Nonlinear Manifold Learning of Protein Folding Funnels from Delay-Embedded Experimental Measurements

蛋白质是执行生命所必需的生物功能的分子引擎。随着蛋白质折叠这一“新观点”的出现，对蛋白质行为的理解出现了一个重要的里程碑。这种观点设想蛋白质的结构、稳定性和动力学由分子水平的景观所支配，这与描述地球表面的地形图没有什么不同。景观上的每个点-类似于纬度和经度-对应于蛋白质原子的特定空间排列。地图上每个点的高度定义了蛋白质的稳定性--不稳定的构象位于山顶，稳定的构象位于谷底。确定这些景观是蛋白质生物学的一个关键目标，因为它们在理解天然蛋白质和将合成蛋白质设计为药物、酶和分子机器方面很有用。使用计算机模拟来计算小蛋白质的这些景观是相对简单的，但从实验测量中无法做到这一点。这项研究的主要目标是将动力系统建模中的数学工具与机器学习方法相结合来分析高维数据集，以直接从实验数据确定近似的蛋白质折叠景观。该方法将首先在已知折叠情况的小蛋白质的计算机模拟中得到验证。理论分析将界定近似地貌与真实地貌的接近程度，并为确定它们所需的实验数据设定条件。最终，该方法将应用于结核病蛋白的实验测量。计算分析工具将以用户友好的软件形式发布，供公众免费下载。积极的研究经验对本科生的成功和留住有很大的好处，这个奖项将支持夏季和学年的研究机会。为了提高中学生和高中生对材料科学和工程的认识，将为伊利诺伊大学的工程开放日开发新的教育宣传材料。这项工作的目的是将非线性流形学习与动力系统理论相结合，从测量单个可观测系统的实验时间序列重建蛋白质折叠景观。蛋白质折叠的“新观点”彻底改变了对折叠的理解，将其理解为一种构象搜索，它是对由少数新出现的集体变量参数化的崎岖和漏斗状的自由能景观的研究，对于理解和设计蛋白质作为药物、酶和分子机器具有变革性的意义。现在，从已知所有原子坐标的计算机模拟中确定多维折叠景观是相对常规的，但通过限于少量粗粒度可观察到的蛋白质动力学实验测量来确定多维折叠景观是不可能的。这项研究项目将Takens的延迟嵌入与使用扩散映射的非线性流形学习相结合，首先将实验可测可观测对象中的单变量时间序列投影到动力学与真实空间中的动力学等价的高维空间，然后从该空间提取折叠漏斗的拓扑和几何等价重建，该重建将由所有原子坐标的知识确定。重建的地貌保留了真实漏斗的拓扑结构--亚稳态构型和折叠路径--但地形可能会受到干扰，即自由能峰和谷的高度和深度。这项工作的三个主要目标是：(I)在已知真实景观的小蛋白质的分子动力学模拟中验证该方法，(Ii)在稳健的景观恢复的实验测量中对采样分辨率和信噪比施加条件，以及(Iii)将该方法应用于实验性单分子Forster共振能量转移(SmFRET)测量结核分枝杆菌蛋白酪氨酸磷酸酶(Mtb-PtpB)的开盖和关闭动力学。

Proteins are the molecular engines that perform biological functions essential to life. A major milestone in the understanding of protein behavior emerged with the advent of the "new view" of protein folding. This perspective conceives of protein structure, stability, and dynamics as governed by a molecular-level landscape not unlike a relief map describing the surface of the Earth. Each point on the landscape - analogous to latitude and longitude - corresponds to a particular spatial arrangement of protein atoms. The altitude of each point on the map defines protein stability - unstable conformations lie on mountaintops and stable conformations within valley floors. Determining these landscapes is a key goal of protein biology since they are useful in the understanding of natural proteins and design of synthetic proteins as drugs, enzymes, and molecular machines. It is relatively straightforward to calculate these landscapes for small proteins using computer simulations, but it has not been possible to do so from experimental measurements. It is the primary objective of this research to combine mathematical tools from the modeling of dynamical systems with machine learning approaches to analyze high-dimensional datasets to determine approximate protein folding landscapes directly from experimental data. The approach will first be validated in computer simulations of small proteins where the folding landscape is known. Theoretical analyses will place bounds on how close the approximate landscapes are to the true landscapes, and place conditions on the experimental data required for their determination. Ultimately the approach will be applied to experimental measurements of a tuberculosis protein. The computational analysis tool will be released as user-friendly software for free public download. Positive research experiences have great benefits for undergraduate success and retention, and this award will support summer and academic year research opportunities. New educational outreach materials will be developed for the University of Illinois "Engineering Open House" to promote awareness of materials science and engineering among middle- and high-school students.The aim of this work is to integrate nonlinear manifold learning with dynamical systems theory to reconstruct protein folding landscapes from experimental time series measuring a single system observable. The "new view" of protein folding revolutionized understanding of folding as a conformational search over rugged and funneled free energy landscapes parameterized by a small number of emergent collective variables, with transformative implications for the understanding and design of proteins as drugs, enzymes, and molecular machines. It is now relatively routine to determine multidimensional folding landscapes from computer simulations in which all atomic coordinates are known, but it has not been possible to do so from experimental measurements of protein dynamics that are restricted to small numbers of coarse-grained observables. This research project integrates Takens' delay embeddings with nonlinear manifold learning using diffusion maps to first project univariate time series in an experimentally measurable observable into a high-dimensional space in which the dynamics are C1-equivalent to those in real space, and then extract from this space a topologically and geometrically equivalent reconstruction of the folding funnel to that which would have been determined from knowledge of all atomic coordinates. The reconstructed landscape preserves the topology of the true funnel - the metastable configurations and folding pathways - but the topography may be perturbed, i.e., the heights and depths of the free energy peaks and valleys. The three primary objectives of this work are to (i) validate the approach in molecular dynamics simulations of small proteins for which the true landscape is known, (ii) place conditions on the sampling resolution and signal-to-noise ratio in experimental measurements for robust landscape recovery, and theoretical bounds on the induced topographical perturbations, and (iii) apply the approach to experimental single-molecule Forster resonance energy transfer (smFRET) measurements on the lid-opening and closing dynamics of Mycobacterium tuberculosis protein tyrosine phosphatase (Mtb-PtpB).