Nonlinear reconstruction of single-molecule free-energy surfaces from univariate time series

Jiang Wang, Andrew L. Ferguson
2016 Physical review. E  
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
more » ... ivalent 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. PHYSICAL REVIEW E 93, 032412 (2016) FIG. 1. Schematic overview of the single-molecule free-energy surface (smFES) reconstruction methodology. (a) Top left: The dynamical evolution of the molecular system proceeds over a low-dimensional manifold M supporting the smFES. The dynamics of the n-tetracosane polymer chain in water considered in this work are contained in a two-dimensional manifold parametrized by the collective variables [ϒ 1 ,ϒ 2 ] that are nonlinear combinations of the molecular degrees of freedom. The smFES maps out the Gibbs free energy of the chain F dedimensionalized by the reciprocal temperature β = 1/k B T as a function of these order parameters. Computing M requires access to the atomic coordinates of the molecule that are typically only available from molecular simulations. Bottom left: Measurements of an experimentally accessible observable v(t) furnish a scalar time series providing a coarse-grained characterization of the single-molecule dynamics. In this work, we consider the head-to-tail distance, , as a quantity measurable by FRET [21]. Assembling d successive measurements separated by a delay time τ produces an n-dimensional delay vector y(t) = [v(t),v(t + τ ),v(t + 2τ ),v(t + 3τ ), . . . ,v(t + (d − 1)τ )]. By computing delay vectors over the entire time series, the scalar time series is projected into an n-dimensional delay space. Bottom right: Under quite general conditions on τ , d, and the observable v(t), Takens' theorem [29] [30] [31] [32] [33] [34] asserts that the manifold (M) containing the delay vectors y(t) is a diffeomorphism to the manifold M containing the real space molecular dynamics, and the variables [ϒ * 1 ,ϒ * 2 ] parametrizing (M) are related by a smooth and invertible transformation to those parametrizing M. Using this approach, topologically and geometrically identical reconstructions of single-molecule free-energy surfaces can be determined directly from experimental measurements. (b) The original and reconstructed manifolds M and (M) exist as low-dimensional surfaces in high-dimensional space. In this work, M is a two-dimensional surface in the 72-dimensional space of Cartesian coordinates of the 24 united atoms of the polymer, and (M) is a two-dimensional surface in the (d = 20)-dimensional delay space. We discover and extract the low-dimensional surfaces using a manifold learning technique known as diffusion maps [3, 6, 39, 40, 81, 82] . Colloquially, this approach may be considered a nonlinear analog of principal components analysis that discovers low-dimensional curved hyperplanes preserving the most variance in the data. As an illustrative example [6], we show the application of diffusion maps to the "Swiss roll" data set comprising a cloud of points in [X,Y,Z] defining a two-dimensional surface in three-dimensional space (top). The diffusion map discovers the latent two-dimensional manifold, and extracts it into the two collective variables [ 2 , 3 ] quantifying, respectively, the location of the points along and perpendicular to the main axis of the spiral (bottom).
doi:10.1103/physreve.93.032412 pmid:27078395 fatcat:szwxoycaifc5pjpzuxxbaey5xu