374 8.5 Advanced In Silico Analysis Tools
(8.110)
g
x y
exp
x
x
y
y
g
x y
exp
x
x
1
1
2
1
2
1
2
2
2
,
,
(
) =
−
−
(
) +
−
(
)
(
)
(
) =
−
−
σ
2
2
2
2
2
2
2
(
) +
−
(
)
(
)
y
y
σ
Their unnormalized overlap can be calculated from the integral of their product:
(8.111)
v
d d
exp
allspace
u
g g
x y
C
r
=
=
−(
)
+
∫∫
1
2
2
1
2
2
2
2
∆
(σ
σ
where
Δr is the distance between the centers of the Gaussians on the xy image plane
C is a normalization constant equal to the maximum overlap possible for two perfectly
overlapped spots so that
(8.112)
(
)
(
)
(
)
∆r
x
x
y
y
2
1
2
2
1
2
2
=
−
+
−
(8.113)
C =
+
2
1
2
2
2
1
2
2
2
πσ σ
σ
σ
The normalized overlap integral, v, for a pair of spots of known positions and widths, is then
(8.114)
v
v
exp
=
=
−(
)
+
u
C
r
∆
2
1
2
2
2
2(σ
σ
The value of this overlap integral can then be used as a robust and objective measure for
the extent of colocalization. For example, for “optical resolution colocalization,” which is the
same as the Rayleigh criterion (see Chapter 3), values of ~0.2 or more would be indicative of
putative colocalization.
One issue with this approach, however, is that if the concentration of one or the both of the
molecules under investigation are relatively high (note, biologists often refer to such cells as
having high copy numbers for a given molecule, meaning the average number of a that given
molecule per cell), then there is an increasing probability that the molecules may appear to be
colocalized simply by chance, but are in fact not interacting. One can argue that if two such
molecules are not interacting, then after a short time their associated fluorescent spots will
appear more apart from each other; however, in practice, the diffusion time scale may often
be comparable to the photobleaching time scale, meaning that these chance colocalized dye
molecules may photobleach before they can be detected as moving apart from each other.
It is useful to predict the probability of such chance colocalization events. Using nearest-
neighbor analysis for a random distribution of particles on a 2D surface, we can determine
the probability p(r) dr that the distance from one fluorescent spot in the cell to another of
the same type is between r and r + dr. This analysis results in effect in a value of the limiting
concentration for detecting distinct spots (see Chapter 4) when averaged over a whole cell
and depends upon certain geometrical constraints (e.g., whether a molecule is located in the
three spatial dimension cell volume or is confined to two dimensions as in the cell membrane