Introduction

Since the discovery of cell in 1653 [1], significant progress has been made in understanding the fundamental components of biological systems and the mechanisms governing them. These developments are summarized in various “omics” of biology [2, 3]. However, recently, the need to have an integrated understanding of how these components work together in a system is being increasingly realized [4].

Immune systems have a high degree of interdependence and interconnection between components. They have a multihierarchical or multiscale organization (Fig. 1), where components at a lower scale are integrated into functional units at the next higher scale [5]. The units are robust and exchange a limited amount of information with their surroundings. Thus, they hide the detailed mechanisms operating at lower scales. At any scale, the system can be studied phenomenologically as long as there are predictable causal relationships between the perturbations and state changes. Indeed, there are separate subdisciplines of biology addressing different phenomenological scales (Fig. 1).

Fig. 1
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Multiscale organization of immune systems and the subdisciplines of biology at various scales

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However, there are certain scenarios where phenomenological descriptions are not sufficient and an integrated description of phenomena occurring at various scales is unavoidable. For example, host–pathogen interactions occur on all scales: from the molecular scale (recognition of pathogen molecular patterns by immune cells) to the cellular scale (phagocytosis of the pathogen by macrophages) and up to the population scale (spread of the pathogen through host population and emergence of different strains through different host conditions) [6]. Elements at different scales interact: for example, host–pathogen interactions at the population level determine the selection pressure on both host and pathogen at the genetic level. Thus, a full understanding of host–pathogen interactions requires the integration of phenomena at various scales. Such integration of different spatial and time scales has been called by the term systems immunology [7, 8].

Another dimension of integration is transcale, which involves aggregating and connecting information from different disciplines of study at the same biological scale. For instance, cells are studied from molecular, biochemical, geometrical, biomechanical and various other perspectives. These properties are interlinked: for example, cytokine TGF-β increases cell stiffness and leads to an elongated cell shape. Reciprocally, cells such as chondrocytes secrete TGF-β in response to mechanical stimulation (mechanotransduction). Transcale integration brings together such interdisciplinary information.

To achieve this integration, systems immunology complements empirical and experimental approaches with modeling and simulation. A model is an abstract representation of a real system. Simulation refers to operating a model under a configuration of interest to “simulate” the system’s behavior. A model has some realism when abstract entities in the model correspond to real components in the system and mathematical or computational rules governing the model correspond to real physical laws [9].

Ideally, a model in systems immunology should give a virtual description of an immune system in the same way as a Google® map describes a physical landscape. The specialty of Google map is its multiscale structure. Zooming in or out of the map not only changes its scale but also displays different features and annotations corresponding to different resolutions. A model of an immunological system should similarly give an integrated description of its components at various scales. Such a model can be easily used to represent the state of the system and infer its dynamics while going from an initial to a final state.

A model therefore serves to synthesize existing knowledge and data about a system via multiscale and transcale integration. The knowledge and data could be static (without temporal information) or dynamic. As a consequence, missing information or gaps in understanding may be revealed, which could initiate further experimentation and guide data acquisition policies. A model can also serve to generate new knowledge about the system by simulation, which is technically known as prediction. Simulation can help in understanding system-level behavior over large ranges of parameter values. This may be difficult to test in vivo due to technological and financial constraints. Modeling and simulation may also demonstrate surprising or counterintuitive consequences of system organization. Indeed, a model can sometimes serve to provide mechanistic explanations about the intricate working of the system [10].

Several formalisms have been proposed in the literature for systems immunology modeling. Each formalism has its particular characteristics that make it suitable for some specific applications. There is no and there will not be any global “best” modeling formalism. Most likely, the only way to model something as complex as the immune system is to blend (compound or aggregate) different formalisms in a common multiscale transcale framework. We can see a trend toward such “hybrid” or multiformalism modeling[11–14].

This paper is an attempt at describing the principal formalisms that currently exist in the literature for systems immunology modeling from an application point of view. It is directed toward immunologists who are interested in using systems immunology modeling but do not have a theoretical or technical exposure to the field. It covers theoretical principles, immunological applications and software tools associated with the formalisms. Some recent reviews have focused on only one or a few of these formalisms and only one or a few biological scales. Thus, an overall view of the modeling field as it exists today is missing. There is no review of available software. Moreover, there are no general guidelines to decide which formalism will be suitable for a given application. This review tries to fill these gaps. The presentation has been kept as much as possible nontechnical for the understanding of immunologists. Technical details are referred to existing relevant literature.

We first provide an introduction to computational modeling and simulation, presenting the key concepts and rationales. Then, formalisms are described in brief, covering their basic principles and modeling strategy. Formalisms covered in this paper include differential equations, cellular automata and agent-based systems. Additional, but less prominent, modeling formalisms are also introduced. For each modeling formalism, we describe typical immunological applications and available computational tools. Finally, we identify and discuss the future research directions where the intricate adaptive nature of immune systems would be considered.

Modeling and simulation

A model is a representation of a real system in mathematical or algorithmic terms. A model cannot possibly include all details of a system. It is an abstraction suitable enough to capture the essential mechanism of the real system. Building a model minimally requires knowledge of the entities in the system such as genes, proteins, cells, organs etc., the characterization of these entities either by a quantity (such as protein concentration, number of cells, etc.) or by a discrete state (such as on/off state of a gene, an individual being uninfected/infected/chronically infected, etc.), and the interactions among different entities such as regulation of genes by proteins, protein–protein interactions, cell–cell interactions, etc. Additional knowledge such as the description of compartments or other subspaces in the system can also be useful.

In addition to knowledge of entities and interactions, a model requires a mathematical or algorithmic structure called formalism for its description [15]. Formalism regulates modeling assumptions, model structure, governing laws, nature of interactions, how a model is simulated, etc. For example, one of the simplest formalisms is a static graph which has traditionally been used for visualizing networks of cell–cell interactions, gene expression or signal transduction [16]. In a static graph, entities are represented by nodes, their quantities or states by a weight or color and an interaction between two entities as a (possibly weighted) edge connecting the nodes. Such a graph can provide some insight into the possible behaviors of a system. However, it does not enable quantitative predictions. More descriptive formalisms can enable quantitative analysis.

From an immunologist’s point of view, the important question is which formalism is suitable for modeling an immunological application at hand. Although there are no readymade answers to this question, some important considerations, such as the ones below, can guide the choice of an appropriate modeling formalism. A glossary of some technical terms in this connection is given in Table 1.

  1. 1.

    Objectives of the study—Different modeling formalisms have different modeling capabilities. The choice of formalism depends on the questions to be answered. For example, when exact numerical quantities of entities and timings of events are of interest, quantitative formalisms are appropriate. On the other hand, when quantitative information is inessential or missing, qualitative models may be appropriate.

  2. 2.

    Scale of the model—Some formalisms are more appropriate for modeling at certain biological scales. For example, intracellular networks are suitably modeled by formalisms having a static network structure, whereas tissue, organ or individual scales where the network is usually dynamic or nonexistent are sometimes more appropriately modeled by formalisms supporting a dynamic network or agent-based modeling. Figure 2 gives a summary showing biological scales at which most applications of a certain formalism are concentrated.

  3. 3.

    Size of the model—Models involving large number of entities are easier to describe and use in compositional formalisms which allow hierarchical organization of entities and implicit definition of entity subtypes.

  4. 4.

    Nature of available data—The nature and amount of data required to construct a model and learn its parameters varies for different formalisms. For example, constructing a model in a quantitative formalism may require comprehensive quantitative experimental data. On the other hand, models can be constructed in qualitative formalisms with little or no quantitative data.

  5. 5.

    Availability of software tools—Some formalisms are popular for certain modeling applications because high-quality software tools are readily available. For example, Petri nets are frequently used for modeling intracellular processes since a well-developed tool Cell Illustrator [17] is available for the purpose.

Table 1 Glossary
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Fig. 2
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Typical scales of application of various formalisms

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This review discusses the basic modeling approach and characteristics of a few important formalisms for systems immunology modeling. Some important characteristics of various formalisms are compared in Table 2. For further technical details, pointers are given in Table 2 to recent reviews and helpful references. The first step toward choosing a suitable modeling formalism is to explore which formalisms have already been used in the literature for modeling similar systems. This review attempts to review salient modeling applications of each formalism.

For implementation of systems immunology models, a wide variety of software tools is available. Models can be implemented by programming in general purpose programming or scripting languages like C++, Matlab, Mathematica, R, Python, etc. Otherwise, one can use specialized tools that are preprogrammed to implement models in a particular formalism. General purpose programming languages provide flexibility of implementing any conceivable model, but programming can be tedious and prone to errors and may involve duplication of efforts in rebuilding what has already been implemented by others. Specialized tools, though limited in functionality, can save time and effort and allow one to focus on modeling rather than programming issues.

Table 2 Characteristics of various formalisms for systems biology modeling
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For immunologists beginning to take interest in systems immunology, a good starting point would be hands-on experience with existing tools that are user-friendly and well documented and come with a set of example models. This review describes a sample of available software tools, which are in this sense ‘biologist-friendly’.

In terms of usage, software tools can be of various types such as a working environment, an application, a plugin, shell or a library. These terms are explained in Table 3. Usually, working environments are a good starting point as they provide a complete set of tools for a typical modeling workflow including model construction, parameter estimation, model checking, simulation, visualization and typical analyses on a model.

Table 3 Categories of software tools
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Databases are very important part of modeling software, which organize and curate experimental data and models. However, in this review, we will not discuss about databases. We refer the reader to some recent reviews on this topic [18–20].

Differential equations

Formalism

Differential equation models give a complete quantitative description of system dynamics in continuous time and space. In a differential equation model, all entities in a system are represented by some real-valued quantitative attributes such as their total numbers, concentrations, masses, etc. All events in the system such as production, degradation and transport of entities and interactions among entities are expressed in the form of chemical reactions. The rate of change of the quantity of products is represented as a continuous function of the quantities of reactants. The set of differential equations (or rate equations) constitutes the model, and its parameters are called rate constants. The rate constants are estimated from experimental data. Ordinary differential equations (ODEs) are used when only temporal dynamics are considered, whereas partial differential equations (PDEs) are used for dynamics in both time and space. Delay differential equations (DDE) are used in systems with delays or lags such as gestation times or transport delays.

Differential equation models can be deterministic or stochastic. Deterministic differential equation models are used when quantities of entities can be precisely measured and the network of interactions among entities is static. On the other hand, when the entities are present in small numbers (of the order of hundreds or less) or there are dynamic interactions or external factors such as different initial states, timing variability of responses, random external inputs, etc., stochasticity can no longer be ignored. Stochastic models such as chemical master equation (CME), stochastic differential equations (SDE), continuous-time Markov chains (CTMC) and Bayesian dynamics model (BDM) are used when these effects have to be explicitly accounted [21].

To simulate a differential equation model, the set of equations is integrated with given initial and boundary conditions to yield exact numerical quantities of all entities in continuous time. Analytical solution is seldom feasible; therefore, simulation is usually performed numerically using computational algorithms.

Applications

Differential equations models of the immune system are reviewed in [22–24]. Systems of ODEs have been extensively used for modeling intracellular (genetic, signal transduction and metabolic) networks. Precise continuous-time dynamics of small networks containing up to tens of entities (genes, proteins or metabolites) is frequently studied using ODE models. However, for larger networks, qualitative formalisms such as Boolean networks are preferred due to ease of parameter estimation and analysis. An important analysis performed with an ODE model of an intracellular network is the exploration of its steady states, which is equivalent to understanding its equilibrium states or homeostasis [26, 40, 41].

Models have been constructed to study the signaling events in macrophages triggered by AvCystatin [25], the IFN-related regulatory network [26] and NF-kappaB signaling [27, 28]. Dynamics of gene expression in response to activators and inhibitors is described using usual rules of reaction kinetics along with Michaelis–Menten enzyme kinetic equations and Hill function. These equations can reproduce the characteristic sigmoid shaped dynamics of gene transcription. ODE-based approaches have also been used to study cancer–immune interactions [29–31], natural killer cell response [32], B cell memory [33, 34], the role of inflammation in atherosclerosis [35], interaction of HIV with CD4+ T cells [36], virus-neutralizing immunoglobulin response [37], and cytotoxic T cell proliferation [38] and activation [39].

Stochastic modeling is common at the intracellular scale for gene expression and signaling [42]. Lysis-lysogenic fate choice of λ-phage-infected Escherichia coli [43] is a classic example of a system where stochastic kinetic modeling of gene expression is required to predict probabilistic outcomes. Here, regulatory proteins Cro and CI compete at low concentrations to control the switching of gene regulatory network between lysogenic and lytic fates, resulting in a probabilistic pathway selection. Other examples of stochastic modeling include terminal differentiation of B lymphocytes [44], variations or oscillations in NFkB pathway [45], optimization of immunoglobulin substitution therapy [46], and viral replication in cells where few virus particles can initiate infection [47].

Simulation tools

Modeling a biological system with differential equations involves multiple related tasks including model construction, verification, calibration (parameter learning), analysis, validation, simulation and visualization. Currently there is no single software tool, which can perform all of these tasks equally well. Therefore use of multiple tools is imperative. Table 4 lists some software tools for working with differential equation models.

Table 4 Comparison of software tools for deterministic differential equation modeling
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A usual problem in using different software tools is that each tool has its own way of handling data, which makes it difficult to share information across different tools. Efforts toward integration of systems biology software tools has resulted in projects such as Systems Biology Markup Language (SBML) [48], CellML [49], Systems Biology Workbench (SBW) [50], and so forth. SBML and CellML define common formats for sharing model structure, parameters and simulation results across different software tools. Almost all software tools today are compatible with SBML. SBW is a working environment, which integrates many software applications and makes it easy to share information across them. Within the SBW, CellDesigner, JDesigner, Copasi and JSIM are useful graphical or scripting tools for constructing and editing models in SBML format. Model parameters can be estimated from an experimental dataset or optimized over a given range using JSim and Copasi. Analyses on a model, such as steady state analysis, sensitivity analysis, mass conservation analysis, etc. are available in JSim, Copasi, Jarnac, Roadrunner, AutoC#, Frequency Analysis, and Jacobian viewer. No tool for model verification exists. However, partial verification is possible in Copasi. Time course simulation can be performed in JSim, Copasi, Gillespie simulator, Jarnac and Roadrunner. Through SBW the same model can be passed around to different tools so that all of these tasks can be performed within a single working session.

Apart from SBW, there are a number of independent software tools that support systems biology modeling and simulation as shown in Table 4. Being independent, each tool tends to support as many features as possible within it. For advanced users, Matlab, Mathematica, Python and C++ are general purpose programming environments. Biology specific capabilities such as support for SBML format, graphical interface for model construction, stochastic simulation, etc. are available through toolboxes such as SimBiology and SBToolBox2 for Matlab, MathSBML for Mathematica, PySCes library for Python and libSBML and SBML ODE solver libraries for C++.

Examples of differential equation models for immunology are stored in the BioModels database. These can be imported into SBW using the Biomodels importer and serve as a useful guide for immunologists.

Automata and statecharts

Formalism

In contrast with differential equations, which describe the immune system in continuous time and space, automata provide a discretized approach where the immune system is regarded as a system having a finite number of states, with one of these states being the current system state. The automaton receives inputs and ‘jumps’ to the next state depending upon its current state and the input. A transition function defines a mapping to a next state given the current state and an input. An automaton is frequently visualized as a graph in which each state is a node and possible state transitions are indicated by directed edges between the nodes. Edges are marked with the inputs or conditions, which enable that transition.

An automaton, which contains a finite number of states and receives a string of inputs in discrete time is called a finite state automaton. A cellular automaton [51] is a special finite state automaton where states, or ‘cells’, are arranged in a grid in any finite number of dimensions. It is useful when spatial organization or patterns are important in a model. Cellular automata have received significant attention due to their ability to exhibit complex system-level emerging behavior, which often strikingly reflect real biological phenomena, using relatively simple local transition rules. A hybrid automaton is a mixed formalism with both discrete states and continuous variables. A statechart is an extension of a finite state automaton to include support for additional features such as nested state hierarchy, concurrency and event broadcasting, which makes models more powerful and scalable.

Applications

Cellular automata have been used in several immunological applications for investigating specific mechanisms of the immune system such as affinity maturation and hypermutation in the humoral immune system [52] and tolerance to pathologic rheumatoid factors [53]. Cellular automata are suitable for studying self-organization and colony formation of tumor cells and effect of the environment and stroma on tumor progression. Thus cellular automata and hybrid variant have been extensively used for modeling the dynamics of various diseases such as tumor growth and invasion [54–64]. In these computer simulations, specific diseases’ factors (e.g., rate of spread, binding affinities of cytokines) can varied to obtain a better understanding of the diseases’ factors and how they may impact the spread dynamics of the disease. Further diseases investigated using cellular automata include: HIV [65, 66], Epstein–Barr [67] and M. tuberculosis [68, 69].

Hybrid automata have been used in two types of applications: which involve a switch like behavior, and those which extend cellular automata to a hybrid (mixed discrete continuous) type. The former are mostly used for modeling gene regulatory and signal transduction networks, while the latter are mostly used for modeling at the tissue level including tumors and immune system.

Kam et al. [70] first applied statecharts for modeling the transition of T cells into active or anergy states depending upon protein tyrosine kinase (PTK) activity, costimulation and inhibition. Subsequently, Efroni et al. modeled T-cell maturation in thymus [71] and thymocyte development [72]. Large numbers of individual T cells expressing different receptors were simulated and their interaction with epithelial cells could be studied in real time by simulation. In another study, Naamah et al. [73] studied how lymph nodes orchestrate the interaction between antigens and various B and T cells bearing receptors for these antigens. States and transitions in the statechart framework were used to simulate the activities in lymph node regions, including immune cell behavior, receptors, interactions and movement of cells, cell proliferation and differentiation.

Simulation tools

Popular applications that supports modeling of various biological systems with cellular automata include DDlab and Cell-Devs. The latter is a programming environment for advanced users. CHARON is a programming environment, which has been extensively used for working with hybrid automata models of biological systems. Bio Sketch Pad is a graphical tool for supporting formal modeling, analysis and simulation of biochemical and cellular networks with CHARON. Since its early introduction in 1992 by Celada and Seiden, the software package IMMSIM-C [74, 75] Footnote 1 has been employed numerous times for immune systems modeling using hybrid cellular automata. In contrast with Cell-Devs, the utilization of IMMSIM-C does not require expertise in computer programming. As such, this modeling platform has been used for educational purposes for undergraduate immunology courses at various universities (e.g., Princeton University, University of Genoa). Also, IMMSIM-C was ported as a public web application[76] Footnote 2 where users can readily implement and simulate immune system models online without requiring any computational skills.

Though many programming environments exist for modeling with statecharts, only Rhapsody has been used so far for systems immunology modeling. Rhapsody provides provisions for drawing statechart models and defining transitions and other actions. It translates the statechart into an executable Java/C++ code. Another software called BioCharts, which is specifically meant for biological system modeling with statecharts, is under development [77].

Agent-based models

Formalism

Agent-based modeling is a distinct paradigm compared to the formalisms discussed above in at least two major ways. Firstly, instead of measuring average or ensemble properties of entities such as concentration and total numbers, agent-based models deal with discrete agents, which could be individual molecules, cells, etc. The agents are autonomous, that is, they are not passively manipulated but can make independent decisions. In some models, the agents can learn and adapt their behavior. Secondly, the interaction topology among the agents is a complex dynamic graph. Agents can be free to move within a space. They contact other agents and the environment by chance and, during these interactions, change their state in accordance with a set of well-defined rules. Agent-based models are pertinent for modeling biological systems involving dynamic interactions among heterogeneous components. The model is capable of re-creating macrolevel phenomena by the actions and interactions of microlevel individual agents. This phenomenon is called emergence. Agent-based models pay attention to the behavior of individual agents, which is not possible in formalisms that characterize ensemble properties of entities.

Applications

ABMs are suitable for modeling biological phenomena at the multicellular level. Cells as agents are a natural metaphor. Thus, ABMs have been commonly used for modeling inflammatory and immune response, tumor formation, tissue morphogenesis, etc. Systemic inflammatory response syndrome (SIRS) is an early and successful application of ABMs [78–80]. The system has complex, nonlinear dynamics driven by multiple feedback loops, due to which analysis of isolated components fails to explain the systemic responses. For instance, nitric oxide is responsible for lethal drop in blood pressure in SIRS, but paradoxically, treatment with nitric oxide improved survival in animal models [81]. The ABM was intended to reproduce systemic behaviors from simple rules known to operate at the cellular level. Despite several simplifications, the model reproduced diverse clinical outcomes in SIRS and provided practical insights into improving the treatment protocol. The model also addressed a controversial issue whether stimulation of pro- and anti-inflammatory cellular responses is concurrent or separated by a time lag. ABM of inflammation continues to be an active area of research [82–85] and has been extended to other related applications as well, such as modeling of Chagas’ disease [86, 87], vocal fold inflammation [88], ischemia [89], leukocyte rolling and adhesion [90], etc. ABMs have been recently been combined with ODE and mechanotransduction models in multiscale modeling of human epidermis [91].

Various aspects of immune system response have been modeled using ABMs. Some studies have modeled immune responses to infections, such as formation of granuloma in tuberculosis [92], formation of atheromatous plaque in hypercholesterolemia [93], and the sites of Epstein–Barr virus infection and persistence [94]. Other studies have focused on the internal mechanisms of the immune system such as the recognition of antigens by T cells [95], B-cell activation following ligand presentation [96], T-cell activation and proliferation in lymph nodes [97], B-cell selection in germinal centers [98], unbalanced differentiation of precursor T helper cells to TH1 and TH2 phenotypes [99], formation and recall of antivirus immunological memory [100], etc. Other studies have attempted to build a comprehensive model of the immune system as a whole [101–105].

Simulation tools

Though many software tools exist for agent-based modeling [106, 107], only a handful are useful for systems biology modeling. All agent-based modeling tools require some programming skills. NetLogo is an easy-to-use platform [108, 109] suited for prototyping agent-based models. However, large-scale simulations would require a more powerful platform. Swarm is a historical ABM platform. Swarm models are written in Objective C, which makes it accessible only to experienced programmers, though it now also supports Java. MASON is a Java library with emphasis on cross-platform portability, fast running simulations and reduced space requirements. It is suitable for batch simulations. Repast is one of the most advanced agent-based modeling platforms. It is well suited for working with large and complicated models. Though it is also built in Java, its user-friendly interface called Repast-symphony helps keep programming minimal. Another useful feature is its automated integration with many external tools such as Matlab, R, SQL, VisAD (scientific visualization), JUNG (network modeling), Excel and GIS software. It supports adaptation and learning of agents through evolutionary algorithms, as well as model optimization over Monte Carlo simulations. It supports large-scale simulations though multithreading and distributed computing.

Other formalisms

The discussion above covered some of the primary formalisms that have been used for modeling and simulation of immune systems. However, there is also literature on various other formalisms that have been used for modeling in immunological applications.

Boolean or multilevel network models are used in applications where there is a fixed network and the entities can take on a discrete number of states, ranging from two (boolean) to multiple (multilevel). For example, in a gene regulatory network, each gene may be represented by two states (OFF or ON) or three states (LOW, MEDIUM or HIGH). Using discrete states instead of numerical values makes the model simple and easy to analyze while capturing its essential dynamics. Boolean and multilevel network models have been used for modeling T-cell differentiation [110, 111], T-cell activation [112], inflammatory signaling [113] and host–pathogen signaling [114].

A limitation of Boolean networks is that the states of entities must be updated one by one. Thus, they cannot perfectly model concurrent or simultaneous events. Immunological systems frequently function as concurrent systems. For example, in humoral immunity B-cell production, antigen presentation, B-cell activation, antibody secretion and virus clearance are all concurrent processes. In signaling networks, protein products of one event dynamically serve as reactants or enzymes for the next set of events, and thus, the production and consumption of proteins are concurrent events. Petri nets [115] and process calculi [116] are two other formalisms based on fixed network and discrete representations of state and time, which are especially suited for modeling concurrent and distributed systems. Petri nets have a graphical representation, while process calculi have a textual representation, so Petri nets can be easier to understand for nonspecialists. These formalisms also provide formal analysis and verification techniques (such as model checking [117]), which makes it possible to analyze all possible model behavior and causal and conflicting relationships. So they can answer questions like is it possible to completely eliminate a pathogen, will the parasite be able to complete its life cycle, etc. An integrated Petri net model of the immune system including innate, humoral and cellular immunity has been reported in [118]. Similarly models of immune system in process calculus, such as activation of helper T cells, are reported in [119, 120].

P-systems or membrane computing is an emerging formalism for modeling concurrent and distributed systems. Its specific advantage is that it explicitly models modules or groups of entities separated by compartments and together performing a specific function. Rules specify the evolution and transport of entity groups. This makes it easier to model some phenomena such as selective transport of entities across membranes. P-systems have been used to model the life cycle of infectious viruses [121, 122].

Toward integrative modeling simulation platforms and evolutionary systems immunology

In the previous sections, the principal modeling formalisms and associated simulation tools were presented. In addition, a range of relatively less prominent and more recent modeling approaches were briefly described. Although this survey is far from being exhaustive, it can be readily noted that a wide variety of modeling formalisms exists where there is no single formalism that can be clearly identified as the universal best. This is also reflected in the literature, where new modeling variants are continuously being developed, addressing specific modeling needs and immunological research questions. Although this may appear to be a diverging and ultimately counter-productive research effort, this diversity of modeling formalisms is actually regarded as necessary and desirable by the authors. What we believe is most important, and currently not developed enough in systems immunology, is the access and utilization of modular and integrative frameworks to facilitate the coupling of differing modeling formalisms. This also implies the need for meta-models (and public repository for such models) accounting for the multilevel, hierarchical nature of the immune system. Modularity is here a key feature for such a framework, which would enable the ‘plug-in’ of the different modeling formalisms and simulation tools while preserving coherency of the different modeled entities in both time and space [123, 124].

We propose an extension to in silco systems immunology to investigate the ability of the immune system to adapt to environmental changes through an evolutionary process. Such an evolutionary approach to systems immunology may be of interest due to the natural ability of pathogens (e.g., HIV) to adapt through evolution [125, 126]. To date, a number of theoretical studies have been conducted toward this direction [127–131] where the adaptive properties of artificial immune systems have been examined while involving the ‘innovation’ process through mutations and selections. These algorithms have been shown to be efficient when applied to high-dimensional search or optimization problems [132] and have inspired novel techniques in computer science to realize computer intrusion detection systems [133]. However, when considering the modeling and simulation of natural immune systems evolution, relatively less studies have been reported in the literature [134, 135]. To explore further this research direction, significant challenges would have to be addressed. Particularly, preserving biological plausibility would be a hard constraint to ensure that the in silico evolutionary exploration of immune systems remains within acceptable and reasonable boundaries. A strong focus should thus be placed on model validation. For instance, tracing the in vivo evolution of pathogens [136] is an example of data useful for validating in silico predictions. This proposed evolutionary systems immunology approach may then, ultimately, provide a complementary and potentially useful tool for immunological and epidemiological studies.

Conclusions

We hope that this review lowers the entry level for biologists to systems biology modeling and simulation. We gave pointers to formalisms and also a list of their biological applications to not only illustrate their power and usage but also help biologists with specific problems at hand. This can serve as a reference to quickly identify which formalism could be of interest. Finally, we believe that providing the list of corresponding software is the key to adoption of modeling and simulation within biology research laboratories. This last and practical section makes this review a good entry point for biologists who wish to not just build a theoretical foundation but to start experimenting with computational models.

We gave insights into preliminary questions whose answers can guide the choice of an appropriate formalism. These questions are listed in Sect. 2 (Modeling and Simulation) and further expanded in Tables 1 and 2. Figure 2 gives an overview of which formalism could be used depending upon the biological level or scale of the problem at hand. Finally, the software part (as illustrated by Table 4 for differential equations) guides one to the right software tools to begin modeling and experimenting in a chosen formalism.

We emphasized the fact that there is no universal best modeling formalism. Modeling something as complex as the immune system will require integration of different formalisms in a common multiscale transcale framework. Thus, we outlined move toward integrative modeling as the next challenge for in silico systems immunology. We also proposed modeling of the adaptive properties of the immune system through evolution as a less studied but promising challenge for systems immunologists.

Finally, we would like to emphasize the fact that there is no modeling without data. Therefore, it is of utmost importance to ensure that accurate and appropriate data are available or that data generation and acquisition is included in the modeling strategy.