Geometric and Topological Data Analysis of Enzyme Kinetics

我们计划在这个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，使其成为当代生物学研究的重要兴趣。虽然这种突变的影响已经在体内观察到，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。目的是获得新的技术来量化遗传扰动，如突变，对ERK动力学的影响。为此，我们将研究基于这种动力学的ODE模型家族，并推导出各种突变体的参数推断。在这种情况下，我们对模型约简技术感兴趣，并通过代数、几何和拓扑工具来描述它们在现实世界数据背景下的效用和实际可识别性。由此产生的各种模型的相对强度可以通过模型比较来检验。目的是利用拓扑数据分析（TDA）的方法，继续研究贝叶斯参数推断产生的分布形状，以便在细胞水平上区分各种遗传扰动。为了实现这一目标，我们确定了三个研究方向：准稳态逼近的代数，其逼近优度的几何特征，以及保证相关拓扑信息可恢复性的TDA理论结果。提出研究方向的动机是在我的硕士论文中通过分析一个模型假设获得的成功结果，该论文随后被写进了一份已经提交并正在修改的手稿中，在这里我们计划以更数学原理的方式分析一个家庭。此外，我们将通过运行涉及合成数据的模拟来研究参数推断的形状，从而研究在细胞遗传扰动水平上可以得出什么结论。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管道推广到涉及更多站点的更复杂的ERK机制模型。在更理论化的方面，开放的问题是加强和扩展现有的关于模型约简的代数和几何特征的结果，例如准稳态近似，旨在更好地理解这些近似的准确性。此外，研究拓扑数据分析工具在不同模型之间的判别能力是值得研究的。可能的合作者是普林斯顿的Shvartsman实验室，他们推动了上述理学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，统计与应用概率和数学生物学。

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.