Department of Physics , University of Illinois at Urbana-Champaign , 1110 West Green Street , Urbana , Illinois 61801 , United States.
Institute for Molecular Engineering , University of Chicago , 5640 South Ellis Avenue , Chicago , Illinois 60637 , United States.
J Phys Chem B. 2018 Dec 20;122(50):11931-11952. doi: 10.1021/acs.jpcb.8b08800. Epub 2018 Dec 6.
The stability and folding of proteins is governed by the underlying single-molecule free energy surface (smFES) mapping the free energy of the molecule as a function of configurational state. Ascertaining the smFES is of great value in understanding and engineering protein structure and function. By integrating tools from dynamical systems theory and nonlinear manifold learning, we describe an approach to reconstruct the multidimensional smFES for a protein from a time series in a single experimentally measurable observable. We employ Takens' delay embeddings to project the time series into a high-dimensional space in which the projected dynamics are C-equivalent to the true system dynamics and employ diffusion maps to recover a low-dimensional reconstruction of the smFES that is equivalent to the true smFES up to a smooth and invertible transformation. We validate the approach in molecular dynamics simulations of Trp-cage, Villin, and BBA to demonstrate that landscapes recovered from univariate time series in the head-to-tail distance are topologically identical-they precisely preserve the metastable states and folding pathways-and topographically approximate-the free energy barrier heights and well depths are approximately preserved-to the true landscapes determined from complete knowledge of all atomic coordinates. We go on to show that the reconstructed landscapes reliably predict temperature denaturation and identify point mutations and groups of mutations critical to folding. These results demonstrate that protein folding funnels can be reconstructed from experimentally measurable time series and used to understand and engineer folding.
蛋白质的稳定性和折叠由潜在的单分子自由能表面（smFES）控制，该表面将分子的自由能映射为构象状态的函数。确定 smFES 对于理解和设计蛋白质结构和功能具有重要价值。通过整合动力系统理论和非线性流形学习工具，我们描述了一种从单个可实验测量的可观测量的时间序列中重建蛋白质多维 smFES 的方法。我们采用 Takens 延迟嵌入将时间序列投影到一个高维空间中，其中投影动力学与真实系统动力学 C 等价，并采用扩散映射来恢复 smFES 的低维重建，该重建与真实 smFES 在平滑和可逆变换下是等价的。我们在 Trp-cage、Villin 和 BBA 的分子动力学模拟中验证了该方法，证明从从头至尾距离的单变量时间序列中恢复的景观在拓扑上是相同的——它们精确地保留了亚稳态和折叠途径——并且地形上近似——自由能势垒高度和势阱深度近似保留到从所有原子坐标完全了解确定的真实景观。我们继续表明，重建的景观可以可靠地预测温度变性，并确定对折叠至关重要的点突变和突变群。这些结果表明，可以从可实验测量的时间序列中重建蛋白质折叠流形，并用于理解和设计折叠。
