Chemotaxis Models in Biology and Texture Development in Materials: Numerical Methods, Analysis, and Modeling

生物学中的趋化模型和材料中的纹理发展：数值方法、分析和建模

• 批准号: 1112984
• 负责人: Yekaterina Epshteyn
• 金额: $ 14.96万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1112984&HistoricalAwards=false
• 关键词: Chemotaxis; Models; Biology; Texture; Development

The focus of the proposed project is to develop new numerical and analytical techniques to study problems in Biology and Materials Science. These problems give rise to challenging issues for analysis, modeling, and simulations. In this project two areas have been selected for the research directly from potential applications: chemotaxis and chemotaxis models in Biology and texture development and evolution of microstructure in Materials. Chemotaxis refers to mechanisms by which cellular motion occurs in response to an external stimulus, usually a chemical one. Chemotaxis is an important process in many medical and biological applications, including bacteria/cell aggregation and pattern formation mechanisms, as well as tumor growth. Mathematical models of the biological systems are an important tool used in the study of these patterns. Although there is an extensive literature on this subject, only a few numerical methods have been proposed for chemotaxis models. Chemotaxis systems are usually described by highly nonlinear time-dependent partial differential equations. Therefore, development of accurate and efficient numerical methods is crucial for the modeling and analysis of the chemotaxis. Furthermore, a common property of all existing chemotaxis systems is their ability to model a concentration phenomenon that mathematically results in solutions, which rapidly grow in small neighborhoods of concentration points/curves. The solutions may blow up or may exhibit a very singular, spiky behavior. This blow-up represents a mathematical description of a cell concentration phenomenon that occurs in real biological systems. In either case, capturing such solutions numerically is a very challenging problem. The goal of the first project is to design, implement and analyze novel accurate and efficient numerical methods, as well as develop new analytical methods to study chemotaxis models along with closely related problems in physics and biology. The goal of the second project is to study texture development and evolution of microstructure in Materials. Cellular networks are ubiquitous in nature. They exhibit behavior on many different length and time scales and are generally metastable. Most technologically useful materials are polycrystalline microstructures composed of a myriad of small crystallites, called grains, separated by interfaces, called grain boundaries. The energetics and connectivity of the grain boundary network plays a crucial role in determining the properties of a material across a wide range of scales. A central goal of research in materials science is to develop technologies capable of producing an arrangement of grains -a texture- appropriate for a desired set of material properties. The main objective of the second project is to understand the role of energy in material texture development. For this, a recently discovered Grain Boundary Character Distribution is introduced and investigated by the use of a large scale simulation and mathematical analysis. Grain Boundary Character Distribution is a new characterization of the texture which is found to be strongly correlated to the interfacial energy. This research will lead to new analytical/computational tools to study grain networks, along with a better understanding of grain boundary distributions, grain boundary properties, and how they evolve during materials processing.The path to new scientific discoveries lies through new areas of research, interdisciplinary collaboration, and new opportunities. The proposed projects will involve interdisciplinary research and will enhance infrastructure through the development of new analytical and computational tools. The first part of the proposed work, will make fundamental contributions to the development of new numerical and analytical methods which will be used in solving biomedical problems, for example in developing a better understanding of cancer, as well as initiating new collaboration among disciplines. In the second part of the proposal, both new knowledge and new tools will emerge from this part of the project, which will be used to increase the reliability of materials used, for example in aircraft, microprocessors, and many other devices. Educational activities for the proposed projects include the mentorship of graduate and undergraduate students and the development of new modeling course.

拟议项目的重点是开发新的数值和分析技术来研究生物和材料科学中的问题。这些问题给分析、建模和模拟带来了挑战性的问题。在该项目中，直接从潜在应用中选择了两个领域进行研究：生物学中的趋化性和趋化性模型以及材料中微观结构的纹理发展和演化。趋化性是指细胞响应外部刺激（通常是化学刺激）而发生运动的机制。趋化性是许多医学和生物应用中的重要过程，包括细菌/细胞聚集和模式形成机制以及肿瘤生长。生物系统的数学模型是研究这些模式的重要工具。尽管关于这一主题有大量文献，但仅提出了几种用于趋化性模型的数值方法。趋化系统通常用高度非线性的时间相关偏微分方程来描述。因此，开发准确有效的数值方法对于趋化性的建模和分析至关重要。此外，所有现有趋化系统的一个共同特性是它们能够对集中现象进行建模，从而在数学上产生解决方案，该解决方案在集中点/曲线的小邻域中快速增长。解决方案可能会崩溃，或者可能表现出非常奇异、尖锐的行为。 这种放大代表了真实生物系统中发生的细胞浓度现象的数学描述。无论哪种情况，以数值方式获取此类解决方案都是一个非常具有挑战性的问题。第一个项目的目标是设计、实施和分析新颖、准确、高效的数值方法，并开发新的分析方法来研究趋化模型以及物理和生物学中密切相关的问题。第二个项目的目标是研究材料中纹理的发展和微观结构的演化。蜂窝网络在自然界中无处不在。它们在许多不同的长度和时间尺度上表现出行为，并且通常是亚稳态的。大多数技术上有用的材料都是多晶微观结构，由无数称为晶粒的小微晶组成，由称为晶界的界面分隔开。晶界网络的能量和连通性在确定材料在广泛尺度上的性能方面发挥着至关重要的作用。 材料科学研究的中心目标是开发能够产生适合所需材料特性的颗粒排列（纹理）的技术。第二个项目的主要目标是了解能量在材料纹理发展中的作用。为此，通过大规模模拟和数学分析引入并研究了最近发现的晶界特征分布。晶界特征分布是一种新的纹理特征，被发现与界面能密切相关。这项研究将带来新的分析/计算工具来研究晶粒网络，并更好地理解晶界分布、晶界特性以及它们在材料加工过程中的演变方式。新的科学发现之路在于新的研究领域，跨学科合作和新机遇。拟议的项目将涉及跨学科研究，并将通过开发新的分析和计算工具来加强基础设施。拟议工作的第一部分将为开发新的数值和分析方法做出根本性贡献，这些方法将用于解决生物医学问题，例如更好地了解癌症，以及启动学科之间的新合作。在提案的第二部分中，该项目的这一部分将出现新知识和新工具，这些新知识和新工具将用于提高飞机、微处理器和许多其他设备中所用材料的可靠性。拟议项目的教育活动包括研究生和本科生的指导以及新建模课程的开发。