Anomalous diffusion in biological fluids

Scott McKinley (June 21, 2012)

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Rapid recent progress in advanced microscopy has revealed that nano-particles
immersed in biological
uids exhibit rich and widely varied behaviors. In some
cases, biology serves to enhance the mobility of small scale entities. Cargo-laden
vesicles in axons undergo stark periods of forward and backward motion, inter-
rupted by sudden pauses and periods of free di usion. Over large periods of time,
the motion is e ectively that of a particle with steady drift accompanied a di u-
sive spread greater than what can be explained by thermal
uctuations alone. As
another example, E. coli and other bacteria are known to respond to the local con-
centration of nutrients in such a way that they can climb gradients toward optimal
locations. Again, the e ective behavior is drift toward a desired" location, with
enhanced di usivity.
In other cases, biological entities are signi cantly slowed. Relatively large parti-
cles di using in
uids such as mucus, blood, bio lms or the cytoplasm of cells all
experience hinderances due to interactions with the polymer networks that consti-
tute small-scale biological environments. Researches repeatedly observe sublinear
growth of the mean-squared displacement of particle paths. This signals to theo-
reticians that the particles are not experiencing traditional Brownian motion. In-
terestingly, many viruses are actually small enough to avoid this type of hinderance
when moving through human mucus. However, the body's immune response in-
cludes teams of still smaller antibodies that can immobilize virions by serving as an
intermediary creating binding events between virions and the local mucin network.
Underlying the mathematical description of all these phenomena is a modeling
framework that employs stochastic di erential equations, hybrid switching di u-
sions and stochastic integro-di erential equations. We will begin with the Langevin
model for di usion. This is the physicist's view of Brownian motion, derived from
Newton's Second Law. We will see how the traditional mathematical view of Brow-
nian motion arises by taking a certain limit. The force-balance view permits a
variety of generalizations that include particle-particle interactions, the in
uence of
external energy potentials, and viscoelastic force-memory e ects. We will use sto-
chastic calculus to derive important statistics for the paths of such particles, develop
simulation techniques, and encounter a number of unsolved theoretical problems.