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.