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Graduate Student Supervision
Doctoral Student Supervision (Jan 2008 - May 2021)
Fluorescence microscopy has provided cellular biologists with quantifiable data, that can be paired with mathematical models to discover the mechanics of the imaged processes. We developed mathematical models to analyze data from two fluorescence techniques: direct Stochastic Optical Reconstruction Microscopy (dSTORM) and fluorescence recovery after photobleaching (FRAP). dSTORM is a super-resolution technique that uses photo-switchable fluorophores to achieve nanometer resolution images, allowing us to visualize the organization of proteins at nano-scales. However, dSTORM images can suffer from recording a single photo-switchable fluorophore multiple times, possibly creating artificial features. This is specially relevant in the analysis of membrane B-cell receptors clustering, where spatial clustering might relate to immune activation. I developed a protocol to estimate the number of unique fluorophores present in the experiment by coupling their temporal (with a Markov-chain model) and spatial (with a Gaussian mixture model) dynamics within a maximum likelihood framework. Previous studies have used the temporal information, but they have not coupled it with the spatial information (both localization and localization estimation error). I tested my protocol on simulated data, well-characterized DNA origami data and B-cell receptor data with positive results. My model is general enough to apply to other biological systems besides B-cell data and will enhance a microscopy technique that is widely used in biological applications.FRAP can be used to quantify the mobility of membrane proteins. We used it on live Drosophila organisms to study the outside-in pathway in cell adhesion to the extracellular matrix (ECM). We developed an ODE model to describe the recycling of the membrane protein, integrin, in charge of the adhesions. We found that both integrin and ECM ligands stabilize outside-in signalling and that relevant chemical treatments do not balance mutant integrin activation but stabilize the adhesions in control organisms. We also analyzed inside-out activation with a similar ODE model and by labeling the cytosolic protein talin. We found that talin is sensitive to increases and decreases in applied force. Disruptions of the intracellular force negatively affected adhesion stability. Increasing the force resulted in a faster assembly of new adhesions, whereas decreased forces increased the talin turnover.
We develop and apply mathematical models to obtain insights into the dynamics of HIV and malaria infection. We consider three case studies.1. The duration of the time between exposure and detectability of HIV infection is difficult to estimate because precise dates of exposure are rarely known. Therefore, the reliability of clinical HIV testing during the first few weeks of infections is unknown, creating anxiety among HIV-exposed individuals and their physicians. We address this knowledge gap by fitting stochastic models of early HIV infection to detailed viral load time-courses, taken shortly after exposure, from 78 plasma donors. Since every plasma donor in our data eventually becomes infected, we condition our model to reflect this bias before fitting to the data. Our model prediction for the mean eclipse period is 8-10 days. We further quantify the reliability of a negative test t days after potential exposure to inform physicians about the value of initial and follow-up testing.2. The recently launched Get Checked Online (GCO) program aims at increasing the HIV testing rate in the Vancouver men who have sex with men population by facilitating test taking and result delivery. We develop mathematical models and extract parameter values from surveys and interviews to quantify GCO's population-level impact. Our models predict that the epidemic is growing overall, that its severeness is increased by the presence of a high-risk group and that, even at modest effectiveness, GCO might avert 34-66 new infections in the next five years.3. Metarhizium anisopliae is a naturally occurring fungal pathogen of mosquitoes that has been engineered to act against malaria by effectively blocking onward transmission from the mosquito vector. We develop and analyse two mathematical models to examine the efficacy of this fungal pathogen. We find that, in many plausible scenarios, the best effects are achieved with a reduced or minimal pathogen virulence, even if the likelihood of resistance to the fungus is negligible. The results depend on the interplay between two main effects: the ability of the fungus to reduce the mosquito population, and the ability of fungus-infected mosquitoes to compete for resources with non-fungus-infected mosquitoes.
A critical step in mounting an immune response is antigen recognition by T cells. This step proceeds by productive interactions between T cell receptors (TCR) on the surface of T cells and foreign antigen, in the form of peptide-major-histocompatibility-complexes (pMHC), on the surface of antigen-presenting-cells (APC). Antigen recognition is exceedingly difficult to understand because the vast majority of pMHC on APCs are derived from self-proteins. Nevertheless, T cells have been shown to be exquisitely sensitive, responding to as few as 10 antigenic pMHC in an ocean of tens of thousands of self pMHC. In addition, T cells are extremely specific and respond only to a small subset of pMHC by virtue of their specific TCR.To explain the sensitivity of T cells to pMHC it has been proposed that a single pMHC may serially bind multiple TCRs. Integrating present knowledge on the spatial-temporal dynamics of TCR/pMHC in the T cell-APC contact interface, we have constructed mathematical models to investigate the degree of TCR serial engagements by pMHC. In addition to reactions within clusters, the models capture the formation and mobility of TCR clusters. We find that a single pMHC serially binds a substantial number of TCRs in a TCR cluster only if the TCR/pMHC bond is stabilized by coreceptors and/or pMHC dimerization. In a separate study we propose that serial engagements can explain T cell specificity. Using Monte Carlo simulations, we show that the stochastic nature of TCR/pMHC interactions means that multiple binding events are needed for accurate detection of foreign pMHC.Critical to our studies are estimates of TCR/pMHC reaction rates and mobilities. In the second half of the thesis, we show that Fluorescence Recovery After Photobleaching (FRAP) experiments can reveal effective diffusion coefficients. We then show, using asymptotic analysis and model fitting, that FRAP experiments can be used to estimate reaction rates between cell surface proteins, like TCR/pMHC. Lastly, we use FRAP experiments to investigate how the actin cytoskeleton modulates TCR mobility and report effective reaction rates between TCR and the cytoskeleton.
Master's Student Supervision (2010 - 2020)
This project presents a mathematical framework for identifying partially permeable biological boundaries, and estimating the rate of absorption of diffusing objects at such a boundary based on limited experimental data. We used partial differential equations (PDEs) to derive probability distribution functions for finding a particle performing Brownian motion in a region. These distribution functions can be fit to data to infer the existence of a boundary. We also used the probability distribution functions together with maximum likelihood estimation to estimate the rate of absorption of objects at the boundaries, based on simulated data. Furthermore, we consider a switching boundary and provide a technique for approximating the boundary with a partially permeable boundary.
Patients with Herpes Simplex Virus-2 (HSV-2) infection face a significantly higher risk of contracting HIV-1. This marked increase is thought to be due not only to herpetic lesions serving as an entry point for the HIV-1 virus, but also to the increase in CD4+ T cells in the human genital mucosa during HSV-2 lesional events. By creating a stochastic, spatial, mathematical model describing the behaviour of the HSV-2 infection and immune response in the genital mucosa, I first capture the dynamics that occur during the development of an HSV-2 lesion. I then use this model to quantify the risk of acquiring HIV-1 in HSV-2 positive patients upon sexual exposure, and determine whether antivirals meant to control HSV-2 can decrease HIV-1 infectivity. While theory predicts that HSV-2 treatment should lower HIV-1 infection probability, my results show that this may not be the case unless a critical dosage of HSV-2 treatment is given to the patient. These results help to explain the conflicting data on HIV-1 infection probability in HSV-2 patients and allow for further insight into the type of treatment HSV-2 positive patients should receive to prevent HIV-1 infection.
Nowdays, HIV infection can be controlled by anti-retroviral drug therapy (ART). However, a persistent viral reservoir in treated patients prevents the eradication of HIV infection. H-iART is an innovator treatment that consists of regular ART and the drugs Maraviroc and Darunavir, and H-iART was enforced with Auranofin. The drug Maraviroc (MRV) was proved to be a good CCR5 inhibitor, which is a HIV correceptor. The drug Auranofin has been shown to accelerate the activation rate of latent cells and also alters the kinetics of viral rebound when drug treatment is interrupted. Recent studies on monkeys infected with SIV have shown a complete suppression of the viral load during H-iART with Auranofin treatment and a persistent suppression of it in the absence of ART. Motivated by the results of the experiments I present deterministic and stochastic models of HIV after treatment interruption. For H-iART treatment, the ODE models were used as a start point to create three different continuous time multi-type branching process. From equations for the probability generating function we use analytic solutions, numerical approximations, or numerical simulations to extract the probability of observable viral blips. We compare our results with the data of two rhesus macaques. We find that more than one latent cell needs to activate in order to observe the data blips, and that the net reproductive number of virus must be very close to one. Since this is unlikely, these results suggest that the viral dynamics must be more complex than our model allows for. For the ART+Auranofin treatment, I will present an ODE model of HIV population dynamics including drug treatment and the immune response to model the viral rebound at treatment interruption.
T cells are part of the immune system and as such play a very important role in keeping us healthy. One crucial step in the complex process which is the immune response to pathogens is T cell activation. The general goal of my thesis is to mathematically describe the migration patterns followed by T cells while waiting to be activated in the lymph node. Insight into these migration patterns could lead to better knowledge of the strategies T cells take to make activation such an efficient process.In order to fulfill my goal I have used two different approaches: one mainly computational and the other mainly theoretical.On the computational side, I analyzed three-dimensional microscopic movies of mice lymph nodes inside of which labelled T cells are moving. From the movies I extracted the trajectories of the cells. I studied movies from two experimental frameworks, exogenous and endogenous. On the former, more frequent type of experiment, T cells are labelled outside the mouse and then transferred in. The endogenous experiments, on the contrary, involve genetically modified mice whose T cells are born labelled. I concluded that there is a significant difference in labelled T cell motion between the two experimental frameworks. This suggests that previous results from exogenous experiments should be treated with caution due topossible errors introduced by the methods specific to that type of experiment.On the theoretical side I studied the time it takes for a model T cell to be activated under different scenarios regarding the characteristics of the lymph node as well as of the other cells in it. Since T cells become activated after establishing contact with a specific cell among many similar ones which also move within the lymph node, what I effectively computed was the mean first passage time for a model T cell to reach a defined target within the model lymph node.
Single particle tracking is a powerful technique often used in the study of dynamic mechanisms on the cell surface such as binding, confinement and trafficking. Experimental trajectories can be used to detect changes in the lateral mobility of individual molecules over time and space. Therefore, a potential problem in the analysis of single particle trajectories is to account for transitions between modes of mobility. Here we present two coupled statistical methods which characterize particle mobility that is temporally and spatially heterogeneous. The first method detects periods of drift diffusion or reduced mobility within single trajectories due to transient associations with other biomolecules. The second locates spatial domains which have higher or lower concentrations of these associating molecules. The trajectory is modeled as the outcome of a two-state Hidden Markov model parameterized by the diffusion coefficients and drift velocities of each state and the rates of transitions between them (which may change in space). Transitions between states arise from association and disassociation with a binding partner, either membrane-associated or cytosolic. These associations lead to either reduced Brownian diffusion or drift diffusion. An adapted Markov chain Monte Carlo algorithm was used to optimize parameters and simultaneously select the most favorable model of lateral mobility (transient reduced mobility or transient drift diffusion) and to locate spatial domains. Analysis of simulated particle tracks with a wide range of parameters successfully distinguished between the two models, gave accurate estimates for parameters and accurately located spatial domains.
- Stochastic Dynamics of the Latently Infected Cell Reservoir During HIV Infection (2019)
Bulletin of Mathematical Biology, 81 (1), 131-154
- A novel Bayesian approach to predicting reductions in HIV incidence following increased testing interventions among gay, bisexual and other men who have sex with men in Vancouver, Canada. (2018)
Journal of the Royal Society, Interface,
- Distribution of take-home opioid antagonist kits during a synthetic opioid epidemic in British Columbia, Canada: a modelling study (2018)
The Lancet Public Health, 3 (5), e218-e225
- Effects of spatiotemporal HSV-2 lesion dynamics and antiviral treatment on the risk of HIV-1 acquisition (2018)
PLoS Computational Biology, 14 (4)
- Limitations of Qdot labelling compared to directly-conjugated probes for single particle tracking of B cell receptor mobility. (2017)
- On the duration of the period between exposure to HIV and detectable infection. (2017)
- Sustained Reduction in Sexual Behavior that May Pose a Risk of HIV Transmission Following Diagnosis During Early HIV Infection Among Gay Men in Vancouver, British Columbia. (2017)
AIDS and behavior,
- Applied stretch initiates directional invasion through the action of Rap1 GTPase as a tension sensor (2016)
Journal of Cell Science, 130 (1), 152--163
- Conditions for eradicating hepatitis C in people who inject drugs: A fibrosis aware model of hepatitis C virus transmission (2016)
Journal of Theoretical Biology, 395, 31-39
- In vivo regulation of integrin turnover by outside-in activation (2016)
Journal of Cell Science, 129 (15), 2912-2924
- Conditional mean first passage times to small traps in a 3-D domain with a sticky boundary: Applications to T cell searching behavior in lymph nodes (2015)
Multiscale Modeling and Simulation, 13 (4), 1224-1258
- Design Parameters for Granzyme-Mediated Cytotoxic Lymphocyte Target-Cell Killing and Specificity (2015)
Biophysical Journal, 109 (3), 477-488
- Erratum: Correction: The Impact of Implementing a Test, Treat and Retain HIV Prevention Strategy in Atlanta among Black Men Who Have Sex with Men with a History of Incarceration: A Mathematical Model (PLoS ONE (2015) 10:5 (e0128734) doi:10.1371/journal.po (2015)
PLoS ONE, 10 (5)
- Erratum: Toll-like receptor ligands sensitize B-cell receptor signalling by reducing actin-dependent spatial confinement of the receptor (2015)
Nature Communications, 6
- In vivo quantitative analysis of Talin turnover in response to force (2015)
Molecular Biology of the Cell, 26 (22), 4149-4162
- Probability of a false-negative HIV antibody test result during the window period: a tool for pre- and post-test counselling (2015)
International Journal of STD and AIDS, 26 (4), 215-224
- Sir-network model and its application to dengue fever (2015)
SIAM Journal on Applied Mathematics, 75 (6), 2581-2609
- The impact of implementing a test, treat and retain HIV prevention strategy in Atlanta among black men who have sex with men with a history of incarceration:A mathematical model (2015)
PLoS ONE, 10 (4)
- Toll-like receptor ligands sensitize B-cell receptor signalling by reducing actin-dependent spatial confinement of the receptor (2015)
Nature Communications, 6
- Assessing the optimal virulence of malaria-targeting mosquito pathogens: A mathematical study of engineered Metarhizium anisopliae (2014)
Malaria Journal, 13 (1)
- Asymptotic Analysis of First Passage Time Problems Inspired by Ecology (2014)
Bulletin of Mathematical Biology, 77 (1), 83-125
- Regulatory vs. helper CD4+ T-cell ratios and the progression of HIV/AIDS disease (2014)
- Stochastic analysis of pre-and postexposure prophylaxis against hiv infection (2013)
SIAM Journal on Applied Mathematics, 73 (2), 904-928
- Mechanical force regulates integrin turnover in Drosophila in vivo (2012)
Nature Cell Biology, 14 (9), 935-943
- Mechanical modulation of receptor-ligand interactions at cell-cell interfaces (2012)
Biophysical Journal, 102 (6), 1265-1273
- A biophysical model of cell adhesion mediated by immunoadhesin drugs and antibodies (2011)
PLoS ONE, 6 (5)
- A stochastic model of latently infected cell reactivation and viral blip generation in treated HIV patients (2011)
PLoS Computational Biology, 7 (4)
- Antigen potency and maximal efficacy reveal a mechanism of efficient T cell activation (2011)
Science Signaling, 4 (176)
- Modeling effect of a γ-secretase inhibitor on amyloid-β dynamics reveals significant role of an amyloid clearance mechanism (2011)
Bulletin of Mathematical Biology, 73 (1), 230-247
- Modeling Membrane Domains (2011)
Cellular Domains, , 71-84
- Reverse engineering an amyloid aggregation pathway with dimensional analysis and scaling (2011)
Physical Biology, 8 (6)
- Vaccination against 2009 pandemic H1N1 in a population dynamical model of Vancouver, Canada: Timing is everything (2011)
BMC Public Health, 11
- Dependence of T Cell Antigen Recognition on T Cell Receptor-Peptide MHC Confinement Time (2010)
Immunity, 32 (2), 163-174
- Dynamic regulation of CD45 lateral mobility by the spectrin-ankyrin cytoskeleton of T Cells (2010)
Journal of Biological Chemistry, 285 (15), 11392-11401
- Microelastohydrodynamics of swimming organisms near solid boundaries in complex fluids (2010)
Quarterly Journal of Mechanics and Applied Mathematics, 63 (3), 267-294
- The space and time frames of T cell activation at the immunological synapse (2010)
FEBS Letters, 584 (24), 4851-4857
- A hidden Markov model for single particle tracks quantifies dynamic interactions between LFA-1 and the actin cytoskeleton (2009)
PLoS Computational Biology, 5 (11)
- A role for rebinding in rapid and reliable T cell responses to antigen (2009)
PLoS Computational Biology, 5 (11)
- Analysis of membrane-localized binding kinetics with FRAP (2008)
European Biophysics Journal, 37 (5), 627-638
- Analysis of serial engagement and peptide-MHC transport in T cell receptor microclusters (2008)
Biophysical Journal, 94 (9), 3447-3460
- Effects of intracellular calcium and actin cytoskeleton on TCR mobility measured by fluorescence recovery (2008)
PLoS ONE, 3 (12)
- Improving parameter estimation for cell surface FRAP data. (2008)
Journal of biochemical and biophysical methods, 70 (6), 1224-1231
- Kinetic proofreading model (2008)
Advances in Experimental Medicine and Biology, 640, 82-94
- Evaluating the importance of within- and between-host selection pressures on the evolution of chronic pathogens (2007)
Theoretical Population Biology, 72 (4), 576-591
- Modeling within-host evolution of HIV: Mutation, competition and strain replacement (2007)
Bulletin of Mathematical Biology, 69 (7), 2361-2385
- Quantification and modeling of tripartite CD2-, CD58FC chimera (Alefacept)-, and CD16-mediated cell adhesion (2007)
Journal of Biological Chemistry, 282 (48), 34748-34757
- A theoretical and experimental study of competition between solution and surface receptors for ligand in a biacore flow cell (2006)
Bulletin of Mathematical Biology, 68 (5), 1125-1150
- Analysis of peptide/MHC-induced TCR downregulation: Deciphering the triggering kinetics (2006)
Cell Biochemistry and Biophysics, 46 (2), 101-111
- Evolution of virulence: Interdependence, constraints, and selection using nested models (2006)
Theoretical Population Biology, 69 (2), 145-153
- T cell activation: Kinetic proofreading, serial engagement and cell adhesion (2005)
Journal of Computational and Applied Mathematics, 184 (1), 121-139
- T cell receptor binding kinetics required for T cell activation depend on the density of cognate ligand on the antigen-presenting cell (2005)
Proceedings of the National Academy of Sciences of the United States of America, 102 (13), 4824-4829
- An age-structured model of hiv infection that allows for variations in the production rate of viral particles and the death rate of productively infected cells. (2004)
Mathematical biosciences and engineering : MBE,
- Effects of the geometry of the immunological synapse on the delivery of effector molecules (2004)
Biophysical Journal, 87 (4), 2215-2220
- Equilibrium thermodynamics of cell-cell adhesion mediated by multiple ligand-receptor pairs. (2004)
Biophysical journal, 86 (3), 1408-1423
- Optimizing within-host viral fitness: Infected cell lifespan and virion production rate (2004)
Journal of Theoretical Biology, 229 (2), 281-288
- Optimal Viral Production (2003)
Bulletin of Mathematical Biology, 65 (6), 1003-1023
- Activated TCRs remain marked for internalization after dissociation from pMHC (2002)
Nature Immunology, 3 (10), 926-931
- Erratum: Activated TCRs remain marked for internalization after dissociation from pMHC (Nature Immunology (2002) vol. 3 (926-931)) (2002)
Nature Immunology, 3 (11)
- Periodic Chirality Transformations Propagating On Bacterial Flagella (2002)
Physical Review Letters, 89 (11)
- Calculations Show Substantial Serial Engagement of T Cell Receptors (2001)
Biophysical Journal, 80 (2), 606--612
- The influence of transport on the kinetics of binding to surface receptors: Application to cells and BIAcore (1999)
Journal of Molecular Recognition, 12 (5), 293-299