A DNA algorithm-based logic gate provides an efficient platform for generating various patterns through self-assembly. Self-assembly algorithms using *M*-input *N*-output logic gates are easily implemented in DNA tiles. The patterns generated by a 3-input 1-output logic gate show interesting features, such as demonstrations of mathematical functions, physical phenomena observed in nature, and logic operators. We notice that among the 3-input 1-output logic rules, the algorithmic lattices generated by R30 show interesting Big Bang-like patterns. A pattern generated by R30 and specific initial values shows expanding characteristics during the growth of lattices that resemble the Big Bang expansion of the universe. In this study, we demonstrate Big Bang-like patterns using simulations generated by R30 and analyze pattern sizes as a function of growth step number. We discuss pattern sizes and pattern-size-expansion-speeds, both of which are heavily influenced by perturbed initial values. We examine eight different perturbed initial values that induce Big Bang-like patterns with the generation of multiple pattern sizes during the growth of patterns. In addition, we fabricate patterns using DNA algorithmic self-assembly generated by the R30 logic rule with a 3-input 1-output logic operation. The generated algorithmic patterns are visualized by an atomic force microscope. Our method allows the generation and analysis of naturally occurring patterns, such as those found on lizard skin and Zelkova serrata lenticel patterns.

## I. INTRODUCTION

DNA, which is known as a genetic information carrier in biological species, is a useful nanomaterial for developing physical and chemical science and engineering as well as biological and medical applications.^{1} In addition, DNA is an efficient molecule for generating complex algorithmic patterns through logic implementation in DNA base sequences and for storing huge amounts of information in DNA base sequences.^{2–6} In the last few decades, researchers have developed biological logic gates composed of DNA molecules followed by the construction of various algorithmic patterns on DNA lattices. Algorithmic assembly using *M*-input *N*-output logic gates is easily implemented in DNA tiles with carefully designed binding domains. Consequently, DNA tile-based algorithmic assembly provides an efficient design scheme of *M*-input *N*-output logic gates (where *M* = 1, 2, 3 and *N* = 1, 2), the formation of robust algorithmic patterns sequentially by well-defined rules, and the verification of algorithm patterns via an atomic force microscope (AFM). By using *M*-input *N*-output logic gates, we can demonstrate logic operators [such as NOT, COPY (with *M* = 1, *N* = 1), AND, OR, XOR (*M* = 2, *N* = 1), binary counter (*M* = 2, *N* = 2), line-like and triangle-like patterns (*M* = 3, *N* = 1), and full-adder (*M* = 3, *N* = 2)], mathematical functions [such as enumeration, remainder, power, and ceiling function (*M* = 3, *N* = 1)], and patterns shown in nature [e.g., lizard’s skin and Zelkova serrata lenticel (*M* = 3, *N* = 1)].^{7–23}

Patterns generated by a 3-input 1-output logic gate (one of the most versatile *M*-input *N*-output logic rules) have interesting properties, such as demonstrating mathematical functions [e.g., f(n) = 3n and f(m) = 2^{m}] and physical phenomena revealed in nature (e.g., matter–antimatter annihilation, e^{−} + e^{+} → γ) as well as logic operators.^{8} 3-input 1-output logic gates have a total of 256 (=2^{8}) rules (i.e., R000−R255), which provide distinct 256 patterns on DNA lattices. We notice that among 256 rules, algorithmic lattices generated by R30 (with specific initial values) show interesting Big Bang-like patterns on them. For R30, an 8-digit outcome with corresponding 3-digit inputs [i.e., 111_{(2)}, 110_{(2)}, 101_{(2)}, 100_{(2)}, 011_{(2)}, 010_{(2)}, 001_{(2)}, and 000_{(2)}] is determined as 00011110_{(2)} (here, rule number 30, i.e., R30, is obtained by using 2^{7} × 0 + 2^{6} × 0 + 2^{5} × 0 + 2^{4} × 1 + 2^{3} × 1 + 2^{2} × 1 + 2^{1} × 1 + 2^{0} × 0, which corresponds to the decimal number 30). A pattern generated by R30 with specific initial values shows evolving characteristics during the growth of lattices that resemble the Big Bang expansion of the universe. By analyzing R30 patterns, we can estimate the expansion speed of the Big Bang-like pattern size at a given step (roughly related to recessional speed *v* in Hubble’s law, i.e., *v* = *H* ⋅ *r*, where *H* and *r* stand for the Hubble constant and proper distance from the galaxy to the observer, respectively).

In this study, we demonstrate Big Bang-like patterns through simulations generated by R30 (implementing a 3-input 1-output logic gate) with various initial values and analyze pattern sizes as a function of growth step number. In addition, we fabricate patterns through DNA algorithmic self-assembly ruled by R30 and discuss the feasibility of constructing patterns shown in physical phenomena in nature.

## II. EXPERIMENTS

### A. Construction of DNA algorithmic lattices

We use high-performance liquid chromatography (HPLC)-purified synthetic oligonucleotides (Integrated DNA Technologies, IA, USA). By using the two-step annealing method, DNA lattices possessing Big Bang-like patterns are obtained.^{6} Individual RU (i.e., RU000, RU110, RU011, and RU101) and OP (i.e., OP0011, OP0101, OP1000, and OP1100) tiles are obtained by mixing a stoichiometric quantity of each strand in buffer, 1 × TAE/Mg^{2+} [Tris-acetate-EDTA: 40 mM Tris, 1 mM EDTA (pH 8.0), 12.5 mM magnesium acetate]. A final concentration of 1 μM is achieved. For the first-step annealing (a phase changed from single strands to tiles through facilitating the hybridization process), the sample test tubes are placed in a Styrofoam box with 2 L of boiling water followed by cooling from 95 to 25 °C over a period of at least 24 h. For the second-step annealing (phase changed from tiles to lattices through the hybridization process), equal amounts of annealed four RU and four OP tiles are mixed together in a new test tube. A final tile concentration of 100 nM is obtained. The sample test tube is cooled from 40 to 25 °C over a period of at least 12 h by placing it in 2 L of water in a Styrofoam box to facilitate further hybridization.

### B. AFM imaging

AFM measurement is performed in the Scanasyst fluid mode of the Digital Instruments Nanoscope V8 (Bruker, MA, USA). Five μL of annealed DNA lattices were spotted onto a mica substrate (with a size of 5 × 5 mm^{2}) followed by incubation for 30 s. Additionally, 35 *µ*l of 1 × TAE/Mg^{2+} buffer is dispensed on mica. Another 20 *µ*l of 1 × TAE/Mg^{2+} buffer is pipetted to dispense on the oxide-sharpened silicon nitride AFM tip (Bruker, MA, USA).

## III. RESULTS AND DISCUSSION

The truth table and schematic representation of the 3-input 1-output logic gate for R30 are shown in the first and second columns in Fig. 1. Eight different 3-digit inputs [from 000_{(2)} to 111_{(2)}] in 3-input 1-output logic gates provide eight possible outcomes, which make an 8-digit binary number, N_{1}N_{2}N_{3}⋯N_{8(2)} (where N_{i} = 0 or 1). The following formula can be used to calculate the rule number: 2^{7} × N_{1} + 2^{6} × N_{2} + 2^{5} × N_{3} + 2^{4} × N_{4} + 2^{3} × N_{5} + 2^{2} × N_{6} + 2^{1} × N_{7} + 2^{0} × N_{8}. For instance, the 8-digit number 00011110_{(2)} corresponds to the decimal number 30 (which is rule number 30, i.e., R30 obtained through 2^{7} × 0 + 2^{6} × 0 + 2^{5} × 0 + 2^{4} × 1 + 2^{3} × 1 + 2^{2} × 1 + 2^{1} × 1 + 2^{0} × 0). For implementing the 3-input 1-output logic gate, two different types of building blocks [i.e., rule (RU) and operator (OP) tiles] are introduced. RU tiles possess either 0-bit or 1-bit information (marked as pink and magenta, respectively), and OP tiles (placed between RUs; shown as gray) deliver information bits obtained from previous RUs to the next RUs.

A simulation pattern (cell size of 128 × 140) generated by R30 is displayed in Fig. 1. This pattern shows a Big Bang-like pattern (embedded with various sizes of triangles, which are created and diffused during the growth of lattices) at a given disturbed initial value (which is placed on the first layer). A lattice with an undisturbed initial value (of repeated 10, i.e., ⋯ 101010 ⋯) ruled by R30 generates no disturbed pattern on a lattice, which serves as a background. It means that identical bit information is copied from one layer to the next (because 010_{(2)} and 101_{(2)} inputs in R30 produce 1_{(2)} and 0_{(2)} outputs, respectively). Consequently, by applying a minute disturbance (e.g., 1000) to an initial value of ⋯ 10101 1000 10101 ⋯ (shown in a zoomed-in image with a cell size of 5 × 10), we create a Big Bang-like pattern-embedded lattice, which clearly differentiates from the background.

A Big Bang-like pattern generated by R30 and analysis of pattern sizes as a function of growth step number are shown in Fig. 2. A Big Bang-like pattern (cell size of 128 × 100 cells) and a magnified pattern (cell size of 6 × 15) generated by R30-implemented computer simulation are shown in Figs. 2(a) and 2(b). We can obtain detailed pattern sizes (S_{n}) (obtained by counting the number of cells) at a given growth step number (n), which is heavily dependent on perturbed initial values. In order to clarify bit information in RU tiles within a Big Bang-like pattern on a lattice, we mark 0-bit and 1-bit information as yellow and brown (instead of pink and magenta), respectively. By comparison with unperturbed values (i.e., ⋯ 10101010 ⋯), a domain of perturbed initial values can be easily determined. Within the black and red boxes, a row with perturbed initial values and a grown Big Bang-like pattern are placed, respectively. The initial step number (i.e., n = 1) with corresponding Big Bang-like pattern size (S_{n} = 4) is marked with black in the table, and the growth step numbers with corresponding pattern sizes generated with given perturbed initial values (i.e., 10101 1000 101010 ⋯) are marked with red. In Fig. 2(c), we display the graph of the pattern size as a function of step number, i.e., S_{n}(n). Here, a linear slope obtained by linear fitting of the original curve having minute fluctuation indicates the dimensionless pattern-size-expansion-speed (V_{E} of 0.70), which is defined by ΔS_{n}/Δn (where ΔS_{n} indicates the fluctuation of the pattern size, which is varied with n). Consequently, we anticipate a larger V_{E} when the difference of S_{n} at a given unit n is drastically larger.

An analysis of Big Bang-like patterns generated with various perturbed initial values and Big Bang-like pattern sizes as a function of growth step number is shown in Fig. 3. In order to create Big Bang-like patterns, unperturbed initial values of repeating 10 (i.e., ⋯ 10101010 ⋯) have to be perturbed. Eight different perturbed initial values (P1–P8), which contain No. of 0 s in perturbed domains in P# with corresponding sizes (S_{1}) and Big Bang-like patterns (cell size of 128 × 256), are shown in Figs. 3(a) and 3(b). With unperturbed initial values, i.e., P0, S_{1} has to be zero, which means that no patterns can be generated. In contrast, perturbed initial values, such as P1 (with S_{1} = 2) and P5 (with S_{1} = 6), provide the initiation of Big Bang-like patterns with the generation of different S_{n} during the growth of patterns. The graph of S_{n}(n) with initial values of P1–P8 is shown in Fig. 3(c). Although S_{n} increases as n increases, the degrees of fluctuation (ΔS_{n}) vary with perturbed initial values. We adopt a linear fitting in two different ways, i.e., the original curves without (Fitting 1) and with (Fitting 2) adjusting S_{n} = 0 at n = 0. The linear fitting graphs of S_{n}(n) are shown in Fig. 3(d). Based on them, pattern-size-expansion-speeds (V_{E}) (slopes of fitted lines) are calculated based on them [shown in Fig. 3(e)]. Among eight different perturbed initial values, the maximum and minimum values of V_{E} for both fittings are obtained by perturbing the initial values of P1 and P6, respectively. This happened due to the characteristics of R30. Due to the larger values of S_{1} in P6, P7, and P8 among the perturbed initial values, larger differences in V_{E} between Fitting 1 and Fitting 2 are noticed.

We introduce rule (RU) and operator (OP) tiles made of DNA double-crossover (DX) building blocks to show the DNA lattices formed by R30 [shown in Fig. 4(a)].^{6} A rectangular-shaped DX building block (which has dimensions of 12.6 × 4.0 nm^{2}) comprises two duplexes connected through two crossover junctions. Here, RU tiles deliver binary-bit-information (either 0 or 1), possessing input and output binding domains, to OP tiles, which are placed between RU tiles. In order to implement three inputs in DX building blocks, OP tiles are required. Core sequences of RU tiles without hairpins (i.e., possessing 0-bit information) and with hairpins (1-bit) are indicated in RUXX0 and RUXX1, respectively. XX and 0 (or 1) in the name of the RU tile indicate 2-input (XX) and 1-output 0 (or 1) bit information, respectively. Sticky-end sequences of RU tiles are placed in RU000 and RU110 for RUXX0, and RU011 and RU101 for RUXX1. Similarly, core sequences of OP tiles are indicated in OPXXXX (XX on the left and right indicates 2-input and 2-output bit information) and sticky-end sequences of OP tiles are placed in OP0011, OP0101, OP1000, and OP1100. In this study, outputs in RU and OP are determined by a given rule, i.e., R30. A table of sticky-end sequences with corresponding sticky-end names is shown in Fig. 4(a). Here, unprimed and primed sticky-end names complement each other. For instance, the 5 nt-long output sticky-end of s3 (i.e., ACGTC; 5′ → 3′) in RU000 binds to the input sticky-end of s3′ (i.e., TGCAG; 3′ → 5′) in either OP0011, OP0101, or OP1000.

We present DNA lattices with embedded triangle-shape patterns constructed by a 3-input 1-output logic rule, R30 [Figs. 4(b) and 4(c)]. An example of the lattice built to the 3rd step using given initial values of 11010 placed in the first row is shown in Fig. 4(b). Three RU tiles for input, a single RU tile for output, and two OP tiles are required to connect a 3-input to a single one-output. For instance, inputs of 110 and outputs of 1 require three RU tiles (RU011, RU101, and RU110) for input, a single RU tile (RU101) for output, and two OP tiles (i.e., OP1100 and OP0101) (marked in yellow). Representative AFM images of DNA lattices with various scan sizes are shown in Fig. 4(c). The overlaid guidelines in AFM images are embedded to aid understanding of algorithmic pattern formation. Here, RU tiles with hairpins are marked with blue dots. Errors are marked with yellow crosses. An average error rate is ∼2.5% [which is calculated as (total number of errors/total number of cells) × 100]. Our DNA lattices formed by R30 show patterns with embedded triangles and no background. It means that there are certain limitations on measuring the size of a pattern at a given growth step number. A robust boundary structure with well-defined initial values has to be designed and constructed to study DNA lattices possessing Big Bang-like patterns (which can be done in the near future).

## IV. CONCLUSIONS

In conclusion, we show Big Bang-like patterns generated by simulations of R30 and analyze pattern sizes as a function of growth step number. We discuss pattern sizes and pattern-size-expansion-speed, which are affected by perturbed initial values. Eight different perturbed initial values that provide the initiation of Big Bang-like patterns with the generation of different pattern sizes during the growth of patterns were examined. In addition, we fabricate patterns using DNA algorithmic self-assembly, which is generated by a 3-input 1-output logic rule, R30. The generated algorithmic patterns are visualized by an AFM. On average, an error rate of ∼2.5% is obtained. The DNA algorithm gives an effective way to construct logical patterns including patterns existing in nature. Consequently, our method might provide the clue of understanding of certain physical phenomena metaphorically by implementing certain rules and analyzing occurring patterns.

## ACKNOWLEDGMENTS

This work was supported by the Postdoctoral Research Program of Sungkyunkwan University (2022).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.