Eric Cytrynbaum

The self-organization of ordered cortical microtubule arrays plays an important role in the development of plant cells. This is observed to emerge from a combination various factors such as microtubule-microtubule interactions, nucleation, and localization of microtubule-associated proteins. Distilling this process into the interaction of one-dimensional bodies on the two-dimensional cortex, quantitative models have been proposed to emulate array formation. These have assisted in understanding the importance of each aspect in addition to identifying potential avenues of experimentation. Until recently, the direct mechanical influence of cell geometry on the constrained microtubule trajectories have been largely ignored in computational models. Modelling microtubules as thin elastic rods constrained on a surface, it has been found that microtubules shapes may differ significantly from the previously assumed geodesics. Restricting to cylinders, we formulate a model incorporating the mechanics of curvature-induced deflection and examine its implications. First, we introduce a minimal anchoring process necessary to describe the geometry of individual microtubules as they grow and become fixed to the cortex. Curvature-induced deflection is governed by the extent to which microtubules can explore the surface curvature, as prescribed by the segment lengths between anchoring. This, in turn, can be modulated by the anchoring kinetics proposed. We implement this model as an event-driven simulation in Python, accommodating for the curvilinear shapes prescribed by the anchoring process. In doing so, ambiguities between differing models are identified. Although these are not resolved here, these provide a possible explanation to the reported conflicting results among differing models. Based on some preliminary simulations using our model, we find that the curvature mechanics provides strong influence for longitudinal alignment with realistic anchoring rates. In particular, catastrophe-inducing edges are unable to overcome this influence to reproduce the observed transverse arrays during the cell elongation phase. This simulation provides the opportunity for further exploration into mechanical influences on array formation and their regulation by the anchoring process.

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Epithelial-mesenchymal transition (EMT), a process in which immotile cells that line surfaces in the body become motile mesenchymal cells, plays a crucial role in major processes such as wound healing, embryo development, and cancer growth. Therefore, examining the dynamics behind individual and collective cell migration would allow for a better understanding of these processes. It has been previously observed that the protein YAP is activated by external mechanical stimuli and affects the expression and activation of the proteins E-cadherin and Rac1, which are involved in intercellular adhesion and migratory ability respectively. It has also been demonstrated that the mechanical stimulation of expanding cell sheets leads to the formation of finger-like projections and EMT, as well as quantitative differences in properties between cells near the sheet edge and cells away from it. Such cell sheets can be simulated using a Cellular Potts Model simulation. I propose an ODE model for YAP/Rac1/E-cadherin dynamics, implement it in a 2D computation of cells in a cellular Potts model, and demonstrate that the predictions are consistent with experimental observations of epithelial sheets grown on topographic features in vitro.

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Human milk production is controlled by a variety of internal and external factors, including hormones, neurons, suckling stimulus and milk removal. One method for increasing milk production suggested to mothers who want to produce more milk is cluster feeding: splitting one feeding into a cluster of multiple feedings. Phenomenological models have been proposed to describe average weekly milk production rates in dairy cattle, but these models do not take into account the effect of the milk removal schedule used. In this thesis, two ordinary differential equation models for describing milk production and milk removal are presented: Model 1, which assumes a linear rate of milk removal during feeding, and Model 2, which uses physical models of fluid flow through a system of ducts to estimate the rate of milk removal. These models are qualitatively very similar, but Model 2 allows for examination of differences in milk dynamics between alveoli at different depths in the mammary gland. 24 hour milk production was then predicted for each model version using various cluster feeding schedules. Feeding schedules which had the most frequent and regular feedings elicited small increases in milk production during a day compared to less frequent or regular schedules. This small increase in milk production suggests that cluster feeding may not be an effective method of increasing production, as other factors that decrease production, like stress, may override this small effect. More work needs to be done to test these models in order to determine an effective method for increasing milk production.

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Recent data measured in nanodiscs conflicts with the standard theory of maltose transport in the MalE-MalFGK₂ uptake system found in E. coli. Nanodisc fluorescence quenching data suggest an alternate pathway in which unliganded MalE binds the P-open transporter, facilitating maltose acquisition. Nanodisc data also indicate that MalE regulates maltose uptake at high concentrations. We analyzed four mathematical models of the maltose uptake system: the distinct standard and alternate models, and two integrated models. Nanodisc fluorescence quenching data and nonlinear regression analysis were used to fit equilibrium constants and kinetic rates. The flux through each pathway in an integrated model was calculated using asymptotic analysis and fit parameter values. We conclude that it is likely that transport occurs when liganded MalE associates to a P-open conformation of MalFGK₂, rather than binding to the P-closed transporter as suggested by the standard model. The standard pathway was calculated to be negative, i.e. to occur in reverse as a means of regulating maltose uptake at high concentration. This analysis conflicts with the standard model in which liganded MalE binds to a closed transporter and triggers an opening of the transporter proteins which in turn open the liganded MalE. The analysis also found that a relatively small amount of maltose transport may occur through the alternate pathway involving unliganded MalE.

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Few biological systems, showing complex pattern formation that spans multiple spatial and temporal scales, have been reduced in understanding to several components. The Min system in Escherichia coli, consists of three proteins, MinD, MinE, and MinC. Through ordered, cyclic, membrane binding and unbinding, facilitated by ATP hydrolysis, the Min system regulates the site of cell division in vivo. The Min system is tightly coupled with cell growth and division. Various mathematical models have been proposed to describe specific biological phenomena, arising from the Min system, but no model has been tested in a biologically relevant space, which grows and divides. In this thesis, I develop a computationally efficient method for solving reaction-diffusion equations, in a growing and dividing geometry, which approximates growth and division in a rod-shaped bacterium.

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An excitable medium has two key properties: a sufficiently large stimulus provokes an even bigger response (excitability), and immediately following a stimulus the medium cannot be excited (refractoriness). A large class of biological systems from cardiac tissue to slime mold are examples of excitable media. FitzHugh Nagumo (FHN) is the canonical model of excitable media. Its two variables are the state of excitation and refractoriness of the one- or two-dimensional medium. Although one of the simplest models, FHN exhibits complex dynamics that have not been fully explored. For example, it supports a stable traveling pulse solution. However, this pulse can be destabilized by large perturbations. In Chapter 2 I explore a one-dimensional example where the perturbation is an increasing refractory profile. This perturbation can lead to collapse of the pulse depending on the steepness of the profile, as conjectured by Keener [Keener, J. (2004) J Theo. Bio. 230(4):459-73]. In Chapter 3 I consider a perturbation in two dimensions which can cause the stable traveling pulse to wrap around the perturbation and generate self-sustaining spiral activity.The one-dimensional example for exponential refractory profiles is explored numerically for a piecewise linear FHN system. Steep profiles lead to collapse while milder profiles allow propagation. The exponential profiles are used as bounds for more general profiles to predict where collapse and propagation will occur. I also make use of a singular FHN system in the limit ε→ 0 to provide insight into the behaviours of the full FHN system for small ε and small diffusion. I conclude this chapter by showing analytically that, in contrast to the full system, a wave in the singular system will propagate for any exponential refractory profile.The two-dimensional case is explored numerically in a FHN system. The use of a temporarily refractory region as a perturbation is a novel mechanism for generating spiral activity. Moreover, it is shown to be robust for refractory regions of a large area. This situation models the appearance of abnormal electrical activity in the heart. In particular, it models the appearance of abnormal electricity activity in undamaged cardiac tissue.

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