Xiao-Bo Li, PhD candidate
David R. Cheriton School of Computer Science
Protein structure elastic network models (ENMs) model the interaction of nearby atoms with fictitious springs using a Hookean potential. The Hookean potential has been used to efficiently calculate the normal modes of a protein structure. The Hookean potential has also been used to interpolate transitional protein structures between a given beginning and ending protein structure. The Hookean potential is a function of distances. The square of the distance is called quadrance. Changing the Hookean potential to use quadrance instead of distance gives a PSD potential energy function on the rank 3 positive semidefinite (PSD) matrix manifold.
Classical mechanics formulates the equations of motion on a Riemannian manifold. Since the rank 3 PSD matrix manifold is a Riemannian manifolds, it can be used to model the dynamics of ENMs. This idea is explored in this thesis. Firstly, we show the PSD potential gives normal mode fluctuations in agreement with the original Hookean potential. Then, enforcing inter-atomic quadrance constraints is discussed. Thereafter, the problem of interpolating transitional conformations between a given beginning and ending conformation is used as an example to demonstrate modelling the dynamics of ENMs as a rank 3 PSD matrix manifold optimization problem.
The rich mathematical structure of the rank 3 PSD matrix manifold can describe the mathematics of modelling ENMs. For example, a protein structure is invariant to rotation and translation, this invariance can be described using a quotient geometry and PSD facial reduction. Inter-atomic quadrance constraints can be enforced using the concept of a retraction, and both the PSD potential energy and constraint enforcing potential are formulated using quadrance consistently. In addition, the PSD potential is not related to springs and no fictitious springs need to be introduced.
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