Immunology and mathematics: crossing the divide

Abstract

‘It's high time molecular biology became quantitative, it cries out to a physicist … for modeling. Modeling isn't a crutch, it's the opposite; it's a way of suggesting experiments to do, to fill gaps in your understanding.’ John Maddox, Editor of Nature 1966–73, and 1980–95.

Introduction

A not uncommon view held by cell and molecular biologists (including immunologists) is that the biological sciences and the physical sciences, particularly mathematics and statistics, are somehow fundamentally different and that crossing the boundaries is only likely to lead to confusion. This view of the biological sciences is frequently reinforced by the artificial separation in schools as well as in higher education of those who do biology and those who do mathematics and physics. It is a view that continues to hold currency in spite of major drives towards interdisciplinary science, particularly in mathematics and biology, that are evident from recent developments in systems biology bringing biologists together with physical scientists (mathematicians. computer scientists and physicists) and by the number of recent articles and reviews in Science, Nature and Current Biology that cross this divide.

Mathematical modelling of immune processes such as T-cell receptor (TCR) signalling and the immune synapse^1,2 and T-cell turnover and homeostasis^3–5 has aided considerably our understanding of the biology. Despite this gradual change in perception it is astonishing that at the most recent International Immunology Congress held in Montreal this year there was not one workshop devoted to the many new developments in modelling of the immune system. This review puts forward the argument that mathematics is not just useful but essential for understanding biological systems and that the immune system more than most needs a mathematical approach if our understanding is to move forward beyond the purely descriptive.

Mathematics has many important uses in biology and all biologists employ some degree of mathematical and statistical techniques, for example to work out dilutions, molarities means and standard deviations. This is taken for granted, and is not what will be considered here. Rather, it is the process of modelling where mathematics is used to investigate the behaviour of a biological (immune) system that will be considered.⁶ Before looking at particular examples though, it is necessary to understand what features of the immune system require a mathematical approach for their understanding.

The immune system as a biological network

The immune system consists of an extraordinarily complex network of interacting cells and molecules that work together to ensure an appropriate response is elicited to a particular infection or pathogen. This may be for example the generation of cytotoxic T cells that kill virally infected cells, generating a T helper (Th2) response to helminth parasites or the production of immunoglobulin G2 (IgG2) antibody to Gram-negative bacteria. Producing the right response to an invading micro-organism can be a matter of life and death. The cell and molecular interactions that determine the type of response include cytokines produced at the site of infection; activation and migration of antigen presenting (dendritic cells); presentation of antigen to T cells; and T-cell collaboration with B cells. The enormous effort made over the past 10 or 20 years to identify and characterize the interactions concerned has revealed a staggering number of cytokines, cell surface and signalling molecules that mediate the cellular interactions required to mount an effective immune response. Coming to grips with the scale of the immune system is in itself a daunting task and most research today is still focused on examining the details of particular molecular or cellular components in isolation without taking into account how they work in the context of the intact immune system. But the sheer number of components is only a part of the problem. It is not just the properties of the individual components but the way in which the components interact that is most important for understanding how the immune system functions. Interactions between the molecular and cellular components of the immune system (or any biological system for that matter) are non-linear and often include positive and negative feedback. Complex non-linear systems typically have unusual and non-intuitive properties that may include chaotic behaviour. As a result, their behaviour cannot be predicted or understood just by knowing about the properties of the individual components. Non-linearity is at the heart of the problem and is the reason why mathematical modelling in combination with the more traditional experimental approach is required for a proper understanding of how the immune system works. In order to understand why this is, it will be necessary first to consider what is meant by non-linearity and then explore what the significance of non-linear interactions has for the behaviour of the immune system.

What is a non-linear interaction?

In simple terms, a non-linear system is one in which the output is not proportionally related to the input. All cytokine, surface signalling and cell interactions in the immune system are non-linear. A typical example would be T-cell responses to different concentrations of interleukin-2 (IL-2) (Fig. 1). Below certain IL-2 concentrations, the T cells do not respond. As the IL-2 concentration increases, there is an exponential increase in the T-cell response followed by a plateau maximum or even reduced response at high cytokine concentrations. All immunologists are familiar with dose—response curves like this but few are likely to be aware of the implications this has for the behaviour of the system as a whole. Non-linear interactions can give rise to effects even in quite simple systems that are counterintuitive and in larger systems such as the cytokine network, non-linear interactions may result in quite unexpected behaviour that is not always recognized or taken into account. For example the unpredictable outcomes of cytokine gene knockout experiments such as a functional immune system in IL-2^−/− mice^7,8 and the unexpected effect of leukaemia inhibitory factor (LIF) knockouts on embryo implantation with little effect on haematopoietic development⁹ may be the result of the non-linearities of the cytokine network. A brief digression here is appropriate. The failure of cytokine knockout experiments to result in the predictable phenotype is sometimes considered to be a consequence of redundancy in the immune system such as IL-15 replacing the action of IL-2 by binding to the IL-2R. However, closer analysis makes this argument untenable since only T cells make IL-2 whereas almost every cell type except T cells produces IL-15. It is therefore inconceivable that IL-15 can act in vivo as an IL-2 substitute. It is more likely that that the cytokine network of which IL-2 is a part is stable and can function fairly normally without IL-2. This property of network stability is well recognized. In addition, the often-unpredictable nature of gene knockout experiments may arise from the non-linearities of the system under study.

Figure 1.

Non-linear response of IL-2-dependent T-cell line to increasing doses of IL-2. At low doses, there is little or no response. As the IL-2 concentration increases, T-cell proliferation increases exponentially and then plateaus and in some cases decreases at high concentrations of IL-2.

One of the most important features of non-linear systems is that small changes in the initial conditions can be rapidly magnified to give effects much greater than expected from the initial variation. This is referred to as sensitive dependence. Sensitive dependence on initial conditions can make the long-term behaviour of the system difficult to understand and impossible to predict. This does not mean however, that the behaviour is random or non deterministic. Rather it means that immeasurably small variations in the initial conditions (cytokine concentration or cell numbers for example) can give rise to very different and often unpredictable behaviour known as chaos. To illustrate this property of non-linear systems, a brief digression away from immunology is appropriate to consider a classical example of sensitive dependence in non-linear systems known as the three-body problem. This describes the motion of three planets or celestial bodies all conforming to Newton's laws of motion and is the system in which chaos was first discovered some 100 years ago by Poincaré. In this system, immeasurably small variations in the starting conditions of one or more of the three bodies can give rise to very large differences in their trajectories. In Poincare's own words ‘a small perturbation in the initial state such as a slight change in one body's initial position might lead to a radically different later state than would be produced by the unperturbed system. If the slight change isn't detectable by our measuring instruments, then we won't be able to predict which final state will occur'. In this way Poincaré showed that the problem of determinism and the problem of predictability are distinct problems. A gentle introduction to these ideas can be found in books by Ian Stewart¹⁰ and Edward Lorenz.¹¹ An illustration of the three-body problem following the trajectories of two identical satellites starting from slightly different positions is given in Fig. 2. Rapid divergence of trajectories as shown in the figure can occur even if the differences in the initial starting conditions are too small to be measured. In this case, the outcome is unpredictable. Since we can never measure the starting conditions of any complex system (such as concentrations of cytokines or expression of cell surface molecules) to an infinite degree of accuracy, the long-term behaviour of such a system is essentially unpredictable even though it is deterministic. In this sense, non-linear systems are so complex that they defy complete understanding. Sensitive dependence in chaotic systems can however, bring advantages as well as disadvantages. If a chaotic system is understood well enough, it may be possible to control it using only very small actions. This has been amply demonstrated in a wide range of experimental systems ranging from cardiac contractions to lasers.¹²

Figure 2.

The three body problem. The trajectories of two satellites (A and B) beginning near the right hand planet with similar but not identical starting conditions under the influence of two planets (•). Initially the two trajectories are quite similar and proportional to the difference in starting position but they soon diverge dramatically and in an unpredictable manner. This illustration of sensitive dependency shows how very large effects can result from very small differences in initial conditions. In fact, immeasurably small (approaching infinitely small) differences in the starting conditions can lead to the same divergence in trajectories.

Good biological examples of how changes in initial conditions can have disproportional effects in outcome abound in immunology. As a simple example, doubling the strength of a signal such as antibody binding to T-cell or B-cell receptor may have no effect on the response or may result in a response many times higher than the increase in signal. This can be seen in almost any dose—response experiment. Perhaps less obvious are gene dose effects such as reported for genetic control of tumour necrosis factor (TNF) production.¹³ In these experiments, TNF-α secretion in response to lipopolysaccharide (LPS) was compared in wild type (TNF-α^+/+) mice; heterozygous (TNF-α^+/−) mice in which one of the alleles coding for the TNFα and lymphotoxin (LTα) genes (both are found in tandem on chromosome 17) was deleted; and mice in which both alleles had been deleted (TNF-α^−/−). As expected, LPS administered in vivo induced high serum levels of TNF-α in wild type mice and no detectable levels in homozygous (−/−) mice. Surprising, only very low levels of TNF-α were detected in the heterozygotes (60 times lower than the wild type) with a corresponding reduction in LPS induced mortality, despite the fact that gene dosage was reduced by only a half. Furthermore, the difference in TNF-α production by monocytes from wild type+/+and heterozygous⁺-mice increased with the dose of LPS and over time from about twofold at 4 hr to between 20- and 100-fold at 18 hr, which is characteristic of the non-linearities of a positive feedback loop.

At first sight, chaotic behaviour of non-linear systems seems far removed from the requirements of the immune response which must perform in a reliable way and cannot be susceptible to every tiny perturbation in the cells and molecules that control it. It is important to appreciate however, that whereas individual behaviour in a chaotic system (expression of cell surface molecules or production of cytokines for example) may be very sensitive to initial conditions, the overall statistical properties of a chaotic system can be quite robust. For example, even though it may not be possible to predict the behaviour of an individual T cell contained in a population of interacting T cells, it is usually possible to obtain robust and predictable responses from the whole population. Furthermore, there can be significant advantages to operating in a regime where non-linearities give rise to sensitive dependence on initial conditions as small but properly timed interventions can give rise to large-scale desired effects that might include rapid expansion of effector T cells exposed to relatively small increases in pathogen load and/or activation signals delivered by antigen-presenting cells (APC).

As the immune system is made up of components that interact in a non-linear fashion, it is likely that complex behaviour can occur from even a small number of interactions. One interesting example is the repeated rise and fall in TNF-α levels over many days in the aqueous humour of the eye following allogeneic corneal transplantation.¹⁴ Although it seems reasonable to assume that such complex behaviour must arise from very complex control mechanisms, this may not be the case. A simple mathematical model including only effector cells making TNF-α, regulatory cells producing an inhibitor of TNF-α such as sTNFR and an activation signal from the allogeneic transplant gave rise to fluctuations in TNF production similar to those found experimentally.¹⁴ Even such a simple model may help explain spiking temperature during infection and the relapsing/remitting nature of some inflammatory diseases such as juvenile chronic arthritis or familial Mediterranean fever through cytokine feedback regulation.

How does modelling help?

There is a great deal of confusion within the general immunology community about what mathematical modelling is and how it can help understand complex non-linear biological systems. Perhaps the main misconception arises from the distinction between modelling by simulation and what might be called analytical modelling.¹⁵ There are a growing number of attempts to use computational methods to simulate immune responses using models that attempt to include as many of the components of the immune system as possible. These computer programs may give realistic outputs consistent with experiment (and may be useful in seeing what the immune system can do) but they do not usually take into account the nature of the interactions between the various cellular and molecular components nor do they usually give much biological insight into how the system works. An alternative modelling approach attempts to exclude as many of the extraneous components as possible but still capture the essential behaviour of the system under study. In the latter, mathematical tools are used to investigate the behaviour of the system in much the same way that experimental systems do by excluding many of the known components or interactions. Biologists often worry when a mathematical model does not include all or most of the known variables (components) but it is possible to gain important information about the behaviour of the system from simplified models. This is not so different from experiment in which only two or three variables are considered (e.g. number of cells of a given type, concentration of stimulus and time). It is often the case that a simple model giving a reasonable approximation to the real world is more useful than a complicated model because it is possible to understand how it works. Alternatively, models can be useful when they show what cannot work rather than what does. Although more complex models may give a better fit to the available data, the key processes giving rise to the behaviour being investigated may be more difficult to identify. The hope is that modelling will allow a better understanding of the underlying principles that govern the behaviour of the biological system in the same way that Newton's laws enabled planetary motion to be understood from basic principles rather than inferred from detailed descriptions based purely on observation.

In order to illustrate how mathematical modelling can be used to investigate the behaviour of the immune system three important immunological problems will be considered, namely T-cell receptor signalling, Th1 and Th2 differentiation and T memory cell homeostasis. For the most part the discussion will not resort to any mathematical equations.

T-cell receptor signalling

The usual biological paradigm for T-cell receptor (TCR) activation shown in Fig. 3(a) is of a single TCR binding to peptide expressed on major histocompatibility complex (MHC) class I or II on the surface of an APC. This cartoon is very useful but it does not take into account the facts that T cells express many receptors of the same specificity and APC express an array of different peptide MHC complexes, many with self peptides. It has been estimated that CD4⁺ T cells can recognize and respond to as few as 100–200 foreign peptide—MHC complexes out of a total of 10⁵−10⁶ peptide—MHC complexes expressed on an APC^16–18 and CD8⁺ T cells may be able to respond to a single foreign peptide MHC complex.¹⁹ This is important because the TCR binds to all of the different peptide MHC complexes expressed by the APC with some finite, albeit in some cases low, affinity (Fig. 3b). Yet the T cell can discriminate between a few interactions with high affinity and the majority of low affinity interactions. Because mature dendritic cells may present up to 10⁵−10⁶ peptide—MHC complexes, each individual TCR must have a false positive rate of less than 1 in 10³−10⁵ if it is to distinguish pathogen signals from self-antigens. This specificity is clearly seen by the ability of T cells to discriminate between altered peptide ligands differing by only a single amino acid residue.²⁰

Figure 3.

T-cell receptor recognition of peptide MHC complexes on antigen presenting cells. Typical model in immunology text books is of a single TCR binding to a high affinity peptide MHC complex on APC (a). A more realistic model is of multiple TCR binding to a range of different peptide MHC complexes with different affinities, some low and some high (b). Discrimination between high and low affinity binding and the contribution to T-cell activation of low affinity binding is taken into account by mathematical models.

Binding of individual TCR to an activating peptide—MHC complex is of quite low affinity compared to antibody—antigen interactions (1000-fold less) and most experiments have suggested that the important feature for T-cell activation is the length of time the TCR is bound measured by the dissociation rate or k_off and not affinity per se. Experiments with peptides altered at single amino acids have shown that the longer the binding continues (low k_off) the more active the peptide although there is also data supporting an optimal duration of binding.^21,22 It has been shown that small changes in binding time of about 30% (equivalent to a 3-s decrease in t_1/2 can result in a 1000-fold difference in the biological potency of the ligand.²³

One of the first significant successes in using a mathematical approach to understand TCR signalling is the kinetic proofreading hypothesis.²⁴ In this model, a signal is not generated immediately the TCR binds to a peptide—MHC complex. Rather, there are a series of intermediate steps beginning with receptor dimerization. Although not specified in the original model, the subsequent steps might include phosphorylation of tyrosine residues followed by docking of proteins containing the src homology 2 (SH2) region. The series of separate signalling events results in a time delay between TCR binding and downstream signal transduction. Peptide—MHC complexes that bind a TCR for a short period (high k_off) fail to complete the sequence and the TCR does not signal, while peptide—MHC complexes that bind for a longer time (low k_off)) lead to T-cell activation. Mathematical analysis of this model showed that the process gives good discrimination between peptide MHC ligands that bind for a short time and those that bind for a long time. Variants of this model allowing for inhibition of signalling by peptide—MHC complexes that dissociate before full TCR activation show that this enhances the ability to discriminate between ligands with low and high off rates.^25,26

TCR discrimination may also arise from non-linear feedback between kinase and phosphatase molecules resulting in a threshold for TCR activation.²⁷ Positive feedback of protein phosphorylation events activated by the TCR can result in a phenomenon called hysteresis with profound implications for understanding T-cell receptor signalling by enabling the TCR to act as a bistable switch (Fig. 4).²⁷ Hysteresis is a very important concept to understand. It means that the behaviour of a system at first remains constant or doesn't change much as the system is subjected to an increasing stimulus and then the behaviour makes a sudden jump at a particular value of the stimulus. Then, if the stimulus is decreased, the jump back to the original behaviour does not occur until a much lower value. In the region between the two jumps, the system is bistable, which means it can exist in either the activated or inactivated state with the same strength of signal. In the case of TCR activation, the presence of hysteresis means that ligand engagement with the TCR must reach a critical threshold level before the cell is activated. The cell will then remain activated if the level of TCR engagement is decreased until a threshold lower than that required for the initial activation is reached (Fig. 4). As a result, low-affinity TCR engagement may be sufficient to keep the cell in an activated state after an initial high affinity activation signal. As with kinetic proofreading, this model provides a possible explanation for the ability of the TCR to discriminate between ligands with high specificity and sensitivity and suggests that TCR interactions with low affinity self peptide MHC complexes as shown in Fig. 3 may contribute to TCR activation and significantly enhance T-cell signalling. It also suggests that autoantigen activation could prolong an autoimmune response after an initial high-affinity activation signal to an antigen that cross reacts with a self antigen has gone. This factor is rarely taken into consideration in biological models of TCR signalling and may be extremely important for understanding thymic selection and ‘self’—‘non-self’ discrimination. Further refinements include the notion of activation-threshold tuning where the threshold of activation is modified according to the cells' previous exposure to self antigens. Modelling of this process has shown how it could contribute to self-non self discrimination and development of the T-cell repertoire.²⁸

Figure 4.

Hysteresis of TCR activation. Hysteresis is an important concept to understand and is defined in the text. In the example shown here, as the TCR signal strength increases from left to right, there comes a point (↑₂) when T-cell activation, which could be measured for example by measured by phosphorylation of T-cell signalling proteins such as Lck, suddenly increases and the cell becomes activated (A). In the reverse direction, the T cell remains activated even when the TCR signal has decreased below the threshold point at (A) until the lower threshold point (B) is reached (↑₁) when the T cell returns to its resting state. This shows how prolonged T-cell activation could occur in response to low affinity ligands that generate an activation signal above (↑₁) but below the level of (↑₂) induced by the initial activation event with a high affinity ligand. See reference ²⁷.

Th1 and Th2 differentiation

The factors that determine Th1 and Th2 differentiation have been the subject of intense investigation since the discovery of these two functionally important T-cell subpopulations. A large number of cytokines and cell surface signalling molecules as well as intracellular transcription factors have been identified but the overall process is still not well understood. A simplified diagram of some of the factors involved in Th1 and Th2 differentiation is shown in Fig. 5. Differentiation of uncommitted T cells into Th1 and Th2 subpopulations depends on intracellular events controlling expression of transcription factors T-bet and GATA-3 and on interactions between cells mediated by cytokines, particularly IL-4 and interferon-γ (IFN-γ). A great deal is known about both the intracellular and extracellular events involved in Th1 and Th2 differentiation, but how these are integrated in T-cell populations or indeed why extracellular cytokine control is required for commitment to Th1 or Th2 after a decision has been made at a transcriptional level is not at all understood. Without a more complete understanding of these processes it will be difficult to predict the outcome of any particular response and perhaps more importantly how to manipulate the response therapeutically in Th1 and Th2 diseases.

Figure 5.

Cytokines acting between cells help determine Th1 and Th2 differentiation. This is not by any means a complete illustration of the cytokines involved but it makes the point that cytokines produced by differentiating Th1 and Th2 cells work in both positive and negative feed back loops to influence the differentiation process.

It is clear even from the simplified model of Th1 and Th2 responses shown in Fig. 5 that non-linear cytokine interactions with positive and negative feed back loops such as IL-4 inhibition of IL-12 production^29,30 and expression of IL-12Rβ2,^31,32 and IFN-γ enhancing expression of IL-12Rβ2^31,32 make the dynamics of this system very difficult to understand. In this case mathematical modelling has a clear role to play. To illustrate this, the intracellular control of T-bet and GATA-3 expression and how the expression of these transcription factors might be integrated with extracellular cytokine signals to regulate Th1 and Th2 differentiation will be considered.

IFN-γ and IL-4 exert their effects on Th1 and Th2 differentiation by controlling the expression of transcription factors T-bet³³ and GATA-3.³⁴ These transcription factors interact with each other and the cytokines that induce them through feedback loops³⁵(Fig. 6). IL-4 binding to its receptor on Th precursor (Thp) cells activates the signalling factor signal transducer and activator of transcription-6 (STAT6), which then translocates to the nucleus and rapidly induces the expression of GATA-3.^34,36 Other signalling pathways such as CD28³⁷ and OX40³⁸ may also induce GATA-3. Expression of GATA-3 is followed by induction of the transcription factor c-MAF, a potent IL-4 gene specific activator.^39,40 This forms a positive feedback loop for IL-4 induction of GATA-3 and Th2 differentiation. At the same time, GATA-3 induction inhibits Th1 differentiation both by increasing IL-4 production, and by inhibiting the master Th1 transcription factor T-bet. Conversely, IL-12 and IFN-γ signalling activate STAT1 which leads to expression of T-bet.^33,41 T-bet activates the IFN-γ gene by chromatin remodelling leading to secretion of IFN-γ and increases expression of the IL-12Rβ2 chain, further enhancing both IFN-γ and IL-12 signals.^32,42 Either directly or indirectly T-bet also inhibits GATA-3 expression and IL-4 production. It therefore provides positive feedback for Th1 development and negative feedback for Th2 development. At a population level, the fate of T cell differentiation therefore depends crucially on both the dynamics of GATA-3 and T-bet expression within individual T cells and the effect of cytokines particularly IFN-γ and IL-4 produced by the differentiating Th subsets and acting on the population of differentiating Th cells.

Figure 6.

Signalling events that determine T-bet and GATA-3 expression in a single T cell. The essential features that determine the outcome of the differentiation process are the positive and negative feed back loops indicated in the diagram. Mathematical modelling of these show hysteresis enabling the cell to behave as a bistable switch as shown in Fig. 7.

The non-linearities in both transcriptional control and cytokine regulation of Th1 and Th2 differentiation make the biology here very difficult to understand. In order to see how intracellular T-bet and GATA-3 expression combine to control Th1 and Th2 differentiation, a mathematical model in which the non-linear interactions and feedback loops shown in Fig. 6 were described by a system of ordinary non-linear differential equations.⁴³ It does not necessarily matter if some of the details of the signalling pathways involved are not included in the model as long as the essential features that determine the biological behaviour are considered. This is no different to an in vitro experimental approach in which only those components the experimenter wants to consider are included. Analysis of the dynamical behaviour of this model showed that the T-bet—GATA-3 interactions act as a bistable switch in which the cell can express either a Th1 phenotype or a Th2 phenotype but not both. To illustrate this, GATA-3 expression in a single cell responding to Th2 promoting signals will be described but the conclusions apply to T-bet as well. Figure 7 shows schematically the steady state expression of GATA-3 in a single Th cell activated by antigen in the presence of extrinsic Th2 polarizing stimuli but in the absence of any Th1 stimuli. At low levels of extrinsic Th2 promoting signals, the cell is in a steady sate with constant low levels of GATA-3 and T-bet expression. This may be interpreted as a Th0 state in which a recently activated cell is beginning to produce both Th1 and Th2 cytokines at low levels. This Th0-like state will continue until the extrinsic Th2 promoting signal is increased to a critical threshold level (↑₂) when the cell shifts very quickly from one steady state of low-level GATA-3 expression to another steady state of high level expression. It is important to note that the positive feedback through autoactivation of GATA-3 is essential for the existence of this high level state. The cell will then remain in this stable state producing high levels of GATA-3 and no T-bet even if the initial Th2 signal is reduced below the original threshold level (↑₂). Only when the Th2 signal is reduced to a much lower level shown in the figure by (↑₁) below the original activation threshold will GATA-3 expression fall back to the lower Th0 state. This simple model shows that the cell acts as a bistable switch, requiring a threshold of extrinsic Th2 signalling such as IL-4 after TCR activation to reach a high level of GATA-3 expression. This is then insensitive to small reductions in the Th2 signal. An entirely analogous picture holds for the relation between T-bet levels and the stimulation of T-bet production by IL-12, IFN-γ or IFN-α, when GATA-3 levels are low and no Th2 stimuli are present.

Figure 7.

A schematic diagram of how a single cell responds to a Th2 signal by expressing GATA-3. The curve represents the steady state GATA-3 expression reached by the cell in response to Th2 promoting signals. The x-axis represents the strength of the external stimuli that induces expression of GATA-3 such as TCR signalling in the presence of IL4. Here there is no Th1 stimulus and so T-bet expression remains low. Recently activated cells start with low levels of T-bet and GATA-3 expression. Increasing stimulation by extrinsic signals up to a threshold level (↑₂) increases GATA-3 levels slowly (region A). At the threshold, the cell rapidly approaches a state of high GATA-3 expression (region C) in which autoactivation of GATA-3 occurs at its maximum rate. In this state, the level of GATA-3 is relatively insensitive to fluctuations in the external stimulus. However if the stimulus is reduced below the lower threshold (↑₁) the cell reverts to low level GATA-3 expression. Continued extrinsic signalling at a level greater than (↑₁) but below the upper threshold (↑₂) is able to sustain high levels of GATA-3 during the first few rounds of division.

Biologically, this means that once a cell has been induced to express GATA-3 or T-bet by an initial Th2 or Th1 signal it will continue to express GATA-3 or T-bet at high levels even when the original extrinsic stimulus has been reduced as long as the signal strength remains above the lower threshold level (↑₁). This may give an impression in experiments that Th2 and Th1 differentiation is irreversible, but this is not necessarily true. If STAT-mediated or other Th2 or Th1 signalling is reduced below the lower threshold level (↑₁), the steady state characterized by high GATA-3 or T-bet expression is lost and the cell returns to a Th0 state. Feedback through IL-4 production for example may keep the Th2 cell in an activation state above the lower threshold and prevent reversion to the uncommitted state until the switch can be made permanent by chromatin rearrangement.^44,45 A similar model describing bistable expression of GATA-3 has been described,⁴⁶ although in this case the authors argue that once the high-level state of GATA-3 expression is reached it can be sustained solely by autoactivation of GATA-3 and so may be irreversible without subsequent modification of the rates of transcription or decay of GATA-3.

Experimental work has indicated that GATA-3 and T-bet expression is mutually exclusive in a single cell.⁴⁷ The dose-dependence and strength of the cross-inhibition of GATA-3 expression by T-bet and vice versa are critical for the behaviour of the model in this regard. The mathematical model also shows how a cell expressing high levels of GATA-3 maintained by continued signalling through the IL-4R above the lower threshold level (↑₁) is unable to increase T-bet expression even in the presence of a strong Th1 stimulus because of direct negative feedback of GATA-3 on T-bet and by down-regulation of IL12Rβ2.³⁶ This would give the appearance of irreversible commitment unless the original Th2 signal was reduced to a level well below that required to trigger the Th2 response in the first place. Essentially the same argument applies for Th1 responses. One important prediction that follows from this model is that it may only be possible to shift between committed Th1 and Th2 responses by reducing the initial activation signal. This is supported by experiments where a healing Th1 response to Leishmania induced by IL-12 in mice required reduction in antigen load by treatment with Pentosam.⁴⁸

An interesting outcome of the cross-regulation between T-bet and GATA-3 is that alterations of the parameters governing cross regulation by, for example, polymorphisms in these transcription factors could be responsible for inappropriate or mixed Th1 and Th2 responses. These parameters have not yet been determined experimentally but methods to measure them could be designed. It would also be interesting to compare these parameters in BALB/c mice, which have a predisposition to Th2 responses.

T memory cell homeostasis

Models do not need to be complicated or involve high-level mathematics in order to give insight into a biological problem. Good examples of this in immunology are models of T-cell homeostasis. A recent elegant study has shown that decreased thymic output with age is required but not sufficient to explain the observed decline in TCR rearrangement excision circles (TRECS) in naïve T cells and homeostatic increase in renewal rate or in average cellular lifespan is also required.⁴⁹ Other models have considered antigen stimulation of T cells and the generation of T-cell memory.⁵⁰ It is a remarkable fact that the number of T memory cells remains roughly constant throughout the life of an animal from adulthood to old age despite the continuous inflow of new memory cells from antigen activation of naïve cells or reactivation of already existing memory cells. It is also remarkable that specific clones can be long lived and not lost by dilution from entry of new clones. CD4 and CD8 memory compartments appear to be maintained independently⁵¹ with approximately constant cell numbers throughout adult life.⁵²

It has been suggested that the numbers of T memory (Tm) cells are maintained by low-level reactivation with persistent or cross-reacting antigens but there is now persuasive evidence for the maintenance of specific memory in the absence of TCR stimulation.^53,54In vivo labelling experiments in mice and humans have shown that approximately 1–5% of both CD4 and CD8 memory cells are in cycle at any one time.^55–57 This background proliferation has been termed homeostatic proliferation and is required to maintain the Tm compartment.^58,59 Homeostatic proliferation of CD8 Tm cells occurs in response to IL-15^59–61 and a similar (cytokine-driven) mechanism, probably IL-7 is thought to be responsible for CD4 memory T-cell proliferation.⁶²

Homeostasis of the Tm pool clearly depends on balancing the number of cells entering the pool through antigen stimulation and homeostatic proliferation against the number leaving the pool through death or terminal differentiation. This balancing trick by the Tm compartment is illustrated in Fig. 8.

Figure 8.

Homeostasis of the T memory cell compartment. A stable memory T-cell compartment requires that the input is exactly balanced by the output, but how is this achieved?

The biological mechanism for Tm homeostasis has not been clearly established but there are a several possibilities that can be considered.

1. A quorum-sensing device that can count the number of Tm cells and then ensure removal of surplus cells or increase proliferation if the number of cells is insufficient. This is a theoretical possibility but it is hard to envisage a biological mechanism that can actually do this.

2. Feedback between output and input could work, but again this is an unlikely mechanism because input into the Tm compartment depends on unpredictable exposure to antigen.

3. Competition for resources or space. There is some evidence that competition for MHC class I or II occurs for naïve T-cell homeostasis^63,64 but this mechanism does not seem to be involved in Tm cell homeostasis. Competition for a limiting supply of cytokines such as IL-15 or IL-7 in homeostatic proliferation of Tm cells could be important.

4. Density dependent death of Tm cells by example fratricide through Fas-mediated apoptosis is also a potential mechanism for Tm homeostasis.⁶⁵

One of the important uses of mathematical modelling in biology is to show whether a proposed mechanism is feasible and/or to distinguish between apparently distinct mechanisms. A comparison of competition and fratricide models of Tm homeostasis is a good example.

First, consider the fratricide model in which activated T cells kill each other through a Fas/FasL mechanism as described previously.⁶⁵ The model explicitly considers resting (non-dividing) and cycling Tm cells as separate compartments and fratricide by Fas/FasL apoptosis or a similar mechanism is confined to the proliferating compartment. This is justified by evidence that apoptosis of Tm shown by Annexin V staining is largely confined to the small proportion (∼1%) of the activated/cycling cells (P.C.L. Beverley, personal communication). A detailed mathematical description of the model is reported elsewhere.⁶⁶ In both cases, a simple model will be considered in which there is no flow into the Tm compartment from antigen activated naïve cells as illustrated in Fig. 9. The role of antigen in these models has been considered previously^65,66 and does not alter the general conclusions but is omitted here for simplification.

Figure 9.

T-cell fratricide model of homeostasis. In this simplified model, only the resting and proliferating Tm cell compartments are considered. Input from antigen-activated naïve cells or memory cells has been considered elsewhere and does not alter the general conclusions.⁶⁶ In this simplified model, differential equations are used to describe the rate of change in the resting and proliferating Tm compartments as a consequence of resting Tm cells undergoing homeostatic proliferation induced for example by IL-15 or IL-7 (a), reversion back to the resting state (r), T cell proliferation within the cycling compartment (c), death or loss of cells within the resting compartment (d) and fratricide by T cells within the cycling compartment (fy²).

In this model, homeostasis means that at equilibrium there is no change over time in either the resting or cycling Tm compartments. That is, the rate of change in the resting Tm compartment denoted (x) and in the cycling compartment (y) is zero. This is easily expressed by two coupled ordinary differential equations:

(Note: in this simple model c = 0, which is consistent with recent observations that cells receiving a homeostatic proliferation signal divide only once before re-entering the resting compartment; A. Yates and B. Seddon unpublished observations.)

Where dx/dt and dy/dt are the rate of change of the resting and cycling Tm cell compartments, a is the rate at which resting cells undergo homeostatic proliferation, r is the rate at which cycling cells revert back to the resting state, d is the rate at which the resting cells are lost or die and f is the rate at which the cycling cells undergo fratricide by a Fas/FasL like mechanism. The death term for fratricide (fy²) is non-linear and gives rise to a cell density-dependent death rate, which is very important for homeostasis. To illustrate how this works, assume that the cells are well mixed and meet at random. The rate at which they meet each other with the possibility of inducing Fas-mediated cell death will increase as the number of cells increases. If the cells double in number there will be twice as many that any one cell could potentially encounter. But there are twice as many cells, each of which has double the chance of encountering another cell. The total number of encounters therefore quadruples. In general, the rate at which cells will encounter each other is proportional to y². Assuming that there is a fixed rate of Fas mediated fratricide for each such encounter, the overall rate of apoptosis is denoted as fy² where f is the rate constant for apoptosis.

The solution of these equations at equilibrium (i.e. when both dx/dt and dy/dt equal zero) is easy and shows that the homeostatic level is dependent on the non-linear death rate (fratricide f). To see how this non-linear death term can maintain homeostasis of the Tm compartment consider what happens when the number of T cells is low. Because the cell density will be low, it is unlikely that T cells will meet and fratricide will be rare. Proliferation will therefore win over death and the number of cells will increase. This will continue until the equilibrium is reached when the rate of proliferation equals the rate of death. If the number of Tm cells exceeds the number at equilibrium, the death rate will be greater than proliferation and the numbers will be reduced until equilibrium is again reached. The number of Tm cells at equilibrium is determined simply by the overall proliferation rate and the Fas-mediated death rate and is independent of antigen.

The competition model is similar to the fratricide model except that the non-linear death term fy² of the fratricide model is replaced by a linear death rate (δy) and the activation term ax is replaced by the non-linear activation term ax/(k + x) which has the effect of limiting the rate of activation as the number of Tm cells (x) increases to a plateau maximum rate (a) (Fig. 10).

Figure 10.

Model of Tm homeostasis by competition for resources. Again, only the resting and proliferating Tm cell compartments are considered. Input from antigen activated naïve cells or memory cells does not alter the general conclusions. As with the fratricide model, differential equations are used to describe the rate of change in the resting and proliferating Tm compartments but a saturating term ax/(k + x) is used to describe the rate at which resting Tm cells undergo homeostatic proliferation induced for example by IL-15 or IL-7 and death by fraticide is replaced by the linear term δy.

The differential equations that describe the model now are:

(Note: again in this simple model c = 0, which is consistent with recent observations that cells receiving a homeostatic proliferation signal divide only once before re-entering the resting compartment; A. Yates and B. Seddon unpublished observations.)

Although these coupled equations may appear on the surface to be quite different from the equations describing the fratricide model, mathematically they both describe a Tm density-dependent mechanism for maintaining homeostasis, which may be difficult to distinguish experimentally. Further analysis does however, predict different behaviour when additional Tm cells are added to the memory compartment.

As shown in Fig. 11, the behaviour of the memory compartment in response to adoptive transfer of additional resting Tm cells is different in the fratricide and competition models. In principle, this could be investigated by transfer of different numbers of resting Tm cells and plotting the number of transferred cells against the number of circulating cycling Tm cells after one or two days. The fratricide model predicts a slope of this curve greater than 0 whereas the competition model predicts a slope of 0, assuming the pool is saturated. Although this may be difficult to achieve experimentally because the increase in cycling cells is likely to be small (×1·6 on doubling the number of resting cells) it is nevertheless a good example of how mathematics can suggest an experimental approach to distinguish between two different biological models that is not at all obvious or intuitive from an experimental/biological point of view.

Figure 11.

The fratricide and competition models can be distinguished by following the number of cycling cells in the circulation after adoptive transfer of resting Tm cells. In the fratricide model, addition of Tm cells () results in a rapid increase in the number of cycling cells and then a slow return of the cycling and resting cells back to the normal equilibrium point (black filled circle). In contrast, in the competition model, addition of Tm cells does not significantly increase the number of cycling cells and the number of resting cells slowly returns back to the equilibrium point (•).

Conclusions

The non-linear interactions between the molecular and cellular components of the immune system may give rise to non-intuitive behaviour that cannot be predicted from the properties of the individual components. Insight can be gained by mathematical analysis of the interactions. It is important to appreciate that mathematical modelling is a tool that can be used to investigate the properties of the immune system and does not have to be used to build a virtual immune system by including all the components to simulate the biological behaviour. Deciding which biological components to include in a model is an important skill and depends on the particular questions being asked and the system under investigation. It is important to use mathematical modelling in conjunction with experimental data. Mathematical models used in this way can help determine whether a biological concept is feasible and/or can be used to distinguish between alternative biological models. As immunology (biology) moves beyond the taxonomic imperative of the molecular era and into what is now commonly referred to as systems biology, it will become more important to focus on how components interact to give rise to a typical immune response rather than describing the properties of individual components. This interdisciplinary approach combining experimental immunology with mathematics is rapidly becoming the norm rather than the exception.