TRANSFORMATION MODEL: COMPARISON BETWEEN SINGLE GENE AND DOUBLE GENE SYSTEM

By Ved and Amogh

We make a mass action model for transformation for single gene system

\begin{align}
\require{mhchem}
\ce{ \varnothing &->[growthrate \cdot A \cdot \left( 1 - \frac{A + B + B1 + C}{N_{0}} \right)] A}\
\ce{ \varnothing &->[growthrate \cdot B \cdot \left( 1 - \frac{A + B + B1 + C}{N_{0}} \right)] B}\
\ce{ \varnothing &->[growthrate \cdot B1 \cdot \left( 1 - \frac{A + B + B1 + C}{N_{0}} \right)] B1}\
\ce{ \varnothing &->[growthrate \cdot C \cdot \left( 1 - \frac{A + B + B1 + C}{N_{0}} \right)] C}\
\ce{ A &<=>[0.01][0.02] B}\
\ce{ A &<=>[0.01][0.02] B1}\
\ce{ A &->[0.001] \varnothing}\
\ce{ B &->[0.001] G}\
\ce{ C &->[0.001] \varnothing}\
\ce{ B1 &->[0.001] G}\
\ce{ G &->[0.002] \varnothing}\
\ce{ C + G &->[0.05] B1}
\end{align}

Here:
A = Cell in which GI is incorporated in chromosome
B = Cell in which GI is present in the cytoplasm
B1 = Transformed Cell
C = Empty Cell
GI = Genomic Island

Converting the model into differential equation

\begin{align}
\frac{dA(t)}{dt} =& growthrate \cdot A \cdot \left( 1 - \frac{A + B + B1 + C}{N_{0}} \right) - 0.01 \cdot A + 0.02 \cdot B - 0.01 \cdot A + 0.02 \cdot B1 - 0.001 \cdot A \
\frac{dB(t)}{dt} =& growthrate \cdot B \cdot \left( 1 - \frac{A + B + B1 + C}{N_{0}} \right) + 0.01 \cdot A - 0.02 \cdot B - 0.001 \cdot B \
\frac{dB1(t)}{dt} =& growthrate \cdot B1 \cdot \left( 1 - \frac{A + B + B1 + C}{N_{0}} \right) + 0.01 \cdot A - 0.02 \cdot B1 - 0.001 \cdot B1 + 0.05 \cdot C \cdot G \
\frac{dC(t)}{dt} =& growthrate \cdot C \cdot \left( 1 - \frac{A + B + B1 + C}{N_{0}} \right) - 0.001 \cdot C - 0.05 \cdot C \cdot G \
\frac{dG(t)}{dt} =& 0.001 \cdot B + 0.001 \cdot B1 - 0.002 \cdot G - 0.05 \cdot C \cdot G
\end{align}

Now, we make a mass action model for double gene system

\begin{align}
\require{mhchem}
\ce{ \varnothing &->[growthrate \cdot A \cdot \left( 1 - \frac{A + B + B1 + B2 + B3 + C}{N_{0}} \right)] A}\
\ce{ \varnothing &->[growthrate \cdot B \cdot \left( 1 - \frac{A + B + B1 + B2 + B3 + C}{N_{0}} \right)] B}\
\ce{ \varnothing &->[growthrate \cdot B1 \cdot \left( 1 - \frac{A + B + B1 + B2 + B3 + C}{N_{0}} \right)] B1}\
\ce{ \varnothing &->[growthrate \cdot B2 \cdot \left( 1 - \frac{A + B + B1 + B2 + B3 + C}{N_{0}} \right)] B2}\
\ce{ \varnothing &->[growthrate \cdot B3 \cdot \left( 1 - \frac{A + B + B1 + B2 + B3 + C}{N_{0}} \right)] B3}\
\ce{ \varnothing &->[growthrate \cdot C \cdot \left( 1 - \frac{A + B + B1 + B2 + B3 + C}{N_{0}} \right)] C}\
\ce{ A &<=>[0.01][0.02] B}\
\ce{ A &<=>[0.01][0.02] B1}\
\ce{ A &<=>[0.01][0.02] B2}\
\ce{ A &->[0.001] \varnothing}\
\ce{ B &->[0.001] G1 + G2}\
\ce{ B1 &->[0.01] G1}\
\ce{ B2 &->[0.01] G2}\
\ce{ B3 &->[0.01] G1 + G2}\
\ce{ C &->[0.001] \varnothing}\
\ce{ C + G1 &->[0.05] B1}\
\ce{ C + G2 &->[0.05] B2}\
\ce{ B1 + G2 &->[0.05] B3}\
\ce{ B2 + G1 &->[0.05] B3}
\end{align}

Converting the double gene system model into ODEs

\begin{align}
\frac{dA(t)}{dt} =& growthrate \cdot A \cdot \left( 1 - \frac{A + B + B1 + B2 + B3 + C}{N_{0}} \right) - 0.01 \cdot A + 0.02 \cdot B - 0.01 \cdot A + 0.02 \cdot B1 - 0.01 \cdot A + 0.02 \cdot B2 - 0.001 \cdot A \
\frac{dB(t)}{dt} =& growthrate \cdot B \cdot \left( 1 - \frac{A + B + B1 + B2 + B3 + C}{N_{0}} \right) + 0.01 \cdot A - 0.02 \cdot B - 0.001 \cdot B \
\frac{dB1(t)}{dt} =& growthrate \cdot B1 \cdot \left( 1 - \frac{A + B + B1 + B2 + B3 + C}{N_{0}} \right) + 0.01 \cdot A - 0.02 \cdot B1 - 0.01 \cdot B1 + 0.05 \cdot C \cdot G1 - 0.05 \cdot B1 \cdot G2 \
\frac{dB2(t)}{dt} =& growthrate \cdot B2 \cdot \left( 1 - \frac{A + B + B1 + B2 + B3 + C}{N_{0}} \right) + 0.01 \cdot A - 0.02 \cdot B2 - 0.01 \cdot B2 + 0.05 \cdot C \cdot G2 - 0.05 \cdot B2 \cdot G1 \
\frac{dB3(t)}{dt} =& growthrate \cdot B3 \cdot \left( 1 - \frac{A + B + B1 + B2 + B3 + C}{N_{0}} \right) - 0.01 \cdot B3 + 0.05 \cdot B1 \cdot G2 + 0.05 \cdot B2 \cdot G1 \
\frac{dC(t)}{dt} =& growthrate \cdot C \cdot \left( 1 - \frac{A + B + B1 + B2 + B3 + C}{N_{0}} \right) - 0.001 \cdot C - 0.05 \cdot C \cdot G1 - 0.05 \cdot C \cdot G2 \
\frac{dG1(t)}{dt} =& 0.001 \cdot B + 0.01 \cdot B1 + 0.01 \cdot B3 - 0.05 \cdot C \cdot G1 - 0.05 \cdot B2 \cdot G1 \
\frac{dG2(t)}{dt} =& 0.001 \cdot B + 0.01 \cdot B2 + 0.01 \cdot B3 - 0.05 \cdot C \cdot G2 - 0.05 \cdot B1 \cdot G2
\end{align}