# Data-driven Langevin Modeling

Biomolecular dynamics involves processes occurring at time scales ranging
from picoseconds to seconds. In practice, slow
conformational changes that take place on long time scales often represent the dynamics
of interest.
The data-driven Langevin algorithm (dLE) provides a systematic approach to capture the slow dynamics
by projection of the systems free energy landscape along the coordinates of interest.
Based on the local information provided by a time series (obtained for example from MD simulations), the dLE aims to
construct a low-dimensional dynamical model of the underlying system.
Furthermore, it recovers information about the physical properties of the reduced coordinate.

Multidimensional
Langevin modeling of biomolecular dynamics,
J. Chem. Phys. 130, 034106 (2009)

## Performance of the dLE

Adopting various simple model problems of biomolecular dynamics, we studied
the theoretical virtues and limitations as well as the practical applicability and
performance of the dLE.
The robustness of the modeling depends on the model parameters and local sampling.
Given sufficiently sampled input
data, the Langevin modeling is shown to successfully recover the correct statistics (such as the probability distribution) and the dynamics (such as the position autocorrelation function) of all considered
problems.

A simple two state model (left) is suited to test the dLE applicability. The dLE (blue) fairly recovers populations (center) and position autocorrelations (right fig.) of the MD reference (black).

## Overdamped versus nonoverdamped dynamics

When dealing with physical data as provided by, e.g., all-atom
molecular dynamics simulations, effects due to small damping may be important to correctly describe
the dynamics along the free energy landscape. To account for these effects
in a dynamical model, an algorithm that propagates a second-order Langevin scheme has been derived, which
facilitates the treatment of multidimensional data. Adopting extensive molecular dynamics simulations of a
peptide helix, we constructed a five-dimensional model that successfully forecasts the complex structural
dynamics of the system. Neglect of small damping effects on the other hand is shown to lead to significant
errors and inconsistencies.

Transition rates (left) and metastabilities (right) of the network model of AIB9 shown above: Comparison of MD results (black) with second (red) and first order dLE (green).

## Enhanced sampling

Long time scales are difficult to access using standard all-atom computer simulations.
A common strategy to circumvent the problem that molecular dynamics simulations are is to use techniques that enhance
sampling the free energy landscape. Since the propagation of the dLE requires only local
information, the input data are neither required to be Boltzmann weighted nor to be a continuous
trajectory. Similar to a Markov state model, the dLE approach therefore holds the promise of
predicting the long-time dynamics of a biomolecular system from relatively short trajectories which
can be run in parallel.

The practical applicability of the approach is shown to be mainly limited by the
initial sampling of the systemâ€™s conformational space obtained from the short trajectories. Adopting
extensive molecular dynamics simulations of the unfolding and refolding of the short peptide helix AIB9,
it is shown that the dLE approach is able to describe microsecond conformational dynamics from a
few hundred nanosecond trajectories. In particular, the dLE quantitatively reproduces the free energy
landscape and the associated conformational dynamics along the chosen five-dimensional reaction
coordinate.

Communication: Microsecond peptide dynamics from nanosecond trajectories: A Langevin approach,
J. Chem. Phys. 141, 241102 (2014)