Calculation of π and Classification of Self-avoiding Lattices via DNA Configuration

Numerical simulation (e.g. Monte Carlo simulation) is an efficient computational algorithm establishing an integral part in science to understand complex physical and biological phenomena related with stochastic problems. Aside from the typical numerical simulation applications, studies calculating numerical constants in mathematics, and estimation of growth behavior via a non-conventional self-assembly in connection with DNA nanotechnology, open a novel perspective to DNA related to computational physics. Here, a method to calculate the numerical value of π, and way to evaluate possible paths of self-avoiding walk with the aid of Monte Carlo simulation, are addressed. Additionally, experimentally obtained variation of the π as functions of DNA concentration and the total number of trials, and the behaviour of self-avoiding random DNA lattice growth evaluated through number of growth steps, are discussed. From observing experimental calculations of π (πexp) obtained by double crossover DNA lattices and DNA rings, fluctuation of πexp tends to decrease as either DNA concentration or the number of trials increases. Based upon experimental data of self-avoiding random lattices grown by the three-point star DNA motifs, various lattice configurations are examined and analyzed. This new kind of study inculcates a novel perspective for DNA nanostructures related to computational physics and provides clues to solve analytically intractable problems.

A multitude of analytically intractable problems in various disciplines are addressed by performing numerical simulations that employ a computational model of a system to describe its complex behaviour over a time period by incorporating given variables. One such commonly used model is Monte Carlo (MC) simulation 1 that refers to an effective computational algorithm adopted to perform an underlying stochastic and random sampling experiment on a computer to calculate various outcomes. MC simulation is used in science and engineering to understand complex physical phenomena, generate useful mathematical functions, and predict complicated algorithmic processes. Interestingly, the MC method has also been effectively used to understand complex biological process mechanisms such as the biological self-assembly behaviour, biomolecule dynamics, and the interaction between biomolecules and nanomaterials  .
Among typical MC simulation applications, there are two interesting ones; calculating π (one of most important mathematical constants defined as the ratio of a circle's circumference to its diameter), and interpreting a self-avoiding walk (an abstract model describing the behaviour of chain like entities where no two points can occupy the same place) 24 . Several approaches have been adapted to calculate π, among which the famously used one is Buffon's needle approach 25 . The MC method is also used to enumerate the characteristics of the self-avoiding walk, to interpret the possibility to estimate proper paths.
The fabrication of various dimensional DNA nanostructures is well established due to the programmability of DNA base sequences and the stability of DNA molecules. Although these artificially designed DNA nanostructures find various applications as physical, chemical, or biomedical devices and sensors [26][27][28][29][30][31][32] , calculating mathematical constants and incorporating abstract modeling via DNA nanostructures are rarely discussed.

Results
Calculation of π value. The representative schematics for π calculation with a different number of dots in a square having a quadrant of a circle are shown in Fig. 1a. For acquiring an calculated numerical value of π (=π est , where est stands for estimation), a random event needs to be considered which can be defined as drawing uniformly distributed dots (like throwing darts randomly at a board) over a square bounding box within the region Figure 1. Calculation of π using Monte Carlo simulation. (a) The representative schematics for π calculation with a different number of dots in a square. Calculated numerical value of π (=π est , where est stands for estimation) is defined as (N D-in /N D ) × 4, where N D-in and N D stand for the number of dots inside a quadrant of a circle (with a radius of R) and the total number of dots in a square (with a length of R). By definition, π est with four different N D , i.e. 10, 50, 100, and 1000 are calculated to be 2.40 (=6/10 × 4), 2.72, 2.88 and 3.09 respectively, showing that roughly larger N D gives a more accurate known value of π (π known ≈ 3.14). (b) A flow chart depicting algorithmic steps to obtain π est with various N D and total number of trials (n T ). (c) π est as a function of n T at a given N D (e.g. 10, 50, 100, or 1000). In general, π est approaches to π known with the increasing n T at relatively larger N D values, as expected. (d) π est as a function of N D at a fixed n T (e.g. 1, 5, 10, 50, or 100 marked as a dotted line in (c). From observation, π est approaches to π known with the increasing N D at relatively smaller n T but π est is roughly independent with N D at relatively larger n T . (e) A representative graph of π est as a function of N D . As N D is increased, fluctuation of π est from π known tends to decrease. Insets show tendencies of fluctuation of π est in the two different ranges of N D .
Scientific RepoRts | (2019) 9:2252 | https://doi.org/10.1038/s41598-019-38699-0 whose area is to be determined. By considering a quadrant of a circle with a radius R bounded by a square with a length R, the ratio of the quadrant area to the square area is approximately equal to the ratio of the total number of dots falling inside the quadrant (N D-in , marked as blue) to the total number of dots inside the square (N D ) due to the uniformly distributed dots within the square. Therefore, π est can be defined as (N D-in /N D ) × 4. By definition, representative π est with four different N D (i.e. 10, 50, 100, and 1000) are calculated to be 2.40 (=6/10 × 4), 2.72, 2.88 and 3.09 respectively. This shows that a roughly larger N D gives a relatively more accurate known value of π (π known ≈ 3.14). Consequently, the magnitude (i.e. 0.060 = |3.2-3.14|, 0.020, 0.019, and 0.001) of the deviation of π est from π known (∆π exp = |π est − π known |) will be smaller as N D (10, 50, 100, and 1000) increases at a given optimum N D-in (i.e. 8, 39, 79, and 785, which provides the most accurate π est compared to π known at a given N D ). Figure 1b shows a flowchart representing algorithmic steps in order to obtain π est as a function of either N D, or the total number of trials (n T ). By assigning an initial input of N D with the unit-step increment of j, dots are randomly sampled in a square. Then, N D-in are counted until j reaches to N D followed by evaluation of π est calculated as (N D-in /N D ) × 4. Similarly, when the unit-step increment of i reaches an initial input of n T , summed π est is divided by n T to get the average π est .
By using the algorithm for π est , numerical values of the π est as functions of N D and n T can be obtained and analyzed. π est as a function of n T at four different N D values (i.e. 10, 50, 100, and 1000) are obtained, which approaches π known with the increasing n T at any given N D values, as expected (Fig. 1c). The π est with varying N D at a fixed n T (e.g. 1, 5, 10, 50, or 100 marked as a dotted line in Fig. 1c) are extracted in order to evaluate the trend of π est as a function of N D which shows that π est heavily relies on N D at relatively smaller n T but it is roughly independent of N D at larger n T (Fig. 1d). A representative graph of π est as a function of N D is shown in Fig. 1e. As N D is increased, the fluctuation of π est from π known tends to decrease. Insets show the fluctuation tendency of π est in the two different ranges of N D (i.e. between 0~20 and 980~1000), which clearly shows that fluctuation of π est from π known tends to decrease with the increase in N D , as expected. In addition, the differentiation of π est per unit number of dots (=Δπ est /ΔN D ) as a function of N D is shown in Supplementary Fig. 1. Differences in the π est per unit number of dots tend to decrease with the increase in N D because π est at a relatively larger N D has a greater chance to give an accurate value of π.
Experimental observation of π using DNA nanostructures. Experimental observation of π (π exp ) is demonstrated by constructing two types of DNA nanostructures, i.e. double crossover (DX) DNA lattices 33,34 and DNA rings [35][36][37] (Fig. 2). Two sets of DX DNA motifs (i.e. PR and PS) are designed for construction of DX DNA lattices. Here, P stands for Pi (π) and R/S indicate opposite helical directionalities of the duplexes within the motifs (See Supplementary Fig. 2, Supplementary Tables 1 and 2). Each set has two DX motifs, without and with hairpins marked as PR(S)0 and PR(S)1, respectively (Fig. 2a). A DX motif having hairpins ~3.5 nm long protruding up and down is called DXH (i.e. PR1 and PS1). DX and DXH motifs, having identical sets of sticky ends in each set with the equal probability of binding (two exemplified binding sites are indicated by question marks in Fig. 2b), can hybridize to form a DX lattice with the aid of complementary colour-coded and shape-coded sticky ends. In addition, DNA rings comprised of T motifs (non-crossover based DNA motifs having three double-stranded domains connected through single strands. See Fig. 2c, Supplementary Fig. 3, and Supplementary Tables 3 and 4) are fabricated in order to obtain π exp . A ring with inner and outer diameters of 13 nm and 29 nm is constituted through the complementary base-pairs of the sticky ends in T motifs (Fig. 2d).
Representative structural configurations of DX DNA lattices and DNA rings are shown in Fig. 2e,h, respectively. Atomic force microscope (AFM) images of DX lattices with different concentrations of DXH (0, 25, 50, 100, 150, and 200 nM symbolized as DXH 0 , DXH 0.25 , DXH 0.5 , DXH 1.0 , DXH 1.5 , and DXH 2.0 , respectively) were annealed in free solution. An arc (shown in blue) in each image is drawn representing the first quadrant in a circle. π exp (0.00, 3.26, 3.27, 3.50, 2.98, 3.32, and 3.13) through images in Fig Similarly, π exp and ∆π exp as functions of [T] and n T analysed from DNA rings are discussed. As observed from the bar graph of π exp in Fig. 3d, the standard deviation of an error bar roughly decreases as [T] increases and ∆π exp is approximately independent with [T], which might be due to the uniform distribution of the DNA rings on a Self-avoiding random lattice growth. A self-avoiding random walk path (called a lattice configuration) constructed by a unit building block is demonstrated via MC simulation in order to understand the feasibility to predict proper paths. A self-avoiding random lattice has a growth path on a lattice configuration that does not visit the same place more than once. Schematics of various lattice configurations constructed by a three-point star motif having single blunt-end (3PS B ) are represented in Fig. 4a. A blunt-end in a 3PS B , which is introduced to generate asymmetric self-avoiding random lattices, is marked with a black (serves as a seed), a red (grown to the left), or a green dot (grown to the right).   Supplementary Fig. 4. In order to predict applicable numbers of self-avoiding lattices, available lattice configurations at a given N S are analyzed. There are two types of available lattice configurations, i.e. an open, marked as a hollow circle and a blocked lattice configuration marked as either a half-filled (with red for left-blocked or green for right-blocked configurations) or a fully-filled circle as shown in Fig. 4a. Open, half-blocked, and full-blocked  Fig. 5a-d (Supplementary Figs 5 and 6). Two-dimensional self-avoiding random lattices are self-assembled through the subsequent 3PS B bindings to a seed tile of 3PS B , which has two binding sites, left and right leading the paths of the red and green, respectively. Here, open, half-blocked (growth blocked on either the left (a red path) or right (a green) side of the lattice), and full-blocked configurations are symbolized by a hollow, half-filled and fully-filled circle, respectively.  (Fig. 5f). As mentioned, D becomes 0 at N S of 9.12 and magnitude of D increases with increasing or decreasing N S from the cross point at N S = 9.12.
Experimental observation of self-avoiding random lattices. Three different DNA nanostructures (a honeycomb lattice, a hexagonal ring, and a three-point star dimer) are constructed by slightly modified three-point star DNA motifs in order to test their applicability in the growth of self-avoiding random lattices (See Fig. 6, Supplementary Fig. 7, and Supplementary Table 5). Figure 6a shows a schematic of a three-point star DNA motif (3PS HL ) for construction of a honeycomb lattice (a simplified one shown at a right bottom) and its representative AFM image of a honeycomb lattice. A 3PS HL is comprised of 7 strands (marked as #1~#7) with palindromic self-complementary sticky-end sequences (indicated as S1, S2, and S3) located at the end of each arm 41,42 . Schematics and representative AFM images of three-point star DNA motifs with a single (3PS HR , for fabrication of a hexagonal ring) and double blunt ends (3PS D , for formation of a 3PS dimer) are shown in Fig. 6b and c. A 3PS HR (a black dot in simplified 3PS HR indicates a blunt end arm as shown in Fig. 6b) and a 3PS D (two black dots in simplified 3PS D represent the blunt end arms in Fig. 6c) need 6 strands (strand #7 removed from 3PS HL ) with two sets (S1 and S2) of palindromic self-complementary sticky-end sequences, and 5 strands (#6 and #7 removed from 3PS HL ) with a single set (S1) of palindromic self-complementary sticky-end sequences, respectively. From the observation of the AFM images, honeycomb lattices, hexagonal rings, and 3PS dimers are well formed in agreement with the design schemes with relatively higher production yields than cross-tile lattices made of four-point star motifs 43 . Figure 6d-s show the representative experimental results and analysis of self-avoiding random lattices grown by the 3PS DNA motifs (3PS B ). In 3PS B , a #6 strand from 3PS HL is removed and self-complementary sticky-end sequences in #7 are replaced from S3 to S1. A blunt-end in a simplified 3PS B shown in the right bottom of Fig. 6d is marked with either a black (served as a seed), a red (grown to the left), or a green dot (grown to the right) in order to easily evaluate the lattice configurations. Representative AFM images with the lattice configurations (either an open, a half-blocked or a full-blocked configuration at a given step number) of self-avoiding random lattices comprised of 3PS B are displayed in Fig. 6e-p. Simplified 3PS B motifs are overlaid on AFM images to enhance the visibility of lattice configurations. Figure 6r,s display percentages of the total number of 3PS B motifs (α) in specific ranges (i.e. below 10, 11-20, 21-30, and above 30) and percentages of total number of open, half-blocked and full-blocked lattice configurations (β) obtained from the AFM data. Although it would be difficult to form relatively larger self-avoiding lattices due to the existence of a blunt end in a 3PS B, as we anticipated, interestingly we observe that lattices having more than 31 numbers of 3PS B are dominant (38.5% among all evaluated lattices). In addition, the percentages of lattice configurations in the range of 3 to 49 of N S are examined. Open (blocked) lattice configurations are dominant below (above) N S = 9.12, which agree well with the simulation results discussed in Fig. 5e,f.

Discussion
We discuss methodologies to calculate the numerical value of π and to evaluate a possible number of self-avoiding walk paths with the aid of computational MC simulation. Additionally, we demonstrate the calculation of π and evaluation of applicable self-avoiding walk paths by distinct DNA nanostructures. Finally, we analyze the trend of numerical variations of π as functions of DNA concentration and the total number of trials for π calculation, and the behaviour of self-avoiding random DNA lattice growth evaluated through number of growth steps for the self-avoiding walk path. From observation of experimental calculations of π (π exp ) demonstrated by constructing two different types of DNA nanostructures (i.e. double crossover DNA lattices and DNA rings), fluctuation of π exp from known π tends to decrease as either DNA concentration or the number of trials increases. Based upon experimental observation of self-avoiding random lattices grown by the three-point star DNA motifs, the percentage of lattice configurations is examined. Open (blocked) lattice configurations are dominant below (above) the step number of 9.12 (at this step number obtained by simulation, numbers of open and blocked configurations are the same). This in depth study of numerical calculation of mathematical constants and characteristic estimation of abstract models via DNA provides a novel perspective for the applicability of DNA in the field of science and engineering. Figure 6. Experimental observation of self-avoiding random lattice growth with the three-point star DNA motif. (a) A schematic of a three-point star DNA motif (3PS HL ) for construction of a honeycomb lattice and its representative AFM image (scan size of 500 × 500 nm 2 ) of a honeycomb lattice. Seven strands constituting 3PS HL are numbered as #1~#7, where palindromic self-complementary sticky-end sequences located at the end of each arm are indicated as S1, S2, and S3. A simplified 3PS HL and a magnified honeycomb lattice (100 × 100 nm 2 ) are shown at the right bottom corners of them. (b) A schematic of a three-point star DNA motif with a single blunt end (3PS HR ) for fabrication of a hexagonal ring and its AFM image. Six strands (strand #7 removed from 3PS HL ) and two sets (S1 and S2) of palindromic self-complementary sticky-end sequences are required. A black dot in simplified 3PS HR indicates a blunt end arm. Inset in AFM image is 3-dimensional visualization of a hexagonal ring. (c) A schematic of a three-point star DNA motif with double blunt ends (3PS D ) for formation of a 3PS dimer and its AFM image. Five strands (#6 and #7 removed from 3PS HL ) and single set (S1) of palindromic self-complementary sticky-end sequences is required. Inset in AFM image is 3-dimensional visualization of 3PS dimers. (d) A schematic of a three-point star DNA motif with a blunt end (3PS B ) for demonstration of a self-avoiding random lattice. Strand #6 is removed from 3PS HL and self-complementary sticky-end sequences in #7 are modified. A blunt-end in a simplified 3PS B is marked with a black (served as a seed), a red (grown to the left), or a green dot (grown to the right) in order to easily analyze the lattice configurations.  = 75, 100, 25, and 0 nM; 50, 100, 50, and 0 nM; 25, 100, 75, and 0 nM; 50, 50, 50, and 50 nM; 0, 50, 100, and 50 nM; 0, 0, 100, and 100 nM) were prepared to construct DXH 0.25 , DXH 0.5 , DXH 0.75 , DXH 1.0 , DXH 1.5 , and DXH 2.0 DNA lattices, respectively. Second step annealing was performed by placing sample test tubes in a Styrofoam box containing 2 L of water (initial temperature, 40 °C) and cooling them from 40 °C to 25 °C for about 24 hours to obtain DX DNA lattices. (Fig. 2, Supplementary  Fig. 2, Supplementary Tables 1 and 2) DNA rings were formed by mixing a stoichiometric quantity of each strand in a buffer containing a mica substrate (size of 5 × 5 mm 2 ). This strand mixture with mica was annealed in a test tube by slowly cooling from 95 to 25 °C in a Styrofoam box. Eventually, DNA rings formed on the mica surface with different coverages depending upon the concentration of a T motif. DNA rings with a five different T motif concentrations of 2, 5, 8, 10 and 20 nM were prepared and analyzed. (Fig. 2, Supplementary Fig. 3, Supplementary Tables 3 and 4) Honeycomb lattices, hexagonal rings, 3PS dimers, as well as self-avoiding random lattices were constructed by specific three-point star motifs; 3PS HL , 3PS HR , 3PS D , and 3PS B motifs. They were formed by mixing stoichiometric quantities of each strand in the buffer by cooling from 95 °C to 25 °C in a Styrofoam box. Final concentrations of 3PS for all DNA nanostructure configurations were 200 nM. (Fig. 6, Supplementary Fig. 7, Supplementary Table 5) AFM imaging. 5 μL of DNA nanostructures (i.e. DX lattices, honeycomb lattices, hexagonal rings, 3PS dimers, and self-avoiding random lattices) in buffer solution prepared via the free-solution annealing method were dropped on a freshly cleaved mica surface. A 30 μL of 1 × TAE/Mg 2+ buffer solution was then placed onto the mica, and another 20 μL was placed onto the silicon nitride AFM tip (NP-S10, Veeco Inc., CA, USA). To image DNA rings fabricated through the MAG method, a mica substrate with preformed DNA rings was taken from a test tube and placed on a metal puck. Then, 30 μL of buffer was pipetted onto the mica substrate, and another 20 μL was dispensed onto an AFM tip. Corresponding AFM images were then obtained using a Multimode Nanoscope (Veeco Inc., CA, USA) in the fluid-tapping mode (Figs 2 and 6).