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We describe a computational model for studying the complexity ofself-assembled structures with active molecular components. Our model capturesnotions of growth and movement ubiquitous in biological systems. The model isinspired by biology's fantastic ability to assemble biomolecules that formsystems with complicated structure and dynamics, from molecular motors thatwalk on rigid tracks and proteins that dynamically alter the structure of thecell during mitosis, to embryonic development where large-scale complicatedorganisms efficiently grow from a single cell. Using this active self-assemblymodel, we show how to efficiently self-assemble shapes and patterns from simplemonomers. For example, we show how to grow a line of monomers in time andnumber of monomer states that is merely logarithmic in the length of the line.  Our main results show how to grow arbitrary connected two-dimensionalgeometric shapes and patterns in expected time that is polylogarithmic in thesize of the shape, plus roughly the time required to run a Turing machinedeciding whether or not a given pixel is in the shape. We do this while keepingthe number of monomer types logarithmic in shape size, plus those monomersrequired by the Kolmogorov complexity of the shape or pattern. This work thushighlights the efficiency advantages of active self-assembly over passiveself-assembly and motivates experimental effort to construct general-purposeactive molecular self-assembly systems.

Active Self-Assembly of Algorithmic Shapes and Patterns in Polylogarithmic Time