A Novel Delay Differential Equation Model of the Germinal Center Reaction and an Algorithm for Minimum Length Surveillance Paths
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The humoral adaptive immune system in vertebrates includes a process called the germinal center reaction in which B-cells rapidly increase their binding affinity to an antigen that is part of a pathogen. A fraction of germinal center B cells differentiates into plasma cells that secrete antibodies. Antibodies bind to the pathogen and neutralize it. In a secondary immune response to the same pathogen, memory B-cells and long-lived plasma cells generated during the primary immune response encode higher affinity antibodies and are able to fight the pathogen more efficiently. We develop a delay differential equation model of the germinal center reaction incorporating known physical mechanisms. We find that secondary immune responses including low affinity seeder B-cells outperform those seeded only with higher affinity. This helps to explain a recent laboratory observation that a high fraction of seeder B-cells in a secondary immune response are naive. Further, two mechanisms of antibody feedback are explored, where antibodies produced in the reaction interact with the reaction itself. Negative feedback occurs via epitope masking, which is consistent with experimental data. Positive feedback occurs via improved antigen presentation on follicular dendritic cells, which is a mechanism we propose. Additionally, we propose a novel path optimization algorithm. Given a path connected environment, our proposed algorithm finds the shortest paths from which surfaces in the environment are surveyed under a limited visibility constraint. We further explore how this is related to the inradius problem in classical geometry: What is the shortest curve whose convex hull includes the unit sphere? The solution is known for closed curves, but not for open curves. Our algorithm seems to converge numerically to the true solution for closed curves and to the best-known conjecture for open curves. This offers validation of our method and evidence for the open path conjecture.