Geometric principles of second messenger dynamics in dendritic spines

Dendritic spines are small, bulbous protrusions along dendrites in neurons and play a critical role in synaptic transmission. Dendritic spines come in a variety of shapes that depend on their developmental state. Additionally, roughly 14~19% of mature spines have a specialized endoplasmic reticulum called the spine apparatus. How do the shape of a postsynaptic spine and its internal organization affect the spatiotemporal dynamics of short timescale signaling? This question is central for understanding the beginnings of synaptic transmission, learning, and memory formation. In this work, we used mathematical modeling using reaction-diffusion equations in idealized geometries (ellipsoids and spheres) to characterize the effect of spine and spine apparatus geometries on the spatio-temporal dynamics of second messengers. Our analyses and simulations showed that in the short timescale, spine size and shape coupled with the spine apparatus geometries govern the spatiotemporal chemical dynamics of second messengers within the cell. We showed that the curvature of the geometries gives rise to pseudoharmonic functions, which predict the locations of maximum and minimum concentrations. Furthermore, we showed that the lifetime of the chemical gradient can be fine-tuned by localization of fluxes on the spine head and varying the relative curvatures and distances between the spine apparatus and the spine head. Thus, we identified some of the key geometric determinants of how spine head and spine apparatus may regulate the short timescale chemical dynamics of small molecules.