# Gas station without pumps

## 2013 April 21

### Noise in nanopores

Filed under: Uncategorized — gasstationwithoutpumps @ 16:32
Tags: , , , ,

I’ve been trying to understand the sources for noise in current measurements through nanopores, so that I can better understand the signals that are generated in the nanopore lab.  I’ve not studied noise in small signals before, so I’ve been having to rely on Wikipedia for information about sources of noise.  I’ve probably missed some important ones and would appreciate those more knowledgeable pointing out important noise sources that I missed.

I think that there are two main noise sources:  the nanopore itself and the resistors used in converting the nanopore current to a voltage in the first stage of amplification.  Each of these noise sources has two types of white noise: thermal noise (dependent on temperature) and shot noise (independent of temperature).

#### Thermal noise

The RMS current for thermal noise is $i_{\mbox{\scriptsize thermal}} = \sqrt{\frac{4k_B T \Delta f}{R}}$ for a resistance of R at temperature T, with kB being Boltzmann’s constant, 1.3896593E-23 J/°K, and $\Delta f$ being the bandwidth of the filter looking at the noise.  Normally, nanopore scientists don’t report the resistance of the nanopore, but the bias voltage and the DC current through the pore, but we can use Ohm’s law to rewrite the noise formula as $i_{\mbox{\scriptsize thermal}} = \sqrt{\frac{4k_B T I \Delta f}{V}}$.  A typical setup for a nanopore may have a voltage of 180mV and 60pA (for an open channel in 0.3M KCl), and a temperature of 25°C = 298.15°K, for $4 k_B T = \mbox{16.573E-21 J}$, and noise of $\mbox{2.35E-15} \sqrt{\Delta f} A/\sqrt{\mbox{Hz}}$.

#### Shot noise

The RMS current for shot noise is $i_{\mbox{\scriptsize shot}} = \sqrt{2 q I \Delta f}$, where q is the magnitude of the charge of the carriers (here 1.60217646E-19 C, since the charge carriers are K+ and Cl), and I is the DC current.  Again $\Delta f$ is the bandwidth of the filter looking at the noise.  For a current of 60pA, the shot noise would be $\mbox{3.259E-15} \sqrt{\Delta f} A/\sqrt{\mbox{Hz}}$, slightly more than the thermal noise.

#### Combined noise in nanopore

The combined noise in the nanopore from both shot noise and thermal noise should then be $i_{\mbox{\scriptsize pore}} = \sqrt{\left(2 q + \frac{4k_B T}{V}\right)I \Delta f}$, which for 180mV and 60pA would be $\mbox{4.02E-15} \sqrt{\Delta f} A/\sqrt{\mbox{Hz}}$.  If a 1kHz low-pass filter is used, that makes an RMS noise level of 0.127pA, and with a 10kHz low-pass filter, 0.4pA.

Increasing the ionic concentration would provide a larger DC current (current proportional to concentration), and the noise current only grows with the square root of the DC current, so the signal to noise ratio also grows with the square root of the ionic concentration.

#### Amplifier noise

These noise levels are lower than what is actually observed, of course, because we haven’t taken into account noise generated in the first stage of amplification.

In the UCSC nanopore lab, they use an Axon Axopatch 200B Capacitor Feedback Patch Clamp Amplifier, which costs about $2000 used (if you have to ask the new price, you can’t afford it—you can get a quote, but there is no list price). The web site claims “By introducing active cooling of components in the headstage to well below 0°C, the open-circuit noise in patch mode has been reduced to unprecedented levels, as low as <15 fA (RMS) at 0.1–1 kHz.” But the lab uses a resistive headstage (so as to be able to make DC current measurements), with a 500MΩ feedback resistor in the current-to-voltage converter, which introduces thermal noise of about $\mbox{5.76E-15} \sqrt{\Delta f} A/\sqrt{\mbox{Hz}}$, larger than the nanopore noise. The shot noise for the resistor in the I-to-V converter should be the same as as the shot noise for the nanopore, since they have the same DC current. Combining the thermal and shot noises for both the nanopore and the resistor at 180mV and 60pA, I estimate $\mbox{7.74E-15} \sqrt{\Delta f} A/\sqrt{\mbox{Hz}}$, which would be 0.244pA at 1kHz, 0.547pA at 5kHz, and 0.774pA at 10KHz. The noise levels observed in the signals are close to those levels, so there is not much that can be done to improve the amplifiers to get a better signal-to-noise ratio. We could be interested in lower-cost amplifiers than the$2k lab instrument, but a very low noise off-the-shelf op amp (like the AD8432 ) has a noise level of about $2.0pA\sqrt{\Delta f}/\sqrt{\mbox{Hz}}$, which is much larger than what is observed in the lab.  Designs have been done at UCSC for much lower noise amplifiers (Gang Wang; Dunbar, W.B., “An integrated, low noise patch-clamp amplifier for biological nanopore applications,” Engineering in Medicine and Biology Society (EMBC), 2010 Annual International Conference of the IEEE , vol., no., pp.2718,2721, Aug. 31 2010-Sept. 4 2010 doi: 10.1109/IEMBS.2010.5626570) claiming “input referred noise of the amplifier is 0.49 pA RMS over a 5 kHz bandwidth”, from simulation (I don’t know whether they have fabricated and tested the amp yet).

UPDATE 2013 Nov 13:  A student showed me a data sheet for an off-the-shelf ($3–$4) op amp that has low enough noise for this purpose: the AD8625 (or AD8626 or AD8627).  They have only a 0.25pA bias current and a current noise density of $0.4 fA\sqrt{\Delta}/\sqrt{\mbox{Hz}}$. That’s actually better than the AxoPatch 200B, whose manual reports about 0.1 pA at 10kHz, which is $1fA \sqrt{\Delta}/\sqrt{\mbox{Hz}}$.   Even better specs are available for  the AD549L (about \$40) with a 60fA bias current and a current noise density of $0.11 fA\sqrt{\Delta}/\sqrt{\mbox{Hz}}$.   Of course, once the amplifiers are this good, most of the noise is coming from thermal noise in the feedback resistor (about $6 fA\sqrt{\Delta}/\sqrt{\mbox{Hz}}$), unless one uses capacitive feedback (which precludes measuring DC currents).

I don’t know why I missed these op amp before, as they have been around since 2002 or 2003.