Linear Systems Analysis of Proteins using Molecular Dynamics

Most proteins reduce the complexity of atomic motion to stable and coherent structures. Molecular dynamics (MD) has provided swaths of trajectory data of proteins. We analyze these stochastic trajectories using the methods of stochastic signal analysis, well established and utilized by engineers. The coherence function, from linear systems analysis, says an input and output are linearly related if and only if coherence equals one. That is, the coherence function provides the existence of a frequency function (also known as a transfer function) between the input and output. We choose to model this linearity as a spring linking the two atoms. We are motivated to explore experimentally well-defined interactions like hydrogen bonds where we see coherence near one. Other atomic interactions, such as salt bridges and covalent bonds, are analyzed too with similarly high coherence. Furthermore, coherence between substructures, defined as averages of atomic positions, such as turns in the alpha helix, provide high coherence. We present the versatility and applicability of the method by analyzing all pairwise interactions within the protein. Additionally, with hierarchical clustering, we can identify these substructures. Potential mechanical models are proposed, and a pairwise coherence matrix is compared to a harmonic mechanical model of the protein.