Mathematical Sciences: RUI: Modeling Biological Systems and Delay Differential Equations

数学科学：RUI：生物系统建模和时滞微分方程

• 批准号: 9208290
• 负责人: Joseph Mahaffy
• 金额: $ 10.2万
• 依托单位: San Diego State University Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1992
• 资助国家: 美国
• 起止时间: 1992-08-15 至 1997-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9208290&HistoricalAwards=false
• 关键词: Mathematical; Sciences; RUI; Modeling; Biological

The principal investigator continues research on cellular control systems and delay differential equations, focussing on three main topics. The first area is an interdisciplinary study done in collaboration with Dr. J. W. Zyskind on the modeling of initiation of DNA replication in Escherichia coli. Using mathematical models, they attempt to explain which are the most important steps in the beginning of the DNA replication cycle. Several other regulatory processes for growing cells are examined, providing additional insight into cellular control systems and introducing new dynamical systems to study. The second area of inquiry analyzes the stability region for a linear differential equation with two delays. Both analytical and numerical techniques are used to determine how the stability region varies with the parameters in this equation. The stability region varies dramatically for even small changes in some parameters, so studying these anomalies is very important from a modeling point of view. The final area of investigation examines mathematical models for erythropoiesis and thrombopoiesis. Previous models have included two delays linking this study to the previous one. More recently, models have been developed using either state-dependent delays or age-structured modeling. The principal investigator has developed a new age-structured model that more accurately reflects the biology in hematopoietic systems. He will undertake mathematical studies to determine the qualitative behavior of this model and to examine how it compares to models with either multiple delays or state-dependent delays. Parameter identification methods may provide quantitative information for how these systems can become unstable in certain disease states. The principal objective of this project is to develop mathematical models that correspond to known biological behavior. The models will provide insight into the mechanisms that govern the specific biological problem. DNA replication marks the beginning of the cell cycle in bacteria. An understanding of the controls underlying the cell cycle is important for other areas of research, including genetic engineering and cancer research. The mathematical studies provide support for certain biological experiments, suggest further avenues of experimentation, and show how some biological theories are impossible. The computer simulations of cellular processes require less time to perform than experiments and are significantly less costly. However, it is the interaction between the experimenter and the mathematical modeler in this study that will be most important to the advancement in understanding the cell cycle. Mathematical models for erythropoiesis or red blood cell production can elucidate the effects of certain changes in this complicated process that begins with simple undifferentiated cells in the bone marrow and proceeds to mature red blood cells, which carry oxygen throughout the body. The models may explain the mechanisms underlying certain blood diseases and suggest possible therapeutic procedures. The models could aid in developing new procedures to efficiently collect blood, which is significant for patients who choose to provide their own blood supply for elective surgery and avoid risks of HIV infection. The work includes several novel mathematical techniques that will advance fundamental research tools available for future mathematical studies. These tools will be applicable to a wide range of problems in applied mathematics.

首席研究员继续研究细胞控制系统和延迟微分方程，重点关注三个主要主题。第一个领域是与J. W. Zyskind博士合作进行的跨学科研究，关于大肠杆菌DNA复制起始的建模。利用数学模型，他们试图解释DNA复制周期开始时最重要的步骤是什么。研究了生长细胞的其他几个调节过程，为细胞控制系统提供了额外的见解，并引入了新的动力系统来研究。研究的第二个领域分析了具有两个时滞的线性微分方程的稳定区域。本文采用解析和数值两种方法来确定稳定区域随方程参数的变化情况。即使某些参数的变化很小，稳定区域也会发生巨大变化，因此从建模的角度研究这些异常非常重要。研究的最后一个领域检查了红细胞生成和血小板生成的数学模型。之前的模型包含了将这项研究与之前的研究联系起来的两次延迟。最近，使用状态相关延迟或年龄结构建模开发了模型。首席研究员开发了一种新的年龄结构模型，更准确地反映了造血系统中的生物学。他将进行数学研究，以确定该模型的定性行为，并检查它如何与具有多重延迟或状态相关延迟的模型进行比较。参数识别方法可以为这些系统如何在某些疾病状态下变得不稳定提供定量信息。该项目的主要目标是建立与已知生物行为相对应的数学模型。这些模型将提供对控制特定生物学问题的机制的洞察。DNA复制标志着细菌细胞周期的开始。了解细胞周期背后的控制因素对其他研究领域很重要，包括基因工程和癌症研究。数学研究为某些生物学实验提供了支持，提出了进一步的实验途径，并表明一些生物学理论是不可能的。计算机模拟细胞过程所需的时间比实验要少，而且成本也低得多。然而，在这项研究中，实验者和数学建模者之间的相互作用对于理解细胞周期的进步是最重要的。红细胞生成或红细胞生成的数学模型可以阐明这个复杂过程中某些变化的影响，这个过程始于骨髓中简单的未分化细胞，然后发展到成熟的红细胞，红细胞将氧气输送到全身。这些模型可以解释某些血液疾病的潜在机制，并提出可能的治疗方法。这些模型可以帮助开发新的方法来有效地采集血液，这对那些选择为选择性手术提供自己的血液供应并避免艾滋病毒感染风险的患者来说意义重大。这项工作包括一些新的数学技术，这些技术将为未来的数学研究提供基础研究工具。这些工具将适用于应用数学中广泛的问题。