Chapter 8 – Theoretical Biophysics 375
or one dimension as for translocation of molecular motors on a filament). As a rough guide,
this is equivalent to ~1017 fluorescent spots per liter for the case of molecules inside the cell
volume, which corresponds to a molar concentration of ~100 nM. For interested students,
an elegant mathematical treatment for modeling this nearest-neighbor distance was first
formulated by one of the great mathematicians of modern times (Chandrasekhar, 1943) for a
similar imaging problem but involving astrophysical observations of stars and planets.
For example, consider the 2D case of fluorescent spots located in the cell membrane. The
probability p(r)dr must be equal to the probability that there are zero such fluorescent spots
particles in the range 0−r, multiplied by the probability that a single spot exists in the annulus
zone between r and r + dr. Thus,
(8.115)
p r
r
p r
r
rn r
r
( )
=
−
′
( )
′
⋅
∫
d
d
d
1
2
0
π
where n is the number of fluorescent spots per unit area in the patch of cell membrane
observed in the focal plane using fluorescence microscopy. Solving this equation and using
the fact that this indicates that p → 2πrn in the limit r → 0, we obtain:
(8.116)
p r
rn
r n
( ) =
−(
)
2
2
π
π
exp
Thus, the probability p1(w) that the nearest-neighbor spot separation is greater than a dis
tance w is
(8.117)
p w
p r
r
rn
r n
r
w n
w
w
1
0
0
2
2
1
1
2
( ) =
−
( )
=
−
−(
)
=
−
∫
∫
d
exp
d
exp
π
π
π
(
)
The effective number density per unit area, n, at the focal plane is given by the number of
spots Nmem observed in the cell membrane on average for a typical single image frame divided
by the portion, ΔA, of the cell membrane imaged in focus that can be calculated from a know
ledge of the depth of field and the geometry of the cell surface (see Chapter 3). (Note in the
case of dual-color imaging, Nmem is the total summed mean number of spots for both color
channels.)
The probability that a nearest-neighbor spot will be a distance less than w away is given
by 1 − p1(w) = 1 − exp(−πw2Nmem/A). If w is then set to be the optical resolution limit (~200–
300 nm), then an estimate for 1 − p1(w) is obtained. Considering the different permutations of
colocalization or not between two spots from different color channels indicates that the prob
ability of chance colocalization pchance of two different color spots is two-thirds of that value:
(8.118)
p
w N
A
chance
mem
=
−
−(
)
(
)
2 1
3
2
exp
/
π
It is essential to carry out this type of analysis in order to determine whether the putative
colocalization observed from experimental data is likely due to real molecular interactions or
just to expectations from chance.
8.5.4 CONVOLUTION MODELING TO ESTIMATE PROTEIN COPY NUMBERS IN CELLS
Fluorescence images of most cells containing which specific mobile molecule has been
fluorescently labeled inevitably are composed of a distinct spot component, that is, distinct
fluorescent spots that have been detected and tracked and a diffusive pool component. The
latter may be composed of fluorescent spots that have not been detected by automated code,