State reconstruction from low-dimensional measurements

Takens’ Delay Embedding Theory is a remarkable result from dynamical systems theory that prescribes how a time series in a single observable of a dynamical system can be used to reconstruct the high-dimensional system state. We combined this result with tools from statistical thermodynamics, manifold learning, artificial neural networks, and rigid graph theory to establish Single-molecule TAkens Reconstruction (STAR) as a technique to reconstruct molecular configurations from single molecule Förster resonance energy transfer (smFRET) measurements. We developed and validated the approach in applications to synthetic smFRET data extracted from molecular dynamics simulations of small proteins to demonstrate reconstruction accuracies to Angstrom-level resolution even in the presence of experimentally realistic noise.

We are pursuing the following projects in this theme:

  • Empirical limits on the Jacobian determinant of the diffeomorphism linking the two landscapes
  • Application to real experimental smFRET data
  • Optimal integration of multiplexed low-dimensional observations
  • Extension beyond molecular systems: mechanical systems, ecological systems, financial markets

Representative Publications

100.  M. Topel, A. Ejaz, A.H. Squires, and A.L. Ferguson* “Learned reconstruction of protein folding trajectories from noisy single-molecule time series” J. Chem. Theory Comput. (in press, 2022) [ http://dx.doi.org/10.1021/acs.jctc.2c00920 ]

→ Invited article for Machine Learning for Molecular Simulation special issue

75.   M. Topel and A.L. Ferguson* “Reconstruction of protein structures from single molecule time series” J. Chem. Phys. 153 194102 (2020) [ https://doi.org/10.1063/5.0024732 ]

→ Invited submission to the “2020 JCP Emerging Investigators in Science Collection”

52.  J. Wang and A.L. Ferguson* “Recovery of protein folding funnels from single-molecule time series by delay embeddings and manifold learning” J. Phys. Chem. B 122 50 11931–11952(2018) [ https://doi.org/10.1021/acs.jpcb.8b08800 ]

→ Invited submission to the “Deciphering Molecular Complexity in Dynamics and Kinetics from the Single Biomolecule to Single Cell Levels” special issue

22.  J. Wang and A.L. Ferguson* “Nonlinear reconstruction of single-molecule free-energy surfaces from univariate time series” Phys. Rev. E 93 032412 (2016) [ http://link.aps.org/doi/10.1103/PhysRevE.93.032412 ]