Conjugation Model

By Ved and Amogh

We make a mass action model for conjugation

\begin{align}
\require{mhchem}
\ce{ \varnothing &->[growthrate \cdot R \cdot \left( 1 - \frac{R + T + D}{N_{0}} \right)] R}\
\ce{ \varnothing &->[growthrate \cdot T \cdot \left( 1 - \frac{R + T + D}{N_{0}} \right)] T}\
\ce{ \varnothing &->[growthrate \cdot D \cdot \left( 1 - \frac{R + T + D}{N_{0}} \right)] D}\
\ce{ R + D &<=>[kf1][kr1] interrd}\
\ce{ T + R &->[kf2] interrt}\
\ce{ interrt &->[kr2] R + T}\
\ce{ interrd &->[k1] T + D}\
\ce{ interrt &->[k2] 2 \cdot T}
\end{align}

Here:
$R\;=\; Reciepient\; Cells$
$T\; =\; Transconjugants$
$D\; =\; Donors$
$interrd\; =\; Mating\; pair\; between\; recipient\; and\; donor$
$interrt\; =\; Mating\; pair\; between\; recipient\; and\; transconjugant$

Converting the model into Differential equations we get:

\begin{align}
\frac{dR(t)}{dt} =& growthrate \cdot R \cdot \left( 1 - \frac{R + T + D}{N_{0}} \right) - kf1 \cdot R \cdot D + kr1 \cdot interrd - kf2 \cdot T \cdot R + kr2 \cdot interrt \
\frac{dT(t)}{dt} =& growthrate \cdot T \cdot \left( 1 - \frac{R + T + D}{N_{0}} \right) - kf2 \cdot T \cdot R + kr2 \cdot interrt + k1 \cdot interrd + 2 \cdot k2 \cdot interrt \
\frac{dD(t)}{dt} =& growthrate \cdot D \cdot \left( 1 - \frac{R + T + D}{N_{0}} \right) - kf1 \cdot R \cdot D + kr1 \cdot interrd + k1 \cdot interrd \
\frac{dinterrd(t)}{dt} =& kf1 \cdot R \cdot D - kr1 \cdot interrd - k1 \cdot interrd \
\frac{dinterrt(t)}{dt} =& kf2 \cdot T \cdot R - kr2 \cdot interrt - k2 \cdot interrt
\end{align}