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A fast method is developed to efficiently compute three-dimensional Brinkman flows induced by triply-periodic arrays of points forces and regularized forces. For point forces, we decompose the periodic Brinkman velocity into the sum of two series: one in real space and one in Fourier space. To do the splitting, we create a regularized solution with special decay properties so that both summands will decay in a Gaussian manner. For regularized forces, the same methodology is used to split the regularized velocity, and again, Gaussian decay of the summands is achieved. When there are N forces ( N periodic arrays), the overall complexity is O ( N 2 ) . We discuss different ways to reduce the complexity to O ( N 3 / 2 ) and to O ( N log N ) . Finally, we present two sets of numerical results. The first validates the computational complexity of the algorithm and the second illustrates how this method can be used to study microscopic flows of organisms in a porous medium. A simple dumbbell model of swimmers is implemented that exhibits a large scale flow that varies based on the number of swimmers and the resistance within the porous medium. [ABSTRACT FROM AUTHOR]

Blood coagulation is a network of biochemical reactions wherein dozens of proteins act collectively to initiate a rapid clotting response. Coagulation reactions are lipid-surface dependent, and this dependence is thought to help localize coagulation to the site of injury and enhance the association between reactants. Current mathematical models of coagulation either do not consider lipid as a variable or do not agree with experiments where lipid concentrations were varied. Since there is no analytic rate law that depends on lipid, only apparent rate constants can be derived from enzyme kinetic experiments. We developed a new mathematical framework for modeling enzymes reactions in the presence of lipid vesicles. Here the concentrations are such that only a fraction of the vesicles harbor bound enzymes and the rest remain empty. We call the lipid vesicles with and without enzyme TF:VIIa + and TF:VIIa − lipid, respectively. Since substrate binds to both TF:VIIa + and TF:VIIa − lipid, our model shows that excess empty lipid acts as a strong sink for substrate. We used our framework to derive an analytic rate equation and performed constrained optimization to estimate a single, global set of intrinsic rates for the enzyme–substrate pair. Results agree with experiments and reveal a critical lipid concentration where the conversion rate of the substrate is maximized, a phenomenon known as the template effect. Next, we included product inhibition of the enzyme and derived the corresponding rate equations, which enables kinetic studies of more complex reactions. Our combined experimental and mathematical study provides a general framework for uncovering the mechanisms by which lipid mediated reactions impact coagulation processes. • We developed a new mathematical framework for lipid-dependent reactions • We formulated an analytic rate law for lipid-dependent enzyme kinetics • We estimated a single set of rate constants from experimental data with varied lipid • Our model is applicable to a diverse set of enzyme reactions • Our framework serves as a foundation for advanced coagulation models [ABSTRACT FROM AUTHOR]