nanoHUB-U Principles of Nanobiosensors/Lecture 4.4: Selectivity - Noise in Transducers
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[Slide 1] Welcome back. We are talking about selectivity of nanobiosensors. And in this series you have already heard two types of selectivity issues. And first of all I have-- tried to explain very well that selectivity is one of the most important and fundamental concerns for nanobiosensors. Many sensors are highly sensitive and often they can detect molecules at very low concentration. But the reason many of the new technologies never make it is because they are not really selective enough. So today I'm going to talk about the third universal selectivity concern of nanobiosensors. And that has to do with the noise in the transducers.
[Slide 2] So I will explain why noise is such a big concern, such a big selectivity problem. And it turns out that it is a concern for all different types of biosensors. Be it, potentiometric, amperometric, or even cantilever-based. So therefore this is something that will be present no matter what technology you use. I will explain different types of noise sources that are important. And then in particular I will explain the math associated with it or the physics associated with the white noise and the pink noise. And then once we describe what the noises are, just like we have done previously, we'll try to see if there is any way to improve or improve the signal to noise ratio. Or can we suppress the noise somehow so we can have improved detection and better selectivity?
[Slide 3] Briefly, you may remember from the previous two slides that when the biomolecules land on the sensor's surface, then of course we have the signal damping rate. It means the correct biological molecule has attached to the correct receptors and the conjugation is what we are really looking for. But of course there can be wrong biomolecules attached to the receptor. In that case that's the blue region. So that's a selectivity problem. Of course there may be the wrong biomolecules may be sitting in the empty space on the sensor's surface. That will also produce a noise. That's the typical kind of noise we have talked about. And if you express the fraction of the correctly captured molecule with respect to the total number of molecules captured, biomolecules captured, call that alpha, and call beta to be all the parasitic capture events, then we saw that the biosensors can be described by a small matrix. Where the alpha on the diagonal, they dictate or they allow you to say what fraction of the molecules are correctly identified. And beta are all the incorrectly sort of identified, false positive, associated with this sensor. And so therefore if you come with 1001, let's say two proteins present and the two being absent. It turns out sometimes you will say that the protein is present while it is actually not. And sometimes you will miss the presence of protein because alpha plus beta is not equal to one. So in that case you sometimes may actually miss it. Whereas in reality the protein is actually there. So this is the issue about false positive and false negative that we have already discussed. What I'd like to say now is that there are a few additional terms that contribute to beta.
[Slide 4] So where do those new terms come from? Where does this additional noise come from? Well you remember that the whole system, if you look at the entire biosensor system, there are all these red biomolecules and different types of biomolecules, lets say in the blood. We capture only a certain one. And of course there can be a false capture. But even if the capture was perfect, every biomolecule with 100 percent specificity get connected with the corresponding capture molecule. Unless somehow by magic we could eliminate all incorrect binding whatsoever. Even in that case, the detection will not be perfect. The reason is that there are environmental noises. There will be fluctuation in the fluid itself for example. That can produce a signal even though there is no molecules present. So that gives you a false positive. Similarly, there will be noise associated with this transducer itself. So for example, we may have captured three molecules, but because of the noise-- sort of going in the opposite direction, we may report that we have only captured one. So therefore, both this transducer and environmental noise can lead to false positive or false negative. That we really do not want.
[Slide 5] And I have alluded to this noise problem before and in various lectures. For example when we were talking about how to beat the NARS limit, super NARS sensors, pH based sensors, potentiometric sensors. In that case we discussed about these various noises shown here in the red line. And I said that compared to the signal, which is the blue, the noise levels can be different even for this type of structure. And similarly, if you look back in the subsequent lectures we'll see that the genome sequences will also in that case we will see there is a certain amount of noise associated with it. We wrote a formula like this. So today what I'd like to do is to explain where this comes from. And in the process then you will be able to understand probably these things which were not very clear in the previous lecture.
[Slide 6] Let's quickly review that there are three types of sensors that we have discussed in this course. Potentiometric sensor that converts the charge of the biomolecule, this yellow biomolecule, to the current of a channel current of a MOSFET. And then there are amperometric sensors. In the amperometric sensor the chemical affinity associated with the yellow molecule is converted, the current in the circuit, to a redux reaction. And finally the mass gets converted to frequency in the mechanical sensors, or cantilever-based sensors. All three have noises. What it means is that even if the biomolecule were not present, let's say there is no biomolecule, so therefor you should not be detecting anything. The current should be flat. All the responses should be completely time independent. But if you go and measure, then what you will see, even when the yellow molecules are absent, there is a current flow, fluctuating current flow. There is a current flow in the output circuit in here. And there is an irreducible oscillation of the cantilever beam even when it has not captured the biomolecule. So let's see where these noises are coming from. What is the physical origin of these noises?
[Slide 7] So here I show five examples. The first example is, let's look at the entire potentiometric sensor when the electrons go in and out of the fluid gate. As the electrons go in and out, there is a corresponding noise. Because sometimes 50 electrons will go in and sometimes 48 electrons will go in. And, then, therefore-- it is never a constant number. It's constantly fluctuating about the mean value. So there is a noise associated with it. We'll call this noise electron noise and we'll use the symbol A to classify it. So we'll say it's a type A noise. Then if you look inside, although the fluid looks perfectly calm, these are the salt molecules, looks perfectly calm. PH hydrogen protons and OH atoms perfectly calm. But if you sort of took a microscope, a really small microscope that can see water molecules, within a given volume you will see that the molecules are running around all the time. And sometimes the box will contain X number of particles. Sometimes X plus five number of molecules. Sometimes less. And so this would be called an electrolyte noise. Because of the electrolyte noise, of course there will be corresponding fluctuation in the channel current. Similarly, you can have, once the molecule comes in and gets absorbed, it may not stay there forever. It can get dissolved after a while. Remember the kF and kR: the association and dissociation waves. So there can be an absorption, desorption noise. That's possible. But even if all this stops and is completely calm, no noise, even then this will not be noise free. The channel will not be noise free. Why? Because as the electrons are trying to come from source to the drain, the number of electrons which is shown here in the green circle, the number of holes in a given volume, that number fluctuates because it's interacting with the thermal environment. And finally as the electrons are coming in, as I will explain a little bit later, it can be trapped by traps within the oxide. That is shown here in green. So sometimes let's say 50 electrons are coming in. Three got trapped. So 47 moves along. And this trapping and detrapping correspondingly leads to a fluctuation in noise. So there are three types of noise that we have tried to classify. One is this electrode noise, electrolyte noise and channel noise. These we will call white noise. I'll explain why I do so or call them so. And this type of trapping/retrapping we'll call it pink noise, or one over f noise. And then of course there is a third category, absorption desorption noise. This for the time being we will not consider further.
[Slide 8] The same thing happens of course. The same thing happens for the amperometric sensors. Amperometric sensors use the electrolyte. So there's the electrolyte noise, electrode noise, absorption desorption noise. And of course as the electrons through the redox reaction, electrons are coming in and out. Then there is a corresponding noise source associated with it.
[Slide 9] Now if you go back and see for the cantilever, once again there will be noise associated with it. For example, if you look at this, there is constant air molecules constantly coming in and bumping across this cantilever. So the cantilever is oscillating like this, even though the biomolecule may not be present. But of course there also could be absorption/desorption noise associated with the biomolecule itself. So the bottom line is no matter what sensor you have, there will be always these fundamental noises associated with the sensor itself.
[Slide 10] So there are two types. This is the top one. What we called white noise is type A. If you go and review the previous three slides you will see that anywhere you see type A noise we will call that a white noise. And the white noise you can see is going up and down within a more or less constant amplitude. And the reason it is called white because if you fully transform this signal, then you will see equal amplitude for the signal for all frequencies. That's why, just like white noise-- white light contains frequencies 00:13:41,076 --> 00:13:46,146 of various frequencies of equal proportion. Similarly, since this frequency-- this noise contains all frequencies, hence the term white noise. The pink noise is essentially slightly different. This is the time spectrum associated with it. Sometimes very large, sometimes quite low. And if you look at the transform of the frequency component, on low frequency there will be very big amplitude. So you can see very big amplitude there. And in between the higher frequencies have a relatively lower concentration, fewer. This amplitude is smaller. So these two types of noise, let's see where they come from.
[Slide 11]
[Slide 12] White noise is very easy. You see, you think about a resistor and we have added a little capacitor across it. Let's say this resistor has let's say seven electrons and let's say seven holes. So the charge is neutral. And therefore there is no charge across the capacitor because you see there is no battery or anything so you would think that across the capacitor there would be no charges or no voltage that would be generated. But in general that's only on average. Because sometimes it may happen that a certain number of electrons, these blue electrons, may sort of go in one side and the white holes may go in the other side temporarily, statistically. And so what will happen, this terminal will become positive. That terminal will become negative, so there will be a little bit of positive voltage generated for a transient moment And then of course a little while later there may be holes on this side and electrons on the other side. And so this terminal will become positive and this terminal will become negative. So you can see if you add a voltometer across here, it will read sometimes positive, sometimes negative and it will keep fluctuating around. That's noise. Now this is how it is going to look like. And it is very easy to calculate what the strength of this noise is. Generally the energy across this capacitor is half CV squared. Now this energy is coming essentially from the environment and therefore it will equal kT over 2 because one degree of freedom. And so therefore in one degree of freedom from thermodynamics people say that it will be equal to kT over 2. Let's just assume that's the case. And so you can immediately see that the Vn squared will be inversely proportional to the capacitance. But we also know that for a resistor/capacitor combination the bandwidth or the cut of frequency is 1 over 2 pi RC. Take this C, insert it in here and you will see that the amplitude of the noise is given by 2 pi, which is something we know. KT, k is the bozeman constant, T is the temperature. Let's say we are doing it at room temperature so 300 degrees Kelvin. And resistance associated with it and the bandwidth. The bandwidth meaning the range of frequency's that we are interested in. And if you do a real calculation, instead of 2 pi, which is about 6 or so, we'll have 4. But other than that, as soon as you know the resistance, other things are very easy to calculate and you can see how noisy your particle resistor is going to be. Now often what people do, they want to know per unit frequency what is the noise. And so they will divide Vn squared over B. And therefore it is 4kBTR. R is the resistance, T is the temperature. Okay, now what does this resistor have to do with biosensors? It turns out that a lot.
[Slide 13] For example, if you want to look at this electron coming in and out in the case of the potentiometric sensor for example. Electrons coming in and out of the electrode. In that case the noise that we will have 4kT. But we simply replace the R with the contact resistance. That's something you can easily measure. Pump the current through this small electrode, find out what the resistance is. And then you put it in. That will tell you what type of fluctuation to expect from this electrode. Now what is this? This one is the salt molecules moving up and down. They're getting in and out of the box, sodium chloride atoms. Again, all you have to do is replace R, the resistance, with a solution, the resistance of the solution. How do you calculate that? Well, you know the molar concentration and then you know the mobility associated with sodium or chlorine and from that you can calculate or solve. Put it in so you know the electrolyte noise. Very easy, you see. And finally if you wanted to know channel noise as the electrons are going from the source to drain in the MOSFET channel, once again all you have to do is calculate the channel resistance, put it in and you get the noise. So type A white noise is easy to calculate. Just find out the resistance associated with the corresponding process and you are done. And then eventually you can add them all up.
[Slide 14] Now what about the pink noise? Well the pink noise is a little bit more complicated but it's not too complicated. You see the drain current is proportional to the number of electrons you have and the mobility of the electron multiplied by the electric field. So change in the drain current is, if you take a log on both sides and take a differential, is proportional to the change in the number of electrons and change in the mobility. So you see what happens. That let's say three electrons has started coming in. It comes in up to a certain point and then gets captured by the white trap. So now two electrons are going in. So if you look at the number of electrons in that instant, you will see the number of electrons has gone down from three to two. Of course, if it gets trapped farther in, then it is going to stay there longer before it comes back. And so therefore it will still go from three to two, but it will stay there longer before it comes back. So that's the electron number issue, that there will be a fluctuation in electron number, this delta n. Moreover, when the charge is trapped they are going to scatter the other two electrons, which are going on their own way. And that's the change in the delta mu. So once you account for this, these two processes, there will be a corresponding noise, pink noise, associate with it.
[Slide 15] And so people have calculated what this noise is. But I just show you how it comes about. Generally, what the general interpretation is, if you look at a textbook you will get a formula like this. It has a 1 over f dependence. Smaller the frequency, higher the magnitude. And the number of fluctuation when things get trapped is given by this force factor multiplied by one. And the mobility fluctuation, because this capturing of the free electrons by their trapped counterparts is given by this mobility fluctuation. Now typically for biosensors, alpha will be equal to one because we are always in the linear regime. The current is very, very small. And so therefore all one has to do, that if you wanted to know how much noise you have between frequency f1 to frequency f2, all you have to do is integrate this noise up and that will give you the total amount of fluctuation that will happen because of trapping and retrapping. See, even though the biomolecules are not producing anything, this noise itself can give false positive or false negative.
[Slide 16] You can easily calculate this number for example if you have your oxide capacitance of a certain amount, different density. The tunneling length into the oxide. You can put these numbers in and you can easily calculate that the noise can indeed be significant. This is V squared. this ten to the power minus 4. So V would be 10 to the power minus 2. So that's quite significant. It's in the millivolt per hertz range. So it can be a significant amount of noise. But the most important point I wanted to emphasize is that WL. This is the curse of nanobiosensors, that as you make things smaller and smaller and smaller, the WL becomes small. And therefore the noise becomes very large. Because in small structures you have only a few electrons. So if you get trapped, the disruption is more significant. That's why nanobiosensors pays dearly in terms of noise enhancements, especially 1 over f noise enhancement.
[Slide 17] Okay, so this is a fundamental issue, this type of noise. How are you going to beat it? Is there any way, or do we have to live with it?
[Slide 18] It turns out there is a very interesting way to beat this noise. You see the noise essentially is contaminating our detection, as I explained in the last slide. So let me explain to you how we can actually reduce the noise by averaging.
[Slide 19] You see, if you had a signal S which is produced by the biomolecule, and if you had the noise associated either with the electrolyte or with the pink noise trapping, retrapping. And let's say sometimes it is going positive, sometimes it is going negative. Sometimes though-- there will be a constant fluctuation up and down. One thing you immediately realize, that if you were just measuring one sample, then this S divided by n3, s naught divided by n3, that will be your signal to noise ratio. And anytime the S, the green, falls below the n3, then we are done. We cannot detect it. But you see if instead of just measuring one sample, the signal was changing slowly as it often happens in biosensing, and if you can make many measurements. Then if you add them all up, because sometimes they are positive and sometimes they are negative, the total noise will go down after averaging. So if you sum the five samples up and then compare with five signals-- five samples, do this ratio that will be significantly better. In fact, more the number of samples, noise will keep going down and signal to noise ratio will keep improving. And this is essentially the same statement shown here on the bottom.
[Slide 20] So let me give you an example. For example, here for the white noise we have a spectrum and for the pink noise let's say we have a spectrum. Pretty bad. Growing magnitude is quite high. But instead of staying with it and being sort of frustrated that the signal to noise ratio is defined by the maximum value of the noise associated with the signal. What we can do is that we can take the average of 100 samples or 200 samples, 500 samples, the window. In that case, as soon as you average it, because it's going up and down, this noise will be considerably suppressed. And the signal to noise ratio will improve. And the same for the pink noise. In that case also the signal to noise ratio would be considerably enhanced. All right, so let's see, after a proper calculation how much gain can we expect?
[Slide 21] So here is an example. The blue line is a white noise. The X axis is the sample size. You do average over 20. If you average over 20 then it would be at this point. If you average over 500, then this is 100, this is 500. So you would be at this point. Now in this case, even if your signal was a factor of 10 smaller, compared to noise, let's say in that case you wouldn't be able to detect anything classically. However, as soon as you take about 50-100 samples, you can see the white noise. Essentially the signal becomes larger than the white noise. And white noise takes a little bit longer. It takes a couple of hundred samples. But if you do take 1,000 samples, sample it at the kilohertz range for the signal that is changing on the second landscape, in that case you can see even if the signal was buried inside the noise, you can have a proper signal to noise ratio and faithful detection of the signal itself. 00:27:35,046 --> 00:27:36,696
[Slide 22] So let me conclude then. I told you about two types of selectivities. One has to do with the biomolecule. That was the last two lectures. This one has to do with transducers, which is the noise associated with the transducers themselves. Even if the biomolecules are perfect, this noise would remain. Now of all the noise sources, if you do a proper calculation, it turns out the 1 over f or the pink noise is dominant for a potentiometric sensor. Something has to be done. You can ignore all the electrolyte noise as well as other noise sources. But 1 over f noise is something that cannot be ignored. And so therefor understanding the noise sources is very important. On the other hand, thermal noise, this white noise plays a very important role in cantilever sensing because after all there is no trapping/detrapping. So in that case this is not present and in its absence the thermal noise becomes very important. Now because the analyte concentration, generally in the fluid flowing in our blood, how frequently is the glucose concentration going to change? It's not going to change every second, hopefully. And if it sort of stays on a one-minute, two-minute length scale, then you can take many, many samples. And in that case, you can improve the signal to noise ratio considerably and you can detect very minute signal levels. Finally, if you consider the noise properly and analyze them properly, interpret them properly and design the system accurately, only then you can expect a very good biosensor platform that will be useful and will not have significant amount of false positive or false negative associated with these noise sources. Let me end here. In the next couple of lectures we are going to bring everything together and then interpret everything that we have learned so far in terms of a genome sequencing problem.