SummaryThe circuit tile assembly model (cTAM) consists of a voltage source and resistive circuit tiles, configured as a voltage divider, that attach to form resistive ladders or grids if the voltage on the boundary is greater than or equal to a threshold. The model produces a family of circuits whose composition and properties change with time. As growth proceeds, the voltage decreases until it no longer exceeds a threshold, which causes growth to cease. This is referred to as self‐controlled growth, because the properties of the assembly itself are the primary determinant in controlling the extent of the assembly. The model augments tile assembly models, which are programmable through specific chemical interactions, with an alternative electrical mechanism. The ladder and grid assemblies have bounded size and unique shape that are determined by parameters from the electrical network. Using the harmonicity of the electric potential, the shape of the grid assembly is shown to be symmetric around the main diagonal. Finally, two models of growth, differentiated by whether the voltage is measured before or after attachment, are equivalent. The model and analysis have potential application to self‐assembly of nanostructures, as well as to networks whose structure changes over time.

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By guiding cell and chemical migration and coupling with genetic mechanisms, bioelectric networks of potentials influence biological pattern formation and are known to have profound effects on growth processes. An abstract model that is amenable to exact analysis has been proposed in the circuit tile assembly model (cTAM) to understand self-assembled and self-controlled growth as an emergent phenomenon that is capable of complex behaviors, like self-replication. In the cTAM, a voltage source represents a finite supply of energy that drives growth until it is unable to overcome randomizing factors in the environment, represented by a threshold. Here, the cTAM is extended to the axon or alternating cTAM model (acTAM) to include a circuit similar to signal propagation in axons, exhibiting time-varying electric signals and a dependence on frequency of the input voltage. The acTAM produces systems of circuits whose electrical properties are coupled to their length as growth proceeds through self-assembly. The exact response is derived for increasingly complex circuit systems as the assembly proceeds. The model exhibits complicated behaviors that elucidate the interactive role of energy, environment, and noise with electric signals in axon-like circuits during biological growth of complex patterns and function.

Tile self-assembly models have proved to be important theoretical tools for studying nanoscale manufacturing techniques, and have provided insight into the computational capabilities of systems inspired by molecular biology. A tile assembly model (rcTAM), whose tiles are composed of simple electric circuit components, exhibits three important properties akin to those found in living organisms. First, it grows from a seed tile by self-assembly of component tiles. Second, it autonomously stops growth when a maximum size is reached, as determined by parameters associated with the tiles. Third, when the circuit assembly has reached the limit of its growth, it generates an identical copy of the original seed, which then will grow to replicate a copy itself. The size of the assembled circuit is controlled by values of the circuit components (voltage sources, resistors), and a threshold voltage for tile attachment. Since every new tile attachment instantly changes the circuit properties, such as voltage drops and currents across resistors, as well as equivalent resistances, the model exhibits instantaneous distant communication and cooperation between components, as well as dynamic behavior. Proofs are given for a bound on growth, the self-replicating property, and possible aging phenomena that produce a stable circuit population. The model might have application to electrochemical growth processes at the nanoscale, and provides insight into self-replicating systems that are not necessarily composed of organic materials. In addition, it models certain features of bioelectric networks that contribute to pattern formation in collections of cells.