We assert that the relationship between renaturation and
concentration of single-stranded DNA (c) obeys
second-order kinetics, i.e.
dc/dt = -k2c2
We can rewrite this as
-c-2dc = k2dt
Integrate both sides, with
∫ un du
= (un+1) / (n + 1) + Q
where Q is a constant that we will determine from the boundary
condition.
The integral on both sides gives us
{-(-1)}c-1 = k2t - Q
or 1/(k2t - Q) = c
What is the boundary condition? It's that all of the DNA
is single-stranded to begin with, i.e.
c = c0 at time t = 0.
Therefore when t = 0,
-1 / Q = c; but that value of c is, in fact,
c0, so Q = -1 / c0, and
1 / (k2t + 1 / c0)
= c
Multiplying numerator and denominator on the left by by c0,
c0 /
(k2c0t + 1) = c,
or
(k2c0t
+ 1)-1 = c / c0
To show that the half-time for dissociation is
t1/2 =
(k2c0)-1:
The half-time appears when c = c0 / 2, so,
from the formula for c / c0,
(k2c0t1/2
+ 1)-1 = 1/2, or, inverting,
2 = k2c0t1/2 + 1
Thus 1 =
k2c0t1/2,
i.e. t1/2
= (k2c0)-1