Figure 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 supporting the smFES. The dynamics of the -tetracosane polymer chain in water considered in this work are contained in a two-dimensional manifold parametrized by the collective variables that are nonlinear combinations of the molecular degrees of freedom. The smFES maps out the Gibbs free energy of the chain dedimensionalized by the reciprocal temperature as a function of these order parameters. Computing 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 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 successive measurements separated by a delay time produces an -dimensional delay vector . By computing delay vectors over the entire time series, the scalar time series is projected into an -dimensional delay space. Bottom right: Under quite general conditions on , and the observable , Takens' theorem [29, 30, 31, 32, 33, 34] asserts that the manifold containing the delay vectors is a diffeomorphism to the manifold containing the real space molecular dynamics, and the variables parametrizing are related by a smooth and invertible transformation to those parametrizing . 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 and exist as low-dimensional surfaces in high-dimensional space. In this work, is a two-dimensional surface in the 72-dimensional space of Cartesian coordinates of the 24 united atoms of the polymer, and is a two-dimensional surface in the (-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 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 quantifying, respectively, the location of the points along and perpendicular to the main axis of the spiral (bottom).
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