Fig. 4-2-1. Maz System Gene Circuit

Expressions

• 1.1. Cell Population

  $$ \frac{dP_{Snow White}}{dt} = g \frac{E_{DiMazF}}{E_{DiMazF}+[DiMazF]}\left(1- \frac{P_{Snow White}+P_{Queen}+P_{Prince}}{P_{max}} \right) P_{Snow White} $$
  $$ \tag{4-2-1-1} $$

  $$ \frac{dP_{Queen}}{dt} = g \frac{E_{DiMazF}}{E_{DiMazF}+[DiMazF]}\left(1- \frac{P_{Snow White}+P_{Queen}+P_{Prince}}{P_{max}}\right) P_{Queen}$$
  $$ \tag{4-2-1-2} $$

  $$ \frac{dP_{Prince}}{dt} = g\left(1- \frac{P_{Snow White}+P_{Queen}+P_{Prince}}{P_{max}}\right) P_{Prince} \tag{4-2-1-3} $$

  The equations above describe how cells grow in the culture. Equations (4-2-1-1), (4-2-1-2) and (4-2-1-3) describe the populations of Snow White, the Queen and the Prince. (4-2-1-3) is described by the logistic growth equation, but (4-2-1-1) and (4-2-1-2) 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 low, and its value converges to 1 when the concentration of MazF dimers is high.

• 1.2. Maz System

  • 1.2.1. Expression of Maz System

    After translation, MazE and MazF each form an stable dimer which can be activated to exert its 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 Maz System

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

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

    Snow White

    $$ \frac{d[mRNA_{MazF}]}{dt} = leak_{P_{lux}} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}}+ [C12]^{n_{Lux}}} \\ 　　　　　　　- d[mRNA_{MazF}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazF}}})[mRNA_{MazF}][DiMazF] $$
    $$ \tag{4-2-2-1} $$

    $$ \frac{d[MazF]}{dt} = \alpha [mRNA_{MazF}] - 2k_{Di_{MazF}}[MazF] + 2k_{-Di_{MazF}}[DiMazF] - d_{MazF}[MazF] $$
    $$\tag{4-2-2-2}$$

    $$ \frac{d[DiMazF]}{dt} = k_{Di_{MazF}}[MazF] - k_{-Di_{MazF}}[DiMazF] - 2k_{Hexa}[DiMazE][DiMazF]^2 \\ 　　　　　　　+ 2k_{-Hexa}[MazHexamer] - d_{DiMazF}[DiMazF] $$
    $$ \tag{4-2-2-3} $$

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

    $$\frac{d[MazE]}{dt} = \alpha [mRNA_{MazE}] - 2k_{Di_{MazE}}[MazE] + 2k_{-Di_{MazE}}[DiMazE] - d_{MazE}[MazE]$$
    $$\tag{4-2-2-5}$$

    $$ \frac{d[DiMazE]}{dt} = k_{Di_{MazE}}[MazE] - k_{-Di_{MazE}}[DiMazE] - k_{Hexa}[DiMazE][DiMazF]^2 \\ 　　　　　　　+ k_{-Hexa}[MazHexamer] - d_{DiMazE}[DiMazE]$$
    $$\tag{4-2-2-6} $$

    $$\frac{d[MazHexa]}{dt} = k_{Hexa}[DiMazE][DiMazF]^2 - k_{-Hexa}[MazHexa] - d_{Hexa}[MazHexa]$$
    $$ \tag{4-2-2-7}$$

    Queen

    $$ \frac{d[mRNA_{MazF}]}{dt} = leak_{P_{lux}} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}}+ [C12]^{n_{Lux}}} \\ 　　　　　　　- d[mRNA_{MazF}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazF}}})[mRNA_{MazF}][DiMazF] $$
    $$ \tag{4-2-2-8} $$

    $$ \frac{d[MazF]}{dt} = \alpha [mRNA_{MazF}] - 2k_{Di_{MazF}}[MazF] + 2k_{-Di_{MazF}}[DiMazF] - d_{MazF}[MazF] $$
    $$\tag{4-2-2-9}$$

    $$ \frac{d[DiMazF]}{dt} = k_{Di_{MazF}}[MazF] - k_{-Di_{MazF}}[DiMazF] - 2k_{Hexa}[DiMazE][DiMazF]^2 \\ 　　　　　　　+ 2k_{-Hexa}[MazHexamer] - d_{DiMazF}[DiMazF] $$
    $$ \tag{4-2-2-10} $$

    $$ \frac{d[mRNA_{MazE}]}{dt} = k - d[mRNA_{MazE}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazE}}})[mRNA_{MazE}][DiMazF] $$
    $$ \tag{4-2-2-11} $$

    $$\frac{d[MazE]}{dt} = \alpha [mRNA_{MazE}] - 2k_{Di_{MazE}}[MazE] + 2k_{-Di_{MazE}}[DiMazE] - d_{MazE}[MazE]$$
    $$\tag{4-2-2-12}$$

    $$ \frac{d[DiMazE]}{dt} = k_{Di_{MazE}}[MazE] - k_{-Di_{MazE}}[DiMazE] - k_{Hexa}[DiMazE][DiMazF]^2 \\ 　　　　　　　+ k_{-Hexa}[MazHexamer] - d_{DiMazE}[DiMazE]$$
    $$\tag{4-2-2-13} $$

    $$\frac{d[MazHexa]}{dt} = k_{Hexa}[DiMazE][DiMazF]^2 - k_{-Hexa}[MazHexa] - d_{Hexa}[MazHexa]$$
    $$ \tag{4-2-2-14}$$

    Eq. 4-2-2. Differential Equations of Maz System

    Equations (4-2-2-1) and (4-2-2-8) 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 dependent on the concentration of C12/C4, terms of original degradation and terms of degradation from cleavage at ACA sequences by MazF dimers. Since Equations (4-2-2-2), (4-2-2-3), (4-2-2-5), (4-2-2-6), (4-2-2-7), (4-2-2-9), (4-2-2-10), (4-2-2-12), (4-2-2-13) and (4-2-2-14) describe the concentrations of complexes, mainly they comprise terms of production and terms of binding and dissociation.

  • 1.2.2. Cleavage by MazF dimers

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

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

    $$\frac{d[mRNA_{RFP}]}{dt} = k - d[mRNA_{RFP}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{RFP}}})[mRNA_{RFP}][DiMazF] $$
    $$ \tag{4-2-3-1} $$

    $$ \frac{d[mRNA_{RhlI}]}{dt} = leak_{P_{lux}} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}} + [C12]^{n_{Lux}}} - d[mRNA_{RhlI}] - F_{DiMazF}$$
    $$ \tag{4-2-3-2} $$

    $$\frac{d[mRNA_{MazF}]}{dt} = leak_{P_{lux}} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}}+ [C12]^{n_{Lux}}} \\ 　　　　　　　- d[mRNA_{MazF}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazF}}})[mRNA_{MazF}][DiMazF] $$
    $$\tag{4-2-3-3}$$

    $$\frac{d[mRNA_{MazE}]}{dt} = k - d[mRNA_{MazE}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazE}}})[mRNA_{MazE}][DiMazF]$$
    $$ \tag{4-2-3-4} $$

    Queen

    $$\frac{d[mRNA_{GFP}]}{dt} = k - d[mRNA_{GFP}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{GFP}}})[mRNA_{GFP}][DiMazF] $$
    $$ \tag{4-2-3-5} %%

    $$ \frac{d[mRNA_{LasI}]}{dt} = leak_{P_{rhl}} + \frac{\kappa_{Rhl}[C4]^{n_{Rhl}}}{K_{mRhl}^{n_{Rhl}} + [C4]^{n_{Rhl}}} \\ 　　　　　　　- d[mRNA_{LasI}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{LasI}}})[mRNA_{LasI}][DiMazF] $$
    $$ \tag{4-2-3-6} $$

    $$\frac{d[mRNA_{MazF}]}{dt} = leak_{P_{lux}} + \frac{\kappa_{Lux}[C12]^{n_{Lux}}}{K_{mLux}^{n_{Lux}}+ [C12]^{n_{Lux}}} \\ 　　　　　　　- d[mRNA_{MazF}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazF}}})[mRNA_{MazF}][DiMazF] $$
    $$\tag{4-2-3-7}$$

    $$\frac{d[mRNA_{MazE}]}{dt} = k - d[mRNA_{MazE}] - F_{DiMazF}(1-(1-f)^{f_{mRNA_{MazE}}})[mRNA_{MazE}][DiMazF]$$
    $$ \tag{4-2-3-8} $$

    Eq. 4-2-3. Differential Equations of mRNAs

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

• 1.3. Signal Molecules

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

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

  After translation, C12AHL and C4 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 C12AHL / C4 synthesis rate per cell is estimated to be proportional to the LasI and RhlI concentrations.

  C4 decreases from original degradation meanwhile C12AHL decreases from both original degradation and degradation by AmiE, which Prince products.

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

  Equations (1), (4) and (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 dependent on the concentration of C12/C4, 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 (3), (6) describe the concentrations of C4 C12AHL in the whole culture medium.