Bioinformatics Advance Access originally published online on November 22, 2007
Bioinformatics 2008 24(2):209-217; doi:10.1093/bioinformatics/btm560
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Petri net-based method for the analysis of the dynamics of signal propagation in signaling pathways
Department of Computer Engineering, École Polytechnique de Montréal, P.O. Box 6079, Station Centre-Ville, Montréal, H3C 3A7, Canada
*To whom correspondence should be addressed.
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ABSTRACT |
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 TOP ABSTRACT 1 INTRODUCTION2 METHODS3 RESULTS AND DISCUSSION4 CONCLUSIONACKNOWLEDGEMENTSREFERENCES |
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Motivation: Cellular signaling networks are dynamic systems that propagate and process information, and, ultimately, cause phenotypical responses. Understanding the circuitry of the information flow in cells is one of the keys to understanding complex cellular processes. The development of computational quantitative models is a promising avenue for attaining this goal. Not only does the analysis of the simulation data based on the concentration variations of biological compounds yields information about systemic state changes, but it is also very helpful for obtaining information about the dynamics of signal propagation.
Results: This article introduces a new method for analyzing the dynamics of signal propagation in signaling pathways using Petri net theory. The method is demonstrated with the Ca2+/calmodulin-dependent protein kinase II (CaMKII) regulation network. The results constitute temporal information about signal propagation in the network, a simplified graphical representation of the network and of the signal propagation dynamics and a characterization of some signaling routes as regulation motifs.
Contact: simon.hardy{at}polymtl.ca
Supplementary information: Complete data of the Petri net model of the CaMKII regulation pathway available at http://www.polymtl.ca/rgl/Downloads.php
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1 INTRODUCTION |
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TOPABSTRACT1 INTRODUCTION2 METHODS3 RESULTS AND DISCUSSION4 CONCLUSIONACKNOWLEDGEMENTSREFERENCES |
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With high-throughput technologies, cell signaling research has produced a great deal of data on the signaling networks of cells. To understand how complex biological processes perform and how biological information is processed, some researchers are investigating how multiple components interact in a systemic manner. Part of this understanding will come from models developed by computational biologists, and a number of modeling and simulation tools have been developed in the last two decades specifically for biological applications. Some mathematical approaches are aimed at developing quantitative models that are numerically simulated (Eungdamrong and Iyengar, 2004). Other approaches are based on graph theory and ignore quantitative information to focus on the topology of the interactions between biological compounds in cellular networks (Mason and Verwoerd, 2007).
Petri net theory is used in another group of theoretical approaches to studying biological systems. This mathematical and graphical formalism was created by Petri (1962) to study systems with causal, concurrent processes. Since then, Petri nets have been further developed and applied to different types of systems, such as communication and industrial processes. Reddy et al. (1993) first applied this theory to biological systems. Meanwhile, a number of the theoretical tools and methods provided by Petri net theory have been used to study metabolic networks, signal transduction pathways and gene regulation networks (Goss and Peccoud, 1998; Li et al., 2006; Matsuno et al., 2006; Sackmann et al., 2006; Steggles et al., 2007; Voss et al., 2003). Because of the versatility of Petri net theory, Petri net approaches can be used to complete either qualitative or quantitative studies.
This article presents a new Petri net-based method for the analysis of the dynamics of signal propagation in the quantitative models of the signal transduction networks. This method combines invariant analysis (a structural analysis of the dynamic properties of Petri net models) and topological analysis to analyze simulation data. This method gives temporal information about signal propagation, produces a simplified graphical representation of the network and of the signal propagation dynamics, and it characterizes some signaling routes as regulation motifs. This method can be part of a unified Petri net framework used by computational biologists for modeling, simulation and simulation data analysis.
We apply this method to the Ca2+/calmodulin-dependent protein kinase II (CaMKII) regulation network, which is composed of several interconnected signaling pathways. The next section briefly presents the model of the CaMKII regulation network, gives a short introduction to Petri net basics, presents the signal dynamic analysis method and provides an example of it with a model of the calmodulin and calcineurin pathways. Section 3 presents the results of the analysis of the signal dynamics of the CaMKII regulation network.
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2 METHODS |
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TOPABSTRACT1 INTRODUCTION2 METHODS3 RESULTS AND DISCUSSION4 CONCLUSIONACKNOWLEDGEMENTSREFERENCES |
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2.1 The model of the Ca2+/calmodulin-dependent protein kinase II regulation pathway
The model of the CaMKII regulation pathway that is analyzed in this article is one of the three developed for a theoretical study of long-term potentiation in the hippocampal CA1 neuron (Bhalla and Iyengar, 1999). In the cited study, several kinetic models of signaling pathways were connected to form complex networks in order to investigate their signal processing capabilities. The experimentally observed link between persistently activated CaMKII in the postsynaptic neuron and increased synaptic responses was one of the motivations for conducting this quantitative study. The cyclic adenosine monophosphate (cAMP), protein kinase A (PKA) and protein phosphatase 1 (PP1) signaling pathways were connected to the calmodulin (CaM), calcineurin (CaN, also known as protein phosphatase 2B, PP2B) and CaMKII signaling pathways to create a single network, because earlier studies had shown evidence that the cAMP pathway gated CaMKII signaling through the regulation of protein phosphatases (Blitzer et al., 1998; Iyengar, 1996). The kinetic parameters of the model were derived from experimental observations, and are now available in the Database of Quantitative Cellular Signaling (Sivakumaran et al., 2003). The simulation data were analyzed with concentration plots to work out the dynamic behavior of the network. The modeled CaMKII regulation pathway is shown in Figure 4a.
We provide an example of the Petri net-based signal analysis method in this section using a subset of the CaMKII regulation pathway model. This subset model is composed of the CaM and CaN modules. The results section presents the conclusions of the signaling dynamic analysis of the complete model of the CaMKII network.
2.2 Petri net modeling
A Petri net is a directed, bipartite graph. It contains two kinds of nodes, places and transitions, connected by directed arcs. In biochemical Petri net models, places represent biological compounds, like enzymes or metabolites, and transitions represent chemical reactions. Arcs can only link one node of each kind. They indicate the causal relations between biological compounds and chemical reactions. These relations are weighted. In biochemical Petri net models, arcs are labeled with weights corresponding to the stoichiometric parameters of the reaction equations. Places can contain dynamic objects, called tokens, which represent a certain quantity of a chemical compound. The mark of a place corresponds to its number of tokens, thus to the number of molecules of the biological compounds.
Tokens are produced and consumed through the firing of transitions. A transition is able to fire if the marks of all its input places satisfy the token amount required by the weights of the arcs connecting these input places with the considered transition. This is the precondition of a transition. If a transition fires, it means that the chemical reaction has occurred. The weights of the outgoing arcs indicate the token amount that is added to the output places of the considered transition. This is the postcondition of a transition.
The distribution of tokens in the Petri net places is called the marking of the net and indicates the state of the system. For a biochemical model, the marking is the distribution of compounds. The initial marking is the marking of the start state.
The theoretical description of Petri nets given above is the description of the original theory (Petri, 1962). These nets are also called place/transition nets. In these, places contain a discrete quantity of tokens (marks are integers) and the firings of transitions are discrete events. Different extensions have been added to the original theory to augment the modeling possibilities of this formalism. In this article, we also use elements with continuous properties. Continuous places contain a continuous quantity; and the marks of continuous places are real numbers. The firing of continuous transitions is a continuous flow, and continuous transitions have a specified speed. A kinetics model can be modeled with continuous Petri net elements: the mark of continuous places represents the concentration of chemical compounds, and the marking-dependant speed of the continuous transitions can correspond to the reaction-rate equations of chemical reactions. For more detailed information on using Petri nets for biochemical modeling, see Hardy and Robillard (2004).
The model of the CaM and CaN modules of the CaMKII regulation pathway, shown in Figure 1, is an example of a biochemical Petri net model incorporating kinetic parameters. In this model, every place and transition is continuous. Tables 1 and 2 present the specifications of the Petri net model of the CaM and CaN modules. The continuous places are drawn as double circles, and the continuous transitions as white rectangles. The input places of a transition indicate that these biological compounds are the substrates of the chemical reaction, and the output places its products. This continuous model is mathematically equivalent to a model of ordinary differential equations, but with a formal graphical representation and numerous theoretical tools for model analysis.
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2.3 Invariant analysis
Invariants are dynamic properties of Petri nets, and are obtained from a structural computation. In biochemical models, they usually have a mass conservation meaning or represent a cyclic pathway. This subsection presents the theoretical background on Petri net invariants and the result of the invariant analysis of the CaM-CaN model.
The structure of a Petri net model, i.e. the arrangement of places, transitions and arcs, can be expressed in linear algebra by a two-dimensional matrix. This is the incidence matrix W. One dimension of the incidence matrix is the number of places of the model and the other dimension is the number of transitions. In place-transition nets, each element wij of the matrix indicates the token change at place i after the firing of transition j. In continuous nets, each element wij of the matrix indicates the flow from transition j to place i. The element wij is negative if the flow direction is from a place to a transition. The flow is the product of the speed of a transition and the weight (or factor) of the arc between the place and the transition. From the incidence matrix, it is possible to determine structural properties of Petri nets like invariants. Among all the attainable states of a model, called the reachable markings of a Petri net model, some quantities do not change, even when transitions are fired. This first type of invariant property is the marking invariant (p-invariant). Every p-invariant of a Petri net model is a positive vector x that is a solution of the following equation:
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A p-invariant characterizes a conservation component of the model. A conservation component is a set of places over which the weighted sum of the tokens is constant for every reachable marking. Relationships between p-invariants and biochemical modeling concepts have already been discussed. p-invariants express conservation relations of metabolites in metabolic pathway models (Voss et al., 2003; Zevedei-Oancea and Schuster, 2003). In signaling pathway models, p-invariants can represent a different kind of conservation relation (Sackmann et al., 2006). Biochemical processes in signaling pathways are performed by enzymes that change state to transmit a signal. The total concentration of all forms of an enzyme is modeled as a constant quantity. This quantity is a marking invariant of a Petri net model. Thus, the p-invariants and their associated conservation components identify all the places representing a specific form of an enzyme. The four conservation components of the CaM-CaN model are listed in Table 3. An abridged graphical representation of a Petri net model of a signaling pathway can be obtained by displaying its conservation components and their relationships as a graph. To create this representation, the conservation components and the places that are not part of any component are drawn as nodes. Then, if places from two different conservation components or individual places are linked through a transition in the Petri net model, a link is drawn between their associated nodes in the simplified graph. This representation based on conservation components will be used in the data analysis method presented in this article to display a dynamical portrait of the simulation data. For example, Figure 3 shows the simplified graph of the Petri net model of Figure 1. This graph is built using the conservation components listed in Table 3 plus the individual place Ca. The PKC component is not represented in Figure 3 because the information flux analysis of the simulation data will show that this component is inactive.
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Among all the possible firing sequences of a model, some can repeat themselves, creating a cycle of successive states. This second type of invariant is the firing invariant (t-invariant). The t-invariant of a Petri net model is a positive vector y that is a solution of the following equation:
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A t-invariant characterizes a repetitive component of a model. A repetitive component is a set of transitions causing a return to a previous state of a model. In other words, all the firings of the transitions of a repetitive component together have a null effect on the marking of the model. In Petri net models of signaling networks, the t-invariant property has been used for the identification methods of functional units (Sackmann et al., 2006) and of transduction activation components (Li et al., 2006). These last two model validation methods are similar: they decompose a discrete Petri net model into biologically significant subnets, thus providing insights into the architecture of signaling pathways. This qualitative validation is done during modeling using only the net structure. In other words, the models validated with these methods do not have to include the kinetic data necessary for simulation. Comparatively, the method presented in this article also uses the t-invariant property, but it is designed to be used subsequently to the simulation of models, during the analysis of the simulation data.
Because of the continuous components of the continuous and hybrid Petri nets, the t-invariant definition has to be adapted. The firing of a continuous transition is a continuous flow, not a discrete event. The conventional repetitive component corresponds to a sequence of discrete firings causing a Petri net to return in its initial state. But because continuous flows cannot be repeated like discrete events, the concept of a repetitive component having a periodic behavior makes no sense for most models. In discrete Petri net models of biomolecular systems, like in Sackmann et al. (2006) and Li et al. (2006), tokens represent molecules and the arc weights usually relate to the reactions stoichiometry. For these models, a repetitive component corresponds to a sequence of chemical reactions. In continuous Petri net models of biomolecular systems including kinetic parameters, like the model presented in this article, transitions have speeds corresponding to reaction rates and several transitions fire simultaneously. Instead of the repetitive component concept, we suggest using another concept also derived from t-invariant analysis: the steady component. From an unstable state, and in certain conditions, a continuous Petri net model evolves to a steady state. In a stable steady state, without any external perturbations, the marking of the model no longer changes. In this stable state, a subset of the solutions of Equation (2) lists the steady components of a model, which are the sets of transitions for which the total flow is null. To be part of this subset of solutions, these special t-invariants must be vectors composed only of 0 and 1 (a flow can either exist or not, it cannot be multiplied). Hence, the non-zero elements of these t-invariants correspond to the transitions composing a steady component. In the case of a continuous Petri net with marking-dependent speeds, Equation (2) must be solved with the incidence matrix filled with the steady state flow values in order to find the steady components. Linear combinations of the t-invariants are also solutions of Equation (2); however, only the vectors that do not include other solutions are searched. These are the minimal steady components. Definition 1 is the formal definition of the concept of a steady component.
Definition 1. Let K be a set of continuous transitions. The set K is a steady component if and only if OG-firings [U] (on go-firings: simultaneous firing of several transitions whose characteristic vector is u) exist such that:
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A simpler way to describe a steady component is to state that it is a set of continuous transitions for which the cumulative effect of their simultaneous firing on the marking of the Petri net is null. The minimal steady components of the CaM-CaN model are listed in Table 4.
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2.4 Identification of the information flux segments and nodes of a signaling network
To detect signal propagation through a signaling network, places and transitions must be identified to serve as measuring points. These are called information flux nodes and segments. Information flux segments are the portions of a pathway composed of places and transitions that might transmit or route a signal in the network. Information flux segments are positioned between two groups of one or more places described as information flux nodes. In other words, a segment is a subnet of the Petri net model. The flow circulating on an information flux segment is bidirectional. At steady state, the flow in each direction, on every possible pathway linking the nodes, is equal. This is the equal flow condition that will be used in Algorithm 1 to identify information flux segments and nodes. This condition is true only at steady state for the places and transitions of information flux segments. As we will see in the next subsection, a comparison of the flow in one direction with the flow in the opposite direction on a segment reveals the direction of a signal. The steady components of a Petri net model provide the initial subnets used in the search for the information flux segments of the signaling network.
The search for the information flux nodes and segments can be accomplished with the following algorithm:
Algorithm 1. Search for the information flux segments and nodes of a steady component.
- Let SC be a given steady component and SPN the subnet of a given Petri net with only the transitions of SC and all the places directly linked to these transitions. Generate B, the set of all the pairs of places from SPN. Initialize the set of information flux nodes IFN
Ø. IFN is a set containing pairs of sets of places.
- If B
Ø, then pull the pair <{b1},{b2}> from B and perform steps (i)–(v), otherwise go to (3).- Let D1 be the set of all the paths in SPN from b1 to b2 and D2 be the set of all the paths in SPN from b2 to b1. A path is a succession of places and transitions linked by arcs. The paths with a transition having an outgoing arc towards the initial place of the path are rejected. Initialize cumulative flow values F1-in = F1-out = F2-in = F2-out = 0. If D1 or D2 is empty, stop.
- Add to F1-in the value of the speeds of every transition directly linked to b1 in the paths from D2 (i.e. every transition ty that is part of the last place-transition-place triplet px
ty b1 of a path). For triplets where px is not b2 (i.e. for indirect paths), withdraw from F1-in the values of the speed of any transition tz in SPN in a place-transition-place triplet b1 tz px where place px is the same as one of the triplets of the first category.
- Add to F1-out the value of the speeds of every transition directly linked to b1 in the paths from D1 (i.e. every transition tm that is part of the last place-transition-place triplet b1 tm py of a path). For triplets where py is not b2 (i.e. for indirect paths), withdraw from F1-out the values of the speed of any transition tn in SPN in a place-transition-place triplet py tn b1 where place py is the same as one of the triplets of the first category.
- Add to F2-in the value of the speeds of every transition directly linked to b2 in the paths from D1 (i.e. every transition ty that is part of the last place-transition-place triplet px ty b2 of a path). For triplets where px is not b1 (i.e. for indirect paths), withdraw from F2-in the values of the speed of any transition tz in SPN in a place-transition-place triplet b2 tz px where place px is the same as one of the triplets of the first category.
- Add to F2-out the value of the speeds of every transition directly linked to b2 in the paths from D2 (i.e. every transition tm that is part of the last place-transition-place triplet b2 tm py of a path). For triplets where py is not b1 (i.e. for indirect paths), withdraw from F2–out the values of the speed of any transition tn in SPN in a place-transition-place triplet py tn b2 where place py is the same as one of the triplets of the first category.
- If F1-in = F2-out, F1-out = F2-in and these values are not null, then add <{b1},{b2}> to IFN. Go to (2).
- Let D1 be the set of all the paths in SPN from b1 to b2 and D2 be the set of all the paths in SPN from b2 to b1. A path is a succession of places and transitions linked by arcs. The paths with a transition having an outgoing arc towards the initial place of the path are rejected. Initialize cumulative flow values F1-in = F1-out = F2-in = F2-out = 0. If D1 or D2 is empty, stop.
- If two pairs <Bx, By> in IFN share an identical element (Bx and By are sets of places identified as information nodes at step (2), determine whether or not the two pairs also share the same transitions. If they do, form the union of the two different sets of places and replace the two pairs in IFN by the new unified pair.
In step 1 of the above algorithm, the main operation is the construction of the set B of all possible pairs of places. These places are part of the subnet SPN corresponding to the steady component SC. Also, the set IFN that will contain the information flux nodes of SC identified by the algorithm is initialized.
In step 2, each pair of sets of places <{b1},{b2}> of set B is tested with the equal flow condition to determine whether or not they are part of an information flux segment. In step 2i, the sets D1 and D2 are constructed to contain every possible path linking places b1 and b2 in SPN. In steps 2ii–2v, four cumulative flow values are computed; these values are the flows going in and out of places b1 and b2. In step 2vi, the cumulative flow values are compared to verify the equal flow condition. If it is verified, the pair is added to IFN. The execution of step 2 is illustrated in Figure 2a.
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In step 3, the pairs of sets of places in INF sharing an identical set are merged into a single pair only if they are part of the same segment. The execution of step 3 is illustrated in Figure 2b. Table 5 shows the results of the search for the information flux nodes and segments of the steady components of the CaM-CaN model. The segments of Table 5 are the subnets formed by the information nodes identified with Algorithm 1 and the path sets D1 and D2. A steady component can contain more than one segment, like steady component SC5 that has two information flux segments, S5 and S6.
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2.5 Analysis of the information flux of a signaling network
Once the information flux segments and nodes have been identified, these measuring points can be used, together with the topology of the network, to characterize the dynamics of a signal. At steady state, the difference between the flows in each direction on an information flux segment is zero. However, after a perturbation, this difference indicates the direction of the propagation of a signal, which can then be followed throughout the structure of the model. With formal definitions about the pathway structure of regulation motifs, such as positive and negative feedback loops, different types of signal routes can also be identified.
Here, we complete the introduction of the Petri net-based signal analysis method by presenting an analysis of the information flux caused by the inflow of calcium in the CaM-CaN model. The CaM-CaN pathway was modeled and simulated with the software Genomic Object Net (GON, commercially known as Cell Illustrator) (Nagasaki et al., 2003). The simulation period lasts 10 s. The calcium inflow occurs at the first second and lasts 1 s. This inflow perturbs the CaM-CaN model and a propagation of a signal ensues.
The speed differential of an information flux segment is defined as the difference between its cumulative flow values F1-in and F2-in. Figure 3 presents the normalized speed differential plots of the information flux segments computed with the data generated by the simulation of the CaM-CaN model. The plots show an increase in calcium binding with calmodulin and calcineurin, an increase in the dephosphorylation of neurogranin and a decrease followed by an increase in the binding of calmodulin with neurogranin. The speed differential plots give a first indication of the consequences of the calcium inflow, but with Petri net properties and the topological data of the network, a deeper analysis of the dynamics of the signal propagation can be performed and a graphical representation of the signal propagation can be generated.
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The initial data needed for the signal analysis of a Petri net model is the source of the signal and the duration of the stimulation. For the CaM-CaN model, the Ca place is the source and the stimulation lasts 1 s. At every simulation step of the stimulation period, the differential speeds of the information flux segments linked to the source of the signal are checked. For the CaM-CaN example, the speed differentials of segments S1, S2, S3, S4 and S7 are verified, because the Ca place is part of these segments. When the relative difference between F1-in and F2-in, is above an arbitrary threshold (10% in this example), the segment is considered to be propagating a signal. When this occurs, the speed differentials of other segments connected to the nodes of the activated segment are examined to determine whether or not the propagation continues. In our example, following signaling activation of segments S1, S2, S3, S4 and S7 due to the calcium inflow, the speed differentials of segments S8, S9 and S10 are evaluated. This assessment of the speed differentials of the connected information flux segments continues as the signal is propagated. In our example, following signaling activation of segments S8, S9 and S10, the activation of segment S6 and eventually of segment S5, is observed.
More than just observing successive signaling activations, the Petri net-based signal analysis method can yield temporal information about these activations and produce a simplified graphical representation of the network and of the signal propagation dynamics, as well as characterizing some signaling routes as regulation motifs. These three features of the method are shown in Figure 4. In this figure, the precise moments of the signaling activation of the information flux segments are indicated. Also, instead of displaying the detailed model of the network, this figure shows a simpler structure. As explained previously in Section 2.3, the results of the p-invariant analysis of the model provide the information to generate this view: the nodes of this abridged graph correspond to the conservation components of the Petri net model. Finally, the definitions of the graph structure of two regulation motifs, positive and negative feedback loops, are used to detect special signaling pathways. A positive feedback loop is identified when a signal is propagated to an information flux segment that was previously activated by the same signal. The signal then promotes the activation of the segment in the same direction in which it was initially activated. In other words, the signal loops back to sustain the signaling activation of an already activated segment. A negative feedback loop is identified in two situations. The first situation is when a signal is propagated to a segment that was previously activated by the same signal, and the signaling flux of the segment is then reversed. The second situation is when a signal is propagated to a segment in such a way that it competes with a connected segment already activated by this signal. In both situations, the signal loops back to hinder the signaling activation of an already activated segment. A negative feedback loop was identified in the CaN-CaM model and is shown in Figure 4d. The activation of the segment S6, (CaM + Ng CaM-Ng), competes with the activated segment S1 (CaM + 2 Ca CaM-Ca2) because they both share the CaM place. This process is known as the inhibition of calmodulin by neurogranin. In our example, this inhibition occurs at 2.5, 0.5 s after the end of the calcium inflow.
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3 RESULTS AND DISCUSSION |
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TOPABSTRACT1 INTRODUCTION2 METHODS3 RESULTS AND DISCUSSION4 CONCLUSIONACKNOWLEDGEMENTSREFERENCES |
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This section presents the results of the signal dynamic analysis of the CaMKII regulation pathway with a Petri net-based method. Due to the size of the specifications of the complete model, it is only available online. See the Supplementary Material section for hyperlink. The Petri net model of the CaMKII regulation pathway was simulated with GON for 800 s. A calcium inflow reproducing a tetanic stimulation is induced at the 60 s and lasts 1 s. A p-invariant analysis of the model resulted in the identification of 15 conservation components. Also, using the concentration values of the steady state of the model, 30 information flux segments have been identified. Figure 5 shows the dynamics of the signal propagation in the network. This figure is a simplified graphical representation of the Petri net model of the network produced with the p-invariant analysis data. It gives temporal information about the signaling activation of the information flux segments of the network. The initial and final states are not shown because they are steady states in which no signal is propagated (no arrows displayed). Figure 5 also shows the activity of three feedback loops. The first loop, the negative feedback loop leading to the inhibition of calmodulin by neurogranin, has already been discussed and is shown in an active state in Figure 5c. Another negative feedback loop has been detected: the increase of the degradation of cAMP into AMP because of the activation of phosphodiesterase (PDE) by PKA. This regulation motif is shown in an active state in Figure. 5c and d. The third loop is a positive feedback loop. It corresponds to the autophosphorylation cycle of CaMKII. This motif is active immediately following calcium stimulation at 61 s and lasts for more than 500 s. This loop is represented as a circular arrow on the CaMKII conservation component in every subfigure of Figure 5. The details of this motif and the cyclic activation of two information flux segments are hidden, because both segments are enclosed in the CaMKII conservation component.
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Figure 5 illustrates key moments of the signal propagation due to the Ca inflow in the CaMKII regulation network. Figure 5b shows the propagation of the signal immediately after the end of the stimulus. At 61 s, the CaM activation initiates the production of cAMP through the activation of adenylyl cyclases 1 and 8 (AC1/8), which in turn immediately starts activating PKA. CaM activation also starts the autophosphorylation cycle of CaMKII and with CaN, dephosphorylates Ng. Two seconds later, as shown in Figure 5c, the Ng and PKA negative feedback loops are active. Also, CaN has activated PP1 and the inhibited Inhibitor-1 protein (I1), which in turn has a deactivation effect on CaMKII. Figure 5d shows that the PP1 activation by CaN is overcome by the activation of PKA, because it causes the phosphorylation of I1 and therefore the inhibition of PP1. Figure 5e shows that the positive feedback loop of the CaMKII is still active, for a long period of time after the stimulus has been withdrawn.
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4 CONCLUSION |
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TOPABSTRACT1 INTRODUCTION2 METHODS3 RESULTS AND DISCUSSION4 CONCLUSIONACKNOWLEDGEMENTSREFERENCES |
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This article describes a method derived from Petri net theory to analyze the dynamics of signal propagation in a signaling network. The Petri net formalism is useful for depicting biochemical networks and performing simulations, but, more importantly, it provides techniques for the validation of the model and for the analysis of the simulation data. Because of its theoretical richness, Petri net theory offers a unified framework to computational biologists. The techniques that have been exploited in this article are the calculation of p-invariants (conservation components) and a special case of t-invariants (steady components) of a system. The method has been applied to the Ca2+/calmodulin-dependent protein kinase II (CaMKII) regulation network, an important pathway in the synaptic plasticity of neurons, and resulted in a new portrayal of the dynamics of the signal propagation in the network. We believe that this Petri net-based method will be of great value to computational biologists who need to rapidly interpret the simulation data of their signaling network models and gain insights into the systemic dynamics of signal processing components.
Understanding the dynamics of signal propagation in signaling network models is usually achieved with the analysis of the concentration plots of the simulation data. However, studying the concentration variations of a model yields more information about the changes of state of the model components than the dynamics of the signal propagation, and, as a result, the signal propagation dynamics have to be deduced from the data. The Petri net-based method presented in this article is a new analysis tool, which highlights, in a comprehensive way, the propagation of a signal in cellular network models and facilitates a system-level understanding.
The study of this method has been limited to models with a single signal source. Its application to the signaling dynamics of networks with simultaneous signals from multiple sources has yet to be performed.
New definitions of the graph structure of regulation motifs such as scaffolds and bifans can be added to this method. This will enrich its analytical possibilities and enhance its ability to help in deciphering the signal processing functions of cellular networks. Also, in the future, this method will be adapted to some particularities of hybrid Petri nets.
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ACKNOWLEDGEMENTS |
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TOPABSTRACT1 INTRODUCTION2 METHODS3 RESULTS AND DISCUSSION4 CONCLUSIONACKNOWLEDGEMENTSREFERENCES |
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This work was supported in part by NSERC grant A-0141 and an NSERC Postgraduate Scholarship. We are especially grateful to one reviewer for very useful comments.
Conflict of Interest: none declared.
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FOOTNOTES |
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Associate Editor: Martin Bishop
Received on July 8, 2007; revised on October 9, 2007; accepted on November 5, 2007
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REFERENCES |
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TOPABSTRACT1 INTRODUCTION2 METHODS3 RESULTS AND DISCUSSION4 CONCLUSIONACKNOWLEDGEMENTSREFERENCES |
|---|
Bhalla US, Iyengar R. Emergent properties of networks of biological signaling pathways. Science (1999) 283:381–387.
Blitzer RD, et al. Gating of CaMKII by cAMP-regulated protein phosphatase activity during LTP. Science (1998) 280:1940–1943.
Eungdamrong NJ, Iyengar R. Computational approaches for modeling regulatory cellular networks. Trends Cell Biol. (2004) 14:661–669.[CrossRef][Web of Science][Medline]
Goss P.JE, Peccoud J. Quantitative modeling of stochastic systems in molecular biology by using stochastic Petri nets. Proc. Natl Acad. Sci. USA (1998) 95:6750–6755.
Hardy S, Robillard PN. Modeling and simulation of molecular biology systems using petri nets: modeling goals of various approaches. J. Bioinform. Comput. Biol. (2004) 2:595–613.[CrossRef][Medline]
Iyengar R. Gating by cyclic AMP: expanded role for an old signaling pathway. Science (1996) 271:461–463.[CrossRef][Web of Science][Medline]
Li C, et al. Structural modeling and analysis of signaling pathways based on Petri nets. J. Bioinform. Comput. Biol. (2006) 4:1119–1140.[CrossRef][Medline]
Mason O, Verwoerd M. Graph theory and networks in Biology. Syst. Biol. IET (2007) 1:89–119.[CrossRef]
Matsuno H, et al. A new regulatory interaction suggested by simulations for circadian genetic control mechanism in mammals. J. Bioinform. Comput. Biol. (2006) 4:139–153.[CrossRef][Medline]
Nagasaki M, et al. Genomic Object Net: I. A platform for modeling and simulating biopathways. Appl. Bioinformatics (2003) 2:181–184.[Medline]
Petri CA. Kommunikation mit Automaten (in German). In: Ph.D. Thesis (1962) Bonn: Institut für Instrumentelle Mathematik.
Reddy VN, et al. Petri net representation in metabolic pathways. In: Proceedings of the First International Conference on Intelligent Systems for Molecular Biology (ISMB)—Hunter L, et al, eds. (1993) Bethesda, MD, USA: AIII Press. 328–336.
Sackmann A, et al. Application of Petri net based analysis techniques to signal transduction pathways. BMC Bioinformatics (2006) 7:482.[CrossRef][Medline]
Sivakumaran S, et al. The Database of Quantitative Cellular Signaling: management and analysis of chemical kinetic models of signaling networks. Bioinformatics (2003) 19:408–415.
Steggles LJ, et al. Qualitatively modelling and analysing genetic regulatory networks: a Petri net approach. Bioinformatics (2007) 23:336–343.
Voss K, et al. Steady state analysis of metabolic pathways using Petri nets. In Silico Biol. (2003) 3:367–387.[Medline]
Zevedei-Oancea I, Schuster S. Topological analysis of metabolic networks based on Petri net theory. In Silico Biol. (2003) 3:323–345.[Medline]
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