magnetophoretic separation model. F

magnetophoretic separation model. For mathematical convenience, we used radii of the magnetic beads ( r b = d b 2 −1 ) instead of diameter in modeling equations. To explain the binding effi ciency of beads to bacteria f ( r b ), we employed a colloidal model of collisions (perikinetic) due to Brownian motion. [ 3,7 ] We can safely neglect a colloidal model (orthokinetic) of collisions driven by gravity because of the extremely small size of the beads, ranging from nanometer to micrometer scales. [ 3,8 ] To simplify the binding reaction between the bead and the bacteria, we assumed that this binding reaction is a pseudo-fi rst-order reaction, where most of magnetic beads are free in solution and their concentration does not change as bacteria are bound with the beads. [ 8,15 ] Because cell surface proteins bind to a ligand rapidly (within milliseconds), [ 16,17 ] we assumed that the binding reaction is determined by the rate of collision between bacteria and magnetic beads, not by the kinetics of lectin binding to carbohydrate ligands on cell surface. In the model, we assumed that the surface density of the FcMBL protein on the beads is constant, and the binding of the beads to bacteria is not reversible. [ 3,7,8 ] For modeling the magnetic separation effi ciency g ( r b ), we employed a model based on the geometry of the system and the magnetic forces applied on the bead–bacteria complex. We reasoned that the total magnetic separation effi ciency h ( r b ) would theoretically be proportional to the product of these two modeled processes: h ( r b ) ∝ f ( r b ) g ( r b ). Below, we described our models for the functions f ( r b ) and g ( r b ), and also included remarks about the experiment protocols that relate to the modeling. Among the various parameters affecting the magnetic separation effi ciency, we parameterized the effi ciency in a function of magnetic bead sizes, and then used this model to theoretically determine the optimal size of magnetic beads that should yield the most effi cient magnetic separation in our experimental condition.