Combining DEVS simulation and ontological modeling for hierarchical analysis of the SARS-CoV-2 replication

2.1. Life cycle of SARS-CoV-2

As illustrated in Figure 1, the SARS-CoV-2 genome consists of a positive-sense single-stranded RNA, about 30 kilobases coated with N protein and covered by a lipid bilayer containing spike S, membrane M, and envelope E proteins.^14,15 The SARS-CoV-2 life cycle includes four main stages: the virus entry, translation of the viral replicases, genome replication and transcription, structural and accessory proteins’ expression, virion assembly and release: ¹⁶

• Virus entry: the virus enters the target cell either by endocytosis or through viral/plasma membrane fusion. In both cases, this entry is handled by the attachment of the spike glycoprotein (spike S) present on the surface of the virus envelope to the membrane of the host cell through its angiotensin-converting enzyme 2 (ACE2) receptors. ¹⁷

• Translation of viral replication machinery & replication: once released in the host cell cytoplasm, the viral genome is translated by host ribosomes to produce two large replicase polyproteins (pp1a and pp1ab), which generate through the proteolytic process, sixteen non-structural proteins (nsp1 to nsp16) that have a critical role in viral RNA synthesis, and structural proteins. These 16 non-structural proteins form the viral replication and transcription complex $(\mathrm{RTC})$ and the viral RNA-dependent RNA polymerase $(\mathrm{RdRp})$ $\mathrm{nsp}12$ in association with its two viral co-factors $\mathrm{nsp}7$ and $\mathrm{nsp}8$ as the catalytic core of RNA synthesis. The function of the RTC is to replicate a negative full-length genome and transcribe a set of negative-sense subgenomic $(\mathrm{sg})$ RNAs. In turn, these will serve as templates to produce a new positive-sense RNA genome and nine subgenomic messenger ribonucleic acid mRNAs $(\mathrm{sgmRNAs})$ encoding structural and accessory proteins. ¹⁵

• Production of the viral structural proteins and virus egress: at late stages of infection, the structural proteins in spike $(S)$ , envelope $(E)$ , membrane $(M)$ , and nucleocapsid are translated from the sg mRNAs by the cellular ribosomes, in the endoplasmic reticulum $(\mathrm{ER})$ and retained at the site of budding in the ERGIC. The viral RNA genome coated with N protein ultimately bud into ERGIC membranes, which are decorated with M, E, and S proteins, producing new virions. ¹⁵

• Egress of Virions: Newly enveloped virions leave the cell by the exocytic pathway or by cell lysis at the ERGIC compartment, they exit the infected host cell by exocytosis through the lysosomal trafficking pathway, budding, or by cell lysis. ¹⁵

Accordingly, the SARS-CoV-2 life cycle and its interaction with the human organism can be considered as a complex multicomponent hierarchical dynamical system composed of different scales, the micromolecular scale comprising the proteomic and metabolomic entities, while the macromolecular scale consisting of the genomic and transcriptomic entities. These entities can be transferred from one scale to another one, making their modeling and simulation very complex.

2.2. Modeling and simulation: DEVS formalism

The modeling and simulation theory (M&S) is used to mimic the operation of an existing or proposed system to understand and analyze its dynamic behavior in a simplified replica within a risk-free environment. ¹⁸ This theory involves two separate activities. The first modeling activity focuses on making a representation of a system from an observer’s point of view. It answers all the questions the observer may have about the structure and function of the system. While the second simulation activity focuses on executing the model to produce its behavior by modifying its inputs and parameters. ¹⁸ Various M&S formalisms have been developed, such as the Differential Equation System Specifications (DESS) for traditional differential equation systems or the Discrete Time System Specifications (DTSS) for automata. In our study, we choose to use the Discrete Event System Specification (DEVS) formalism that was initially proposed by Bernard P. Zeigler ¹⁹ in the 70s as a discrete-event modeling specification formalism. This formalism supports the modeling and simulation of complex systems by defining them as hierarchical, modular models that can be easily reused. Accordingly, a system is defined as a composite of sub-models, each of them being behavioral (atomic) or structural (coupled), as seen in Figure 2. Coupled models can be embedded in a model hierarchy. Both models consist of a time base, inputs, states, outputs and functions to compute the next states and outputs.

As shown in Figure 2(a), a DEVS atomic model is defined by this seven-tuple equation (1):

where $X$ the input events set, $Y$ the output events set, S the state set, ${\delta }_{\mathrm{int}}$ the internal transition function $(S\to S)$ , ${\delta }_{\mathrm{ext}}$ the external transition function $(Q\times X\to S)$ , with $Q=\{(s,e),s\in S\mathrm{and}e\in [0,\mathrm{ta}(s)]\}$ , $\lambda$ the output function $(S\to Y)$ , and $\mathrm{ta}$ the elapsed time function $(S\to R_0^{+}\cup \infty )$ .

A DEVS coupled model, as illustrated in Figure 2(b), may include several sub-models (atomic and/or coupled models) to define the hierarchical structure of the system. It is defined by equation (2):

where: $X$ the set of input events, $Y$ the set of output events, $D$ an index for the components of the coupled model, $M_d$ a basic DEVS model $d\in D$ (atomic or coupled) defined by $M_d=〈X_d,Y_d,S_d,{\delta }_{\mathrm{int}},{\delta }_{\mathrm{ext}},\lambda ,\mathrm{ta}〉$ . $\mathrm{EIC}$ the external input coupling, $\mathrm{EOC}$ external output coupling, $\mathrm{EIC}$ the internal coupling, and $\mathrm{select}$ the tie-break selector.

Various extensions of DEVS-based simulators have been proposed, ²⁰ including the fwkDEVS ²¹ a DEVS/GLE simulation framework designed in java and supporting GDEVS simulations. DEVSJAVA environment ²² designed in java supports parallel execution on a uni-processor. This simulator can also be linked with DEVS/HLA and DEVS/CORBA frameworks. VLE (for Virtual Laboratory Environment) is a multi-modeling and simulation platform. It uses C++ to create simulation models and Python to use simulation as a scripting language for data analysis. ²³ JDEVS ²⁴ is implemented in java and enables discrete event, object-oriented, GIS-connected, and visual simulation model development. PythonPDEVS ²⁵ is a Python-based simulation framework for classic and parallel DEVS. ADEVS ²⁶ is also distinguished by its support for both discrete event and continuous dynamical systems. CD++  ²⁷ is a general framework for modeling and simulation of classic, parallel and cellular DEVS models. Written in C++, the CD++ toolkit has been widely used to model and simulate a variety of applications. ²⁸ Moreover, CD++ was revised and extended several times.^29,30 A more complete list of DEVS-based simulators is available on the Advanced Real-Time Simulation Laboratory web page: https://arslab.sce.carleton.ca/index.php/devs-tools/. As can be seen, there is no better simulation tool, and they cannot be compared. Since each tool uses its own language and can be used for specific purposes. A possible way to compare them is by implementing an appropriate test model in the different simulation frameworks and then comparing the performance of each simulator, but it remains an expensive and time-consuming task. The proposed DEVS models in this study will be implemented in the CD++ simulator. ²⁷

2.3. Ontologies and SWRL rules

Over the last few decades, there has been increasing use of ontological modeling in different domains, especially in health and biological data management. Ontologies are historically rooted in philosophy as the study of being, including their main categories and relations. In computer science, an ontology is a way of showing the properties of a subject area and how they are related, by defining a set of concepts and categories that represent the subject. ³¹ Concepts represent explicit classes, used to describe the main entities of a domain of interest. They are usually organized in taxonomies through which inheritance mechanisms can be applied. Properties represent a type of association between those concepts. An ontology uses axioms, which are assertions (including rules) in a logical form that together comprise the overall theory that the ontology describes in its domain of application. It also contains instances that are used to represent elements or individuals, the basic objects of the ontology.

Formally, an ontology is defined by this quadruplet ³² equation (3):

where, $C$ is the set of classes describing the concepts of the domain of interest, $P$ is the set of properties describing the individuals of the classes $C$ , $\mathrm{Sub}$ is the subsumption relation function $(C\to 2^C)$ with $2^C$ the power set of $C$ , and $\mathrm{Ass}$ the classes-properties (object and data properties) association function $(C\to 2^P)$ .

All of these components are well-defined by the domain expert. The formalization of knowledge representation ontologies can be done using different formalisms or languages, such as first-order logic (FOL), eXtensible Markup Language (XML), Resource Description Framework (RDFS) or the World Wide Web Consortium (W3C) standard Web Ontology Language (OWL). ³³ Thus, a formal ontology is computer-readable, allowing the computer to “understand” the relationships, the “formal semantics,” of the ontology. By incorporating expert rules, written as antecedent-consequent pairs, ontologies can generate logical inferences of relevant knowledge. ³³ These knowledge rules can be constructed using the Semantic Web Rule Language (SWRL). ³⁴ In this study, we use the OntoRepliCov ontology, ¹¹ an ontology dedicated to the description of the coronavirus replication process, to provide semantic knowledge about the CoV genome organization, SARS-CoV-2 sequences and the different viral proteins produced.

4.1. The DEVS module

The first module of our proposed approach is based on the DEVS framework to mathematically model and simulate the SARS-CoV-2 replication process. This DEVS module determines the structures, behaviors, and dynamic evolution of the molecular entities involved in the SARS-CoV-2 replication process, and uses atomic and coupled models to model them. This module consists of two submodules (Figure 3), the DEVS model and DEVS simulation engine. As stated in Section 2.2, the proposed DEVS module is based on the CD++ modeling and simulation toolkit. ²⁷ Based on the abstract simulator concepts proposed by, Zeigler et al. ¹⁹ CD++ provides an environment to (1) design discrete events conceptual models and (2) simulate them. It consists of main classes Models and Processors. The first class defines the conceptual DEVS models. Using instance variables, the Model can: identify its associated processor, identify the link to the coupled model containing this model (parent), and specify the interaction of the model with other models (inport and outport). ⁷⁰ The atomic model is implemented with its own methods namely the internal transition, external transition, output, and time advancement methods. In our case, as the aim of our work is to combine DEVS formalism with ontological modeling, these methods are related to the SWRL rules engine. The aim is that these functions should interact with the inference module of the ontological module and request to execute the rule(s) that define the desired behavior (or process). A coupled model is defined by determining its components, and their coupling relationships. The second Processor class implements the simulation mechanism. We found different types of simulation processors, namely Simulators, Coordinators, and the Root-coordinator. The processors ensure the execution of the abstract simulation procedures through the implementation of the theoretical DEVS concepts. These processors are related to the models: a simulator is associated with an atomic model, and a coordinator with a coupled model. In this way, both simulators and coordinators ensure the functioning of the atomic and coupled models, respectively. The root processor manages the simulation. Indeed, it ensures the beginning and end of the simulation, the connection of the simulator with the environment in terms of passing external events/outputs from/to the environment, and the advancement of the simulation clock. ⁷⁰ More details about these classes and DEVS algorithm simulation can be found in Wainer and colleagus.^27,70

The SARS-CoV-2 replication can be seen as a complex system involving a number of cellular and viral components at different scales. We, therefore, propose a hierarchical DEVS model of the SARS-CoV-2 replication process, consisting of a coupled model representing the host cell. This coupled model is itself formed by three coupled models, each one representing a level of the cell, respectively, the cell membrane, the cytoplasm, and the secretory pathways. An overview of the DEVS hierarchical representation of the SARS-CoV-2 replication machinery is displayed in Figure 4. Each of these coupled models is composed of a set of atomic models. The ten atomic models forming the entire SARS-CoV-2 replication system will be detailed in the next section (Section 5).

4.2. The ontological module

This module includes a rule-based system. It consists of an ontology, the OntoRepliCov ontology, ¹¹ and a set of expert reasoning rules. The OntoRepliCov ontology has been specially designed and developed to describe the micro-level of the replication machinery. More specifically, it handles the translation and the transcription sub-processes of the machinery where chains of proteins and codons (amino acids: a combination of three nucleotides) are produced. Modeling these sub-processes requires a description of the virus organization at the micro-level. Therefore, OntoRepliCoV provides a hierarchical formalization of elements representing this micro-level organization, such as codons and nucleotides. This formal model is also enriched with rules to infer knowledge (nucleotides, codons, polyproteins) resulting from the different stages of the sub-processes that consider elements from the virus organization. The Inferences are based on symbolic artificial intelligence and developed for semantic web technologies. We exploit these inferences to enrich the DEVS simulation engine with a micro-level description.

As described in Table 1, the OntoRepliCov ontology consists of eighty-four classes. Biologically, these classes can be categorized into two distinct groups: those describing the hierarchical organization of the genome represented by the concept Genome, and those describing the replication elements of the virus represented by the concept Replication $\_$ element.

The first group, describing the viral genome, consists of seven subclasses: Global $\_$ codon, Accessory $\_$ protein, Open $\_$ reading $\_$ frame, Polyprotein, Stem $\_$ loop, Non $\_$ structural $\_$ protein, and Structural $\_$ protein, Nucleobase. The second group of classes, represented by the Replication $\_$ element concept, describes all elements needed for the replication process. This concept is further decomposed in Subgenomic $\_$ RNA, Replication $\_$ Transcription $\_$ Complex, m $\_$ RNA, t $\_$ RNA, Subgenomic $\_$ Positive $\_$ RNA, Subgenomic $\_$ Negative $\_$ RNA, Amino $\_$ acid, and so on. All of these classes and subclasses are shown in Table 1.

Moreover, the OntoRepliCov ontology defines the semantic relations among the previously defined classes through object and data properties. Each property links two concepts, and each data property links a concept with a specific datatype. Both are defined by a binary property. Among these 15 properties, the following can be mentioned:

• has $-$ first $-$ base (Codon, Nucleobase): a property defining the first nucleobase of the codon sequence.

• has $-$ second $-$ base (Codon, Nucleobase): a property defining the second nucleobase of the codon sequence.

• has $-$ third $-$ base (Codon, Nucleobase): a property defining the third nucleobase of the codon sequence.

• has $-$ next (Nucleobase, Nucleobase): a property associating each nucleobase to its next nucleobase.

• has $-$ next (Codon, Codon): a property associating each codon to its next codon.

• has $-$ beginning $-$ base (Stem-loop, Nucleobase): a property specifying the beginning nucleobase of the stem-loop.

• has $-$ ending $-$ base (Stem-loop, Nucleobase): a property specifying the ending nucleobase of the stem-loop.

• sequence $-$ polyprotein $-$ pp1ab(Codon, CodonFS): a property specifying the polyprotein pp1a.

• sequence $-$ polyprotein $-$ pp1ab(Codon, CodonFS): a property specifying the polyprotein pp1ab.

• has $-$ rank (Nucleobase, xsd: integer): a data property assigning the rank of each nucleobase.

• has $-$ rank (Codon, xsd: integer): a data property assigning the rank of each codon.

• frameshifting (Nucleobase, xsd: boolean): a data property indicating the nucleobase where the ribosome slips back in the translation process.

• is $-$ start(Codon, xsd: boolean): a data property specifying the starting codon of the translation process.

In addition to this knowledge base, the ontological module is semantically enhanced with an SWRL rule engine (Table 2) to describe the translation and the transcription sub-processes of the replication machinery. The inferences of these rules represent elements (amino-acid sequences, polyproteins, etc.) resulting from the sub-processes. Each sub-process includes different stages. For instance, in the translation sub-process, chains of codons (a combination of three nucleotides) are formed to produce amino acids. The translation terminates when a combination of three codons describes the stop codon. Then, a polyprotein pp1a is synthesized from the beginning of the translation to the Stop codon. But sometimes, the translation slips back one nucleotide and a longer polyprotein pp1ab is produced. In collaboration with biologists, a set of expert rules has been defined to compute the resulting elements from the sub-processes described with several stages (several rules): the structure of the viral proteins, ribosome activities, discontinuous transcription of subgenomic mRNAs, and continuous genomic RNA replication. Thus, this module contributes to defining and modeling the biological micro-level of the SARS-CoV-2 RNA genome replication process. For sake of space, the set of SWRL rules will be explained in Section 5.

4.3. The mapping between DEVS and ontological modules

In this section, we explain how the DEVS module is connected with the ontological model. The CD++ simulation process is performed through data transfer. Typically, these data include information about the sending component, the time of the related event, and information about the input/output port and a value. In CD++, there are four kinds of data (or data messages) to communicate (Figure 3): (1) the change of state caused by an internal event denoted by the symbol $*$ , (2) the arrival of an external event by specifying its value and its port denoted by the symbol $X$ , (3) the output of a model denoted by the symbol $Y$ , and (4) the end of a model’s task denoted by the symbol $\mathrm{Done}$ . ⁷⁰

As the aim of our work is to combine DEVS formalism with ontological modeling, we exploit the internal and external transition functions and relate them to the SWRL rules engine. More specifically, the DEVS module requires some inputs describing elements produced through the sub-processes of the replication machinery at micro-level. These elements are described through OntoRepliCov and represent the inferences resulting from SWRL rules combined with some axioms (concepts definitions); hence, each atomic model’s simulator may send queries to the SWRL rule engine to get the elements required to fulfill a specific task in a macro-level of the replication machinery.

As shown in Figure 3, SARS-CoV-2 viral RNA is addressed simultaneously to both the atomic module ACE2, which is the biomolecular component by which the virus will penetrate the host cell as seen in section 2.1, and to the ontological model in which it will be modeled and semantically enriched with expert knowledge. Since the ACE2 atomic model can not identify to which the different groups of amino acids correspond, it appeals to the ontological model which, thanks to the SWRL rules provided by the biology experts, can identify the amino acids—decrypt them—and determine the different role of each amino acid in the viral RNA sequences. Simultaneously, the ontological modeling of the viral RNA will allow enriching it semantically and by means of some expert rules (presented in Table 2), the ontology will decompose the viral RNA sequences and determine the function of each amino acid, a very detailed analysis (at a micromolecular level) that the DEVS simulation module cannot perform. Next, the ontological module provides the ACE2 atomic component with the identified groups of amino acids and their corresponding biological elements. At this stage, the ACE2 component will have enough knowledge to initiate the simulation at a macromolecular scale, a simulation at a scale that the ontology cannot ensure, and will provide a messenger RNA ready to be replicated in the cytoplasm of the host cell as an output. This is the first step in the life cycle of the virus replication process, corresponding to viral entry as described in section 2.1.

Since the simulation process works through the exchange of messages and data among the different processors (Figure 3). The root processor computes the time of the next external and internal state transition to check which one should occur first. Once the upcoming model is identified and the next event is triggered internally, the root processor sends a $*$ -message, through the middle-level coordinators, to the selected simulator. Then, the atomic model simulator is launched by initializing its current time simulation $t$ and the state $s$ of the atomic model. The simulator requests the inference engine of the ontology module as detailed in Algorithm 4. This one executes the rule(s) that corresponds to the request. Then, SPARQL queries (semantic query language) are exploited to fetch the inferences resulting from the reasoning process. When the response is given on schedule, the simulator performs the internal state transition and repeats. Whether the next event is triggered by an external input, the root processor sends an external $X$ -message to the simulator. This one requests the inference engine, through Algorithm 4, to execute the rule corresponding to the requested task. After completing the appropriate rule(s), the reasoner returns the result to the simulator. Then, the simulator handles the external state transition and repeats. Finally, the simulators return $\mathrm{done}$ -messages and $Y$ -messages to the (root) coordinator to notify it that the model has completed its task and send it the model’s output. Both of these $\mathrm{done}$ -messages and $Y$ -messages are in turn converted to new $*$ -messages and $X$ -message, respectively. These steps are repeated until the end of the simulation.

Algorithm 1 describes the operation and function of the Root Coordinator (Root) that is used to drive a Simulator. The simulation process starts by initializing the root coordinator’s simulator using the Initialization() function. As presented in Algorithm 2, this function initializes the initial states and simulation times of atomic models. Then, it executes its ComputeStateTransition() function until the final simulation time is reached. The ComputeState Transition() function computes the next states of the atomic models during a simulation run, as detailed in Algorithm 3. The function starts by selecting the time of the first upcoming atomic model in the priority queue consisting of pairs of atomic models and their time $(d,t)$ . This queue is denoted by the scheduler. Once selected, the atomic model is removed from the scheduler and added to the imminentAMset a set of models in which their next state will be immediately computed.

As described in Algorithm 3, the simulator exploits the ComputeStateTransition() function to compute the next states of the atomic models. It starts by checking if the model’s inputs $x_d$ are not empty and that the model $M_d$ is scheduled for an internal transition, it computes the confluent transition function ${\lambda }_{\mathrm{conf},d}(x_d)$ . If the model’s inputs $x_d$ are empty and that model $M_d$ is scheduled for an internal transition, the simulator computes the internal transition function ${\delta }_{\mathrm{int},d}(s_d)$ by calling the InferenceEngineProcess() function. Otherwise, the elapsed time $e_d$ is found and the model external transition function ${\delta }_{\mathrm{ext},d}(s_d,e_d,x_d)$ is computed using also the InferenceEngineProcess() function. The ComputeStateTransition()simulator’s function finishes by computing the last transition time, and check if the next transition time $t_{n,d}$ is less than infinity, then the model reference $M_d$ and its next event time $t_{n,d}$ are added to the scheduler.

As illustrated in Algorithm 4, the InferenceEngine Process() function cycles through three main sequential steps. In this method, the inference engine starts by finding all SWRL rules (Table 2) that match the given task requested by the state transition function (internal or external). It looks for antecedents, and conditions on the left-hand side of the SWRL rule, that satisfy the given state transition function. This first step is ensured by the Match-rules function. Then, through the Select-rules function, the inference engine prioritizes the different SWRL rules that were matched to determine the order to execute them. Finally, in the third step, the engine executes each matched rule, using the Execute-rules function, in the order determined in step two and then iterates back to step one again. This cycle continues until no new rules are matched to the input values given by the state transition function. Finally, the SWRL rule-based inference engine sends the newly computed state of the atomic model to the DEVS simulator.

Therefore, through this simulation process as described by Algorithms 1–4, the two modules can be aligned, and we can benefit from the finer semantic rules provided by the SWRL rules as well as from the rich knowledge of the OntoRepliCov ontology. This mapping between both modules will be further detailed through examples in the next section.

5.1. Development environment

The different software and tools needed to implement the proposed hybrid approach are:

• Eclipse ² . It hosts the CD++ Builder GUI as a plugin.

• Boost Software License ³ for knowledge base and ontologies management.

• CD++Builder ⁴ for discrete event modeling and simulation based on the DEVS and Cell-DEVS formalisms.

• Cygwin ⁵ to emulate a Unix system under Windows.

• Protégé ⁶ for developing knowledge-based systems and editing ontologies.

• owlcpp ⁷ for parsing, querying, and reasoning with OWL ontologies.

• FaCT++ ⁸ for reasoning on OWL DL-based ontologies.

• Redland Raptor ⁹ to parse and execute Resource Description Framework (RDF) queries.

5.2. Implementation of the proposed hierarchical modeling of the SARS-CoV-2 replication

As described in the previous section (Section 4), to model the structure and behavior of the SARS-CoV-2 replication system, we have used primarily the classic DEVS formalism. Therefore, we have split up the replication process into four coupled models namely, the host cell which itself is divided into three compartments represented by three coupled models: the cell membrane, the cytoplasm and the secretory pathways (Figure 4). The SARS-CoV-2 replication process is ensured by these three coupled models. Each coupled model is composed of a set of atomic models representing its main viral and/or molecular components. The cell membrane coupled model consists of only one component to ensure the virus entry step, the angiotensin-converting enzymes 2 (ACE2) atomic component. The cytoplasm coupled model which ensures the translation and the replication/transcription steps of the SARS-CoV-2 RNA genome is composed by, the first host ribosome (HR1) atomic component, the viral protease (VP) atomic component, the four replication transcription complex (Complex_RTC, RTC1, RTC2, and RTC3) atomic components, the second host ribosome (HR2) atomic component. Finally, the secretory pathways coupled model ensuring virions assembly and their releases outside the host cell consists of the ER-Golgi intermediate compartment (ERGIC) atomic component, the Golgi apparatus (GOLGI) atomic component. Most of the atomic models rely on the inference engine rule (as presented in the previous section) to compute their state changes and behaviors. The inference engine exploits 12 rules, those described in Table 2. These rules are not only used to compute the atomic components’ state changes but also to reach a lower, more microscopic scale, for considering the most basic building blocks and biomolecules (having no changes of states or behavior), that the DEVS formalism cannot capture.

Table 3 shows the correspondence of the cellular and viral components involved in the SARS-CoV-2 RNA genome replication process, as described in Section 2.1 and Figure 1, with their corresponding components in the DEVS simulation framework. It is important to mention that in our case, we suppose that in our modeling there are four classes of replication/transcription complex (RTC). The first one, denoted by Complex_RTC is required to synthesize the genomic and subgenomic RNA transcripts. Consequently, we will give it three different names according to their function. The second class, denoted by RTC1, corresponds to the functional complex RTC that allows the synthesis of negative-strand templates from the positive mRNA genome. The third one, denoted by RTC2, corresponds to the functional complex RTC responsible for synthesizing subgenomic RNAs using Transcription Regulatory Sequences (TRSs). Then, finally, the fourth class, denoted by RTC3 corresponds to the functional complex RTC that ensures the synthesis of positive-stranded subgenomic RNAs from the negative-strand templates.

In the virus entry step, the SARS-CoV-2 attaches to the host cell receptor ACE2 through its spike protein S to enter the cell by the endocytic pathway. This first step occurs in the cell membrane compartment through the ACE2 atomic model, as presented by Figure 5. Once the ACE2 component receives the spike S (via its input port), it triggers the fusion of the viral and cellular membranes, allowing the penetration of the virus into the host cell cytoplasm as a genomic RNA (mRNA). Table 4 presents the formal specification of the cell membrane compartment, and its unique ACE2 component, according to Figure 5.

After virus entry, the viral RNA genome is translocated to the cytoplasm compartment in which two main stages of the replication process occur, the (1) translation of viral replication machinery and (2) translation of viral structure proteins (as described in Section 2.1 and Figure 1). Figure 6 illustrates a DEVS representation of the HR1 atomic component, one of the components involved in the cytoplasmic compartment. Since the representation is the same for the different components of the cytoplasm compartment and for the sake of space, only one of them (the HR1 atomic component) is presented. See Appendix 1 for the entire DEVS representation of cytoplasm compartment (with its seven components), including its formal specification in Table 5 and AppendixTables 8 and 9. These tables present the formal specification of the seven cytoplasm compartments, according to the AppendixFigure 13. During this phase, the host cell translation machinery is hijacked and dedicated to viral component synthesis. Thus, the genomic mRNA is translated by the host ribosomes (denoted here by HR1) to produce two polyproteins $\mathrm{pp}1a$ and $\mathrm{pp}1\mathrm{ab}$ from $\mathrm{ORF}1a$ and $\mathrm{ORF}1a$ , which are then further processed by proteolytic cleavages (Figure 6). As this process is complex and consists of four steps namely the initiation, elongation, termination, and recycling, the host ribosome component uses some specific SWRL rules. Indeed, the DEVS formalism is able to manage the different states, inputs, and outputs of the host ribosome component, but is unable to capture how these four steps are done. Therefore, the state transition function of the host ribosome component uses the adequate SRWL rules set. This process of rules selection is managed by the Algorithm 4. At this stage of the replication (the translation process through the host ribosome component), and as illustrated in Figure 7, the Algorithm 4 is set up by parameters that specify the mRNA_in. Since we are in the translation process the algorithm computes the Match-rules() function and selects (Select-rules()) those that satisfy this stage of transition. As a result of these two functions, eight rules are selected 1, 2, 3, 4, 5, 6, 7, and 8 (Table 2) to compute the outputs in the right order. Once the rules are selected, the Execute-rules() function is considered to execute the rules from 1 to 8 and get the output of the different translation sub-stages. First, rules 1 and 2 are used to ensure the division and definition of the genomic codons. Then, rules 3, 4, and 5 are used for the ribosomal frameshifting. As the ribosome may slip back to translate the ORF1b, it uses these three SWRL rules to identify the pattern responsible for the ribosomal frameshifting and at which nucleotide rank this frameshifting occurs. In 30% of cases, the ribosome by-pass the stop codon of the $\mathrm{ORF}1a$ to translate the $\mathrm{ORF}1b$ . Thus, the polyprotein 1a $(\mathrm{pp}1a)$ is extended to produce the $\mathrm{pp}1\mathrm{ab}$ . This process called post-frameshifting codons is ensured by the SWRL rule number 6. Finally, the SWRL rules 7 and 8 delimit the polyproteins $\mathrm{pp}1a$ and $\mathrm{pp}1\mathrm{ab}$ . These rules ensure the elongation step in which the encoded polyproteins $\mathrm{pp}1a$ and $\mathrm{pp}1\mathrm{ab}$ are assembled until the termination occurs when the elongating ribosome meets the stop codon (Table 2). These polyproteins are the results of several incremental inferences and are considered outputs for the Algorithm 4 and H1 and inputs for VP. Therefore, a SPARQL query is required to get the sequence of these two polyproteins and the answer to this query is returned through the Algorithm 4. An example of a SPARQL query is presented in Figure 8.

Both polyproteins $\mathrm{pp}1a$ and $\mathrm{pp}1\mathrm{ab}$ are cleaved by the viral protein component (VP), through the proteolytic cleavages by two viral proteases, to provide non-structural proteins. The first polyprotein is cleaved into non-structural proteins $\mathrm{nsp}1$ to $\mathrm{nsp}12$ , while the second one is cleaved into additional non-structural proteins, $\mathrm{nsp}12$ to $\mathrm{nsp}16$ . These nsps proteins can be considered as the first viral proteins to be expressed after virus entry. In this process, the translation of the second polyprotein requires a $-1$ frameshifting event. As the DEVS formalism cannot ensure this process, the VP atomic component also calls some specific SWRL rules, namely rules 1, 2, 3, 4, and 5 in Table 2, and axioms in charge of the amino acids’ inference from codons. The SWRL rules 1 and 2 are used for the division and definition of the codons, and rules 3 and 4 are for ensuring the frameshifting during the cleavage process.

Among the non-structural proteins, the $\mathrm{nsp}7$ , $\mathrm{nsp}8$ , $\mathrm{nsp}10$ , $\mathrm{nsp}12$ , and $\mathrm{nsp}14$ are joined together to form the minimal catalytic core of the proofreading RNA polymerase complex. They are modeled in our study by the RTC atomic component. This combination produces a functional RTC that will be used to synthesize the viral genomic and subgenomic RNA transcripts. This functional RTC has a major role in the viral replication process, as it ensures the replication of the RNA genome and the transcription of a set of subgenomic mRNAs. As stated at the beginning of this section, we distinguish in our DEVs modeling three kinds of RTC aside from the functional RTC (the creation of RTC). RTC 1, a functional RTC, that ensures the synthesis of negative-strand templates $-\mathrm{RNA}$ from the genomic RNA. RTC 2, a functional RTC, ensures the synthesis of subgenomic RNAs using Transcription Regulatory Sequences (TRSs). And, RTC3, a functional RTC, ensures the synthesis of positive-stranded subgenomic $+\mathrm{RNA}$ from the negative-strand templates $-\mathrm{RNA}$ . All these functional RTC components (RTC1, RTC2, and RTC3) use also some specific SWRL rules (9, 10, 11, and 12) and the axioms of amino acids to ensure the translation (in both directions) and synthesis of subgenomic RNAs. The rules (9, 10, 11, and 12) are used to ensure the transcription process. These rules are used to produce the complementary copy of the viral mRNA (either positive-strand RNA or negative-strand RNA).

The genomic and subgenomic RNAs are translated by the host ribosome HR2 into structural proteins namely the Spike (S), the Envelope (E), the Membrane (M), and the Nucleocapsid (N). The HR2 atomic component relies on the specific SWRL rules designed for the translation process, 1, 2, 9, 10, 11, and 12. Rules 1 and 2 are used for the division and definition of codons as ribosomes translate a codon (formed by three nucleobases) into an amino acid, constituting the proteins. While the rest of the rules (9, 10, 11, and 12) are used to ensure the transcription process and produce the fourth structural proteins as described in Figure 9(b).

At late stages of infection, new virus particles are formed and released into the extracellular environment (as described in Section 2.1 and Figure 1). This step is ensured by the secretory pathway compartment through both atomic components ERGIC and Golgi, as presented by Figure 10. The ERGIC atomic model ensures the virus assembly process by joining the four structural proteins (M, E, S, and N). This process is totally ensured by the DEVS formalism without the need for SWRL rules. While the Golgi atomic component ensures the release process of the assembled virions outside the cell via a process called exocytosis. When the nucleocapsid is created, the Golgi takes as input the viral nucleocapsid and some regions of the genomic $+\mathrm{RNA}$ , and then releases a mature virion outside the cell. This process is completely ensured by the DEVS formalism without the need for SWRL rules. Once outside the cell, these released virions become extracellular virions and are able to infect other healthy cells. The formal specification of the secretory pathway compartment and both of the ERGIC and Golgi components are presented in Table 6, according to Figure 10.

Obtained simulation results of our proposed hierarchical approach are presented in Figures 9 and 11. Figure 9(a) presents the dynamic state changes and the output of the HR1 atomic component responsible for the translation process. We can see how the HR1 component changes its states and the two polyproteins $\mathrm{pp}1a$ and $\mathrm{pp}1\mathrm{ab}$ are produced at a rate of 70% and 30%, respectively. These polyprotein expression proportions are in accordance with the study of Snijder et al., ⁷¹ as a large part of the ribosomal production (about 70%) is dedicated to the production of the $\mathrm{pp}1a$ and the rest for the production of $\mathrm{pp}1\mathrm{ab}$ . Indeed, the polyprotein pp1ab is translated through a ribosomal frameshifting at the end of ORF1 and which harbors an efficiency of around 30% to extend pp1a in pp1ab. Thus, starting with the SARS-CoV-2 genomic (mRNA), the simulator uses the ontological model to compute and generate two polyproteins $\mathrm{pp}1a$ and $\mathrm{pp}1\mathrm{ab}$ . The figure details the functioning of HR1 during the replication process: at t = 0, the host ribosome atomic model is created and initialized in the passive state. Once it receives the viral genome, at t = 30, its external transition function changes its state in the Translation process. After some time (at t = 40), the internal state function changes the state of the host ribosome from the translation process to proteins release as the HR1 completed the translation process. During this state, the output function transfers the products (here 70% of $\mathrm{pp}1a$ and 30% of $\mathrm{pp}1\mathrm{ab}$ polyproteins) to the next atomic model. After the release state, the internal transition function changes the state of the ribosome to finished (at t = 45), then to the passive state (at t = 46).

Figure 9(b) presents the dynamic state changes and the output of the RTC1 atomic component responsible for the transcription process. It shows the synthesis of negative-strand templates from a positive mRNA genome, a process called transcription. At t = 0, the complex RTC1 atomic model is created and initialized in the passive state. Once it receives the positive viral genome (at t = 1:20), its external transition function changes its state in the replication process. After some time (at t = 1:60), the internal state function changes the state of the RTC1 from the replication process to the RNA release as the RTC1 completed the transcription process. During this state, the output function transfers the negative RNA produced to the next atomic models. After the release state, the internal transition function changes the state of RTC1 to finished (at t = 1:65), then to the passive state (at t = 1:66). Unfortunately, we were unable to compare our results with other approaches, as at present there is no comparable work available with precise data. Furthermore, the purpose of this work was to provide biologists with a tool that would address their need to understand the different phases of SARS-CoV-2 viral replication and the interactions of viral entities with cellular entities at different biological scales. These experiments were carried out with uncertain parameters. At present, biologists face difficulties in obtaining the right values for the parameters of the simulation. Therefore, it must be noted that the simulation parameters used in this simulation are not realistic and are not approved by the biologists. Currently, no study has been done to provide the exact parameters to conduct a precise simulation. The SARS-CoV-2 simulation starts with a single initial virus particle. The reference genome was downloaded from the ^NCBI. As the exact parameters of the SARS-CoV-2 replication machinery are still unknown, we conducted a literature search to collect settings that may meet our needs. The collected parameters used in the simulation are presented in Table 7. The translation step is started with a translation rate of 120 amino acids per minute (aa/min), which depends on the ribosome density per cell set at 0.0125 ribosome per amino acids (ribosome/aa). When a polypeptide is translated, ribosome frameshift is randomly sampled with a probability of 25 %. Once these polypeptides are synthesized, non-structural proteins are randomly produced. According to the literature, we assumed that the RdRp has a constant transcription/replication rate equal to two kb/min without any transcription activation factor. Two RdRp complex are assigned to each negative subgenomic mRNA for transcription. In the absence of precise biological parameters and with the agreement of expert biologists, we assumed that the number of subgenomic mRNAs is equal to 9 (sgmRNA1 to sgmRNA9). In the absence of precise biological parameters and with the agreement of expert biologists, we assume that the packaging ratio for the proteins E, S, N, and M are equals to 20, 1.5, 2.1, and 1.0, respectively. While the packaging ratio for positive viral RNA was fixed to 1.4. The packaging ratios have no units.

This simulation experiment was performed only to illustrate the viral quantification process while taking into account the different replication steps presented by the proposed approach. Figure 11 (a CD++ screenshot) presents a first proof of concept of the viral quantification of an epithelium host-cells with a single SARS-CoV-2 virion, focusing into the different stages of the replication cycle presented in this work. The figure presents the number of SARS-CoV-2 virions leaving host-cell during an infection period of 13.5 days (around 1.160.014 s) over its viral life cycle. As observed, this graphical description reflects the number of virions according to the different phases of the virus life cycle. During the initial stage, a single virus causes the infection by binding and penetrating the cells. Then, we observe in the viral growth curve virions are released from the infected host cells at the same time, creating an exponential growth of produced virions until reaching a peak. It is at this stage that the transcription and translation processes reach their maximum, and therefore triggering a significant assembly and packaging of new virions. Such a process is called a “viral burst.” Finally, when no viable host cells remain, the virus particles begin to slowly decrease until they totally disappear. According to expert biologists, this viral growth curve approximates the real behavior of SARS-CoV-2 infection. It is important to note that our simulation was not conducted on real data. It was only designed to make a viral quantification considering the different stages of the viral replication cycle as detailed in the previous sections. From a biological point of view, the viral evolutionary appearance and structure, as well as its approximate quantification, seems logical—it clearly shows the phases of the SARS-CoV-2 viral replication. Several studies have recently started to investigate these fundamental data of SARS-CoV-2 infection, on which numerous uncertainties remain. This underlines the importance of our proposed hybrid approach which, despite the lack of certain parameter values, approximates the viral replication cycle. Through the experiment, we clearly observe that both main processes of the viral replication, the translation and the transcription, are handled by the DEVS simulator through the ontological module through SWRL rules and axioms. For sake of space, we only display the results of both the translation and the transcription processes.

We conclude that the CD++ simulator is able to provide quantitative information on the different viral molecular components produced during the infectious viral cycle. In other words, we obtained both a qualitative simulation outputting the viral components that are produced during the different stages of the viral cycle (polyprotein cleavages, non-structural proteins production, RNA replication, transcription of the different sg mRNAs, assembly of the structural proteins) and a qualitative simulation outputting the expected number of virions produced as a result of the different stages of the viral cycle. Combining qualitative and quantitative information for simulation-based analysis gives several advantages, which will be discussed in the next section. However, it is important to note that this simulation does not take into account the defenses (innate immunity) that the cell can put in place to fight against the infection.

It is important to notice that, in this work, we are only trying to understand the viral replication and how viral entities interact with the host cell components during the replication. This is why all DEVS atomic components depend on each other and the input values are ignored in the external transitions. Only the atomic component ACE2 considers the input value, as it is activated exclusively by an external transition, the presence of viral RNA. Such input values will be considered in future work, especially when biologists investigate the impact of drugs on viral replication performance, a question not addressed in this work.

5.3. Results and discussion

The results of the implementation, presented in the previous section, illustrate how the proposed hybrid approach is enhancing the understanding of the SARS-CoV-2 cell infection process at the single-cell level. The hybrid approach provided a thorough analysis and understanding of viral replication biological processes occurring in the infected cell and how they influence each other. It allows tracking the evolution and replication of the SARS-CoV-2 through hierarchical cellular compartments and quantifying its release at the individual cell level. The hierarchical approach can be considered as a suitable framework to model heterogeneous populations and simulate the behavior of complex systems with an intrinsic discrete nature, such as the virus replication process and virus–host interaction.

We have demonstrated that, through its modular structure, the DEVS formalism is suitable to be integrated with an ontological-based model within a hierarchical framework that can link the small biomolecules to the cell levels. The approach effectively embeds the viral replication behavior from the simulation of the (1) dynamics of the individual atomic models (e.g. ribosomal translation of the different viral proteins and RNA replication), (2) interaction among atomic components (e.g. the interaction between viral proteins and cellular ribosomes), and (3) micro and macroscopic scales (e.g. interaction among the different cell’s compartments). These simulation results will provide valuable support for new cutting-edge biological approaches to accurately analyze transcriptome dynamics at the single-cell level (Single-Cell RNA-Seq) as well as to quantify viruses released at the single-cell resolution (viro-fluidic method).

With the hybrid nature of the proposed modeling and simulation approach, an interoperable relationship has been established between both DEVS simulation and ontology modules. It takes advantage of the benefits achieved by grouping and combing both domains. As seen in the previous section, both modules of the approach were able to adapt and collaborate together.

Using the DEVS formalism, the hierarchical approach easily captures spatial-related aspects of cell compartments and can integrate viral component behaviors at different scales. The SARS-CoV-2 multiplication process lies on a hierarchical structure with different spatial and time scales: the building biomolecules scale typically spans from nanoseconds to microseconds to the cell level from seconds to hours. It is also used to describe the temporal variation of viral stage state process variables and capture both their temporal and spatial-related evolution.

The OntoRepliCov domain ontology, rich in viral replication semantic knowledge, enriches the simulation model and improves its efficiency. Moreover, the ontological module provides rich knowledge about the microscopic scale of the viral replication process and computes as well the dynamic behavior of micro-level components. Through this module, the approach offers a natural description of the infected host-cell system through the definition of rules governing the viral and cellular component activities (e.g. protein synthesis, assembly of nucleobase, polyprotein production, transcription, and frameshifting). It provides the DEVS module with semantic rules to compute the dynamic state transitions of DEVS models depicting viral and cellular components at a micromolecular level. Such micro-level cannot be modeled by the DEVS model alone. Thus, one of the limitations of DEVS simulation models is their lack of formal semantics and definition of the basic micro-models. For example, in the modeling and simulation of the SARS-CoV-2, there are some basic components representing the micro-level, such as building molecules, which have no states or inputs and outputs. As the ontology was populated with more than 11 million SARS2 sequences from GISAID repository (https://gisaid.org/), it is able to consider the smallest and finest components of the replication process, such as amino acids, nucleobases, etc., which the DEVS model is unable to capture.

Therefore, our proposed hybrid approach offers several advantages, including formalizing a relationship between DEVS simulation components and domain knowledge. This relationship has resulted in enriching the simulation model with semantically rich knowledge that the DEVS simulation alone could not successfully model or simulate. The DEVS formalism is very suitable for modeling and simulating the dynamic behavior of a complex system such as the SARS-CoV-2 viral replication. Nevertheless, it cannot express the semantic interpretation of what is occurring in the executed simulation or in the simulation module. Likewise, it cannot deduce any kind of semantic knowledge about the domain area. Therefore, it is not able to enrich or complete the missing data on viral replication. Moreover, the use of a unique DEVS model can potentially increase the complexity of the simulation model, since the formal semantics of the various parameters used by this DEVS model and their relationship with domain knowledge is often overlooked. This explains the need for an ontological model that would complement the DEVS formalism and semantically enhance it.

Classical approaches for modeling and simulation of viruses and their interaction with host cell components are generally focused on single biological scales. The hybrid hierarchical approach proposed in this study meets the objective of integrating biological knowledge within a simulation formalism in which multiple scales are considered at the same time. As presented in Section 2.1, the characteristics of the viral replication process are self-organization and self-reproduction. As described In Figure 12, in this process, both characteristics are extended from the micromolecular scale including small molecules, building blocks, and biomolecules, to the macromolecular scale including supramolecular structures, organelles, and the whole cell.

As discussed above, the proposed hierarchical modeling and simulation approach was specifically designed to be modular and extensible to support continuous updating and parallel development. Considering the modularity of the formalism DEVS, the proposed approach can be easily scalable and adapted to a change of scale in the viral life cycle (or the viral cell interaction process) and support updating in response to rapid advances in the understanding of infection mechanisms. For example, if we intend to add other cellular or viral components involved in the viral replication cycle in light of scientific discovery (as cell immunity), it would just be enough to add the atomic or coupled components and their SWRL rules describing their behavior. Such changes in the hierarchical model will not impact the original model, which can maintain its functionalities even with the new changes. As such, the proposed approach is sufficiently flexible and generic to be applied to the replication of other viruses belonging to (+)RNA virus families with an entirely cytoplasmic life cycle (e.g. Dengue, Zika, Chikungunya, and Hepatitis E viruses).