Qualitative dynamics semantics for SBGN process description

Abstract

Background: Qualitative dynamics semantics provide a coarse-grain modeling of networks dynamics by abstracting away kinetic parameters. They allow to capture general features of systems dynamics, such as attractors or reachability properties, for which scalable analyses exist. The Systems Biology Graphical Notation Process Description language (SBGN-PD) has become a standard to represent reaction networks. However, no qualitative dynamics semantics taking into account all the main features available in SBGN-PD had been proposed so far.

Results: We propose two qualitative dynamics semantics for SBGN-PD reaction networks, namely the general semantics and the stories semantics, that we formalize using asynchronous automata networks. While the general semantics extends standard Boolean semantics of reaction networks by taking into account all the main features of SBGN-PD, the stories semantics allows to model several molecules of a network by a unique variable. The obtained qualitative models can be checked against dynamical properties and therefore validated with respect to biological knowledge. We apply our framework to reason on the qualitative dynamics of a large network (more than 200 nodes) modeling the regulation of the cell cycle by RB/E2F.

Conclusion: The proposed semantics provide a direct formalization of SBGN-PD networks in dynamical qualitative models that can be further analyzed using standard tools for discrete models. The dynamics in stories semantics have a lower dimension than the general one and prune multiple behaviors (which can be considered as spurious) by enforcing the mutual exclusiveness between the activity of different nodes of a same story. Overall, the qualitative semantics for SBGN-PD allow to capture efficiently important dynamical features of reaction network models and can be exploited to further refine them.

Keywords: Automata networks; Modeling of dynamics; Qualitative dynamics; Reaction networks; SBGN-PD.

Fig. 1

Reference card of the SBGN-PD language from [7]. Every glyph of SBGN-PD is associated to the biological concept it represents

Fig. 2

An example of asynchronous automata network and its transition graph. Top: an asynchronous automata network composed of the 3 automata a, b and c. Automata are represented by labeled boxes, and their local states by circles identified with the ticks. For instance, the circle ticked 1 in the automaton a is the state 1 of a, noted a ₁. Local transitions are represented by directed labeled edges, where the labels indicate the set of conditions that have to be satisfied for firing the transition. The local states in blue represent a potential global state of the automata network: the state a ₀,b ₁,c ₀. Bottom: the transition graph of the asynchronous automata network, from the global initial state represented in blue. This graph represents all transitions that can be successively fired from the global initial state. For example, from the global initial state, it is possible to fire the transition labeled $l_{a_1}$ or the transition labeled $l_{\overline{b}}$. One of these two will be fired non-deterministically. Firing transition $l_{a_1}$ will change the state of a from 0 to 1, hence replacing a ₀ with a ₁ in the global state of the network, becoming a ₁,b ₁,c ₀. Firing transition $l_{\overline{b}}$ will change the state of b from 1 to 0, hence replacing b ₁ with b ₀ in the global state of the network, becoming 〈a ₀,b ₀,c ₀〉

Fig. 3

A SBGN-PD process modeled by an asynchronous automata network under the general semantics. Top: An example of SBGN-PD map. The legend of the map is given by the SBGN-PD reference card reproduced in Fig. 1. Bottom: the asynchronous automata network modeling the SBGN-PD map under the general semantics, with the different automata and for each transition, its firing conditions. The global initial state <a ₁,a t p ₁,b ₁,m ₁,a P ₀,a d p ₀,c ₀,p ₀,q ₀> is represented in blue

Fig. 4

Stories of an SBGN-PD map. Top: The SBGN-PD map of Fig. 3. Bottom: the set of possible (non-singleton) stories respecting constraints (i-iv) and the maximally valid sets of stories. Final sets are colored in blue, and epn-maximal sets are colored in green. Each of the constants a, b, c, adp and atp denotes the EPN whose label equals that constant. Constant aP denotes the phosphorylated macromolecule labeled “a”

Fig. 5

AT _1AR-mediated ERK activation map. This map represents the two main pathways responsible for the AT _1AR-mediated (and more generally 7TMRs receptors-mediated) ERK activation. The AT _1AR receptor activates the (classical) G protein pathway to reach ERK but also the less known β-arrestins pathway. These pathways are tightly regulated by the presence of molecules called the G-protein coupled receptor kinases (GRK2/3 and GRK5/6), which act directly on the phosphorylation of the receptor. This map is represented using the SBGN-PD language. EPNs with bold borders constitute the initial state of the map. Every colored EPN belongs to a story, and each color is assigned to a different story. The story in pink focuses on the receptor HR and comprises seven different physical states of this receptor: unbound, phosphorylated (on either of two sites), bound to β-arrestin 1 or β-arrestin 2. The other stories focus on ERK (in yellow), on protein G (green), on PIP2 (blue), and on PKC (gray)

Fig. 6

A SBGN-PD map modeled by an asynchronous automata network under the stories semantics. Top: the SBGN-PD map from Fig. 3. We chose a final set of two stories $\mathfrak{𝔰}$ and $\mathfrak{𝔱}$, whose EPNs are colored in yellow or blue, respectively. Bottom: The corresponding asynchronous automata network using the stories semantics with the stories $\mathfrak{𝔰}=\{a,\mathit{\text{aP}},c\}$ and $\mathfrak{𝔱}=\{\mathit{\text{adp}},\mathit{\text{atp}}\}$. Each of the constants a,c,a d p and atp denotes the EPN whose label equals that constant. Constant aP denotes the phosphorylated macromolecule labeled “a”. The global initial state is represented in blue. Note that here, p # q. Therefore q ₀∈l _p and p ₀∈l _q

Fig. 7

Transition graph for a dynamical model built under the general semantics. Transition graph of the asynchronous automata network of Fig. 3, modeling the SBGN-PD map of Fig. 3. Each node represents a global state of the asynchronous automata network. There is a directed edge from a state S to a state S ^′ iff S ^′ is reachable from S. Circled states are point attractors. States colored in blue are all states present in the transition graph of the asynchronous automata network modeling the same map under the stories semantics (Fig. 8)

Fig. 8

Transition graph for a dynamical model built under the stories semantics. Transition graph of the asynchronous automata network of Fig. 6, modeling the SBGN-PD map of Fig. 6. Each node represents a global state of the asynchronous automata network. There is a directed edge from a state S to a state S ^′ iff S ^′ is reachable from S. Circled states are point attractors

Fig. 9

RB/E2F map. This map represents the regulation of the cell cycle by E2F/RB. The cell cycle is a succession of four phases (G1, S, G2 and M phases) that are tightly regulated by so-called pocket proteins, whose main representative is the RB protein. The RB protein major function is to inhibit transcription factors belonging to the E2F family, and in particular the E2F1 protein. Diverse cyclin dependent kinases (CDKs) play a key role in the regulation of the cell cycle. In particular, CDKs’ function is to phosphorylate the RB protein, decreasing its inhibiting effect on E2F transcription factors. This map is represented using the SBGN-PD language. EPNs with bold borders constitute the initial state of the map. Every colored EPN belongs to a story, and each color is assigned to a different story

Fig. 10

Workflow of the method. Rectangles represent objects and ellipses tasks. Pink processes are those developed as part of this work; blue processes are users’ interventions; yellow processes are tools publicly available that were used for this study