To simulate the cell-cell communication system, we developed an ordinary differential equation model. The following sentences describe how the equations were developed with the Maz system.

Fig. 4-2-1. the Maz system gene circuit

Expressions

• 1. Cell Population

  $$ \frac{dP_{Snow White}}{dt} = g \frac{E_{DiMazF}}{E_{DiMazF}+[DiMazF]}\left(1- \frac{P_{Snow White}+P_{Queen}+P_{Prince}}{P_{max}} \right) P_{Snow White} $$
  $$ \tag{1-1} $$

  $$ \frac{dP_{Queen}}{dt} = g \frac{E_{DiMazF}}{E_{DiMazF}+[DiMazF]}\left(1- \frac{P_{Snow White}+P_{Queen}+P_{Prince}}{P_{max}}\right) P_{Queen}$$
  $$ \tag{1-2} $$

  $$ \frac{dP_{Prince}}{dt} = g\left(1- \frac{P_{Snow White}+P_{Queen}+P_{Prince}}{P_{max}}\right) P_{Prince} \tag{1-3} $$

  Eq.1. Differential Equation of Cell Population

  The equations above describe how cells grow in the culture. Equations (1-1), (1-2) and (1-3) describe the population of Snow White coli, the Queen coli and the Prince coli. (1-3) is described by the logistic growth equation, but (1-1) and (1-2) are represented by the growth inhibition by MazF dimers. This factor is designed so that its value is small when the concentration of MazF dimers is high, and its value converges to 1 when the concentration of MazF dimers is low.

• 2. the Maz system

  • 2.1. Expression of the Maz system

    After translation, MazE and MazF each form a dimer which can be activated to exert their function.

    
    

    Two MazE dimers sandwich the MazF dimer, forming MazF2-MazE2-MazF2 heterohexamers and suppressing the toxicity of the MazF dimers.

    

    Fig. 4-2-2. Reaction of the Maz system

    The mRNAs of Snow White coli and the Queen coli decrease because of their original degradation and the cleavage at ACA sequences by MazF dimers.

    Applying mass action kinetic laws, we obtain the following set of differential equations.

    Snow White

    $$ \frac{d[mRNA_{MazF}]}{dt} = leak_{Plux} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}}+ [C12]^{n_{Lux}}} \\ 　　　　　　　- d[mRNA_{MazF}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazF}}})[mRNA_{MazF}][DiMazF] $$
    $$ \tag{2-1} $$

    $$ \frac{d[MazF]}{dt} = \alpha [mRNA_{MazF}] - 2k_{DiMazF}[MazF] + 2k_{-DiMazF}[DiMazF] - d_{MazF}[MazF] $$
    $$\tag{2-2}$$

    $$ \frac{d[DiMazF]}{dt} = k_{DiMazF}[MazF] - k_{-DiMazF}[DiMazF] - 2k_{Hexa}[DiMazE][DiMazF]^2 \\ 　　　　　　　+ 2k_{-Hexa}[MazHexamer] - d_{DiMazF}[DiMazF] $$
    $$ \tag{2-3} $$

    $$ \frac{d[mRNA_{MazE}]}{dt} = k - d[mRNA_{MazE}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazE}}})[mRNA_{MazE}][DiMazF] $$
    $$ \tag{2-4} $$

    $$\frac{d[MazE]}{dt} = \alpha [mRNA_{MazE}] - 2k_{DiMazE}[MazE] + 2k_{-DiMazE}[DiMazE] - d_{MazE}[MazE]$$
    $$\tag{2-5}$$

    $$ \frac{d[DiMazE]}{dt} = k_{DiMazE}[MazE] - k_{-DiMazE}[DiMazE] - k_{Hexa}[DiMazE][DiMazF]^2 \\ 　　　　　　　+ k_{-Hexa}[MazHexamer] - d_{DiMazE}[DiMazE]$$
    $$\tag{2-6} $$

    $$\frac{d[MazHexa]}{dt} = k_{Hexa}[DiMazE][DiMazF]^2 - k_{-Hexa}[MazHexa] - d_{Hexa}[MazHexa]$$
    $$ \tag{2-7}$$

    Queen

    $$ \frac{d[mRNA_{MazF}]}{dt} = leak_{Plux} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}}+ [C12]^{n_{Lux}}} \\ 　　　　　　　- d[mRNA_{MazF}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazF}}})[mRNA_{MazF}][DiMazF] $$
    $$ \tag{2-8} $$

    $$ \frac{d[MazF]}{dt} = \alpha [mRNA_{MazF}] - 2k_{DiMazF}[MazF] + 2k_{-DiMazF}[DiMazF] - d_{MazF}[MazF] $$
    $$\tag{2-9}$$

    $$ \frac{d[DiMazF]}{dt} = k_{DiMazF}[MazF] - k_{-DiMazF}[DiMazF] - 2k_{Hexa}[DiMazE][DiMazF]^2 \\ 　　　　　　　+ 2k_{-Hexa}[MazHexamer] - d_{DiMazF}[DiMazF] $$
    $$ \tag{2-10} $$

    $$ \frac{d[mRNA_{MazE}]}{dt} = k - d[mRNA_{MazE}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazE}}})[mRNA_{MazE}][DiMazF] $$
    $$ \tag{2-11} $$

    $$\frac{d[MazE]}{dt} = \alpha [mRNA_{MazE}] - 2k_{DiMazE}[MazE] + 2k_{-DiMazE}[DiMazE] - d_{MazE}[MazE]$$
    $$\tag{2-12}$$

    $$ \frac{d[DiMazE]}{dt} = k_{DiMazE}[MazE] - k_{-DiMazE}[DiMazE] - k_{Hexa}[DiMazE][DiMazF]^2 \\ 　　　　　　　+ k_{-Hexa}[MazHexamer] - d_{DiMazE}[DiMazE]$$
    $$\tag{2-13} $$

    $$\frac{d[MazHexa]}{dt} = k_{Hexa}[DiMazE][DiMazF]^2 - k_{-Hexa}[MazHexa] - d_{Hexa}[MazHexa]$$
    $$ \tag{2-14}$$

    Eq. 2. Differential equations of the Maz system

    Equations (2-1) and (2-8) describe the concentration of mRNAs under the AHL inducing promoters. Thus, they comprise terms of production by leaky expressions of promoters, terms of production by Hill function depending on the concentration of C4 and C12, terms of original degradation and terms of degradation from cleavage at ACA sequences by MazF dimers. Since Equations (2-2), (2-3), (2-5), (2-6), (2-7), (2-9), (2-10), (2-12), (2-13) and (2-14) describe the concentrations of complexes, mainly they comprise terms of production and terms of binding and dissociation.

  • 2.2. Cleavage by MazF dimers

    MazF dimers recognize and cleave ACA sequences in mRNAs, thus acting as Toxin.

    We estimated the rate of recognitions of ACA sequences by MazF dimers at \(1-(1-f)^n\), where n is the number of ACA sequences in mRNA.

    Then, we expressed the rate of degradation by MazF dimers in \(F(1-(1-f)^{f_{mRNA}})\) and obtain the following set of differential equations.

    Snow White

    $$\frac{d[mRNA_{RFP}]}{dt} = k - d[mRNA_{RFP}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{RFP}}})[mRNA_{RFP}][DiMazF] $$
    $$ \tag{3-1} $$

    $$ \frac{d[mRNA_{RhlI}]}{dt} = leak_{Plux} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}} + [C12]^{n_{Lux}}} - d[mRNA_{RhlI}] - F_{DiMazF}$$
    $$ \tag{3-2} $$

    $$\frac{d[mRNA_{MazF}]}{dt} = leak_{Plux} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}}+ [C12]^{n_{Lux}}} \\ 　　　　　　　- d[mRNA_{MazF}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazF}}})[mRNA_{MazF}][DiMazF] $$
    $$\tag{3-3}$$

    $$\frac{d[mRNA_{MazE}]}{dt} = k - d[mRNA_{MazE}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazE}}})[mRNA_{MazE}][DiMazF]$$
    $$ \tag{3-4} $$

    Queen

    $$\frac{d[mRNA_{GFP}]}{dt} = k - d[mRNA_{GFP}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{GFP}}})[mRNA_{GFP}][DiMazF] $$
    $$ \tag{3-5} $$

    $$ \frac{d[mRNA_{LasI}]}{dt} = leak_{Prhl} + \frac{\kappa_{Rhl}[C4]^{n_{Rhl}}}{K_{mRhl}^{n_{Rhl}} + [C4]^{n_{Rhl}}} \\ 　　　　　　　- d[mRNA_{LasI}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{LasI}}})[mRNA_{LasI}][DiMazF] $$
    $$ \tag{3-6} $$

    $$\frac{d[mRNA_{MazF}]}{dt} = leak_{Plux} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}}+ [C12]^{n_{Lux}}} \\ 　　　　　　　- d[mRNA_{MazF}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazF}}})[mRNA_{MazF}][DiMazF] $$
    $$\tag{3-7}$$

    $$\frac{d[mRNA_{MazE}]}{dt} = k - d[mRNA_{MazE}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazE}}})[mRNA_{MazE}][DiMazF]$$
    $$ \tag{3-8} $$

    Eq. 3. Differential Equations of mRNAs

    The equations above comprise terms of production, terms of only original degradation and terms of degradation from cleavage at ACA sequences by MazF dimers.

• 3. Signal Molecules

  

  Fig. 4-2-3. Reaction of Signal Molecules

  Snow White coli expresses RhlI under Plux induced by C12, the Queen coli expresses LasI under Prhl induced by C4 and the Prince coli expresses AmiE under Plux induced by C12.

  The mRNAs of Snow White coli and the Queen coli decrease from original degradation and the cleavage at ACA sequences by MazF dimers. On the other hand, those of the Prince coli don’t have any MazF gene so they decrease from only original degradation.

  After translation, C4 and C12 are enzymatically synthesized by LasI and RhlI from some substrates respectively. For simplicity, we assumed that the amount of substrates is sufficient so that the C4 and C12 synthesis rate per cell is estimated to be proportional to the LasI and RhlI concentrations.

  C4 decreases from original degradation meanwhile C12 decreases from both original degradation and degradation by AmiE, which the Prince coli produces.

  Applying mass action kinetic laws, we obtain the following set of differential equations.

  $$ \frac{d[mRNA_{RhlI}]}{dt} = leak_{Plux} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}} + [C12]^{n_{Lux}}} - d[mRNA_{RhlI}] - F_{DiMazF}f[mRNA_{RhlI}][DiMazF] $$
  $$\tag{4-1}$$

  $$\frac{d[RhlI]}{dt} = \alpha [mRNA_{RhlI}] - d_{RhlI}[RhlI] \tag{4-2}$$

  $$ \frac{d[Rhl AHL]}{dt} = p_{Rhl}[RhlI]P_{Snowwhite} - d_{RhlAHL}[RhlAHL] \tag{4-3} $$

  $$ \frac{d[mRNA_{LasI}]}{dt} = leak_{Prhl} + \frac{\kappa_{Rhl}[C4]^{n_{Rhl}}}{K_{mRhl}^{n_{Rhl}} + [C4]^{n_{Rhl}}} - d[mRNA_{LasI}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{LasI}}})[mRNA_{LasI}][DiMazF] $$
  $$\tag{4-4}$$

  $$\frac{d[LasI]}{dt} = \alpha [mRNA_{LasI}] - d_{LasI}[LasI] \tag{4-5}$$

  $$\frac{d[LasAHL]}{dt} = p_{Las}[LasI]P_{Stepmother} - d_{LasAHL}[LasAHL] - D[LasAHL][AmiE]$$
  $$\tag{4-6}$$

  $$\frac{d[mRNA_{AmiE}]}{dt} = leak_{Plux} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}} + [C12]^{n_{Lux}}} - d[mRNA_{AmiE}]$$
  $$\tag{4-7}$$

  $$\frac{d[AmiE]}{dt} = \alpha [mRNA_{AmiE}]P_{Prince} - d_{AmiE}[AmiE] \tag{4-8} $$

  Eq. 4. Differential Equations of Signaling Molecules

  Equations (4-1), (4-4) and (4-7) describe the concentrations of mRNAs under the AHL inducing promoters. Thus, they comprise terms of production by leaky expressions of promoters, terms of production by Hill function depending on the concentration of C4 and C12, terms of original degradation and terms of degradation from cleavage at ACA sequences by MazF dimers.

  The other ODEs describe how the concentrations of materials change in individuals, on the other hand (4-3), (4-6) describe the concentrations of C4 and C12 in the whole culture medium.