Geometric and Topological Data Analysis of Enzyme Kinetics

我们计划在本研究项目中进行的研究涉及细胞外信号调节激酶（ERK）动力学。影响ERK的突变与人类癌症和发育缺陷等疾病有关，使其在当代生物学研究中具有重要意义。虽然已经在体内观察到这种突变的影响，但它们对ERK激活机制的影响，即描述这种现象的数学模型的定量变化，迄今为止仍然未知。我们的目标是获得新的技术，量化的影响遗传扰动，如突变，对ERK kinetics.To为此，我们将调查家庭的ODE模型的基础上，这样的动力学和推导参数推断各种突变。在这方面，我们感兴趣的模型简化技术和表征其实用性和实际的可识别性的背景下，现实世界的数据，通过代数，几何和拓扑的工具。然后，可以通过模型比较来检验所产生的各种模型的相对强度。我们的目标是，然后继续研究贝叶斯参数推断的分布形状，使用拓扑数据分析（TDA）的方法，以区分各种遗传扰动的细胞水平。我们确定了三个研究方向，我们的目标是调查，以实现这一目标：准稳态近似的代数，几何表征他们的近似的善良，并从TDA保证恢复相关的拓扑信息的理论结果。所提出的研究方向的动机是成功的结果，从分析一个模型的假设在我的MSC论文，随后已被写入手稿已提交并进行修订，在这里，我们计划分析一个家庭在更多的数学原理的方式。此外，我们将研究通过运行涉及合成数据的模拟来研究参数推断的形状，从而在细胞遗传扰动水平上得出什么结论。据我们所知，这将是一项新的成就。此外，我们的目标是将硕士论文中使用的推理流水线推广到涉及大量sites.On更多理论方面的ERK机制的更复杂模型，开放问题是加强和扩展现有的结果代数和几何特征的模型简化，如准稳态近似，旨在更好地理解这些近似的准确性。此外，这将是值得调查的工具的区分能力的拓扑数据分析比较不同的模型。可能的合作者是Shvartsman实验室在普林斯顿大学，谁激发了上述硕士项目，并提供了测量数据。该项目属于EPSRC研究领域，代数，几何与拓扑，统计和应用概率和数学生物学福尔斯。

The research we plan to conduct in this research project is concerned with Extracellular Signal Regulated Kinase (ERK) kinetics. Mutations affecting ERK are associated with diseases such as human cancer and developmental defects, making them of significant interest in contemporary biological research. While the effects of such mutations have been observed in vivo, their effects on the mechanism of ERK activation, i.e. quantitive changes to mathematical models describing such phenomena, have remained unknown so far. The aim is to derive novel techniques for quantifying the effect genetic perturbations, such as mutations, have on ERK kinetics.To this end, we will investigate families of ODE models based on such kinetics and derive parameter inferences for various mutants. In this light, we are interested in model reduction techniques and characterising their utility and practical identifiability in the context of real-world data through tools of algebra, geometry and topology. The relative strength of the various models arising can then be tested by means of model comparison. The objective is to then go on to study the shape of distributions resulting from Bayesian parameter inferences, using methods of Topological Data Analysis (TDA), in order to distinguish between various genetic perturbations at the level of cells. We identify three directions of research we aim to investigate in order to achieve this goal: The algebra of Quasi-Steady-State Approximations, geometric characterisations of their goodness of approximation, and theoretical results from TDA guaranteeing recoverability of relevant topological information. The proposed directions of research are motivated by successful results obtained from analysing one model assumption during my MSC thesis, which has subsequently been written into a manuscript been submitted and undergoing revision, here we plan to analyse a family in a more mathematically principles manner. Moreover, we will look into investigating what conclusions can be made at the level of genetic perturbations in cells by studying the shape of parameter inferences through running simulations involving synthetic data. To the best of our knowledge, this would constitute a novel achievement. Furthermore, we aim to generalise the inference pipeline used in the MSc thesis to more complex models of ERK mechanisms involving a larger number of sites.On the more theoretical side, open questions are to strengthen and extend existing results on algebraic and geometric characterisations of model reductions, such as Quasi-Steady-State Approximation, aiming to understand better the accuracy of these approximations. Moreover, it would be worthwhile to investigate how the discriminative power of tools of Topological Data Analysis compares between different models.Possible collaborators are the Shvartsman Lab in Princeton, who motivated the MSc project mentioned above and supplied the measurement data.This project falls within the EPSRC Research areas, Algebra, Geometry & Topology, Statistics and Applied Probability and Mathematical biology.