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Bacteria such as Rhodobacter sphaeroides use a single flagellum for propulsion and change of orientation. These types of simple organisms have inspired microrobotic designs with potential applications in medicine, which motivates this work. In this paper, an elastic model for a single-flagellum micro-structure is presented and followed by an analysis of the system based on optimization. The model is based on the method of Regularized Stokeslets which allows for a discretization of the system into particles connected by spring forces. The optimization analysis leads to the design of an optimal elasticity distribution that maximizes the mean forward speed of the structure. These elasticity coefficients are obtained through the use of adjoint-based optimization. The results are illustrated through simulations showing improvement on the swimming pattern of the micro-structure.