In literature, most researches related to the understanding of the behaviour of the cell networks focus only on some parts of the biomolecular network. However, to completely understand their behaviour over time we have to integrate the different parts of the biomolecular network and analyse them together. The objective of the present study is to propose to the biologist a platform to simulate the state changes of biomolecular networks with the hope of steering their behaviours. In this paper, we firstly present an efficient formalism to represent the dynamic behaviour of biomolecular networks. This logical model is based on the three levels of systems theory: structural, functional and behavioural modeling. We then propose a semantic approach based on four ontologies to formalise the domain knowledge of complex biomolecular networks. Both of these approaches provide the necessary elements to model, analyse and understand the dynamic behaviour and the transition states of these networks.
Systems biology is a comprehensive quantitative analysis of the manner in which all
the components of a biological system interact functionally over time [
Many formalisms have been proposed in recent years for the modeling of
biological networks. Among these formalisms, we can list the ordinary differential equations
[
In this paper, we firstly present a logical approach for modeling the dynamic behaviour of biomolecular networks. This formalism is based on the three levels of analysis of systems theory: structural, functional and behavioural modeling. Second, we propose a semantic approach to formalize the domain knowledge of complex biomolecular networks. This approach based on four ontologies provides the necessary concepts for modeling the dynamic behaviour and the state transitions of these networks.
This paper is organised as follows: Section 2 details the logical formalisation for describing the logical structure, function and behaviour of complex biomolecular networks. Section 3 is devoted to present the semantic approach that provides a rich description for modeling a biomolecular network and its state changes. Finally, in Section 4, we provide conclusions and future works. 2
In this section, we detail the mathematical formalism that will allow to translate, in
logical terms and under certain assumptions, the dynamicity of a complex biomolecular
network. The first parts of this approach have been presented in [
Indeed, a biomolecular network is a dynamical system that can be characterised by
the triple (be, do, become) describing its structure, function and behaviour according to
the systems theory [
The behavioural modeling: to describe how the biomolecular network and its individual components evolve over time.
Thus, the biomolecular network BN can be represented by its structure SR, its function FR and its behaviour CR[t0;tn] that evolves over time t (a simulation interval [t0; tn]). Therefore, mathematically, the biomolecular network BN is defined as follows:
BN = (SR; FR;CR[t0;tn])
For lack of space, we will not detail the structural SR and the functional FR modeling.
We will just remember their basic notation. For more details about these parts, please
see our previous work [
The structure of the biomolecular network SR = (M; I) is a graph defined by:
– M= fm1; m2; : : : ; mng denotes all the molecules composing the network. M is partitioned as: MG the set of genes, MP the set of proteins and MM the set of metabolites. – I= fi1; i2; : : : ; img denotes the set of interactions between the network’s molecules.
Thus, for an edge i 2 I, we note s(i) the starting node and d(i) the destination node. The partition of the graph nodes induces a partition of the different types of interactions. We have three interactions between molecular components of the same type, four interactions between the nodes belonging to different networks and two interactions IGM and IMG are not taken into account because there is no direct interaction between the genes and metabolites and vice versa.
The partition of M and I are detailed in [
Time Discretisation. Complex biomolecular networks are dynamic systems characterised by continuous interactions. The activity of these interactions can be modeled in the form of differential equations. However with the large size of these networks, solving these differential equations in continuous time leads to very significant practical difficulties and heaviness in their implementation.
Thus, in order to model the dynamic evolution of a biomolecular network and reproduce its behaviour over time, we proceed with a behavioural simulation in discrete time. This simulation allows to study the behaviour of biomolecular network through successive transitions. The state of a node (representing a cell component) at the next generation is calculated according to its state and the state of its predecessors at the current generation as well as the possible influence of each one of its incoming edges. The node states evolve synchronously. The results of the simulation will be tested and validated by biologist experts.
State of the Network. The state of the network at a given time if defined by a function en(n; t) which associates to each node its state at the moment t.
en en : (m; t) 7! (Activation 2 fTrue; Falseg [cm(t)] 2 R if m 2 MG: if m 2 MP [ MM : – For all m 2 MP [ MM : en(m; t) = [cm(t)] 2 R where: cm(t): the value of the concentration of the molecule denoted by the node m at a given time t. – For all m 2 MG: en(m; t) = Activation where: Activation 2 fTrue; Falseg.
Associating a gene with a concentration is not meaningful. Instead, a gene may have two specific states, activated or not.
We can define ER(t) the state of the biomolecular network at an instant t with a set representing the states of all components in the network at any given time t.
ER(t) = hen(m1; t); en(m2; t); :::; en(mn; t)i Transition of the Network State. For a node m 2 M, we define ie(m) (resp. oe(m)) the set of incoming edges (resp. outgoing edges) on m, defined as follows: ie(m) = fi j d(i) = mg and oe(m) = fi j s(i) = mg We also define Pred(m) the set of predecessor nodes on the node m such as:
Pred(m) = fn 2 M; 9i 2 I j s(i) = n and d(i) = mg
The state of a node at time t + 1 depends on its state at time t, as well as the possible influence of each one of its incoming edges. This influence obviously depends on the state of the starting node of the arc in question.
For each node m, we define an aggregate function Am (relating to the node m) which calculates the evolution of the node status between two successive instants of the simulation. This aggregate function Am depends on the current state of the node m, the state of its predecessor nodes Pred(m) and the characteristics of its incoming edges ie(m).
en(m; t + 1) = Am(en(m; t); ie(m); en(n; t) ; n 2 Pred(m)) Network Steering. A state transition in the network occurs by changing at least one of its nodes. The changes of a node state (that is, changes in the concentration of the molecule) can occur either by an internal stimulus (for example, due to reactions that are internal to the cell) already seen with the aggregate function (Section 2.1) or by an external stimulus generated outside the cell (for example, because of a medicine taken by the patient). Indeed, a stimulus is an event that can cause changes in the state of the molecule where it operates and therefore to change the state of all the biomolecular network (changing a node automatically modifies other network nodes).
An external stimulus S is a triplet [t; m; Dc], where: – t is the time of introduction of the stimulus S. – m is the node targeted by the stimulus S. – Dc is the change in concentration caused by the stimulus S and which depends on the type of the node:
If m 2 MG, Dc determines the activation or deactivation of a gene: Dc 2 fActivated; Deactivatedg.
Else, if m 2 MP [ MM, Dc represents the change of the concentration caused by the stimulus S: Dc 2 R .
We denote ER(t), with t 2 N, the state of the network at time T (t) = t0 + t:DT (where DT is the time step and t0 the initial time of the simulation).
To simulate the different transition states of a biomolecular network, we give a state ER(0) at time t0 and a time step size DT . Then the successive states ER(t + 1) are calculated from the current state ER(t) according to the interactions and the aggregate functions defined by the network, and the external stimuli.
At a given time t + 1, for each m 2 M we have: – If there are no external stimuli in time t for the node m then: en(m; t + 1) = Am(en(m; t); ie(m); en(n; t)) – Else
If m 2 MG: Else (if m 2 MP [ MM):
en(m; t + 1) = Dc en(m; t + 1) = Am(en(m; t); ie(m); en(n; t) ) + Dc where: n 2 Pred(m) Behaviour. The behaviour of the biomolecular network CR[t0;tn] is given by the sequence of its successive states during the simulation time.
CR[t0;tn] = [ER(0); ER(1); :::; ER(n)] Indeed, the behaviour of the network extends between two distinct instants t0 and tn forming the simulation interval [t0; tn]. 3
Modeling the behaviour of complex biomolecular networks requires, first and foremost,
to formalize the domain knowledge. However, it is not sufficient to simply describe it.
Certainly, the behaviour of biomolecular networks is investigated through appropriate
semantic structures for the description of their components that must not be overlooked.
Thus, the use of a formalized language such as ontologies provides a rich description
but also allows to perform reasoning. To do this, we propose a semantic architecture
composed of four ontologies: three of them already exist in the literature, the Gene
Ontology (GO) [
In this study, the Gene Ontology6 is considered as a core ontology. It ensures the
description and the classification of cellular components. It provides a structured
terminology for the description of gene functions and processes, and the relationships between
these components [
We chose to use the Gene Ontology for the following reasons, (1) it is an initiative of several genomic databases such as the Saccharomyces Genome database (SGD), the Drosophila genome database (FlyBase), etc. to build a generic ontology for describing the role of genes and proteins, (2) it is the most developed and most used in biology (since 2000), and (3) it provides annotation files about large number of cellular entities.
3.2
The Simple Event Model ontology7 proposed by Van Hage et al. [
We chose to use the Simple Event Model ontology because it provides the necessary concepts to describe and model events in various subject domains. 3.3
The Time ontology8 developed by Hobbs and Pan [
The main classes of this temporal ontology can be summarized as TemporalEntity which consists of two sub-classes Instant and ProperInterval, DurationDescription, DateTimeDescription, TemporalUnit, etc. Also, it contains several proprerties such as hasDurationDescription, intervalStarts, hasDateTimeDescription, etc.
We chose to use the Time Ontology because of its basic structure that is not specific to a particular application and because it is simple to adapt it in our context. 3.4
To study the dynamic behaviour and the transition states of a biomolecular network, it is required to model its domain knowledge. Therefore, we developed the Biomolecular Network ontology. This ontology is the major contribution of this paper, it is intended to describe exhaustively the field of complex biomolecular networks by describing the static aspect of its structure. It was defined in collaboration with domain experts.
Figure 1 presents the Biomolecular Network ontology. We use the graphical
notation for OWL ontologies defined by Brockmans et al. [
These types of networks can be connected to the other ontology concepts through three properties, has node that depicts its cellular components, has interaction that describes the interactions linked to its components and the property has node only that specifies exactly the nature and type of its components. – The class Node: This class contains the different types of cellular entities M that constitute the biomolecular network. In fact, we can identify three sub-classes: the Gene which describes the set MG, the Protein which models the set MP and the Metabolite which describes the MM. This class is connected with the Node State through the property has state. – The class Interaction: This class covers all the diverse types of interactions that can be operated among the nodes of the biomolecular network. This class consists of two sub-classes, Intraomic Interactions that covers the interactions between molecular components of the same type and the class Interomic Interaction that describes the interactions between molecular components of the different type. This class is connected to the Node class via two properties, has source and has end. – The class Node State contains the possible states of the nodes. This class is composed of two sub-classes, the Concentration and the Activation. – The class Interaction Type allows to specify the types and the nature of the interaction among cellular components. This class is linked to the BNO:Interaction class through the properties Has type. To successfully integrate the main Interaction ontology concepts (IO:Activity flow and IO:Process) with the Biomolecular Network ontology, we create an abstract BNO UML BNO:Interaction Type to generalise those two Interaction ontology concepts (Figure 1). 3.5
Concepts in the Biomolecular Network ontology are linked to the Gene ontology classes. In fact, the concepts of the Gene Ontology are used to enrich the definitions of the concepts of the Biomolecular Network ontology by an equivalence relation owl:equivalenceClass. For example, as described in Figure 2(b), after inference the concept BNO:Protein will be specialized by the concept GO:beta-galactosidase (GO: 0009341) because the BNO:Node concept is equivalent to the concept GO:cellular component (GO: 0005575). The Biomolecular Network ontology is also linked with the Simple Event Model ontology through the BNO: Node concept, in fact an SEM:event can stimulate a molecular entity (represented by the concept BNO: Node). The Simple Event Model ontology will be used to describe the states of BNO:Node and its behaviour.
Moreover, the Time ontology (TO) has been integrated in the Simple Event Model ontology. The concept sem:Time was made equivalent to the concept TO:TemporalEntity which represents the root of the Time ontology. Hence, the property sem:hasTime will connect the Simple Event Model ontology to the Time ontology and, as a consequence, the diverse types of temporal concepts will be defined as specializations of the class sem:Time. Figure 2(b) shows a use of this principle. Thus, we can exploit the wealth of temporal concepts provided by this temporal ontology to describe the SEM:event class. Using these relationships it is possible to merge these ontologies to formalize the necessary knowledge to study the state changes of the biomolecular network’s behaviour. This paper proposes an effective approach for analysis and understanding the behaviour of complex biomolecular networks over time. This approach combines both of a logical modeling of biomolecular networks and a semantic approach that consists on four ontologies merged together.
We present a logic-based approach for modeling the dynamic behaviour of biomolecular networks. This formalism is based on the three levels of analysis of systems theory: structural, functional and behavioural modeling.
Moreover, the use of a semantic approach based on merging different ontologies can overcome issues of study the state changes of the complex biomolecular networks and their behaviour. We develop the Biomolecular Network Ontology (BNO) to describe the static structure of complex biomolecular networks and merge it with the Gene Ontology (GO) to provide structured terminologies for the description of cellular components. We also chose the Simple Event Model Ontology (SEMO) to describe events and stimuli which can stimulate the network’s components and integrated the Time Ontology (TO) to study the different states of the biomolecular network and its nodes over time.