Chapter 8 – Theoretical Biophysics 355
Under general non-steady-state conditions,
(8.73)
∂
∂
=
−
(
) −
−
n
t
k
n
n
k n
B
T
B
B
1
1
Solving this, substituting and rearranging then indicates
(8.74)
n
t
n
t
f
n
k n
n
n
B
B
B,S
T t
T
B,S
*
*
,
( ) =
( )
−
(
) =
−
−
(
)
−
−
−
1
1
1
1
– exp
where α is the ratio of the bound photoactive component of the labeled biomolecule at zero
time, immediately after the initial confocal volume photobleach to the bound photoactive
component of Ssb-YPet at steady state. Since the fluorescence intensity IB(t) of the bound
biomolecule component is proportional to the number of photoactive biomolecules, we
can write
(8.75)
I
t
I
t
t
B
B
r
( ) =
∞
( )
−
−
(
)
−
1
1
α exp
where tr is the characteristic exponential fluorescence recovery time constant, which in terms
of the kinetics of molecular turnover can be seen to depend on the off-rate but not by the
on-rate since the assumption of rapid diffusion means in effect that there are always available
free subunits in the cytoplasm ready to bind to the molecular complex if a free binding site
is available:
(8.76)
t
n
n
k n
r
T
B,S
T
=
−
−1
In other words, by fitting an experimental FRAP trace to a recovery exponential function, the
off-rate for turnover from the molecular complex can be estimated provided the total copy
number of biomolecules in the cell that bounds to the complex at steady state can be quan
tified (in practice, this may be achieved in many cases using fluorescence intensity imaging-
based quantification; see in silico methods section later in this chapter).
Other examples in biology of reaction-limited processes are those that simply only involve
reaction kinetics, that is, there is no relevant diffusional process to consider. This occurs, for
example, in molecular conformational changes and in the making and breaking processes of
chemical bonds under external force. The statistical theory involves the principle of detailed
balance.
The principle of detailed balance, or microscopic reversibility, states that if a system is in
thermal equilibrium, then each of its degrees of freedom is separately in thermal equilib
rium. For the kinetic transition between two stable states 1 and 2, the forward and reverse
rate constants can be predicted using Eyring theory of QM oscillators. If the transition goes
from a stable state 1 with free energy G1 to a metastable intermediate state I with a higher
free energy value GI, back down to another stable state 2 with lower free energy G2, then the
transition rates are given by
(8.77)
K
G
G
k T
K
G
G
k T
I
B
I
B
12
1
21
2
=
=
−
−
−
−
exp
vexp