Abstract: The recently proposed Wigner function for a particle in an infinite lattice (Hinarejos M, Banuls MC and Perez A 2012 New J. Phys. 14 103009) is extended here to include an internal degree of freedom as spin. This extension is made by introducing a Wigner matrix. The formalism is developed to account for dynamical processes, with or without decoherence. We show explicit solutions for the case of Hamiltonian evolution under a position-dependent potential, and for evolution governed by a master equation under some simple models of decoherence, for which the Wigner matrix formalism is well suited. Discrete processes are also discussed. Finally, we discuss the possibility of introducing a negativity concept for the Wigner function in the case where the spin degree of freedom is included.

Abstract: In small clinical studies, the application of transcranial photobiomodulation (PBM), which typically delivers low-intensity near-infrared (NIR) to treat the brain, has led to some remarkable results in the treatment of dementia and several neurodegenerative diseases. However, despite the extensive literature detailing the mechanisms of action underlying PBM outcomes, the specific mechanisms affecting neurodegenerative diseases are not entirely clear. While large clinical trials are warranted to validate these findings, evidence of the mechanisms can explain and thus provide credible support for PBM as a potential treatment for these diseases. Tubulin and its polymerized state of microtubules have been known to play important roles in the pathology of Alzheimer's and other neurodegenerative diseases. Thus, we investigated the effects of PBM on these cellular structures in the quest for insights into the underlying therapeutic mechanisms. In this study, we employed a Raman spectroscopic analysis of the amide I band of polymerized samples of tubulin exposed to pulsed low-intensity NIR radiation (810 nm, 10 Hz, 22.5 J/cm2 dose). Peaks in the Raman fingerprint region (300-1900 cm-1)-in particular, in the amide I band (1600-1700 cm-1)-were used to quantify the percentage of protein secondary structures. Under this band, hidden signals of C=O stretching, belonging to different structures, are superimposed, producing a complex signal as a result. An accurate decomposition of the amide I band is therefore required for the reliable analysis of the conformation of proteins, which we achieved through a straightforward method employing a Voigt profile. This approach was validated through secondary structure analyses of unexposed control samples, for which comparisons with other values available in the literature could be conducted. Subsequently, using this validated method, we present novel findings of statistically significant alterations in the secondary structures of polymerized NIR-exposed tubulin, characterized by a notable decrease in alpha-helix content and a concurrent increase in beta-sheets compared to the control samples. This PBM-induced alpha-helix to beta-sheet transition connects to reduced microtubule stability and the introduction of dynamism to allow for the remodeling and, consequently, refreshing of microtubule structures. This newly discovered mechanism could have implications for reducing the risks associated with brain aging, including neurodegenerative diseases like Alzheimer's disease, through the introduction of an intervention following this transition.