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

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 ${\mathrm{C}}_{24}{\mathrm{H}}_{50} n$-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.

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 $M$ 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 $[{\mathrm{{\rm Y}}}_1,{\mathrm{{\rm Y}}}_2]$ 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 $\beta =1/k_{B}T$ 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 $v\left(t\right)$ 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, $\ell$, as a quantity measurable by FRET [21]. Assembling $d$ successive measurements separated by a delay time $\tau$ produces an $n$-dimensional delay vector $\vec{y}\left(t\right)=[v\left(t\right),v(t+\tau ),v(t+2\tau ),v(t+3\tau ),...,v(t+(d-1)\tau )]$. 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 $\tau , d$, and the observable $v\left(t\right)$, Takens' theorem [29, 30, 31, 32, 33, 34] asserts that the manifold $\mathrm{\Theta }\left(M\right)$ containing the delay vectors $\vec{y}\left(t\right)$ is a diffeomorphism to the manifold $M$ containing the real space molecular dynamics, and the variables $[{\mathrm{{\rm Y}}}_1^{*},{\mathrm{{\rm Y}}}_2^{*}]$ parametrizing $\mathrm{\Theta }\left(M\right)$ are related by a smooth and invertible transformation $\mathrm{\Theta }$ 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 $\mathrm{\Theta }\left(M\right)$ 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 $\mathrm{\Theta }\left(M\right)$ 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 $[{\mathrm{\Psi }}_2,{\mathrm{\Psi }}_3]$ quantifying, respectively, the location of the points along and perpendicular to the main axis of the spiral (bottom).

Figure 2

Application of diffusion maps to the atomistic simulation trajectory employing a Gaussian kernel of bandwidth $\epsilon =0.04$ specified using the approach in Ref. [83]. (a) Neglecting the trivial leading eigenvalue, ${\lambda }_1=1$, the spectral gap between ${\lambda }_5$ and ${\lambda }_6$ indicated by the horizontal line informs an embedding dimensionality of four into $[{\vec{\varphi }}_2,{\vec{\varphi }}_3,{\vec{\varphi }}_4,{\vec{\varphi }}_5]$. (b) Projection of the data into $[{\vec{\varphi }}_2,{\vec{\varphi }}_3]$ results in an effectively one-dimensional manifold, revealing a functional dependence between these two collective variables. We eliminate this redundancy using hierarchical nonlinear principal components analysis (h-NLPCA) to extract the effectively one-dimensional manifold that we term ${\vec{\mathrm{\Gamma }}}_{23}$. Points are colored by the first principal moment of the gyration tensor ${\xi }_1$ [48].

Figure 3

Embedding of the atomistic simulation trajectory into the top three collective modes $[{\vec{\mathrm{\Gamma }}}_{23},{\vec{\varphi }}_4,{\vec{\varphi }}_5]$ identified by diffusion maps. Projection of the 10 001 simulation snapshots into (a) the ${\mathrm{\Gamma }}_{23}-{\varphi }_5$ projection colored by the first principal moment of the gyration tensor ${\xi }_1$, (b) the ${\mathrm{\Gamma }}_{23}-{\varphi }_5$ projection colored by ${\xi }_2$, and (c) the ${\mathrm{\Gamma }}_{23}-{\varphi }_4$ projection colored by ${\xi }_3$. (d) The smFES $F({\vec{\mathrm{\Gamma }}}_{23},{\vec{\varphi }}_4,{\vec{\varphi }}_5)$ with isosurfaces plotted at $\beta F=3$, 4, 5, 6, 7, 8, 9, where $F$ is the Gibbs free energy and $\beta =1/k_{B}T$. The “kink-and-slide” collapse pathways are indicated by arrows, wherein a kink forms at the head or tail of the chain, the kink migrates towards the center of the chain expelling water molecules from between the arms to form a symmetric hairpin with a dry interior, then the chain condenses into a hydrophobically collapsed right- or left-handed helical coil.

Figure 4

Application of diffusion maps to the spatially symmetrized atomistic simulation trajectory employing a Gaussian kernel of bandwidth $\epsilon =0.03$ specified using the approach in Ref. [83]. (a) The spectral gap between ${\lambda }_5$ and ${\lambda }_6$ indicated by the horizontal line informs an embedding dimensionality of four into $[{\vec{\varphi }}_2,{\vec{\varphi }}_3,{\vec{\varphi }}_4,{\vec{\varphi }}_5]$. (b) Projection of the data into $[{\vec{\varphi }}_2,{\vec{\varphi }}_3]$ results in an effectively one-dimensional manifold, informing a functional dependence between these two collective variables. We eliminate this redundancy using h-NLPCA to extract the effectively one-dimensional manifold that we term ${\vec{\mathrm{\Gamma }}}_{23}$. Points are colored by the first principal moment of the gyration tensor ${\xi }_1$.

Figure 5

Embedding of the spatially symmetrized atomistic simulation trajectory in which the head-to-tail and mirror symmetries of the molecule were removed, into the top three collective modes $[{\vec{\mathrm{\Gamma }}}_{23},{\vec{\varphi }}_4,{\vec{\varphi }}_5]$ identified by diffusion maps. Projection of the 10 001 simulation snapshots into (a) the ${\mathrm{\Gamma }}_{23}-{\varphi }_5$ projection colored by ${\xi }_1$, (b) the ${\mathrm{\Gamma }}_{23}-{\varphi }_5$ projection colored by ${\xi }_2$, and (c) the ${\mathrm{\Gamma }}_{23}-{\varphi }_4$ projection colored by ${\xi }_3$. Elimination of the head-to-tail symmetry collapses together asymmetrically kinked chain configurations, and elimination of mirror symmetry collapses together right- and left-handed helices. (d) The smFES $F({\vec{\mathrm{\Gamma }}}_{23},{\vec{\varphi }}_4,{\vec{\varphi }}_5)$ exists as a two-dimensional surface within the three-dimensional space spanned by $[{\vec{\mathrm{\Gamma }}}_{23},{\vec{\varphi }}_4,{\vec{\varphi }}_5]$. Free-energy isosurfaces are plotted at $\beta F=3$, 4, 5, 6, 7, 8, 9, and the “kink-and-slide” collapse pathway is indicated by an arrow.

Figure 6

Additional dimensionality reduction of the spatially symmetrized diffusion map embedding of the atomistic simulation trajectory (Fig. 5). (a) The three-dimensional diffusion map embedding of the spatially symmetrized atomistic simulation trajectory exists as an approximately two-dimensional surface in $[{\vec{\mathrm{\Gamma }}}_{23},{\vec{\varphi }}_4,{\vec{\varphi }}_5]$. (b) Application of h-NLPCA to the embedding generates a new basis set of three nonlinear principal components, $[{\overrightarrow{\mathrm{{\rm Y}}}}_1,{\overrightarrow{\mathrm{{\rm Y}}}}_2,{\overrightarrow{\mathrm{{\rm Y}}}}_3]$, formed from nonlinear combinations of $[{\vec{\mathrm{\Gamma }}}_{23},{\vec{\varphi }}_4,{\vec{\varphi }}_5]$ and arranged in order of decreasing variance. This figure was generated using the “Nonlinear PCA toolbox for Matlab” developed by Scholz [45, 85]. (c) Plotting the cumulative fraction of variance explained upon incorporating additional nonlinear principal components shows that more than $99.95\%$ of the variance in the embedding resides in the first two principal components, confirming that the manifold is effectively two-dimensional and can be projected into $[{\overrightarrow{\mathrm{{\rm Y}}}}_1,{\overrightarrow{\mathrm{{\rm Y}}}}_2]\in {\mathbb{R}}^2$ with essentially no loss of information.

Figure 7

Embedding of the spatially symmetrized atomistic simulation trajectory into the top two nonlinear principal components $[{\overrightarrow{\mathrm{{\rm Y}}}}_1,{\overrightarrow{\mathrm{{\rm Y}}}}_2]$ identified by sequential application of diffusion maps and h-NLPCA. Projection of the 10 001 simulation snapshots colored by (a) ${\xi }_1$, (b) ${\xi }_2$, and (c) ${\xi }_3$. (d) The smFES $F({\overrightarrow{\mathrm{{\rm Y}}}}_1,{\overrightarrow{\mathrm{{\rm Y}}}}_2)$ over which the “kink-and-slide” collapse pathway is indicated by chevrons.

Figure 8

The scalar time series ${\left\{\ell \left(t_i\right)\right\}}_{i=1}^K$ measuring the head-to-tail distance of the $n$-tetracosane at $K=10001$ points at 10 ps intervals over the course of the 100 ns molecular simulation trajectory.

Figure 9

Application of diffusion maps and h-NLPCA to the 20-dimensional delay embedding constructed from the scalar time series of the chain head-to-tail distance. (a) Diffusion maps were applied using a Gaussian kernel of bandwidth $\epsilon =4.00$ specified using the approach in Ref. [83]. The gap in the eigenvalue spectrum between ${\lambda }_4$ and ${\lambda }_5$ indicated by the horizontal line informs an embedding dimensionality of three into $[{\vec{\varphi }}_2^{*},{\vec{\varphi }}_3^{*},{\vec{\varphi }}_4^{*}]$. (b) Application of h-NLPCA to the embedding generates a new basis set of three nonlinear principal components, $[{\overrightarrow{\mathrm{{\rm Y}}}}_1^{*},{\overrightarrow{\mathrm{{\rm Y}}}}_2^{*},{\overrightarrow{\mathrm{{\rm Y}}}}_3^{*}]$, formed from nonlinear combinations of $[{\vec{\varphi }}_2^{*},{\vec{\varphi }}_3^{*},{\vec{\varphi }}_4^{*}]$ and arranged in order of decreasing variance. (c) Plotting the cumulative fraction of variance explained upon incorporating additional nonlinear principal components shows that more than $99.79\%$ of the variance in the embedding resides in the first two principal components, confirming that the manifold is effectively two-dimensional and can be projected into $[{\overrightarrow{\mathrm{{\rm Y}}}}_1^{*},{\overrightarrow{\mathrm{{\rm Y}}}}_2^{*}]\in {\mathbb{R}}^2$ with essentially no loss of information.

Figure 10

Recovery of the reconstructed intrinsic manifold by the application of diffusion maps and h-NLPCA to the 20-dimensional Takens' delay embedding of the $\ell$ scalar time series. (a) Projection of the 9963 delay embedding vectors into the top three collective modes $[{\vec{\varphi }}_2^{*},{\vec{\varphi }}_3^{*},{\vec{\varphi }}_4^{*}]$ identified by diffusion maps. Points are colored by the first principal moment of the gyration tensor averaged over the 20 molecular configurations constituting each delay embedding vector, ${\mathrm{\Xi }}_1$. (b) Projection of the reconstructed intrinsic manifold in panel (a) into the top two nonlinear principal components $[{\overrightarrow{\mathrm{{\rm Y}}}}_1^{*},{\overrightarrow{\mathrm{{\rm Y}}}}_2^{*}]$ recovered from the application of h-NLPCA to the diffusion map embedding. High (low) ${\mathrm{\Xi }}_1$, extended (collapsed) chain configurations lie at low (high) values of ${\mathrm{{\rm Y}}}_1^{*}$. In the delay embedding vectors selected for visualization, the 10th of the 20 configurations constituting the delay embedding is visualized. (c) Reproduction of panel (b) with points colored by the change in ${\mathrm{{\rm Y}}}_1^{*}$ between consecutive delay embedding vector projections, $\mathrm{\Delta }{\mathrm{{\rm Y}}}_1^{*}\left(t_i\right)={\mathrm{{\rm Y}}}_1^{*}(t_i+\tau )-{\mathrm{{\rm Y}}}_1^{*}\left(t_i\right)$. The delay embedding breaks the symmetry of Newton's equations of motion such that collapse and extension pathways are embedded into different regions of the reconstructed intrinsic manifold. Chain collapse proceeds as indicated by the lower arrow, and extension by the upper, resulting in a net counterclockwise flow.

Figure 11

Recovery of the temporally augmented reconstructed intrinsic manifold by the application of diffusion maps and h-NLPCA to the 20-dimensional Takens' delay embedding of the $\ell$ scalar time series produced by concatenating the delay embeddings resulting from the forward and backward simulations. (a) Projection of the 19 926 delay embedding vectors into the top three collective modes $[{\vec{\varphi }}_2^{*},{\vec{\varphi }}_3^{*},{\vec{\varphi }}_4^{*}]$ identified by diffusion maps. (b) Projection of the reconstructed intrinsic manifold in panel (a) into the top two nonlinear principal components $[{\overrightarrow{\mathrm{{\rm Y}}}}_1^{*},{\overrightarrow{\mathrm{{\rm Y}}}}_2^{*}]$ recovered from the application of h-NLPCA to the diffusion map embedding. (c) Reproduction of panel (b) visualizing only the 9963 data points derived from the forward trajectory colored by $\mathrm{\Delta }{\mathrm{{\rm Y}}}_1^{*}\left(t_i\right)={\mathrm{{\rm Y}}}_1^{*}(t_i+\tau )-{\mathrm{{\rm Y}}}_1^{*}\left(t_i\right)$. (d) Reproduction of panel (b) visualizing only the 9963 data points derived from the backward trajectory colored by $\mathrm{\Delta }{\mathrm{{\rm Y}}}_1^{*}\left(t_i\right)$.

Figure 12

Comparison of the $[{\overrightarrow{\mathrm{{\rm Y}}}}_1^{*},{\overrightarrow{\mathrm{{\rm Y}}}}_2^{*}]$ coordinates of the forward, $\left\{\vec{y}\left(t_i\right)\right\}$, and backward, $\left\{\vec{\nu }\left(t_i\right)\right\}$, delay embeddings in the reconstructed manifold in Fig. 11. The embedding of each forward delay vector, $\vec{y}\left(t_i\right)$, and its backward delay vector partner, $\vec{\nu }\left(t_i\right)$, possess (a) identical values of ${\mathrm{{\rm Y}}}_1^{*}$, and (b) sign-inverted values of ${\mathrm{{\rm Y}}}_2^{*}$, such that $\vec{y}\left(t_i\right)\mapsto [{\mathrm{{\rm Y}}}_1^{*},{\mathrm{{\rm Y}}}_2^{*}]$ and $\vec{\nu }\left(t_i\right)\mapsto [{\mathrm{{\rm Y}}}_1^{*},-{\mathrm{{\rm Y}}}_2^{*}]$.

Figure 13

Embedding of the temporally symmetrized $\ell$ delay embedding into the top two nonlinear principal components $[{\overrightarrow{\mathrm{{\rm Y}}}}_1^{*},{\overrightarrow{\mathrm{{\rm Y}}}}_2^{*}]$ identified by sequential application of diffusion maps and h-NLPCA. Projection of the 9963 delay embedding vectors colored by (a) ${\mathrm{\Xi }}_1$, (b) ${\mathrm{\Xi }}_2$, and (c) ${\mathrm{\Xi }}_3$. (d) The smFES $F({\overrightarrow{\mathrm{{\rm Y}}}}_1^{*},{\overrightarrow{\mathrm{{\rm Y}}}}_2^{*})$ over which the “kink-and-slide” collapse pathway is indicated by chevrons. In the delay embedding vectors selected for visualization, the 10th of the 20 configurations constituting the delay embedding is visualized.

Figure 14

Empirical validation of the existence of a diffeomorphism between the intrinsic manifolds recovered from the atomistic simulation trajectory and the $\ell$ delay embedding. (a) Determinant of the Jacobian, $\det \left({\mathbf{J}}_{\mathrm{\Theta }}\right)$, of the forward mapping, $\mathrm{\Theta }:M\to \mathrm{\Theta }\left(M\right)$, at each point on the 20-point averaged intrinsic manifold, $M$. (b) $\det \left({\mathbf{J}}_{\mathrm{\Theta }}\right)\left(t\right)$ for each 20-point averaged point on $M$ expanded out as a linear time series for clarity of viewing. (c) Determinant of the Jacobian, $\det \left({\mathbf{J}}_{{\mathrm{\Theta }}^{-1}}\right)$, of the reverse mapping, ${\mathrm{\Theta }}^{-1}:\mathrm{\Theta }\left(M\right)\to M$, at each point on the reconstructed manifold, $\mathrm{\Theta }\left(M\right)$. (d) $\det \left({\mathbf{J}}_{{\mathrm{\Theta }}^{-1}}\right)\left(t\right)$ for each point on $\mathrm{\Theta }\left(M\right)$ expanded as a linear time series. Projected points with fewer then 40 neighbors within the bandwidth of the Gaussian kernel were not displayed due to insufficiently many neighbors to return a robust estimation of the Jacobian matrix elements. That both $\det \left({\mathbf{J}}_{\mathrm{\Theta }}\right)$ and $\det \left({\mathbf{J}}_{{\mathrm{\Theta }}^{-1}}\right)$ remain single-signed verifies that the two manifolds are diffeomorphic.

Figure 15

Empirical selection of delay time, $\tau$, and delay embedding dimensionality, $d$. (a) The mutual information in the $\ell$ signal. We choose $\tau =20\text{ps}$ corresponding to the delay time at which $I(\ell \left(t\right),\ell (t+\tau ))$ drops to $1/e$ of its initial maximum, identifying the minimum period of time beyond which subsequent measurements of $\ell$ contain significant “new” information about the system. (b) The characteristic function $E_1\left(d\right)$, at a delay time of $\tau =20$ ps, measures the number of false nearest neighbors of points in the reconstructed embedding as a function of the delay embedding dimensionality. Once the phase space reconstruction has been fully unfolded, $E_1\left(d\right)$ saturates to unity, motivating our choice of $d=20$.

Figure 16

Topology of the autoassociative multilayer perceptron (autoencoder) used to perform h-NLPCA. The input (a), bottleneck (c), and output (e) layers each possess a number of nodes, $p$, equal to the dimensionality of the input data, $\mathbf{x}$. The mapping (b) and demapping (d) layers contain $r>p$ nodes. The input, bottleneck, and output layers employ linear activation functions, while mapping and demapping layers employ the nonlinear $tanh$ activation function. The autoencoder is trained to perform the identify mapping (i.e., the output of the neural network, ${\mathbf{x}}^{\prime }$, is equal to its input, $\mathbf{x}$) by minimizing the hierarchical error, $E_H={\sum }_{i=1}^p{E}_i$, using conjugate gradient descent.

Figure 17

Selection of Gaussian kernel bandwidth $\sigma$ in mesh-free Jacobian estimation for the mapping $\mathrm{\Theta }$. $R(\sigma ;\mathrm{\Delta }\sigma )$ measures the relative change in the Jacobian determinant, $\det \left[{\mathbf{J}}_{\mathrm{\Theta }}({\vec{z}}^{\left(j\right)};\sigma )\right]$, averaged over all observations, ${\left\{{\vec{z}}^{\left(j\right)}\right\}}_{j=1}^K$, under small perturbations to the bandwidth of $\mathrm{\Delta }\sigma /L=0.01$. $L=1.0$ is the characteristic size of the manifold $M$ in Fig. 14. $R(\sigma ;\mathrm{\Delta }\sigma )$ reaches a local minimum at $\sigma /L=0.09$ corresponding to a balance between incorporating sufficiently many neighbors into the estimator to robustly evaluate partial derivates, but not so many as to incorporate irrelevant nonlocal information.

Figure 18

Selection of Gaussian kernel bandwidth $\sigma$ in mesh-free Jacobian estimation for the mapping ${\mathrm{\Theta }}^{-1}$. $R(\sigma ;\mathrm{\Delta }\sigma )$ measures the relative change in the Jacobian determinant, $\det \left[{\mathbf{J}}_{{\mathrm{\Theta }}^{-1}}({\vec{F}}^{\left(j\right)};\sigma )\right]$, averaged over all observations, ${\left\{{\vec{F}}^{\left(j\right)}\right\}}_{j=1}^K$, under small perturbations to the bandwidth of $\mathrm{\Delta }\sigma /L=0.01$. $L=1.0$ is the characteristic size of the manifold $\mathrm{\Theta }\left(M\right)$ in Fig. 14. $R(\sigma ;\mathrm{\Delta }\sigma )$ reaches a local minimum at $\sigma /L=0.11$ corresponding to a balance between incorporating sufficiently many neighbors into the estimator to robustly evaluate partial derivates, but not so many as to incorporate irrelevant nonlocal information.

Figure 19

The Lorenz attractor is a fractal object of correlation dimension ($2.05\pm 0.01$) [94] within the three-dimensional phase space of the Lorenz model [Eq. (B1)]. Points represent samples from a single numerical trajectory of the Lorenz system and are colored according to the value of their $y$ coordinate. We will employ Takens' theorem to demonstrate that topologically equivalent representations of this attractor can be reconstructed from univariate time series in a single system observable.

Figure 20

Reconstruction of the Lorenz attractor from three-dimensional Takens' delay embeddings of the scalar time series $x\left(t\right)$. To assist in comparisons with Fig. 19, each point $\vec{y}\left(t_i\right)=(x\left(t_i\right),x(t_i+\tau ),x(t_i+2\tau ))$, where $\tau =0.429$, is colored according to the $y$ coordinate of the center point, $y(t_i+\tau )$. By Takens' theorem, this reconstruction is topologically identical to the Lorenz attractor in Fig. 19.

Figure 21

Reconstruction of the Lorenz attractor from three-dimensional Takens' delay embeddings of the scalar time series $z\left(t\right)$. To assist in comparisons with Fig. 19, each point $\vec{y}\left(t_i\right)=(x\left(t_i\right),x(t_i+\tau ),x(t_i+2\tau ))$, where $\tau =0.146$, is colored according to the $y$ coordinate of the center point, $y(t_i+\tau )$. The observable $z$ contains a spurious symmetry that is not present in the Lorenz system such that it cannot distinguish between the two butterfly wings. This violates a key assumption for the success of Takens' theorem causing it to fail, and the reconstructed attractor is not topologically equivalent to the Lorenz attractor in Fig. 19.

Figure 22

Two-dimensional Brownian dynamics in a three-well potential energy landscape. The point cloud corresponds to the 10 000 snapshots of the particle location recorded over the course of the numerical Brownian dynamics simulation.

Figure 23

Potential energy surface estimated from 10 000 samples of the system coordinates $(x,y)$ recorded over the course of the Brownian dynamics simulation.

Figure 24

Projection of the 20-dimensional delay embedding of the scalar time series in $x\left(t\right)$ extracted from the Brownian dynamics simulations into the top two collective modes $[{\vec{\varphi }}_2,{\vec{\varphi }}_3]$ identified by diffusion maps. Points are colored by the change in ${\varphi }_2$ between consecutive delay embedding vector projections, $\mathrm{\Delta }{\varphi }_2\left(t_i\right)={\varphi }_2(t_i+\tau )-{\varphi }_2\left(t_i\right)$.

Figure 25

Potential energy surface over the two-dimensional reconstructed intrinsic manifold generated from delay embeddings of the scalar time series in $x\left(t\right)$.

Figure 26

Potential energy surface over the two-dimensional reconstructed intrinsic manifold generated from delay embeddings of the scalar time series in $x\left(t\right)-y\left(t\right)$.

Figure 27

Reconstructed potential energy landscape over the two-dimensional reconstructed intrinsic manifold generated from delay embeddings of the scalar time series in $4\times \sin \left[x\left(t\right)-1.5\right]+3\times \cos \left[y\left(t\right)\right]$.