Petri net Modeling and Property Analysis of C3 plant Photosynthesis and Photorespiration

Metabolic pathway - Modeling and Analysis, is the emerging research area in Biochemistry over the past few years.C3plants are the best carbohydrate producers and also are the most common plants found in the nature. These plants are very much efficient in photosynthesis in cool and wet climate while, in very hot climatic condition it undergoes photorespiration process. In this paper we model the process of making food by C3 plants to study the behavior in different climate conditions.

Acid Metabolism (CAM) plants, which named on the number of carbon present in the first stable compound formed during the photosynthesis process. Three photosynthesis pathways exist among terrestrial plants. Different plants have different networks depending on certain conditions [1,2]. Graph theory results are also available in several biological networks and in photosynthetic effect [3,4]. Several other mathematical models are proposed in order to understand the network of plants photosynthesis and explain the stomatal behavior of leaf-level C 3 photosynthesis but they all are analytical model [5,6]. In this paper we propose a Descriptive mathematical model which approaches through graphical representation and draws some better qualitative and quantitative properties of any biological model or any complex system network. In order to understand the dynamic property and behavior of C 3 -plants Carboxylation process (Photosynthesis) and the Oxygenase process (Photorespiration), we propose a mathematical and graphical approach of modeling and that is Petri Net (PN). Due to its adaptability Petri net is very much efficient to model any biological system and can handle any concurrent and distributed complex system perfectly This model is Petri net (PN) which is a discrete event systemfirst proposed by Carl Adam Petri in early sixties to model the concurrent, asynchronous, distributed system [7,8]. Petri net is also used in several other field of sciences determining binary vectors or can be used as a recommender system in several networks [9,14]. Petri net has also application in biological networks like in Cardiovascular disease or in tuberculosis [15,18] In this paper we focus on C 3 plants metabolic network and will compare the networks with C 4 plants using Petri net model. There are several software available to trace Petri net model and validate the system modeled and provide several results to explain the behavior of network considered. In this paper we are using PIPE v4.3.0 to draw the network and uses the results to explain the C 3 plant network [19] Photosynthesis is the most common and most important biochemical reaction occurring in nature. Almost all types of plants undergo photosynthesis under different conditions. C 3 plants are the most common plants found in nature. Most of the carbohydrate producers are the C 3 plants for example Beans, Rice, Wheat, Potatoetc,whichare alsocalled energy giving plants. The process in which these plants produce energy is Photosynthesis. This process has a network explained in Figure 1, as they are efficient at photosynthesis in cool and wet climates. C 3 plants do photosynthesis using NORMAL CALVIN CYCLE.
Photosynthesis is the process where Reduced Nicotinamide Adenine Diphosphate (NADPH) and Adenosine Diphosphate (ATP) are created in CBB(Calvin Benson Bassham).In the suitable weather CO 2 enters through the tiny pores on leaves called stomata.This carbon dioxide then combines with an enzyme called RuBisCO to produce Sugar. This process produces a molecule made up of three carbon atoms, and that is why C 3 plants named. For C 3 plants, this process takes place in a chloroplast layer, the green cells present in plants which helps in photosynthesis whichmakesany plant green. The cycle continues and with the help of energy in the form of light from the sun, sugar is made as well as RuBp is also produced forthe future use [20]. Sometimes the C 3 plant consumes oxygen too, which can slow down the process of photosynthesis.This process is called Photorespiration also called oxidative photosynthetic carbon cycle, or C 2 cycle. C 2 cycleis named so as the first stable compound formed is atwo carbon molecule. In Figure 2, one can observe the network involved in producing energy in C 3 plants during Photorespiration. It is the process of taking O 2 instead of CO 2 in the process of Photosynthesis. Instead of fixing carbon when Rubisco fixes oxygen under certain conditions results in Photorespiration (PR).Sometimes it is also called anti-photosynthesis process. Under hot conditions RuBisCo has more affinity towards oxygen, so under hot weather when stomata closed so CO 2 can't diffuse in and O 2 can't diffuse out. Earlier it was said that this process is harmful for plants but recent researches shows the importance of this process.The process PR converts 2-phosphoglycolate (2PGP) into 3-phosphoglycerate (3PGA) and is escorted by O 2 intake and CO 2 release. PR plays the role of salvage or metabolic repair process which convert the toxic compound produced i.e., PGP into a useful compound PGA which produced during CBB cycle. PR also leads to the loss of CO 2 and NH 3 . Eventually, PR decreases the rate of photosynthesis by 30% in current atmospheric concentration of CO 2 and O 2 [21] .

Results And Discussion
While modeling the process we are considering the stable intermediate products as  Table 1, the description of places and transitions are shown according to the model.
Here we have modeled both the process of i.e., photosynthesis and photorespiration in a single model. When CO 2 is present in the stomata of the plant cell then the normal photosynthesis process will take place as t 1 will fire and further CO 2 combines with RuBp which is already present in the plant cells to form PGA then G3P followed by the formation of Glucose to produce energy for the plant.
Hence the path of firing of these transitions leads to normal photosynthesis t 0 t 11 t 12 t 13 , t 14 t 15 , t 16 . This cycle of model is validated from PIPE v4.3.0 software and some results follows which can be explained by the properties of Petri net.
The model is bounded, signifies the compounds formed during photosynthesis is in limited number of molecules. There is no accumulation of any product formed during the process either the product has been used for the further process or being used by the plants growth. The number of molecules formed is different for different compound, so this model is not Safe.
In this model we are not concerning after the formation of glucose so this model must have a deadlock.
But the product RuBp is produced again during the process so the process is also Live.
In other case when there is no CO 2 enters through stomata then tokens on p 0 will be zero. The compound RuBp is already present in the plant, So to start the process we are taking the six molecules of RuBP i.e., µ 0 (RuBp) = 6, and for all other places in P, µ 0 (p i ) = 0.
Here, only transition t 1 will fire due to inhibitor arc property and consume O 2 to start the process. Now, O 2 combines with RuBp in the presence of RuBisCo enzyme to form two compounds PGA and PGP. PGA will again start normal Calvin cycle and PGP goes through a certain number of steps to form PGA back to follow Calvin cycle to produce Glucose.
Availability of sufficient amount of CO 2 results in Photosynthesis and insufficient amount of CO 2 results in photorespiration with availability of O 2 . In either case the Petri net is non-terminatingAlso, the process is Livealways. We can infer that the formation of glucose which is very essential for plants will never stop and the production of food given by C 3 plants can be increased by maintaining a suitable atmospheric composition.During modeling of metabolic pathways in photosynthesis and photorespiration, as the starting moleculeis produced back to start the reactions again so the process is always

Conclusions And Scope
We have discussed the modeling of metabolic pathways as discrete-event systems. The With modeling of all these networks, validation and simulation is also very important with Petri net software like PIPE or in MATLAB with Petri Net tool box.

Petri Net (PN)
Petri net is a special kind of directed graph where two kinds of nodes are present, one is place and another is transition. While drawing the graphical representation of any model, places are draw as a circle, and transitions as rectangle. It describes the complex system in graphical manner and gives an easy visualization of the system. Peterson describes further advantages of Petri net as a model of concurrent and conflict systems. The power of Petri net means the properties of the net can help to describe the modeled system in a better way. An advance Petri net analysis of techniques provides systematic and qualitative results of the modeled system. Petri net model analysis is drawing attention because of its simple graphical approach and certain high level Petri nets like timed Petri net, stochastic Petri net and colored Petri nets are proved very useful to study about any complex system. Petri net is 5-tuple set, consisting of a finite set of places, a finite set of transitions, a positive incidence function that denotes the number of arc from any transition to place, a negative incidence function that denotes the number of arc from any place to transition and an initial marking which represents the number of tokens on each place initially.
Mathematically represented as PN = (P, T, I -, I + , µ 0 ). Any transition fires at the given marking if and only if the outgoing arcs from all the places is less than and equal to the present marking of that place. After firing the tokens are moved from the input places and deposited on the output places accordingly.
The Reachability Graph of a Petri net is denoted by R (PN, µ) where, all the markings µ are taken as the vertices and the transition responsible for the corresponding marking is taken as edges. It helps in drawing the results of reachability of one state to another state. One state is reachable from another state if there is a directed connected path found between them [22,23].

Properties of Petri Net
Reachability Graph/ Coverability Graph: The graphical representation of the firing sequences and all possible marking states is called Reachability Graph of the Petri net. Here, nodes are the marking states and the arcs show the corresponding transition which fires to attain that state. In biological system the reachability graphs help to find the possible ways in the formation of the required product.
Boundedness:A Petri net is Bounded if the number of tokens on any of the places involved never exceeds a finite or a fixed amount. In biological system it helps to implies that the number of molecules formed in the product compound is limited. Accumulation of any compound is not taking place. Hence it shows the property of conservation of mass.
Liveness: A Petri net is live for µ (initial marking) , if there exists a firing sequence to reach to one marking from any other marking by a suitable firing sequence . in biological network it implies that the involved reactions will occurs repeatedly which contributes to the development depending on time.
Reversible: A Petri Net is reversible if the initial marking is reachable from all the other reachable markings. In biological system it implies that the compound which starts the reaction is formed or produced again within the process Extended Petri-net: The extended Petri net is somehow an extension of standard Petri net where it has a form of arc called an inhibitor arc. An inhibitor arc is from a place p to a transition t which enables only when there are no tokens on p. This makes the concurrent phenomena much easier to model [