Tau creates spatially moving hotspots of amyloid-$β$ secondary nucleation: a reaction-diffusion model

Tau-amyloid-$β$ cross-seeding creates a spatial paradox. Bioavailable tau can promote A$β$42 nucleation, thereby increasing local fibril number and catalytic-site burden; yet the same fibrils consume the free A$β$ monomer required for homotypic secondary nucleation. The region containing the most tau or fibrillar material therefore need not be the region with the highest secondary-nucleation rate. We test this idea with a reaction-diffusion model that tracks a propagating effective tau catalyst, free A$β$ monomer, elongation-competent fibrils, fibril mass, and the catalytic sites exposed to monomer. The model reduces the spatial competition to two quantities: accessible catalytic sites and free monomer. Their balance yields an exact condition for an interior kinetic maximum: the logarithmic decrease in accessible catalytic sites must equal twice the logarithmic increase in free monomer. Two Damkohler numbers then separate the relevant competitions-elongation versus clearance, and elongation versus diffusive redistribution-while a monomer screening length sets the physical scale over which depletion can persist. Independent transport and turnover estimates place this length near 0.44-0.54 mm. Simulations show that tau first raises secondary nucleation near its source; continued fibril growth then produces a monomer-depleted wake, and the maximum shifts to a region with fewer catalytic sites but more available monomer. Oligomer transport and sustained reversible exchange can weaken or eliminate this contrast, whereas growth-history-dependent site density can strengthen it. The robust prediction is therefore a spatial separation among tau burden, fibril-mass growth, and instantaneous A$β$ secondary-nucleation activity, rather than a universal hotspot speed.