Malarial infection is associated with complex immune and erythropoietic responses in the host. A quantitative understanding of these processes is essential to help inform malaria therapy and for the design of effective vaccines. In this study, we use a statistical model-fitting approach to investigate the immune and erythropoietic responses in Plasmodium chabaudi infections of mice. Three mouse phenotypes (wildtype, T-cell-deficient nude mice, and nude mice reconstituted with T-cells taken from wildtype mice) were infected with one of two parasite clones (AS or AJ). Under a Bayesian framework, we use an adaptive population-based Markov chain Monte Carlo method and fit a set of dynamical models to observed data on parasite and red blood cell (RBC) densities. Model fits are compared using Bayes' factors and parameter estimates obtained. We consider three independent immune mechanisms: clearance of parasitised RBCs (pRBC), clearance of unparasitised RBCs (uRBC), and clearance of parasites that burst from RBCs (merozoites). Our results suggest that the immune response of wildtype mice is associated with less destruction of uRBCs, compared to the immune response of nude mice. There is a greater degree of synchronisation between pRBC and uRBC clearance than between either mechanism and merozoite clearance. In all three mouse phenotypes, control of the peak of parasite density is associated with pRBC clearance. In wildtype mice and AS-infected nude mice, control of the peak is also associated with uRBC clearance. Our results suggest that uRBC clearance, rather than RBC infection, is the major determinant of RBC dynamics from approximately day 12 post-innoculation. During the first 2–3 weeks of blood-stage infection, immune-mediated clearance of pRBCs and uRBCs appears to have a much stronger effect than immune-mediated merozoite clearance. Upregulation of erythropoiesis is dependent on mouse phenotype and is greater in wildtype and reconstitited mice. Our study highlights the informative power of statistically rigorous model-fitting techniques in elucidating biological systems.
Malarial infection of humans is a major cause of morbidity and mortality, continuing to cause around 250 million cases and close to a million deaths annually [1]. The vast majority of severe cases and deaths are due to Plasmodium falciparum, which is endemic in most of sub-Saharan Africa and other tropical areas [2]. Although there is no simple relationship between the pathogenic processes and clinical syndromes, disease only begins once the asexual parasite begins to multiply within the host' s red blood cells (RBCs) [3]. The asexual dynamics depend on a complex interaction between the malaria parasite and the host' s immune and erythropoetic responses [4]. Experimental methods have helped elucidate key aspects of this interaction. Such factors include the parasite' s destruction of RBCs due to reproduction [5], [6], immune-mediated clearance of merozoites and parasitised RBCs (pRBC) [7], [8], and immune-mediated clearance of unparasitised RBCs (uRBC). In particular, there is evidence that loss of uRBCs is responsible for the vast majority of the anaemia [9]–[11]. Suppression of RBC production (dyserythropoiesis) during the acute phase may also contribute to anaemia [12], [13], although recent modelling suggests that, overall, the level of erythropoiesis increases during malaria infection [8], [9], [14]. A full understanding of the infection dynamics requires quantitative analysis of the relative importance of the contributory factors [3]. Such an assessment is vital to help inform malaria treatment and intervention programmes [8], [15], [16]. In particular, the design of effective vaccines and immunotherapies depends largely on our understanding of the innate and adaptive immune responses [2], [17]. In this context, rodent malaria models allow a highly replicable, highly controlled experiment. Although there are important differences between rodent and human malarias, a quantitative understanding of the rodent system, where we can control both host and parasite genetics, should help our understanding of the human case in which controlled experiments are unethical. As in other areas of science, mathematical models can be used to make inferences about complex dynamical systems by fitting them to data. This approach allows us to formally and quantitatively test and compare competing hypotheses, and to make quantitative predictions for future empirical testing. It is the most powerful and rapid way of culling possible, but incorrect, hypotheses. In the mathematical modelling literature on malaria, there are a number of studies that quantitatively fit models to data [6], [8], [9], [14], [15], [18]–[22]. However, the poorly developed statistical, diagnostic and computational methodologies of fitting nonlinear dynamical models to noisy data (see [23] and Discussion) meant that these studies had to focus on particular aspects of the host-pathogen system in isolation. The method used was maximum likelihood. Its application to nonlinear systems is problematic because the nonlinearities create complex multi-dimensional likelihood surfaces. Search algorithms easily become trapped in local maxima, leading to false inferences [24]. Even if one is reasonably sure of having found the global maximum, evaluating parameter confidence intervals and covariances is computationally expensive and laborious, and computing predictive intervals practically impossible [25]. Recent developments in adaptive, population-based Markov chain Monte Carlo (McMC) methods overcome all of the problems associated with maximum likelihood [24], [26]–[30]. The use of a Bayesian framework enables us to incorporate prior knowledge and uncertainty about the parameters. It allows us to quantify our relative belief in one model predicting the data over another, rather than accepting and rejecting models using conventional, but arbitrary, cut-offs. In order to use Bayesian statistics, we need to know the structure and variance of the measurement errors. Fortunately, these are known for our data sets. In this study we develop a set of models to test competing hypotheses describing the asexual stage of the malaria parasite. We fit the models to a set of data on Plasmodium chabaudi infections [31] using an adaptive McMC algorithm. We provide parameter estimates, examine differences between mouse and parasite strains, and make quantitative predictions about the immune and erythropoietic systems' dynamics, and their effects on the RBC population. In modelling the asexual dynamics, there are three general processes we need to consider: (i) the infection of RBCs, (ii) the immune response, and (iii) the response of the erythropoietic system to malaria-induced anaemia. The immune system' s response to malaria is exceedingly complex and there is still much to learn about it qualitatively, let alone quantitatively [17]. Mathematical models have generally represented the immune response either as a single variable functionally linked to parasite density, or as separate innate and adaptive components [8], [21], [32]–[35]. The model of Recker et al. (2004) further discriminates, on the basis of human serologic data, between short-term, partially cross-reactive immune responses and long-term specific responses [36]. These models have given valuable insights into the immune dynamics, but it is important to acknowledge that the immune response consists of multiple arms, each targeting different aspects of the parasite [2]. Here we model the immune system as time-dependent immune-mediated clearance rates of merozoites, pRBCs and uRBCs. This allows us to bypass the debate about the highly interdependent innate and adaptive arms of the immune response, i. e. , when they are activated, what they target, and how they develop over time, and instead focus on the functional consequences in terms of the infection dynamics. We also draw attention to a key aspect of malaria asexual reproduction universally ignored in previous modelling studies. It is established that individual RBCs may be parasitised by more than one merozoite. Multiply-parasitised RBCs are often observed in experiments, but it is not known whether their subsequent behaviour is the same as that of singly-parasitised RBCs; previous models have generally assumed that their dynamics are identical. Here we test that assumption. In particular, we test whether multiply-parasitised RBCs have a greater death rate than other RBCs, and whether they produce a greater number of merozoites than singly-parasitised cells.
We used data obtained from a previous experiment [31]. Briefly, three different mice phenotypes were infected with either of two genetically distinct clones of Plasmodium chabaudi (AS or AJ). Both clones were originally isolated from thicket rats (Thamnomys rutilans) in the Central African Republic [37]. The AS clone is associated with a lower peak density relative to AJ; it also has lower virulence, as measured by anaemia and weight loss [38]. Three different phenotypes of BALB/c mice were used: (i) wildtype mice; (ii) nu/nu mice (“nude mice”; Harlan UK); and (iii) nude mice reconstituted with T cells taken from wildtype mice. The mutation nu is a recessive mutation that blocks the development of the thymus. Nude mice therefore lack mature T cells, whereas heterozygotes (nu/+) have a normal immune system [39]. Nude and reconstituted mice are smaller than the wildtype and are hairless. Mice of each phenotype (wildtype, nude, reconstituted) were innoculated with pRBCs of either AS or AJ. This gave six treatment groups. There were seven mice in both treatment groups for nude mice, and six mice in each treatment group for reconstituted and wildtype mice. Measurements of RBC and parasite density were taken on days 0,2, 4, and then daily until day 18 when the experiment was terminated. Parasite density was measured daily at 08: 00 hrs using quantitative PCR, at which point the asexual merozoites have yet to replicate within pRBCs. Both RBC and parasite densities are expressed in terms of the number per microlitre () of blood. Full details of the experimental methods are given in [31]. We removed a single data point from one of the reconstituted AJ-infected replicates. This mouse had much lower parasite density on day 14 than all the other mice. The data point was therefore considered to be an outlier. The averaged dynamics of each treatment are shown in Fig. 1. The AJ clone (solid line) does not show the normal higher peak density compared to the AS clone (dotted line); however, the AJ clone exhibits a higher density during the exponential growth phase compared to AS. Parasite density tends to level off in reconstituted and nude mice from day 12, but continues declining in the wildtype, presumably because of a stronger immune response in these mice. In reconstituted and wildtype mice, the AJ clone causes greater anaemia than the AS clone. All three mouse phenotypes show an earlier drop in RBC density from days 6–8 when infected with AJ compared to AS. The recovery of RBC density in nude mice is weaker than in reconstituted and wildtype mice. We discuss these observations below in relation to inferences from the model fitting. In Plasmodium chabaudi, pRBCs rupture synchronously every 24 hours, releasing on average 6–8 parasites (merozoites) into the bloodstream [40]. These newly released merozoites infect further RBCs and the cycle repeats. The rupture of pRBCS (schizogony) occurs at approximately midnight under normal lighting conditions [41]. We use a discrete-time formulation to model the dynamics, where each time step corresponds to a single day. The start of day is defined as the point immediately following rupture of pRBCs, before any infection has occurred (i. e. , the point at which merozoites are released into the bloodstream). The densities of merozoites and uRBCs at the start of day are denoted and, respectively. We assume that the processes determining RBC density occur on two non-overlapping timescales. The first corresponds to the short infection phase during which merozoites infect RBCs, which occurs within a few minutes following schizogony. The second and subsequent timescale (the remainder of the day) corresponds to the RBC turnover phase: here the parasites replicate within pRBCs, new uRBCs are released into the bloodstream and, if active, the (non-merozoite) immune responses clear pRBCs and uRBCs. At the end of the RBC turnover phase, surviving pRBCs rupture and release new merozoites. Using the statistical algorithm described in the Supporting Information (Text S1, Figure S1 and Figure S2), we fit to the observed data on RBC and parasite densities. The fitted parameters are given in Table 2, along with their prior distributions. The prior distributions were based either on values taken from the literature, or approximate estimates obtained before the main model-fitting (see Text S1 for details). Our aim is to find a set of minimal adequate models which explain the data well and contain as few parameters as possible. We take as our baseline the model described above. This is denoted and contains 20 fitted parameters. We developed a set of nested and non-nested models in which specific assumptions are made about the immune and erythropoietic responses (outlined in Table 3). All models were fitted to the data. Each model fit was then evaluated relative to that of the baseline. The model fits were compared using Bayes' factors, which naturally penalise overfitting [44] (refer to Text S1 for further details on measurement error, the likelihood function, model fitting, assessment and comparison). We estimate parameters separately for each mouse, rather than across treatments. Even inbred mice are phenotypically different, and these differences result in variability in parasite and RBC dynamics. Immune responses are significant sources of variability in vivo; but we might also expect variation between mice in parameters such as infection rate, because of the multifactorial nature of such processes which involve the interaction of many host and parasite proteins. We therefore make no assumption about which parameters are invariant across mice and instead estimate parameters separately for each mouse. This method allows us to infer parameter (and hence process) variability within and between treatments from the posterior estimates.
We begin by analysing the baseline model, . The fits to all mice are shown in Figs. 2–4. Note that the posterior predictive interval (PPI) of the dynamics widens from around day 15 for some mice because of the lack of data. The model fits appear to adequately explain the data. A more rigorous assessment of the model fits is attained by plotting the overlaid standardised residuals for parasite and RBC densities. Fig. 5 shows the standardised residuals (the blue crosses) for all mice across the six treatments. Poor fits are suggested by outlying and serially correlated residuals. The fits to parasite density are accurate but not perfect. There are no outliers, but the model tends to overestimate parasite density on day 4. This is unexpected because we expect parasite density to be growing exponentially during this time, and indeed this is how the model behaves. This discrepancy suggests that parasite density may initially grow at a slower than exponential rate. Also, the model tends to underestimate parasite density on day 11. Currently, we have no explanation for this. The fits to RBC density are accurate, except on day 8 where RBC density is marginally overestimated. Therefore, the model may not be correctly capturing the trough in RBC density. We calculated the Bayes' factors of models, for, relative to the baseline model. We adopt the scale of interpretation for Bayes' factors proposed by Jeffreys [45] and reproduced in Table 4. For ease of interpretation, the Bayes' factors are converted to deciBans; i. e. , (Bayes' factor). Bayes' factors were calculated for each mouse, giving a total of 38 values for each model comparison. The full list of Bayes' factors is given in Table S1 and Table S2. For conciseness and clarity, we report for each model: (i) the sum of the deciBans for all mice within each treatment, and (ii) the sum of the deciBans for all mice across treatments (Table 5). We also report the standard error of the deciBans. Errors occur because deciBans are estimated from a finite sample of the posterior distribution. Our inferences are conservative; thus, we interpret a deciBan of as “barely worth mentioning” (see Table 4). Statistical comparison of parameter values between treatments was performed using Analysis of Variance (ANOVA) in the R statistical package [46]. The method was as follows. For a given parameter, we first took the mean of the posterior distribution, , for each individual mouse. The mean for a given treatment, , was then calculated as the average of the posterior mean values taken across all mice in the treatment.
The aim of this paper was to provide a quantitative assessment of the immune and erythropoietic responses in Plasmodium chabaudi infections. Hypotheses were drawn from experimental data and the mathematical modelling literature. These were fit to data on malaria infected mice using a Bayesian statistical framework. Crucially, by quantifying the experimental error, we were able to provide a rigorous assessment of the model fit. In particular, we were able to evaluate and compare the accuracy of different models in explaining the data. Models were compared using Bayes' factors, which impose a penalty for additional parameters. We interpreted our results with reference to the product of Bayes' factors (sum of deciBans) within and across treatments. Our results provide very strong evidence that the immune response to P. chabaudi involves clearance of both parasitised and unparasitised RBCs. Both effects were evident during the initial peak of parasite density, implying that control of the peak is regulated by the immune response. Previous modelling studies have suggested that innate or early specific immune responses regulate the initial dynamics of parasite density and anaemia [8], [9], [21], [47]. Our study provides a statistically rigorous analysis in support of this hypothesis. Parasite-infected erythrocyte surface antigens (PIESA) may be an important immune target in both rodent and human malarias. In the case of P. falciparum, a longitudinal study of Kenyan children found that clinical malaria was caused by parasite isolates expressing PIESA variants that corresponded to gaps in the repertoire of antibodies carried by the children before they became ill [48]. We have shown that uRBC clearance by the immune system plays a key role in determining the infection dynamics of P. chabaudi. Experimental studies confirm that in the rodent malaria P. berghei [10], and also in P. falciparum which infects humans [11], the vast majority of anaemia is due to uRBC loss. Our results suggest that the level of anaemia following control of the initial peak (from about day 12), is mediated by the activity of the uRBC targeting response. This provides a mechanistic explanation for the variation in RBC dynamics between individual mice. If uRBC clearance decreases following control of the initial peak, then the increase in erythropoiesis that occurs from approximately day 10 allows the host to recover quickly from anaemia; in contrast, prolonged uRBC clearance is associated with a slow recovery from anaemia. Due to the lack of data to describe parasite density, the role of the pRBC targeting response during this later stage of infection is less certain. Our results suggest that the immune responses targeting pRBCs and uRBCs do not show a high degree of synchronisation. This implies they may be controlled by different effector mechanisms. Our model does not account for the antigenic variation that occurs in P. chabaudi infections [49], [50]. Indeed, our model formulation only permits a single “switching-on” and “switching-off” of each immune component (merozoite, pRBC, uRBC), and therefore does not distinguish between non-specific (innate) versus specific antibody responses to antigenically distinct variants. We modelled the infection up until day 18, at which point antigenic variation may have a significant effect on the dynamics. In some mice, the data show a second drop in RBC density following the recovery from initial anaemia (Fig. 2). Although we only have data on parasite density up to day 14, this second anaemia is commensurate with a recrudescent parasite density. However, it is significant that our model is able to explain the observed dynamics so well without including antigenic variation (Fig. 5). Extending the model up to, for example, day 30 post-infection would require explicit modelling of immune responses to the different antigenic variants. Such modelling would probably need to include both short-lived (innate) and long-lasting antibody responses, and may also need to consider cross-reactivity. The cascade of sequentially dominant antigenic variants seen in P. falciparum infections has recently been explained as the result of short-lived cross-reactive immune responses directed against shared epitopes [36]. Our results are consistent with the observation that T-cell-deficient (“nude”) mice have impaired immune responses, and are unable to resolve malaria infections [2]. T cells play an important role during the early stages of malarial infection, by amplifying the phagocytic and cell-mediated antiparasite responses; later in the infection, they help B cells to produce antibody, and assist in regulating the innate response [51]–[53]. Immune-mediated clearance of uRBCs was necessary to explain the dynamics in all three phenotypes, but nude mice had the higher maximum clearance rate of uRBCs. The infection rate of RBCs with merozoites, as reflected in the parameter, could also be higher in nude mice. As a simple proxy for the real biological system, these results indicate that nude mice are less able to limit the replication of the malaria parasite, and that their less specific immune response is associated with greater destruction of uninfected cells. At the individual mouse level, there was no evidence for immune-mediated clearance of merozoites. However, the cumulative evidence over all mice suggests that it is required to explain our data. Our interpretation of this result is that merozoite clearance is weak during the first few weeks of infection, and that pRBC and uRBC clearance are the major determinants of the dynamics during this time. Previous models have shown that, for a given level of immune activity, merozoite clearance is less effective at controlling parasite density compared to an equivalent response that clears pRBCs [7], [8]. One explanation for this is that the duration of the merozoite infection phase (estimated to be on the order of minutes) is too short for the immune system to effectively target merozoites [7]. However, there is no a priori reason that one or several fast-acting, highly effective immune responses could not operate during this phase. There is considerable empirical evidence that merozoite surface protein one (MSP1) is a target of immune mechanisms in malaria infections [54]. The presence of high levels of naturally acquired IgG antibodies to merozoite surface protein two (MSP2) is also strongly associated with protection against clinical malaria [55]. Recent results suggest that this naturally acquired protection is not specific in relation to the major allelic dimorphisms of MSP2 [56]. We have shown that erythropoiesis upregulates during malarial infection, and that wildtype and reconstituted mice have higher upregulation than nude mice. This may reflect that the erythropoietic response only upregulates to the extent that the host is controlling the parasite. Recent theoretical results have shown that excessive upregulation of erythropoiesis facilitates the growth of the parasite, and may result in greater anaemia and a higher peak of parasite density [57]. We also investigated whether there is a time delay before the upregulation of erythropoiesis, and a lag in the feedback between RBC density and the level of erythropoiesis. The results for the AJ-infected wildtype mice suggest that upregulation of erythropoiesis occurs from day 10. Both reconstituted and nude mice show no evidence of a time delay. Our results also suggest a time lag of 2–3 days in the feedback between RBC density and erythropoiesis in AJ-infected reconstituted mice; however there is no evidence for a time lag in the other treatments. The reasons for this are unclear. One possibility is that our putative time lag is compensating for another process not included in the model. Only analysis of other data sets may reveal what this may be. In summary, our results show that the immune system plays a key role in determining the RBC and parasite dynamics in malaria-infected mice. We have shown that immune-mediated clearance of both parasitised and unparasitised RBCs is necessary to explain the RBC and parasite dynamics. Previous models have examined the implications of RBC age structure for the infection dynamics [14], [22]. In particular, recent work by Mideo et al. (2008) suggests that P. chabaudi may preferentially infect mature RBCs (normocytes), but produce more merozoites in younger cells (reticulocytes) [14]. Future models may need to consider how RBC age structure and immune system dynamics can be combined to obtain a more complete picture of the asexual stage of malaria.
Malaria is a disease caused by a protozoan parasite of the genus Plasmodium. Every year there are around 250 million human cases of malaria, resulting in around a million deaths. Most of the severe cases and deaths are due to Plasmodium falciparum, which is endemic in much of sub-Saharan Africa and other tropical areas. The pathology of malaria is related to the asexual stage of the parasite. Understanding the infection dynamics during this stage is therefore essential to inform malaria treatment and vaccine design. Experimental infections of rodents represent an important first step towards understanding the more complicated human infections. We developed a series of models representing different hypotheses about the main processes regulating the infection dynamics during the asexual stage. Models were fit to data on Plasmodium chabaudi infections of mice, using a Bayesian statistical framework. The accuracy of different models in explaining the RBC and parasite densities was quantified. We identify the role of different types of immune-mediated mechanism, and show that RBC production (erythropoiesis) increases during infection. Differences between mouse phenotypes are explained. Our study highlights the informative power of model-fitting techniques in explaining biological systems.
mathematics/statistics
public health and epidemiology/infectious diseases
immunology/immune response
computational biology
Quantitative Analysis of Immune Response and Erythropoiesis during Rodent Malarial Infection