Platform Color Designs for Interactive Molecular Arrangements

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1 Platform Color Designs for Interactive Molecular Arrangements Jasper Braun, Daniel Cruz, and Nataša Jonoska University of South Florida June 5, / 23

2 Overview 1 Simulation of Cellular Automata 2 Directions of Communication 3 Checkerboard Coloring and Communication 2 / 23

3 Experimental Foundation Only tiles of one species (color) can attach at a time Jonoska, N. and Seeman, N.C. Molecular ping-pong Game of Life on a two-dimensional DNA origami array. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 373 (2046), (2015) 3 / 23

4 Binary Colorings Definition Let C be a finite set of order 2; a mapping ϕ : V C is a binary coloring of V by C. In this presentation, we set C = {0, 1}. 4 / 23

5 Systems of Interactive Molecular Arrangements Definition A system of interactive molecular arrangement over Z 2 (SIMA) is a four-tuple S = (ϕ, Σ, Φ B, Φ W ) where ϕ - binary coloring of Z 2 Σ - finite set of states Φ B, Φ W - local functions. ϕ = 5 / 23

6 Arrangement Definition An arrangement of a SIMA (ϕ, Σ, Φ B, Φ W ) is a map σ C of either black points or white points in Z 2 to Σ. 6 / 23

7 Computation Definition An arrangement of a SIMA (ϕ, Σ, Φ B, Φ W ) is a map σ C of either black points or white points in Z 2 to Σ. 7 / 23

8 Encoding Cellular Automata 8 / 23

9 Encoding Cellular Automata 9 / 23

10 Simulating Cellular Automata Definition A SIMA S = (ϕ, Σ, φ B, φ W ) simulates a cellular automaton C if for any configuration α : Z d A of C σb 0 which encodes α {σ mn C } subsequence of σ0 B, σ0 W, σ1 B,... where σmn C (α) for n 1. G n l encodes 10 / 23

11 Simulating Cellular Automata Strict simulation of CA C by SIMA S. 11 / 23

12 Simulating Cellular Automata Strict simulation of CA C by SIMA S. Lemma For any one-dimensional CA C with a radius 1 neighborhood, there exists a SIMA S which simulates C strictly. Isokawa, T., et al. Universal totalistic asynchonous cellular automaton and its possible implementation by DNA. Lecture Notes in Computer Science 9726, (2016) 12 / 23

13 Communication In a SIMA, only oppositely colored points can interact or communicate. Communication across multiple points requires a path in which each two consecutive points are of opposite color. 13 / 23

14 ϕ-alternating Paths Definition Let G = (V, E) be an undirected simple graph and let ϕ : V {0, 1} be a binary coloring. A path in G is ϕ-alternating if every two consecutive vertices in the path are oppositely colored. 14 / 23

15 Directional Communication Definition For a binary coloring ϕ of an infinite graph G and a vertex v V, let d be the maximum number of ϕ-alternating simple bi-infinite paths whose pairwise intersection is just {v}. If d 1, then ϕ provides communication in d directions at v. 15 / 23

16 1-Perfect Colorings Definition A binary coloring ϕ on an infinite regular graph G = (V, E) is 1-perfect or isotropic if the number of vertices adjacent to v V with color j {0, 1} only depends on the color of v. 1-perfect colorings have associated matrices White - 0; Black - 1 ( ) / 23

17 1-Perfect Colorings Axenovich showed that there are only nine unique matrices associated with 1-perfect colorings on Z 2. Corollary The checkerboard coloring is the only 1-perfect coloring of Z 2 which provides communication in 2 directions at all vertices. Axenovich, M.A. On multiple coverings of the infinite rectangular grid with balls of constant radius. Discrete Mathematics 268 (1-3), (2003) 17 / 23

18 Checkerboard Coloring Definition In a (possibly infinite) graph G = (V, E), a binary coloring ϕ is called a checkerboard coloring if every two adjacent vertices are oppositely colored. 18 / 23

19 Checkerboard Coloring Lemma Let ϕ be a binary coloring of a bipartite graph G. A path is ϕ-alternating if and only if some checkerboard coloring of G colors each vertex in the same manner. 19 / 23

20 Checkerboard Coloring Lemma Let ϕ be a binary coloring of a bipartite graph G. A path is ϕ-alternating if and only if some checkerboard coloring of G colors each vertex in the same manner. Corollary Let ϕ be a binary coloring of a bipartite graph G. If there exists a vertex v which can communicate with every other vertex in G, then ϕ is a checkerboard coloring. 20 / 23

21 Communication in Non-bipartite Graphs Theorem There are uncountably many binary colorings ϕ of the triangular grid T which provide communication in 3 directions at infinitely many, but not all, vertices and which allows any two vertices to communicate. 21 / 23

22 Proof Idea 22 / 23

23 Simulation of Cellular Automata Directions of Communication Checkerboard Coloring and Communication Acknowledgements This work has been supported in part by the NSF grant CCF and NIH grant R01 GM Special thanks to the USF Math/Bio Research Group. 23 / 23