Publication Date


Date of Final Oral Examination (Defense)


Type of Culminating Activity


Degree Title

Master of Science in Mathematics



Major Advisor

Uwe Kaiser, Ph.D


Daniel Fologea, Ph.D


Michal Kopera, Ph.D


Memory is traditionally thought of as a biological function of the brain. In recent years, however, researchers have found that some stimuli-responsive molecules exhibit memory-like behavior manifested as history-dependent hysteresis in response to external excitations. One example is lysenin, a pore-forming toxin found naturally in the coelomic fluid of the common earthworm Eisenia fetida. When reconstituted into a bilayer lipid membrane, this unassuming toxin undergoes conformational changes in response to applied voltages. However, lysenin is able to "remember" past history by adjusting its conformational state based not only on the amplitude of the stimulus but also on its previous its conformational state. The current model is a simple two-state Markov description which may not describe a system with memory. In this respect, this thesis aims to provide a more accurate description of this toxin's memory and response to external stimuli by applying a more rigorous mathematical approach. The traditional setting for investigating the conformational changes of voltage-responsive channel proteins is based on analyzing the ionic currents recorded through one or many channels in response to applied voltage stimuli. However, this approach provides only indirect evidence of the conformational state of the channel, i.e open (conducting) or closed (non-conducting). Although very useful, this setting is seriously limited by the inability of electrical measurements to discern between electrically identical yet conformational different open or closed states. The literature that inspired this thesis topic consider models of diffusion on a path-graph with one open state and an arbitrary number of closed states. The mathematics typically begins with approximations from a continuous model. In this thesis we study the analytic solution of the system of linear homogeneous differential equations which are probability vectors describing the diffusion process; this involves exponential theory of weighted Laplacian graphs. Since the Laplacian matrix of the path graph is well studied, we have access to both eigenvectors and eigenvalues in terms of roots of unity making for a succinct solution. We find that polynomial weights model the hysteresis successfully.