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Complexity in immunology, until 2000, was classically equivalent to mathematical modeling. Mathematical models were used in immunology, already since the middle of the previous century. These models were traditionally based on ordinary differential equations, difference models and cellular automata. Classical models focused on the "simple" dynamics obtained between a small number of reagent types (e.g. one type of receptor and one type of antigen or two T cell populations.).

With the advent of high throughput methods, and genomic data, the shift in complex immunology shifted more toward the informatics side. The main subjects related to complexity currently studied in immunology are now focused around the concepts of high throughput measurements and system immunology (immunomics), as well as the bioinformatics analysis of molecular immunology. At the level of mathematical models, the analysis shifted from mainly ODEs of simple systems to the extensive use of Monte Carlo simulations. The transition to a more molecular and more computer based attitude is similar to the one occurring over all the fields of complex systems.

An interesting additional aspect in theoretical immunology is the transition from an extreme focus on the adaptive immune system (which has more interesting questions from a theoretical point of view) to a more balanced focus taking into account the innate immune system too.

Short immunology basics

The immune system is a decentralized system, whose main role is the destruction of abnormal cells and external pathogens. The immune system can be broadly differentiated into the innate and adaptive immune system. The innate immune system contains all elements built to detect fix patterns. The main cells involved in the innate immune system are macrophages that "swallow" external objects, Dendritic cells that present peptides to the immune system and regulate its function and Natural Killer cells, which kill malfunctioning cells. The adaptive immune system is a group of cell containing receptors specifically adapted against a given pathogen. The main two types of cells (called lymphocytes) in the adaptive immune system are the antibody producing B cells and the T cells. B cells produce antibodies to kill and neutralize mainly extra cellular pathogens, while T cells main role is to kill virus infected cells. The most interesting aspects of B and T cells is there capacity to produce a large repertoire of receptors able to recognize any antigen through random combinations of existing genes.

History

The immune system is composed of a large number of different molecules (antibodies) and cells (among which lymphocytes i.e. B and T cells). Molecules and lymphocytes specifically “recognise” antigens (macromolecules either foreign or self, e.g. of one's own body) by a chemical reaction. The central issues of immunology have been the explanation of the extraordinary specificity of antigen recognition, how come self molecules are not recognised, what is the mechanism of vaccination. Everyone agreed that these properties are due to interactions among the components of the immune system.

Immunology is one of the fields in biology, where the role of mathematical modeling, and mathematical analysis was recognized the earliest. As early as the 60's (Groves et al., 1969; Marchalonis & Gledhill, 1968)and 70's (Bell, 1971; Cristea, 1971; Dibrov et al., 1977; Dietz, 1979; Hiramoto et al., 1973; Hoffmann, 1975; Ishkov & Lishchuk, 1979; Larson et al., 1976; Levi et al., 1973; Mitchison, 1978; Perelson et al., 1976; Streilein & Read, 1976)of the previous century, mathematical models were used in various domains of immunology, such as:

• Antigen-receptor (receptors are the macromolecules at the surface of lymphocytes which specifically recognise the antigen) interaction (Eichmann et al., 1981; Lafferty & Cunningham, 1975; Norcross et al., 1984; Perelson & Wiegel, 1981; Swain et al., 1983; Thomas et al., 1985).
• T and B cell population dynamics (Asquith & Bangham, 2003; Borisova & Kuznetsov, 1997; Fraser et al., 2002; McLean, 1994; Nowak & Bangham, 1996; Utzny & Burroughs, 2001).
• Vaccination (Allen, 1988; Altes et al., 2001; Ferguson et al., 2003; Iwasa et al., 2003; Jacobs et al., 1993; Khan & Greenhalgh, 1999; McLean & Blower, 1995; Mossong & Muller, 2003; Nowak & McLean, 1991)
• Germinal center dynamics (the immune reaction is localised in the germinal centers of lymph nodes)(Iber & Maini, 2002; Kesmir & De Boer, 1999; Meyer-Hermann, 2002)
• Virus dynamics and their clearance by the immune system (Anderson & May, 1979; Bocharov, 1998; Bremermann & Anderson, 1990; Katri & Ruan, 2004; Klenerman et al., 1996; Neumann et al., 1998; Nowak et al., 1990).

Many concepts central to the existing immunological dogma are actually the result of mathematical models. We here describe only one example:

Receptor cross linking and the immunon theory as developed by Alan Perelson and further analyzed by Carla Wofsy(Bhanot, 2004; Dintzis et al., 1976; Sulzer & Perelson, 1996; Sulzer & Perelson, 1997). This theory uses the fact that low valence antigen cannot activate B cells, while high valence antigens (i.e. antigens with repeated motifs) can activate B cells, even at much lower antigen density (3-4 orders). This led Sulzer and Perelson (Sulzer & Perelson, 1996; Sulzer & Perelson, 1997) to develop the theory and the mathematical model arguing that antigen must aggregate B cell receptor in order to activate B cells. This conclusion is one of the cornerstones of B cell immunology

There are many more examples of the contribution of models to immunology, however a special place must be dedicated to the concept of immunological networks and to the question of self-non self selection.

Jerne interaction networks.

The receptor repertoire is supposed to be full i.e. there should be receptors for each existing molecule, and specifically there should be receptors reacting to other receptors. This led to Jerne (Jerne, 1974a; Jerne, 1974b; Jerne, 1974c; Jerne et al., 1974) to propose the existence of regulatory immune networks. An antigen activates lymphocytes, which in turn produce new receptors. These receptors become the antigens for other lymphocytes and so on and so forth. This network concept was very attractive to theoreticians, especially following the concepts of emergence of a cognitive behavior in neural networks. A large number models of immune networks were developed. Some modeled the large scale behavior, using cellular automata and Boolean networks (Antia et al., 1991; Atlan, 1989; Coutinho, 1991; Hiernaux, 1981; Hoffmann, 1975; Lohse et al., 1993; Nicholls, 1979; Parisi, 1990; Perelson, 1989; Roitt et al., 1981; Stadler & Schuster, 1990; Stewart & Varela, 1990; Urbain et al., 1979; Urbain et al., 1981; Varela & Coutinho, 1989; Varela & Stewart, 1990; Vaz & Varela, 1978; Weisbuch et al., 1990), while some modeled local regulatory networks, using mainly ODEs. The interest in Jerne networks is reduced over time, mainly since no good correlation was obtained between the theoretical models of Jerne networks and direct experimental validation  Note that the concept of small regulatory network in the context of CD25 regulatory T cells and of dendritic cell priming is still present, but it does not require the tools statistical physics.

Self-non self selection.

Regulatory networks are actually part of a larger issue in mathematical immunology, which is the self/non self selection. Lymphocytes with self reactive receptors are supposed to be eliminated (negative selection). Most of the negative selection is probably due to central tolerance (i.e inside the thymus for T cells and the bone marrow for B cells in human and mice). The failure of this tolerance mechanism can lead to autoimmunity autoimmune diseases. Multiple approaches were developd to treat self-non self selection. Some very molecular and based on specific selection mechanisms, while some used a more complex attitude, such as the danger model of Poly Matzinger (Fuchs & Matzinger, 1996) or the Homunculus model of Irun Cohen (Cohen et al., 1993; Cohen & Young, 1991; Cohen & Young, 1992). These models represent a true complex system attitude, as they do not approach the problem only through the detection of its components, but they try to develop a larger point of view. Interestingly, up to this day, there is no unchallenged explanation about the way tolerance is obtained and broken (in autoimmune diseases).

Recent developments

The above mentioned models and questions that were classically raised in mathematical immunology are slowly giving way to a more molecular mathematical immunology. The evolution of the theoretical aspects cannot be separated from the evolution of the experimental techniques. Specifically:

• The full sequencing of a large number of genomes
• The huge advances in molecular biology tools and
• The development of high throughput measuring techniques.
• The ability to measure and model the effects of spatial distribution.

Another important element affecting modern theoretical immunology is the advances in computer power and modeling techniques.  Mathematical immunology still treats the classical issues discussed above. However, there are many new fields of research. We hereby discuss some of those:

Immsim, Simmune and other comprehensive models

Probably one of the most daring attempts in immunology is the effort to produce a systematic model of the immune system. The first significant effort to make such a model was the IMMSIM model developed in the late 80's by Philip Seiden at IBM (Kohler et al., 2000). IMMSIM is based on a spatially extended cellular automaton, and it represents the variability of receptors, antigens and MHC molecules using bitstrings. Note that the bitstring representation of antigen and receptor diversity was adopted by many other authors (De Boer & Perelson, 1994; Perelson, 1989; Sulzer et al., 1996). IMMSIM contains all major components of the adaptive immune system i.e. CD4 and CD8 T cells, B cells and their corresponding receptors. MHC class I and MHC class II molecules, as well as some cytokines. It is however, still a very schematic description of the immune system. A few other approaches were developed. The first developed mainly by Martin. Meier-Schellersheim , named Simmune (Meier-Schellersheim, 2005; Meier-Schellersheim & Mack, 1999) attempts to produce a platform wide and complex enough to allow the simulation of practically any process in immunology. It is more a modeling technique or language than a specific model. A second approach attempts to include all possible details of the immune system and to model the dynamics, using either various mathematical methods, such as Monte Carlo simulations (e.g. immunosim (Al-Ubaydli, 2003)), state-charts (Efroni et al., 2003). One of the most impressing attempts in this direction is the work of Sol Eforni. This model attempts to provide a complete simulation of the spatially extended dynamics in the thymus and to study selection using the simulation (Efroni et al., 2003). The advantage of these general simulations is that they include all details of the current knowledge. Their disadvantage is their complexity that does not allow us to understand in details the reason for the observed dynamics, and their sensitivity to parameter changes.

Spatially extended models

At a more molecular level, one of the big advances in the analysis of complex systems in immunology is the measurements of molecule localization in cells. This led to the discovery of immune synapses . Many membrane dynamics models were developed (Li et al., 2004a) to explain the synapses formation, and the molecular dynamics in the synapse . Membrane dynamics models were also developed for B cells (Chakraborty et al., 2002, some of them by our own group (Nudelman & Louzoun, 2004). These models either assume a fix membrane on a 2D lattice, or an free floating membrane. Another emerging aspect is the receptor dynamics and the interaction between receptors and other membrane components, such as Src family kinases and lipid rafts (Li et al., 2004a). All current models in this field are based on extensive numerical simulations.

Another field of spatially extended simulations is the simulation of germinal center dynamics. While classical models treated mainly, the homogenous dynamics of one or two populations, mainly using ODES, modern simulations study the interaction of multiple spatially extended or homogeneous populations, using mainly Monte Carlo simulations (Kleinstein and Singh, 2001; Kleinstein and Singh, 2003; Meyer-Hermann, 2002; Meyer-Hermann and Beyer, 2002; Meyer-Hermann and Maini, 2005), but also some approximations based on ODEs.

Immunogenetics and immunoinformatics

The sequencing of various genomes and the sequence of various alleles led to the development of a comprehensive immunogenetic database. The IMGT (Bromley et al., 2001; Lefranc, 2001). The database contains the sequence of T and B cell receptor genes (V, D and J for H chain in B cells and beta/delta chain in T cells, V and J for L chain/alpha Chain/gamma chain) for a large number of species. It also contain un updated list of MHC molecule sequences (both classical and non-classical). Finally the database contains a large number of lymphocyte receptors rearranged sequences.

The building of, such a large database is accompanied by the development of a large number of immunoinformatics tools. These include tools for junction analysis (Monod et al., 2004), for immunogene alignment and for phylogeny (Lefranc et al., 1999; Ruiz et al., 2000; Ruiz et al., 1999) . All these tools are based on the application of bioninformatics ideas to immunology. The growing importance of immunogenetic databases shows the transition to a more genomic attitude to modeling in immunology.

Evolutionary immunology

The accumulation of immunogenes in multiple organism, as well as the measurment of multiple rearranged B cell receptor sequences, led the rapid advance of immuno-phylogenetics. The main question, currently investigated is the origin of the adaptive immune system (The part of the immune system that can create random gene combinations to adapt to new pathogens). It is obvious that the adaptive immune system has first appeared in the jawed vertebrates before the split of cartilaginous fish (Eason et al., 2004). However, the source of the source of such a complex system is still unknown. The similarity between the T cell receptor domain and the B cell receptor domain, and the critical role of the RAG1 and RAG2 molecules (RAG1 and RAG2 are the molecules actually doing the random gene joining) in the rearrangement process and their physical proximity, lead many researchers to propose that the origin of lymphocyte receptor rearrangement is the lateral transfer of a transposon into a primeval immune receptor (Schatz, 2004). The main tool used in this domain is phylogeny analysis, and all the mathematical models related to it.

Another application of phylogeny concepts and methods is the analysis of Somatic Hyper Mutations (SHM) in B cells. During germinal center reactions, B cells specifically hypermutate their receptor genes, using AID. These hypermutation at an average rate of one mutation per division within the B cell receptor gene accompanied with clonal expansion leads to the creation of mutated clones. These clones represent a micro-evolution, which can be easily studied in the lab. The analysis of the phylogenetic trees of B cells and their relation to other parameters, such as aging and  autoimmunity is currently investigated by a few groups (Kleinstein et al., 2003; Mehr et al., 2004).

Molecular bioinformatics and Epigenetics

As the level of molecular information increases, so does the detail level of the models used in immunology. One of the special aspects of immunology is the need for models combining signal transduction with genetic rearrangements. Models were developed for the rearrangement process in B and T cells in different conditions (Li et al., 2004b; Louzoun et al., 2002a; Louzoun et al., 2002b; Mehr, 2001; Mehr et al., 2004; Mehr et al., 1999) and for lymphocyte signal transduction. Another important molecular aspect of molecular modeling is the analysis of the antigen processing before his presentation to T cells (Hakenberg et al., 2003; Larsen et al., 2005; Louzoun et al., 2005; Mamitsuka, 1998; Margalit & Altuvia, 2003; Peters et al., 2005; Reche et al., 2005).

High throughput research methods

Immunology was classically a hypothesis and dogma driven field. As such it was one of the last field to move to the data driven high throughput methods currently used in other fields of biology. In the last five years a big advance has been done in this domain. This advance was mediated through the adaptation of classical gene expression and localization techniques from other fields of biology, and the development of new techniques, specific to immunology. This field is mainly experimental, but the results it supplies are at the systematic side of biology, and as such, it is highly related to the science of complexity. Gene expression experiments were done for practically every type of immune situation (Argyropoulos et al., 2004; Choi et al., 2004; Eriksson et al., 2003; Ji et al., 2003; Johnson et al., 2003; Qin et al., 2003; Rangel et al., 2004; Zhao et al., 2001).

An interesting advance more specific to immunology is the use of FISH techniques to localize genes during rearrangement (Singh et al., 2003). These measurements allow us to determine the interaction between different parts of the receptors during rearrangement.

Another tool, specific for the immune system is the development of antigen chips. These chips measure the B cell response to hundreds to thousands of antigen simultaneously and provide a systematic representation of the full immune system (Quintana et al., 2004). The main mathematical tools used in this type of analysis are clustering methods.

Immunomics

The jewel in the crown of mathematical immunology is probably the new emerging field of immunomics. This young field already developed its own journal – immunomic research (http://www.immunome-research.com). The main goals of immunomics is to develop a global view of the immune system. This field uses a mixture of experimental and theoretical tools. Some of the projects currently ongoing in immunomics are for example: the detection of all T cell epitopes (Louzoun et al., 2005), the definition of the full B cell antibody repertoire and the way it changes in different situations (Quintana et al., 2004). The detection of all gene loci linked to an autoimmune disease (Jorgensen et al., 2004; Lauwerys et al., 2005).

This emerging field has still limited results, but, there is a good chance that within less than a decade, immunological modeling will move from theoretical issued based on predefined hypotheses to specific modeling based on the full knowledge of the immune system receptors and targets.

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