Daniel Coombs: Professor at Department of Mathematics, UBC Faculty of Science

Professor

Faculty of Science

Research Classification

Mathematics and statistics

Research Interests

Cell Signaling and Infectious and Immune Diseases

Cell biophysics

Disease models

Epidemiology

Immune cell signalling

Mathematical biology

Mathematics

Relevant Degree Programs

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Affiliations to Research Centres, Institutes & Clusters

Biomedical Imaging and Artificial Intelligence

Institute of Applied Mathematics

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Research Methodology

Super-resolution microscopy

Mathematical modelling of cell processes

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Graduate Student Supervision

Doctoral Student Supervision (Jan 2008 - May 2021)

Interpretation of fluorescence microscopy experiments on cell surface receptor dynamics with stochastic and deterministic mathematical models (2020)

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.

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On the Dynamics of HIV and Malaria Infection - Insights from Mathematical Models (2015)

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.

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Mathematical modeling in cellular immunology: T cell activation and parameter estimation (2009)

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.

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Master's Student Supervision (2010 - 2020)

Mathematical Modelling of Partially Absorbing Boundaries in Biological Systems (2016)

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.

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Spatial Stochastic Models of HSV-2 Lesion Dynamics and Their Link with HIV-1 Acquisition (2016)

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.

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Mathematical Models for Immunodeficiency Virus, Post-Treatment and Memory Activation (2012)

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.

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Two Mathematical Approaches to a Study of T Cell Motion and Activation in the Lymph Node (2012)

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.

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Deciphering multi-state mobility within single particle trajectories of proteins on the plasma membrane (2010)

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.

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