Inside biological system, cells are composed by thousands of components that interact in a myriad of ways. Despite this interconnection, it is necessary to classify these networks of cells according to their biological function.

The emerging of systems biology with multi-disciplinary field is involved to the study of the relationships between various parts of a biological system, and modeling method. They are vital role in the drive to present the processes of life. Advancements in experimental technologies in biology and medicine have generated an amount of biological data. Many different molecular cell processes interact and change their behavior quickly. So, we need develop methods for exploring this various data. Much formalism from the fields of biology, mathematics and the computer sciences is used to integrate, represent and analyze the vast amount of biological data.

To understand the functioning of complex biological systems, it is necessary to model the interactions that take place. In fact, the use of a formal method is crucial to prevent ambiguities, uncertainties and even contradictions to appear in dynamic biological systems. Petri Nets allow the analysis of qualitative structural to quantitative behavioral properties. PNs are effective for the modeling of molecular networks [1] . In fact, the mathematical formalism of Petri net theory is able to encompass many of these techniques. Various extensions to the original theory of Petri nets have been used for modeling molecular biology systems and metabolic [2] . Such systems permit to coordinate various molecules. We propose in this work to model this molecule as agent and biologic system as Multi agent system. The specification of biologic system is complex and each entity can interact and communicate in a dynamic environment. Indeed the complexity of the systems studied is increasing. The precision, reliability and the hardiness have become difficult factors to reach. Therefore, the integration of a mathematical tool offers an exact way, in presence of graphic tools, to succeed the conception of these systems, especially the multi agent systems.

This paper focuses on the theoretical foundations of modeling Biological Systems based on Agent Petri Nets. Section 2 introduces various preliminaries, including the advantages notion of Multi Agents System and Agent Petri nets. Section 3 discusses the specification of Biological systems, such the properties as Multi Agent System of this system is introduced. Section 4 shows how to create a general framework to model Biological systems based on Agent Petri nets. Related work is discussed in Section 6. Section 7 concludes the paper.

The formatter will need to create these components, incorporating the applicable criteria that need following.

This section introduces basic concepts related to Multi Agents System and Petri nets. Moreover, we introduce the notion of Agent Petri Nets. This formalism will be used to descript and model Biological systems.

Inside S. epidermidis biofilm, bacterial cells present different behavior than in their planktonic counterpart. Much knowledge is gathered concerning molecular mechanism and cells behavior inside biofilm toward external environment. Biofilm formation in S. epidermidis is a four-step process, it begins with initial cell attachment to native or conditioned a biotic surface, the second step, known as accumulation step, is marked with active cell multiplication and multi-layers population forming, the third step is the biofilm maturation during which biofilm micro-colonies takes a mushroom like form owing to metrical components distribution. The last step is the detachment of cells which regains their planktonic statute [6] . Different genes involved in biofilm formation in S. Epidermidis are under complex regulation in time and in space, indeed, an important genes network interacting together and with different targets involved directly or not in biofilm formation are behind the chronologically organized growth phases as well as the defined structure of S. epiermidis biofilm.

IcaR, the fifth gene of icaoper on, is located upstream to icaADBC genes. This gene is divergently transcribed from the other ica genes [7] .

The Teicoplan in Associated Regulator “TcaR” was reported as a negative regulator of ica transcription since inactivation of this gene enhanced the transcriptional level of icaADBC [8] .

Rbf (regulator of biofilm) is a member of AraC/XylS transcriptional regulators family. This protein was reported to play an important role in biofilm formation in S. epidermidis [9] .

In S. epidermidis, sigma B (σB) alternative factor plays a key role in the relationship of bacterial cell to its external environment (Figure 1). Indeed this factor is activated by numerous environmental stresses including high temperature, high osmolarity, antibiotics, or extreme pH [10] .

Sar proteins were classified in three sub-families basing on their structural properties. The first subfamily contains proteins acting as homodimers and binds DNA with a single DNA binding domain [12] . As conclusion, SarA enables S. epidermidis to switch between mechanisms of biofilm formation, ensuring the adaptation of this bacteriumo hostile environment [13] .

The accessory gene regulator (agr) quorum sensing system is a chromosomal oper on encoding two divergently transcribed transcripts, RNAII and RNAIII [14] (Figure 2).

In formal manner this reaction is defined by:

where:

Ÿ ;

Ÿ : set of tasks;

Ÿ : set of genes;

Ÿ : set of performed tasks;

We can deduce that:.

The set of tasks TS can be performed by a group of Genes A and can obtain a new set of performed tasks noted. Tsj. This change can infect the behaviour and the structure of the set of Genes. The Gene Ai employed to perform a task tsj leading to achieve some Goals. This Gene undergoes behavioural and structural changes.

In formal manner this reaction is defined by:

where:Ÿ

;Ÿ

: set of tasks.

Ÿ : set of genes;

Ÿ : set of performed tasks;

Ÿ : set of genes transformation.

where:

Ÿ N: New relation between genes;

Ÿ D: Destruction of gene;

Ÿ S: Substitution: base is replaced by one of the other three bases;

Ÿ Dl: Deletion: block of one or more DNA pairs are lost;

Ÿ Is: Insertion: block of one or more DNA pairs are added;

Ÿ Iv: Inversion: 180˚ rotation of DNA piece;

Ÿ R: Reciprocal translocation: parts of no homologous chromosomes change places;

Ÿ C: Chromosomal rearrangements: affect many genes at one time.

After perform a Reaction there are a behavioural and a structural changes of Gene noted by: performed/TG. We obtain new instance of Gene Ai (Figure 5):.

In this case:.

Definition1 presents a formal description of performed task (reaction). After firing the transition Performed there are new result (.tsj). This action can’t infect the gene:

But, in Definition 2, after firing the transition there are behavioural and structural changes ofGene Ai. The transformation of gene can be present by one state of the set of transformations TG, where:

In most of Biologic system reaction, there are major changes in behaviour and in structure of gene. So Performed is a particular case of Performed/TG.

We define the function given the set of result of reaction Ts performed by a set of Gene Ai related to the transition Tk by: Perfect (Tk) =.

In Petri Nets model, Perfect (tk) presents a condition of firing the transition Tk.

Figure 6 presents an example of Performed reaction achieved by set of Gene.

We transform the APN model to a Matrix of Gene Transformation. This matrix mention in each line the transformation achieved for the Gene Ai when there are perform of reaction tsj.

For example, and. We deduce that:.

Any reaction tsj can be performed by a set of Gene A. We define

Using the same APN model presented in Figure 7, we can deduce that:

, , , and.

A goal of set of Gene is define as the achievement of all reaction planned for the total of Biologic system (Figure 7): Goal (Gp) =.

In APN model, we use Inhibitor arc to firing the transition T2. In this state all reaction tsj are performed successfully.

We define the migration model of mobile Gene by a following Sub Petri Nets:

where:

Ÿ P: a non-empty finished set of Places;

Ÿ :: an application of front incidence corresponds to environment of departure;

Ÿ :: a back application of incidence environment of arrival;

Ÿ : is a non-empty finished set of Transitions (environment of departure Envi, or of Arrival Envj). Where:

Ÿ n: number of Places;

Ÿ k: number of place in Env1;

Ÿ n-k: number of Places in Env2;

Ÿ : final state of Agents in Env1;

Ÿ : initial state of Agents in Env2;

Ÿ : Output of Env1;

Ÿ : Input of Env2;

Ÿ : Interface between Env1 and Env2.

In this section we present an APN model for modeling Staphylococcus epidermidis Biofilm Formation (Figure 8). We define all reaction that can be performing by all Genes in the reaction system.

Much formalism has been used to model Biological Systems, in part due to the various phenomena that occur in those systems. The second part due to multi-disciplinarily of research groups. Formal Method for Biologists may be more familiar with mathematical modelling and computer scientists. Several works has been discussed the dichotomy between mathematical and computational models elsewhere such in [14] . Indeed, using mathematical models of cellular metabolism, it is possible to automatism test of generate sub-optimal phenotypes for specific applications [9] . Although different formal approaches has been questioned if there is such Petri Nets models. [17] proposes colored Petri Nets to simulate enzymatic reaction process: token is a pair encompassing the name and the concentration of the related substrate. Transitions present a kinetic function. [13] proves that with standard PNs we can modeling the essential components in biochemical pathways, and that PN models can be used to perform a qualitative analysis. In those models, places represent reactants, products or enzymes. Whereas transitions represent reactions [15] , it was elaborated a Physicochemical modeling of cell signaling pathways. They address the model design process, as well as, model verification, interpretation validation, calibration and publication of models. Another recent review on the modeling of signaling networks can be found in [8] . Recently an excellent review was elaborated by [18] for Modeling Signaling Networks with Different Formalisms. In Table 1, [2] summarizes some of the literature references reviewed herein, classified by type of intracellular process implemented.

We present in this paper a formal framework based on Multi agent System and Petri Nets to model Staphylococcus epidermidis Biofilm Formation. We provided a high-level description for this formalization, with a semantics given using the Agent Petri Nets. All process was present rigorously and clearly with APN. This method can be used to describe the reaction among molecules, their interaction and their transformation. With this framework it’s easy to present the behavior and the structure of each entity or the total of biologic system. Such system has a dynamic behavior, quickly generation of result and incomprehensible reaction. The objective of the dynamic model consists in proposing a formal method to understand the functioning of the Staphylococcus epidermidis Biofilm Formation and it is possible to perform formal analyses on environments thus described.

The ability to predict system behavior with an APN helps evaluate model completeness as well as improve our understanding of the function of biological systems. In fact, the meaning facets framework establishes a new methodology for computer-aided collaborative modeling in Systems Biology.

Our meaning facets are also a way for structuring and clarifying our understanding of bio-models. Due to the graphical visualization of biologic system by Petri nets, a bioscientist can intuitively understand the modeled process.

BorhanMarzougui,KamelBarkaoui,Mohamed AmineMakni, (2016) Agent Petri Nets Framework for Modeling Staphylococcus epidermidis Biofilm Formation. E-Health Telecommunication Systems and Networks,05,19-30. doi: 10.4236/etsn.2016.51003