Inferring Stochastic Rates from Heterogeneous Snapshots of Particle Positions.

Many imaging techniques for biological systems-like fixation of cells coupled with fluorescence microscopy-provide sharp spatial resolution in reporting locations of individuals at a single moment in time but also destroy the dynamics they intend to capture. These snapshot observations contain no information about individual trajectories, but still encode information about movement and demographic dynamics, especially when combined with a well-motivated biophysical model. The relationship between spatially evolving populations and single-moment representations of their collective locations is well-established with partial differential equations (PDEs) and their inverse problems. However, experimental data is commonly a set of locations whose number is insufficient to approximate a continuous-in-space PDE solution. Here, motivated by popular subcellular imaging data of gene expression, we embrace the stochastic nature of the data and investigate the mathematical foundations of parametrically inferring demographic rates from snapshots of particles undergoing birth, diffusion, and death in a nuclear or cellular domain. Toward inference, we rigorously derive a connection between individual particle paths and their presentation as a Poisson spatial process. Using this framework, we investigate the properties of the resulting inverse problem and study factors that affect quality of inference. One pervasive feature of this experimental regime is the presence of cell-to-cell heterogeneity. Rather than being a hindrance, we show that cell-to-cell geometric heterogeneity can increase the quality of inference on dynamics for certain parameter regimes. Altogether, the results serve as a basis for more detailed investigations of subcellular spatial patterns of RNA molecules and other stochastically evolving populations that can only be observed for single instants in their time evolution.

Minimal Mechanisms of Microtubule Length Regulation in Living Cells.

The microtubule cytoskeleton is responsible for sustained, long-range intracellular transport of mRNAs, proteins, and organelles in neurons. Neuronal microtubules must be stable enough to ensure reliable transport, but they also undergo dynamic instability, as their plus and minus ends continuously switch between growth and shrinking. This process allows for continuous rebuilding of the cytoskeleton and for flexibility in injury settings. Motivated by in vivo experimental data on microtubule behavior in Drosophila neurons, we propose a mathematical model of dendritic microtubule dynamics, with a focus on understanding microtubule length, velocity, and state-duration distributions. We find that limitations on microtubule growth phases are needed for realistic dynamics, but the type of limiting mechanism leads to qualitatively different responses to plausible experimental perturbations. We therefore propose and investigate two minimally-complex length-limiting factors: limitation due to resource (tubulin) constraints and limitation due to catastrophe of large-length microtubules. We combine simulations of a detailed stochastic model with steady-state analysis of a mean-field ordinary differential equations model to map out qualitatively distinct parameter regimes. This provides a basis for predicting changes in microtubule dynamics, tubulin allocation, and the turnover rate of tubulin within microtubules in different experimental environments.

Minimal Mechanisms of Microtubule Length Regulation in Living Cells.

The microtubule cytoskeleton is responsible for sustained, long-range intracellular transport of mRNAs, proteins, and organelles in neurons. Neuronal microtubules must be stable enough to ensure reliable transport, but they also undergo dynamic instability, as their plus and minus ends continuously switch between growth and shrinking. This process allows for continuous rebuilding of the cytoskeleton and for flexibility in injury settings. Motivated by in vivo experimental data on microtubule behavior in Drosophila neurons, we propose a mathematical model of dendritic microtubule dynamics, with a focus on understanding microtubule length, velocity, and state-duration distributions. We find that limitations on microtubule growth phases are needed for realistic dynamics, but the type of limiting mechanism leads to qualitatively different responses to plausible experimental perturbations. We therefore propose and investigate two minimally-complex length-limiting factors: limitation due to resource (tubulin) constraints and limitation due to catastrophe of large-length microtubules. We combine simulations of a detailed stochastic model with steady-state analysis of a mean-field ordinary differential equations model to map out qualitatively distinct parameter regimes. This provides a basis for predicting changes in microtubule dynamics, tubulin allocation, and the turnover rate of tubulin within microtubules in different experimental environments.

Inferring stochastic rates from heterogeneous snapshots of particle positions.

Many imaging techniques for biological systems - like fixation of cells coupled with fluorescence microscopy - provide sharp spatial resolution in reporting locations of individuals at a single moment in time but also destroy the dynamics they intend to capture. These snapshot observations contain no information about individual trajectories, but still encode information about movement and demographic dynamics, especially when combined with a well-motivated biophysical model. The relationship between spatially evolving populations and single-moment representations of their collective locations is well-established with partial differential equations (PDEs) and their inverse problems. However, experimental data is commonly a set of locations whose number is insufficient to approximate a continuous-in-space PDE solution. Here, motivated by popular subcellular imaging data of gene expression, we embrace the stochastic nature of the data and investigate the mathematical foundations of parametrically inferring demographic rates from snapshots of particles undergoing birth, diffusion, and death in a nuclear or cellular domain. Toward inference, we rigorously derive a connection between individual particle paths and their presentation as a Poisson spatial process. Using this framework, we investigate the properties of the resulting inverse problem and study factors that affect quality of inference. One pervasive feature of this experimental regime is the presence of cell-to-cell heterogeneity. Rather than being a hindrance, we show that cell-to-cell geometric heterogeneity can increase the quality of inference on dynamics for certain parameter regimes. Altogether, the results serve as a basis for more detailed investigations of subcellular spatial patterns of RNA molecules and other stochastically evolving populations that can only be observed for single instants in their time evolution.

Palmitate-mediated disruption of the endoplasmic reticulum decreases intracellular vesicle motility.

Essential cellular processes such as metabolism, protein synthesis, and autophagy require the intracellular transport of membrane-bound vesicles. The importance of the cytoskeleton and associated molecular motors for transport is well documented. Recent research has suggested that the endoplasmic reticulum (ER) may also play a role in vesicle transport through a tethering of vesicles to the ER. We use single-particle tracking fluorescence microscopy and a Bayesian change-point algorithm to characterize vesicle motility in response to the disruption of the ER, actin, and microtubules. This high-throughput change-point algorithm allows us to efficiently analyze thousands of trajectory segments. We find that palmitate-mediated disruption of the ER leads to a significant decrease in vesicle motility. A comparison with the disruption of actin and microtubules shows that disruption of the ER has a significant impact on vesicle motility, greater than the disruption of actin. Vesicle motility was dependent on cellular region, with greater motility in the cell periphery than the perinuclear region, possibly due to regional differences in actin and the ER. Overall, these results suggest that the ER is an important factor in vesicle transport.

Transport of lysosomes decreases in the perinuclear region: Insights from changepoint analysis.

Lysosomes are membrane-bound organelles that serve as the endpoint for endocytosis, phagocytosis, and autophagy, degrading the molecules, pathogens, and organelles localized within them. These cellular functions require intracellular transport. We use fluorescence microscopy to characterize the motion of lysosomes as a function of intracellular region, perinuclear or periphery, and lysosome diameter. Single-particle tracking data are complemented by changepoint identification and analysis of a mathematical model for state switching. We first classify lysosomal motion as motile or stationary. We then study how lysosome location and diameter affects the proportion of time spent in each state and quantify the speed during motile periods. We find that the proportion of time spent stationary is strongly region dependent, with significantly decreased motility in the perinuclear region. Increased lysosome diameter only slightly decreases speed. Overall, these results demonstrate the importance of decomposing particle trajectories into qualitatively different behaviors before conducting population-wide statistical analysis. Our results suggest that intracellular region is an important factor to consider in studies of intracellular transport.

A change point analysis protocol for comparing intracellular transport by different molecular motor combinations.

Intracellular transport by microtubule-based molecular motors is marked by qualitatively different behaviors. It is a long-standing and still-open challenge to accurately quantify the various individual-cargo behaviors and how they are affected by the presence or absence of particular motor families. In this work we introduce a protocol for analyzing change points in cargo trajectories that can be faithfully projected along the length of a (mostly) straight microtubule. Our protocol consists of automated identification of velocity change points, estimation of velocities during the behavior segments, and extrapolation to motor-specific velocity distributions. Using simulated data we show that our method compares favorably with existing methods. We then apply the technique to data sets in which quantum dots are transported by Kinesin-1, by Dynein-Dynactin-BicD2 (DDB), and by Kinesin-1/DDB pairs. In the end, we identify pausing behavior that is consistent with some tug-of-war model predictions, but also demonstrate that the simultaneous presence of antagonistic motors can lead to long processive runs that could contribute favorably to population-wide transport.

Leaf Economics of Early- and Late-Successional Plants.

AbstractThe leaf economics spectrum ranges from cheap, short-lived leaves to expensive, long-lived leaves. Species with low leaf mass per area (LMA) and short leaf life span tend to be fast growing and shade intolerant (early successional), whereas species with high LMA and long leaf life span tend to be slow growing and shade tolerant (late successional). However, we have limited understanding of how different leaf mass components (e.g., metabolically active photosynthetic components vs. structural toughness components) contribute to variation in LMA and other leaf economics spectrum traits. Here, we develop a model of plant community dynamics in which species differ in just two traits, photosynthetic and structural LMA components, and we identify optimal values of these traits for early- and late-successional species. Most of the predicted increase in LMA from early- to late-successional species was due to structural LMA. Photosynthetic LMA did not differ consistently between early- and late-successional species, but the photosynthetic LMA to structural LMA ratio declined from early- to late-successional species. Early-successional species had high rates of instantaneous return on leaf mass investment, whereas late-successional species had high lifetime return. Our results provide theoretical support for the primary role of structural (rather than photosynthetic) LMA variation in driving relationships among leaf economics spectrum traits.

Topological Data Analysis Approaches to Uncovering the Timing of Ring Structure Onset in Filamentous Networks.

In developmental biology as well as in other biological systems, emerging structure and organization can be captured using time-series data of protein locations. In analyzing this time-dependent data, it is a common challenge not only to determine whether topological features emerge, but also to identify the timing of their formation. For instance, in most cells, actin filaments interact with myosin motor proteins and organize into polymer networks and higher-order structures. Ring channels are examples of such structures that maintain constant diameters over time and play key roles in processes such as cell division, development, and wound healing. Given the limitations in studying interactions of actin with myosin in vivo, we generate time-series data of protein polymer interactions in cells using complex agent-based models. Since the data has a filamentous structure, we propose sampling along the actin filaments and analyzing the topological structure of the resulting point cloud at each time. Building on existing tools from persistent homology, we develop a topological data analysis (TDA) method that assesses effective ring generation in this dynamic data. This method connects topological features through time in a path that corresponds to emergence of organization in the data. In this work, we also propose methods for assessing whether the topological features of interest are significant and thus whether they contribute to the formation of an emerging hole (ring channel) in the simulated protein interactions. In particular, we use the MEDYAN simulation platform to show that this technique can distinguish between the actin cytoskeleton organization resulting from distinct motor protein binding parameters.

Renewal Reward Perspective on Linear Switching Diffusion Systems in Models of Intracellular Transport.

In many biological systems, the movement of individual agents is characterized having multiple qualitatively distinct behaviors that arise from a variety of biophysical states. For example, in cells the movement of vesicles, organelles, and other intracellular cargo is affected by their binding to and unbinding from cytoskeletal filaments such as microtubules through molecular motor proteins. A typical goal of theoretical or numerical analysis of models of such systems is to investigate effective transport properties and their dependence on model parameters. While the effective velocity of particles undergoing switching diffusion dynamics is often easily characterized in terms of the long-time fraction of time that particles spend in each state, the calculation of the effective diffusivity is more complicated because it cannot be expressed simply in terms of a statistical average of the particle transport state at one moment of time. However, it is common that these systems are regenerative, in the sense that they can be decomposed into independent cycles marked by returns to a base state. Using decompositions of this kind, we calculate effective transport properties by computing the moments of the dynamics within each cycle and then applying renewal reward theory. This method provides a useful alternative large-time analysis to direct homogenization for linear advection-reaction-diffusion partial differential equation models. Moreover, it applies to a general class of semi-Markov processes and certain stochastic differential equations that arise in models of intracellular transport. Applications of the proposed renewal reward framework are illustrated for several case studies such as mRNA transport in developing oocytes and processive cargo movement by teams of molecular motor proteins.

Antibody-Mediated Immobilization of Virions in Mucus.

Antibodies have been shown to hinder the movement of herpes simplex virus virions in cervicovaginal mucus, as well as other viruses in other mucus secretions. However, it has not been possible to directly observe the mechanisms underlying this phenomenon, so the nature of virion-antibody-mucin interactions remain poorly understood. In this work, we analyzed thousands of virion traces from single particle tracking experiments to explicate how antibodies must cooperate to immobilize virions for relatively long time periods. First, using a clustering analysis, we observed a clear separation between two classes of virion behavior: freely diffusing and immobilized. While the proportion of freely diffusing virions decreased with antibody concentration, the magnitude of their diffusivity did not, implying an all-or-nothing dichotomy in the pathwise effect of the antibodies. Proceeding under the assumption that all binding events are reversible, we used a novel switch-point detection method to conclude that there are very few, if any, state switches on the experimental timescale of 20 s. To understand this slow state switching, we analyzed a recently proposed continuous-time Markov chain model for binding kinetics and virion movement. Model analysis implied that virion immobilization requires cooperation by multiple antibodies that are simultaneously bound to the virion and mucin matrix and that there is an entanglement phenomenon that accelerates antibody-mucin binding when a virion is immobilized. In addition to developing a widely applicable framework for analyzing multistate particle behavior, this work substantially enhances our mechanistic understanding of how antibodies can reinforce a mucus barrier against passive invasive species.

Assessing the Impact of Electrostatic Drag on Processive Molecular Motor Transport.

The bidirectional movement of intracellular cargo is usually described as a tug-of-war among opposite-directed families of molecular motors. While tug-of-war models have enjoyed some success, recent evidence suggests underlying motor interactions are more complex than previously understood. For example, these tug-of-war models fail to predict the counterintuitive phenomenon that inhibiting one family of motors can decrease the functionality of opposite-directed transport. In this paper, we use a stochastic differential equations modeling framework to explore one proposed physical mechanism, called microtubule tethering, that could play a role in this "co-dependence" among antagonistic motors. This hypothesis includes the possibility of a trade-off: weakly bound trailing molecular motors can serve as tethers for cargoes and processing motors, thereby enhancing motor-cargo run lengths along microtubules; however, this introduces a cost of processing at a lower mean velocity. By computing the small- and large-time mean-squared displacement of our theoretical model and comparing our results to experimental observations of dynein and its "helper protein" dynactin, we find some supporting evidence for microtubule tethering interactions. We extrapolate these findings to predict how dynein-dynactin might interact with the opposite-directed kinesin motors and introduce a criterion for when the trade-off is beneficial in simple systems.

ZMapp Reinforces the Airway Mucosal Barrier Against Ebola Virus.

Filoviruses, including Ebola, have the potential to be transmitted via virus-laden droplets deposited onto mucus membranes. Protecting against such emerging pathogens will require understanding how they may transmit at mucosal surfaces and developing strategies to reinforce the airway mucus barrier. Here, we prepared Ebola pseudovirus (with Zaire strain glycoproteins) and used high-resolution multiple-particle tracking to track the motions of hundreds of individual pseudoviruses in fresh and undiluted human airway mucus isolated from extubated endotracheal tubes. We found that Ebola pseudovirus readily penetrates human airway mucus. Addition of ZMapp, a cocktail of Ebola-binding immunoglobulin G antibodies, effectively reduced mobility of Ebola pseudovirus in the same mucus secretions. Topical delivery of ZMapp to the mouse airways also facilitated rapid elimination of Ebola pseudovirus. Our work demonstrates that antibodies can immobilize virions in airway mucus and reduce access to the airway epithelium, highlighting topical delivery of pathogen-specific antibodies to the lungs as a potential prophylactic or therapeutic approach against emerging viruses or biowarfare agents.

Resource-driven encounters among consumers and implications for the spread of infectious disease.

Animals share a variety of common resources, which can be a major driver of conspecific encounter rates. In this work, we implement a spatially explicit mathematical model for resource visitation behaviour in order to examine how changes in resource availability can influence the rate of encounters among consumers. Using simulations and asymptotic analysis, we demonstrate that, under a reasonable set of assumptions, the relationship between resource availability and consumer conspecific encounters is not monotonic. We characterize how the maximum encounter rate and associated critical resource density depend on system parameters like consumer density and the maximum distance from which consumers can detect and respond to resources. The assumptions underlying our theoretical model and analysis are motivated by observations of large aggregations of black-backed jackals at carcasses generated by seasonal outbreaks of anthrax among herbivores in Etosha National Park, Namibia. As non-obligate scavengers, black-backed jackals use carcasses as a supplemental food resource when they are available. While jackals do not appear to acquire disease from ingesting anthrax carcasses, changes in their movement patterns in response to changes in carcass abundance do alter jackals' conspecific encounter rate in ways that may affect the transmission dynamics of other diseases, such as rabies. Our theoretical results provide a method to quantify and analyse the hypothesis that the outbreak of a fatal disease among herbivores can potentially facilitate outbreaks of an entirely different disease among jackals. By analysing carcass visitation data, we find support for our model's prediction that the number of conspecific encounters at resource sites decreases with additional increases in resource availability. Whether or not this site-dependent effect translates to an overall decrease in encounters depends, unexpectedly, on the relationship between the maximum distance of detection and the resource density.

The evolutionary trade-off between stem cell niche size, aging, and tumorigenesis.

Many epithelial tissues within multicellular organisms are continually replenished by small independent populations of stem cells largely responsible for maintaining tissue homeostasis. These continually dividing populations are subject to mutations that can lead to tumorigenesis but also contribute to aging. Mutations accumulate in stem cell niches and change the rate of cell division and differentiation; the pace of this process and the fate of specific mutations depend strongly on niche population size. Here, we create a mathematical model of the intestinal stem cell niche, crypt system, and epithelium. We calculate the expected effect of fixed mutations in stem cell niches and their effect on tissue homeostasis throughout the intestinal epithelium over organismal lifetime. We find that, due to the small population size of stem cell niches, mutations predominantly fix via genetic drift and decrease stem cell fitness, leading to niche and tissue attrition, and contributing to organismal aging. We also explore mutation accumulation at various stem cell niche sizes and demonstrate that an evolutionary trade-off exists between niche size, tissue aging, and the risk of tumorigenesis. Further, mouse and human niches exist at a size that minimizes the probability of tumorigenesis, at the expense of accumulating deleterious mutations due to genetic drift. Finally, we show that the trade-off between the probability of tumorigenesis and the extent of aging depends on whether or not mutational effects confer a selective advantage in the stem cell niche.

The implications of small stem cell niche sizes and the distribution of fitness effects of new mutations in aging and tumorigenesis.

Somatic tissue evolves over a vertebrate's lifetime due to the accumulation of mutations in stem cell populations. Mutations may alter cellular fitness and contribute to tumorigenesis or aging. The distribution of mutational effects within somatic cells is not known. Given the unique regulatory regime of somatic cell division, we hypothesize that mutational effects in somatic tissue fall into a different framework than whole organisms; one in which there are more mutations of large effect. Through simulation analysis, we investigate the fit of tumor incidence curves generated using exponential and power-law distributions of fitness effects (DFE) to known tumorigenesis incidence. Modeling considerations include the architecture of stem cell populations, that is, a large number of very small populations, and mutations that do and do not fix neutrally in the stem cell niche. We find that the typically quantified DFE in whole organisms is sufficient to explain tumorigenesis incidence. Further, deleterious mutations are predicted to accumulate via genetic drift, resulting in reduced tissue maintenance. Thus, despite there being a large number of stem cells throughout the intestine, its compartmental architecture leads to the accumulation of deleterious mutations and significant aging, making the intestinal stem cell niche a prime example of Muller's Ratchet.

Using Computational Modeling To Optimize the Design of Antibodies That Trap Viruses in Mucus.

Immunoglobulin G (IgG) antibodies that trap viruses in cervicovaginal mucus (CVM) via adhesive interactions between IgG-Fc and mucins have recently emerged as a promising strategy to block vaginally transmitted infections. The array of IgG bound to a virus particle appears to trap the virus by making multiple weak affinity bonds to the fibrous mucins that form the mucus gel. However, the antibody characteristics that maximize virus trapping and minimize viral infectivity remain poorly understood. Toward this goal, we developed a mathematical model that takes into account physiologically relevant spatial dimensions and time scales, binding, and unbinding rates between IgG and virions and between IgG and mucins, as well as the respective diffusivities of virions and IgG in semen and CVM. We then systematically explored the IgG-antigen and IgG-mucin binding and unbinding rates that minimize the flux of infectious HIV arriving at the vaginal epithelium. Surprisingly, contrary to common intuition that infectivity would drop monotonically with increasing affinities between IgG and HIV, and between IgG and mucins, our model suggests maximal trapping of HIV and minimal flux of HIV to the epithelium are achieved with IgG molecules that exhibit (i) rapid antigen binding (high kon) rather than very slow unbinding (low koff), that is, high-affinity binding to the virion, and (ii) relatively weak affinity with mucins. These results provide important insights into the design of more potent "mucotrapping" IgG for enhanced protection against vaginally transmitted infections. The model is adaptable to other pathogens, mucosal barriers, geometries, and kinetic and diffusional effects, providing a tool for hypothesis testing and producing quantitative insights into the dynamics of immune-mediated protection.

Minimizing biases associated with tracking analysis of submicron particles in heterogeneous biological fluids.

Tracking the dynamic motion of individual nanoparticles or viruses offers quantitative insights into their real-time behavior and fate in different biological environments. Indeed, particle tracking is a powerful tool that has facilitated the development of drug carriers with enhanced penetration of mucus, brain tissues and other extracellular matrices. Nevertheless, heterogeneity is a hallmark of nanoparticle diffusion in such complex environments: identical particles can exhibit strongly hindered or unobstructed diffusion within microns of each other. The common practice in 2D particle tracking, namely analyzing all trackable particle traces with equal weighting, naturally biases towards rapidly diffusing sub-populations at shorter time scales. This in turn results in misrepresentation of particle behavior and a systematic underestimate of the time necessary for a population of nanoparticles to diffuse specific distances. We show here via both computational simulation and experimental data that this bias can be rigorously corrected by weighing the contribution by each particle trace on a 'frame-by-frame' basis. We believe this methodology presents an important step towards objective and accurate assessment of the heterogeneous transport behavior of submicron drug carriers and pathogens in biological environments.

Modeling of Virion Collisions in Cervicovaginal Mucus Reveals Limits on Agglutination as the Protective Mechanism of Secretory Immunoglobulin A.

Secretory immunoglobulin A (sIgA), a dimeric antibody found in high quantities in the gastrointestinal mucosa, is broadly associated with mucosal immune protection. A distinguishing feature of sIgA is its ability to crosslink pathogens, thereby creating pathogen/sIgA aggregates that are too large to traverse the dense matrix of mucin fibers in mucus layers overlying epithelial cells and consequently reducing infectivity. Here, we use modeling to investigate this mechanism of "immune exclusion" based on sIgA-mediated agglutination, in particular the potential use of sIgA to agglutinate HIV in cervicovaginal mucus (CVM) and prevent HIV transmission. Utilizing reported data on HIV diffusion in CVM and semen, we simulate HIV collision kinetics in physiologically-thick mucus layers-a necessary first step for sIgA-induced aggregation. We find that even at the median HIV load in semen of acutely infected individuals possessing high viral titers, over 99% of HIV virions will penetrate CVM and reach the vaginal epithelium without colliding with another virion. These findings imply that agglutination is unlikely to be the dominant mechanism of sIgA-mediated protection against HIV or other sexually transmitted pathogens. Rather, we surmise that agglutination is most effective against pathogens either present at exceedingly high concentrations or that possess motility mechanisms other than Brownian diffusion that significantly enhance encounter rates.

Hidden semi-Markov models reveal multiphasic movement of the endangered Florida panther.

Animals must move to find food and mates, and to avoid predators; movement thus influences survival and reproduction, and ultimately determines fitness. Precise description of movement and understanding of spatial and temporal patterns as well as relationships with intrinsic and extrinsic factors is important both for theoretical and applied reasons. We applied hidden semi-Markov models (HSMM) to hourly geographic positioning system (GPS) location data to understand movement patterns of the endangered Florida panther (Puma concolor coryi) and to discern factors influencing these patterns. Three distinct movement modes were identified: (1) Resting mode, characterized by short step lengths and turning angles around 180(o); (2) Moderately active (or intermediate) mode characterized by intermediate step lengths and variable turning angles, and (3) Traveling mode, characterized by long step lengths and turning angles around 0(o). Males and females, and females with and without kittens, exhibited distinctly different movement patterns. Using the Viterbi algorithm, we show that differences in movement patterns of male and female Florida panthers were a consequence of sex-specific differences in diurnal patterns of state occupancy and sex-specific differences in state-specific movement parameters, whereas the differences between females with and without dependent kittens were caused solely by variation in state occupancy. Our study demonstrates the use of HSMM methodology to precisely describe movement and to dissect differences in movement patterns according to sex, and reproductive status.