Modeling and Controlling TGF- Pathway Using Standard Petri Nets

Niloofar Nickaeen, Jafar Ghaisari, IEEE Senior Member Dept. of Electrical and Computer Eng.

Isfahan University of Technology Isfahan, 84156-83111, Iran Email: ghaisari@cc.iut.ac.ir

Yousof Gheisari, Shiva Moein Regenerative Medicine Lab, Isfahan Kidney Diseases

Research Center, Isfahan University of Medical Sciences Isfahan, Iran

Email: ygheisari@gmail.com

AbstractCell growth is controlled by some factors known as growth factors. One of these growth factors is a substance called TGF-. Any malfunction in the system which growth factors perform in, results in drastic cell growth thus cell death. The aim of this study is to control the system of TGF- growth factor in the case of fibrosis, a condition created by malfunction in the pathway of TGF-. A model of TGF- pathway is proposed using standard Petri Nets. Then a control is proposed not only to control the amount of TGF- in the pathway but also to observe the law of conservation of matter as the absolute rule in biological systems. The results clearly show the control of TGF- concentration in the pathway. The research may provide a basis for drug design and determination of target reactants to control fibrosis.

Keywords TGF-; Ptri Nets; Fibrosis; Biological Networks; Biological Modeling; P- invariants.

I. INTRODUCTION Particularly, in the last two decades, engineering and

applied mathematics have been widely used in modeling and analysis of biological networks. A pathway is defined by some chemical reactions occurring sequentially so that the product of each reaction is the reactant for the upcoming reaction. TGF- pathway is a pathway responsible for many behaviors of the cell such as cell growth or differentiation. TGF- is a growth factor, a factor which results in the cell growth. If as a result of a malfunction in the pathway the amount of growth factors increase upon some limits, the infected cell grows dramatically until death. One of the malfunctions which may result in this phenomenon is known as fibrosis. In the case of fibrosis, a positive feedback happens in the following way, resulting in cell death: TGF- initializes a pathway which eventually results in a gene being expressed. One of the productions of the very same gene is TGF- itself. Therefore if the cell does not function properly, the amount of TGF- increases with time resulting in the more of the gene being expressed and more of TGF- being produced. So with the increasing amount of TGF-, the cell grows more and more until it dies. Many have tried to model and analyze the behavior of TGF- pathway [1-4]. In [1], authors proposed a model for this system using Ordinary Differential Equations

(ODE). A search was done in the parameter space in order to identify the most important factors involved in oscillatory responses of the pathway. In [2], authors modeled the pathway using ODEs and estimated series of parameters to create certain types of outputs e.g. oscillatory responses. In [3], again an ODE model of TGF- pathway was constructed and robustness analysis was done using the model. In [4], an ODE model of the TGF- pathway is developed and the importance of negative feedback motifs in the pathway is investigated. [1-4] study the dynamic behavior of TGF- pathway using the models constructed based on the literature. Yet, neither has tried to use mathematical approaches to determine the target substances to control malfunctions in this pathway which usually results in drastic conditions.

In this paper, the TGF- pathway is modeled using standard Petri Nets. Gathering data from literature, a more sophisticated model is presented. Several reactions involved in TGF- pathway which have not been noticed in previous studies have been used in this modeling. To decrease the effects of malfunctions in the pathway, a control based on the biological network structure is proposed. The control consists of adding a place representing a drug to the pathway such that a P-invariant is created in the new net. Using this P-invariant, the control is applied to the system. As a result of the control, TGF- density stays under a limited boundary.

II. MATERIAL AND METHODS

A. Petri Nets Standard Petri Nets (PNs) are a modeling language

especially used to model asynchronous, concurrent and distributed systems. Different biological systems have been modeled by various types of PNs [5-7] where each type emphasizes on a special characteristic of a biological system. For example in [5], a pathway is modeled using standard PNs. Then the model is upgraded to timed PNs by attributing reaction speed to its transitions. The same approach of modeling via standard PNs is used in this paper but the goal here, is controlling the pathway.

Formal definition of PNs is as follows: PNs are five tuple set of PN = (P, T, A, W, M0) where:

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P = { p1, p2, ..., pn } is a finite set of places

T = { t1, t2, ..., tm } is a finite set of transitions

A (PT) (TP) is a set of arcs

W : A { 1, 2, ... } is a weight function

M0 : P { 0, 1, 2, ... } is the initial marking

P T = and P T =

There are two sets of nodes present in a PN: places and transitions. Places are shown by circles. They represent system states or conditions. In the modeling of biological systems, places may represent proteins, species, organism, ions, temperature and PH value. Transitions which are illustrated by squares describe the process or the event by which system states change. In biological models they refer to biochemical reactions.

Tokens which are illustrated by dots or numbers within a place, quantify the models. They represent the value of a state or condition. In biological systems, they refer to concentrations of different species, temperature and PH value [8].

A transition has to be executed for the system to change state. The execution of a transition is known as firing the transition. A set of firing rules have been defined for PNs. The rules are as follows:

A transition "t" is enabled in a marking "m" if p t : f (p, t) m(p). p t refers to all the input

places of transition t. f (p, t) is the arc weight between place "p" and transition "t".

A transition "t", which is enabled in "m", may fire. When "t" in "m" fires, a new marking " m' " is reached, with

p P: m' (p) = m (p) f (p, t) + f (t, p). Where f (p, t) represents the weight of the arc linking

place "p" to transition "t" and f (t, p) is the weight of the arc linking transition "t" to place "p".

The firing itself is timeless and atomic [9].

The first step to model TGF- pathway using PNs is to determine transitions, places, arcs and their equivalents in the TGF- pathway. To accomplish this purpose, the biological structure of TGF- pathway needs to be studied.

B. TGF- Pathway As it was mentioned, pathways are several chemical

reactions which happen sequentially. One of many pathways functioning in the cell is TGF- pathway. It is a rather important pathway due to its critical functions such as differentiation (that is a process during which a specialized cell is developed from a basic one) or cell growth. The pathway functions as follows: TGF- as the initializing

element of the pathway, binds to a dimmer of its type two receptor, TGF-R2. The structural formation of the complex induces the recruitment of TGF- type one receptor, TGF-R1. In this complex, TGF-R2 phosphorylates TGF-R1. The activated TGF-R1 then phosphorylates certain proteins called SMAD2 and SMAD3. Two phosphorylated SMADs then bind to a protein known as SMAD4 resulting in a nuclear complex. This trimmer binds to a specific DNA sequence and result in the target gene being expressed. A more detailed description will be presented in the next section including a list of involved reactions [10, 11].

III. MODELING THE TGF- PATHWAY The first step to control the TGF- pathway includes

modeling the pathway. PNs were chosen as the modeling method due to their ability to model certain properties of biological pathways including concurrency and distribution. In the proposed model, places represent substances which function in the pathway. These substances were gathered from the literature. Places used in the model along with their equivalent substances are listed in table 1.

Model transitions represent chemical reactions of the pathway. Their inputs are the reactants and their outputs are products of each reaction. Transitions used in the model with their equivalent reactions are listed in table 2. Transitions number 20, 21 and 22 which illustrate the negative feedback performance of the expressed gene, have not been noticed in previous studies.

TABLE . PLACES IN THE PN MODEL places used in the model with their equivalent substances

Place Substance

P1 TGF-

P2 TGF- Receptor2

P3 TGF- Receptor2/TGF- Complex

P4 TGF- Receptor1

P5 TGF- Receptor1/ TGF- Receptor2/ TGF- Complex P6 Sara

P7 TGF- Receptor1/ TGF- Receptor2/ TGF- /Sara Complex P8 R-Smad

P9 phosphorilated R-Smad

P10 Cosmad

P11 R-Smad /Cosmad Dimmer

P12 R-Smad /Cosmad trimmer

P13 P300

P14 R-Smad /Cosmad Dimmer/P300 Complex

P15 R-Smad /Cosmad trimmer/P300 Complex

P16 R-Smad /Cosmad Dimmer/P300/CBP Complex

P17 R-Smad /Cosmad trimmer/P300/CBP Complex

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places used in the model with their equivalent substances Place Substance

P18 GENE expression

P19 Inactive Rsmad

P20 Smurf

P21 Smad6/7

P22 Snon/c-ski

P23 Inhibited Smad6/7

P24 Inactive Cosmad

P25 Inactive TGF- Complex

P26 SHC_GRB2_SOS_RAS_RAF_MEK1_MEK2_ERK1_ERK2 P27 Inactive Rsmad_cosmad_trimmer

P28 Inactive Rsmad_cosmad_Dimmer

P29 Inactive R-Smad (Rsmad degradation)

P30 Inactive Sara/ TGF- Complex (Sara/ TGF- Complex _degradation)

P31

Inactive Sara/ TGF- Complex (Sara/ TGF- Complex _degradation) (in another location of net other than P30 )

P32 CBP

TABLE . TRANSITIONS IN THE PN MODEL transitions used in the model with their equivalent

reactions Transition Reaction

1 TGF- +2 TGF-R2 TGF- - TGF-R2

2 TGF- + TGF- R2+2 TGF- R1 TGF- complex

3 Sara + TGF- complex Sara_ TGF- complex

4 Smurf+Smad6/smad7+ TGF- complex inactive TGF- complex 5 Rsmad+SmurfRsmad degradation

6 Sara_ TGF- complex + Rsmad Phosphorilated Rsmad

7 Phosphorilated Rsmad+ Cosmad Rsmadcosmad dimmer

8 Phosphorilated Rsmad+2 Cosmad Rsmadcosmad trimmer

9 Snon/ski+ Rsmadcosmad dimmer (degradation)

10 Snonski+Rsmadcosmad trimmer (degradation)

11 Snon/ski+ phosphorilated Rsmad + smurf (degradation)

12 P300+Rsmadcosmad dimmer Rsmad_Cosmad_Dimmer_P300 13 P300+Rsmadcosmad trimmer Rsmad_Cosmad_trimmer_P300 14 Rsmad_Cosmad_Dimmer_P300+CBP Rsmad_Cosmad_Dimmer_P300_CBP

15 Rsmad_Cosmad_trimmer_P300+CBP Rsmad_Cosmad_trimmer_P300_CBP

16 Rsmad_Cosmad_Dimmer_P300_CBP GENEexpression+ Snon/ski + Rsmad +Cosmad+ Smad6/7 + TGF-+ P300+CBP

17 Rsmad_Cosmad_trimmer_P300_CBP

transitions used in the model with their equivalent reactions

Transition Reaction GENEexpression+ Snon/ski + Rsmad +Cosmad+ Smad6/7 + TGF-+P300+CBP

18 TGF- ; (degradation)

19 GENE products; (degradation)

20 GENEexpression+ Sara_ TGF- complex Sara_ TGF- complex degradation

21 GENEexpression + Rsmadcosmad dimmer Rsmadcosmad dimmer degradation

22 GENEexpression + Rsmadcosmad dimmer Rsmadcosmad trimmer degradation 23 Cosmad+ Smad6/7 inactive Cosmad

24 Sara_ TGF- complex+ Smad6/7 Inactive Sara_ TGF- complex 25 Smurf (Smurf Production)

26 Smurf + Smad6/7Inhibited Smad6/7

27 SHC+GRB2+SOS+RAS+RAF+MEK1+MEK2+ERK1+ERK2 (production)

28 SHC+GRB2+SOS+RAS+RAF+MEK1+MEK2+ERK1+ERK2 +Rsmad Rsmad degradation

Using the proposed places and transitions, a model of the pathway is constructed in a software specially designed to model biological systems known as SNOOPY [12]. The model is represented in Fig. 1.

In biological systems, a substance can affect different parts of the system simultaneously. Thus, to avoid repeating the same place in the model, some places known as logic places are used. Logic places are illustrated by grey circles. Different logic places with the same name are actually the same substance performing in different parts of the system. Also for the pathway to function, some substances must be present. Therefore initial conditions were attributed to these places.

IV. CONTROLLING THE PATHWAY The first step to control the cell growth is controlling the

amount of TGF- as the initiator of the pathway. As it was explained previously, based on the structure of the pathway, the more the amount of TGF- increases, the more the pathway occurs.

Before controlling the pathway, we need to make sure that the applied control does not contrast with the law of conservation of matter which is the absolute rule in biological systems. Thus the same approach as [13] was chosen as the control mechanism. In [13], an arbitrary standard PN is considered and a control method based on P-invariants is proposed for the PN. P-invariants are a subset of places in which the weighted sum of tokens is constant. As the tokens represent the amount of matter involved in reactions, P-invariants indicate the law of conservation of matter in a biological system. So using P-invariants as the control approach guarantees the preservation of matter in a system.

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Therefore the method illustrated in [13] was considered to be appropriate for controlling TGF- pathway.

Suppose that the limit value of TGF- is given by:

L*p b (1)

b is a vector which its elements bi1 represent the limit which needs to be set on the place i of the system. P is a vector [p1, p2, , pn] in which pi represents place i. The coefficients of equations summarized in (1) are the elements of a matrix L. Adding C as the control places, we have

L*p +c=b (2)

Defining M as the incidence matrix of the system, it is proved that the incidence matrix of the controlled system is

Mc=-L*M (3)

First, a limit must be set on the number of tokens in the place representing TGF-. Defining b, L will be calculated using (1). The incidence matrix of the controlled system is calculated from (3), defining the structure of the system with control applied on TGF-. To simulate the output, a limit of 500 tokens is defined as the limit on TGF- concentration. Using a software called CHARLI, the incidence matrix of the

Petri model was calculated. Mc then was calculated using (3). Calculated Mc had four nonzero elements indicating four arcs connecting the added control place to the transitions of the net. Mc = [2 0 0 0 0 0 0 0 0 0 0 0 0 0 -100 -100 0 0 0 0 0 0 0 0 1 0 0 0] Element i of MC represents the connection between the added control place and transition i in the system. The new arcs were added using these four nonzero coefficients as the arc weights. The model with control added, is illustrated in Fig. 2. The initial condition of the control was calculated using (4),

C0=b-L*P0 (4)

where P0 is the initial marking of the system. C0 is calculated to be 400 tokens.

Applying the control, four new connections are formed in the system. The following four stoichiometric reactions represent changes according to the control place added. 2TGF-+2ReceptorTGF-/Receptor Complex+2X (5) TGF- TGF- Degradation +X (6) Trimmer-P300-CBP+100XGeneExpression (7) Dimmer-P300-CBP+100XGeneExpression (8) The control place represents an anonymous substance which

Fig. 1 . The Petri net model of TGF- pathway.

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controls the pathway. X is the anonymous factor which stabilizes the system if it participates in the pathway with the calculated stoichiometric coefficients. Because of the law of conservation of matter, substance X must be made of chemical units involved in the reactions with X. So the chemical structure of X is known to some level. As illustrated in Fig. 3 and Fig. 4, in the system with no control, number of tokens in the place representing TGF- breaks the 500 threshold in step 50 whereas in the controlled system the number of tokens stays within the range.

In summary, one approach of drug design may be focused on determining the X factor which participates in (5) to (8) with the calculated stoichiometric coefficients. This factor along with a specified initial condition may control the pathway. The process may be applied to any of the key elements in the pathway which may halt the pathway in their absence.

At last, some points are worth noting. The amount of gene expression differs due to different conditions of the cell. A variety of factors may affect gene expression. To model these factors, the weights of input arcs of the place which represents gene expression may differ due to cell conditions. For this study, the arc weights resulting in gene expression and thus reproduction of TGF- were set in a way so that to make sure the amount of TGF- increases drastically thus recreating fibrosis condition. This way the function of the applied control

mechanism becomes clear. To fulfill this condition, the weights of two input arcs of the place representing gene expression were set to 100 each.

V. CONCLUSION Due to the importance of TGF- pathway activities, malfunctions result in disastrous effects. Therefore controlling the pathway in the case of malfunction is substantial.

In this paper, a PN based model was proposed for TGF- pathway. Several new reactions, compared to prior models, were also included. The model parameters were set in a way so that the output increases dramatically, as it happens in fibrosis condition. A control based on P-invariants was set on the system in order not only to control the pathway but also to observe the law of conservation of matter. The simulation results illustrate the efficiency of this approach. The control was translated to stoichiometric equations; thus describing the structure of the anonymous matter which is able to control the pathway in the case of fibrosis. The results are considered as a basis for drug design and determination of target substances for halting the pathway. A method for selecting target substances to apply this control approach on, may be addressed in future works.

REFERENCES

Fig. 2 . The Petri net model of the TGF- pathway with the control added. The control is specified by the cycle around it.

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[1] K. Wegner et al., Dynamics and feedback loops in the transforming growth factor TGF- signaling pathway, Biophysical Chemistry, vol. 162, 2012 , pp. 22-34.

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[9] B.H. Junker, and F. Schreiber, Analysis of biological networks, John willy, New jersey, 2008.

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[13] K. Yamalidou, J. Moody, M. Lemmon and P. Antsaklis, Feedback control of Petri nets based on place invariants, Technical Report of the ISIS group at the university of Notre Dame, 1994.

Fig. 3. Number of tokens representing TGF- density in a pathway with

malfunction and no control.

Fig. 4 . Number of tokens representing TGF- density in the controlled pathway.

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