Monday, November 5, 2012

Electromigration dispersion & the Burgers equation

Electromigration dispersion is one of several “anamalous dispersion” mechanisms in capillary zone electrophoresis (CZE). It results in strange wedge shaped peaks that appear when sample concentrations are too high. This may be seen in the electropherogram displayed below (reproduced from: Bouskova et al. Electrophoresis 2004, vol. 25, pg. 355-359).

The simplest model is a 3 ion system: a sample ion (of either sign) a positive and a negative ionic species. All three species are assumed to have the same diffusivity (D) but they differ in their charge (z,zp,zn) and electrophoretic mobility (μ,μpn). All components are considered strong electrolytes (fully dissociated).  Using a system of rational approximations, we reduce the system of coupled equations to a single one dimensional nonlinear partial differential equation for the normalized sample concentration ϕ. If the sample concentration is not too large (the weakly nonlinear regime), we show that this equation reduces to Burgers equation.

Burgers equation is one of the few nonlinear equations that admit an exact solution for arbitrary initial conditions. The solution is found by using the nonlinear Cole-Hopf transformation to reduce it to the Fourier heat equation. It provides a mathematical description of a wide variety of physical problems: water waves, shocks in gas dynamics, traffic flow on roads, a one dimensional model of turbulence etc. Now we can add CE to this list of applications.

Many features of the CE signal may now be understood in terms of the properties of Burgers equation. We show for example that either leading edge or trailing edge shocks are possible, depending on whether the sample valence z lies between the valence of the anion and cation or outside this range. A Peclet number related to the "sample loading" may be defined. We show that a full range of peak shapes from slightly skewed Gaussian to a triangular saw-tooth shaped wave may be generated depending on whether this Peclet number is small or large compared to unity.


This research is Supported by the NIH under grant R01EB007596.


1. Ghosal, S. and Chen, Z.  "Nonlinear Waves in Capillary Electrophoresis" Bull. of Math. Biol. (2010), 72(8), 2047
2. Chen, Z. and Ghosal, S. "Electromigration dispersion in Capillary Electrophoresis" Bull. Math. Biol. (2010), 73(12), 346

Thursday, November 1, 2012

(Background) What is Electrophoresis?


1. What is Electrophoresis?

The speed of migration of molecules in an aqueous medium under an applied electric field depends on the size and charge of the molecule. This phenomenon, which is called electrophoresis, may be used to separate a mixture of molecules into its components.

2. What are its applications?

Electrophoresis has a wide range of applications. Gel electrophoresis is a standard procedure in molecular biology. The separation medium is a porous gel permeated by an electrolyte (a salt solution). A drop of the sample is placed on the gel and an electric voltage is applied. The gel may be stained later to make specific components visible. In this way biologists “run a gel” to discover for example which proteins are present or absent when certain genes are expressed. We have all heard of “DNA fingerprinting” or the “Human Genome Project”. These applications use electrophoresis to sort DNA by size (see The figure on left, from Agilent technologies)

3. What is Capillary Electrophoresis?

Capillary Electrophoresis (CE) is the modern way of “running a gel”. In CE instead of using a porous network permeated by an electrolyte as the molecular race track one uses a single micro-capillary filled with an electrolyte that connects reservoirs at either end. The capillary must be very narrow 25-75 micron internal diameter is commonly used. Larger diameters result in poor separation due to excessive Joule heating, convective mixing of the fluid and other effects. A UV light source and photodetector near one end of the capillary picks up the signal as a series of peaks and troughs that correspond to modulation of the UV intensity due to adsorption by sample components.


4. What are the different modes of CE?

The simplest kind of separation in CE is Zone Electrophoresis. Here molecules migrate in response to an applied electric field and separate into zones by virtue of their different electrophoretic mobilitiies. Usually the migration of the molecules is accompanied by a bulk flow of the fluid in the capillary (electroosmotic flow) because of electrostatic charge on the capillary wall. The electroosmotic flow causes both positive and negative components in the sample to drift in the same direction and therefore pass through a single detector near the capillary exit. Another kind of separation that is used for proteins is known as isoelectric focussing. Here an electric field and a pH gradient are simultaneously applied across the capillary. Proteins have the property that their charge depends on the pH of the surrounding medium. Therefore they move to the location where the pH is such that the charge is neutral (the iso-electric point of the protein). In iso-tachophoresis all sample components actually move at the same velocity but arrange themselves in layers with the ordering depending on the mobility of the ions.


5. What are the advantages of CE?

The main advantage of CE is that wet chemistry can be done via fluidic circuits on glass or silicon chips in the same way that digital electronics is done today. Many biochemical protocols (such as genome sequencing) call for a series of chemical operations repeated a vast number of times. Therefore, miniaturization provides the usual advantages of scalability and parallelization that drove the semiconductor revolution in the last century. However, such a “Lab on a Chip” that performs complex biochemistry in a miniaturized automated setting is in its infancy when compared to the vastly developed semiconductor chip. Practical, albeit quite simple on chip CE systems are sold commercially by some biotech companies such as Caliper.


6. What is the role of mathematical modeling in CE?

The design issues in CE are similar to the questions that arise in optical systems such as the microscope. Just as for a microscope one is interested in the minimum achievable angular resolution, in a CE system we would like to know the smallest difference in mobilities of molecules that may be detected. An ideal microscope should be “diffraction limited”, that is, it has the best optical performance that is possible under the constraint that light is a wave and undergoes diffraction. Likewise, an ideal CE system is “diffusion limited” that is, the resolution is as good as it can be, given the constraint that concentration peaks spread due to the diffusivity of molecules. In a microscope, there are a host of phenomena: spherical aberration, chromatic aberration etc. that stands in the way of designing a diffraction limited instrument. Likewise in CE, there are phenomena such as Taylor dispersion, Electromigration Dispersion etc. that stand in the way of achieving a diffusion limited performance. The only difference is that in designing a microscope one needs to understand the physics of light propagation through media whereas in CE the relevant physics has to do with fluid flow and transport in an ionic medium.


7. Where can I learn more about this?

There are a number of textbooks (such as Electrophoresis: a survey of techniques and applications, Handbook of capillary electrophoresis and The dynamics of electrophoresis) devoted to capillary electrophoresis. If you are interested in the mathematical modeling aspects of it you may want to start with the review, Electrokinetic flow and dispersion in capillary electrophoresis (if your home is in engineering) or Fluid mechanics of electroosmotic flow and its effect on band broadening in capillary electrophoresis (if your home is in chemistry). There are a number of broader reviews (like Engineering flows in small devices by H.Stone, Microfluidics: Fluid physics at the nanoliter scale by TM Squires and SR Quake) on the physics and mathematics of microfluidic systems that you may also look into.