Computational Approaches and Modelling of Signaling Processes in

Computational Approaches andNo.4 Modelling of Signaling Proc Indian Natn Sci Acad 74 pp. 187-200 (2008)Processes in Immune System

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Review Article

Computational Approaches and Modelling of Signaling Processes in Immune System M RAHUMAN SHERIFF and RAM RUP SARKAR*

Centre for Cellular and Molecular Biology (CSIR), Uppal Road, Hyderabad-500007, Andhra Pradesh

(Received on 23 September 2008; Accepted on 29 December 2008) In recent times, in addition to wide range of experimental methods, signaling processes in Immune system are also being investigated using computational and modelling approaches. Such theoretical methodologies are profound to provide deeper insight in understanding the immune mechanisms, and analysis of such interactions offers new directions as well as predictions. Mathematical models of tumor and immune cell interaction with drug treatment which are developed using experimental analyses have recently been proved to be effective tools for developing risk free and enhanced therapeutic strategies. Hence, it is inevitable for the researcher to identify and employ appropriate approaches which can assist their study. This review presents various computational and modelling approaches which are used to study intra-cell signaling or signal transduction and inter-cell signaling in immune cells and their out comes. We have discussed a broad range of approaches from qualitative to quantitative, which are available, and are employed in the study of immune signaling. Further we have considered different pathways and explained some methodologies to study the signaling networks in T-cells. We have briefly reported the advantages and disadvantages of different methodologies and have suggested application of useful integrative approaches. These theoretical approaches for analysis of signaling processes in immune system can bring out many drug targets to counteract immunological disorders. With the emerging advancement in Systems Biology, which focuses to develop approaches to combine inter cell and intra cell signaling, such integrative approaches will open enhanced scope for further exploration of immune system and associated disorders.

Introduction Immune system is a remarkably versatile defense system which has evolved to protect animals including human from invading pathogenic microorganisms and cancer. It has both less specific component (innate immunity) and more specific component (adaptive immunity) respectively [1]. It is a complex society of interacting immune cells (such as T and B lymphocytes, NK cells, Macrophages, and Professional Antigen – Presenting Cells (APCs) and their sub classes) as well as molecules (such as Antibodies, Complement Proteins, Cytokines, and pharmacologically active substance like Histamine, etc.) together contributing to the state of protection [2]. T-lymphocytes play a central role in maximizing the capacities and abilities of two types of adaptive immunity viz. Cell mediated immunity and Humoral immunity. Among the subtypes of T-lymphocytes, T- helper cells produce cytokines and are more crucial, as they help in the process of determination of humoral and cell mediated immunity, where as the cytotoxic Tlymphocytes mature into an effector cell that can target cancerous and virally infected cells. Activation of T cells requires the presentation of the processed peptide by the Antigen Presenting Cells (APCs) e.g. dendritic cells, Macrophages, B cells, etc. The peptide is presented with MHC molecule of the APC.

These peptides MHC interacts with the T Cell Receptor (TCR) and induces a cascade of signaling events that finally ends with regulation of the expression of various cytokines. These secreted cytokines bind to their respective ligand in their target cells and further induce cell-signaling cascade in (side) them, which in turn affects the structural and functional response of the cell. For example, activated T-helper cells produce Tumor Necrosis Factors (TNF) and Interferons which induce apoptosis in cancer cells and virally infected cells through their respective receptor signaling. Certain cytokines, like interleukins, assist activation and maturation of other immune cells (Tc Cell, B cells, Macrophages etc). Different immune cells secrete different cytokines assisting one another. Thus there exists a network of interaction between the immune cells, mediated by cytokines, along with the direct cell to cell interaction, as in the case of APC – T lymphocyte interaction by various receptors and ligands. Besides cytokine (TNF and IFN) induced apoptosis, activated Cytotoxic T lymphocyte (CTLs) expressing Fas ligand induces apoptosis in cancer and virally infected cells through Fas receptor expressed by them. The later two are involved in the tumor-immune interaction [1]. Hence, complex immune system consists of highly connected networks of interacting components and cells. Behind the function of T cells there are wide ranges of signaling

* Corresponding Author’s E-mail: [email protected]; Tel./Fax: (040) 27192786 / (040) 27160591

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processes functioning within and between the cells i.e. inter-cellular and intra-cellular signaling. Coordination of various inter- and intra-cellular signaling takes place to accomplish defense against non-self. Immune system is the only system that invades all other system of an organism, as it has to protect all of them. Hence immune system is not just an organ level system but indeed is an organism level system. It deals from the level of gene expression within the cells to interaction between the cells through various receptors signaling at organ level. Various theoretical approaches have been proposed so far in this modern era of systems biology, which work in hands with experimentation and help to deepen the understanding of underlying mechanisms of the immune system. Such a complex network often shows a counterintuitive behaviour that can be better understood with the development of mathematical and computational models. These models, which better represent the system, provide a deeper insight into inter- and intra-cellular signaling, which ultimately controls the body defense mechanism and can also be potential tools to predict the broad spectrum of immunological responses [3]. These studies have opened the door for integration of various immuno-informatics and resulting in an integrative approach of Systems Immunology or Synthetic Immunology. Mathematical models of signaling systems represent the key species and their functional characteristics through a list of mathematical equations which can be solved analytically or numerically to get more information about the system. Computational and mathematical models serve as an organized and controlled framework to study the functionality of such systems [3]. Further, modelling and simulations of signaling may yield insight into various mechanisms and can show new directions for predictions as well as experiments. Using modelling approaches a computational or artificial immune system (AIS) can be developed, which can serve as an analogy of natural immune system. Detailed modelling of interaction with cellular and molecular components can help one to uncover relationships of immunological signaling and responses, which sometimes may not be possible to reveal by experiments [3, 4]. In this review, we discuss some of the methodologies, which have been proposed to model and analyze the signaling within and between the important immune cells, the T-lymphocytes. We broadly classified the entire discussion in two ways: firstly, we discuss the modelling approaches used to study the Intra-cellular signaling pathways and secondly, the Inter-cellular interactions. In each classification, we further divide the discussion in two major parts following the specific approaches of modelling, namely, Qualitative and Quantitative approaches. Qualitative approaches include various

M Rahuman Sheriff and Ram Rup Sarkar

approaches based on Boolean formalism, Network theory, Logical Steady State analysis, Visual formalism and few more. Quantitative approaches include probabilistic and kinetic modelling approach. Among these, differential equations are widely used to model inter-cell signaling between tumor and immune cells. Further, we consider different pathways and explain some methodologies to study the signaling networks in T-cells. We briefly report the advantages and disadvantages of different methodologies and in our final discussion we emphasize the importance of such computational techniques, specifically considering an Integrative Approach to study the signaling processes involved in T-cells. With the emerging advancement in systems biology which focuses to develop approaches to combine inter- and intra-cell signaling, such integrative approaches for studying signaling immune may be useful for further exploration of immune system and countering related disorders. Modelling of Inter- and Intra-cellular Signaling Processes Various qualitative and quantitative systems biology approaches have been proposed and used to model the signaling processes involved in the immune system. In the subsequent sections, we discuss and describe those approaches, which involve inter-cellular (between the cells) and intra-cellular (within the cells) signaling processes. Intra-cellular Signaling Intra-cellular signaling or signal transduction is a cascade of events initiated by the assembly of ligand and the receptor. The continuation of signaling depends on the stability of their interactions. If the bonding is weak, the interaction will be brief: not sufficient to trigger the cascade or it may lead to formation of complexes, which produce inhibitory signals. The ligand-receptor complex formation is dependent of various factors such as the concentration of the ligand and other proteins, which can bind to stabilize the complex, the strengths of the bonds and the degree of the enzymatic activity of various kinases and phosphatases that assist the process [3]. Many theoretical approaches have been introduced and employed to analyze these cell-signaling cascades. There are qualitative as well as quantitative approaches and each of them has it’s own pros and cons. Some of the approaches discussed, though not employed in cell signaling but in metabolic and gene regulatory pathways, can be used to model and analyze the cell signaling pathways as well. Qualitative Modelling: In qualitative methods, Boolean Modelling is one of the popular approaches used for modelling signaling and regulatory networks. In this

Computational Approaches and Modelling of Signaling Processes in Immune System

approach, only two states are considered either ON or OFF state [5]. Highly interconnected signaling network can be easily modeled and analyzed using this approach, which sometimes difficult to model using quantitative approach. In certain cases where the change from one state to another is a random process, some probabilistic features can be added to the Boolean model [6]. Boolean approaches can give some details of the network and alterations in Boolean approaches have also been proposed and are discussed subsequently. Kinetic Logic is also a qualitative approach, which is on the basis of kinetic boolean logic. In this approach the relationship between the regulator and effector are in course of time delay. Thus the state of an entity in the signaling cascade depends on which signal reaches first. Based on it more possibilities of a signaling molecule to get activated or inactivated are accounted. Sarkar and Franza [7] showed a simple model for T cell activation signaling with kinase and Ca++ ions intermediates. The model showed oscillation in the activation and inactivation states of Ca2+ ions initiation by the activated tyrosine kinase and it also showed the increase in signaling with co-stimulatory molecules. Among various qualitative approaches, Pathway Logic is another approach following boolean formalism, based on the rewriting logic and it is implemented in the tool, Pathway Logic Assistant (PLA). It has been used in general for signaling pathways and can thereby used for immune system signaling. Analysis of EGF and FB induced Rac1 activation pathway using PL revealed Src and Rac1-GDP as single element Minimal Intervention Set (MIS) i.e. those single component of the signaling pathway, the inhibition of which will affect Rac1 activation [8]. Various other pathways like EGF -EGFR induced pathway, Protein Kinase C regulation network etc. [9] have also been analyzed using Pathway Logic. Other than Kinetic logic and Pathway Logic, there is one more very useful approach, namely, Logical Steady State Analysis (LSSA), which works on the basis of two formalism viz. Interaction graphs and (logical) interaction hyper graphs and is a modified version of the boolean formalism and graph theory [10]. In this approach, the signaling (interaction) networks are modeled as boolean (or logical) equations describing the interactions. A high degree of interlinked boolean equations form Logical Interaction Hyper graphs, which are analyzed under the steady state signal flows. The whole signaling system can be studied under a userdefined scenario by turning ON or OFF any signaling molecules. Such perturbation allows Insilco experiments beyond the laboratory experiments. LSSA is one of the methodologies which allow functional and structural analysis of the signaling network and keeping in mind the usefulness of this method we have shown in two

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model pathways the applicability of this approach and the outcomes, which can be a suitable candidate for future System Biology approach. Model Pathway-I: First, we consider the T-lymphocyte activation-signaling pathway, which is one of the pathways well-studied using computational approaches (Fig. 1(a)). For initiation of activation of the pathway, the ligand, MHC with peptide from the APC should interact with the T Cell Receptor (TCR), which in turn activates cascade of signaling events that results in the activation of ZAP70 (a protein kinase). cCBL is a ubiquitin ligase which shuts off the TCR signaling by down regulation of TCR, ZAP70 and other key proteins in the signaling cascade by the process of ubiqutination, which is followed by proteosomal degradation [10]. The ZAP70 signaling cascade turns on expression of various genes including cytokines like IL2. We studied this pathway through LSSA approach and from our computation we obtain Fig. 1(b-c), which show the application of this technique using model T-Cell Activation pathway (see [10] for the details of formulation of interaction graph and interaction hypergraph). CellNetAnalyzer (CNA) is the tool into which LSSA is implemented and can be easily used to know the interactions of the network [11]. CNA provides the interdependency of the species (signaling molecules; in subsequent description we will use this terminology) in the form of Dependency Matrix (graphical representation of the inter-relationship between two species in a signaling network) from which the Dependency of the species and the Influence of the species can be found (Fig. 1 (d-f)). This in turns helps to identify the most and least influenced and dependent species in the signaling network. The species may be an Activator or Inhibitor or Ambivalent Factor (regulate both in a positive and negative loop), Total Activator or Total Inhibitor (species ‘A’ is called a total activator of species ‘B’ if ‘A’ can activate ‘B’ irrespective of activation or inactivation by other species; similarly for total inhibitor, if ‘A’ is total inhibitor of species ‘B’, then B is always inhibited if A is active, irrespective of the other species in the network). In this pathway-I, many observations can be drawn, for e.g. the MHCpep is the least dependent species, TCR has most inhibitors and cCBL has many activators, MHCpep is the activator of most of the species, cCBL is the inhibitor of most of the species and so on (Fig. 1 (d-f)). In a large network of signaling, one can rank activator and inhibitors based on their role in the network. Most influenced and most influencing species in a network can be found and they can be subjected to perturbation to study the network using LSSA. The dependency matrix shown in Fig. 1(d) is during the later stage of activation and in the initial stage of T cell activation, there will not be activation of cCBL, and

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hence the pattern of dependency varies if the species and interactions are excluded (figure not shown). Thus by adding more and more details to the signaling pathway, the network behave differently and it reveals actual inter dependency of the species. To show this phenomenon, we modified the pathway introducing more details and used the same methodology in model pathway-II to see the differences in the outcome. Model Pathway-II: The model T cell activation pathwayII shown in Figure 2 (a) is a simplistic model with 9 general components. Unlike the pathway-I, there are two inputs to this pathway, as there is a co-stimulator (Y) in addition to TCR. Pathway-II is different from that of Pathway-I with slight difference as it has more number of species to be compared. The inhibitor signals are produced during the later stage of activations as a negative feedback regulator. The plots of dependency and influence of species (Fig. 2 (c, d) respectively) are extracted from the dependency matrix (Fig. 2 (b)). In the pathway- II, from the dependency of species (Fig. 2 (b, c)), it can be observed that the species ‘S’ is regulated by relatively more ambivalent factors than any other species, ‘K’ acts as ambivalent factor for itself, B is the

M Rahuman Sheriff and Ram Rup Sarkar

least dependent species next to L and Y, which are not dependent on any other species as they are ligands from external cells and so on. From Fig. 2 (b and d), it can be found that K acts as an ambient factor for relatively more (two) species than L, B and I (one species) in the signaling pathway-II. Other than L, B, K all other species have some inhibitory effects. Thus, for a large network, similar properties can be extracted and it will help to identify key species in the signaling pathway and can be subjected to perturbation by LSSA which can yield some valuable insight about the network. It can be observed from the Dependency matrix (Fig.1 (d) and Fig. 2 (b)) of these two pathways, the role of species have changed. Many ambivalent factors (yellow color), activators and inhibitors are coming out from the detail study performed in pathway-II. It also helps to identify various feed back loops within the network. In order to get a realistic picture, the maximal pathways from all available databases and literature should be integrated. Similarly one can study T Cell Receptor (TCR) signaling pathway and its cross talk with other receptor (eg, CD28) in the signaling pathways using LSSA. Interdependency of various signaling molecules can be analyzed through

Fig. 1: Analysis of subset of T-Cell activation pathway-I: (a) Model T cell activation pathway (black line/arrow activation, red line/ arrow inhibition); (b) Matrix representation of Interaction graph; (c) Matrix representation of logical interaction hyper graphs; (d) Dependency Matrix; (e) Dependency of Species; (f) Influence of Species

Computational Approaches and Modelling of Signaling Processes in Immune System

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Fig. 2: Analysis of a model T-Cell signaling pathway-II (a) Model T cell signaling pathway (black line/arrow activation, red line/ arrow inhibition) (b) Dependency Matrix (c) Dependency of Species (d) Influence of Species

this method. By insilico deletion and knock out studies various drug targets can be identified and its effect on network during diseased conditions can be studied further. Recent study reveals that the predictions using LSSA are reliable and are well matching with the experimentation, thus validating methodology [10, 12]. Visual formalism of State charts is another primitive approach used for modelling signaling networks where various processes of cell signaling are represented as charts. Different states in activation and compartments are defined. Figure 3 shows a simple model for T cell activation at different levels (intercellular signaling and cell cycle progression and autocrine action of IL2) using this approach and was demonstrated in details by Kam et al. [13]. Rhapsody tool is used for simulation and it helps to compare the dynamic behaviour of the model to experimental data.

Fig. 3: State chart representation of Co-activation of T and Blymphocytes

Petrinet Theory is a bipartite graph theory with two types of nodes i.e. the place (signaling molecules) and transition (activation/ inactivation). Jayasuryan et al. [14] has constructed a model for TCR mediated T Cell activation pathway which ends in production of IL-2 and

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it provides a basis for an extended model using hierarchical colored Petri nets or hybrid Petri nets. The model was created, and simulated using the simple tool, HPSim [14]. Petrinet in general, is widely used for modelling signaling pathways. Heiner et al. [15] showed a petrinet model for apoptotic signaling induced with two receptors (Fas and TNFR1). Using this approach, in Sacchromyces cerevisiae system, mating pheromone response signaling pathway was modeled. Common pathways and components shared in different pathways in the network are identified and theoretical knock out experiments are carried out [16]. These approaches can be applied to study signaling in immune system based on petrinet modelling of MAP kinase cascade, receptor endocytosis and so on. Quantitative modules like ODEs can be used to upgrade it into a quantitative petrinet which forms another approach. Semantic Network Theory has also been used to model and analyze signaling pathways. Macrophage Cell signaling events during invasion of bacteria was analyzed as Semantic Network (SN) model using BioCAD a visual knowledge application package. The analysis revealed various reported and new potential response [17]. Quantitative Modelling: All the above approaches discussed are qualitative but for quantitative analysis of the signaling processes, ordinary differential equations can be used and it can well describe the kinetics of the signaling interactions. Such differential equations can be modified as delay differential equations and stochastic differential equations in order to make the processes more realistic. Ordinary Differential Equations (ODEs): Differential equations are found to be the best tool for dynamic modelling of the signaling reactions. The modelling of receptor signaling in immune system has been found to provide a new insight in the signaling process [3]. By modelling the kinetics of ligand receptor interaction, McKeithan [18] introduced the concept of Kinetic Proof Reading, which describes that signaling can proceed only if the ligand and receptor remains bound for certain threshold time and it can only happen if the ligand is specific to the receptor. If not specific it will dissociate faster that the time bound will not be sufficient to trigger the signaling cascade [18]. Adding up to this concept later studies also revealed that the different phosphoforms of CD3z are induced by different dwell/bound time of the ligand [19]. Revised model proposed by Rabinnowitz et al. [20] was on kinetic discrimination that described a two step model for kinetic proof reading showing single modified TCR results negative and double modified TCR results in positive signal. Later, Lord et al. [21] proposed a new model in which states that positive and negative signals are

M Rahuman Sheriff and Ram Rup Sarkar

generated by ligand triggered TCR by two different pathways. The concept of kinetic proof reading was also extended to FceRI [22]. On this basis it was found that slowly dissociating ligand produces high level of signaling when compared to that of rapidly dissociating ligand. The model was further extended with incorporation of cytosolic messenger molecule that is activated by the receptor. Such model suggested that the response controlled by the secondary messenger can break out the process of kinetic proof reading if there is saturation of messenger activation by two ligands with different dwell time [23, 24]. Rachmilewitz and Lanzavecchia [25] developed a model that explained the serial engagement phenomenon in TCR activation. This model explained that the peptide- MHC from Antigen presenting cell (APC) interact with a TCR and trigger the signaling cascade and it engages with another TCR and triggers it and this continues till the signal get amplified. It engages with another TCR before the signal in the first fades off. Some studies which have used ODEs for modelling signaling process and have scope for application in immune signaling process are discussed below. Mechanistic Modelling: Ordinary Differential Equations (ODEs) are used for mechanistic modelling of cell signaling pathways. Mechanistic model using ODEs has always been useful to understand the kinetic, signaling cascade’s steady state input-output behavior and it’s sensitivity. Such analysis of Hung-Ferrel’s MAPK cascade model using Monte Carlo algorithm explained the bistability and robustness of the system [26]. Bhalla et al. [27] with relevant experiments in mouse NIH-3T3 fibroblasts modeled the (Mitogen Activated Protein Kinase) MAPK and Protein Kinase C (PKC) controlled network with a system of coupled ODEs defining their kinetics and implemented in the neural simulator GENESIS with the Kinetikit inference. The computational analysis along with experiments helped to understand that the MAPK and PKC can control the network with monostable or bistable states and MAPK plays a crucial role for this flexibility to the network. Similarly, Hayer and Bhalla [28] modeled the signaling (activity-dependent movement of a glutamate receptor, AMPAR and a calcium- dependent kinase, CaMKII) at the synaptic connection and studied their stability. They demonstrated that the maintenance of level of the molecules that undergo high movement is dependent on their self-recruitment cycles, and is regulated by their self recruitment and thus the system is in stable state. Ordinary differential equation (ODEs) for power-law formalism using S-System and Genetic Algorithm are used to analyze the gene regulatory pathways [29, 30] and such approaches can also be extended for signaling pathways.

Computational Approaches and Modelling of Signaling Processes in Immune System

Stochastic Differential Equations (SDEs): Stochastic differential equations that are used to model the robust metabolic pathways [31], can also be used for modelling signaling pathways. Batada et al. [32] used a stochastic model to study the diffusion based protein- protein interaction, which is based on the principle of Brownian movement, accounting the probability of the interaction of proteins that result in continuation of signaling. They showed the need for co-localization of mRNA and proteins and its effect over the interaction. On the basis of Queuing Theory, Wadagedera and Burroughs [33] analyzed TCR signaling pathway involved in T-cell activation using a stochastic differential equation model, where the probabilistic events are accounted for the signal to reach the threshold. By fitting the model to the experimental result, the effect of the density of the ligand to trigger the signaling has been studied [33]. Besides the above mentioned approaches various other methodologies have been proposed for analysis of cell signaling pathways. And they can be implemented in analyzing the cell signaling interaction in the immune cells. Some of them are discussed below. Piece Wise Differential Equation: Piece wise differential equation is another tool available for modelling signaling processes. de Jong and Page [34] demonstrated a Piece wise differential equation model of gene regulatory network implemented in Genetic Network Analyser (GNA). Though this is a qualitative approach but helps to identify all steady states in the network. Elementary Mode and Extreme Pathways: Elementary modes are the sets of minimal number of reactions that can exist as a functional unit. Elementary mode analysis is a tool used to analyze all feasible metabolic pathways based on the Stoichiometric theory. The systematically independent subset of the elementary modes forms the

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extreme pathways [35]. Extreme pathways analysis is helpful to study the regulation in metabolic pathways[36]. There exists a unique set of Elementary modes and Extreme pathways for a given biochemical network [35]. Similar approaches and methods can also be used in signaling pathways considering their interaction equations. Probabilistic: Signaling events within the cell are highly probabilistic process, hence a stochastic modelling based on Continuous time Markov Chain (CTMCs) have been used to model signaling pathway using a tool called Probabilistic Symbolic Model checker (PRISM) [37]. Erk pathway has been analyzed as stochastic process to study probability of change in concentration of the signaling molecules and also in Raf1 pathway, nodal stability of the active form at various rate of binding and activation is accounted [38]. FGF pathway is also modeled using PRISM and the model analysis as well as simulation based on the probability of the signaling molecules to interact, activate another, and their stability [38, 39]. In Table 1 and Table 2, we summarise all the above methods and classify them in two main categories as qualitative and quantitative approaches, respectively, which can be used for mathematical and computational analysis of different cell signaling processes. Moreover, in the tables we briefly report the theories behind such approaches and the tools/softwares which can be used for analysis with suitable references for the benefit of the readers. Inter-cell Signaling or Cell-cell Interaction Communication or flow of signal between two cells is the inter-cell signaling which cross over the boundaries of the cell, i.e. from one cell to another. The ligand that triggers the receptor to initiate signal transduction is secreted by another cell or the signaling cell expressing the ligand over its surface can directly interact with their

Table 1. Approaches for Qualitative modelling and analysis of Intra-cell signaling Major Theory

Approach

Specific theory

Tools/ Software

Ref.

1. Boolean

a) Simple Boolean logic b) Pathway Logic c) Probabilistic Boolean d) Kinetic Logic e) Logical Steady State Analysis (LSSA)

Simple Boolean Theory/Logic Rewriting Logic Probability and Boolean Dynamic Boolean Boolean and steady state

– Pathway Logic Assistant (PLA) – – CellNetAnalyzer (CNA)

[5] [8] [6] [7] [10]

2. Graph theory

a) Qualitative petrinet

Bipartite graph

b) Semantic Network

Network Theory

Snoopy and Integrated Net Analyzer (INA) BioCAD, SN-simulator

[16] [17]

3.

Differential Equation

Piecewise Linear Differential equation (PLDE)

Linear differential equation using discontinuous step function

Genetic Network Analyser (GNA)

[34]

4.

Stoichiometry or Interaction Equation

a) Elementary Mode Analysis

CNA, YANA, MetaTool

[35]

b) Extreme Pathway Analysis

Thermodynamically and stoichiometrically feasible routes Thermodynamically and stoichiometrically feasible boundary routes

ExPA

[36]

State Charts

Object oriented modelling

Rhapsody tool

[13]

4. Visual Formalism

M Rahuman Sheriff and Ram Rup Sarkar

194 Table 2. Approaches for Quantitative modelling and analysis of Intra-cell signaling Major Theory

Approach

Specific theory

Tools/ Software

1. Graph theory and Stoichiometry

Hybrid petrinet

Bipartite graphs simulator :

Designer Petri net editor (PED), [66] PedVisor, and analyser: Integrated Net Analyser (INA)

[67]

2. Probabilistic

Probabilistic Symbolic Model (PRISM)

Continuous-time Markov chain (CTMCs)

Probabilistic Symbolic Model Checker (PRISM), BioSpi, SPiM.

[39]

a) Kinetic modelling.

Coupled ODEs

[27]

b) Queuing Theory (Dynamic) c) Mechanistic kinetic model (Dynamic) d) Stochastic Differential equation (Dynamic) e) 4 S- System f) Genetic Algorithm and S-System

Stochastic (SDEs) Ordinary Differential equations Brownian movement, SDEs

GENESIS with the Kinetikit inference –

3. Differential Eequation

4. Visual Formalism

ODEs for power-law formalism ODEs for power-law formalism

Reactive Animation

Ref.

ODE15S in MATLAB Numerical Simulation Simulation Numerical Simulation PEACE1 in a module of AIGNET using C code.

[33] [26] [32] [29] [30]

Rhapsody tool, State charts using Flash, Matlab

receptor on the other cell. Transient cell – cell interaction is profound in immune response, for e.g., T & B lymphocyte co activation, APC – T lymphocyte interaction, activated CTL – cancerous cell interaction in immunological surveillance and so on, and has an essential role in the functioning of immune system. Immunological surveillance is a process by which immune system recognizes and eliminates the transformed cells in the body. In this process the cell mediated immunity plays a vital role. T lymphocytes are activated against the cancer cells. Activated Cytotoxic T lymphocytes (CTLs) with their Fas ligand interacts with the tumor cells expressing Fas (receptor) which results in activation of cell death signaling cascade and there by induce apoptosis in the cancer and virally infected cells. Mainly modelling approaches are used to model the tumor-immune interaction. Many efforts have been taken to model this inter-cell signaling i.e. the process of CTL and Cancer cell interaction. Mostly the effectiveness of therapy that is specific to cancer cells and general chemotherapeutic agents are compared. Various approaches are employed to study the inter cell signaling, few of them are summarised in Table 3 along with appropriate theories and tools and all are quantitative approaches. In subsequent section we discuss some of them briefly.

Reactive Animation: A novel approach used for modelling inter-cell signaling is Reactive Animation (RA) and it has been used to simulate various biological processes, such as, thymocytes maturation and these are integrated dynamic representations. RA is a quantitative approach based on Visual Formalism of State Charts. Complex dynamic properties of the system can be revealed using such approach. A model for Thymic TCell Maturation, with 288 final decisions of a thymocytes regarding which it may respond, was developed by Efroni et al. [40] and various response like positive selection, negative selection, maturation of Thymocytes, interaction with epithelial cells and movement of thymocytes in the thymus in time scale are simulated using Rhapsody tool and the animation is visualized using flash. The percentage of all thymocytes bearing particular markers that are responsive to CCL25 (TECK) and number of thymocytes stem from one progenitor and number of thymocytes die from neglect or from negative selection are determined. T cells that encounter a specific macrophage throughout its history were also identified. Effect of gene knockouts of CXCR4 and CCR9 are analyzed. It has also been shown that there is competition in MHC presentation and it influence the rate of thymocytes migration and development in to CD4+ and CD8+ cells from double positive precursors [40].

Table 3. Approaches for modelling Inter-cell signaling Nature

Theory

Approach

Tools/ Software

References

Quantitative and visual

State Charts visual formalism

Reactive Animation

Rhapsody tool

[40]

Quantitative

Kinetic modelling

Ordinary Differential Eequations

Numerical Analysis Control theory

[42 – 44, 49, 50]

Delayed kinetics

Delay Differential Equations

Numerical Analysis

[53, 57]

Stochastic modelling

Stochastic Differential Equations

Numerical Simulation, Stochastic Algorithm [56, 58 - 65, 67]

Computational Approaches and Modelling of Signaling Processes in Immune System

The process of tumor-immune interaction (or specifically cell-cell interaction) has been modeled using differential equations and seems to be a promising tool for developing clinical strategies for treatment of leukemia. Kinetic Modelling: Ordinary differential equations are found to be a potential tool for kinetic modelling of tumor-immune interaction [41- 44]. A good summary can be found in [45]. Kuznetsov et al. [46] presented a mathematical model of the cytotoxic T lymphocyte response to the growth of an immunogenic tumor. Through mathematical modelling Kirschner and Panetta [47] have illustrated the dynamics between tumor cells, immune effector cells and interleukin-2 (IL-2). Their efforts explain both short-term tumor oscillations in tumor sizes as well as long-term tumor relapse. They have also explored the effects of adoptive cellular immunotherapy on tumor model and have described under what circumstances the tumor can be eliminated. Kolev [48] presented a mathematical model, showing competition between Tumors and Immune system, considering the role of antibodies. The model is developed with statistical methods analogous to those of the kinetic theory and is expressed in terms of a system of integro-differential equations. In recent times, various mathematical models for tumor immune interaction have been proposed to Chronic Myeloid Leukemia (CML). Nanda et al. [49] used a mathematical model to demonstrate tumor-immune interaction in CML. The dynamics of the disease was given as set of ordinary differential equations which described the interaction between the T lymphocytes and the cancer cells. The model has incorporated two therapies to find the regimens that minimize cancer cells number and the adverse effect of the therapy of the lymphocytes. Parameter values for the model were estimated using numerical methods. Thus the ODE model of T cell and cancer cell interaction seems to be useful tool to suggest individual specific therapy regimens. Adaptive control design approaches were also being used to study such interactions models and found to be effective to suggest treatment strategies [50]. Michor et al. [51] developed a simple model for dynamics of CML with different stages of the cancer cell population and treatment with imatinib and also the evolution of disease resistance. Similar model was also proposed by Roeder et al. [52] with quiescent and proliferating cells and the effect of imatinib over them. These two models further can be extended with immune components to study tumor-immune interaction. Delayed Kinetics: In order to model the kinetics with time delay which arise from the time taken for cell division or the development of immune response (for e.g. activation of naïve T lymphocytes into active

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T lymphocytes), researchers are using Delay Differential Equations to explain the dynamic nature of the system. Recently, Kim et al. [53] developed a mathematical model that incorporated the recent experimental data, which suggests that imatinib promotes anti- leukemic immune response. Interaction of T cells to eliminate cancer cell is incorporated in the model using delay differential equations, where the delay is accounted for the time taken for cell division. The mathematical model based on the experimental data helps to enhance low risk clinical and/or therapeutic strategy for CML treatment with imatinib. Yafia [54, 55] studied Hopf bifurcation and stability of limit cycle in a delayed model for tumor immune system with negative immune response. Recently, Banerjee and Sarkar [57] have approached the problem quite differently. The model they developed for spontaneous tumor regression and progression is an interaction between the immune cells (T cells) that destroy the malignant tumor cells, that is, a prey-predator like relationship (which is a very familiar phenomenon in ecological systems). They modelled the two states of T-lymphocytes cells as hunting (cytotoxic T-lymphocytes (CTLs)) and resting (T-helper cells). They have studied the system under external fluctuations and proposed certain thresholds, which are helpful to control the malignant tumor growth. It is always difficult to model the actual phenomenon in the tumor-immune cell interactions. Cells, which react effectively with malignant cells in combination with appropriate co-stimulation are positively selected (generated by genetic re-arrangements in developing T-cell receptors which recognize the antigen) to divide rapidly and become converted. Most of the resulting cells react to eliminate the malignant tumor cells has the antigen as part of its structure, but some of the resulting cells remain as memory cells. Most CTLs also require cytokines from helper (resting) T-cells in order to be activated efficiently. Helper T-cells release interferon gamma (which activates macrophages) and IL-2 (which stimulates T-lymphocytes into CTLs/killer cells/hunting cells). It is interesting to note that this activation process and conversion of resting (or helper) T-cells into hunting cytotoxic T-cells are not instantaneous but followed by some time lag. This occurs due to several reasons, such as, identification of malignant cells by T-cell receptors, storing information as memory cells, processing the cytolytic information to the T-helper cell for activation and simultaneous costimulation etc. All the above processes requires some time interval to materialize, though small but cannot be ignored (a detail description of the above mechanisms has been given in [41]). Keeping in mind the above biological scenario, a modified model [56] of T-cell and malignant tumor cell interaction is proposed using the discrete time delay in conversion of resting cells to hunting cells as well as in the growth/activation of

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hunting cells (a schematic diagram of their model is shown in Fig. 4(a)). Such delay in the interaction is more realistic as it is possible to incorporate the delay by the lymphocyte to recognize the cancer cell. The effects and interactions of tumor cells and immune cells through a system of nonlinear delay differential equations have been explored. In these dynamics, key roles are played by the activation rate, b, (from resting to hunting stage) of the immune cells, rate at which tumor cells are destroyed (a) and the time delay (t) in conversion of resting cells to hunting cells. This study reveals certain thresholds for the activation rate (b) and the tumor destruction rate (a), which are effective to control the unlimited growth of malignant tumor cells so as to control

M Rahuman Sheriff and Ram Rup Sarkar

the oscillations in the system (reduction in the steady state levels of malignant tumor cells for different activation rate is shown in Fig. 4(b)). Further, this analysis shows that it is possible to reach the tumor-free stable steady state by activating more resting cells into hunting cells. The delay time estimation can be used to identify the time at which subsequent doses should be given to reduce the chance of occurrence of further relapse (oscillations), whereas, the other thresholds (for the parameters a,ÿb etc.) can be used to identify the dosages for Adaptive Cellular Immunotherapy which would best suit the patient so as to control tumor progression. The study also helps to identify the region where one can reduce the oscillations in the tumor cells [57].

Fig. 4: (a) Schematic diagram showing formulation of the model system with the delayed conversion of T-helper cells into cytotoxic T-lymphocytes cells/hunting cells. MHC-I means class I major histocompatibility complex/molecules; (b) Different levels of malignant tumor cells in the steady state level for different activation rate (b) values. b = 0 corresponds to the diseased steady state. It shows that a 1.63-fold increase in b leads to a 7.88-fold decrease in malignant cell levels. The comparison has been made from b = 4.3 × 10-9 onwards satisfying the existence condition of the steady state (the other parameter values and details of the study can be found in the original literature [57]).

Computational Approaches and Modelling of Signaling Processes in Immune System

Stochastic Modeling: During recent decades, the disciplines of cybernetics, non-linear dynamics, stability theory and synergetics have emphasized the importance of small fluctuations which can drastically alter the behavior of systems, whether physical, chemical or biological. The study of such transitions is one of the most fascinating fields of cybernetics and biophysics, especially neurocybernetics and immunodynamics. They have enabled a unified vision of the laws which govern the dynamics, control and instability in biological systems [56, 58 – 61] and have shown the potential use of Stochastic Differential Equations. Using a System Theory approach, Roy [62] has attempted an analysis of tumor destabilization. It is also worth observable that extrinsic and intrinsic environmental fluctuations of different modalities such as temperatures, pH, oxygenation (pO2) and radiation inputs seem to have considerable effect on the microscopic properties of the organisms, such as, extinction of cancer population and tumor instability. In a recent study, Sarkar and Banerjee [56] developed a model for spontaneous tumor regression and progression is an interaction between the immune cells, namely, cytotoxic T-lymphocytes (CTLs) and macrophages, which are natural killer cells that destroy the malignant (tumor) cell. They have contended that spontaneous cancer regression can be taken as fluctuation regression and hence, extended the deterministic system to a stochastic one by allowing random fluctuations around the steady state of the system. They investigated the dynamical behavior of the model by computing the strong solution and the study confirms that stochastic mean square stability is achieved under particular conditions for the intensities. This means that, if the intensities of the stochastic fluctuations remain below some threshold values, the density of the malignant tumor cells decreases to a very low value, that is, there occurs a phase transition from macro cancer focus to micro cancer focus. This corresponds to the regression and elimination of malignancy. Moreover, if one consider external stochastic fluctuations, for examples, (i) Radiation flux, (ii) Cytotoxic chemical flux, (iii) Immune cell concentration, (iv) Tumor temperature, (v) Glucose level of the blood impinging on the tumor, (vi) Oxygen partial pressure, that is, oxygenation level in tumor matrix, (vii) Haemodynamic perfusion of the tumor, and so on, then this procedure may be applied as well to get an estimation of the system parameters like rate of destruction of tumor cells by hunting T-cells and rate of conversion of resting T-cells to hunting T-cells, for controlling the growth of the malignant tumor cells. The model which they developed is a general one. However, they put special emphasis to the therapeutic applicability in some of the tumors which are really difficult to treat conventionally: (i) Radio-resistant Ewing’s bone tumor

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(temperature variation therapy) [63], (ii) Lung Carcinoma oxygenation by endostatin therapy [64], (iii) Neurogranuloma [65]. Discussion Unraveling the mysteries of highly coordinated function of immune system underneath extremely complex network and interaction has always been a question of interest. Efforts are always put forth by the biologists to understand how immune system is very systematically and steadily regulated by intra- and inter-cell-signaling interactions. Recent developments in integrative approaches, mathematical and computational methods, have been found to be inevitable tools in understanding such complex systems. Such models provide a platform for extensive analysis of the immune system at various conditions and paves way for engineering the system. In this paper, we have discussed various approaches which have been used for modelling intra- and inter-cell signaling process. Each approach, besides it advantage has its own limitations. Qualitative approaches based on boolean formalism (like Pathway Logic and Kinetic Logic etc.), network theory (Pertinent and Semantic Network) and visual formalism are easy to model and help to understand the process in a simplistic way, but will not always serve the purpose to understand the function of the system as a whole. But qualitative and quantitative approaches (using various types of differential equations) provide a better understanding of the kinetics and serve as a very promising tool. Kinetic modelling of the immune receptor signaling has formed the basis of the phenomenon Kenetic Proof Reading, which has helped to understand the specificity of ligand receptor interaction. Modelling large signaling networks using differential equations needs more mathematical background and the set of differential equations can be solved using computational tools as shown in [27]. But in this case, only few parameter values for the differential equations are available in literature and hence great extents of experiments are in demand. To gain more at little cost, one can go for modeling using Logical Steady State Analysis (LSSA), which is based on the formalism of logical interaction hyper graphs (a hybrid of Boolean and network theory) and can allow modeling of large scale network using the tool CellNet Analyzer (CNA). Though qualitative, LSSA offers enhanced structural and functional analysis of the signaling networks and has been well demonstrated in T cell signaling. LSSA helps to study the signal flow at steady state, besides the dependency of the signaling network and also allows perturbation of the network using theoretical knockouts that open scopes for drug targets identification. In this paper, we have shown this through two model pathways (I and II) using LSSA approach and identified several

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key species (most influenced and influencing) in the network as well as their role (activator, inhibitor, ambivalent factor etc.). Also, we have shown that interdependency of various signaling molecules can be analyzed through this method. After this study using LSSA, with improved understanding over the signaling network, one can go for the kinetic modeling using differential equations for specific key interactions thereby reducing the complexity of the equations and minimizing the experiments. To establish this we have shown the differential equations, delay-differential equation and stochastic differential equations modeling used by several authors exploring the dynamics of cell-cell interaction. In this context, the studies demonstrated by researchers [56, 57] play significant roles showing the reduction of malignant tumor levels through activation of T-cells, which are also discussed in this paper. Through these detail discussions, we want to emphasize that, beginning with LSSA and then switching to detailed modeling in a systematic way would be an appropriate method to explore interesting key features in the immune system. Integration of LSSA with differential equation modeling in this aspect may be the future prospect of Systems Biology for studying immune system as a whole. We hope these approaches will help to identify suitable preventive measures against cancer, and other immunological disorders, as recent reports show validation also will be helpful for auto-immune of such models through experimental evidences. For instance, Saez-Rodriguez et al. [12] demonstrated that T cell model using LSSA predicted unexpected signaling events and validated them subsequently using antibody mediated perturbation of CD28 and genetic knock out of the kinase, Fyn. Mathematical models with tumor immune interaction have been useful for development of low risk therapeutic strategies like drug dosage, vaccination and so on. Such models when developed based on experimental data seems to be predictive about the dynamics of the cancer. As these models well demonstrate the dynamics of the system, it can also be used as a tool for prognosis of the disease. More over, by fitting the parameter values to an individual’s clinical data, it could be possible to develop individual specific therapeutic regime. Michor et al. [51], Roeder et al. [52] and Kim et al. [53] used experimental data to find the parameter values, rather than assumption and using such parameter values in models seem to show an improved result by better demonstrating the dynamics and such models can serve as a framework to predict the future response of disease to therapy and there by help one to fine tune the treatment accordingly. In general, modeling along with experimentation, is a reliable and potential

tool as these two are back & forth processes, so that one can switch between experiment and model to enhance efficiency of either. Since modelling constitutes a deeper understanding from simple to detail mechanism of any system, modelling approaches discussed here could be helpful to build a realistic and systematic study of the immune system. We therefore tried to review available literatures and summarised different approaches, specific theories and major software used to study the systems, in a tabular format, so as to give an overall idea of available techniques and methodologies in this direction. Our paper aims to explore such possibilities to the researchers for future development of mathematical as well as computational techniques which integrates inter- and intra-cell signaling processes, and is one of the main objectives of systems immunology. Acknowledgements The authors are thankful to the Director, CCMB, Hyderabad, for allowing Mr. M Rahuman Sheriff (currently at NCBS, Bangalore) to work with RRS under the CCMB Summer Fellowship Programme to work with RRS. The authors also deeply acknowledge Dr. Somdatta Sinha, Group Leader, Mathematical Modelling and Computational Biology Group, CCMB, for her continuous inspiration, suggestions, comments and help, which improved the quality of the paper. References 1

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