354 8.4 Reaction, Diffusion, and Flow
that of a single cell, in which total content of the fluorescently labeled biomolecule of interest
is in a steady state. Each specific system may have bespoke physical conditions that need to
be characterized in any mathematical description; however, a typical system might involve
a molecular complex that contains subunits that are either integrated into that complex or
are diffusing free in the cell cytoplasm. Then, the general starting point might involve first
denoting the following:
nF(t) = Number of fluorescently labeled biomolecule of a specific type under investiga
tion, which is free in the cytoplasm at time t (≥0) following initial focused laser bleach
nB(t) = Number of fluorescently labeled biomolecule bound to a molecular complex
identified in the bleach zone
nT(t) = Total number of fluorescently labeled biomolecules in the cell
n
t
B
* ( ) = Number of photoactive fluorescently labeled molecules bound to the molecular
complex in the bleach zone
f = Fraction of photobleached fluorescently labeled biomolecules following initial focused
laser bleach
k1 = On-rate per fluorescently labeled biomolecule for binding to bleach-zone molecular
complex
k−1 = Off-rate per fluorescently labeled biomolecule for unbinding from molecular com
plex If the fluorescently labeled molecules are in steady state, we can say
(8.68)
∂
∂
=
∴
+
=
n
t
n
n
n
T
F
B
0
T
Thus, nT is a constant. The reaction–diffusion equations can be decoupled into separate
diffusion and reaction components:
(8.69)
∂
∂
=
∇
n
t
D
n
T
2
F
(8.70)
∂
∂
=
−
−
n
t
k n
k n
B
1
1
F
B
D is the effective diffusion coefficient of the fluorescently labeled biomolecule in the cell cyto
plasm. Since the typical diffusion time scale τ is set by ~L2/D where L is the typical length
dimension of the cell (~1 μm if a model bacterial organism is used) and D for small proteins
and molecular complexes in the cytoplasm is ~10 μm2 s−1, indicating τ ~10 ms. However,
the turnover in many molecular complexes is often over a time scale of more like ~1–100 s,
at least two orders of magnitude slower than the diffusion time scale. Thus, this is clearly a
reaction-limited regime and so the diffusion component can be ignored. At steady state just
before the confocal volume photobleach, we can say that
(8.71)
∂
∂
= ∴
=
∴
=
−
−
−
n
t
k n
k n
k
k n
n
n
B S
F,S
B,S
B,S
B,S
,
0
1
1
1
1
T
where nB,S and nF,S are the values of nB and nF, respectively, at steady state. If we assume that
the binding kinetics of photoactive and photobleached labeled biomolecules are identical and
that the population of bleached and nonbleached are ultimately well mixed, then
(8.72)
n
n
f
B
B
* =
−
(
)
1