Abstract
Malaria, a disease caused by parasites of the Plasmodium genus, begins when Plasmodium-infected mosquitoes inject malaria sporozoites while searching for blood. Sporozoites migrate from the skin via blood to the liver, infect hepatocytes, and form liver stages which in mice 48 hours later escape into blood and cause clinical malaria. Vaccine-induced activated or memory CD8 T cells are capable of locating and eliminating all liver stages in 48 hours, thus preventing the blood-stage disease. However, the rules of how CD8 T cells are able to locate all liver stages within a relatively short time period remains poorly understood. We recently reported formation of clusters consisting of variable numbers of activated CD8 T cells around Plasmodium yoelii (Py)-infected hepatocytes. Using a combination of experimental data and mathematical models we now provide additional insights into mechanisms of formation of these clusters. First, we show that a model in which cluster formation is driven exclusively by T-cell-extrinsic factors, such as variability in “attractiveness” of different Py-infected cells, cannot explain distribution of cluster sizes in different experimental conditions. In contrast, the model in which cluster formation is driven by the positive feedback loop (i.e., larger clusters attract more T cells) can accurately explain the available data. Second, while both Py-specific CD8 T cells and T cells of irrelevant specificity (non-specific T cells) are attracted to the clusters, we found no evidence that non-specific T cells play a role in cluster formation. Third and finally, mathematical modeling suggested that formation of clusters occurs rapidly, within few hours after adoptive transfer of T cells, thus illustrating high efficiency of T cells in locating their targets in complex peripheral organs such as the liver. Taken together, our analysis provides novel insights into and discriminates between alternative mechanisms driving the formation of clusters of antigen-specific CD8 T cells in the liver.
1 Introduction
Malaria is a life-threatening disease that is a result of red blood cell (erythrocyte) destruction by eukaryotic parasites of the Plasmodium genus. The majority of deaths (estimated to be about 500,000 annually) are among children, who have not yet developed immunity to the pathogen [1, 2]. There are five species that infect humans: P. falciparum, P. vivax, P. malariae, P. ovale, and P. knowlesi [3]. Three species of malaria parasites that are used as animal models for human malaria in mice are P. yeolii, P. berghei, and P. chabauidi [4]. While there are similarities and differences in replication and pathogenesis of Plasmodium species in humans and mice, in this paper we focus solely on infection of mice with Plasmodium parasites.
The infection of the host is started by a mosquito, the vector between mammalian hosts, injecting the sporozoite form of parasites into the skin. Studies have estimated that the initial number of sporozoites entering the host is as low as 10-50 [5, 6], of which only a fraction succeed to migrate to the liver to start an infection of hepatocytes by forming liver stages [7–9]. This liver stage of infection lasts for approximately 6.5 days in humans and about 2 days in mice [10–13]. Because liver stage is asymptomatic, removal of all liver stages prevents clinical symptoms of malaria and thus is highly desirable feature of an effective vaccine. Indeed, previous studies have shown that memory CD8 T cells are required for protection against a challenge with a relatively large number of sporozoites [14, 15] and that vaccination that induces exclusively memory CD8 T cells of a single specificity can mediate sterilizing protection against a sporozoite challenge [16–21]. Antibodies and CD4 T cells may also contribute to protection in some circumstances [22]. Given that mouse liver contains about 1 − 2 × 108 hepatocytes [23–25] and only a tiny proportion of these are infected the ability of memory CD8 T cells of a single specificity to locate and eliminate all liver stages within 48 hours is remarkable. Yet, specific mechanisms by which T cells achieve such an efficiency remain poorly defined.
Recent studies utilizing fluorescently labeled sporozoites and activated Plasmodium-specific CD8 T cells and intravital microscopy revealed clustering of CD8 T cells near the parasite in the mouse livers whereby multiple T cells were located in close proximity (≤ 40 µm) of some liver stages [26–28]. Interestingly, we observed that clustering of T cells near the parasite results in a higher chances of parasite’s death suggesting that clusters may increase the efficiency at which T cells eliminate the infection. Recent in vivo studies also found that the killing of virus-infected cells occurs faster when several T cells are near an infected cell [29] that is consistent with a previous report estimating that killing of targets in vivo follows the law of mass-action [30] (meaning that the rate of killing is directly proportional to the concentration of the killers and targets).
Clustering of T cells around Plasmodium liver stages in mice was not uniform as the majority of parasites had no T cells around them (at 6 hours after T cell transfer), while some parasites were surrounded by 20-25 T cells [26]. We have developed three alternative mathematical models aimed at explaining this observed variability in cluster formation and by fitting the models to a subset of the data concluded that the data are best explained by a model in which formation of clusters is driven by a positive feedback loop — clusters of a large size attract more T cells to the site of infection [26]. Analysis of T cell movement in the liver suggested that there may be a bias towards the infection site [31]. Additional experiments revealed a significant correlation between the rate at which new T cells locate the infection site and the number of T cells found in the cluster and independence of the per capita rate at which T cells leave the cluster from the cluster size — both observations were consistent with the “density-dependent” recruitment model.
Yet, our previous analysis did not investigate several other important issues of cluster formation. In particular, we did not fully determine the role of the environment in the formation of clusters around Plasmodium liver stages. For example, the observed correlation between entry rate into a cluster and cluster size could simply arise because some parasites may accidentally “attract” more T cells (e.g., due to higher induced inflammation or a higher blood flow rate). In addition, we have observed that transfer of activated T cells, specific to Plasmodium, and T cells of irrelevant specificity resulted in co-clustering of T cells of two types [26]; however, whether “non-specific” T cells contributed to the formation of clusters was not determined. Finally, our previous analyses did not determine the kinetics of the cluster formation by assuming that cluster size reaches a steady state by the time of imaging. In this paper we analyze novel and re-analyze some of the previously published data to further define mechanisms by which clusters of activated CD8 T cells around Plasmodium liver stages are formed.
2 Materials and methods
2.2 Data
In our analyses we mainly used data from previously published work [26, 32]. Data were generated using an experimental system with Py-specific TCR transgenic T cells. For most of our analyses, data were from experiments involving infection of Balb/c mice with variable doses of Plasmodium yoelii (Py) sporozoites, expressing GFP; location of T cells around GFP-expressing liver stages was then visualized using spinning disk confocal microscopy [26]. Following our previous work we consider T cells located within 40 µm distance from the parasite as being close enough to recognize the infection; this, all T cells within 40 µm from the parasite are called to form a “cluster”. How well the length of 40 µm represents the size of hepatocytes in mice remains unclear. 2D images of mouse hepatocytes suggested the diameter of 40-80 µm [33]; however, measurements of the total volume of mouse hepatocytes of about Vh = 104 µm3 [34] suggest a radius for a “spherical” hepatocyte or cube edge length for a “cubical” hepatocyte. A classical textbook on human liver anatomy cites the human hepatocyte volume of 104 − 6 × 104 µm3, corresponding to a cube edge of about 40 µm [35, p. 13]. Despite these inconsistent estimates we consider T cells within 40 µm from the parasite to be close enough for recognition of the parasite.
Clustering of CD8 T cells around the parasite was measured in several alternative experiments. In one set of experiments, clustering of CD8 T cells around Py liver stages was performed by immunizing mice with radiation-attenuated Py sporozoites (RAS). In another set of experiments, clustering of T cells was observed following transfer of activated T cell receptor (TCR) transgenic CD8 T cells, specific to epitope located in Py circumsporozoite (CS) protein (CS280-288: SYVPSAEQI) denoted as PyTCR cells, or of TCR transgenic CD8 T cells, specific to epitope in chicken ovalbumin (OVA257-264: SIINFEKL) denoted as OT1. Following infection with Py, PyTCR recognize the infection while OT1 cells serve as a control (non-specific to Py) T cells. PyTCR and OT1 cells were activated in similar in vivo experiments (using Vaccinia virus expressing CS or OVA epitopes) [26]. Experiments involving co-transfer of PyTCR and OT1 cells were done in CB6 mice (F1 cross of Balb/c and C57BL/6 mice) [26].
In summary, the following datasets were used in the analysis. Dataset #1: fluorescently labeled PyTCR cells (’PyTCR alone’) and PyTCR cells pre-treated with pertussis toxin (’PyTCR+PT’) and transferred into mice infected with Py. Dataset #2: naive or RAS-immunized mice that were infected with Py-GFP and clustering around the parasite was imaged using anti-CD8 antibody. Dataset #3: labeled PyTCR and OT1 cells transferred alone into individual mice (‘PyTCR alone’ and ‘OT1 alone’) or as 1:1 mixture (‘PyTCR mix’ and ‘OT1 mix’) to Py-infected mice. This dataset has information on co-clustering of PyTCR and OT1 cells around the same parasite in the co-transfer experiments. Dataset #4: labeled PyTCR cells transferred into Py-infected mice and imaged at two different time points after T cell transfer. All datasets are made available as an online supplement to this paper to facilitate further independent analyses.
2.2 Mathematical models
2.1.1 Basic mathematical model for clustering of one cell type
Following our previous work we propose a standard “birth-death” model to describe formation of clusters around Plasmodium liver stages [26]. This model assumes that infection of hepatocytes by Plasmodium sporozoites occurs independently. This is likely to be justified given that in our experiments i) in general ~ 105 sporozoites are injected i.v. into mice, ii) only a fraction of these is expected to reach the liver [7–9], and iii) mouse liver contains 1 − 2 × 108 hepatocytes [23–25]. Because in general in our experiments the number of Plasmodium-specific T cells exceeds the number of liver stages by 10 to 30 fold, we assume that clustering of T cells around one parasite does not interfere or compete with T cells clustering around another parasite. In the model describing formation of clusters around Plasmodium liver stages by T cells of a single specificity we denote Pk(t) as the probability to observe k T cells around the parasite at time t with k = 0, 1, 2,…kmax. Increase in cluster size occurs at the “birth” (or entry) rate λk (k = 0, 1, 2,…, kmax) and decline in cluster size occurs due to “death” (or exit) rate µk (k = 1, 2,…, kmax). The mathematical model describing the change in the probability Pk(t) with time is given by the system of differential equations:
By assuming different specific forms for the T cell entry (λk) and exit (µk) rates (e.g., see Figure 1 and below) the model can be solved numerically and fitted to the data using maximum likelihood method (see below). For some analyses we made a simplifying assumption that the distribution of cluster sizes reaches a steady state, and the steady state values for the probability to observe k CD8 T cells near a given liver stage is given by where is found by normalizing the total probability to one. By assuming steady state solutions it is in general impossible to estimate individual values for the rates of T cell entry into the cluster and exit from the cluster but we can estimate the ratio of the entry and exit rates, which we define as the relative entry rate θk = λk/µk.
Mechanisms explaining the clustering of T cells around Plasmodium parasites in the liver can be broadly divided into two categories: T cell-intrinsic and T cell-extrinsic mechanisms. In the T cell-intrinsic mechanism, the formation of clusters is driven exclusively by T cells and thus this mechanism ignores any potential differences in the variability in local liver environment. In the T cell-extrinsic mechanism, formation of clusters is driven exclusively due to variability in the liver environment near individual parasites, for example, due to a higher blood flow to some liver stages or a higher degree of inflammation that individual parasites may induce. It is possible that ultimately both mechanisms may contribute to the cluster formation.
2.2.2 Sub-models assuming T cell-intrinsic clustering mechanisms
We consider several alternative models of how T cells may mediate formation of clusters around Plasmodium liver stages in mice (Figure 1A-C). Some of these models have been presented in our previous publication [26]. Our simplest random entry/exit (Poisson model) assumes that entry into the cluster and exit from the cluster occur randomly, i.e., λk = λ0 and µk = kµ where λ0 and µ are constants (Figure 1A). Solving eqn. (3), the probability to observe k T cells around a parasite according to this random entry/exit model is then given by the Poisson distribution: where θ0 = λ0/µ.
We have shown previously that the Poisson model is often unable to describe distribution of cluster sizes of Plasmodium-specific CD8 T cells in the liver [26]. One potential mechanism proposed to describe formation of large clusters is a “retention” model in which T cells which recognize the infection, are retained near the parasite. One version of such a model is a density-independent (DI) exit model (Figure 1B) in which the rate of T cell exit from a cluster declines with the number of T cells in the cluster, i.e., µk = kµ/k = µ for k > 0 and λk = λ0 for all k. Solving eqn. (3), the probability to observe k T cells around a parasite according to the DI exit model is given by a geometric distribution: where θ0 = λ0/µ. There are other ways in which the total rate of T cell exit from the cluster µk could decline with cluster size k and in our additional analyses we tested two of such alternative models: a powerlaw model in which µk = kαµ (defined for k > 0 with α and µ being constant) and an exponential model in which µk = kµe−αk (defined for k > 0 with α and µ being constant). When fitting these retention models to experimental data we did not dervice the steady state solutions but instead used numerical solutions of the basic mathematical model (eqns. (1)–(2)).
An alternative mechanism for the formation of large clusters of CD8 T cells around the infection is an “attraction” model in which the rate of T cell entry into the cluster depends on cluster size. In this density-dependent (DD) recruitment model (Figure 1C) the entry rate into the cluster is given by λk = λ0 + λ1k while the total exit rate is density-dependent µk = kµ. Solving eqn. (3), the probability to observe k T cells around a parasite according to the DD recruitment model at the steady state is calculated numerically: where θ0 = λ0/µ and θ1 = λ1/µ and is found by normalizing eqn. (6) assuming the maximal cluster size to be kmax. In general, and therefore, the sum must be taken for a finite number of terms due to this reason [26].
To understand dynamics of cluster formation in the Poisson and DD recruitment models it is also useful and possible to derive the model describing the change in the average number of T cells around the parasite (average cluster size), using standard methods of physical chemistry [36] which is a standard birth-death process with immigration which for 〈k〉(0) = 0 has the solution
In cases when λ1 > µ the average cluster size grows indefinitely with time. When λ1 < µ, which is often found in our analyses (see Main text), average cluster size at the steady state is given by where θ0 and θ1 are defined after eqn. (6).
2.2.3 Sub-models assuming T cell-extrinsic clustering mechanisms (environment)
An alternative mechanism for the formation of T cell clusters around Plasmodium-infected hepatocytes is proposed in this paper, namely, that the formation of clusters is driven simply due to variability in the ability of different parasites to “attract” T cells. For example, some parasites while traveling from the blood to hepatocyte or while replicating in the hepatocyte may induce higher degree of inflammation than other parasites, thus, potentially increasing the chance of finding such “inflamed” sites by T cells.
We consider two versions of the “environment” model in which T cell recruitment to sites is determined by the variability in parasite’s “attractiveness”. In one such version, a two population model, we assume that there are parasites of two types found at frequencies f and 1 − f, and these parasites differ in the rate at which T cells find them (Figure 1D). The formation of clusters around parasites of a given parasite type is given by random entry/exit model with rates λ01 and λ02 while the rate of exit of T cells from the cluster is µk = kµ. Then assuming a steady state the probability to observe clusters of size k is given by where θ01 = λ01/µ and θ02 = λ02/µ. Alternatively, the rate at which T cells find parasites could be given by a continuous function, and we tested a model in which entry rate into the cluster is given by a gamma distribution , i.e., the probability for T cells to have an entry rate in the interval (λ0, λ0 + dλ0) is g(λ0; α, β)dλ0. The probability to observe a cluster of size k given that clustering around a parasite “attracting” T cells at a rate λ follows a Poisson model is given by an integral where α is the shape parameter and β is the rate parameter of the Gamma distribution and Γ(α) = (α − 1)!.
2.2.4 Basic mathematical model for clustering of two cell types
In some of our experiments we tracked clustering of T cells of two specificities: one type of T cells was specific to Plasmodium sporozoites (PyTCR) and another type of T cells was specific to irrelevant antigen (OT1). To quantify the kinetics of clustering of Plasmodium-specific (PyTCR) and non-specific T cells (OT1) around Plasmodium-infected hepatocytes, we extended our basic model (eqns. (1)–(2)) to include two types of cells, t1 and t2, in the cluster to formulate a co-clustering model (Figure 1F). We define Pij(t) as the probability to observe i cells of type 1 and j cells of type 2 in a given cluster. Then the rate at which new T cells of type x, where x = t1, t2 enter the cluster with i T cells of type 1 and j T cells of type 2 is . Similarly, is the rate of exit of T cell of type x from a cluster with (i, j) T cells. The dynamics of the probability to observe a cluster with (i, j) T cells is given by equations
The dynamics of the probability Pij(t) can be simulated by assuming different functional forms for the entry and exit rates (Table 1). For example, when the entry rate into the cluster is independent of the cluster size or cell type, , and the exit rate is dependent on the number of T cells of a given specificity present near the parasite, and where µ = const, clustering of T cells is independent and is described by the Poisson distribution (results not shown). Another model is when the entry rate of T cells into the cluster is dependent only on the number of specific T cells (t1) already in the cluster: with exit rates being similar to the random entry/exit model described above.
2.2.5 Stochastic simulations
We simulated cluster formation using the Gillespie algorithm as previously described [37]. In short, for every iteration we first determined randomly the time of the change in cluster size which is determined by the total rate at which clusters could increase or decrease in size (e.g., in the DD recruitment model this rate for a cluster of size k is λ0 + kλ1 + kµ). The second step was to then choose at random which of two events (cluster size increase or decrease) occurs; this is determined by the relative value of the entry rate into the cluster (e.g., λ0 + kλ1) or exit from the cluster (e.g., kµ).
2.2.6 Statistics
Our clustering data are given as the number of T cells found in the 40 µm radius of a given parasite following intravital microscopy imaging [26], i.e., the data are simply a column of integers representing T cell numbers per parasite. (In co-clustering experiments the data also represent the number of Plasmodium-specific and non-specific T cells found per parasite.) As the data shows, in many cases majority of parasites have no T cells associated with them [26, and see main text].
To estimate parameters of mathematical models we used a likelihood approach where the likeli-hood represents the product of probabilities to observe clusters of different sizes where Pk is the mathematical model-predicted probability of observing a cluster of size k according to a set of parameter values, x(k) is the number of clusters of size k in the data, and kmax is the maximal cluster size in the data. In this procedure, the probability Pk(t) can be either given analytically as a steady state solution (e.g., eqn. (4)) or can be found by numerically solving the basic mathematical model predicting Pk(t) at a particular time (e.g., eqns. (1)–(2)). When fitting numerical solutions of the model to experimental data in some cases we fixed the rate of exit of T cells from clusters µ to different values because we found that it is generally impossible to accurately estimate both entry and exit rates simultaneously (see Results section).
The models were fitted by calculating negative log-likelihood and using routine FindMinimum in Mathematica version 11. When alternative models were fitted to the same dataset, we compared quality of the model fits to the data by comparing Akaike weights w based on the corrected Akaike Information Criterion (AIC) [38]: where p is the number of model parameters and N is the number of data points (parasites). AIC provides a score for each model based on its maximum likelihood value and the number of model parameters. The model with the lowest AIC score is considered to be best relative to the tested models. Weights of a given model can be treated as a likelihood of the model in the list of the tested models. As a rule of thumb, models with w < 0.05 can be considered to be inconsistent with experimental data in favor for models with higher weights. In addition, when comparing nested models, where one model is a special case of the other, we used the likelihood ratio test.
3 Results
3.1 Pertussis toxin-treated PyTCR cells form clusters randomly
By using intravital microscopy two recent studies have observed that activated CD8 T cells, specific to Plasmodium antigens, often localize near Plasmodium liver stages [26, 27]. In our experiments we used GFP-expressing Plasmodium yoelii (Py) sporozoites and fluorescently-labeled activated Py-specific CD8 T cells (PyTCR cells) [26]. A peculiar feature of these observations was a variable number of T cells found near individual parasites: in many experiments multiple parasites had no T cells while some parasites had more than 20 T cells in a 40 µm radius [26]. We have developed three alternative mathematical models that assume that the formation of clusters of CD8 T cells around Plasmodium liver stages is T-cell-intrisitic and is mediated exclusively by T cells. These models include 1) the null model predicting formation of clusters by chance (“random entry/exit” or Poisson model, eqn. (4)), 2) a model in which T cells find parasites randomly but are retained near the parasite (“density-independent (DI) exit” model, eqn. (5)), and 3) a model in which T cells actively recruit other T cells to the infection site (“density-dependent (DD) recruitment” model, eqn. (6)). By fitting these models to experimental data on clustering we concluded that the DD recruitment model is best consistent with data on clustering of PyTCR cells [26]. By performing experiments in which either PyTCR cells or PyTCR cells, treated with pertussis toxin (PyTCR+PT cells), were transferred into mice, previously infected with Py, we found that PT-treated PyTCR cells form smaller clusters that untreated T cells (Figure 2). PT treatment inactivates G-protein-coupled receptors (GPCRs) in the cells, and in particular, makes PT-treated cells to be unresponsive to chemokine gradients [39]. Fitting three basic mathematical models, predicting steady state distribution of T cells around the liver stages, to the data on clustering of PyTCR cells (see Materials and Methods for more detail on how models were fitted to data) confirmed our previous result that DD recruitment model provided the best fit of clustering data (Figure 2B). The random entry/exit model did not fit these data adequately, while the DI exit model provided a visually reasonable fit which was not supported by the Akaike weights or log-likelihood (result not shown). Interestingly, plotting the data and model fits on the log-scale revealed that even the best fit model did not accurately predict formation of a large cluster with 21 PyTCR cells (Figure 2B). Indeed, comparing the prediction of the DD recruitment model with data using a goodness-of-fit χ2 test showed that the model described the clusters until size 8 well (, p = 0.99), but failed at describing all clusters including one with 21 cells (, p < 0.001). According to the prediction of the mathematical model, the average cluster size of T cells around the liver stages at the steady state is rather small, 〈k〉∗ ≈ 1.5 (see eqn. (9)), because most parasites were not found by T cells within 6 hours (Figure 2A).
Importantly, however, all three models could accurately describe the formation of clusters by PT-treated PyTCR cells (Figure 2C) suggesting that the formation of these clusters is most likely due to a random encounter between T cells and the Py-infected hepatocyte. Indeed, the estimated relative entry rate θ0 was similar for two datasets (see legend of Figure 2) and was only slightly higher than the relative rate estimated in our previous work from other experiments involving co-transfer of PyTCR and OT1 cells [26, see Materials and Methods and below]. Thus, this result further supports the conclusion that GPCR-mediated signaling is important in the formation of CD8 T cell clusters around Plasmodium liver stages.
3.2 Clustering of endogenous CD8 T cells does not allow to discriminate between T-cell-intrinsic and T-cell-extrinsic models of cluster formation
Our previous analyses so far attempted to explain mechanisms behind the formation of clusters around Plasmodium liver stages from the T-cell-centric point of view; namely, we assumed that cluster formation is dependent on the presence of T cells (e.g., DD recruitment model, see eqn. (6)). However, it is possible that a very different alternative mechanism drives the formation of clusters, which is T-cell-extrinsic. In this case, variable clustering of T cells around the liver stages is driven by variability in the environment, for example, due to the level of “attractiveness” of individual parasites. This may arise because individual parasites may induce different degrees of inflammation as they migrate from the blood into the liver parenchyma, or some parasites may infect hepatocytes which are located in liver parts with a larger blood flow, increasing the chance of T cells to locate such parasites [35, 40].
To investigate whether a T-cell-extrinsic mechanism can be sufficient to explain the formation of clusters in our experiments we formulated two alternative mathematical models predicting the formation of clusters of different sizes: in the first model we assumed that there are two populations of parasites with different levels of attractiveness/rate of entry λ01 and λ02 (“2 population” model, Figure 1D and eqn. (10)), and in the second model we assumed that there is a distribution in the level of attractiveness of parasites given by a continuous Gamma distribution (“gamma” model, Figure 1E and eqn. (11)). To test whether models assuming T-cell-intrinsic or T-cell-extrinsic mechanisms of cluster formation perform better, we fitted the models to previously published data on the clustering of endogenous CD8 T cells around Py liver stages in mice [26]. In these experiments, mice were left naive or were immunized with radiation-attenuated sporozoites (RAS) and then 10 days later infected with wild-type Py expressing GFP (Figure 3A). Clustering of CD8 T cells around GFP-expressing liver stages was visualized by injecting CD8-binding antibody. We fitted five mathematical models to these data using a likelihood approach (eqn. (16)). This analysis showed that all mathematical models could accurately describe the lack of formation of large (k > 5) clusters around Plasmodium liver stages in naive (unimmunized) mice, and the simplest (null) random entry/exit model was favored by the Akaike weights (Figure 3B). While all models provided similar likelihood of the model given the data, lower weights for 2 population and gamma models was due to a larger number of fitted parameters (3 in 2 population/gamma models vs. 1 in the Poisson model).
Consistent with our previous analysis [26], the random entry/exit (Poisson) model could not adequately describe the distribution of cluster sizes in RAS-immunized mice (Figure 3C). Interestingly, both 2 population and gamma models did not provide an improved fit of these clustering data as compared to DD recruitment or DI exit models which fitted the data with similar quality (Figure 3C). This was surprising given that the DD recruitment and DI exit models were not able to accurately describe the two peaks in the cluster size distribution (at 0 and 7 T cells/parasite). A closer inspection revealed that the DD recruitment, DI exit, 2 population, and gamma models provided fits of nearly identical quality as based on the negative log-likelihood values (𝓛 ≈ 168), and lower weights were selected for models with more parameters. All models except the Poisson and 2 population models could accurately describe the data (based on goodness-of-fit χ2 test); the 2 population model deviation was due to its inability to accurately predict the formation of one cluster with 19 cells (results not shown). Importantly, the 2 population and gamma models could fit other clustering data relatively well, e.g., data on clustering of PyTCR cells or PyTCR cells treated with PT (Figure 2, results not shown). Taken together, these results demonstrate that several clustering datasets do not allow to discriminate between T-cell-intrinsic and T-cell-extrinsic mechanisms of formation of CD8 T cell clusters around Plasmodium liver stages.
3.3 Environmental variability is not the main driver of cluster formation
To discriminate between T-cell-intrinsic and T-cell-extrinsic mechanisms of formation of CD8 T cell clusters around Plasmodium liver stages we turned to additional experimental data generated previously [26]. In these experiments, Py-specific T cells and T cells of irrelevant specificity (OT1) were transferred either separately or together into mice previously infected with Py-GFP, and the formation of clusters around Py liver stages was measured by intravital microscopy (Figure 4A). We have previously shown that the DD recruitment model describes best (based on Akaike weights) the data on the clustering of PyTCR cells when transferred alone or data on the clustering of PyTCR and OT1 cells when transferred together [26]. In contrast, the clustering of OT1 cells alone was best described by the random entry/exit model [26, Figure 4B-C]. Therefore, these data indicate that the clustering of T cells, which are not specific to Plasmodium depends on the presence of Py-specific T cells suggesting that variability in parasite’s “attractiveness” alone cannot explain these data. We formally tested if the 2 population or gamma models can describe the clustering data of OT1 cells in the following way. We fitted the 2 population model to the data on clustering of OT1 cells alone or in the mixture with PyTCR cells simultaneously. We therefore fitted the models by allowing all three parameters of the model (θ01, θ01, and f, see eqn. (10)) to be different for the two datasets or by allowing only the fraction of parasites with different attractiveness level f to vary between two datasets while keeping other parameters the same between datasets (Figure 4B-C). In this way, we tested the hypothesis that the clustering of OT1 cells is driven exclusively by factors which are independent of Py-specific CD8 T cells. Because two fits are from nested models, comparing the quality of the fits revealed that the model assuming PyTCR-cell-independent environment fits the two datasets significantly worse (, p < 0.001). Fitting the gamma model to the same two datasets assuming either identical or variable parameters between the two datasets also suggested that the model with constant parameters fits the data significantly worse (results not shown). Thus, these results strongly suggest that the T-cell-extrinsic models of cluster formation are not consistent with the data on different clustering patterns of OT1 cells in the absence or presence of Py-specific T cells.
It is important to note that the use a specific mathematical model (e.g., 2 population model) simply allows to formally test if distributions of cluster sizes of OT1 cells are different in two different conditions. This can be also done using a χ2 test [41] which showed that these distributions are only marginally different (, p = 0.04). Thus, the use of models allows to obtain much stronger statistical power at falsifying the T-cell-independent (“environment”) hypothesis as the sufficient mechanism of cluster formation.
3.4 Several alternative retention models poorly describe data on clustering of PyTCR cells
While our experiments of clustering of OT1 T cells either alone or in presence of PyTCR T cells argue against T-cell-extrinsic clustering model, they do not allow to fully discriminate between alternative T-cell-intrinsic clustering models. Fitting the steady state prediction of the DI exit and DD recruitment model to clustering of PyTCR T cells (Figure 2) or clustering of OT1 T cells in the presence of PyTCR cells [26] favored the DD recruitment model (based on Akaike weights). However, it is possible that a specific form of the retention model, i.e., that the per capita exit rate is inversely proportional to the cluster size, was an accidentally poor choice. Therefore, we tested two alternative models of how exit rate from a cluster could depend on cluster size with µk = kαµ or µk = kµe−αk. We fitted the numerical solution of the basic mathematical model (eqns. (1)–(2)) to the clustering of PyTCR T cells (Figure 2A) using a likelihood approach. Both alternative retention models still described the data worse than the DD recruitment model (w < 0.001 and w = 0.02 for the two models, respectively, results not shown) suggesting limited support for the hypothesis that retention of T cells plays the major role in cluster formation. Therefore, in our following analyses we focus exclusively on the DD recruitment model.
3.5 No evidence that T cells of irrelevant specificity influence clustering
In our previous analysis we showed that the DD recruitment model-based fit of the data on the clustering of PyTCR and OT1 cells in the co-transfer experiments (Figure 4A) predicted similar relative recruitment rate parameters θ0 and θ1 (see Table S1 in Cockburn et al. [26]). However, the previous analysis treated clustering of PyTCR and OT1 cells in the co-transfer experiments independently, and here we extend this analysis by considering potential mechanisms behind co-clustering of these two cell populations. First, we found that there was no significant difference in the number of PyTCR or OT1 T cells clusters around Plasmodium liver stages with similar proportions of parasite having more PyTCR or OT1 cells (Figure 5A). To investigate whether the data on the co-clustering of T cells may provide evidence of OT1 T cells assisting in cluster formation we developed a mathematical model tracking the dynamics of co-clustering of two types of cells (see eqns. (12)–(15) in Material and Methods) and fitted that model to the co-clustering data (dataset #3) using a likelihood approach. As we show in the next section, our clustering data do not allow to identify both the rate of T cell entry into the cluster and exit rate from the cluster from measuring clusters at one time point. Therefore, in this analysis we fixed the per capita exit rate µ = 0.5/h and estimated entry rates. Our overall results were robust to several other tested values of the exit rate such as µ = 0.1/h or µ = 3/h even though estimates of the entry rates were strongly dependent on the assumed exit rate (results not shown and see next section).
Using the DD recruitment model we next tested several different mechanisms of how specific and non-specific T cells may participate in cluster formation (see Table 1). Despite the highly correlated numbers of the Py-specific and non-specific T cells around Plasmodium liver stages (Figure 5A), different roles of these two CD8 T cell types seem to be inherent in the data (Table 2). Specifically, the model in which PyTCR T cells attract all cells into the cluster was statistically better at describing these data as compared to any other model tested (based on Akaike weights); interestingly, an alternative model in which OT1 cells exclusively drive cluster formation could not fit the co-clustering data well (model “Only OT1 cells recruit” in Table 2).
In two separate models we tested whether OT1 cells “help” in the formation of clusters which is driven by Py-specific T cells. Perhaps unsurprisingly in both models (“PyTCR and OT1 cells recruit at different rates” and “ PyTCR and OT1 cells recruit at different rates towards different cell types”) we found no evidence that OT1 cells enhance cluster formation (Table 2 and Figure 5B-C).
In contrast, parameter estimates suggested that OT1 cells may inhibit cluster formation because the estimated OT1-driven recruitment rates λ1 were negative (Table 2); however, improvements of the fits of these two more complicated models were not supported by the likelihood ratio test (p > 0.1, see Figure 5B-C). Thus, our results suggest that non-specific T cells are “passive” participants in the clusters and do not significantly promote or impede the formation of clusters. A similar result was obtained recently using another Plasmodium experimental system [28].
Predictions of our best mathematical model in which only PyTCR cells recruit all activated T cells to the site of infection can be shown as the distribution of cluster sizes for each cell type (e.g., Figure 5B-C) as well as the probability to observe a cluster with i PyTCR and j OT1 cells (Figure 5D). Careful examination of this fit revealed that the model predicts a slight bias towards having more PyTCR cells per cluster than OT1 cells (Figure 5E). Linear regression analysis of the co-clustering data indeed suggests that there may be bias towards having more PyTCR cells than OT1 cells per cluster (Figure 5F); however, this result is not fully consistent with another analysis (e.g., Figure 5A), and the application of linear regression to data with round numbers may not fully appropriate. While the existence of such a bias is indeed in line with the analytical analysis of the steady state distribution of cluster sizes (Harshana Rajakaruna, ms. in preparation), this bias is small (perhaps one extra PyTCR cell in clusters of a total size of 20), and biological relevance of such a bias for the killing of the parasite is most likely limited.
3.6 Clusters of Py-specific CD8 T cells around Py-infected hepatocytes are formed rapidly
Our analyses so far made an assumption that clusters around Plasmodium liver stages reach a steady state by 6-24 hours after T cell transfer. To understand potential limitations of this approach, we therefore performed several analyses using previously published experimental data.
Because our main mathematical model of T cell clustering (eqns. (1)–(2)) can be solved numerically and thus fitted to experimental data assuming a specific clustering mechanism (e.g., DD recruitment model), we investigated if the rates of T cell entry into the cluster (λ0 and λ1) and rates of exit from the cluster (µ) can be estimated from data in which PyTCR cell clusters around Py-infected hepatocytes were observed at 6 hours after T cell transfer. Interestingly, fitting the DD recruitment model (eqns. (1)–(2)) to data on the clustering of PyTCR cells transferred alone (Figure 2B or Figure 5A) revealed that model fits favored very high entry and exit rates, e.g., rates exceeding 20-30/h (results not shown). By fixing the exit rate from the cluster to multiple values we found that estimates of the absolute and relative values of the entry rate depended strongly on the exit rate values, and the relative entry rates (θ0 and θ1) approached constant values at high exit rates (Table 3). Importantly, all the fits of models with dramatically different exit rates were of nearly identical quality as based on negative log-likelihood suggesting that data on clustering of T cells at one time point are not sufficient to estimate entry and exit rates simultaneously. These results were confirmed for two independent datasets (experiments with PyTCR cells alone as shown in Figure 2A and Figure 5A) although exact values of parameter estimates such as λ0 did slightly vary between two sets of experiments (see Figure 2 and estimates in Table S1 in [26]).
To gain further insights into the kinetics of T cell cluster formation we analyzed additional data in which the same parasites (n = 32) were followed after T cell transfer over time and cluster sizes at different time points were recorded (Figure 6A and Figure S2). In these experiments imaging started between 4 and 8 hours and followed for about 4 hours [26]. As expected there was a variable and statistically significant increase in the number of T cells found around individual Py-infected hepatocytes between T cell transfer and start of imaging (tstart, Figure 6B). In contrast, in the following ∼ 4 − 8 hours there was a minor change in cluster sizes (tend, Figure 6B). However, because imaging of CD8 T cell cluster formation started at different time points after T cell transfer there may be biases associated with the simple analysis of the data which takes into account only start and end time points of the clusters (e.g., Figure 6B). To obtained more accurate insights we further analyzed these data using mathematical models of cluster formation.
To take full advantage of these “longitudinal” data in which T cell cluster formation was followed over time for individual parasites (Figure S2), we divided the data into individual “paths”, i.e., the number of T cells found near the parasite at sequential time points. For example, a parasite which did not have any T cells nearby and for which measurements were done at 0, 4 and 6.2 hours after T cell transfer, the path is “0 → 0 → 0”. For the parasite that was surrounded by most T cells in these experiments, the path is “0 → 8 → 9” for times 0, 4, and 6.2 h post T cell transfer (Figure 6B). A mathematical model of cluster formation can then be used to calculate the likelihood of a particular path by assuming that individual “sub-paths” along the path are independent (and thus by multiplying likelihoods of the model for individual sub-paths). For example, the probability to observe the path “0 → 8 → 9” at times (0, 4, 6.2) h is simply the product of the probability to observe 9 T cells in the cluster at 6.2 hours given that at 4 hours there were 8 T cells in the cluster and the probability to observe 8 T cells in the cluster at 4 hours given that at 0 hours there were 0 T cells in the cluster: where the probability Pk(t|i, t0) was calculated using the basic model (see eqns. (1)–(2)) with initial conditions Pi(t0) = θij and θij is Kronicker delta (θij = 1 if i = j and θij = 0, otherwise). Fitting the DD recruitment model to these “longitudinal” data subdivided into “paths” resulted in the following entry/exit rates λ0 = 0.14/h, λ1 = 0.16/h, µ = 0.09/h. Additional analysis by fixing exit rate µ to different values and then comparing quality of the model fit using likelihood ratio test revealed that estimate of the parameters are relatively robust (i.e., fixing the exit rate to much lower or much higher values resulted in fits lower quality as judged by likelihood ratio test). Furthermore, by resampling the paths with replacement we found relatively small confidence intervals for the estimated parameters suggesting that measurement of T cell clusters longitudinally allows for a relatively accurate estimates of all three parameters of the DD recruitment model determining the kinetics of cluster formation (results not shown).
Parameter estimates of the model fitted to “longitudinal” (paths) data suggest that rates of entry into the cluster and exit from the cluster are relatively small, and this appears to contradict the formation of relatively large clusters already in 4 hours after T cell transfer (Figure 6C). Indeed, model fits did not accurately predict formation of clusters with > 5 T cells (results not shown). In addition, while the estimate of θ0 = λ0/µ was reasonable, the estimate of relative recruitment rate θ1 = λ1/µ was too low when compared with model estimates for clustering of T cells at 6 hours after transfer (e.g., Table 3 for µ = 0.1/h).
The major caveat of the analysis is the assumption that the parameters determining T cell clustering are constant over time. Our data indicate that formation of clusters may be slowing down over time (Figure 6B). Therefore, we fitted the DD recruitment model to the longitudinal/path data assuming that parameters determining kinetics of cluster formation depend on the time since T cell transfer. Given how the data were collected (Figure S2) for our analysis we made the simplest assumption that the rates are constant in two time intervals: (0 − 4) h and (4 − 12) h but may be different between the time intervals. Assuming that in the DD recruitment model recruitment rates λ0 and λ1 are time-dependent and the exit rate µ is time-independent, the model fitted the data significantly better than the DD recruitment model with constant parameters (likelihood ratio test, , p < 10−6). Parameter estimates suggest a six fold reduction in both λ0 and λ1 4 hours after T cell transfer (see legend of Figure 6 for actual parameter estimates). A similar decline in both λ0 and λ1 at 4 hours after T cell transfer was confirmed by fitting the model in which both rates declined by the same amount α; such a model fitted the data with a similar quality as the model that allowed for different decline in the two rates with time since T cell transfer (likelihood ratio test, , p = 0.89). Because the distribution of cluster sizes was measured experimentally at different time points it was not possible to visualize the model fits of the data. However, because model predictions suggested little change in cluster size distributions between 4 and 12 hours after T cell transfer (Figure 6C), the model predicted well the distribution of cluster sizes for each of the parasite at the end of imaging (Figure 6D, , p = 1). Interestingly, this analysis indicated an extremely slow rate of T cell exit from clusters at 4-12 hours after T cell transfer suggesting that nearly every cell that enters the cluster after 4 hours post T cell transfer remains in the cluster which is an indirect support for the “retention” model.
An alternative DD recruitment model is one in which recruitment rates into the cluster remain constant over time but exit rates from the cluster change with time. This model did slightly improve the fit of the as compared to the model with constant parameters (likelihood change of 3.32, , p = 0.01) and predicted that constant recruitment rates are λ0 = 0.14/h and λ1 = 0.19/h, and exit rate for the first 4 hours is µ = 0/h and for after 4 hours is µ = 0.19/h. This model suggests an alternative interpretation of the cluster formation dynamics — namely that T cells are recruited into the cluster and retained during the first four hours after T cell transfer — but after the initial time additional recruited T cells have a high chance of leaving. Because the quality of this model fit of the data was significantly worse than that of the model with time-dependent recruitment rates (∆AIC = 21 or Akaike weight w < 0.001 for time-dependent exit rate model), our data appear to be more consistent with the time-dependent recruitment and constant exit. This suggests that the best explanation of the longitudinal clustering data is that the formation of clusters is driven by the DD recruitment model in which the rate of T cell recruitment into the cluster declines over time. Parameter estimates also suggest that the formation of clusters around Py-infected hepatocytes occurs mainly during the first 4 hours after T cell transfer.
The dynamics of change in the number of T cells near the parasite between 4 and 12 hours were followed by time-lapse intravital microscopy which allowed to calculate the number of T cells entering the cluster and leaving the cluster in this time period [26, Figure 6E-F]. Analysis showed that both entry and exit rates were strongly dependent on the cluster size k even though there was large variability in the number of T cells entering and exiting individual clusters. Interestingly, the slopes of the dependence of recruitment and exit rates was 2-3 fold higher for experimentally measured rates as compared to the parameters found by fitting DD recruitment model to longitudinal data (Figure 6E-F). One potential explanation of this difference is that perhaps not all cells that come near the parasite (i.e., within 40 µm distance) recognize the infection and leave, thus, increasing the overall observed T cell surveillance rate. In contrast, the model only accounts for T cells which actually formed clusters and thus most likely have recognized the parasite.
4 Discussion
Studies from two independent groups showed that activated Plasmodium-specific CD8 T cells form clusters around Plasmodium-infected hepatocytes and that such clusters are correlated with elimination of the Plasmodium liver stages [26–28]. Application of mathematical models to data on the distribution of the number of Py-specific CD8 T cells around randomly chosen parasites suggested that formation of the clusters is not a random process; the model in which activated T cells of different specificities are attracted at a rate proportional to the number of Py-specific T cells already present near the parasite, described the data with best quality [26]. More recent work also suggested that formation of CD8 T cell clusters around Plasmodium-infected hepatocytes depends on CD11c+ cells and that activated CD8 T cells, specific to irrelevant antigens, do not appear to play a significant role in protection against Plasmodium challenge [28]. We analyze previous and additional data with the use of mathematical modeling to provide further insights into potential mechanisms of the formation of clusters around Py-infected hepatocytes.
First, we found that several independent experimental datasets are fully consistent with the model in which variability in the number of activated Py-specific CD8 T cells located near the parasite-infected hepatocytes is driven by variability in the “environment” around the infected hepatocytes, providing indirect support for the T cell-extrinsic mechanism of cluster formation (e.g., Figure 3). These results suggested that data on clustering of Py-specific T cells alone are simply insufficient to discriminate between T cell-intrinsic and T cell-extrinsic mechanisms of cluster formation [42]. A key experiment, rejecting the “variability of the environment” hypothesis as the sufficient mechanism explaining distribution of cluster sizes is one involving either transfer of only OT1 T cells (which are not specific to Py antigens) or OT1 T cells together with PyTCR T cells – only in the latter case, OT1 T cells form co-clusters with Py-specific T cells (Figure 4 and see [26, 28]). The mathematical model assuming fixed yet variable (between individual parasites) environment was not able to accurately explain such data (Figure 4).
Second, while OT1 T cells of irrelevant specificity are found in clusters together with Py-specific CD8 T cells, we found no evidence that OT1 improve cluster formation (Figure 5). If anything, OT1 T cells may in fact reduce the rate of recruitment of other T cells into the cluster as indicated by the negative values for the recruitment rate λ1 (Table 2); however, this value was not significantly different from zero. Mathematical modeling also suggested that there may be a slight bias in the clusters to have more Py-specific T cells than T cells of irrelevant specificity per cluster but the biological relevance of such a small bias remains unclear. The unimportant role of T cells of irrelevant specificity in the formation of T cell clusters in Py-infected mice is consistent with the observation that transfers of large numbers of activated CD8 T cells with irrelevant specificity into Plasmodium-infected mice did not impact efficiency at which parasites were killed by Plasmodium-specific T cells [28]. Interestingly, and perhaps surprisingly, this result contradicts a recent observation of suppression of development of T-cell-driven type 1 diabetes by islet-non specific CD8 T cells [43].
Third, by following longitudinal changes in the number of CD8 T cells around individual parasites over time we found that T cell clusters are formed rather rapidly, at least within the first 4 hours after T cell transfer (Figure 6B), and mathematical modeling predicted recruitment of T cells to the parasite and retaining of the T cells in the cluster in that time period. Interestingly, the rates of entry into and exit from the clusters declined after the 4 hours six fold further supporting the conclusion that clusters are formed rapidly and few cells enter and exit the cluster after 4 hours since T cell transfer (Figure 6C-D). Stochastic simulations of the formation of clusters assuming DD recruitment model with different entry/exit rates also suggested that between 4 and 8 hours post T cell transfer, entry and exit rates cannot be large (Figure S3). This is because when these rates are large, changes in the cluster size in the 4-8 hour time period are highly variable with some clusters growing in size exponentially while other clusters nearly disappearing (e.g., Figure S3C&F). This, however, was not observed in experimental data (shown by dotted histogram in Figure S3D-F and see Figure 6B and Figure S2). Rapid recruitment of CD8 T cells to the liver stages in the first 4 hours after T cell transfer may be the result of the specific experimental set-up as it is expected that immediately after intravenous injection, large numbers of T cells would be passing through the liver increasing chances of T cells finding the infection site [44], and that the number of liver-resident CD8 T cells tends to reach a steady state at 2-3 hours after T cell transfer [45, James O’Connor and Ian Cockburn (ms. in preparation)].
Interestingly, there was some discrepancy between the estimated rates of T cell entry into the cluster and exit from the cluster measured experimentally and predicted by the model (Figure 6E-F) most likely indicating that not all T cells that were observed to come in close proximity with the parasite recognize it. We also found a strong correlation between experimentally measured rates of T cell entry into and exit from the clusters (Figure S4) which may indicate that in addition to T cell-intrinsic mechanisms of clustering, some parasites may be more “attractive” to T cells. Indeed, none of our tested models could well explain the formation of extremely large T cell clusters around Py-infected hepatocytes (e.g., with 15 or more T cells, see Figure 2B) which could indicate the need for future models to include both DD recruitment and variability in parasite’s attractiveness.
In this paper we analyzed a number of different datasets that involve different cell types, different experimental set-ups, and different mice. We found it encouraging and perhaps a bit surprising that some of these datasets were in a way “consistent”. Specifically, we observed similar clustering of CD8 T cells in naive mice (Figure 3A), PT-treated PyTCR T cells (Figure 2B), or activated OT1 T cells of irrelevant specificity (Figure 4A) and the random entry/exit model described these data with nearly identical parameters (likelihood ratio test, , p = 0.07). The DD model could describe the distribution of cluster sizes of PyTCR T cells in three different experiments (Figure 2A and data in [26]) with identical parameters (, p = 0.51). However, the clustering of CD8 T cells following immunization with radiation-attenuated sporozoites (Figure 3B) did not match well the clustering of the mixture of PyTCR and OT1 T cells (, p = 0.002) perhaps highlighting potential differences between active and passive immunizations (the latter involving transfer of pre-activated CD8 T cells).
In multiple analyses we found that a DI exit (retention) model did not describe well the clustering data. However, a poorer fit of the data (as compared to other tested models) does not necessarily falsify a model [42], and additional experiments will be needed to formally rule out this model. Fitting the DI exit model to the “longitudinal” data on change in cluster size around individual parasites (e.g., Figure S2) revealed that this model could not accurately describe the data assuming a constant entry and time-dependent exit rates based on likelihood of the model (𝓛 = 100.2 vs. 𝓛 = 83.0 of the DD recruitment model with time-dependent recruitment and constant exit rates, results not shown). In addition, if the rate of T cell exit from the clusters found in the DD recruitment model is constant over the course of the first 12 hours since T cell transfer, it would suggest that T cells mostly enter the clusters and rarely exit them (given µ = 0.028/h corresponding to the residence time of T cells in the cluster of about 1/µ ≈ 36 h), providing some indirect support for the retention model.
Conversely, our result that the DD recruitment model describes most of the data with best quality does not prove that this model is the true mechanism of the formation of large clusters around Py-infected hepatocytes. Future experiments will have to test the major prediction of the model — that clusters of a large size attract more T cells per unit of time. Such experiments may involve measurement of T cell movements in the liver using intravital microscopy and estimating bias in T cell shift towards the parasite. Indeed, our recent work suggested that there is a bias in PyTCR cell movements towards Py-infected cells [31] but more analyses are needed to evaluate whether such a bias depends on the number of T cells already present at the parasite and whether a small bias is sufficient to explain the formation of larger clusters (with k ~ 5 − 10 of CD8 T cells per parasite) within few hours after T cell transfer. Detecting a bias in T cell movement towards the infection site may be complicated as our current analysis predicts that “attraction” seems to be present only during the first 4 hours after T cell transfer (Figure 6C-D).
Our analysis has several potential limitations. The biggest issue is that by using numerical solutions of the DD recruitment model we showed that the distribution of cluster sizes at a single time point does not allow to accurately estimate the rates of T cell entry into the clusters and T cell exit from the clusters, and thus, most of our analysis were restricted to estimating relative recruitment rates. Ongoing analysis has also demonstrated this point using analytical derivations (Harshana Rajakaruna, ms. in preparation). While the estimated values of the recruitment rates λ0 and λ1 in the DD recruitment model directly depend on the assumed exit rate µ (see Table 3) we showed that the likelihood of the model fit to data assuming a steady state or dynamics for clusters at a given exit rate µ were nearly identical strongly suggesting that our results on best fit models obtained assuming steady states are robust. However, the actual values of the entry and exit rates cannot be found with certainty as these depend on the actual value of the assumed exit rate (Table 3).
Another complexity in the analysis comes from our finding that rates of T cell entry into the cluster are time-dependent (Figure 6). To investigate whether this impacts our selection of best fit models assuming steady state solutions we did the following. We fitted the DD recruitment model to the clustering data at one time point by assuming that early recruitment rates λ0 and λ1 are unknown and that late entry rates are fixed to values found from the analysis of longitudinal data (Figure 6) and that the exit rate µ is constant. Under these minimal assumptions the model fit was of nearly identical quality as the model fit of the data assuming a steady state (results not shown). Therefore, even for time-dependent parameters our results determining which models are not consistent with clustering data remain valid.
An important experimental limitation of our data is the way of how experiments were performed whereby pre-activated T cells were transferred into mice that had been already infected with Plasmodium sporozoites (e.g., Figure 2A). This sequence of events does not fully match the physiological situation in which activated or memory CD8 T cells are already present at the time of sporozoite challenge. In fact, the rapid predicted decline in the rates of T cell recruitment into clusters with time suggests that it may be an artifact of the experimental system. Whether change in the experimental protocol will lead to support of the same mathematical models of cluster formation remains to be determined (and is the focus of our ongoing experiments and analyses).
Mathematical methodologies used in this work provided deeper understanding of how CD8 T cells form clusters around Plasmodium-infected hepatocytes. While formation of such clusters was a novel observation in malaria infection of the liver, clusters of immune cells have been observed in multiple systems including herpes simplex virus (HSV) [46] and Mycrobacterium tuberculosis (Mtb) [47]. In fact, formation of granulomas in Mtb-infected animals and humans is a classical example of T cell clustering around the infection site. Interestingly, both Mtb-specific CD4 T cells and CD4 T cells of irrelevant specificities were found in granulomas of Mtb-infected monkeys [48] which could be explained by the DD recruitment model extended in this work. Movement of neutrophils towards an injury site may also depend on the number of neutrophils that have already reached the site [49]. It may be useful to combine mathematical modeling tools for deeper understanding of the mechanisms of formation of clusters of immune cells in these and other systems.
While our work provides some clarification regarding mechanisms of CD8 T cell cluster formation around Plasmodium-infected hepatocytes, many questions remain. In particular, while clusters appear to be important for the death of the parasite [26, 27], whether clusters of a larger size kill the parasites faster remains unknown. Classical work involving killing of chromium-labeled target cells by cytotoxic T lymphocytes (CTLs) suggested a faster killing of targets bound by multiple CTLs [50], and in vivo, death of peptide-pulsed targets is directly proportional to the number of peptide-specific CTLs [30]. Recent work also suggested that the probability of death of virus-infected cells in skin in vivo was higher when the infected cell was contacted by several antigen-specific CD8 T cells [29]. Whether the same relationship holds for T cells killing Plasmodium parasites in the liver remains to be determined.
Our results suggest that activated CD8 T cells of irrelevant specificities do not play a major role in cluster formation, and elegant experiments demonstrated that large numbers of non-specific T cells do not impair the ability of Plasmodium-specific T cells to eliminate the parasites [28]. However, the latter result was found by using only two different ratios of Plasmodium-specific and non-specific T cells and 3 mice per group, so it remains to be determined if competition between such cells for the access to infected cells occurs at higher ratios, e.g., as has been observed in another system [43]. In natural settings we do expect that Plasmodium-specific T cells will be likely outnumbered by memory T cells specific to other infections, and therefore, deeper understanding of such competition may be of relevance to malaria vaccines, inducing liver-resident memory CD8 T cells for protection [51].
Accumulation of large numbers of CD8 T cells around Plasmodium-infected cells raises an intriguing possibility that parasites may in fact attempt to attract T cells. While this may be detrimental to an individual parasite, as a population this may give an advantage if attracting many T cells to one site prevents T cells from effectively locating parasites at other sites. To cause blood-stage infection, there is a need for only one liver stage to mature and release differentiated parasites (merozoites) into the blood stream. Indeed, it should be noted that in most of our experiments, many of surveyed parasites did not have a T cell nearby at 6-8 hours after T cell transfer. Future studies may be needed to investigate whether such a strategy is indeed evolutionarily advantageous.
Taken together, here we illustrated the power of combining the use of detailed quantitative experimental data with mathematical modeling, and limitations that come from inability to make solid conclusions from extensive yet limited experimental data. The field of immunology will likely benefit from closer collaborations between experimentalists and modellers where experimentalists being involved in data analyses and modeling, and modellers are cooperating with experimentalists in designing experiments to test and potentially falsify alternative mathematical models.
6 Author contributions
RKK, IAC, and VVG designed the study. RKK and VVG run most of analyses and wrote first drafts of the paper. IAC performed additional data analyses and provided feedback on data interpretation and modeling. The final version of the paper was written by VVG but all authors contributed to the writing the final version of the paper via comments and suggestions.
5 Acknowledgements
We would like to thank Harshana Rajakaruna for useful feedback on previous versions of the paper. This work was supported by the NIH grant (R01 GM118553) to VVG.