The wheels of the colloid goes round and round ....

Fluid jets are all around us. They range in size from astrophysical jets several million light years long shooting out of the heart of galaxies to tiny liquid jets in ink jet printers that may have generated the text that you now read. Here on earth, liquid jets such as the "Old Faithful" geyser in Yellowstone National Park fill us with wonder. Not surprisingly, scientists who are interested in fluid phenomena have turned their attention to jets.

An important landmark in the modern study of jets is Lord Raleigh's (1878) explanation of the break up of a liquid jet into drops, a classic example of an "instability" in fluid flow. In 1944, the Russian physicist Lev Landau  (who later went on to receive the Nobel Prize for his work on superfluidity) presented an exact solution of the equations of fluid mechanics for a "submerged jet from a point source." A submerged jet is what you have when you relax in your hot tub or when you blow out your birthday candles - it is a jet within a bath of the same fluid. The equations of fluid mechanics (the "Navier-Stokes equations") are nonlinear and it is extraordinarily difficult to derive exact solutions, though approximate ones can be found; using computers, for example. Landau's solution is one of a handful of exact solutions that are known. In 1951, an Englishman, H.B. Squire published the same solution, quite unaware of Landau's work. This is perhaps excusable as communication between Russia (the USSR then) and the west wasn't exactly smooth. In the fluid mechanics community this solution has become known as "The Landau-Squire solution". Scientists are not good in history though, they almost always get it wrong. The so called "Landau solution" was actually discovered by Slezkin (Slezkin, N.A. 1934 "On an exact solution of the equation of viscous flow."  Moscow State Univ. Uchenie Zapiski, vol. 2) right on the other side of town and then "re-discovered" twice; approximately once every decade!

We have created in the laboratory what is perhaps the world's tiniest jet. The diameter of our jets range from 40-150 nano meters, that is, a few hundred water molecules lined up in a row. Remarkably, the Landau-Squire solution works almost perfectly for our tiny jet and the theory allows us to calculate the volume flux from it. The Navier-Stokes equations and everything derived from it is supposed to go awry as you approach molecular scales, but no one knows how far down one can push before it breaks. It is a bit like the the next mega earthquake in LA, it is a certainty but nobody knows when it will call. We found that it all works very nicely at the 100 nanometer scale, which is nice. The flow rates turn out to be in the range of tens of pico liters per second. At this rate if you started to fill a 2 liter soda bottle about the time when the first Pyramid was being built in Egypt, your bottle will be about half full now!

The set up for our experiment is shown in the figure below (Panel A). At the heart of the apparatus is a glass "nano capillary" that is fabricated by heating an ordinary glass capillary with a laser and gently pulling it in a machine until it breaks making a fine tip.  The SEM (Scanning Electron Microscope) image on the left shows the tip of one such capillary, the white bar is 100 nano meter long or about a fifth of the wavelength of light. The flow is generated by applying an electric voltage across the capillary which generates an electro-osmotic jet. To "see" the jet we build a tiny "anemometer" the spinning wheels that meteorologists mount on tall buildings to measure wind speed. Our anemometer is a polystyrene (a kind of plastic) bead one fiftieth the width of a human hair. The tiny anemometer is mounted on a spindle made of light - a fine laser beam that constitutes an "optical trap". When the bead is positioned in front of the jet, it spins. Panel B is a light microscope image of an experiment in progress (the white bar is 5 microns - a micron is a thousandth of a millimeter).

In order to measure the rotation rate, the bead is manufactured so as to have a little dimple on one side - kind of like a ping pong ball that has been stepped on! An SEM image of the bead is shown in Panel D (the white scale bar is 1 micron). As it spins, a video camera picks up the tiny fluctuations in light (Panel D & E) coming from the dimpled ball. Click the arrow below to see a video of this. You will see three passes of the capillary tip with the voltage set successively to +1 V, -1 V and 0 V. Notice the direction of spin of the bead. The poor quality of the image is not due to bad equipment but because we are at the limit of resolution of the optical microscope! Our technique of measuring the rotation rate from fluctuations of light intensity is in fact quite similar to the technique astronomers use to determine the rotation period of binary stars that are too far away to resolve in telescopes. The measured rotation rate can be compared to the one predicted on the basis of the Landau-Squire solution and the flow rate determined.

The measurements also reveal something that we had not expected. If you reverse the voltage, the flow direction reverses, that is, the capillary now sucks in the fluid. This is of course expected. However, the flow rate at these negative voltages is much lower. In other words, the capillary behaves like a flow rectifier similar to a semiconductor diode, except, it is fluid rather than electrons that are flowing. If you are perhaps reading this blog on your computer, then billions of semiconductor diodes and transistors are working silently making your activity possible. What applications can a "nano fluidic diode" have? I am afraid, I do not have a clue but I am reminded of the inventor who gave us (almost) all things electrical, Michael Faraday. Faraday  was once visited by a delegation of government dignitaries who were shown his electric motors and other demos. One of them said "This is all very interesting, but of what possible use are these toys?" Faraday responded: "I cannot say what use they may be, but I can confidently predict that one day you will be able to tax them."

Acknowledgement: The experimental part of the work was carried out in the laboratory of my colleague Ulrich Keyser at the Cavendish Laboratories, Cambridge University in the UK where I was a Leverhulme Visiting Professor. I am grateful to the Leverhulme Trust for making this possible. I would also like to thank the NIH in the US for support. The SEM image of the nanocapillary was taken by Lorenz Steinbock at the Keyser lab.

REFERENCE

Our paper was published in Nano Letters

[1]  A Landau-Squire nanojet authors: Nadanai Laohakunakorn, Benjamin Gollnick, Fernando Moreno-Herrero, Dirk Aarts, Roel PA Dullens, Sandip Ghosal, Ulrich F Keyser. Web publication date: 2013/10/14  Journal: Nano Letters

Press Reports: News from McCormick

And three is a crowd....

Figure 1: Phone booth stuffing

Figure 1 is an image from the 1950's fad of "Phone Booth Stuffing". The Guinness book of world records reports N=27. We had been doing something similar - stuffing DNA into nanopores. Actually DNA exists in nature in a very compacted state. We carry about 3 meters of DNA in each of our cells but all of this is contained in a cell nucleus about 6 microns in diameter. This is nicely illustrated in Figure 2 showing a "DNA spill" from a ruptured E. Coli cell.  Multiple DNA strands interacting in a confined environment is encountered in various other situations e.g. the packaging of DNA in phages and the movement of DNA through the pores of a gel during electrophoresis.  How DNA behaves in crowded environments is therefore a subject of some interest.

Figure 2: A DNA spill from E.Coli

It is actually not difficult to stuff many DNA molecules into a nanopore, the hard part is to keep count of how many got in! Figure 3 shows an experimental set up where this can be done in a controlled way. We use nanocapillaries with diameters in the range 20 - 200 nm that can be made very cheaply using a commercial pipette puller. The inside of the capillary is kept at a positive voltage relative to the bath. A DNA coated polystyrene bead held in a Laser Optical Trap is very gently moved towards the nanopore. As the bead approaches the pore, DNA is yanked into the pore by the strong electric field at the entrance region. The capture of each DNA strand is observable as a change in the force acting on the bead (measurable through its displacement in the optical trap) as well as through a change in the measured current. A typical data set is shown in Figure 4 where we "see" N=1,2,3,4,5, .... DNA being yanked into the pore. 

Figure 3: Experimental
set up 

Figure 4: The sequential capture of DNA
strands in the nanopore

The data from such an experiment is shown in Figure 5. It is seen that the force on the bead is proportional to the applied voltage, which is expected. However, the force is not proportional to the number of DNA molecules occupying the pore. In fact, the force per DNA strand is observed to decrease with the number of DNA strands occupying the pore. 

Figure 5: The force scales linearly with
the applied voltage but not with the number
of DNA strands in the pore. 

The explanation for this behavior lies in the hydrodynamic coupling between the individual DNA strands. When the electric field is switched on, the negatively charged DNA moves in a direction opposite to the field. Simultaneously, there is a gush of electroosmotic flow in the direction of the applied field driven by the positively charged counter-ions that surround the DNA as well as the capillary wall. This flow creates a hydrodynamic drag that slows the DNA down. Adding more DNA to the pore increases this electroosmotic flow as each DNA acts as an "electroosmotic pump". Thus, even though the electric force on an individual DNA is not changed by the presence of neighbors, the hydrodynamic drag is. This idea can be transformed into a scaling law according to which the force per strand is a linear function of ln N / ln (R/a) where R is the pore radius and a is the DNA radius. It is shown that the experimental data is in accord with this scaling law. 

REFERENCES

1. Laohakunakorn N., Ghosal S., Otto O., Misiunas K. & Keyser U. "DNA Interactions in Crowded Nanopores" Nano Letters (2013), 13 (6), 2798–2802 

Catch a colloid by the tail !

About five years ago, an interesting twist [1] on the resistive pulse technique emerged from the Dekker lab at TU Delft (Netherlands). Their set up is shown in the sketch on the left. One end of the DNA was attached to a polystyrene bead which is held in place by a spot of laser light: a very useful innovation in nanotechnology known as a Laser Optical Trap (LOT) (aka Optical Tweezers). When a voltage is applied and the DNA starts to translocate, the LOT holds it back and "frustrates" the translocation. As the DNA pulls at its tether, the bead is displaced slightly from its equilibrium position which gives a way to directly measure the force acting on the DNA. Instead of yielding the translocation time this method yields the "tether force" together with the conductance change of the pore. This set up can be used to test the idea of the "hydrodynamic origin" of the resistive force discussed in my last Blog. I was able to calculate [2] the tether force using a modification of the theory for translocation times discussed earlier, and this simple analytical formula could be compared with the tether force measurements [3].

This is shown in the panel to the left. The top sub figure shows that the tether force is proportional to the applied voltage. The force per unit applied voltage is plotted in the lower figure against the pore radius (R). The data shows quite clearly the 1/ln R dependence of the force predicted by theory. The dashed curve is obtained if the DNA "bare charge" is used. The solid curve is obtained if the DNA "effective charge" is considered less than the bare charge by a fixed ratio q, considered here as a fitting parameter.

The figure on the right shows a more recent [4] experimental test of the hydrodynamic theory emerging out of the Keyser Lab at the Cavendish.  The Keyser lab has capitalized on an astoundingly simple (= cheap!) way of producing nanopores. They heat a glass microcapillary with a laser and use the traditional glass pulling technique to obtain a sharp tip. Internal diameters of 10-100 nanometers can be obtained in this way. The experiment shows the force signal as a trapped DNA is gradually pulled out of the pore. The force can be calculated using the lubrication theory [5,6] for electrokinetics and is shown by the red dots in the lower panel.

These experiments give us confidence that the principal resistive force in DNA translocation does indeed arise primarily from hydrodynamic drag in the pore region. The challenge now is to use this knowledge to evolve new tools for characterizing DNA such as the base sequence, interactions with proteins and fundamental questions related to the behavior of DNA as a charged polymer. This is still an open book with many exciting possibilities in basic science as well as in nascent technologies. Already, a new term is being used in this context "DNA Force Spectroscopy"!

Acknowledgement:  My research in this area has been supported by the NIH (USA) and by the Leverhulme Trust (UK).

References

[1] Optical tweezers for force measurements on DNA in nanopores (2006) UF Keyser, J van der Does, C Dekker & NH Dekker Rev. Sci. Instruments 77 (10)

[2] Electrokinetic-flow-induced viscous drag on a tethered DNA inside a nanopore (2007) S Ghosal Physical Review E 76 (6), 061916

[3] Origin of the electrophoretic force on DNA in solid-state nanopores (2009) S van Dorp, UF Keyser, NH Dekker, C Dekker & SG Lemay Nature Physics 5, 347 - 351

[4] Single macromolecules under tension and in confinement  (PhD thesis, Cambridge University) Oliver Otto 2011

[5] Lubrication theory for electro-osmotic flow in a microfluidic channel of slowly varying cross-section and wall charge (2002) S Ghosal Journal of Fluid Mechanics 459, 103-128

[6] Electrophoresis of a polyelectrolyte through a nanopore (2006) S Ghosal Physical Review E 74 (4), 041901
 
[7] Effect of salt concentration on the electrophoretic speed of a polyelectrolyte through a nanopore (2007) S Ghosal Physical review letters 98 (23), 238104

How fast do DNA zip through nanopores?

The resistive pulse technique provides two basic observables from which to infer the properties of the translocating molecule. These are (a) the blockade time (b) the relative conductance change.  The first of these is a measure of the speed (v) of the translocating molecule.

The force driving the translocation is the electrical force acting on the part of the DNA inside the pore. This force is the voltage drop across the membrane times the charge on the polymer within the pore. For typical parameters, this amounts to about 20 pico Newton (pN). Since inertia plays no role at such small scales, to calculate the speed we need to set this driving force to equal a resistive force that will depend on the speed of translocation (v). What is the origin of this resistive force? I made the hypothesis that the resistive force arises primarily from the hydrodynamic drag on the part of the polymer that occupies the pore. The frictional drag from the polymer coil outside the pore as well as the "entropic force" required to straighten the polymer against thermal fluctuations are both small compared to this hydrodynamic drag from the pore. If the primary drag force arose from within the pore, the speed v would be independent of polymer length. This appears to hold to a good approximation from the experimental data.

The hydrodynamic drag on a rod (the DNA) translating through a slightly larger pore (of arbitrary cylindrically symmetric shape) may be calculated using the classical "lubrication theory" of fluid mechanics. In our problem there is some additional physics not encountered in the classical problem and it is this: both the DNA and the wall of the capillary are charged and there is an applied electric field. As explained by Peter Debye all charged macromolecules carry a cloud of mobile counter-ions (positive ions or cations in the case of DNA) that "shield" their electrical interactions with other objects. The electric field acts on these counter-ions to create a body force that drives a jet of fluid upwards through the gap as the DNA moves downwards. This "electroosmotic" effect must be included in the calculation of the drag. Nevertheless, even with this additional effect the problem can be solved and an expression for the velocity obtained in closed form. The graph to the left shows how this formula measures up against the experimental data. The interesting thing about this graph is that it does not involved any "adjustable" parameters even though reasonable approximations (detailed in the papers cited below) need to be made. The upper dashed curve is calculated using the DNA bare charge and the lower curve uses the DNA "effective" charge in accordance with the Manning theory of counter-ion condensation.

press report

References

1. Electrophoresis of a polyelectrolyte through a nanopore S Ghosal (2006) Physical Review E 74 (4), 041901

2. Effect of salt concentration on the electrophoretic speed of a polyelectrolyte through a nanopore (2007) S.Ghosal Physical review letters 98 (23), 238104

(Background) How polymers cross membranes

A cell is enclosed in a lipid bi-layer membrane which serves as a reaction chamber for the biochemical processes of life. Within the cell there are various organelles such as the nucleus, mitochondria and so on which too are surrounded by membranes. In order for the cell to function, some but not all bio-molecules must be able to cross the membrane. This usually happens through proteins that form pores on these membranes. These pores are very small, often a nanometer or so in  size. They serve as border "check points" for the intra-cellular and intra-organelle "traffic".  Many important bio-molecules are long chain polymers e.g. DNA, RNA and proteins. The physical process by which such polymers cross membranes is of importance for understanding how cells function.

As in other areas of science, real insight is often gained by studying an effect in isolation free of the influences of non-essential phenomena. Biologists call this "in vitro" experiments (in vitro = in the test tube as opposed to inside a living organism or "in vivo"). In 1996, Kasianowicz et al. published a very beautiful in vitro experiment that mimicked the natural process of polymer translocation across membranes. A version of the experiment (figure taken from the later paper by Meller et al.) is shown in the left. A natural protein that goes by the name of "alpha-hemolysin" was extracted from the "staph" bacteria Staphylococcus aureus. This protein is a heptamer - it comes in seven parts like Leggo pieces. When the pieces are absorbed on to a lipid membrane, they self assemble forming a pore. In the experimental set-up the lipid membrane formed the partition between two baths containing salt water across which an electric voltage was applied. The assembly of the pore was signaled by the current that would start to flow as soon as a conducting path through the membrane was established. When a small amount of DNA was added to the negative side of the bath, every now and then, the electric field would shoot a DNA molecule through the pore (DNA being negatively charged). Each time this happened, the pore would be transiently blocked creating a dip in the current signal. The current is "quantized" that is, it only takes one of two values the low one when the DNA is in the pore and the high one when it is out. The signal contains information about the length of the DNA strand, the density of DNA in the cell and perhaps even the identity of the bases of the DNA.

This pioneering paper has led to a flood of experimental work refining the technique which has come to be known as "the resistive pulse technique". The idea that the method can be refined to directly read the base sequence of DNA has led to a nanotechnology gold rush for the "Thousand Dollar Genome", the goal of sequencing a person's DNA at a cost of less than a thousand dollars (the Human Genome Project cost 3 billion dollars). Perhaps on a less grand scale, it raises some interesting physics questions such as how fast does the DNA go through? Can this speed be controlled and so on. I will be posting on these issues in future blogs .... so don't go away!

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,z_p,z_n) and electrophoretic mobility (μ,μ_p,μ_n). 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.

Acknowledgements

This research is Supported by the NIH under grant R01EB007596.

Reference

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

(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.