Stochastic models of reaction-diffusion systems are important for the study of bio- chemical reaction networks where species are present in low copy numbers or if reac- tions are highly diffusion limited. In living cells many such systems include reactions and transport on one-dimensional structures, such as DNA and microtubules. The cytoskeleton is a dynamic structure where individual fibers move, grow and shrink. In this paper we present a simulation algorithm that combines single molecule simula- tions in three-dimensional space with single molecule simulations on one-dimensional structures of arbitrary shape. Molecules diffuse and react with each other in space, they associate to and dissociate from one-dimensional structures as well as diffuse and react with each other on the one-dimensional structure. A general curve embed- ded in space can be approximated by a piecewise linear curve to arbitrary accuracy. The resulting algorithm is hence very flexible. Molecules bound to a curve can move by pure diffusion or via active transport, and the curve can move in space as well as grow and shrink. The flexibility and accuracy of the algorithm is demonstrated in four numerical examples.
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