MODELING AND SIMULATING IP3R SUBUNITS OF SIX-STATES GRAPHICALLY

MODELING AND SIMULATING IP3R SUBUNITS OF

SIX-STATES GRAPHICALLY

Amr Ismail¹ ; Mostafa Herajy¹ and Elsayed Atlam²

¹ Department of Mathematics and Computer Science, Faculty of Science, Port Said University, 42521 - Port Said, Egypt

² Mathematics Department, Faculty of Science, Tanta University, Egypt College of Computer Science and Engineering, Taibah University, Yanbu, KSA.

ABSTRACT.

Inositol 1,4,5-trisphosphate receptor ( 14IP3R"> ) is a calcium channel that consists of four identical subunits. They are grouped inside clusters in the Endoplasmic Reticulum. They are a prime stochastic process in regulating the calcium dynamics in the cell. Moreover, the opening and closing the channel occurs randomly based on the number of active subunits. The current models of ( 14IP3R"> ) are modeled and simulated based on stochastic reactions with neglecting the opening and closing the channel. Therefore, stochastic Petri nets ( 14SPN"> ) and coloured stochastic Petri nets ( 14SPNC"> ) are a good structure to deal with challenge. 14SPN">  and 14SPNC">  are two Petri nets classes that are used to model and simulate the stochastic processes. In this paper, we present two graphical models for the subunits of Inositol 1,4,5-trisphosphate receptor of 6-states based on the 14SPN">  and 14SPNC"> . Moreover, we simulate the 14SPNC">  model based on the direct semantic generations and compare these results with the unfolding method.

1-    INTRODUCTION

Signals of calcium organize many cellular functions. The inositol 1,4,5 triphosphate ( 14IP3"> ) receptor ( 14IP3R"> ) is one of intracellular calcium channels that plays a vital role in the releasing of calcium from the Endoplasmic Reticulum (ER) into the cytosol [1]. This type of calcium channels are found inside clusters on the ER membrane with tens of numbers about 10-30 channels. The structure of IP3R consists of four identical subunits [2]. Every subunit has three binding sites, one for IP3 while the other two binding sites are related with 14Ca2+"> : one activation and the other inhibitory. The binding sites of 14Ca2+ "> location is a unknown but when the IP3 is bounded it increases the chance of the subunit to binding 14Ca2+"> at the activation site [3]. Therefore, the primary binding site, which activate the subunit, is 14IP3"> . Moreover, the opening and closing the channel is based on the number of active subunits.

Subsequently, many classical models have been proposed such as DeYoung- Keizer model [4] to study the stochastic processes. Although, the correctness of these models and the efficiency of the corresponding stochastic simulation, they did not show how can the channel open or close. Therefore, to overcome this problem, we depend on the framework of stochastic Petri nets ( 14SPN"> ) coloured stochastic Petri nets ( 14SPNC"> ) [5].

14SPN">  and 14SPNC">  are have been got a lot of application in system biology such as in [5]. One of the attractive advantage of 14SPNC">  for creating biological models supports increasing the model size that contains a lot of similar components only be increasing the number of colors (adding colors). Therefore, 14SPNC">  can solve the previous problem. However, Simulation of 14SPNC">  is based on the unfolding method which a time consuming. The main aim of unfolding method is converting 14SPNC">  models into stochastic Petri nets 14SPN">  models. Therefore, we simulate SPNC model via a direct simulation approach to simulate the 14SPNC">  model of 14IP3R">  without passing on the unfolding step.

In this paper, we introduce graphical models for the subunits of 14IP3R">  channels based on SPN. Also, we consider how to encode the 14SPN">  model using the colors to show the flexibility of 14SPNC">  in modeling. The organization of the paper as follows. In section 2, presents a background about the calcium channel 14IP3R">  as well as the framework of SPNC. Section 3 sketch the various models of 14IP3R">  subunits by using SPN and SPNC. In Section 4, discus the results of the models. Finally, section 5 presents the conclusions.

2-    BACKGROUND

2.1 14IP3R">  subunits

14IP3R">  channels are grouped inside clusters in the Endoplasmic Reticulum (ER) [6] and have an important role in  intracellular calcium dynamics that is involved in many cellular processes such as neurons degeneration [7]. 14IP3R">  channels consisting of four identical subunits [4]. The channel opens when there exist at least three subunits in the active state [3] . The activation of subunits is depending on the [ 14IP3"> ] and 14Ca2+"> as discussed in [8]

Many of approach have been introduced to model and simulate IP3R channels [9] such as the deterministic model of De Young-Keizer model [4] and a comparison stochastic model [8] But all these models do not clearly show the arrangement of the IP3R channels geometrically relative to each other. This problem is solved by introducing a general approach using colored stochastic Petri nets in [5] however this approach not study the opening and closed channels depending on the active subunits for the channel and the simulation of the model was depend on the unfolding method that is a time consuming.

2.2 Colored Stochastic Petri Nets

Colored Stochastic Petri Nets ( 14SPNC"> ) group between the colored Petri net ( 14PNC"> ) definition and the structure of stochastic Petri nets ( 14SPN"> ). 14SPNC">  is de_ned with a set of places, set of transitions, set of arcs, set of color sets, acolor function that assigns color set for each place, a guard function that gives an expression of type Boolean for each transition, an arc function that assigns the arcs expression and a function that assign the initial marking for each color

for more details [5].

14SPNC">  are introduced to improve the appearance of the 14SPN">  by grouping the similar parts in only one part and distinguish between them by colors. The classical tools of 14SPN">  is not efficient to model the large stochastic models so that it extended with the colors definitions to catch the rapid progress in the system biology. 14SPNC">  has been got a lot of application in different fields such as retrial systems [10], production systems [11] and systems biology [5].

3-                MATERIALS AND METHODS

The inositol-1, 4, 5 triphosphate ( 14IP3"> ) receptor is a type of intracellular calcium channels. Every IP3R consists of four identical subunits. Therefore, we indicate to subunits with variables 14subn">  where 14 n = 1…4">  in 14SPN">  model. In each subunit found three binding sites: An activation site for 14Ca2+"> , an inhabitation site for 14Ca2+"> , and binding site for 14IP3"> . Therefore, we indicate to states by variables sijk in the SPNC model where the index i indicates to the 14IP3">  site state, j indicates to 14Ca2+">  activation site state and k is the inhabitation 14Ca2+">  site state. Every index has only two values 1 or 0 where 1 indicates to the ion is bound and 0 otherwise. According to the experimental studies 14Ca2+">  can not bind to the activation site if 14IP3">  is not bound. Therefore, binding of 14Ca2+">  to the activation site is sequential as it only binds once 14 IP3 "> is bound to the receptor. States s010 and s011 mean 14IP3"> is not bound to the receptor. Therefore, these two states can be removed from each subunit and the number of states for each subunit is reduced to six states.

3.1 Modeling 14IP3R">  Subunits of 6-states with 14SPN">

Each subunit contains six stats of three binding sites. Figure 1 illustrates the 14SPN">  model that describes 14IP3R">  subunits with six-states.

In Figure 1, we model the 14IP3R">  as four equal and independent subnets. Each subnet represents a subunit. Each subnet consists of 6 places that are named with variables 14subn">  where 14n = 1…4">  and 15 transitions. In addition to, two places 14Xact">  and 14 opchans "> to check the number of active subunits and the states of the channel, respectively.

Figure 1. The Model of 14IP3R">  subunits of 6-stats using stochastic Petri net.

3.2 Modelling IP3R Subunits with 14SPNC">

In this subsection, we show the power of 14SPNC">  in modelling the models which contains a repetition of components without changing the structure of the model.

For more illustration, consider the 14SPN">  model in Figure 1 which consists of four identical subnets each one contains of six places. Therefore, we consider one of these subnet and we define a color set 14chan">  contains four colors where each colour indicates to a subnet.

Figure 2 introduce the 14SPNC"> model of 14IP3R">  subunits of 6-states. Every place in the model related with the colour set chan instead of places 14 xact">  and 14opchans"> . They are related with the color set dot which contains only one color.

Figure 2: The Model of 14IP3R">  subunits of6-stats using colored stochastic Petri net.

Table 1 provides the rates of transitions of the 14SPN">  models as well as values of binding constants and dissociation constants are introduced in Table 2.

Table 2 Transitions rates: 6-states 14SPN">  model in Figure 1 and   14SPNC"> model Figure 2

Table 3 Values of (binding(ai)/ dissociation(bi)) constants of the receptor

4-    RESULTS AND DISCUSSIONS

In this section, we introduce the models results. We also discuss the run–time due to the direct semantics generation and the unfolding method.

4.1 Simulation Procedures

  The created Petri nets models have been designed with Snoopy and simulated with it. Snoopy is a free software that can be used to model and implement different classes of Petri nets such as 14SPN "> and 14 SPNC"> .

Opening and closing the channel is depending on the number of active subunits where the channel opens when the number of active subunits at least three. Therefore, we run the models to test its validations which results are given in Figure 3. Figure 3(a) presents the number of active subunits over the time while Figure 3(b) illustrate the opening and closing the channel over the time.

Figure 3 Numer of Active subunites with opening and closing the channel. (a) gives the number of active subunits over the simulation time and (b) gives the channel states (open/close) over the simulation time

In Snoopy 14 SPN">  models are simulated scholastically based on the stochastic simulation algorithms such as Gillespie's algorithms (first reaction method) while simulation of SPNC models are simulated by unfoldi