Chapter 8 – Theoretical Biophysics 361
diffusion for a certain distance before being captured by a “pure” absorber—a pure
absorber is something that models the effects of many interactions between ligands and
receptors such that the on-rate is significantly higher than the off-rate, and so within a
certain time window of observation, a particle that touches the absorber is assumed to
bind instantly and permanently. By simplifying the problem in assuming a spherically
symmetrical geometry, it is easy to show that this implies (see Berg, 1983) that a par
ticle released at a radius r = b in between a pure spherical absorber at radius r = a (e.g.,
defining a the surface of a cell organelle) and a spherical shell at r = c (e.g., defining
the surface of the cell membrane), such that a < b < c, the probability P that a particle
released at r = b is absorbed at r = a is
(8.92)
P
a c
b
b c
q
=
−
(
)
−
(
)
So, in the limit c → ∞, P → a/b.
General analysis of the shape of experimentally determined mean square displacement
relations for tracked diffusing particles can be used to infer the particular mode of diffusion
(Figure 8.6). For example, a parabolic shape as a function of time interval over large time
scales may be indicative of drift, indicating directed diffusion, that is, diffusion that is direc
tionally biased by an external energy input, as occurs with molecular motors running on a
track. Caution needs to be applied with such analysis however, since drift can also be due to
FIGURE 8.6 Different diffusion modes. (a) Schematic cell inside of which are depicted four
different tracks of a particle, which exhibits four different common modes of diffusion or anom
alous, Brownian confined and directed. (b) These modes have distinctly different relations on
mean square displacement with time interval, such that directed is parabolic, Brownian is linear,
confined is asymptotic, and anomalous varies as ~τα, where τ is the time interval and α is the
anomalous diffusion coefficient such that 0 < α < 1.