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 h 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 h, 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 liver stages, 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 CD8 T cells) can accurately explain the available data. Second, while both Py-specific CD8 T cells and T cells of irrelevant specificity (non-specific CD8 T cells) are attracted to the clusters, we found no evidence that non-specific CD8 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 CD8 T cells, thus illustrating high efficiency of CD8 T cells in locating their targets in complex peripheral organs, such as the liver. Taken together, our analysis provides novel insights into and attempts to discriminate between alternative mechanisms driving the formation of clusters of antigen-specific CD8 T cells in the liver.

Pathogen-specific CD8 T cells face the problem of finding rare cells that present their cognate Ag either in the lymph node or in infected tissue. Although quantitative details of T cell movement strategies in some tissues such as lymph nodes or skin have been relatively well characterized, we still lack quantitative understanding of T cell movement in many other important tissues, such as the spleen, lung, liver, and gut. We developed a protocol to generate stable numbers of liver-located CD8 T cells, used intravital microscopy to record movement patterns of CD8 T cells in livers of live mice, and analyzed these and previously published data using well-established statistical and computational methods. We show that, in most of our experiments, Plasmodium-specific liver-localized CD8 T cells perform correlated random walks characterized by transiently superdiffusive displacement with persistence times of 10–15 min that exceed those observed for T cells in lymph nodes. Liver-localized CD8 T cells typically crawl on the luminal side of liver sinusoids (i.e., are in the blood); simulating T cell movement in digital structures derived from the liver sinusoids illustrates that liver structure alone is sufficient to explain the relatively long superdiffusive displacement of T cells. In experiments when CD8 T cells in the liver poorly attach to the sinusoids (e.g., 1 wk after immunization with radiation-attenuated Plasmodium sporozoites), T cells also undergo Lévy flights: large displacements occurring due to cells detaching from the endothelium, floating with the blood flow, and reattaching at another location. Our analysis thus provides quantitative details of movement patterns of liver-localized CD8 T cells and illustrates how structural and physiological details of the tissue may impact T cell movement patterns.

Pathogen-specific CD8 T cells face the problem of finding rare cells that present their cognate antigen either in the lymph node or infected tissue. To optimize the search for rare targets it has been proposed that T cells might perform a random walk with long displacements called Levy walks enabling superdiffusive behavior and shorter search times1–3. Many agents ranging from cells to large animals have been found to perform Levy walks3–5 suggesting that Levy walk-based search strategies may be evolutionary selected6,7. However, whether random walk patterns are driven by agent-intrinsic programs or shaped by environmental factors remains largely unknown8. To address this problem we examined the behavior of activated CD8 T cells in the liver where both the movement of the cells and the underlying structural constraints can be clearly defined. We show that Plasmodium-specific liver-localized CD8 T cells perform short displacement, Brownian-like walks and yet display superdiffusive overall displacement, the cardinal feature of efficient Levy walks. Because liver-localized CD8 T cells are mainly associated with liver sinusoids, simulations of Brownian or Levy walkers in structures derived from the liver sinusoids illustrate that structure alone can enforce superdiffusive movement. Moreover, Brownian walkers require less time to find a rare target when T cells search for the infection in physiologically-derived liver structures. Importantly, analysis of fibroblastic reticular cell networks on which CD8 T cells move in lymph nodes also allows for superdiffusion in simulations, though this is not observed experimentally, suggesting that structure is not the only factor determining movement patterns of T cells. Our results strongly suggest that observed patterns of movement of CD8 T cells are likely to result from a combination of cell-intrinsic movement programs, physical constraints imposed by the environmental structures, and other environmental cues. Future work needs to focus on quantifying the relative contributions of these factors to the overall observed movement patterns of biological agents.

Mathematical modeling provides a rigorous way to quantify immunological processes and to discriminate between alternative mechanisms driving specific biological phenomena. It is typical that mathematical models of immunological phenomena are developed by modellers to explain specific sets of experimental data after the data have been collected by experimental collaborators. Whether the available data are sufficient to accurately estimate model parameters or to discriminate between alternative models is not typically investigated. While previously collected data may be sufficient to guide development of alternative models and help estimating model parameters, such data often do not allow to discriminate between alternative models. As a case study we develop a series of power analyses to determine optimal sample sizes that allow for accurate estimation of model parameters and for discrimination between alternative models describing clustering of T cells around Plasmodium liver stages. In our typical experiments, mice are infected intravenously with Plasmodium sporozoites that invade hepatocytes (liver cells), and then activated CD8 T cells are transferred into the infected mice. The number of T cells found in the vicinity of individual infected hepatocytes at different times after T cell transfer is counted using intravital microscopy. We previously developed a series of mathematical models aimed to explain highly variable number of T cells per parasite; one of such model, the density-dependent recruitment (DDR) model, fitted the data from preliminary experiments better than the alternative models, such as the density-independent exit (DIE) model. Here we show that the ability to discriminate between these alternative models depends on the number of parasites imaged in the analysis; analysis of about n = 50 parasites at 2, 4, and 8 hours after T cell transfer will allow for over 95% probability to select the correct model. The type of data collected also has an impact; following T cell clustering around individual parasites over time (called as longitudinal (LT) data) allows for a more precise and less biased estimates of the parameters of the DDR model than that generated from a more traditional way of imaging individual parasites in different liver areas/mice (cross-sectional (CS) data). However, LT imaging comes at a cost of a need to keep the mice alive under the microscope for hours which may be ethically unacceptable.We finally show that the number of time points at which the measurements are taken also impacts the precision of estimation of DDR model parameters; in particular, measuring T cell clustering at one time point does not allow accurately estimating all parameters of the DDR model. Using our case study, we propose a general framework on how mathematical modeling can be used to guide experimental designs and power analyses of complex biological processes.

11Copyrighted by the AAI (abstract was published as a part of AAI annual meeting). 12 1 Introduction 13Malaria is a life-threatening disease that is a result of red blood cell (erythrocyte) destruction by 14 eukaryotic parasites of the Plasmodium genus. The majority of deaths (estimated to be about 500,000 15 annually) are among children, who have not yet developed immunity to the pathogen [1, 2]. There are 16 five species that infect humans: P. falciparum, P. vivax, P. malariae, P. ovale, and P. knowlesi [3]. 17 Three species of malaria parasites that are used as animal models for human malaria in mice are P. 18 yeolii, P. berghei, and P. chabauidi [4]. While there are similarities and differences in replication and 19 pathogenesis of Plasmodium species in humans and mice, in this paper we focus solely on infection 20 of mice with Plasmodium parasites. 21 The infection of the host is started by a mosquito, the vector between mammalian hosts, injecting 22 the sporozoite form of parasites into the skin. Studies have estimated that the initial number of 23 sporozoites entering the host is as low as 10-50 [5, 6], of which only a fraction succeed to migrate to 24 the liver to start an infection of hepatocytes by forming liver stages [7][8][9]. This liver stage of infection 25 lasts for approximately 6.5 days in humans and about 2 days in mice [10][11][12][13]. Because liver stage is 26 asymptomatic, removal of all liver stages prevents clinical symptoms of malaria and thus is highly 27 desirable feature of an effective vaccine. Indeed, previous studies have shown that memory CD8 T 28 cells are required for protection against a challenge with a relatively large number of sporozoites 29 [14, 15] and that vaccination that induces exclusively memory CD8 T cells of a single specificity 30 can mediate sterilizing protection against a sporozoite challenge [16][17][18][19][20][21][22][23]. Antibodies and CD4 T 31 cells may also contribute to protection in some circumstances, for example, following inoculation of 32 sporozoites by mosquitoes in the skin [24, 25]. Given that mouse liver contains about 1 − 2 × 10 8 33 1 hepatocytes [26][27][28] and only a tiny proportion of these are infected the ability of memory CD8 T 34 cells of a single specificity to locate and eliminate all liver stages within 48 hours is remarkable. Yet, 35 specific mechanisms by which T cells achieve such an efficiency remain poorly defined. 36 Recent studies utilizing fluorescently labeled sporozoites and activated Plasmodium-specific CD8 37 T cells and intravital microscopy revealed clustering of CD8 T cells near the parasite in the mouse 38 livers whereby multiple T cells were located in close proximity (≤ 40 µm) of some liver stages [23,[29][30][31][32][33][34][35][36][37][38][39] 31]. Interestingly, we observed that clustering of T cells near the parasite results in a higher chances 40 of parasite's death suggesting that clusters may increase the efficiency at which T cells eliminate 41 the infection. Recent in vivo studies also fou...