# Simulation

## Bistable System

Using the bistable system, you can realize two different states, state "only A" and state "only B". State "only A" has many staple strands of type A and few of type B. In contrast, "only B" has few staple strands of type A and many of type B.

Please note that we'll represent state "only X" as "X", and staple strands of type X as X.

How do concentrations of A and B change? In this section, we present the results of the simulations.

### Bistable Simulation Without "Switch"

$\frac{d[A]}{dt} = \frac{K_{1}[A]}{K+c[A]+c'[iA]}-K_{d}[A]$
$\frac{d[B]}{dt} = \frac{K_{1}[B]}{K+c[B]+c'[iB]}-K_{d}[B]$
$\frac{d[iA]}{dt} = \frac{K_{2}[B]}{K'+[B]}-K_{d}[iA]$
$\frac{d[iB]}{dt} = \frac{K_{2}[A]}{K'+[A]}-K_{d}[iB]$
$K_{1}=K_{2}=1 \quad K_{d}=0.1 \quad K=K'=1 \quad c=1 \quad c'=2$
$[A]_{0}=3.0 \quad [B]_{0}=2.9 \quad [iA]_{0}=0 \quad [iB]_{0}=0$

As you can see, even with only a small difference between the initial concentrations of A and B, the appropriate state is reached.

#### How to Set up Equations

The diagram below shows all of the reactions. However, as we explain later, this is a simplified model. For an introduction about these reactions, please refer to Idea section.

$A\xrightarrow{T_{A}}2A \quad A\xrightarrow{T_{iA}}A+iB$
$B\xrightarrow{T_{B}}2B \quad B\xrightarrow{T_{iB}}B+iA$
$A \to 0 \quad iA \to 0$
$B \to 0 \quad iB \to 0$

For example, left of the top reaction formula shows that A is doubled by template TA. Please note that A, B, iA, and iB naturally decrease because exonuclease gradually decomposes these staple strands.

To set up the equations, let's have a closer look at some reactions.

First of all, let's consider the reactions among A, iA and TA

TA has three states.

 TA :normal state ATA :TA hybridizing with A iATA :TA hybridizing with iA

The sum is constant.

TA + ATA + iATA = n

We considered A increases in proportion to $\frac{AT_{A}}{n}$

$A+T_{A} \rightleftarrows AT_{A} \to 2A+T_{A}$

$iA+T_{A} \rightleftarrows iAT_{A}$

[TA] + [ATA] + [iTA] = n

$\frac{[A][T_{A}]}{[AT_{A}]}=L_{1} \quad \frac{[iA][T_{A}]}{[iAT_{A}]}=L_{2}$

The conclusion below is derived from these equations.

$[AT_{A}]=\frac{n[A]}{L_{1}+\frac{L_{1}}{L_{2}}[iA]+[A]}$

$\frac{d[A]}{dt} = \frac{K_{1}[A]}{K+c[A]+c'[iA]}-K_{d}[A]$

This is the equation we showed at the top of this subsection. The other equations can be obtained in the same way.

### Bistable Simulation with "Switch"

The bistable system magnifies even small differences of concentration. If you want to change the state from "A" to "B", all you have to do is to add more B than A when the state is in equilibrium. But, in this case, there are also many iB, so you have to add enough amount of B.

This graph shows such operations at t=400 and t=800.

## The Change between Two Pictures

The next simulation is about the process of changing the surface of the DNA tablet between the two states; "A" and "B". Let's note a hybridized strand as "1", and a non-hybridized strand strand as "0". On the surface of the tablet, two types of reaction occur; 1→0 and 0→1. Because denaturation occurs at a constant rate, the event probability of "1 to 0" does not change, while that of "0 to 1", i.e. hybridization, is proportional to the concentration of the complementary strands existing around the tablet.

There are two simulations below: one is very simple, with a small tablet, only 5×5 pixels, and the other is a larger one, 9×12. In the simulation movies, the blue points represent the hybridized strands "1". In these simulations, the concentration curves of every element are the same to those in the simulation of the bistable system which we solved in the last section.

### 5x5 Origami Simulation

Please watch the simulation video carefully.

As you can see from the movie above, the tablet shows two pictures, because the bistable system change the state from "A" to "B" and from "B" to "A".

### 9x12 Origami Simulation

This is the simulation of the DNA tablet with the actual design. One picture is "Tablet Boy" and the other is "I love DNA".

## DNA Tablet with N-stable System or N-oscillator System

### N-stable System Simulation

• Tristable System

In this system, we can control the state by the input chemicals. For example, we can change state from "A" to "B" by putting enough amount of B and also from "A" to "C" by putting enough amount of C. If we combine the system with the modified origami, we can make the tablet which shows three pictures and change them responding to the surroundings.

• Quintistable System

This is also the similar system as the tristable one. For example, we can change state from "A" to each of four other states by putting enough amount of each DNA. If we combine the system with the modified origami, we can make the tablet which shows five pictures and change them responding to the surroundings.

Therefore, theoretically, we can make the tablet which shows n pictures responding to the surroundings. This result of the simulation confirms that we can make the DNA tablet which show n pictures.

### N-oscillator System Simulation

• Trioscillator System

In this system, the state continues to change as the circuit of "A" -> "B" ->"C" ->"A" and so on. Therefore, if we combine this system with the modified origami, the table can show a movie composed by three pictures.

In this simulation movie, there are three pictures appearing and disappearing in line; one which shows vertical line in the left side, one which shows the same line in the middle and one which shows it in the right side. Thanks to the trioscillator system, the concentration change of the simulation above is applied so that the state changes as the circuit "A" -> "B" -> "C" -> "A" and so on. This simulation video confirms that we can make the DNA tablet which shows a short and simple movie by combining the trioscillator system and modified origami.

• Quintioscillator System

In this system, the state continues to change as the circuit of "A" -> "B" ->"C" ->"D" ->"E" -> "A" and so on. Therefore, if we combine this system with the modified origami, the table can show a movie composed by four pictures. We can also design Quintioscillator system so that we can make the tablet which show a movie composed by four pictures.

In this simulation movie, the tablet shows five pictures; three horizontal lines, a square, three vertical lines, X, and nine dots. As you can see from the movie, the tablet can also show a movie composed by five pictures.

Therefore, theoretically, we can make the tablet which shows a movie composed by n pictures. This result of the simulation confirms that we can make the DNA tablet completely.