Source localization of reaction-diffusion models for brain tumors

We propose a mathematically well-founded approach for locating the source (initial state) of density functions evolved within a nonlinear reaction-diffusion model. The reconstruction of the initial source is an ill-posed inverse problem since the solution is highly unstable with respect to measurement noise. To address this instability problem, we introduce a regularization procedure based on the nonlinear Landweber method for the stable determination of the source location. This amounts to solving a sequence of well-posed forward reaction-diffusion problems. The developed framework is general, and as a special instance we consider the problem of source localization of brain tumors. We show numerically that the source of the initial densities of tumor cells are reconstructed well on both imaging data consisting of simple and complex geometric structures

Local structural preferences in shaping tau amyloid polymorphism

Tauopathies encompass a group of neurodegenerative disorders characterised by diverse tau amyloid fibril structures. The persistence of polymorphism across tauopathies suggests that distinct pathological conditions dictate the adopted polymorph for each disease. However, the extent to which intrinsic structural tendencies of tau amyloid cores contribute to fibril polymorphism remains uncertain. Using a combination of experimental approaches, we here identify a new amyloidogenic motif, PAM4 (Polymorphic Amyloid Motif of Repeat 4), as a significant contributor to tau polymorphism. Calculation of per-residue contributions to the stability of the fibril cores of different pathologic tau structures suggests that PAM4 plays a central role in preserving structural integrity across amyloid polymorphs. Consistent with this, cryo-EM structural analysis of fibrils formed from a synthetic PAM4 peptide shows that the sequence adopts alternative structures that closely correspond to distinct disease-associated tau strains. Furthermore, in-cell experiments revealed that PAM4 deletion hampers the cellular seeding efficiency of tau aggregates extracted from Alzheimer’s disease, corticobasal degeneration, and progressive supranuclear palsy patients, underscoring PAM4’s pivotal role in these tauopathies. Together, our results highlight the importance of the intrinsic structural propensity of amyloid core segments to determine the structure of tau in cells, and in propagating amyloid structures in disease