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We will call the amount (concentration multiplied by volume) of the bacteria A, and the amount of binding site on the DNA (concentration multiplied by volume multiplied by binding sites per DNA sequence) as B.
The rate of change of A= the rate of unbinding-the rate of binding.
These rates are proportional to the amount of unbinded or binded molecules there are.
By letting
We can solve this differential equation using separation of variables.
Where χ is an unknown constant. To find χ we set A=A_0 at t=0.
Note that 4ac-b^2 is negative, so the square root cannot be calculated. We now rearrange to cancel out imaginary parts.
Therfore
Substituting the values for a, b, and c.
Let
Then