The discretized flow on domains of attraction: a structural stability result

It is well known that, for stepsize sufficiently small, compactattractors of ordinary differential equations persist underdiscretization. The present paper describes the structure ofthe discrete-time dynamical system obtained via discretizationon A(M_h)\M_h where M_h is the approximate attractor and A(M_h)is its domain of attraction. The existence of a smooth embeddinginto a continuous-time parallelizable flow is proved. The constructioncan be used to define sections for discretizations and can beinterpreted as a justification of the method of modified equations.     

Dealing with misfolded proteins: examining the neuroprotective role of molecular chaperones in neurodegeneration

Human neurodegenerative diseases arise from a wide array of genetic and environmental factors. Despite the diversity in etiology, many of these diseases are considered "conformational" in nature, characterized by the accumulation of pathological, misfolded proteins. These misfolded proteins can induce cellular stress by overloading the proteolytic machinery, ultimately resulting in the accumulation and deposition of aggregated protein species that are cytotoxic. Misfolded proteins may also form aberrant, non-physiological protein-protein interactions leading to the sequestration of other normal proteins essential for cellular functions. The progression of such disease may therefore be viewed as a failure of normal protein homeostasis, a process that involves a network of molecules regulating the synthesis, folding, translocation and clearance of proteins. Molecular chaperones are highly conserved proteins involved in the folding of nascent proteins, and the repair of proteins that have lost their typical conformations. These functions have therefore made molecular chaperones an active area of investigation within the field of conformational diseases. This review will discuss the role of molecular chaperones in neurodegenerative diseases, highlighting their functional classification, regulation, and therapeutic potential for such diseases.