From a computational viewpoint, the efficient representation of
molecules is an important issue
that has attracted considerable attention in the past decade.
The Modeling
The degrees of freedom (DOF) of a molecule are the number of parameters needed
to specify a placement of the molecule, called a conformation.
Typically, one thinks of the DOF of a molecule in terms of
bond lengths, bond angles, and dihedral or torsional angles.
Bond lengths and bond angles exhibit small variations
and it is very common to consider them constant in
calculations. Torsional angles, however,
vary significantly and they affect the three-dimensional (3D) shape
of the molecule.
Here is a simple
illustration of how a molecule moves when its torsional bonds change.
The energy associated with a conformation measures the likelihood that the
molecule will achieve that conformation in nature (lower energy states
are more likely to occur). Empirical energy functions are commonly used. These energy functions account for bond-length deviations,
changes in bond angles, torsion-angle
deformations, van der Waals potentials, Coulomb, and external potentials.
The constants involved are derived by a combination of
quantum mechanics, vibrational methods, and experimental data.
In our work, we avoid any assumptions on the particular form of the
energy function and the constants involved. Thus, a variety
of energy functions could be successfully substituted.
From a computational viewpoint, the efficient representation of
molecules is an issue worth investigating, especially when a large number of
conformation and their energies are calculated. Depending on the
representation, it may be faster to compute energies and it may also be
possible to solve certain conformational problems fast.
Our Approach
We adopt a robotics-based parameterization for treating
molecular geometry and optimize it for
molecular kinematics and energy calculations.
There is a direct analogy between torsional angles of molecules and
revolute joints of robots. When bond lengths and bond angles
are considered fixed, a molecular chain with n torsions can be viewed as
an articulated robotic mechanism with n revolute joints.
An illustration is offered in the figure below.
The terms forward and inverse kinematics can be used.
In robotics, forward
kinematics compute the spatial configuration of a robot given the values of
its DOF. Inverse kinematics compute the values of the DOF for the robot to achieve
a given configuration. Similarly, one can define forward and inverse kinematics
for molecules.
The Problems We Study
