Department of Physics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA.
Department of Materials Science and Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, USA.
Phys Rev E. 2016 Mar;93(3):032412. doi: 10.1103/PhysRevE.93.032412. Epub 2016 Mar 21.
The stable conformations and dynamical fluctuations of polymers and macromolecules are governed by the underlying single-molecule free energy surface. By integrating ideas from dynamical systems theory with nonlinear manifold learning, we have recovered single-molecule free energy surfaces from univariate time series in a single coarse-grained system observable. Using Takens' Delay Embedding Theorem, we expand the univariate time series into a high dimensional space in which the dynamics are equivalent to those of the molecular motions in real space. We then apply the diffusion map nonlinear manifold learning algorithm to extract a low-dimensional representation of the free energy surface that is diffeomorphic to that computed from a complete knowledge of all system degrees of freedom. We validate our approach in molecular dynamics simulations of a C(24)H(50) n-alkane chain to demonstrate that the two-dimensional free energy surface extracted from the atomistic simulation trajectory is - subject to spatial and temporal symmetries - geometrically and topologically equivalent to that recovered from a knowledge of only the head-to-tail distance of the chain. Our approach lays the foundations to extract empirical single-molecule free energy surfaces directly from experimental measurements.
聚合物和生物大分子的稳定构象和动力学涨落受其单分子自由能表面的支配。通过整合动力系统理论和非线性流形学习的思想，我们从单个粗粒系统可观测量的单变量时间序列中恢复了单分子自由能表面。利用 Takens 延迟嵌入定理，我们将单变量时间序列扩展到一个高维空间中，其中的动力学与真实空间中分子运动的动力学等效。然后，我们应用扩散映射非线性流形学习算法，提取自由能表面的低维表示，该表示与从对所有系统自由度的完整知识计算得出的自由能表面具有微分同胚。我们在 C(24)H(50)n-烷烃链的分子动力学模拟中验证了我们的方法，证明从原子模拟轨迹中提取的二维自由能表面——受空间和时间对称的约束——在几何和拓扑上与仅从链的头尾距离知识中恢复的自由能表面是等价的。我们的方法为直接从实验测量中提取经验单分子自由能表面奠定了基础。
