Team:UT-Tokyo/Sudoku modeling


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4C3leak switch

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



1.We treated DNA as continuous variabl, 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).


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)


k0 : mRNA translation rate

k1 : protein degradation rate

k2 : mRNA translation rate

k3 : cre protein degradation rate

l0 : transcription speed per unit concentration of RNA polymearase

l1 : mRNA degradation rate

l2 : cre mRNA degradation rate

K : recombinase binding constant

v : recombinase reaction rate

pT7 : concentration o T7 polymerase

p : terminator leak probability



Eqs cs.jpg


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


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

The time course of DNA concentration

Sim res.JPG

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



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 minimum value. In this case it is 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.


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

MS2 phage infection

We made a model for a indifferent sudoku cell.

We traverse the amount of phages and see wether the cell react properly against given signals.



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


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


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

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

Equations used in this modeling


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)


Parameters we can vary voluntarily 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.



MS2 2.png



Whole system

What's Sudoku?