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Published online 2016 Oct 6. doi: 10.1007/s11538-016-0214-9
PMID: 27714570
This article has been cited by other articles in PMC.
Abstract
The advances in genetics and biochemistry that have taken place over the last 10 years led to significant advances in experimental and clinical immunology. In turn, this has led to the development of new mathematical models to investigate qualitatively and quantitatively various open questions in immunology. In this study we present a review of some research areas in mathematical immunology that evolved over the last 10 years. To this end, we take a step-by-step approach in discussing a range of models derived to study the dynamics of both the innate and immune responses at the molecular, cellular and tissue scales. To emphasise the use of mathematics in modelling in this area, we also review some of the mathematical tools used to investigate these models. Finally, we discuss some future trends in both experimental immunology and mathematical immunology for the upcoming years.
Keywords: Mathematical immunology, Advances since 2006 and future trends, Innate and adaptive immunity, Multiscale interactions
Introduction
The immune system is subdivided into two main subsystems, the innate system and the adaptive system, which are connected via the action of various cells (e.g., dendritic cells), cytokines, antibodies, etc.; see Fig. Fig.1.1. These two subsystems generally cooperate to ensure the protection of the host (Meraviglia et al. ). The innate immune system focuses on the physical and chemical barriers formed of cells and molecules that recognise foreign pathogens. The adaptive immune system focuses on the lymphocytes’ action to clear these pathogens. The innate immune dendritic cells (DCs), which connect the two immune subsystems, recognise pathogen molecules via invariant cell-surface receptors and then display their antigens on their surface to be recognised by the T cells of the adaptive immune response (Murphy 2012). In addition to the DCs, the two subsystems can be also connected via the action of a particular type of T cell, called the γδ T cells, which are considered both a component of adaptive immunity (since they develop memory) and of innate immunity (since some of their alternative T cell receptors may be used as pattern recognition receptors) (Meraviglia et al. ). We remark here that the notion of immune memory has been associated for a long time with only the adaptive immune response (as mediated by the lymphocytes). However, very recent experimental results have shown also the existence of a type of innate immune memory associated with macrophages (Yoshida et al. ) or with NK cells (Borghesi and Milcarek ). Another distinction between the innate and adaptive immunity is related to specificity: the innate immune response is considered to be non-specific (relying on a large family of pattern recognition receptors), while the adaptive immune response is considered to be very specific (relying on clonally distributed receptors for antigens, which allow cells to distinguish between, and respond to, a large variety of antigens). Finally, both the innate and adaptive immunity include humoral components (e.g., antibodies, complement proteins and antimicrobial peptides) and cell-mediated components (that involve the activation of phagocytes and the release of various cytokines); see Fig. Fig.11.
Brief description of various components of the innate and adaptive immune responses. Both the innate and adaptive immunity include humoral aspects (e.g., antibodies) and cell-mediated aspects (e.g., cytokines)
Many of the complex interactions between the innate and adaptive immune systems and the pathogens that trigger the immune responses (interactions which occur via complex networks of cytokines and chemokines) have started to be revealed in the last 10–15 years, especially because of the advances in genetics, high-throughput methods, biochemistry and bioinformatics. A 2011 review in Nature Reviews Immunology (Medzhitov et al. ) highlighted some of the fundamental advances in immunology since 2001: e.g., improved understanding of Toll-like receptor signalling, improved understanding of immune regulation by regulatory T cells, improved understanding of myeloid-derived suppressor cells. In particular, one of the most cited immunology papers over the last 10 years is a review of monocyte and macrophages heterogeneity by Gordon and Taylor (). Other significant advances made in the last 10 years were in the areas of cancer immunology and immunotherapy (Chen and Mellman ; Kalos and June ), inflammation (Kim and Luster ), autoimmunity (Farh et al. ), infection (Rouse and Sehrawat ; Romani ), and metabolism (Mathis and Shoelson ; Finlay and Cantrell ).
These recent advances in immunology have led to the development of a large number of mathematical models designed to address some of the open questions unravelled by these advances. Particular interest was given to mathematical models for the activation of T cells, models for the molecular pathways involved in the activation, migration and death of various immune cells (e.g., T cells, B cells, neutrophils), models for cancer–immune interactions, as well as models for the immune response against various infectious diseases such as HIV, malaria, tuberculosis, etc. Over the last 10 years, some of these mathematical models have been summarised and reviewed in various contexts: choosing the correct mathematical models for describing an immune process (Andrew et al. 2007), reviewing models for T cell receptor signalling (Coombs et al. 2011), models for various intracellular signalling networks (Janes and Lauffenburger ; Cheong et al. ; Kholodenko ), the evolution of mathematical models for immunology (Louzoun ), non-spatial models of cancer–immune interactions (Eftimie et al. ), agent-based models of host–pathogen interactions (Bauer et al. ), multiscale models in immunology (Kirschner et al. ; Germain et al. ; Cappuccio et al. ; Belfiore et al. ). This large number of reviews of various types of mathematical models, published in both immunology and mathematical journals, is a testimony of the great interest and fast advances in this research field.
In this study, we aim to give a review of mathematical immunology over the past 10 years (i.e., since 2006). To this end, we will cover the breadth of progress rather than any particular research area in great detail. Nevertheless, given the spread of this field, we will only offer a brief description of some of the mathematical models. To ensure minimal overlap with previous reviews published since 2006, we will focus on the most recent models, the techniques developed to investigate these models, and the potential impact of the mathematical results to designing new experimental studies. Since a brief PubMed search showed that a relatively equal number of papers have been published in the last 10 years on either innate or adaptive immune cells (see Fig. Fig.2a),2a), we decided to include in our review mathematical models for both innate and adaptive immune responses. In addition, since the immunological research over the past decade covered a variety of immune responses associated with basic immune activation (via T cell and B cell receptors), viral and bacterial infections, immune response to cancers, inflammation, autoimmunity, etc. (see Fig. Fig.2b;2b; and our previous discussion on recent advances in immunology), we will review mathematical and computational models that were derived to address questions regarding these various immune aspects. Moreover, we will discuss future trends in mathematical immunology, as well as emphasise areas where mathematical immunology methods may be applied beyond their original context.
a Pie-chart description of the number of papers published on PubMed between 2006 and 2016, which focus on different types of cells belonging to the innate and adaptive immunity. b Number of papers published between 2006 and 2016 on PubMed, which deal with various aspects of the immune response: from cancer immunology, to viral and bacterial immunology, immune pathways, etc. The data used to create these figures were obtained from the PubMed database, using the words that appear on the figures labels as the search words. For the red bars (grey on black/white prints) shown in b, we also added “mathematical model” to the search words. Note that the mostly experimental studies described by the black bars and the theoretical/mathematical studies described by the red bars follow similar patterns: a larger number of studies on inflammation and on virus and bacterial immunology, and a much lower number of studies on T cell and B cells receptors, or on single cell transcription (Color figure online)
Role of mathematical models in immunology There are many viewpoints in regard to the purpose of developing mathematical models to describe immunological phenomena: from explaining existing observations and generating new hypotheses that can be tested empirically (Ankomah and Levin ), to understanding which assumptions in the model are useful and generate outcomes consistent with data [and thus help discriminate between different immune hypotheses (Antia et al. ) to uncover basic mechanisms driving some phenomenon (Shou et al. 2015)], organising data resulting from experiments (Shou et al. 2015), offering a selection criteria for ideas that could be tested experimentally in vivo or in vitro (thus reducing the cost and the time associated with performing large numbers of experiments) (Seiden and Celada 1992), evaluating the feasibility of an intuitive argument (Shou et al. 2015), or making theoretical contributions to the knowledge related to immunological systems [by demonstrating the possibility of some outcomes as a results of specific interactions in a particular type of environment, and by suggesting further theoretical problems (Caswell 1988)]. Caswell (1988) distinguished two general purposes for mathematical models: to offer some general theoretical understanding for a theoretical problem (and this understanding does not need to depend on model validation), and to help make predictions (which depends on model validation).
Model validation Throughout this review, whenever we refer to “model validation” we actually mean [as discussed in Oreskes et al. ()] that models are partially confirmed by showing agreement between observation and prediction [complete confirmation of biological models being impossible (Oreskes et al. )].
As stated in Rykiel (1996), the belief that complete model validation is impossible is based on the idea that model falsification should be critical for science. However, Karl Popper’s falsifiability criterion (Popper 1965) (i.e., a theory is scientific only if it makes predictions that can be falsified), which has been already challenged by other philosophers and scientists (Thagard 1988; Mentis 1988; Rykiel 1996), cannot be easily applied to the subtleties of modelling biological phenomena, where many unobservable quantities (e.g., interaction rates) cannot be easily quantified, thus leading to models that cannot be rejected directly (Rykiel 1996) (at least not with our current knowledge). Moreover, as emphasised by Caswell (1988), experimentalists recognise that no experiment represents the last word on the subject, and that an experiment can be usually understood in the context of other experiments that manipulate different factors (and thus might contradict the original empirical experiments), making it difficult to validate mathematical and computational models in immunology.
Parameter estimation In mathematical and computational immunology, many researchers use parameters published in the literature to justify the results of their simulations (both parameters measured experimentally, and parameters taken from other published mathematical and computational models). However, this represents a major issue, since very few laboratories measure and estimate kinetic parameters; see, for example, the studies in Boer and Perelson (), Gadhamsetty et al. (), and their discussion on the difficulty of interpreting kinetic data. Moreover, even in this case, the parameters are estimated for specific experimental systems/models and might differ from study to study (depending on the estimation method used, and on the characteristics of the experimental model, e.g., the inbreed strain of the laboratory mouse used in experiments, or the cell line used in experiments) (Boer and Perelson ; Laydon et al. ). The only rigorous approach (very expensive and time consuming), which could lead to results that could have predictive power, is to estimate in a laboratory all parameters required by a mathematical/computational model (describing a specific system). For simplicity, throughout the next four sections, whenever we refer to models for which parameters were obtained from the literature (in contrast to parameters calculated experimentally) we actually mean that those parameters were not estimated in a rigorous manner and thus they might not depict accurately the kinetics of the system. The studies where kinetic parameters were measured in a laboratory will be emphasised separately throughout this review [see, for example, Sect. 5, where we discuss the computational and theoretical approaches in Zheng et al. () and Henrickson et al. ()].
Multiscale aspects of mathematical models in immunology To capture the complex multiscale dynamics of the immune responses, the review will cover both innate and adaptive immunity across the molecular/genetic scale, cellular scale, and tissue/organ scale (see also Fig. Fig.3).3). We emphasise that in addition to these spatial scales, immunological processes also span a range of temporal scales: from nanoseconds for peptide binding, to seconds/minutes for the production and degradation of cytokines involved in immune cells communication, and to days and months for the proliferation and death of some long-lived immune cells (e.g., memory T cells). However, throughout this study we will neglect the temporal scale (since many of the mathematical models neglect it), and we will focus mainly on the spatial scale. At each of the spatial scales, we will review some mathematical models derived to address some of the questions that have dominated the immune research over the past 10 years. For example, at the molecular scale, the past years have seen the immunology research being focused on: (i) understanding the mechanisms for Tcell receptor binding to peptide major histocompatibility complex (MHC) molecules and B cell receptor binding to antigens, and (ii) understanding the different signalling pathways involved in the activation and functionality of immune cells. This translated into a wide range of mathematical models that have been developed to investigate these aspects in the context of adaptive immune cells [both (i) and (ii)] and innate immune cells [mainly (ii)]. At the cellular scale, the mathematical models followed the advances in the immunology research of diseases (both viral and bacterial), autoimmunity and cancer. The review will summarise models that investigate (i) only the role of innate immune cells, (ii) only the role of adaptive immune cells, and (iii) models that combine both innate and adaptive immunity. At the tissue scale, the few mathematical models for the immune response focused mainly on the immunological aspects of wound healing and scaring, as well as on the immune cells distribution inside solid tumours or granuloma. Finally, we discuss multiscale models, which investigate immune processes that take place across various spatial scales. The variety of mathematical models derived to capture all these different immunological processes is depicted in Fig. Fig.44 (with the models briefly described and compared in “Appendix 1”). For completeness and accessibility, we also added a glossary of mathematical terms in “Appendix 2”.
(Colour figure online) Caricature description of examples of immune processes at molecular, cellular and tissue levels. A different classification of multiscale immune processes focuses on the spatial ranges at which these processes take place: microscale, mesoscale and macroscale. Note the overlap between cellular- and tissue-level processes with the mesoscale spatial level. This is the result of migration of cells between different tissues (e.g., from the lymphoid tissue where cells get activated to the peripheral tissue where pathogens reside). Immunological processes also vary across temporal scales: from nanoseconds (for some molecular processes) to days and even years (for some cellular and tissue-level processes)
(Colour figure online) Schematic description of various types of mathematical models derived to investigate immune dynamics (see also “Appendix 1”). In many cases, these types of models are combined; for example CA models can be coupled with PDE models (which are discretised), PDE models can be coupled with ODE models, CA models can be combined with AB models, etc. There are also many other types of models not depicted here; e.g., probability models (e.g., quantifying the probability of encounters between T cells and dendritic cells (Celli et al. )), algebraic models describing the binding and unbinding of B cell receptors (Fellizi and Comoglio ). All these models are usually coupled with ODEs, to describe multiscale immunological phenomena. For a review of various modelling frameworks in immunology see Kim et al. ()
We start each subsection by presenting a list of references emphasising the variety of studies published after 2006 on that particular topic. Then, we discuss in more detail two arbitrarily chosen studies: one study which emphasises the power or limitations of experimentally validated models and one study which offers a theoretical understanding of a model derived to simulate immunological phenomena.
Note that while we review only research published after 2006, we will also refer to a few types of general papers published before 2006: (i) older papers that put forward or emphasised general ideas regarding the importance and multiple roles of mathematical models in biology; (ii) older papers on the philosophy of science, which we refer to when discussing our take on model validation; (iii) older experimental papers that put forward an important immune concept that we need to refer to (especially in the context of evolution of experimental research).
The article is structured as follows. In Sect. 2 we review mathematical models that address questions regarding the molecular-level immune interactions. In Sect. 3 we review mathematical models for cellular-level immune interactions. In Sect. 4 we review mathematical models for tissue-level immune interactions. In Sect. 5 we give an overview of some of the models derived to investigate immunological phenomena that takes place between different scales. We conclude with Sect. 6, where we discuss the applicability of these models to a broader immunological context, and possible future trends.
Models for the Molecular-Level Immune Dynamics
Two areas in molecular immunology where progress has been made in the past 10 years (see also Fig. Fig.2b),2b), and which generated the development of various mathematical models, are: (i) the mechanisms for T cell receptor (TCR) binding to peptide MHC molecules and B cell receptor (BCR) binding to antigens; (ii) the different signalling pathways involved in cell functionality. Note that while models (i) are developed in the context of adaptive immunity, models (ii) are developed mainly for innate immunity. In the following, we will briefly review the types of mathematical models derived to address these immune aspects.
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The majority of models discussed in the previous paragraphs are described by relatively low numbers of equations. However, there are models that try to incorporate all components of the signalling networks, thus being described by hundreds and even thousands of equations (Danos et al. 2007). These complex models are investigated numerically with the help of software such as BioNetGen, COPASI, Kappa or NFsim—the last one generalising an agent-based kinetic Monte–Carlo method (Faeder et al. ; Sekar and Faeder ; Danos et al. 2007; Sneddon et al. ; Tóth et al. ; Hoops et al. ).
Next, we will discuss in more detail one such complex model, which can offer mainly a theoretical understanding of the system. In addition, we also present a (slightly simpler) model which was validated against some experimental data and further used to make predictions (in the absence of experimental understanding) regarding the synergy between the two components of a signalling pathway and its effect on immune response to infection.
Finally, we remark that the majority of mathematical models studying molecular-level processes are described by non-spatial ODEs. While the use of these equations renders the investigated problem more tractable and the model more easy to investigate, it may not capture all biological phenomena. For example, Chaplain et al. (2015) showed that, while a 2-equation ODE model of the Hes1 transcription factor cannot exhibit the experimentally observed oscillations in both mRNA and protein concentration levels, a spatially-explicit PDE version of the model can account for these oscillations (via the Hopf bifurcation it exhibits). Therefore, more research is necessary to discern between the types of mathematical models that can be applied to model specific biological phenomena.
Models for Cellular-Scale Immune Dynamics
To overview the mathematical models derived to describe cell-level immune dynamics, we will present separately some models that investigate (i′) only innate immune responses, (ii′) only adaptive immune responses, and (iii′) immune responses involving Dendritic Cells (DCs), which act as a bridge between the innate and adaptive immunity. We will also briefly summarise some models that investigate the interplay between innate and adaptive immune responses (without the explicit incorporation of antigen-presenting cells). For this cellular-scale dynamics, in addition to the models describing direct cell–cell interactions, we will also focus on models describing the interactions between cells and cytokines, antigens and viruses (since, despite the molecular action of antigens/cytokines/viruses, the majority of mathematical models treat them as object similar to cells, where the interactions are averaged). We emphasise here that in contrast to the models discussed in Sect. 2, the models for cellular-scale dynamics are described by fewer equations. This allows for a more detailed mathematical investigation of the models, as will be discussed at the end of this section.
Finally, we emphasise that there are many more mathematical models that investigate the cell-level dynamics of the interactions between the innate and adaptive immune responses following pathogen stimulation, following trauma, or following the injection of cancer cells [see, for example, Vodovotz et al. (), Mallet and Pillis (), Hancioglu et al. (), Marino et al. (), Eftimie et al. (), Ankomah and Levin (), Cao et al. (), Pappalardo et al. () and the references therein]. Other models investigate the interactions between the innate/adaptive immune responses and the pharmacokinetics and pharmacodynamics of specific drugs (Ankomah and Levin ). The complex interactions between the innate and adaptive immunity leads to difficulties in parametrising appropriately the models. Next, we will discuss in more detail two models (an experimentally validated model and a theoretical model) that investigate in an integrated manner the innate/adaptive immune responses, to provide some mechanistic understanding of some of the experimentally observed complex immune dynamics.
Since many models for cell-level dynamics are described by relatively few equations, it is easier to investigate them using analytical tools (in addition to the numerical simulations). For example, the complex dynamics between some of the components of the adaptive and/or innate immune responses, or between immune cells and tumour cells, has been investigated with the help of stability and bifurcation theory; see for example Webb et al. (), Liu et al. (2009) and Foryś (2009). These analytical techniques helped address questions regarding the existence of particular types of states (e.g., periodic solutions that arise via Hopf bifurcations), or questions regarding the possible immunological mechanisms behind the transitions between various states.
Models for Tissue-Scale Immune Dynamics![]()
In addition to immunological processes that occur inside cells (at molecular level) and between immune cells (at cellular level), there are also immunological processes that occur at tissue level where cells assemble themselves into multicellular structures. Since these tissue-level processes involve interactions between cells, there is sometimes a very fine line between cell-level and tissue-level models (see also Fig. Fig.3).3). The mathematical models for tissue-level processes are mainly described by PDEs, agent-based or cellular automata models, or hybrid models that combine both PDEs and agent-based approaches—to incorporate the spatial effects of the immune cells on the tissues [see, for example, Su et al. (2009), Sun et al. (), Kim and Othmer (), Kim and Othmer ()]. Nevertheless, there are also a few ODE models that investigate tissue-level processes by ignoring the spatial aspects of these processes and measuring the accumulation of immune cells in the tissues (which can sometimes lead to tissue damage and organ failure, as emphasised by Shi et al. () in a model for immune response to Salmonella infections).
The most common immunological aspects that have been investigated at tissue level are: wound healing (Sun et al. ; Cumming et al. ; Sun et al. ; Adra et al. ), tumour-immune dynamics (Su et al. 2009; Kim and Othmer , ), the formation of granulomas (Su et al. 2009; Clifone et al. ; Fallahi-Sichani et al. ), or the formation of micro-abscesses following bacterial infection (Pigozzo et al. ). Next we discuss in more detail two mathematical models that emphasise the lack of data (at tissue level) to parametrise models, and the potential use of mathematical techniques (e.g., asymptotic analysis) to gain a deeper understanding of the transitions between different regimes in the dynamics of a biological system.
We emphasise that many of the models that describe tissue-level dynamics of immune cells are actually multiscale models, since processes that occur in the tissue are the result of molecular and cellular interactions. (We will return to this discussion in the next section.) Due to the complex nature of these models, it is usually very difficult to estimate model parameters, especially since in tissue there are mechanical forces that act among cells and which are never measured and accounted for in these models. The studies that do parametrise these mathematical models generally use parameter estimates done in isolation, via single experiments, or use parameters estimated for different diseases, cell lines and animal models (Flegg et al. ). Thus, the results of these models are mostly qualitative.
Models for Multiscale Immune Dynamics
As mentioned in the previous section, many of the mathematical models that describe tissue-level dynamics of the immune response are multiscale models, since they focus on the role of molecular-level dynamics—such as changes in the components of various signalling pathways, or in the number of cell receptors – on controlling the formation of cellular aggregation structures inside tissues. However, in addition to the models discussed in the previous section, there are many other models that focus on the macro-scale dynamics of the immune cells. For example, in a 2007 review on the multiscale aspect of antigen presentation in immunity, Kirschner et al. () emphasised that while antigen presentation appears to occur only at molecular and cellular scales, the outcome can be affected by events that occur at other scales (e.g., by increased/reduced trafficking of T cells inside the lymph nodes (LN), which might enhance/reduce the opportunity for antigen presentation by DCs). Since multiscale models are being used more frequently to explore the interconnected pathways that control immune responses across different scales (Kidd et al. ), in this section we expand the discussion on multiscale models started in Sect. 4, by also including multiscale models that focus on the formation of spatial aggregation structures inside tissues. For a more in-depth review of multiscale modelling in immunology—but with a focus on immunological processes that take place at macroscopic level, which includes both tissue-level models and multicompartment models that describe the movement of cells between organs/tissues/compartments—see Cappuccio et al. ().
The majority of multiscale mathematical models in immunology have been developed to investigate phenomena that occur at molecular scale but influence the cell-level dynamics (e.g., cell proliferation, death, cell size, etc.). For example, models have been developed to study the maturation of CD8 T cells in the lymph node as a result of the molecular profile of these cells (as described by TCR and caspase activation, IL-2 production and activation of IL2 receptor, and Tbet protein levels) (Prokopiou et al. 2014); to study the inflammatory response associated with burn injuries (as described by the release of TNF cytokine due to the burn injury, the activation of NF-κB pathway, which triggers early, intermediate and late immune responses associated with increased expression of cytokines) (Yang et al. ); to study the regulation of NFκB signals in the context of macrophage response to M. tuberculosis (Fallahi-Sichani et al. ); to investigate the movement and activation of immune cells in response to receptor levels and antigen levels (Zheng et al. ; Malkin et al. ); to study how the balance between IL-10 and TNF-α (and the binding and trafficking of their receptors) influences the formation of granuloma (comprising macrophages and T cells) following M. tuberculosis infections (Linderman et al. ); or to study the interactions between metabolism (as determined by levels of glucose and insulin produced by β-cells) and the autoimmune response (caused by macrophages) that lead to the loss of pancreatic β-cells (Marino et al. ).
Another class of multiscale models focused on connecting within-host immunological processes following viral infections to between-host epidemiological models for the spread of the infection throughout a population, thus aiming to understand the effect of population immunity on epidemiological patterns (Feng et al. 2012, ; Numfor et al. 2014).
Finally, a completely different class of multiscale models is represented by the kinetic models for active particles (Bellomo and Delitala 2008; Bellomo and Forni ; Bianca 2011; Bellouquid et al. 2013; Bianca and Delitala 2011; Kolev et al. 2013; Bellouquid 2014). These models (given by integro-differential equations or partial integro-differential equations) describe the time evolution of heterogeneous populations of cells that have a certain microscopic state (continuous or discrete), which can represent, for example, the degree of activation of a cell, or the degree of cell functionality. In the context of immunology, they have been used mainly to investigate tumour-immune interactions that involve different types of immune cells, as well as mutated (cancer) cells (Bellomo and Forni ; Bellouquid et al. 2013; Bianca and Delitala 2011; Bellouquid 2014). However, more recent models have been used to study cytotoxic T lymphocytes (CTL) differentiation (Kolev et al. 2012, 2013) or wound healing (Bianca and Riposo 2015). The complexity of these models makes it difficult to quantify them by fitting the model parameters to the data (since at this moment it is difficult to quantify, for example, the flux/death/proliferation of cells that belong to a subpopulation i and have an activity state j). Moreover, the complexity of these models does not allow for intensive numerical simulations to investigate large regions of the parameter space. Nevertheless, these kinetic models could be suitable to describe qualitatively the type of experimental data that cannot be quantified at this moment (e.g, data obtained via immunoblotting techniques)—although, to our knowledge, this has not been done yet mainly due to the lack of immunological knowledge of researchers who develop these kinetic models.
Next we discuss in detail two studies of multiscale dynamics for immune responses: one study that combined modelling approaches with experimental approaches to propose a mechanistic framework for the decision of T cells to make extended contacts with DCs and one theoretical study that investigated the link between HIV transmission in a population and the immunity level in a host, and showed how optimal control theory can be used as a tool to reduce the infection at the level of individuals and at the level of population.
Summary and Further Discussion
Mathematical models can provide a valuable framework to organise in a systematic manner immunological concepts, to show the range of outcomes of various immunological hypotheses that cannot be yet tested experimentally, and to generate new mechanistic hypotheses (based on assumptions made regarding the nonlinear interactions among the various components of the complex systems), hypotheses which can then be attempted to be tested experimentally. In this review, we aimed to offer a broad overview of the progress in mathematical immunology over the past 10 years. Due to the extremely large numbers of mathematical models developed during this time, and the large variety of immunological aspects investigated by these models, it was impossible to provide a detailed description of all these models and the subjects covered. Rather, we aimed to emphasise some immunology areas that have been investigated mathematically, the types of mathematical models developed, and the methods used to understand the dynamics of these models. In terms of mathematical models, we remark a shift from simple ODE models to more complex (and sometimes very large) systems of ODEs, stochastic models that require intensive Monte–Carlo simulations, and hybrid and multiscale models that combine ODEs with PDEs and agent-based approaches (Louzoun ). However, increased model complexity leads to difficulties in model calibration and model use for quantitative predictions, as well as difficulties to analytically investigate these models. Nevertheless, we need to emphasise that the last 10 years have also seen a shift from a qualitative investigation of immunological processes to a more quantitative investigation of these processes. The development of high-throughput methods to generate new data, as well as the development of immunological methods to quantify available data [e.g., quantification of antigen molecules with flow cytometry (Moskalensky et al. ), or detection of antigen-specific T cells (Andersen et al. )] have led to more complex mathematical and computational models that investigate large numbers of interactions (among cells, antigens, cytokines) which occur at different spatial and temporal scales. However, due to the complexity of these new mathematical models they cannot always be fully validated, and the hypotheses generated with their help still have a large qualitative component.
In spite of the very large number of mathematical models developed over the last decade, there are still many immunological aspects not investigated with the help of these models. For example, the recently discovered γδ T cells [which can be considered a component of both innate and adaptive immunity (Meraviglia et al. )] have not been yet the subject of mathematical modelling and investigation. There are also no mathematical models to investigate the type of innate immune memory associated with macrophages (Yoshida et al. ), as well as a few other aspects related to immunological memory [e.g., the role of tissue-resident memory T cells (Mueller and Mackay ), regulatory T cell memory (Rosenblum et al. ), or the effect of antigen load on memory expansion (Kim et al. )]. There are, of course, many other research directions in immunology where mathematical models could propose hypotheses regarding mechanistic understanding of biological phenomena (and we will mention some of them below, in Sect. 6.3).
For a better understanding of the impact of mathematical models in immunology, in the following we discuss: (1) the benefits of mathematical immunology to date; (2) the opportunities to broaden the applicability of some of the models and analytical methods mentioned in this review; (3) the anticipated trends.
The Benefits of Mathematical Immunology to Date
Over the last 10 years, various theoretical models have been able to explain existing experimental observations and to generate new hypotheses regarding different immunological phenomena. These theoretical models have ranged from models for T cell receptor signalling and T cell activation (Coombs et al. 2011), to models for T cell and B cell turnover (Boer and Perelson ), and models for the immune response during specific infections and their associated therapies (Perelson and Guedj ; Canini and Perelson ). For example, Canini and Carrat () used a simple ODE model for the kinetics of human influenza A/H1N1 infections and the anti-viral innate immune response mediated by cytokines and NK cells. The model was fitted to individual influenza virus kinetics data obtained from 44 infected volunteers, and the results of the model predicted that the NK cell activity would peak 4.2 days after inoculation (the authors specified that they had no prior data on cytokine or cellular responses, only viral shedding data). Interestingly, an experimental study published in the following year (Pommerenke et al. ) confirmed that the NK cell activity during influenza infections peaked around day 5. Since the data were only shown for specific days (e.g., days 3, 5, 8; see Fig. 3 in Pommerenke et al. ()) the match between theoretical predictions and data observations seems reasonable. Overall, the majority of mathematical models that have influenced immunology research over the past 10 years were simple models (usually described by ODEs) that could be easily calibrated to experimental data. Nevertheless, also models more difficult to calibrate were beneficial to immunology. For example, the various qualitative models for T cell receptor signalling, such as the kinetic proofreading model that explains pMHC discrimination based on TCR/pMHC bond off-rate (Coombs et al. 2011), have proposed mechanistic hypotheses to shed light on the complex spatial and non-spatial receptor dynamics involved in T cell activation and receptor signalling. Due to a lack of data, these models cannot be confidently parametrised for now (Coombs et al. 2011). Other types of model that have been beneficial to immunology research, despite a lack of model calibration in the absence of relevant data, are the complex systems immunology models that attempt to simulate very large cell signalling pathways (Perley et al. 2014), or models describing complex nonlinear interactions between large numbers of immune cells, cytokines and chemokines (Bianca et al. ; Pappalardo et al. ; Carbo et al. ; Halling-Brown et al. ), with the purpose of achieving a global understanding of the possible outcomes of the immune response following small changes in the components (understanding which is difficult to be obtained experimentally due to high costs).
The continuous advances in quantitative experimental techniques (Bandura et al. ; Andersen et al. ; Newell and Davis ), combined with the demand to interpret ever larger and complex data sets to gain a more mechanistic understanding of the immunological phenomena, will eventually lead to new investigative (modelling and analysis) approaches that have better predictive power and will be more readily accepted by the immunological community when designing new studies. On the other hand, the development of new mathematical and computational models (e.g., kinetic multiscale models of active particles) that can use existent data will also help inform the design of new experiments. It is envisaged that mathematical modelling will become more and more intertwined with experimental immunology, in an attempt to answer fundamental questions about how immune system works and evolves over time [thus following the path taken by theoretical ecology, which now relies on sophisticated mathematical- and computer-based models in addition to traditional fieldwork (Otto and Day 2007)].
Broader Applicability of Some Methods and Models
When thinking about broader applicability of mathematical approaches in immunology, there are two aspects that we need to discuss: (i) broader applicability of certain types of mathematical models and (ii) broader applicability of analytical methods used to investigate specific models.
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Mathematical immunology is a rapidly evolving field, which continues to follow the development of experimental immunology and at the same time tries to influence it by providing qualitative and quantitative assessments of various immune processes. While the power of modelling and computational approaches in immunology has been recognised in various review studies published in high-impact journals (Chakraborty et al. ; Goldstein et al. ; Morel et al. ; Chakraborty and Das ), from an impact point of view the results are still not very encouraging, since these models did not influence significantly the work of experimental immunologists (Andrew et al. 2007).
Despite the fast expansion of mathematical immunology, there are a set of factors that have limited its progress. These factors range from unavailable data to be used by mathematical models, to unavailable models that can interpret existent types of data (e.g., data resulting from Western blots), or more computation power for the numerical simulations of complex models [e.g., 3D agent-based models for particle/cell/protein movement, which sometimes incorporate stochastic rules that require repeated runs to obtain statistical significance (Thorne et al. )]. The progress of mathematical immunology was also limited by an overall lack of interactions between experimentalists and mathematicians. As remarked 10 years ago in Callard and Yates (), there is confusion within the general immunology community about how mathematical models can help understand complex nonlinear interactions. Unfortunately, ten years later this confusion still persist (although at a reduced level). On the other side, mathematicians are not always aware of the most recent developments in various immunology subfields that can benefit from modelling, or of the “hidden” questions in immunology that need an immediate answer to be able to move the subfields forward. Neither are they always aware of the amount and type of data that could be available. This lack of awareness might prevent modellers from asking the right questions which, in turn, creates confusion about the value of modelling. Also, when it comes to using data to parametrise mathematical models, mathematicians are often confronted with a multitude of seemingly similar experimental studies which often hold contradictory results. The variety and interpretation of many immunological observations from in vitro and in vivo experiments was also acknowledged by Zinkernagel (2005). Therefore, discussions with experimental immunologists are crucial in this case to decide which data are most appropriate to use for the validation of the model under consideration. In recognition of this necessary approach, recently there have been suggestions to change graduate programmes in immunology to incorporate training in quantitative and computational biology (Spreafico et al. ).
It is expected that by removing the limiting factors related to data availability, as well as by tightly integrating the efforts of immunologists and modellers would accelerate the progress in mathematical immunology as well as in experimental and clinical immunology. In particular, this approach will lead to:
The changes we mentioned previously in the context of progress in mathematical immunology will be supported by changes in computational and experimental capabilities. The expected increase in computational power over the next few years will lead to a rapid development of 2D and 3D simulations of immune response in tissues and organs—even for large numbers of components of the immune response (using agent-based, cellular automata, PDE models, or hybrid combinations or these approaches). Comparison between these in silico simulations and imaging studies of the immune response [e.g., from lymphocyte activation (Balagopalan et al. ), to tracking immune cells in vivo (Ahrens and Bulte ), or phenotyping immune cells (Mansfield et al. 2015)] will increase the quantitative understanding of spatio-temporal processes in immunology. The increase in computational power will also allow the incorporation into the models of extremely large numbers of possible complexes that can arise in signalling cascades following the multiple ways proteins can be combined and modified (Goldstein et al. ). Finally, possible step changes in the progress of mathematical immunology will likely be associated with the evolution of experimental techniques (Schnell et al. ; Köbig et al. ; Winter et al. ) (e.g., new experimental techniques that could quantify protein levels would lead to a multitude of models for the molecular-level dynamics of these proteins, whose predictions could be tested experimentally).
Since mathematical immunology will continue to follow the developments in immunology, many of the research directions in mathematical immunology that will become most prominent over the next 10 years will follow the main research topics in immunology. A 2011 review article in Nature Reviews Immunology (Medzhitov et al. ) highlighted some of the future research directions in immunology: understanding the complexities in the development and heterogeneity of macrophages, dendritic cells and T helper cells, as well as understanding the immune processes involved in diseases such as cancer (and their escape mechanisms). Other research directions in immunology that are expected to become prominent in the next years will focus on the development of new vaccines for diseases that do not usually induce robust resistance in infected individuals (Germain ) or of vaccines for new infectious diseases, on understanding of metabolic pathways in immune cell activation and quiescence (Pearce and Pearce ; Pearce et al. ), or on understanding how the immune system is integrated with the endocrine and nervous systems (Kourilsky ). Another research direction that will likely become prominent in the next decades is the application of nanotechnology in the field of immunology to improve treatment of various infectious and non-infectious diseases (Smith et al. ). The interactions between nanoparticles and various components of the immune system have been shown in some cases to trigger undesirable effects such as immunostimulation or immunosuppression, and more research will be necessary to improve our understanding of these interactions (Zolnik et al. ). Also, the upcoming years will see immunology research attempting to integrate the controlled environmental conditions associated with the laboratory experiments, into the variable complex world outside the laboratory (Maizels and Nussey ), thus starting addressing questions about the evolutionary processes responsible for observed immunogenic variation, and the importance of the environmental context in various diseases (from parasitic infections, to autoimmunity and cancer). Therefore, from an immunological perspective, it is expected that next decades will see the development of new mathematical and computational models that investigate qualitatively and quantitatively various open questions associated with these prominent research directions in immunology.
Godfather 2 full movie download. From a more mathematical perspective, the research in the next years will likely focus on a few directions, which will include:
To conclude, we emphasise that mathematical immunology is one of the fastest growing subfields of mathematical biology, and the forthcoming years will see this subfield becoming more interlinked with experimental (and eventual clinical) immunology research.
Acknowledgments
Funding for this work was provided by the UK Ministry of Defence under contract DSTLX-1000097390. In addition, R.E. Tamed dino melee dmg per stat ark. would like to acknowledge an Engineering and Physical Sciences Research Council (UK) First Grant (EP/K033689/1).
Appendix 1: Similarities and Differences Between the Main Mathematical Modelling Approaches in Immunology
Appendix B: Glossary of Mathematical TermsA Transition To Mathematics With Proofs Cullinane Download Torrent Download
References
Chapter 1
Logic And Proofs
Chapter 2
Sets And Induction
Chapter 3
Relations And Partitions
Chapter 4
Functions
Chapter 5
Cardinality
Chapter 6
Concepts Of Algebra
![]() Chapter 7
Concepts Of Analysis
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