The evolution over time of the number of signal in the environment for a population of 300 bacteria

Fig. 10

The evolution over time of the number of quorated bacteria in a population of 300 bacteria

At the beginning a few bacteria get quorated and the signal accumulates in the environment. While a threshold is reached the process that makes the population in a coordinated way is accelerated. In Figs. 11 and 12, the number of the signal OHHL in the environment and the number of quorated bacteria for a population of 400 bacteria are shown.

Fig. 11

The evolution over time of the number of signal in the environment for a population of 400 bacteria

Fig. 12

The evolution over time of the number of quorated bacteria in a population of 400 bacteria

Regarding the time evolution of Figs. 9, 10 and 11 can see that as the number of bacteria of colony increase, the number of quorated bacteria increase as well soon. But, when the number of quorated bacteria exceeds a specific threshold the number of the signal inside the environment is about the same for the two simulations. This is due to the fact that for each bacterium the probability to send out a signal and to incorporate it from the environment does not depend on the whole number of bacteria in the colony. The environment play a role of recipient for the signal OHHL, and it is responsible for the degradation process. Observe that at a high cell density the signal accumulate in the environment until reached some thresholds, and then the colony of bacteria start to produce it in a coordinate way.

Given the inherent randomness in such a system, the types of evolution where the environment can send only one signal per step to a non-deterministically chosen bacterium is quite unrealistic. As well as is the case for a dynamical probabilistic P system that uses a parallelism at the level of objects consumption inside the environment presented and studied in (Cazzaniga et al. 2006; Pérez-Jiménez and Romero-Campero 2006). These models attribute the rule for diffuse the signal OHHL into the bacterium to the environment. That makes the probability to send a signal inside the bacterium proportional to the whole number of bacteria in the colony. And that allows some ones of bacteria, which is randomly chosen, to guess wrong the size of the population and got quorated when the colony density is not enough, in this case the signal does not accumulate in the environment.

Pérez-Jiménez and Romero-Campero (2006) proposed a new developed dynamical probabilistic P system that puts the environment in the position to send more than one signal per step to the bacteria of the colony. By this means, a high number of signal are send inside the bacteria which is the reason why the bacteria got quorated although the density of the colony is too low to start produce light. In this last model, the bacteria of the colony evolve in sequential way. However inside the environment a parallel consumption of objects is considered. From the results of this paper it can be see that the behavior of the colony is stochastically well simulated and more accurate. Compared with the behavior obtained in nature, these results give a more realistic approximation of the system’s evolution where the approximated steady state takes a significant range of values. It is more efficient than the steady state in the others models which not agrees with the inherent randomness in such a system.

6 Conclusion

A modified version of multi-compartmental Gillespie algorithm is used to show the effect of triggering several evolution rules in parallel and in each membrane region under the assumption of detecting the quorum sensing inside a bacteria colony. Thus, according to the achieved results, the performance of the proposed improvement has enabled to easily map the communication between the bacteria of the colony through their environment in a scalable way. By this means, we can keep a close look on the quantitative description of the system. Additionally, the independent evolution of the membranes still fundamental to improve legitimacy of this strategy; it leads to a simulation that gets closer to reality and gives a more realistic representation of the system’s evolution.

Notes

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Computer Science DepartmentUniversity of Badji MokhtarAnnabaAlgeria

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