"Dynamics of biochemical, genetic and neuronal networks"

同时测量许多基因表达的技术的最新发展带来了大量的数据，但也非常需要基因调控的“系统理论”来为理解表达模式的动态提供框架。参数（如反应速率和降解速率）通常不知道任何精度，并且确定它们的实验可能是困难和昂贵的，因此开发不包括所有生理细节但保留相互作用结构的定性数学模型已被证明是有用的。其中一种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”允许相当数量的分析（通过将相空间划分为“盒子”），但由于方程中的不连续而导致额外的数学困难。这个模型框架也是对基因网络进行自动定性分析的计算工具的基础，这些计算工具已经被使用，但对其中一些数学困难的处理不足。我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。基因调控是由细胞中的蛋白质进行的，细胞本身通过生化反应的“途径”受到外部信号的影响。这些信号传导途径的详细数学模型可以帮助对干预的效果做出定量预测，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。然而，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。方程式不同，但原理相似。例如，这与用于治疗帕金森氏症的刺激器的效果有关。数学技术有时会解决问题，甚至解决领域问题，我还建议探索将分段线性方法应用于与粒子链中被称为“呼吸者”的持续振荡有关的物理问题。

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.