Trying to measure ionic current through small holes

Two of the bigger biomolecular engineering research groups at UCSC rely on the same basic concept: measuring ionic current through tiny holes. Mark Akeson’s nanopore research uses proteins (porins like alpha hemolysin or MspA) to punch a single tiny hole in a lipid bilayer, about 1.2nm in diameter.  Nader Pourmand’s group makes nanopipettes with orifices about 50nm in diameter and uses them for a variety of different sensors, as well as for manipulating biomolecules (like extracting DNA from mitochondria).

I’ve been thinking that it would be good to have a simple model of these systems in the circuits course, so that students have some understanding of how they work without the exacting lab techniques needed to make and handle nanopores or nanopipettes. (Students who are interested in those can join Mark’s or Nader’s labs—they both have large numbers of undergrads doing research with them.)

My idea was to use off-the-shelf disposable pipette tips (which cost only pennies) in a saline solution and measure the current through the tip and the capacitance of the tip.  Unfortunately, it is a little difficult to get specs on orifice sizes from the pipette-tip manufacturers (except for wide-orifice tips designed for moving whole cells without breaking them), as the crucial spec for most applications is the volume of the tip, not the diameter of the orifice.  In fact, it is hard to find any dimensional information other than volume.

I got a selection of tips from David Bernick to experiment with.

Pipette tips to experiment with (sorry, I don’t have any idea which tips these are—if one of them works well, then I’ll have to ask where to get more from).

The tiny tips seem designed to mount on a 1/8″ or 3mm OD tube from the pipetter, the next size up for 4mm, then 5mm, then 7mm. The outside diameters of the tips at the orifice seem to be 0.8mm, 1mm, 1mm, and 1.5mm.  I had a harder time estimating the orifice sizes, since I couldn’t use the calipers directly.  I got sizes about 0.25mm, 0.25mm, 0.5mm, and 1mm. The lengths are about 31.3mm, 45.6mm, 52.5mm, and 76mm.

If my measurements are at all accurate, they imply fairly large differences in the wall thickness.  I believe that all these pipette tips are polypropylene, which has a fairly low dielectric constant of 2.2–2.6.

How much resistance should we expect through the pipette tips, and what does it depend on?  The simplest model to use is a bulk conductivity of the solution in the pipette tip and some geometry.  Sea water (43g salt / liter water at 20°C) has a bulk resistivity of about 0.2 Ωm (or conductivity of 5 S/m), according to Wikipedia.  The resistivity of NaCl solutions depends on both concentration and temperature, which is why conductivity meters need temperature sensors in them to be useful for measuring concentration of ions.

According to a Defense Documentation Center report from 1964, resistivity for low concentration is easily found from reference sources, but high concentrations needed to be measured.  That report gives the resistivity of 4.9% NaCl solutions (about 0.8 M) as varying from 12 Ωcm at 30°C to 21 Ωcm at 5°C.  These are roughly consistent with the Wikipedia value for sea water (the graphs in the report show about 0.17 Ωm rather than 0.2 Ωm).

Conductance Measurements, Part 1: Theory has a simplified description of the theory of conductance and how conductance is measured, including some about why AC rather than DC measurements are done. Their explanation for using time-varying excitation waveforms involves a series capacitance in the cell (possibly from insulating layers building up on the electrodes?) rather than nebulous “electrolytic effects” mentioned in other sources. It gives the standard formula for the conductivity of a solution: , where is the limiting molar conductivity, C is the molar concentration and K is Kohlrasuch’s coefficient. The units for K must be weird: S M^-3/2 m^-1, since molar conductivity is S m⁻¹ M⁻¹ and concentration C is M. But the reference above tabulated limiting molar conductivity for several ions and gives the most commonly used temperature correction but doesn’t give K for NaCl.

In All-electronic biosensing in microfluidics: bulk and surface impedance sensing, we can find an estimate for K:

Within the framework of this model, K depends mostly on the valency of the ions in the electrolyte,and for aqueous NaCl, K ≈ 0.83. The molar conductivities at infinite dilution of many ions are tabulated [25], those of Na⁺ and Cl⁻ are 5.011 S m⁻¹ M⁻¹ and 7.634 S m⁻¹ M⁻¹ respectively. Using a conductivity meter to measure the conductivity of 1 M NaCl, we find σ ≈ 0.85 S/m (Chapter 4). If we were to assume a linear dependence of conductance on ion concentration (K = 0 in Equation 2.3), we would expect σ ≈ 1 S/m.

I don’t trust that source, though, as no units are given for K and the limiting molar conductivities of the ions are off by a factor of 1000 from other sources (which have 5.011 mS m⁻¹ M⁻¹ and 7.634 mS m⁻¹ M⁻¹).  Also, the resistivity measurements reported elsewhere give about 0.12–0.14 Ωm for 1M NaCl (depending on temperature), for a conductivity of 7.1–8.3 S/m, not 0.85 S/m.

Image of resistivity as a function of concentration for NaCl. Copied from http://www.hendersonpetrophysics.com/Images/rw25.jpg
Note that until the solution is almost saturated, this is essentially a straight line with slope -1 (resistivity is inversely proportional to concentration).

I made up an approximately 1M NaCl solution in my kitchen (using 58±1g salt in 1000±2g tap water), which should have a resistivity of about 0.13 Ωm, since my house is at about 20°C.  So what should the resistance of a pipette tip be filled with 1M NaCl?

If we simplify the shape of  the pipette tip to a truncated cone, with orifice diameter d₁, base diameter d₂, and length L, we can get the cross-sectional area of the cone at any place along it .  This gives us the resistance as , which is . Using ρ=130 Ω mm, and my crude measurements of the pipette sizes, I get 6.9 kΩ, 1.9kΩ, 1.7kΩ, and 1.2kΩ for the different pipette tips.

Unfortunately, making a measurement of the resistance will be difficult, unless I can make electrodes that fill the top of the pipette tip—otherwise, the geometry of the electrode will affect the measurement. Just dangling a wire in the pipette tip, as I was initially thinking of doing, is not going to give any useful information.

If we plug the top of the pipette tip with an electrode, then filling the pipette tips with solution is a problem also.  It is not too difficult to fill the tips with water even without a syringe or pipettor, just by dunking them and shaking a bit to force the air out, but if the top is plugged, filling without bubbles could be a problem.

And what about the saline solution on the other side of the pipette tip?  That will make another cone, expanding from the orifice to the electrode.  We can make the area of the electrode large and position the pipette tip close to it, so that the resistance is much lower than the resistance inside the pipette tip, but the geometry still needs to be controlled.

I’m beginning not to like the idea of using pipette tips.

I could probably fill the smallest pipette tip with salt water, then push it onto the end of a 1/8″ stainless steel welding rod to make a firm connection. I could then put the tip of the pipette a short distance above a flat electrode (or a stainless steel cup) and fill with salt water to the fill line on the pipette tip.  I’d need some sort of (nonconductive) clamp to hold the welding rod steady. This should give fairly repeatable measurements, at least until crud builds up on the stainless steel.

Another approach would be to use two pipette tips, one for each electrode.  Then the stainless steel rods could be firmly mounted to a plastic holder, and the tips dipped into salt water up to their marks (after first filling the tips and squeezing them onto the rods to get rid of air bubbles).  This could give a consistent setup without needing too much fussing.

Another approach

What else could I use?  How about a plastic bag with a pin prick?  I could use a couple fairly large electrodes then, one inside the bag, one outside the bag.  Let’s say I made a 1mm hole in a bag that was 1mil (25 μm) thick, which is the thickness of Ziploc sandwich bags (as measured with my micrometer), and stuck a large electrode in the bag and outside the bag pressing them together with the bag in between. Then I’d be essentially measuring the current through the cylindrical hole 25 μm long and 1mm across (a resistance of about 4Ω).  For testing I could use quarters, which are about 24mm in diameter.

Of course, I should probably first test the bag without a hole, to make sure there isn’t much leakage current through the bag.

OK, I tried the bag, and I’m not happy with it either.  The resistance with no hole is quite high, but putting a tiny pin prick in the bag reduces the resistance way down.  Unfortunately, using quarters as electrodes and using an ohmmeter does not produce usable results—there are galvanic currents generated that produce readings anywhere from 1 kΩ to –4 kΩ (yes, negative resistance readings). OK, that was expected—there are good reasons for doing AC resistance measurements to get the conductivity of solutions.

Small movements of the electrodes changed the readings substantially, so some sort of clamp is going to be needed. Worse, it is extremely easy to spill salt water out of the bag, which is not something we want happening in the electronics lab.  We want any corrosive, conductive fluids contained neatly without much chance of spilling on the electronics equipment.

What’s the point of the lab

I need to back off a bit and rethink the point of the lab.  The nanopore setup measures currents of a few picoamps with a DC bias across the lipid membrane, while the nanopipette setup usually uses a small AC voltage and looks at the resulting (often rectified) currents.  Conductivity meters use AC (or pulsed) waveforms to avoid electrolytic effects and series capacitance. Note: there are hobbyist versions of conductivity meters on the web, and the biggest problem seems to be making a good probe.  I think that we might want to use the bipolar pulse technique (measuring the current at the end of a quick, symmetric charge-discharge pulse) rather than an AC measurement, though that would require using the Arduino to both provide the pulse and measure the current.

What do we want students to learn? The messiness of working with electricity and salty solutions at the same time?  How (and why) to measure conductivity with AC or pulse waveforms rather than DC? Something about small holes and ionic currents?

It looks like I’m going to have to give this whole idea more thought—perhaps even giving up on the “small hole” idea and having them build circuits to measure conductivity with a commercial conductance probe.  The probes are sold commercially as TDS probes (for “Total Dissolved Solids”). Unfortunately, replacement TDS probes are way more expensive than cheap TDS meters, so this seems like a silly thing to do (especially since the replacement probes have no documentation—they are intended for specific meters, not as generic probes, unlike pH probes, for example). I’d have to be convinced that the students would learn a lot from the lab that wouldn’t get from just reading about it or using an existing meter.