Velocity and geometry of propagating fronts in complex convective flow fields

Quantifying the relationship between spreading depolarization and perivascular cerebrospinal fluid flow

Recent studies have linked spreading depolarization (SD, an electro-chemical wave in the brain following stroke, migraine, traumatic brain injury, and more) with increase in cerebrospinal fluid (CSF) flow through the perivascular spaces (PVSs, annular channels lining the brain vasculature). We develop a novel computational model that couples SD and CSF flow. We first use high order numerical simulations to solve a system of physiologically realistic reaction-diffusion equations which govern the spatiotemporal dynamics of ions in the extracellular and intracellular spaces of the brain cortex during SD. We then couple the SD wave with a 1D CSF flow model that captures the change in cross-sectional area, pressure, and volume flow rate through the PVSs. The coupling is modelled using an empirical relationship between the excess potassium ion concentration in the extracellular space following SD and the vessel radius. We find that the CSF volumetric flow rate depends intricately on the length and width of the PVS, as well as the vessel radius and the angle of incidence of the SD wave. We derive analytical expressions for pressure and volumetric flow rates of CSF through the PVS for a given SD wave and quantify CSF flow variations when two SD waves collide. Our numerical approach is very general and could be extended in the future to obtain novel, quantitative insights into how CSF flow in the brain couples with slow waves, functional hyperemia, seizures, or externally applied neural stimulations.

Social evolution of shared biofilm matrix components

Biofilm formation is an important and ubiquitous mode of growth among bacteria. Central to the evolutionary advantage of biofilm formation is cell-cell and cell-surface adhesion achieved by a variety of factors, some of which are diffusible compounds that may operate as classical public goods-factors that are costly to produce but may benefit other cells. An outstanding question is how diffusible matrix production, in general, can be stable over evolutionary timescales. In this work, using Vibrio cholerae as a model, we show that shared diffusible biofilm matrix proteins are indeed susceptible to cheater exploitation and that the evolutionary stability of producing these matrix components fundamentally depends on biofilm spatial structure, intrinsic sharing mechanisms of these components, and flow conditions in the environment. We further show that exploitation of diffusible adhesion proteins is localized within a well-defined spatial range around cell clusters that produce them. Based on this exploitation range and the spatial distribution of cell clusters, we constructed a model of costly diffusible matrix production and related these length scales to the relatedness coefficient in social evolution theory. Our results show that production of diffusible biofilm matrix components is evolutionarily stable under conditions consistent with natural biofilm habitats and host environments. We expect the mechanisms revealed in this study to be relevant to other secreted factors that operate as cooperative public goods in bacterial communities and the concept of exploitation range and the associated analysis tools to be generally applicable.

Model of glymphatic clearance of aggregating proteins from the brain interstitium

A growing body of evidence suggests that cerebrospinal fluid circulates through the brain to sweep away high-molecular-weight solutes. Multiple studies demonstrate that flow through this pathway, often referred to as the glymphatic system, is most active during sleep. We numerically model the clearance of amyloid-β (a high-molecular-weight protein connected to Alzheimer's disease) from the brain interstitium by combined diffusion and glymphatic advection. We first compare the clearance for a range of different flow conditions and quantify the relation between the clearance rates and Péclet number Pe. We then simulate protein buildup using a reaction-advection-diffusion equation based on the Smoluchowski aggregation scheme and quantify the buildup for different Pe. We find that for flows with Pe≳1, the rate of accumulation of heavy aggregates decreases exponentially with Pe. We finally explore the effect of the sleep-wake cycle by incorporating a variation in the flow speed motivated by experimental measurements. We find that periods of sleep lead to better clearance of intermediate protein aggregates and deter the buildup of large aggregates in the brain. In a conservative estimate, for Pe≈1, we find a 32% reduction in the buildup rate of heavier protein aggregates compared to purely diffusive clearance.

Propagating fronts in fluids with solutal feedback

We numerically study the propagation of reacting fronts in a shallow and horizontal layer of fluid with solutal feedback and in the presence of a thermally driven flow field of counterrotating convection rolls. We solve the Boussinesq equations along with a reaction-convection-diffusion equation for the concentration field where the products of the nonlinear autocatalytic reaction are less dense than the reactants. For small values of the solutal Rayleigh number the characteristic fluid velocity scales linearly, and the front velocity and mixing length scale quadratically, with increasing solutal Rayleigh number. For small solutal Rayleigh numbers the front geometry is described by a curve that is nearly antisymmetric about the horizontal midplane. For large values of the solutal Rayleigh number the characteristic fluid velocity, the front velocity, and the mixing length exhibit square-root scaling and the front shape collapses onto an asymmetric self-similar curve. In the presence of counterrotating convection rolls, the mixing length decreases while the front velocity increases. The complexity of the front geometry increases when both the solutal and convective contributions are significant and the dynamics can exhibit chemical oscillations in time for certain parameter values. Last, we discuss the spatiotemporal features of the complex fronts that form over a range of solutal and thermal driving.