"Dynamics of biochemical, genetic and neuronal networks"

最近发展的技术，同时测量许多基因的表达，导致了大量的数据，但也非常需要一个“系统理论”的基因调控，以提供一个框架，了解动态的表达模式。参数（如反应速率和降解速率）通常不知道任何精确度，实验来确定它们可能是困难和昂贵的，所以它已被证明是有用的开发定性数学模型，不包括所有的生理细节，但保留相互作用的结构。一种这样的方法使用分段线性微分方程，其中相互作用项被粗略地建模为阶跃函数。这些“玻璃网络”允许大量的分析（通过将相空间划分为“盒子”），但由于方程中的不连续性，导致额外的数学困难。该模型框架也是计算工具的基础，用于对基因网络进行自动化定性分析，基因网络已经被使用，但其中一些数学困难处理不当。我的研究的一个主要目的是提高我们对这些问题的理解，以及依赖于它们的计算工具。基因调控由细胞中的蛋白质进行，这些蛋白质本身通过生化反应的“途径”受到外部信号的影响。这些信号通路的详细数学模型可以帮助对干预措施的效果进行定量预测，例如使用“嵌合蛋白”将细胞增殖途径与细胞死亡途径联系起来，目的是杀死癌细胞。我使用网络理论来理解细胞过程的另一个背景是具有特定结构的神经元网络以及它们如何对各种类型的电刺激做出反应。公式不同，但原理相似。例如，这与用于治疗帕金森病的刺激器的效果有关。数学技术有时桥梁的问题，甚至领域，我还建议探索应用程序的分段线性方法的问题，在物理学有关的持续振荡称为“呼吸”的链的粒子。

The recent development of technology for simultaneously measuring expression of many genes has led to a profusion of data, but also a great need for a 'systems theory' of gene regulation to provide a framework for understanding dynamics of expression patterns. Parameters (such as reaction rates and degradation rates) are typically not known with any precision, and experiments to pin them down can be difficult and expensive, so it has proven useful to develop qualitative mathematical models that do not include all physiological details, but retain interaction structure. One such approach uses piecewise-linear differential equations in which interaction terms are modelled crudely as step functions. These 'Glass networks' allow a considerable amount of analysis (by a division of phase space into 'boxes'), but lead to additional mathematical difficulties because of the discontinuities in the equations. This model framework is also the basis for computational tools to carry out automated qualitative analysis of gene networks, which are already being used, but which suffer from an inadequate handling of some of these mathematical difficulties. One main thrust of my research is to improve our understanding of these issues, and the computational tools that depend on them. Gene regulation is carried out by proteins in cells, which are themselves influenced by external signals via 'pathways' of biochemical reactions. Detailed mathematical models of these signalling pathways can help to make quantitative predictions about effects of interventions, such as the use of 'chimeric proteins' to link cell proliferation pathways to cell death pathways, with the aim of killing cancer cells. Yet another context where I use network theory to understand cellular processes is in networks of neurons with particular architectures and how they respond to various types of electrical stimuli. The equations are different but principles are similar. This is relevant, for example, to the effects of stimulators used to treat Parkinson's disease. Mathematical techniques sometimes bridge problems or even fields, and I also propose to explore applications of the piecewise-linear approach to problems in Physics relating to sustained oscillations called 'breathers' in chains of particles.