In a simple fluid medium, microscopic particles undergo familiar Brownian
motion or “random walks”: for a collection of particles, the
mean-square displacement (MSD) from their initial positions increases linearly
with time and the distribution of the displacements is Gaussian at any given
moment of time. On the other hand, in complex,
disordered, “crowded” media *anomalous* diffusion is often found,
where the MSD changes more slowly than linearly with time. Recently,
Steve Granick's group
at the University of Illinois has found that in a number of
systems, colloidal particles diffuse in such a way that the MSD is linear in
time, as in normal diffusion, but the displacement distribution is not Gaussian,in fact, the tails of the distribution are close to exponential.
They have called this behaviour “anomalous yet Brownian” diffusion. We argue that
“anomalous yet Brownian” diffusion should be observed when particles
have very short memory of the direction of their motion, but much longer memory
of the rate of diffusion, so that long periods of fast diffusion alternate with
periods of slow diffusion. Just as in Granick's experiments, in our
“diffusing diffusivity” model of this situation we obtain
displacement distributions that are close to exponential for a variety of
conditions. A variant of our model can also produce “usual” anomalous diffusion. In fact, one does not need a complex medium to observe “diffusing diffusivity” effects; we are currently collaborating with
the group of Prof. John Bechhoefer at Simon Fraser University that studies diffusion of a particle in a simple fluid near a wall and sees similar effects. Our paper on the “diffusing diffusivity” model was recently published in Phys. Rev. Lett. (publication
27) and a longer paper is planned.

Electrophoresis is the motion of microscopic charged objects in a fluid under
the influence of an applied electric field. Electrophoresis of DNA,
in particular, is widely used in molecular biology for separating mixtures of
DNA molecules of different lengths. This separation does not occur in free
solution and traditionally, DNA electrophoresis has been carried out in gels or
entangled polymer solutions. Another possibility that has been studied is using
microfabricated arrays of obstacles instead of a gel. Yet another attractive
option is getting rid of a gel-like medium altogether, *but* attaching
identical uncharged or oppositely charged polymers (that serve as
“parachutes”) to DNA molecules that need to be separated. I have worked or am currently working on a
number of projects in this area.

**Electrophoresis of rigid rods in obstacle arrays** Electrophoretic motion
of *flexible* polymers in arrays of obstacles (modeling nanoporous media,
such as a gel, an entangled polymer solution, or a microfabricated array of
obstacles) has been studied extensively in computer simulations. However, the
double-stranded DNA can be stiff on the length scale of a single pore, and there
are even stiffer objects of interest, such as rodlike viruses. Using Brownian
Dynamics simulations, we have studied electrophoresis of *stiff rods* in
arrays of obstacles. We find a nonmonotonic dependence of the velocity of the
rods on their length. As the length increases, the rods first move more slowly
as they collide with obstacles more often. However, as the rod length increases
further, they orient along the direction of their motion and pass through the
obstacle array more easily. A simple scaling theory that we have developed
predicts this behaviour. This research has not been published yet, but see
presentation slides here.

**Free-solution electrophoresis of composite objects** As mentioned, using
electrophoresis, DNA fragments can be separated by length in a nanoporous
medium, such as a gel or a polymer solution. Gel-free (free-solution)
electrophoresis is simpler and faster, but in free solution the electrophoretic
velocities of all but the shortest DNA fragments are the same and thus
separation is impossible. Attaching identical neutral or positively
charged “labels” (or “drag-tags”) to the DNAs should
enable their separation, as smaller DNAs are slowed down more than larger ones.
This method is called end-labeled free-solution electrophoresis (ELFSE).
An existing simple theory of ELFSE neglecting hydrodynamic
interactions (HI) between different parts of the DNA–drag-tag complex predicts
a linear dependence of the migration time on the inverse DNA length, regardless
of the properties of the drag-tag. While experiments seem to agree with this at
first glance, we have argued that this agreement is often illusory.
Calculations taking HI into account show that the dependence can be strongly
nonlinear and qualitatively different depending on the properties of the
drag-tag (whether it is linear or branched or globular, stiff, flexible or semiflexible,
etc.) and the point at which the DNA is attached to it. Our results are also
potentially important for understanding other electrophoretic techniques
involving composite objects, such as some variants of affinity electrophoresis. We have published two papers on this subject recently
(publications 26 and 28) and
several more are in preparation. See also presentation slides here and here.

**Separation matrix inhomogeneities in capillary electrophoresis** In their
work to improve the polymer matrix for DNA separation in microfluidic devices,
Prof. Annelise Barron's group
at Stanford has discovered an interesting
phenomenon: in two different polymer solutions, the DNA velocities are nearly
the same, yet the amount of dispersion (or peak broadening) differs by up to an
order of magnitude. We have studied a possible role of matrix inhomogeneities
in this effect.

**Optimizing the accuracy of lattice Monte Carlo** Monte Carlo simulations on
a lattice provide a faster alternative to continuum Monte Carlo methods and
Molecular Dynamics, but are generally less accurate. For the simple case of
simulations of diffusion of point particles, we have shown that introducing the
possibility for the particle to stay put at a given step of the algorithm and
reducing the time step accordingly can increase the accuracy of the simulation
and that for a given lattice step there is a value of the time step for which
the accuracy of the algorithm is optimal. This may make it possible to increase
the accuracy of lattice Monte Carlo methods and make them more competitive
compared to the alternatives. See publications
24 and 23.

**Lattice Monte Carlo for inhomogeneous systems** Another project (with
Hendrick de Haan from UOIT)
involves constructing lattice Monte Carlo methods for simulating diffusion in a
system where diffusivity varies in space, with either sharp or gradual
boundaries between regions with different diffusivities. Diffusion in an inhomogeneous medium is a tricky
problem, since the solution of the corresponding stochastic differential
equation depends on how the noise term is interpreted. Our algorithms take this
into account and work for different interpretations of the noise term.
Some of our results are described in publication 25 (see also presentation slides here).

When a patient takes some medication, it is important to know the rate at which the drug is released into the patient's body. In the simplest model, the drug can be assumed to simply diffuse out of the drug delivery system, such as a pill, and immediately get absorbed. In this case, it is common to describe the drug release profile (the amount of drug released by a given time, as a function of that time) by the stretched exponential (or Weibull) function with two parameters, a characteristic time and a stretching exponent that is sometimes assumed to characterize the fractality of the medium of the pill. With Maxime Ignacio, a postdoc in our group, we have proposed an alternative, physically motivated drug release function that likewise contains two parameters, but these parameters are the two times characterizing the short- and long-time drug release, respectively. We have compared our function to the Weibull function for the simplest case of a homogeneous medium with different initial loading profiles of the drug inside it and showed that it consistently outperforms the Weibull function. Moreover, the best fits with the Weibull function produce nontrivial values of the stretching exponent that depend on the loading profile and thus cannot characterize the medium itself. A paper describing this research is in preparation.

Studies of diffusion in porous media are important for various applications ranging from oil exploration to DNA separation. One of the simplest models of porous media is a periodic array of cavities. In this work with Francis Torres, we have obtained some analytical and numerical results for diffusion of pointlike particles in such arrays in two and three dimensions. We plan to publish these results in the near future.