Publications

Mechanisms regulating cell movement are not fully understood. One feature of cell movement that determines how far cells displace from an initial position is persistence, the ability to perform movements in a direction similar to the previous movement direction. Persistence is thus determined by turning angles (TA) between two sequential displacements and can be characterized by an average TA or persistence time. Recent studies documenting T cell movement in zebrafish found that a cell's average speed and average TA are negatively correlated, suggesting a fundamental cell-intrinsic program whereby cells with a lower TA (and larger persistence time) are intrinsically faster (or faster cells turn less). In this paper we confirm the existence of the correlation between turning and speed for six different datasets on 3D movement of CD8 T cells in murine lymph nodes or liver. Interestingly, the negative correlation between TA and speed was observed in experiments in which liver-localized CD8 T cells rapidly displace due to floating with the blood flow, suggesting that other mechanisms besides cell-intrinsic program may be at play. By simulating correlated random walks using two different frameworks (one based on the von Mises-Fisher (vMF) distribution and another based on the Ornstein-Uhlenbeck (OU) process) we show that the negative correlation between speed and turning naturally arises when cell trajectories are sub-sampled, i.e. when the frequency of sampling is lower than frequency at which cells typically make movements. This effect is strongest when the sampling frequency is of the order of magnitude of the inverse of persistence time of cells and when cells vary in persistence time. The effect arises in part due to the sensitivity of estimated cell speeds to the frequency of imaging whereby less frequent imaging results in slower speeds. Interestingly, by using estimated persistence times for cells in two of our datasets and simulating cell movements using the OU process, we could partially reproduce the experimentally observed correlation between TA and speed without a cell-intrinsic program linking the two processes. Our results thus suggest that sub-sampling may contribute to (and perhaps fully explains) the observed correlation between speed and turning at least for some cell trajectory data and emphasize the role of sampling frequency in the inference of critical cellular parameters of cell motility such as speeds.

Substantial guidance is available on undergraduate quantitative training for biologists, including reports focused on biomedical science. Far less attention has been paid to the graduate curriculum and the particular challenges of the diversity of specialization within the life sciences. We propose an innovative approach to quantitative education that goes beyond recommendations of a course or set of courses or activities, derived from analysis of the expectations for students in particular programs. Due to the plethora of quantitative methods, it is infeasible to expect that biomedical PhD students can be exposed to more than a minority of the quantitative concepts and techniques employed in modern biology. We collected key recent papers suggested by the faculty in biomedical science programs, chosen to include important scientific contributions that the faculty consider appropriate for all students in the program to be able to read with confidence. The quantitative concepts and methods inherent in these papers were then analyzed and categorized to provide a rational basis for prioritization of those concepts to be emphasized in the education program. This novel approach to prioritization of quantitative skills and concepts provides an effective method to drive curricular focus based upon program-specific faculty input for science programs of all types. The results of our particular application to biomedical science training highlight the disconnect between typical undergraduate quantitative education for life science students, focused on continuous mathematics, and the concepts and skills in graphics, statistics, and discrete mathematics that arise from priorities established by biomedical science faculty. There was little reference in the key recent papers chosen by faculty to classic mathematical areas such as calculus which make up a large component of the formal undergraduate mathematics training of graduate students in biomedical areas.

Cytotoxic T lymphocytes (CTLs) are important in controlling some viral infections, and therapies involving the transfer of large numbers of cancer-specific CTLs have been successfully used to treat several types of cancers in humans. While the molecular mechanisms of how CTLs kill their targets are relatively well understood, we still lack a solid quantitative understanding of the kinetics and efficiency by which CTLs kill their targets in vivo. Collagen-fibrin-gel-based assays provide a tissue-like environment for the migration of CTLs, making them an attractive system to study T cell cytotoxicity in in vivo-like conditions. Budhu.et al. systematically varied the number of peptide (SIINFEKL)-pulsed B16 melanoma cells and SIINFEKL-specific CTLs (OT-1) and measured the remaining targets at different times after target and CTL co-inoculation into collagen-fibrin gels. The authors proposed that their data were consistent with a simple model in which tumors grow exponentially and are killed by CTLs at a per capita rate proportional to the CTL density in the gel. By fitting several alternative mathematical models to these data, we found that this simple "exponential-growth-mass-action-killing" model did not precisely describe the data. However, determining the best-fit model proved difficult because the best-performing model was dependent on the specific dataset chosen for the analysis. When considering all data that include biologically realistic CTL concentrations (E≤107cell/mL), the model in which tumors grow exponentially and CTLs suppress tumor's growth non-lytically and kill tumors according to the mass-action law (SiGMA model) fit the data with the best quality. A novel power analysis suggested that longer experiments (∼3-4 days) with four measurements of B16 tumor cell concentrations for a range of CTL concentrations would best allow discriminating between alternative models. Taken together, our results suggested that the interactions between tumors and CTLs in collagen-fibrin gels are more complex than a simple exponential-growth-mass-action killing model and provide support for the hypothesis that CTLs' impact on tumors may go beyond direct cytotoxicity.

Cytotoxic T lymphocytes (CTLs) are important in controlling some viral infections, and therapies involving transfer of large numbers of cancer-specific CTLs have been successfully used to treat several types of cancers in humans. While molecular mechanisms of how CTLs kill their targets are relatively well understood we still lack solid quantitative understanding of the kinetics and efficiency at which CTLs kill their targets in different conditions. Collagen-fibrin gel-based assays provide a tissue-like environment for the migration of CTLs, making them an attractive system to study the cytotoxicity in vitro. Budhu et al. [1] systematically varied the number of peptide (SIINFEKL)- pulsed B16 melanoma cells and SIINFEKL-specific CTLs (OT-1) and measured remaining targets at different times after target and CTL co-inoculation into collagen-fibrin gels. The authors proposed that their data were consistent with a simple model in which tumors grow exponentially and are killed by CTLs at a per capita rate proportional to the CTL density in the gel. By fitting several alternative mathematical models to these data we found that this simple "exponential-growth-mass-action-killing" model does not precisely fit the data. However, determining the best fit model proved difficult because the best performing model was dependent on the specific dataset chosen for the analysis. When considering all data that include biologically realistic CTL concentrations ( E ≤ 10 7 cell/ml) the model in which tumors grow exponentially and CTLs suppress tumor's growth non-lytically and kill tumors according to the mass-action law (SiGMA model) fitted the data with best quality. Results of power analysis suggested that longer experiments (∼ 3 - 4 days) with 4 measurements of B16 tumor cell concentrations for a range of CTL concentrations would best allow to discriminate between alternative models. Taken together, our results suggest that interactions between tumors and CTLs in collagen-fibrin gels are more complex than a simple exponential-growth- mass-action killing model and provide support for the hypothesis that CTLs impact on tumors may go beyond direct cytotoxicity.

One of the goals of vaccination is to induce long-term immunity against the infection and/or disease. However, evaluating the duration of protection following vaccination often requires long-term follow-ups that can conflict with the desire to rapidly publish results. Arunachalam et al. JCI 2023 followed individuals receiving third or fourth dose of mRNA COVID19 vaccines for up to 6 months and in finding that the levels of SARS-CoV2-specific antibodies (Abs) declined with similar rates for the two groups came to the conclusion that additional boosting is unnecessary to prolong immunity to SARS-CoV-2. However, this may be premature conclusion to make. Accordingly, we demonstrate that measuring Ab levels at 3 time points and only for a short (up to 6 month) duration does not allow to accurately and rigorously evaluate the long-term half-life of vaccine-induced Abs. By using the data from a cohort of blood donors followed for several years, we show that after re-vaccination with vaccinia virus (VV), VV-specific Abs decay bi-phasically and even the late decay rate exceeds the true slow loss rate of humoral memory observed years prior to the boosting. We argue that mathematical modeling should be used to better optimize sampling schedules to provide more reliable advice about the duration of humoral immunity after repeated vaccinations.

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.

Mathematical modeling provides a rigorous way to quantify immunological processes and discriminate between alternative mechanisms driving specific biological phenomena. It is typical that mathematical models of immunological phenomena are developed by modelers 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 CD8 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 models, 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 [Formula: see text] parasites at 2, 4, and 8 h 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.

Viruses are major pathogens of agricultural crops. Viral infections often start after the virus enters the outer layer of a tissue, and many successful viruses, after local replication in the infected tissue, are able to spread systemically. Quantitative details of virus dynamics in plants, however, are poorly understood, in part, because of the lack of experimental methods which allow the accurate measurement of the degree of infection in individual plant tissues. Recently, a group of researchers followed the kinetics of infection of individual cells in leaves of Nicotiana tabacum plants using Tobacco etch virus (TEV) expressing either Venus or blue fluorescent protein (BFP). Assuming that viral spread occurs from lower to upper leaves, the authors fitted a simple mathematical model to the frequency of cellular infection by the two viral variants found using flow cytometry. While the original model could accurately describe the kinetics of viral spread locally and systemically, we found that many alternative versions of the model, for example, if viral spread starts at upper leaves and progresses to lower leaves or when virus dissemination is stopped due to an immune response, fit the data with reasonable quality, and yet with different parameter estimates. These results strongly suggest that experimental measurements of the virus infection in individual leaves may not be sufficient to identify the pathways of viral dissemination between different leaves and reasons for viral control. We propose experiments that may allow discrimination between the alternatives. By analyzing the kinetics of coinfection of individual cells by Venus and BFP strains of TEV we found a strong deviation from the random infection model, suggesting cooperation between the two strains when infecting plant cells. Importantly, we showed that many mathematical models on the kinetics of coinfection of cells with two strains could not adequately describe the data, and the best fit model needed to assume (i) different susceptibility of uninfected cells to infection by two viruses locally in the leaf vs. systemically from other leaves, and (ii) decrease in the infection rate depending on the fraction of uninfected cells which could be due to a systemic immune response. Our results thus demonstrate the difficulty in reaching definite conclusions from extensive and yet limited experimental data and provide evidence of potential cooperation between different viral variants infecting individual cells in plants.

Liver-resident CD8+ T cells can play critical roles in the control of pathogens, including Plasmodium and hepatitis B virus. Paradoxically, it has also been proposed that the liver may act as the main place for the elimination of CD8+ T cells at the resolution of immune responses. We hypothesized that different adhesion processes may drive residence versus elimination of T cells in the liver. Specifically, we investigated whether the expression of asialo-glycoproteins (ASGPs) drives the localization and elimination of effector CD8+ T cells in the liver, while interactions with platelets facilitate liver residence and protective function. Using murine CD8+ T cells activated in vitro, or in vivo by immunization with Plasmodium berghei sporozoites, we found that, unexpectedly, inhibition of ASGP receptors did not inhibit the accumulation of effector cells in the liver, but instead prevented these cells from accumulating in the spleen. In addition, enforced expression of ASGP on effector CD8+ T cells using St3GalI-deficient cells lead to their loss from the spleen. We also found, using different mouse models of thrombocytopenia, that severe reduction in platelet concentration in circulation did not strongly influence the residence and protective function of CD8+ T cells in the liver. These data suggest that platelets play a marginal role in CD8+ T cell function in the liver. Furthermore, ASGP-expressing effector CD8+ T cells accumulate in the spleen, not the liver, prior to their destruction.

Malaria is a disease caused by Plasmodium parasites, resulting in over 200 million infections and 400,000 deaths every year. A critical step of malaria infection is when sporozoites, injected by mosquitoes, travel to the liver and form liver stages. Malaria vaccine candidates which induce large numbers of malaria-specific CD8 T cells in mice are able to eliminate all liver stages, preventing fulminant malaria. However, how CD8 T cells find all parasites in 48 h of the liver stage lifespan is not well understood. Using intravital microscopy of murine livers, we generated unique data on T cell search for malaria liver stages within a few hours after infection. To detect attraction of T cells to an infection site, we used the von Mises-Fisher distribution in 3D, similar to the 2D von Mises distribution previously used in ecology. Our results suggest that the vast majority (70-95%) of malaria-specific and non-specific liver-localized CD8 T cells did not display attraction towards the infection site, suggesting that the search for malaria liver stages occurs randomly. However, a small fraction (15-20%) displayed weak but detectable attraction towards parasites which already had been surrounded by several T cells. We found that speeds and turning angles correlated with attraction, suggesting that understanding mechanisms that determine the speed of T cell movement in the liver may improve the efficacy of future T cell-based vaccines. Stochastic simulations suggest that a small movement bias towards the parasite dramatically reduces the number of CD8 T cells needed to eliminate all malaria liver stages, but to detect such attraction by individual cells requires data from long imaging experiments which are not currently feasible. Importantly, as far as we know this is the first demonstration of how activated/memory CD8 T cells might search for the pathogen in nonlymphoid tissues a few hours after infection. We have also established a framework for how attraction of individual T cells towards a location in 3D can be rigorously evaluated.