Possible application of modelling methods to bovine immune responses to Theileria parva
Immune responses and pathogenesis of bovine trypanosomiasis
Modelling of host-parasite interactions and their influence on the course of infection in tolerant and susceptible animals
Potential applications of modelling in the bovine genome project
Use of mathematical modelling for elucidating trypanotolerance: Preliminary considerations
International Laboratory for Research on Animal Diseases
P.O. Box 30709
Theileria parva is a haemoprotozoan parasite of cattle that gives rise to an acute and usually fatal lymphoproliferative disease. The life cycle of the parasite is complex and involves stages in the ixodid tick Rhipicephalus appendiculatus as well as the bovine host. Cattle become infected following the inoculation of sporozoites by an infected nymphal or adult tick. The severity of the disease has been shown to be related to the quantity of sporozoites inoculated. Sporozoites rapidly enter lymphocytes, where they develop to schizonts in the cytosol. It is this stage of the parasite that is responsible for the major pathology associated with the disease. Schizont-infected cells divide in an apparently uncontrolled manner and invade a variety of non-lymphoid tissues, disrupting their function. A striking feature of the disease is an extensive lympho-destruction, which is not confined to parasitized cells. This results in a net depletion of circulating lymphocytes, which is exacerbated by the invasion of the bone marrow by parasitized cells. Further disruption of immune function is believed to result from the elaboration by infected cells of powerful immune modulators. A significant component of the disease is therefore the result of perverted host immune mechanisms.
It is generally accepted that survival of an infected animal is dependent on successful control of the schizont parasitosis. Considerable research effort has been focused on the identification of those immune mechanisms that protect immune cattle. Immune animals almost invariably exhibit a transient schizont parasitosis before resolving infections, suggesting that protective mechanisms are directed at the schizont-infected cell. A number of observations have indicated that parasite-specific class I MHC-restricted cytotoxic T lymphocytes (CTL) are responsible for eliminating challenge infections in immune cattle. However, this activity is not detected in a significant number of immunized animals during the resolution of infection. It is likely that these alternative responses are mediated initially by CD4+ T cells with possible input from other effector populations. These possibilities are currently being explored.
It is clear that immunity to a parasite that infects and subverts the immune system presents a considerable problem to the modeller. The exact nature of the mechanisms responsible for protection remain to be defined, and although much information is available in this area it is not clear whether it constitutes a sufficient basis for a model. Nonetheless, a number of parameters have been defined that relate to the interaction between the parasite and the bovine host. The quantity of sporozoites-inoculated has been correlated with the severity of disease, and the number of schizont-infected cells required to produce fatal disease has been reasonably defined. In addition, immune responses to a defined antigen of the sporozoite surface have been shown to provide 70-100% protection. These findings might be incorporated in preliminary models to provide estimates of the possible impact of new vaccines.
D.J.L. Williams and L. Logan-Henfrey
International Laboratory for Research on Animal Diseases
P.O. Box 30709
Trypanosome infections in cattle usually result in a chronic disease, characterized by anaemia, leukocytopaenia, immunosuppression and cachexia. The parasites live in the bloodstream and evade immunological control by continuously varying their surface glycoprotein coat. Antibody to the surface-exposed epitopes of the variable surface glycoprotein coat (VSG) mediates the removal of a particular variant, but new variants expressing antigenically different VSGs appear throughout the infection. In cattle, following tsetse-fly transmitted infections, a fluctuating parasitaemia is observed, with-several variants expressed at any one time. Some breeds of cattle such as the N'Dama (Bos taurus) have evolved the ability to control both parasitaemia and anaemia. The control of these two traits does not appear to be linked. Trypanotolerant cattle remain more productive in trypanosomiasis-endemic areas than B. indicus breeds (such as the Boran), but can succumb to the disease when exposed to very high challenge or stress.
ILRAD's immunopathology program aims to elucidate the mechanisms of trypano-tolerance and to identify trypanosome antigens which cause either protective or pathological responses in infected cattle. These antigens will be used in the design of novel vaccines which elicit protective anti-parasite immune responses or to block pathogenic responses.
Results from studies on immune responses following T. congolense infections have shown that N'Dama cattle produce higher levels of IgG1 antibodies to invariant antigens (specifically, a trypanosome cysteine protease and a heat shock protein) and variant antigens; they have earlier and higher T-cell proliferative responses and their monocytes secrete higher levels of costimulatory cytokines (IL1/IL6) early in infection compared to trypanosusceptible cattle. A population of IgM antibodies which bind to non-trypanosome antigens has been identified in Borans but not in N'Damas. This population of antibodies also binds to trypanosome VSG, suggesting that they are polyspecific in origin. Studies are currently under way into differential epitope recognition and antibody avidity between the two breeds. There appear to be no differences between breeds in the titre or isotype of antibody specific for the surface-exposed epitopes of VSG. Both N'Dama and Boran cattle experience a profound macrophage-mediated immunosuppression in the lymph nodes early in infection. The antigen which induces this suppressor activity has been purified and is being characterized. Also a massive increase in the numbers of CD5+ B cells and in serum IgM levels has been described in all infected cattle. It has been suggested that the CD5+ B-cell response is the result of a T-independent response to VSG.
The primary pathological feature of bovine trypanosomiasis is anaemia. The drop in red cells is due to the removal of both mature and immature erythrocytes by cells of the mononuclear phagocytic system (MPS) in the spleen and bone marrow. Results from in vivo and in vitro experiments suggest that erythrophagocytosis is higher in Borans than N'Damas. Data have also been obtained which indicate that in addition to the massive removal of erythrocytes by the MPS, there is insufficient red blood ceil replacement into the circulation. Analysis of erythroid progenitor colonies in bone marrow from trypanosusceptible cattle has shown that there is a suppression of both early (BFU-E) and late (CFU-E) progenitor cells early in infection. As the infection progresses the CFU-E progenitors recover, but the numbers of BFU-E's remain depressed.
In summary, we have accumulated a database in which differences in the pathological consequences of trypanosome infection have been quantified in trypanotolerant and trypanosusceptible cattle. Other factors which affect the outcome of disease are being addressed, such as the role of specific cytokines or the direct pathogenic effects on host molecules, cells or processes, of certain trypanosome antigens. Can we use the data we have to develop models to test the effect of altering variables such as aberrant cytokine production or the failure to develop T-dependent antibody responses? Can we test how these different immunological and haematological processes interact to affect the outcome of disease? Is it possible to model the interactions between innate host factors versus specific immune responses in control of parasite growth and the role of the immune and erythroid responses in the control of pathogenesis?
International Laboratory for Research on Animal Diseases
P.O. Box 30709
Trypanosomes which undergo complete cyclical development are capable of controlling their growth rates at different points within their life cycle. In the tsetse fly vector, mature metacyclic forms, which are infective for the mammalian host, are non-dividing. When an infected fly takes a blood-meal from a mammalian host, the infective metacyclic forms which pass into the host begin to differentiate into actively dividing bloodstream-forms. These actively proliferate in the vascular system. Studies on Trypanosoma brucei brucei in rodents have shown that when an infection has matured, the trypanosomes are capable of undergoing a differentiation event to become non-dividing. These non-dividing forms can be cleared by the host immune system through recognition of the surface coat of the parasite. The clearing of a wave of parasitaemia allows trypanosomes which have a different surface coat, generated through a process termed antigenic variation, to become established in the vascular system. The mechanisms of antigenic variation by which trypanosomes evade the host immune system are well documented. At present, there is no information available on how growth of the parasite is controlled, although there is evidence to suggest that a decrease in growth rate can allow the host to control and eliminate the infection. The available information on infections in trypanotolerant and trypanosusceptible cattle suggests that trypanotolerant breeds control the infection, at least in part, by reducing parasite load in the first wave of parasitaemia, with subsequent waves showing marked reductions until the infection is eliminated. Susceptible animals show only slightly higher parasite load in the first wave but are incapable of controlling subsequent waves. Immune dysfunction is evident in the susceptible animals following the first wave of parasitaemia but not in the tolerant animals.
The consensus opinion at present is that the control of parasitaemia occurs prior to the first peak of parasitaemia and prior to control of parasite numbers through immune recognition of parasite molecules. We believe that there must be signalling between the host and parasite, between parasites and between parasite and host which influence the course of the infection. In exotic breeds these signals are clearly wrong, and an unchecked trypanosome infection eventually results in the death of the animal. We wish to understand how these signals work on parasite proliferation in order to identify the signals and the parasite surface receptors they bind to. In the establishment of an infection and in the first wave of parasitaemia, the differences in parasite numbers could be due to control of parasite growth rates (e.g. cell division cycle) or death rates (e.g. programmed cell death).
We are not sure whether either of these possibilities would influence the modelling of an infection and whether it is important, or even possible, to determine experimentally which of the two is occurring. Towards the peak of the first wave of parasitaemia there is an immune response elicited by the host against the surface coat of the parasite which eliminates that antigenic type from the bloodstream and allows re-invasion of the vascular system with parasites having a different surface coat. At this time, susceptible animals start to display immune dysfunction whereas tolerant animals do not.
A further feature of parasite-host interactions is 'carrier status' in which trypanosomes are not detectable in the vascular system of the host, but if the host is subjected to stress the parasites actively proliferate and often kill the host. It has been suggested that in carrier status, parasites might sequester to 'privileged' sites in the host where they can avoid immunological attack. Again, we would like to know how parasite numbers are controlled in carrier status and what triggers apparent active proliferation following stress in the host. Can carrier status be accounted for by assuming that the parasites are in a dormant state and non-dividing or that they are dividing normally but dying off more quickly so that the numbers do not reach detectable levels in the blood? Should we be establishing whether there are host factors controlling this state?
International Laboratory for Research on Animal Diseases
P.O. Box 30709
Trypanotolerance, the heritable ability of some cattle to remain productive in the face of trypanosome challenge, offers a sustainable means of reducing the impact of trypanosomiasis on productivity. Trypanotolerant cattle are relatively few in number and intensification of their breeding and dissemination is required. Introgression of trypanotolerance into susceptible cattle types is also a possible approach to production of improved cattle fitted to the humid and subhumid tropics.
Approaches to the improvement and dissemination of trypanotolerance genes would benefit from the availability of markers. Ultimately, with cloning of the genes, it will become possible to attempt production of transgenic trypanotolerant cattle designed to be productive under trypanosome challenge.
The bovine genome project is attempting to find genetic markers of trypanotolerance. The genetic basis of trypanotolerance is not well understood and there are, theoretically, an almost unlimited number of possible genes involved with any number of possible interactions. It is known to be a highly heritable trait and crosses between trypanotolerant N'Dama cattle and susceptible Boran cattle show an intermediate level of tolerance. The approach to identifying genes and estimating their number is to construct a large set of genetic markers which will be applied to a population segregating trypanotolerance. There are two possible areas where modelling could make a valuable contribution.
MODELLING THE GENETIC CONTROL OF TRYPANOTOLERANCE
A model could set up a given number of genes on a given number of chromosomes, each of which makes a given contribution to the phenotype and which interact in a given way. What then would be the expected outcome of breeding a trypanotolerant animal with a trypanosusceptible animal and then either back crossing or inter crossing to construct the second generation? Such a model would allow the estimation of the number of animals required for such an experiment and assist in the interpretation of its outcome. There is essentially no data available to assist in the construction of such a model. It would amount to a purely arbitrary, but highly useful, examination of the consequences on the resultant phenotype of a genetic mechanism of arbitrary complexity.
MODELLING MARKER DISTRIBUTIONS
The bovine genome has been estimated to contain up to 3 x 107 polymorphisms, all of which are potential markers. The technique of random amplification of polymorphic DNA (RAPDPCR) allows access to an effectively unlimited number of markers and makes possible the search for genes controlling traits of interest in outbred populations. However, if a given number of animals has a given set of alleles at each marker locus, what fraction of markers will give a spurious association between a marker and genes controlling the trait of interest? A modelling approach could address this question by defining one or more populations of a given number of individuals. This definition must include the possibility of multiple alleles at each locus and a range of allele frequencies as well as allowing for population-specific alleles or allele frequencies. There are some data available which will allow the estimation of likely levels of heterozygosity and allele frequencies. Unfortunately, these are largely based on protein polymorphisms and may not be applicable to RAPD markers. However, this again amounts to a theoretical study which needs little support in hard data. It will allow the establishment of the 'worst-case' conditions under which such an approach would fail and thus allow investigations to be designed to test whether or not these conditions occur in reality.
Z. Agur*+ and R. Mehr*
* Department of Mathematics and Computer Science
Weizmann Institute of Science
76100 Rehovoth, Israel
+ Department of Zoology
South Parks road
Oxford OX1 3PS, UK
African trypanosomes, the causative agents of sleeping sickness in humans and related diseases in livestock, escape immune destruction by sequentially altering the antigenic composition of their surface coat. Since the repertoire of the surface coat antigens is quite extensive, it is assumed that African trypanosomes are invulnerable to vaccination.
In a previous work, a mathematical model was presented for the interactions between the antigenically varying parasite and the humoral immune response in the naive host. Ability of the model to retrieve real-life parasitaemia profiles is crucially dependent on two assumptions: i) that antigenic variation involves an intermediate, possibly very brief, episode in which the old and the new antigens are simultaneously expressed on the parasite's coat; and ii) that the double expressers that vary in the antigenic composition of their coats also vary in their susceptibility to the immune response. Experimental evidence supporting these assumptions are warranted for further development of the model and for examining the prospects of anti-trypanosome vaccine. Now the model is extended to allow for different types of immunity. We see how parasitaemia profiles that characterize different host species or intra-specific variation in trypanotolerance can be retrieved if we vary the parameters of specific antibody response.
Since antigenic variation in African trypanosomes involves very intricate dynamics on the molecular, cellular and cell population levels of the parasite and the host immune system, it is highly likely that important features of parasitaemia are determined by currently undetected factors. It is for the investigation of these complex dynamics that we employ the tool of mathematical modelling, aiming at identifying the crucial subprocesses in the system. Updated information about details of inter-specific differences in the immune response to trypanosomiasis in general, and in affinity maturation in particular, is essential for improving the predictive ability of the mathematical model.
African trypanosomes are protozoan parasites transmitted by the tsetse fly to people and wild and domestic animals in Africa. Infection in domestic livestock results in weight loss, impaired immune responses, and haematopoietic and reproductive disorders. However, a few breeds of livestock that are indigenous to Africa, such as the N'Dama cattle (Bos taurus) of West Africa, are able to tolerate trypanosomes well and, in many cases, appear to suffer no ill effects from infections. This property, shared by many of Africa's wild ruminants, is known as trypanotolerance. The more common livestock breeds, in which trypanosomiasis readily develops, such as zebu cattle (Bos indicus), are referred to as trypanosusceptible (e.g., Murray, 1987).
Understanding trypanotolerance seems to be a necessary step in the development of control methods for these diseases. Thus, a series of experiments was carried out in which N'Dama and a zebu breed called Boran were infected with Trypanosoma congolense and parasitaemia was monitored in the two infected breeds. Initially, the N'Dama and Boran showed similar levels of parasitaemia, but as the infections progressed, high numbers of parasites persisted in the Boran whereas parasitaemia in the N'Dama decreased with time (Figure 1). These results clearly show that the trypanotolerant N'Dama can control parasitaemia, but the mechanism responsible for this control remains obscure (Paling et al., 1991).
Figure 1. Changes in parasitaemia during a primary infection (a) and a rechallenge infection (b) with Trypanosoma congolense ILNat 3.1 in a group of N'Dama (- - -) and a group of Boran cattle (-). Reproduced from Paling et al., 1991.
The difficulty in understanding trypanotolerance is due to the intricate host-pathogen dynamics in African trypanosomiasis. These dynamics involve many non-linear processes at the molecular, cellular and cell population levels in the parasite and in the host. We studied this problem using mathematical modelling techniques, analysing the major processes at the different organizational levels and their interactions. Subsequently, we used the model to examine the effect of variation in different parameters on the overall infection dynamics. We employed a model that retrieves the characteristics of parasitaemia quite accurately and, to date, is the only model that accounts for African trypanosomiasis in the individual host (Agur et al., 1989).
Modelling Antigenic Variation in African Trypanosomes
Trypanosomes can proliferate in blood and can also invade other systems. Due to an efficient antibody-mediated immune response, the sharp increase in parasite numbers is followed by an abrupt decline. However, blood trypanosomes can escape the host's immune responses by undergoing antigenic variation of their unique cell surface coat protein, the variant cell surface glycoprotein (VSG). Since every coat usually consists of a single type of VSG, trypanosomes can change their antigenic identity by switching to the expression of a new VSG gene, thereby expressing a new coat. Each parasitaemia wave consists of a population of parasites, most of which display one kind of VSG on their surface. As the repertoire of potential VSG coats is very large (about 1,000 in T. brucei), blood-based infections are characterized by relapsing parasitaemia waves, which can progress for long periods of time (see Barry and Turner, 1991, for a review).
To explain the course of parasitaemia, a mathematical model was developed to study the interaction between the antigenically varying parasite and the host's immune system. The model focuses on the genetic events that underlie antigenic variation, and assumes that the transition from the expression of one VSG gene to the next is a random event, which can either be instantaneous or gradual. In the latter case, intermediates are presumed to exist, expressing two serologically distinct VSGs on their surface; these parasites are denoted double expressers (DEs). In this model the DE state may be a very brief episode in the parasite's life cycle, so that even if it is obligatory, the proportion of DEs in the population at any given moment may be very low.
The model was studied by numerical simulations for a large range of parameter values, and for different assumptions about the underlying processes. It succeeded in obtaining roughly ordered parasitaemia waves only if it assumed that i) the majority of parasites undergoes the DE transition episode, and that ii) different DE combinations vary in their susceptibility to the immune response against the precursor single expressers (SEs). Under these assumptions parasitaemia can be ordered even if individual parasites switch completely at random. Note that the precision in the order of parasitaemia and the regularity of waves depends on the proportion of parasites undergoing the DE stage and on the variability in the intrinsic growth of different SEs or DEs (Agur et al., 1989).
The aim of this preliminary work was to identify host mechanisms that are responsible for the observed differences in parasitaemia profiles between trypanosusceptible and resistant cattle. The model will be used for checking which host factors may control the length of parasitaemia and the structure of waves. In addition, we will attempt to retrieve the reduction in parasite levels, observed in the trypanotolerant N'Dama cattle in late stages of the primary infection and throughout the rechallenge infection (see Figure 1).
It will be shown that parasitaemia profiles are relatively robust to changes in some ecological parameters, e.g., the parasite intrinsic growth rate, which traditionally is assumed to control population growth. In contrast, temporal parameters, such as the delay period prior to the production of high affinity antibodies, appear to have a significant effect on parasitaemia profiles.
General differences between hosts can be realized in the model in the parasite intrinsic growth rate, r, and in the carrying capacity of the host, K. In addition, hosts may vary in nonspecific immune responses, e.g. immune cell general proliferation rate and mortality rate, as well as in antibody secretion rate, due to differences in cytokine secretion levels (Williams and Logan-Henfrey, this volume). The potential effect of these factors is examined below. Note that since our focus in this work is on parasite dynamics, we ignore the possibility of host mortality, as it may be only indirectly related to the parasite load.
Effect of Host-Difference in Carrying Capacity, K
The carrying capacity, K, measures the maximal density of parasites in the blood. A large K means that the parasite population can grow to large numbers before density begins to suppress population growth. Displayed below are simulation results of parasitaemia in two hosts, one in which maximal parasite density is K = 104 per ml (Figure 2a), and one in which maximal parasite density is K= 107 per ml (Figure 2b). One can note in these figures that carrying capacity has a striking effect on parasitaemia profiles. An upper limit of 104 on parasite density generates, for a host with 20 litres of blood (e.g. a cow), an ordered parasitaemia which progresses for an extended period. In contrast, the higher upper limit on parasite's density, K = 107, leads to a very acute early parasitaemia with very dense high peaks, containing most of the antigenic repertoire. The reason is that a large K enables a fast increase in total parasite numbers to levels that permit an almost synchronous emergence of many variants. The sharp decline in parasite load after about one month of infection is due to the elimination of all the faster growing variants by the immune system. The latter effect is due to our assumption of a perfect immune memory: no mortality of specific B cells is allowed for, so that re-emergence of variants cannot occur. This assumption is relaxed in a further study (see below).
Figure 2. Effect of parasite density on parasitaemia profile. Parasitaemia obtained in simulations using the following parameters. Blood volume is 20 litres. Two parasites of a single VSG are initially introduced. The growth coefficient for single expresser variants is 0.85 per time unit, and for double expresser variants 0.2 per time unit; each simulation time unit represents six hours. B cell growth coefficient is 0.52 per time unit. Maximum antibody secretion rate is 170 per time unit. t is three days. (a) K =107; (b) K = 104. For other parameters, see Agur et al., 1989.
Effect of Host-Differences in Parasite's Intrinsic Growth Rate, r
Reduction in the growth rate of all the parasites in the system has a relatively small effect on parasitaemia (not shown). Results suggest that for K = 104, a 20% reduction in the parasites' intrinsic growth rate has a negligible effect on the height of peaks, but parasitaemia is slightly less dense, as some of the very slow growing variants are now below detection level. When K = 107, a similar reduction in r reduces the peaks by about one order of magnitude, and increases the duration of parasitaemia from about one month to about three months.
Increasing the differences in variant-specific intrinsic growth rate has the effect of upsetting the order of parasitaemia and the characteristic structure of peaks, but it does not significantly alter the height of parasitaemia peaks (results not shown).
Effect of Host-Differences in Immune Cell Birthrate
Increasing B-cell replication rate by 20% does not have a significant effect, except for some dilution of parasitaemia (results not shown). This effect is similar to the above-mentioned effect of decrease in the intrinsic growth rate of the parasites.
Effect of Host-Differences in Antibody Secretion Rate
A hundred-fold increase in antibody secretion rate narrows the peaks but has no effect on their height or on the course of parasitaemia apart from a slight dilution of the slowly growing variant peaks. This is so since, even now, antibody secretion rate is not sufficiently large enough to prevente the rapid rise of the fastest growing parasites (Figure 3a). In contrast, a two hundred-fold increase in antibody secretion rate-has the effect of shortening the parasitaemia course: now the decline of the parasitaemia waves is so rapid that the probability of a successful antigenic switch is much reduced (Figure 3b).
Effect of Host-Differences in Immune Cell Death Rate
Significant B-cell mortality appears to lengthen parasitaemia, due to the reappearance of early variants. Variation in B-cell mortality per se cannot account for the N'Dama-Boran observed differences.
Effect of Host-Differences in the Time Lag Between Antigen Stimulation and the Onset of Specific Antibody Secretion
A theory of population dynamics in perturbed environments asserts that population persistence depends on the relation between the characteristic periodicity of the population and that of the environment (Agur, 1985; Agur and Deneubourg, 1985). Based on this theory we hypothesized that long-term persistence of the parasite population depends on the characteristic time scale of the immune response, that is, on the lag between antigen stimulation and specific antibody secretion (see also Barry and Turner, 1991; Agur 1991). Such host differences may reside in the pre-immune repertoire or in the properties of affinity maturation.
Figure 3. Effect of antibody secretion rate. All parameters as in Figure 2b, except that antibody secretion rate is (a) 17 x 103 per time unit, (b) 34 x 103 per time unit.
To check the above hypothesis, parasitaemia was simulated using different values for the mean and the variance in the time-delay, t, between antigen stimulation of specific B-cell proliferation and the onset of antibody secretion. Results suggest that this time-delay may indeed have a significant effect on parasitaemia profiles. However, the effect is remarkably different in hosts with different K's.
When K = 107 the general pattern of parasitaemia appearing in Figure 2a is maintained. This is so as long as for some variants, at least, the time-delay before antibody secretion is t > 2 days. Such a delay ensures that in a host with 20 litres of blood, the number of parasites becomes so large as to enable the quasi-synchronous emergence of most of the available variants. A significantly different parasitaemia occurs if t £ days (Figure 4a) for all the variants. Now the acute parasitaemia, in which all parasites appeared early in the infection, is replaced by less acute, but long and ordered parasitaemia, due to the suppression of parasite numbers early in the infection.
The picture is quite different when K = 104. We saw in Figure 2b that t = 3 days yields ordered parasitaemia. The same profile was obtained when the efficiency of the immune response was further decreased, so that t = 4, 5 days. We note, then, that a marked decrease in efficiency of the immune response does not alter parasitaemia profile when maximum parasite density is relatively low. The reason is that the depressed K prevents the increase in parasite numbers to the levels that allow for the co-emergence of many antigenic types. If variance in t is increased, so that now t = 3 ± 1 days, parasitaemia is somewhat diluted but the effect is not very significant. A completely different profile is obtained for t = 2 days. Now parasitaemia is significantly shorter, due to the increased efficacy of the immune response. Figure 4b is similar to Figure 2b in displaying simulation results of parasitaemia in a host having K = 104, except that now the time-delay preceding the onset of antibody secretion is not constant, but rather a random variable t = 2 ± 1. One may note here a brief ordered parasitaemia, during which the peaks are reduced by about one order of magnitude. A further decline in parasitaemia peaks is obtained when t = 2 ± 2 days, so that for some variants there is no delay in the onset of antibody secretion (they may be similar to previously encountered antigens). For other variants the delay may be as long as four days or more. Note that if specific antibodies are already present in the blood when the new VSG is detected (Agur, 1991) then the situation is equivalent to negative.
The main conclusion from the present study is that a model that can retrieve the roughly ordered, persistent parasitaemia (characterizing African trypanosomiasis) cannot be very resilient. In such a model a reasonable variation in most parameter values will either have an insignificant effect, or it will completely upset the structure of parasitaemia Our study points to the time-lag between antigenic stimulation and the onset of antibody secretion as a parameter which can modulate parasitaemia. In the present work we assumed that this lag is either constant, i.e., the same in all specific responses, or that it is a normally distributed random variable. The possibility that the time lag may progressively decrease during primary and rechallenge infection, by some kind of a "reaming" mechanism, is implied by the parasitaemia profiles of T. congolense in N'Dama cattle (Paling et al., 1991). This possibility will be further explored (Z. Agur and R. Mehr, in preparation). Support for our theoretical conclusions is provided by the observation that in mice, virulence of T. congolense is correlated with a late and transient protective antibody response; negligible virulence is correlated with an early protective antibody response (Roelants and Pinder, 1987).
Figure 4. Effect of the time delay between antigenic stimulation and the onset of antibody secretion. All parameters as in Figure 2, except t (a) K = 107, t = 2 ± 0 days; K = 104, t = 2 ± 1 days.
At the present stage it is essential to obtain further information about real life parasitaemia in resistant and susceptible cattle. Thus we would like to know the structure of waves, i.e., the number of variants in each wave and their identity. In addition, it is important to evaluate the total number of parasites in individual hosts, rather than group averages. More information about affinity maturation in trypanosomiasis is also essential. Such information will hopefully validate our assertion that trypanotolerance resides in the immune response. Note, however, that the possibility of host differences in affinity maturation implies that cell-cycle genes may be involved. Theoretical results suggest that the onset of hypermutation depends on the number of B cells in the proliferating clone, and it has been suggested that the mechanism of monitoring this number may be related to the mitotic clock (Agur et al., 1991). Another work suggests that the mitotic clock, i.e. cell-cycle duration and the number of cell divisions in a cell-lineage, may be modulated through small changes in the activity of some cell-cycle genes, such as cdc25 or wee1 (Norel and Agur, 1991). The implication of this is that even small host differences in the activity of genes that are not directly connected with the response to parasites can be responsible for differences in susceptibility to African trypanosomes.
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AGUR, Z. 1991. Mathematical models of African trypanosomiasis. Parasitology Today 8(4): 128-130.
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AGUR, Z., ABIRI, D. and van der PLOEG, L.H.T. 1989. Ordered appearance of antigenic variants of African trypanosomes explained in a mathematical model based on a stochastic switch process and immune-selection against putative switch intermediates. Proceedings of the National Academy of Sciences of the USA 86: 9626-9630.
AGUR, Z., MAZOR, G. and MEILIJSON, I. 1991. Maturation of the humoral immune response as an optimization problem. Proceedings of the Royal Society of London 245: 147-150.
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MURRAY, M. 1987. Trypanotolerance, its criteria and genetic and environmental influences. In: Livestock Production in Tsetse Affected Areas of Africa. Proceedings of a Meeting Held 23-27 November 1987, Nairobi, Kenya. Nairobi: ILCA/ILRAD, pp. 133-151.
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MODELLING OF THE IMMUNE SYSTEM
Following the introductory presentations of ILRAD's needs, it was agreed that modelling of the immune system per se is complex, and if interactions with parasites are superimposed, the complexity increases enormously. This complexity can be reduced considerably by focusing on components of the immune system that are relevant to the parasite in question. The discussion addressed the different contributions and approaches of modellers and experimentalists. On the one hand, it was stressed that experimentalists need to be tolerant of first generation abstract models; these invariably evolve as fresh information is incorporated. On the other hand, it was suggested for example that it would be useful if trypanosomologists could manipulate infection challenges to mimic antigenic waves and observe the immune responses that ensue.
A more specific discussion developed following Dr. Agur's paper on some aspects of host differences in trypanotolerance. It was emphasized that it is not known whether waves of antigenic variation occur in cultured trypanosomes. The model does not have affinity maturation of antibody responses incorporated in it. The view was expressed that there may be more than two VSG expression sites within a given trypanosome and that the basis of ordered expression of VSG variants is likely to be the siting of their genes within the genome.
The presentation by Dr. Kemp of the approach being taken at ILRAD to map bovine trypanotolerance genes included two questions to the assembly of modellers. These were:
· Can bulk segregant analysis be applied productively to outbred cattle populations?
· Can the segregation of multiple genes, each with defined contributions to a given trait, be modelled within outbred populations?
Neither question was satisfactorily answered in the ensuing brief discussion.