Springcraft - Investigate molecular dynamics by elastic network models¶

Springcraft is a Biotite extension package, that allows the analysis of AtomArray objects via Elastic Network Models (ENMs). An ENM can be thought of as a system that connects residues via springs: Interaction of nearby residues is governed by a harmonic potential, with the native (input) conformation representing the energy minimum. Normal mode analysis allows the researcher to investigate global functional movements of a protein in a fast coarse-grained manner. For a deeper dive into the theory of ENMs please refer to literature, such as a method review from 2010.

Installation¶

Springcraft can be installed via

$pip install springcraft  or $ conda install -c conda-forge springcraft


You can also install Springcraft from source on GitHub. The package uses Poetry for building distributions. Via PEP 517 it is possible to install the package from local source code via pip:

$git clone https://github.com/biotite-dev/springcraft.git$ pip install ./springcraft


Usage¶

To compute an ENM a ForceField is required, that defines the spring force constant for each pair of atoms via force_constant(). A variety of ForceField subclasses is available, each bringing its own concept for parameterization.

import biotite.structure.io as strucio
import springcraft

# All pairs of atoms within a cutoff distance obtain the same force constant
ff = springcraft.InvariantForceField(cutoff_distance=13.0)

# The force constant is inversely proportional to the squared distance
ff = springcraft.ParameterFreeForceField()

# The force constant is read from tabulated values
# based on residue types and distance
# For the residue type the underlying CA-trace is required
atoms = atoms[(atoms.atom_name == "CA") & (atoms.element == "C")]
ff = springcraft.TabulatedForceField.sd_enm(atoms)


The functions compute_kirchhoff() and compute_hessian() use atom coordinates at the presumed minimum energy conformation, which is usually simply the experimentally determined conformation, and the ForceField to compute the Kirchhoff and Hessian matrix of the molecular system, respectively. As byproduct it also returns the pairs of atoms that were found within cutoff distance given by the ForceField.

ff = springcraft.InvariantForceField(cutoff_distance=13.0)
hessian, pairs = springcraft.compute_hessian(atoms.coord, ff)


One type of ENMs are Gaussian Network models (GNM). They do not include directional information, but can be used to investigate whether atoms move collectively and which atoms are involved in global movements. A GNM is created using an AtomArray representing the structure model and a ForceField. Internally, both are given to compute_kirchhoff() to obtain the Kirchhoff matrix, which can be accessed with the kirchhoff attribute. Actually, GNM is only a thin wrapper around compute_kirchhoff(), that conveniently provides common operations performed on the Kirchhoff matrix. This includes the calculation of eigenvectors and eigenvalues among other things. A useful quantity is the correlation between the displacements of two atoms. This value is contained in the covariance matrix (the covariance attribute), which is the pseudo-inverse of the Kirchhoff matrix.

Note

Note that Springcraft uses arbitrary units, i.e. factors like the Boltzmann constant or the temperature are generally not applied, if not stated otherwise.

The kirchhoff and covariance matrix can also be modified to alter the network model. Changing one attribute also updates the other attribute.

gnm = springcraft.GNM(atoms, ff)
eigen_values, eigen_vectors = gnm.eigen()


Analogous to GNM, the class for Anisotropic Network models ANM is a wrapper around the Hessian matrix, which in contrast to the Kirchhoff matrix also includes the directionality of oscillation modes. Hence, its shape is $$3n \times 3n$$ since it comprises the three spatial dimensions for each atom $$(x_1, y_1, z_1, x_2, ...)$$. Similar to GNM, the Hessian as well as its pseudo-inverse, the covariance matrix, are accessible via the hessian and covariance attributes. The added spatial information allows an ANM the depiction of atom oscillations in normal modes or the application of Linear Response Theory to investigate conformational changes upon ligand binding.

import numpy as np

anm = springcraft.ANM(atoms, ff)
force_vector = np.zeros((atoms.array_length(), 3))
force_vector[42, 0] = 10
displacements = anm.linear_response(force_vector)