Demonstration of elementary functions via DNA algorithmic self-assembly

Target-oriented cellular automata with computation are the primary challenge in the field of DNA algorithmic self-assembly in connection with specific rules. We investigate the feasibility of using the principle of cellular automata for mathematical subjects by using specific logic gates that can be implemented into DNA building blocks. Here, we connect the following five representative elementary functions: (i) enumeration of multiples of 2, 3, and 4 (demonstrated via R094, R062, and R190 in 3-input/1-output logic rules); (ii) the remainder of 0 and 1 (R132); (iii) powers of 2 (R129); (iv) ceiling function for n/2 and n/4 (R152 and R144); and (v) analogous pattern of annihilation (R184) to DNA algorithmic patterns formed by specific rules. After designing the abstract building blocks and simulating the generation of algorithmic lattices, we conducted an experiment as follows: designing of DNA tiles with specific sticky ends, construction of DNA lattices via a two-step annealing method, and verification of expected algorithmic patterns on a given DNA lattice using an atomic force microscope (AFM). We observed representative patterns, such as horizontal and diagonal stripes and embedded triangles, on the given algorithmic lattices. The average error rates of individual rules are in the range of 8.8% (R184) to 11.9% (R062), and the average error rate for all the rules was 10.6%. Interpretation of elementary functions demonstrated through DNA algorithmic patterns could be extended to more complicated functions, which may lead to new insights for achieving the final answers of functions with experimentally obtained patterns.