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

主要研究人员继续研究细胞控制系统和时滞微分方程，主要集中在三个主题上。第一个领域是与J·W·齐斯金德博士合作进行的一项跨学科研究，对大肠杆菌中DNA复制的启动进行建模。利用数学模型，他们试图解释DNA复制周期开始时哪些步骤是最重要的。研究了细胞生长的其他几个调控过程，提供了对细胞控制系统的更多洞察力，并引入了新的动力学系统进行研究。第二部分分析了一类具有两个时滞的线性微分方程解的稳定域。用解析和数值两种方法确定了稳定区域随方程中参数的变化情况。对于某些参数的微小变化，稳定区域变化很大，因此从建模的角度研究这些异常是非常重要的。研究的最后一个领域是研究红细胞生成和血小板生成的数学模型。以前的模型包括了两次延迟，将这项研究与前一项研究联系起来。最近，已经使用状态依赖的延迟或年龄结构的建模来开发模型。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，并检查它与具有多个延迟或依赖于状态的延迟的模型相比如何。参数识别方法可以为这些系统在某些疾病状态下如何变得不稳定提供定量信息。这个项目的主要目标是开发与已知生物行为相对应的数学模型。这些模型将提供对管理特定生物问题的机制的洞察。DNA复制标志着细菌细胞周期的开始。理解细胞周期背后的控制对于其他领域的研究很重要，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，并表明了一些生物学理论是不可能的。与实验相比，细胞过程的计算机模拟需要更少的时间，而且成本要低得多。然而，在这项研究中，实验者和数学建模者之间的相互作用将对促进对细胞周期的理解最为重要。红细胞生成或红细胞生成的数学模型可以解释在这个复杂的过程中某些变化的影响，这个复杂的过程始于骨髓中的简单未分化细胞，然后进入成熟的红细胞，后者在全身携带氧气。这些模型可能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有效收集血液的新程序，这对选择为择期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具有重要意义。这项工作包括几种新的数学技术，这些技术将促进可用于未来数学研究的基础研究工具。这些工具将适用于应用数学中的广泛问题。

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.