A granuloma is a collection of immune cells that contains bacteria or other foreign material. An example is provided by the granulomas of Mycobacterium tuberculosis, a bacteria that infects a third of the world’s population. Although 90% of Tuberculosis cases are latent, 10% result in active infection. I will present a simple model of a generic granuloma and discuss efforts to discover why granulomas breakdown to cause active infections.
The time it takes a cell to divide, or intermitotic time (IMT), is highly variable, even under homogeneous environmental conditions. I will present a multistep stochastic model of the cell cycle and discuss how the model can be used to explain variability in IMT distributions and study the effect of drug treatment.
A stochastic interpretation of spontaneous action potential initiation is developed for the Morris-Lecar equations. Initiation of a spontaneous action potential can be interpreted as the escape from one of the wells of a double well potential, and we develop an asymptotic approximation of the mean exit time using a recently-developed quasi-stationary perturbation method. Using the fact that the activating ionic channel's random openings and closings are fast relative to other processes, we derive an accurate estimate for the mean time to fire an action potential (MFT), which is valid for a below-threshold applied current. Previous studies have found that for above-threshold applied current, where there is only a single stable fixed point, a diffusion approximation can be used. We also explore why different diffusion approximation techniques fail to estimate the MFT.
Shape-based regularization has proven to be a useful method for delineating objects from noisy images encountered in many applications when one has prior knowledge of the shape of the targeted object. When a collection of possible shapes is available, the specification of a shape prior using kernel density estimation is a natural technique. This process transforms the problem of shape-regularized image segmentation into an optimization problem involving a nonlinear energy functional.
In this talk I will present a framework for minimization of this energy functional with application to segmentation of still images. I will then present an extension of this approach to the identification of traveling fronts in image sequences.
Somitogenesis is a process for the development of somites which are transient, segmental structure that lies along the anterior-posterior axis (AP axis) of vertebrate embryos. The pattern of somites is governed by the segmentation clock and its timing is controlled by the clock genes which undergo synchronous oscillation over adjacent cells in the posterior presomitic mesoderm (PSM), oscillation slowing down and traveling wave pattern in the traveling wave region, and the oscillation-arrested in the anterior PSM, called determined region.
In this talk, I will focus on mathematical models which depict the kinetics of the zebrafish segmentation clock genes subject to direct autorepression by their own products under time delay, and cell-to-cell interaction through Delta-Notch signaling. First, for a basic two-cell system with delays, a sequential-contracting technique is employed to derive the global convergence to the equilibrium. This scenario corresponds to the oscillation-arrested for the cells in the determined region. Applying the delay Hopf bifurcation theory, the center manifold theorem, and the normal form method, we derive the criteria for the existence of stable synchronous oscillations for the cells in the tail bud of the PSM. These analytical results can be extended to a specific N-cell model. Hence, we provide an explanation for how synchronous oscillations are generated for the cells in the posterior PSM and how oscillations are arrested for the cells in the anterior PSM. Based on these results of two-cell system, we further construct a non-autonomous lattice delayed system to generate synchronous oscillation, traveling wave pattern, oscillation slowing-down, and oscillation-arrested in each corresponding region in the embryo to fit the experimental observations.
Binocular rivalry is the alternation in visual perception that can occur when the two eyes are presented with different images. Hugh Wilson proposed a class of neuronal network models that generalize rivalry to multiple competing patterns. The networks are assumed to have learned several patterns, and rivalry is identified with time periodic states that have periods of dominance of different patterns. In this talk we will use the theory of coupled cell systems to identify conditions under which networks with two learned patterns reduce to certain recent models of binocular rivalry where much of the dynamics are organized by a Takens-Bogdanov singularity. We also show that Wilson networks support patterns that were not learned, which we call derived. This is important because there is evidence for perception of derived patterns during several binocular rivalry experiments in the literature. We construct Wilson networks for these experiments and use symmetry breaking to make predictions regarding states that a subject might perceive.
An important feature of locomotion in cats, rats, and humans is that changes in speed occur due to a shortening of the stance (extensor) phase, while the swing (flexor) phase duration remains relatively constant. We have analyzed a simplified locomotor model that can replicate this key feature through feedback control. In this model, a central pattern generator (CPG) establishes a rhythm and controls the activity of a pendular limb, with afferent feedback signals closing the loop. Using dynamical systems methods, we analyze the mechanisms responsible for rhythm generation in the CPG, both in the presence and absence of feedback. We exploit our observations to construct a reduced model that is qualitatively similar to the original but tractable for rigorous discussion. We prove the existence of a locomotor cycle in this reduced system using a novel version of the Melnikov function, adapted for discontinuous systems. Finally we utilize our understanding of the model dynamics to explain its performance under various modifications, including recovery of oscillatory behavior after spinal cord injury and response to changes in load.
Vertebrate seed dispersers and seed predators, insect seed predators, and pathogens are known to influence plant survival, population dynamics, and species distributions. My research investigates the importance of these groups of organisms in the sequential stages of early plant recruitment (i.e. from fruit developing in the crown to seedlings on the ground) in Neotropical forests. I use a combination of experimental and theoretical studies to investigate the influence of vertebrates, insects, and pathogens on plant survival and spatial patterns and the role of plant traits in mediating these interactions.
We simulate cellular automaton with a set of individual-based rules to reproduce spatiotem- poral dynamics arising from local interactions of ants in colony. Ants deposit diffusible chemical pheromone that modifies local environment for succeeding passages. Individual ants, then, re- spond to conspecific/heterospecific pheromone gradients, for example, by altering their direction of motion, or switching tasks. We describe ant’s movement by reinforced random walk, and study patterns (foraging trails, territoriality, etc.) emerging from ‘microscopic’ interactions of individual ants. We derive macroscopic PDEs by considering continuum limits of the mechanistic microscopic dynamics, and compare the two models.
The Cdc42 GTPase plays a key role in cell polarization in budding yeast. Although previous studies in budding yeast suggested positive feedback loops whereby Cdc42 becomes polarized, these mechanisms do not include spatial cues, neglecting the normal patterns of budding. In this talk, we present a two-equation reaction-diffusion model of cell polarization with a general function form of positive feedback. In the first part, we perform linear stability analysis, in particular Turing stability analysis to the model to derive conditions of parameters for which cell polarity emerges without any spatial cue. In the second part, we combine live cell imaging and mathematical modeling to understand how diploid daughter cells establish polarity preferentially at the pole distal to the previous division site despite the presence of two landmark cues at distal pole and proximal pole. We report that both spatial landmarks and GTP hydrolysis of Cdc42 by Rga1 controls the robust Cdc42-GTP polarization in diploid daughter cells.
Three-dimensional (3D) chromatin organizations play an important role in transcription regulation and can be used to define chromatin signatures. There is a possibility of long-range (i.e. 3D) epigenetic controls instead of the usual paradigm of promoter-cis control mechanism in distal regulation of gene. Such newly discovered mechanism may reveal novel epigenetic biomarkers for distinguishing between different cell types, for example, normal cells vs. cancer cells.
The current state-of-the-art experimental technique in 3D architecture inference is Hi-C, which was evolved from the earliest experimental protocol, Chromosome Conformation Capture (3C). In this talk, I will propose a flexible random effect model for inference from Hi-C data on constructing 3D architecture of chromatin. The model has advantages over conventional models thanks to its capability of incorporating both correlation structure and unknown sources of error into the model. Properties of the model will be presented, which will be followed by numerical simulations and applications to Hi-C human genome data.