Browsing by Subject "Pattern formation"

Negative differential conductivity (NDC) is a nonlinear property of electronic transport for high electric field strength found in materials and devices such as semiconductor superlattices, bulk GaAs and Gunn diodes. In spatially extended systems, NDC can cause rich dynamics such as static and mobile field domains and moving charge fronts. In this thesis, these phenomena are studied theoretically and numerically for semiconductor superlattices. Two classes of models are considered: a discrete model based on sequential resonant tunneling between neighboring quantum wells is used to described charge transport in weakly-coupled superlattices, and a continuum model based on the miniband transport is used to describe charge transport strongly-coupled superlattices.

The superlattice is a spatially extended nonlinear system consisting a periodic arrangement of quantum wells (e.g., GaAs) and barriers (e.g., AlAs). Using a discrete model and only considering one spatial dimension, we find that the boundary condition at the injecting contact has a great influence on the dynamical behavior for both fixed voltage and transient response. Static or moving field domains are usually inevitable in this system. In order to suppress field domains, we add a side shunting layer parallel to the growth direction of the superlattice. In this case, the model includes both vertical and lateral spatial degrees of freedom. We first study a shunted weakly-coupled superlattice for a wide range of material parameters. The field domains are found to be suppressed for superlattices with small lateral size and good connection between the shunt and the quantum wells of the superlattice. As the lateral size of the superlattice increases, the uniform field configuration loses its stability to either static or dynamic field domains, regardless of shunt properties. A lower quality shunt generally leads to regular and chaotic current oscillations and complex spatio-temporal dynamics in the field profile. Bifurcations separating static and dynamic behaviors are characterized and found to be dependent on the shunt properties. Then we adopt the model to study the shunted strongly-coupled superlattice with the continuum model. Key structural parameters associated with both the shunt layer and SL are identified for which the shunt layer stabilizes a uniform electric field profile. These results support the possibility to realize a SL-based THz oscillator with a carefully designed structure.

Another important behavior of the static field domains in the weakly-coupled superlattice is bistability, i.e., two possible states (i.e., electric field configurations) for a single voltage. Noise can drive the system from one of these states (the metastable state) to the other one (the globally stable state). The process of escape from the metastable state can be viewed as a stochastic first-passage process in a high-dimensional system that possesses complex stability eigenvalues and for which a global potential energy function does not exist. This process is simulated using a stochastic differential equation system which incorporates shot noise. The mean switching time τ is fitted to an exponential expression e(Vth-V)α/D, where V_th denotes the voltage at the end of the current branch. The exponent α in the fitting curve deviates from 1.5 which is predicted for a generic one dimensional system. We develop an algorithm to determine an effective locally valid potential. Principal component analysis is applied to find the most probable path for switching from the metastable current state.

Sorted bedforms are spatial extensive (100 m-km) features present on many inner continental shelves with subtle bathymetric relief (cm-m) and localized, abrupt variations in grain size (fine sand to coarse sand/gravel). Sorted bedforms provide nursery habitat for fish, are a control on benthic biodiversity, function as sediment reservoirs, and influence nearshore waves and currents. Research suggests these bedforms are a consequence of a sediment sorting feedback as opposed to the more common flow-bathymetry interaction. This dissertation addresses three topics related to sorted bedforms: 1) Modeling the long-term evolution of bedform patterns, 2) Refinement of morphological and sediment transport relations used in the sorted bedform model with `machine learning'; 3) Development of a new sorted bedform model using these new `data-driven' components.

Chapter 1 focuses on modeling the long term evolution of sorted bedforms. A range of sorted bedform model behaviors is possible in the long term, from pattern persistence to spatial-temporal intermittency. Vertical sorting (a result of pattern maturation processes) causes the burial of coarse material until a critical state of seabed coarseness is reached. This critical state causes a local cessation of the sorting feedback, leading to a self-organized spatially intermittent pattern, a hallmark of observed sorted bedforms. Various patterns emerge when numerical experiments include erosion, deposition, and storm events.

Modeling of sorted bedforms relies on the parameterization of processes that lack deterministic descriptions. When large datasets exist, machine learning (optimization tools from computer science) can be used to develop parameterizations directly from data. Using genetic programming (a machine learning technique) and large multisetting datasets I develop smooth, physically meaningful predictors for ripple morphology (wavelength, height, and steepness; Chapter 2) and near bed suspended sediment reference concentration under unbroken waves (Chapter 3). The new predictors perform better than existing empirical formulations.

In Chapter 3, the new components derived from machine learning are integrated into the sorted bedform model to create a `hybrid' model: a novel way to incorporate observational data into a numerical model. Results suggest that the new hybrid model is able to capture dynamics absent from previous models, specifically, the two observed end-member pattern modes of sorted bedforms (i.e., coarse material on updrift bedform flanks or coarse material in bedform troughs). However, caveats exist when data driven components do not have parity with traditional theoretical components of morphodynamic models, and I address the challenges of integrating these disparate pieces and the future of this type of `hybrid' modeling.

During embryogenesis, a single zygote gives rise to a multicellular embryo with distinct spatial territories marked by differential gene expression. How is this patterning process organized? How robust is this function to perturbations? Experiments that examine normal and regulative development will provide direct evidence for reasoning out the answers to these fundamental questions. Recent advances in technology have led to experimental determinations of increasingly complex gene regulatory networks (GRNs) underlying embryonic development. These GRNs offer a window into systems level properties of the developmental process, but at the same time present the challenge of characterizing their behavior. A suitable modeling framework for developmental systems is needed to help gain insights into embryonic development. Such models should contain enough detail to capture features of interest to developmental biologists, while staying simple enough to be computationally tractable and amenable to conceptual analysis. Combining experiments with the complementary modeling framework, we can grasp a systems level understanding of the regulatory program not readily visible by focusing on individual genes or pathways.

This dissertation addresses both modeling and experimental challenges. First, we present the autonomous Boolean network modeling framework and show that it is a suitable approach for developmental regulatory systems. We show that important timing information associated with the regulatory interactions can be faithfully represented in autonomous Boolean models in which binary variables representing expression levels are updated in continuous time, and that such models can provide direct insight into features that are difficult to extract from ordinary differential equation (ODE) models. As an application, we model the experimentally well-studied network controlling fly body segmentation. The Boolean model successfully generates the patterns formed in normal and genetically perturbed fly embryos, permits the derivation of constraints on the time delay parameters, clarifies the logic associated with different ODE parameter sets, and provides a platform for studying connectivity and robustness in parameter space. By elucidating the role of regulatory time delays in pattern formation, the results suggest new types of experimental measurements in early embryonic development. We then use this framework to model the much more complicated sea urchin endomesoderm specification system and describe our recent progress on this long term effort.

Second, we present experimental results on developmental plasticity of the sea urchin embryo. The sea urchin embryo has the remarkable ability to replace surgically removed tissues by reprogramming the presumptive fate of remaining tissues, a process known as transfating, which in turn is a form of regulative development. We show that regulative development requires cellular competence, and that competence is lost early on but can be regained after further differentiation. We demonstrate that regulative replacement of missing tissues can induce distal germ layers to participate in reprogramming, leading to a complete re-patterning in the remainder of the embryo. To understand the molecular mechanism of cell fate reprogramming, we examined micromere depletion induced non-skeletogenic mesoderm (NSM) transfating. We found that the skeletogenic program was greatly temporally compressed in this case, and that akin to another NSM transfating case, the transfating cells went through a hybrid regulatory state where NSM and skeletogenic marker genes were co-expressed.

Patterns are ubiquitous in living organisms. However, the mechanisms driving self-organized pattern formations are not well understood. Due to the complexity of natural systems, many confounding factors complicate quantitative experiments and data interpretation, often making it difficult to draw definitive conclusions. Therefore, a limited number of experimental systems could enable precise perturbation and quantification of pattern formation. In comparison, the synthetic system serves as well-defined model systems to elucidate ‘‘design principles’’ of biological networks. In the past sixteen years, engineering pattern formation is a major endeavor in synthetic biology. However, there are only two studies about the generation of programmed self-organized pattern formation in growing cells based on coordinated dynamics in a population.

Intrigued by the challenge, my colleagues and I programmed E. coli with a synthetic gene circuit to generate self-organized pattern formation. Two implications of this engineered pattern-forming system were illustrated in my Ph.D. thesis.

First, the synthetic system provides a well-defined context to probe principles underlying the scaling property of self-organized pattern formation. Our mechanism underscores the importance of temporal control in generating scale-invariant patterns. The fundamental premise of this approach is that the principles defined in such engineered systems can be generally applicable to natural examples.

Second, the synthetic system serves as a foundation to generate structured materials with well-defined physical properties. Diverse natural biological systems can form structured materials with well-defined physical and chemical properties spontaneously. However, these natural processes are not readily programmable. By taking the synthetic biology approach, we demonstrate here the programmable, three-dimensional (3D) material fabrication using pattern-forming bacteria growing on top of permeable membranes as the structural scaffold. We equip the bacteria with an engineered protein that enables the assembly of gold nanoparticles into a hybrid organic-inorganic dome structure. The resulting hybrid structure functions as a pressure sensor that responds to touch. We show that the response dynamics are determined by the geometry of the structure, which is programmable by the membrane properties and the extent of circuit activation. Taking advantage of this property, we demonstrate signal sensing and processing using one or multiple bacterially assembled structures.

Rho-GTPases are master regulators of polarity establishment and cell morphology. Positive feedback enables concentration of Rho-GTPases into clusters at the cell cortex, from where they regulate the cytoskeleton. Different cell types reproducibly generate either one (e.g. the front of a migrating cell) or several clusters (e.g. the multiple dendrites of a neuron), but the mechanistic basis for unipolar or multipolar outcomes is unclear. The design principles of Rho-GTPase circuits are captured by two-component reaction-diffusion models based on conserved aspects of Rho-GTPase biochemistry. Some such models display rapid winner-takes-all competition between clusters, yielding a unipolar outcome. Other models allow prolonged co-existence of clusters. We investigate the behavior of a simple class of models and show that while the timescale of competition varies enormously depending on model parameters, a single factor explains a large majority of this variation. The dominant factor concerns the degree to which the maximal active GTPase concentration in a cluster approaches a “saturation point” determined by model parameters. We further show that the Rho-GTPase polarity machinery in the budding yeast S. cerevisiae, which normally generates only one bud through competition, can be manipulated to generate multiple buds

in ways consistent with this theoretical framework. We suggest that both saturation and the effect of saturation on competition reflect fundamental properties of the Rho-GTPase polarity machinery, regardless of the specific feedback mechanism, which predict whether the system will generate unipolar or multipolar outcomes.

Diverse mechanisms have been proposed to explain natural pattern formation processes, such as slime mold aggregation, feather branching, and tissue stratification. Regardless of the specific molecular interactions, the vast majority of these mechanisms invoke morphogen gradients, which are either predefined or generated as part of the patterning processes. However, using E. coli programmed by a simple synthetic gene circuit, I demonstrate here the generation of robust, self-organized ring patterns of gene expression in the absence of an apparent morphogen gradient. Interestingly, modeling and experimental tests show that the temporal dynamics of the global morphogen concentration serve as a timing mechanism to trigger formation and maintenance of these ring patterns, which are readily tunable by experimentally controllable environmental factors. This mechanism represents a novel mode of pattern formation that has implications for understanding natural developmental processes. In addition, the system can be coupled with inkjet printing technology and metabolic engineering approaches to develop future complex patterned biomaterials.

Drylands, comprising arid and semi-arid areas and the dry subtropics, over some 40% of the world's land area and support approximately 2 billion people, including at least 1 billion who depend on dryland agriculture and grazing. 10-20% of drylands are estimated to have already undergone degradation or desertification, and lack of monitoring and assessment remains a key impediment to preventing further desertification. Change in vegetation cover, specifically in the spatial organization of vegetation may occur prior to irreversible land degradation, and can be used to assess desertification risk. Coherent spatial structures arise in the distribution of dryland vegetation where plant growth is localized in regular spatial patterns. Such "patterned vegetation" occurs across a variety of vegetation and soil types, extends over at least 18 million ha, occurs in 5 continents and is economically and environmentally valuable in its own right.

Vegetation patterning in drylands arises due to positive feedbacks between hydrological forcing and plant growth so that the patterns change in response to trends in mean annual rainfall. Mathematical models indicate that vegetation patterns collapse to a desertified state after undergoing a characteristic set of transformations so that the condition of a pattern at any point in time can be explicitly linked to ecosystem health. This dissertation focuses on the mathematical description of vegetation patterns with a view to improving such predictions. It evaluates the validity of current mathematical descriptions of patterning for the specific case of small-scale vegetation patterns and proposes alternative hypotheses for their formation. It assesses the significance of seed dispersal in determining pattern form and dynamics for two cases: vegetation growing on flat ground with isotropic patterning, and vegetation growing on slopes and having anisotropic (i.e. directional) patterning. Thirdly, the feedbacks between local biomass density and infiltration capacity, one of the positive feedbacks believed to contribute to patterning, are quantified across a wide range of soil and climatic conditions, and new mathematical descriptions of the biomass-infiltration relationship are proposed. Finally the influence of land surface microtopography on the partitioning of rainfall into infiltration and runoff is assessed.

Cell proliferation is one of the elementary operations involved in building and maintaining the bodies of organisms, and animal development employs diverse regulatory strategies to ensure that it happens in the correct spatial and temporal arrangements. This dissertation is a study of some of the mechanisms involved in timing the early cell cycles of the embryo of Drosophila melanogaster. In chapter 1, we introduce many of the important concepts and provide the reader with background on developmental regulation of the cell cycle. In chapter 2, we turn our focus to the problems associated with the cell-cycle transitions that accompany the maternal-to-zygotic transition. Specifically, it had been shown that slowing of the cell cycle following the initial rapid cleavage divisions is linked to the downregulation of protein phosphatase Cdc25/Twine activity. We pursue this problem with a structure-function analysis of Cdc25/Twine. In chapter 3, we turn our attention to the fourteenth round of cell divisions, which form exquisite spatio-temporal patterns called mitotic domains. Six heterochronic genes (btd, ems, kni, slp1, h, and hkb) had been identified that have dosage-sensitive effects on the timing of cell division in mitotic domain 2 (MD2). We tag two of these factors with GFPs using BAC trangenesis and measure their dynamics in MD2 and other head domains. We find that btd is expressed in a gradient that anticipates the mitotic schedule of MD2, and that slp1 is a powerful repressor of mitosis in the head domains. We conclude that these two factors contribute to the timing of MD2 via a mixed hourglass model that involves both activator-accumulation and repressor-depletion.