Helminths,immunologyand equations Nicky Schweitzer and Roy Anderson Althougb the immune system is becoming better characterized, it is by no means becoming easier to predict the outcome of activation given the many potential influences upon the ultimate expression of an immune response. ln this article, Nicky Schweitzer and Roy Anderson investigate the application of mathematics to this highly nonlinear system, and show how it can complement experimentation from both a predictive and interpretative point of view. Recent advances in molecular and cellular biology are enabling immunologists to classify more and more finely the various types of cell and chemical that constitute, direct and regulate the vertebrate immune system. Immunological processes are very complex and involve many components. Many of these interact with each other and with invading infectious agents in highly nonlinear ways, such that responses to stimuli from an antigen or from other cell types often reveal threshold or saturation effects as the intensity of each stimulus increases. In the face of such complexity, many biologists must wonder what possible advantages mathematics can bring to improving understanding beyond that provided by manipulative experiments carried out with live cells and infectious agents. Mathematics simply offers an alternative method for formulating or thinking about problems precisely. For example, as the ability to classify the numerous components of the immune system improves, the number of potential explanations for observed events has in turn tended to increase. In these circumstances, as in any other scientific field, mathematical models provide a powerful method for assessing whether particular hypotheses can indeed generate the observed pattern, both in qualitative and quantitative terms’. Furthermore, these models help to focus attention on what must subsequently be measured by experimentation or observation. The main reason why most immunologists are wary of the use of mathematics is that it apparently results in an oversimplification of the complexity of the immune system, not forgetting our incomplete understanding of this complexity. We stress that the mathematical approach is akin to that of the experimental scientist: starting with simple assumptions, and slowly incorporating complexity. One or a few factors are allowed to vary while others are held constant within the experimental design. This process, particularly when applied to complicated nonlinear systems, allows us to assess sequentially the major factors that generate the observed pattern. In this paper, the combined question of immunoregulation and heterogeneity in response between hosts in the context of parasitic infections, particularly helminth, is addressed. The aim is not to construct precise models of the entire response of the immune system to infection. Instead, we start with simple sets of assumptions to understand the extent to which the degree of complexity in the model is related to the ability to predict, in qualitative terms, a diverse array of outcomes as

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Immunoregulation and heterogeneity The understanding of immunoregulation has improved considerably over the past few years as new experimental techniques have emerged. A good example of this is provided by the CD4+ T-cell subset dichotomy (in this issue). The characterization of CD4+ cells into T,l-cell and T,2-cell subsets in terms of cytokine production and growth/inhibition requirements3-H has suggested the existence of a mutually regulatory and apparently self-contained pathway of differential development, which appears to account for the functional diversity of CD4+ T-cell-mediated immune responses observed in viva. This diversity can be classified loosely into T,l-cell-type responses, which are mediated by gamma-interferon (delayed-type hypersensitivity, cellular cytotoxicity), and T,2-cell-type responses, which direct antibody-mediated mechanisms, the most notable being interleukin 4 (IL-4)-controlled IgE production and IL-S-controlled eosinophilia. This dichotomy has caught the attention of parasitologists, since high levels of IgE production and eosinophilia, often associated with suppressed cellular reactivity, are characteristic features of chronic helminth infections. Conversely, a variety of examples of immunity to helminths, whether natural or vaccination-induced, are associated with strong T,I-type responses9*i0. It has been shown that preferential activation of one subset or the other can be influenced by many factors including genetic makeup 9,11,the manner in which antigen is administered i”*iL,i3, the type of antigen-presenting celli and the dynamics of stimulation in vitro’S.16. In addition, a differential association between antigen dose and affinity, and the requirements for activation of the respective subsets, has been suggestedi7*lx. Resistance or susceptibility to helminth infections is also influenced by a similar range of factors: genetic make up9vi9, the route of infection and subsequent migration of larval stages, the development of the parasite within the host’O and the size of the infecting dosei9Jr. When we began to look at the problem of differential T-cell activation in relation to helminth infections particular emphasis was placed on the temporal dynamics of the host’s exposure to parasite antigen. Parasite input into the host can be considered at a variety of levels: (1) the persistent exposure of individual hosts to the infective stage, which subsequently matures to the adult

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form; (2) the continuous production of antigen (secretory/excretory metabolites) by the adult; (3) the continuous production of larvae or eggs by the same adult stage and/or (4) excretory products of larvae or eggs. Initial investigations looked specifically at the interface between the immune system and the parasite under conditions of constant exposure to antigens produced by long-lived adult parasites (for example, schistosomes and the filarial worms).

An immunosuppressive parasite The heterogeneity that could be generated within a host population if the parasite had an inhibitory effect on the activation of an immune response was considered first. This form of suppression has been implicated in a number of nematode/helminth infections”. The hypothetical immune system in this example is highly simplified and consists initially of a circulating naive cell pool. On contact with specific antigen, naive cells become activated to proliferate. Activated cells are cytotoxic to the parasite (or indirectly stimulate antibody attack) but the parasite can also suppress and therefore deplete the activated cell pool. The set of three differential equations describing this are shown in Box 1. Note that even these very simple assumptions give rise to a system containing three variables and nine parameters. In terms of the immune system, it is assumed that T,lcell-type responses only are important for immunity against the parasite population with which we are dealing; the apparent role of T,,2-cell-type responses in protection against a subsequent exposure to infective stages (concomitant immunity) is ignored. This is not an attempt to prove the existence of a particular immunological mechanism but to gain insight into the importance of particular features of infection and immunity. Since the capacity of the model to produce different outcomes is being examined, it is appropriate to look for the maximum number of steady states, or equilibria (representing final outcomes), that can be generated by the equations. This is explained in Box 2. The simple model possesses three equilibria, only two of which are stable. The two stable states represent high immunity with low parasite load and low immunity with high parasite load. These two stable states are separated by an unstable equilibrium through which a boundary passes, separating the areas of attraction to each stable state. Perturbations to the system created by drug treatment, further stimulation of proliferating cells (immunotherapy/vaccination) or changes in exposure to infection (to use three examples), can induce a sudden switch from low to high immunity concomitant with a transition from high to low parasite load.

If the suppressive influence of the parasite is eliminated, the steady state representing low immunity and high parasite burden is lost. When the host is given only a single inoculum of the potentially inhibitory parasitic stimulus, immunity is always stimulated. In contrast, under conditions of repeated exposure, the state to which

the system is attracted varies according to the level of exposure to infection. High exposure moves the system to a state of low immunity and high parasite load and, conversely, low exposure leads to the state of high immunity and low parasite load (Fig.1). However, a

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Box 1. Basic model A differential equation describes the rate of change in the size of a population with respect to time. Each term reflects one factor (for example, birth or production, death or migration) that will cause the population to increase (positive terms) or decrease (negative terms). The assumptions from which our initial set of differential equations (below) were constructed are illustrated by the flow diagram where the. variables are assigned to the three boxes, and parameter values (or, rate constants) determine the extent to which the various factors affect these variables. Inhibitory effects are shown by broken lines and birth, death or migration processesare shown by solid lines. Under certain circumstances, a mathematical expression is used to account for a phenomenological observation or assumption, exemplified here by the saturation term cX2 which causes the activated T-cell population to proliferate to an upper maximum. Naive T cells dMldt=A-PM-yMP ’ (M(d) dX/dt = yMP + rX - cXZ - PXP Activated and proliferating T cells (X(t)) Parasite dP/dt = A - aP - sXP population within the host VW A Constant rate of production of new T cells p Death rate of naive unactivated cells (k-1: life expectancy) y Rate at which naive cells are activated by contact with parasite antigens r Net rate of proliferation of activated cells (birth rate minus death rate) c Density-dependent constraint on cell population p Rate at which parasite deactivates-or removes activated T cells A Rate of immigration (infection) of young parasites into the host OL Death rate of parasite (CY-l: parasite life expectancy) s Rate at which parasites are killed by activated T cells In the absence of the parasite the equilibrium density of naive T cells is given by AF-I. In the absence of an immune response the equilibrium burden of parasites is given by A a-l.. Production A

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Box 2. Equilibrium,

stability and alternative steady states

What is meant by an equilibrium for the parasite popuIation within a single host? In mathematical models consisting of differential equations, an equilibrium is defined when the rate of change of the parasite population with respect to time (host age) is equal to zero (if P(t) representsthe number of parasitesat time t, then a parasite population in equilibrium resultswhen dP(t)ldt = 0). When many variables interact in a system of equations (parasitesinteracting with immune cells) the equilibrium relationship between variables can be representedon a phaseplane. A phase plane enablesthe relationship between the equilibrium state of one variable (in this instance activatedTcells, X) in relation to another variable (parasites, P) within the model to be investigatedgraphically. Thus, if dP/dt = f(X,P) then f(X,P) can be plotted against a range of valuesof X and P. The line that is generatedis called an isocline. A similar plot can be made for dX(t)l dt = g(X,P) = 0. In this model the problem is considered in two dimensionsonly (eachdimension representinga variable) by assumingthat the naive cell population (M) always returns to equilibrium rapidly following any change in the parasite stimulus (P). M = h(P) is substituted into dX(t)ldt, such that dX(t)ldt = g(X,P) and not g(M,X,P). The figure showsthe phaseplane plotted for activated cells X and parasite burden P. Note that the isoclines intersect at three points. The relationship between activated T cells and parasiteswithin an individual host (solid lines), the boundary region determining the areas of attraction to each state (dashedline) and the trajectories of the variables with respectto one another over time (dotted lines) are shown. The number of intersections and their preciselocationsdependon the parameter values of the system(the rate of host responseto infection, cell proliferation rate, the rate at which the parasite

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suppresses cell proliferation, parasitelife expectancy and others). Clearly, for many values of P, there are two possible equilibrium statesfor X, and vice versa. In other words, when dP(t)ldt = 0, dX(t)ldt may not be equal to zero. An equilibrium or steady state occurs only when dP(t)l dt = dX(t)ldt = 0, which is satisfied when the two isoclinesintersect in the P,X plane. Nevertheless,equilibrium does not necessarilymean stability. The stability of an equilibrium dependson the behaviour of the system in the vicinity of that equilibrium. If an equilibrium is locally stable, the variables will converge on that point over the courseof time, from all local regions on the phase plane. Alternatively, if it is unstable,the trajectories of the variableswill move away from the potential equilibrium. The two outer equilibria are stable, and the one in-between is unstable.The convergence of the model system upon one or other of the stable steady states is dependent on initial conditions, which are perhaps determined by past experience of infection.

perturbation to either of these steady states (perhaps owing to changesin exposurewith host age)can switch the system to the alternative equilibrium. Initial contact with the parasitewould occur during early childhood in an endemicarea.For many helminthsin humancommunities, there is most likely a decline in exposure in adult life which, paradoxically in this simple model, would result in a switch from low immunity to high immunity. Adding complexity to the immune system The precise relationship between the T,l-cell and T,2-cell subsetsand the conditions required for the activation of one asopposedto the other isthe subjectof continuing debate17*23*24. However, attempts are being madeto draw up regulatory circuits involving the various phenotypesobserved”-26and someof the current observations and hypotheses concerning the regulation of differentiation and proliferation of T,l and T,2 cells5~7~8~‘o~24~~ have beenincorporated into a set of assumptions that have been used to construct a more

complex mathematical model. The interrelationships involved are representedin a flow chart which forms the basisof this model, asshown in Fig. 2. In this case, the hypothetical immune system again involves a naive circulating T-cell pool that becomes activated following contact with parasite antigen. The equation describingthe parasitepopulation is identical to that detailed in Box 1. However, this time, instead of simply proliferating following contact with parasiteantigens,the naive cell beginsto produce cytokines. These cytokines regulate the bias of differentiation and drive the proliferation of differentiated and activated T cellsZ4. In the first instance,the processof parasitesuppressionof cell activation and proliferation isexcluded aswe wanted to seewhether the various patterns observed in natural infections could be accountedfor by the intrinsic biological properties of our hypothetical immune system. Heterogeneity, in either the level of parasitaemiaor the host’s immune status within a host population, is not causedonly by absolute differencesbetween individuals

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but can also reflect dynamic changes within an individual where parasites are lost or gained as the host ages. Typically, the heterogeneity observed between humans infected with filarial worms (Brugiu, Onchocerca spp) has been thought to reflect the existence of distinct classes of individual. However, there is evidence that something more dynamic may be occurring (this issue); some individuals may appear susceptible for many years and then relatively suddenly clear infection and exhibit immunity. This was elegantly demonstrated in experimentally infected cats (Fig. 3)“. With the help of computer simulations, it can be shown that four different classes of exposed individual could be generated (Fig. 4b), simply by exposing the model immune system of our host to various levels of exposure to the parasite (antigen, egg or larval input). These classes are as follows: (1) individuals in which exposure is insufficient to generate a T,l-cell response and also insufficient for the establishment of a significant infection; (2) individuals that become immune shortly after the first encounter with the parasite; (3) apparently immunosuppressed individuals that harbour a patent infection for varying but prolonged periods, but that suddenly progress to the immune state and clear infection; and (4) individuals that remain immunosuppressed indefinitely. A similar diversity of responses has been characterized in recent immigrants to an area endemic for IilariasiP. Comparing the relationship between exposure and the progress of infection in this model shows that immunosuppression is most intense in individuals exposed to smaller numbers of parasites. As the dose increases, the time before transition to immunity decreases exponentially. Note that in the first model, increased dose is positively associated with immunosuppression (Fig. 1). The new model therefore suggests that active inhibition of cell activation or proliferation is not necessary for the generation of apparent immunosuppression. In this more complex system it can result from regulatory processes mediated by cytokines in response to the stimulus of infection. What happens if both cytokines and active inhibition of the immune response by parasites are included? The increased degree of immunological complexity in this second model allows a more precise definition of where and how the parasite suppresses the effectiveness of the immune response directed against it. For illustration we chose the IL-2-driven proliferation of the T,l-cell subset, since the responsiveness of T cells to IL-2 is suppressed in Brqia-infected jirds19, while IL-2 production is diminished in chronic mouse Schistosoma munsoni infections30 and impairment of IL-2-driven proliferation has been observed in several examples of tolerance induction”iJ’. This version of the more complex model again generated the four classes of exposed individual seen in the previous version with cytokines but without parasite suppression of activated cells (see Fig. 4c in comparison with 4b). However, closer examination revealed that, as well as showing a more gradual spectrum of change through these classes, the association between the level of exposure and the duration of immunosuppression was now reversed. This latter relationship was similar to that

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Helminths, immunology and equations.

Although the immune system is becoming better characterized, it is by no means becoming easier to predict the outcome of activation given the many pot...
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