# Team:UT-Tokyo/Sudoku modeling

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UT-Tokyo

# Sudoku

## 4C3leak switch

We modeled our 4C3 leak switch as deterministic ODE system.

### Modelling

#### Assumption

1.We treated DNA as continuous variable, since there could be hundreds of plasmid in one E.coli (multicopy plasmid). There are 32 kinds of variables that stand for DNA concentration corresponding to their different internal states.

2.We assumed that cre recombinase operate as tetramer, and other recombinase as dimer (but this seems not to be relevant).

3.We assumed that association and dissociation of DNA recombinase to DNA is sufficiently fast so that equilibrate in the timescale of the whole switch.

4.We ignored the reverse reaction of DNA recombinase because we used irreversible sequence (lox66, lox71 etc).

#### Variable

Basically, "pn" denotes protein concentration and "rn" denotes mRNA concentration

p1,p2,p3,p4 : concentration of 4 recombinase protein

r1,r2,r3,r4 : concentration of 4 recombinass mRNA

u1,u2,u3,u4 : concentration of input mRNA

pc,rc : concentration of cre recombinase protein and mRNA

ps,rs : concentration of SP6 polymerase protein and mRNA

Variables that denote concentration of DNA in each state. "a","b","c","d","e" means there are four,three,two,one,zero recombinases in the DNA, respectively. And the subscripts specify which recobinases remain, while the numbers in bracket mean whether the DNA has been cut by cre recombinase( (1) has not been cut, (2) has been cut).

 all 123 124 134 234 not cut by cre a(1) b123(1) b124(1) b134(1) b234(1) cut by cre a(2) b123(2) b124(2) b134(2) b234(2)

 12 13 14 23 24 34 1 2 3 4 none c12(1) c13(1) c14(1) c23(1) c24(1) c34(1) d1(1) d2(1) d3(1) d4(1) e(1) c12(2) c13(2) c14(2) c23(2) c24(2) c34(2) d1(2) d2(2) d3(2) d4(2) e(2)

#### Parameters

k0 : mRNA translation rate

k2 : mRNA translation rate

k3 : cre protein degradation rate

l0 : transcription speed per unit concentration of RNA polymearase

l2 : cre mRNA degradation rate

K : recombinase binding constant

v : recombinase reaction rate

pT7 : concentration o T7 polymerase

p : terminator leak probability

### Methods

We used our original python program (4 th order explicit Runge-Kutta algorithm) to solve thess equations.

### Results

We input signal 1,2,4 and simulated the time evolution of the system.

The time course of DNA concentration

The time course of mRNA concentration. Note that the output of this system is not protein but mRNA.

### Discussion

It is confirmed that the system works as expected with a certain combination of parameters.

In addition, we changed many parameters, especially,l0,v(whose values are unclear) and p(leakiness of terminator) , to see whether the device is able to operate within a large part of parameter space.

Although increasing p basically increases the concentration of correct output mRNA, it also increases incorrect mRNA. On the other hand, decreasing p generally decreases both the concentration of correct mRNA and incorrect mRNA. Besides, too low value of p cannot leads to failure operation because cre protein concentration cannot get high enough. Careful test have shown that there is certain value of p that [correct mRNA]/[incorrect mRNA] takes maximum value. In this case, the optimal value is approximately p = 0.001 and the ratio is 10^(-6).

Changing the value of v and l0 doesn't affect the function of the circuits qualitatively although it changed the time the systems takes to reach steady state.

As a result, the circuit can function correctly within the range of 0.001< p < 0.1, 0.001 < v <1.0, 0.01 < l0 < 1.0 , demonstrating its robustness. This robustness may come from the assumption that DNA recombination is irreversible, which is supported by our design of the system.

### conclusion

The correct operation of the switch was confirmed. By changing parameter exponentially, the robustness of the system was shown.

## An indifferent sudoku cell simulation

including MS2 phage infection-------
What's Sudoku?

We made computer simulations to confirm that an empty sudoku cell get signals from other cells and differentiate to the correct answer in 4*4 sudoku.

#### Modeling

Variable

U: uninfected coli

I: coli which got '1' signal

Io1: other once infected coli

I2: coli which got '12' signal

Io2: other second infected coli

I3: coli which got '123' signal

Io3: other third infected coli

B4: '4' coli which began phage making

Bo: other coli which began phage making

P4: free phage of '4'

Po: other free phage

S1: signal virus with information of number '1' which come from other sudoku cell

S2: signal virus with information of number '2' which come from other sudoku cell

S3: signal virus with information of number '3' which come from other sudoku cell

S4: signal virus with information of number '4' which come from other sudoku cell

Parameters

g: growth rate of coli

k1: rate of infection

k2: death rate of uninfected

k3: lysis rate of infected

k4: rate of phage producing

k5: phage death rate

k6: terminator leak rate

Given parameters are unit value

• g /min 0.0113

reference:Investigation of Bacteriophage MS2 Viral Dynamics Using Model Discrimination Analysis and the Implications for Phage Therapy Rishi Jain, Andrea L. Knorr, Joseph Bernacki, and Ranjan Srivastava(2006)

• k2 /min 0

reference:Investigation of Bacteriophage MS2 Viral Dynamics Using Model Discrimination Analysis and the Implications for Phage Therapy Rishi Jain, Andrea L. Knorr, Joseph Bernacki, and Ranjan Srivastava(2006)

• k3 /min 0.0167

reference:Investigation of Bacteriophage MS2 Viral Dynamics Using Model Discrimination Analysis and the Implications for Phage Therapy Rishi Jain, Andrea L. Knorr, Joseph Bernacki, and Ranjan Srivastava(2006)

• k5 /min 0.25

reference: Dynamical Determinants of Drug-Inducible Gene Expression in a Single Bacterium Thuc T. Le, Thierry Emonet, Sebastien Harlepp, Călin C. Guet, and Philippe Cluzel(2006)

Arbitrary parameters we can change are

```   * k1(rate of infection) : k1 relates to the specificity of antisenseRNA.
```
```   * k4(rate of phage producing) : k4 relates to the strength of promoter.
```
```   * k6(terminator leak rate) : k6 express terminator leak.
```

We change these parameters, and confirmed its effect.

Equations

We assume that cell lysis caused by MS2 phage is sufficiently rapid than coli death rate.

#### Method

We used original matlab program.

We set conditions as following.

```-when t = 10(min), signal of number '1' comes.(step input)
-when t = 30(min), signal of number '2' comes.(step input)
-when t = 50(min), signal of number '3' comes.(step input)
```

#### Results

The initial condition of uninfected coli concentration is set to be 1, and observed time evolution.

brue(U): initial concentration is 1. After got signal '1', it decrease suddenly.

green: I1 begins increasing after getting first signal '1'.

k1:0.4, k4:0.7, k6:0.1 red: The time course of virus concentration which has the information of number '4'.

blue: The time course of virus concentration which has the information of number '1'or'2'or'3'.

k1:0.4, k4:0.7, k6:0.01

red: The time course of virus concentration which has the information of number '4'.

blue: The time course of virus concentration which has the information of number '1'or'2'or'3'.

As terminator leak rate is less than the previous trial, it takes longer time to start increasing. the concentration of Virus '4' clearly increases comparing to others.

k1:0.4, k4:0.1, k6:0.01

k1:0.8, k4:0.7, k6:0.01

#### Discussion & Conclusion

It is confirmed that when one cell get '123'signals in order, the cell differentiate into '4'.

The rate of infection(k1) corresponds to antisense RNA specificity. Intuitively, when k1 is too high, coli become sensitive to not only target signals but also wrong signals. Our simulation also indicate when k1 becomes higher, the concentration of Virus'4' decrease.

The rate of phage producing(k4) corresponds to the strength of promoter. If promoter is too strong, the amount of the phage increase but it also raises terminator leak. However in our simulation, we confirm the dependence between the rate of terminator leak and k1 is rather small.

The rate of terminator leak relates to the time from receiving signals to the differentiation, and to the accuracy of finding the correct answer. According to the simulation, we find the clear difference between Pharge'4' and other Pharge'123'in 100-300 minutes, when the terminator leak rate is around 0.01.

## Whole system

finally, we made a simulation of solving whole 4*4 sudoku.

This simulation is based on 1 cell simulation written above.

The result of this simulation can be seen below.