About AIGLETools

AIGLETools is a package written in Python, designed to minimize the effort required to build ab initio generalized Langevin equation (AIGLE) & ab initio Langevin equation (AILE) models for multi-dimensional time series data.

What is Generalized Langevin equation (GLE)

GLE is a non-Markovian equation of motion describing the time evolution of a system with generalized coordinates ${\mathbf{x}, \mathbf{p}}$:

$\dot{\mathbf{p}}(t) = -\nabla_{\mathbf{x}}G(\mathbf{x}(t)) + \int_0^t \mathbf{K}(s)\mathbf{p}(t-s) ds + \mathbf{R}(t) + \mathbf{\xi}(t);$

$\dot{\mathbf{x}}(t) = \mathbf{M}^{-1}\mathbf{p}(t)$.

For n-dimensional generalized position $\mathbf{x}$ and n-dimensional generalized momentum $\mathbf{p}$, $G(\mathbf{x})$ is the effective potential energy of the system. $\mathbf{\xi}(t)$ and $\mathbf{F}(t)$ are respectively the external driving force and the environmental noise. $\mathbf{K}(s)$ is a $n\times n$ memory kernel matrix, describing how the system at time $t$ responds to its historical state at time $t-s$. $M$ is a static $n\times n$ matrix, describing the inertia of the system.

What is Langevin equation (LE)

The Langevin equation(LE) is the Markovian limit of GLE, given as

$\dot{\mathbf{p}}(t) = -\nabla_{\mathbf{x}}G(\mathbf{x}(t)) - \eta \mathbf{p}(t) + \mathbf{w}(t) + \mathbf{\xi}(t);$

$\dot{\mathbf{x}}(t) = \mathbf{M}^{-1}\mathbf{p}(t)$.

Here, $\eta$ is a static $n\times n$ matrix, mimicking a friction acting on the system. $\mathbf{w}(t)$ is a white noise.

What is AIGLE and AILE?

AIGLE is the generalized Langevin equation extracted from the history of $\mathbf{x}(t)$ with few or no ad hoc assumptions. AILE is taken as the Markovian limit of AIGLE. Both AIGLE and AILE suppose to recover dynamical properties of $\mathbf{x}(t)$, while AIGLE is expected to be more faithful to the time series data than AILE.

Highlighted features

• Deal with high-dimensional and heterogeneous time-series data

• Integrate data processing, model training and simulation

  • User-friendly training of GLE model; Expertise in GLE not required.
  • Built-in multi-dimensional GLE/LE simulator
  • OPENMM plugin (Python interface)

• Exact enforcement of second fluctuation-dissipation theorem for long-term simulation

Credits

Please cite Xie, Pinchen, and Weinan E. "Coarse-Graining Conformational Dynamics with Multidimensional Generalized Langevin Equation: How, When, and Why." Journal of Chemical Theory and Computation (2024). (doi.org/10.1021/acs.jctc.4c00729) for general purposes.

Installation

pip install .

Use AIGLETools

The training and simulation of AIGLE/AILE model follow the steps below

/examples/harmonic_polymer demonstrates the whole workflow for particle-based coarse-graining. /examples/alaine_dipeptide demonstrates the whole workflow for collective variable-based coarse-graining.

Examples

Check /examples for Jupyter notebook demonstration of AIGLETools.

Questions?

AIGLETools is under active development. More examples will be posted. If you have any question, send email to pinchenx at math dot princeton dot edu.