nanoHUB-U Principles of Nanobiosensors/Lecture 2.5: Settling time Beating the Limits – Barcode Sensors
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[Slide 1] Welcome back. This is the second lecture in week two, and we are discussing how long does it take for the biomolecules to get to the sensor surface. there is a problem of settling time. Now-- by now, I may have scared you a little bit by simply suggesting that the sensors have these-- all these fundamental limits and then there is-- you cannot really, if you look at the previous graphs carefully, you will see that it's very difficult to go below a few, let's say a fraction of a picomolar because of this diffusion limit.
[Slide 2] But you see, one thing I didn't really mention at that time, is that this limit is fundamental in the sense that diffusion is fundamental. But this limit is not fundamental in the same sense as Heisenberg's uncertainty principle is fundamental. There are ways to get around it. And in the next three lectures, I'm essentially going to tell you how people are thinking about getting around to this fundamental limit. One more thing before I move on. This diffusion limit that I told you about, although we did it in the context of the biosensors, but you see, this is no different for, in terms of a jelly fish catching the food around it. It uses a complex sort of "sensor geometry" in order to efficiently capture food. And this is no different also from saying that within a cell, a protein trying to find its target. So therefore, these relationships are very fundamental, although we are doing it in the context of nanobiosensors. I hope you will realize that these are very general principles that we are talking about that may have wide-- very wide range of applications. But returning back to the sensors, the question we ask, how do we beat the diffusion limit which sort of sets the minimum analyte density that we can absorb?
[Slide 3] I will tell you about three approaches. Not that there are only three, but the three dominant ones that I know of, that people had been thinking about. And I will-- The first one will be based on an approach called the biobarcode approach. It has been around for a couple of years, almost seven, eight years. I'll explain how it works and then I will explain how it essentially enhances the detection limit or how it allows you to detect even at a lower concentration, and finally I'll conclude.
[Slide 4] Now, you remember this particular picture, and as I said, getting to femtomolar using standard diffusion-limited based sensors, of course gets difficult. Now, in order to get to even lower density or major densities with more precision, we can have other approaches. The biobarcode one that I've shown here is one. And the other will be based on the evaporation of a droplet. Both can actually give fantastic accuracy and very low level of density. I'm calling this table sort of like a Mendeleev table. In the sense just like in Mendeleev's case, a few organizing principles, essentially allowed you to organize all the elements that are around. This focus on the shape- and the shape of the sensor gives us the same organizing principle. And hence, the analogy to the Mendeleev table.
[Slide 5] Now, there are three ways you can beat the diffusion limit. As I said, the first one, is you see the diffusion limit simply says that if you have a box of size L and if you have the molecule or diffusion coefficient D in a certain fluidic environment, then it takes on average tau seconds before it moves from one end of the box to other on average. So if you wanted to detect the particle faster, you know, you don't want to wait untill tau seconds before it sort of randomly walks around and arrives at your doorstep. What would you do? One way to do it would be, instead of having the sensor sit at one point in a single sensor, you throw a bunch of sensors looking for the single blue analyte. So the rates of sensors, instead of having one sitting on the bottom of the beaker, you put billions of them in the solution looking for the sensor. And the way that so-- so now the effective box, although the big box still remains, the effective box which determines L is now somewhat smaller. The other way you could do this-- reduce the L and thereby reduce tau is essentially compress the box. How do you compress the box? Well, one way to compress the box would be to put things in a big droplet of fluid where there are a certain number of particles and then allow the droplet to evaporate. The water will go away, but the concentrated biomolecule will come to the sensor. Here, you have one sensor. But, the molecule-- the shape of the box, the L, is sort of getting compressed and effectively enhancing the concentration of the biomolecule. And finally, what you can do is you could generate very locally to the sensor and at the same time, you could do something with the diffusing species. The molecules are lighter. In that case, you have replaced both-- reduced both L because you generated locally, not from L-- molecules are not coming from a long distance. And you also enhanced D, because you replaced the biomolecule with a somewhat smaller cell counterpart. I will come to that in a little bit later. Today, the focus is on magnetic biobarcode sensor. This approach on the very right.
[Slide 6] So this is the basic idea. Instead of sitting on the bottom, a sensor sitting on the bottom, and hoping that this analyte molecule will do a random walk and eventually land on the sensor surface allowing detection. You see, we can throw the sensors in the solution and thereby, making the sensor closer to where the analyte molecule is. And what I have done in this bottom picture is essentially rotated the thing 90 degrees. And this red molecule is essentially the analyte molecule that we're hoping to capture. These are-- would be the capture probes which will catch this molecule. And let's say as the molecules are moving around, eventually it gets captured. I haven't allowed these particles to move assuming that these are heavier so they move, but they do not move as quickly. And eventually, what will happen, I said, throw in another set of particles, and only when this particle also binds to the same target. And therefore, you can see it has now been sandwiched. The target now has been sandwiched with the magnetic field now because if this particle is magnetic and hence the name magnetic biobarcode sensor, you can pull it in and sense on the sensor surface. So it has been sandwiched, sort of sandwiched between the two probes, sort of it has been handcuffed by them. And then brought to the sensor surface for detection. And this can be extremely fast because there are millions of these yellow particles looking for the rare analyte. So you can think about it as if it's a police-thief story. In this particular case, the police is sort of sitting in the police station hoping that the thief will give itself up after a drunken walk and eventually show up on the station. Now, of course, that doesn't really make very much sense. It may actually take a long time before that happens. Instead, the police can go out in the streets, and if you have hundreds of thousands of policemen, then the probability of catching the thief is of course much, much faster.
[Slide 7] So let me tell you the math behind it. It's actually very elegant and very simple. Assume that these are the policemen or these are the capture probes and you have a relatively large number of analyte. And this is one extreme where the analyte concentration is larger than the number of capture probes. In the other extreme, you have few analyte but a lot of capture probe is seen-- shown here in these larger circles. We'll get two results. In one case, the settling time or how long it takes for the molecules to get captured by this sensor is inversely proportional to the target molecule. The more target molecule you have, the easier for this capture probe to capture them because they will be closer. And this occurs only when the number of magnetic particles or number of capture probe is much smaller than the number of analyte. On the other extreme, you have the settling time, sort of independent of the number of analytes, but inversely proportional to the particles that is trying to capture them. So these two limits, let me explain how they rise.
[Slide 8] So let's start with a case where the probes are sparsely dispersed and they're only a few. In this case, you can as if think of the particles are stationary, the capture probes are stationary. It's like a classical case of a nanosphere sensor which we saw just sitting there, just like this one. Because after all, you will be capturing only NS number of particles, let's say 20 particles or 30 particles biomolecules. In that case, if they're far apart then each one of them has more than enough to capture for itself, so therefore having a few scattered around makes no difference as far as capture time is concerned. So how would you do it? We'll do it exactly the way we had solved the problem before. Diffusion equation, perfect capture, making sure that the analyte density is close to zero in the sensor surface. We do the diffusion equivalent capacitance formula that we have seen before. And we know that this is a spherical sensor, and therefore, the diffusion equivalent capacitance, there should be a 4 pi here, will be 1 over a naught. This a naught is a radius of that capture probe and square root of Dt, after a while the square root of Dt will go away at longer time. So this capacitance will be a constant 4 pi D a naught, you put it in and that gives you the formula for detection time. So that capture time is inversely proportional to rho. Now you have seen this many times before, remember the spherical sensor? This zero dimensional spherical sensor and I said it goes as rho to the power of 1, same formula, you can check it out from before. So nothing has happened so far.
[Slide 9] But, if you sort of make them the number equal, the number of sensors and probes equal, you can send the same problem exactly the same way as I did before. But remember, now, there's a one-to-one relationship. For every capture probe, there is one analyte molecule. So this is sort of the transition point. And therefore you can calculate it the same way on average. And this, Ns, the number of particle captured per capture probe is about 1, approximately, because of course some may accidentally capture 2 but some will essentially have nothing. But in general, on average, the sensing-- the response time is inversely proportional to the rho, the density of analytes.
[Slide 10] Now, if you go to this other extreme, where you have actually many more capture probes and a few, few analyte molecule. So lots of policemen is looking for two suspects. In this case, the-- from a suspect perspective, from this perspective, it has a diffusion equation, but there is also-- now it's supplemented by the probability that it will be captured. The analogy is something like this. Assume that you have a piece of semiconductor, just an analogy. You have a piece of semiconductor and there are lots of traps in it. The question we are asking is that if you have some minority carriers throw some electrons in there, how long will it be before it finds the traps? Of course, everybody who has a background in semiconductor immediately knows that this is given by diffusion and capture equation, right? This would be a Shockley-Read-Hall recombination for example, just an analogy. Now, how do you calculate? Because this particular capture probability essentially says that it can be captured by this. If it's not captured by that, this molecule, it can be captured by the next one. As it is walking around, it will be captured by one or the other, depending on the trajectory of the target molecule. What is this value of tau? How do we calculate it? Well, turns out, it's very easy. Remember that each one of these capture probes can essentially capture the amount of flux, the number of particles it captures, is given by 4 pi D a naught, rho t. This is from the previous slide. You can check it out, that how a spherical sensor captures particle. This is the flux rate. Now if you wanted to know that what is the rate of capture by individual particles, you will simply-- it's a linear response, so you will simply do DnDt or Nt divided by t, and therefore you will say that it is directly proportional to the number of analytes that are moving around, because then, that will be, give you the rate at which these particles can capture the biomolecules. Now, of course, it's not one bio-- one target molecule which is looking for the analytes, there are many. And so therefore, total capture rate is given by this quantity for individual one multiplied by whatever number of magnetic particles you have. These capture probes you have in the solution. And so, therefore, if you wanted to know from this analytes' perspective, what is the rate of disappearance, then you will simply make it equivalent to rho divided by tau. And this tau is the rate at which the molecules are captured. And so therefore, you can see this is inversely proportional to the number of magnetic particles. Does this make any sense? You see, it's not complicated. If you have a certain number of magnetic particles looking for this analyte molecule, you have hypothetical box around it. You can think about there is a hypothetical box around it, and it will be captured by one or the other within the box. If you increase it, increase the magnetic particle concentration by a factor of 10, let's say, the box will now shrink because there are more magnetic particles close by. And therefore, it will be captured somewhat faster. And this is exactly what the implication is. More the number of policemen, more the number of particles, more the number of capture probes, shorter is the time for capture.
[Slide 11] So, once you have this particular capture time there, in fact, you can write out the rate, the entire solution of how they are going to disappear as a function of time. If it starts from R equals 0, what is the probability that it will survive up to time to a distance R and at a time t, is given by a very simple formula which is the solution of this diffusion capture problem. And the survival rate, the probability that it will still survive beyond a certain point, you simply have to integrate how many have survived and how many started out at time t equals 0. 4 pi r square essentially gives you this spherical shell at a distance R, where the particle is at a given time t and it will simply be given by an exponential decay divided by tau. There's the survival probability. And the bottom line is, approximately, within tau seconds, the whole thing, the particle will surely be captured and the tau is inversely proportional to the number of magnetic particles.
[Slide 12] So let's summarize the result. Let's say in the same plot as we have done before, how does this new result look like? This is the classical result when you have only a very few sensors and captured molecules and there are lots of analyte, it is as if the other magnetic particles don't exist. And you can just view it as-- in terms of individual nanospherical sensors. And the result is, something that you have already known, that it essentially has a slope of one, duration between ts and rho-t if you take log on both sides, you can see the slope will be minus 1. However, if you take millions of capture probes to capture a certain number or fewer number of analyte molecules, in that case, the response is somewhat different. You see what will happen, that so long the analyte density is larger than the number of capture probes, the response would be the same as in this case. However, once the number has exceeded, so for example, here we have 10 to the power 15 analyte capture probes, once the number has exceeded, in that case you see, it's solely determined by the capture probes themselves. And it doesn't matter what is the concentration of the particles, because if your concentration of the particles is low, true, that you don't have this one but it doesn't matter. It will be captured by the other ones. And so therefore, it becomes independent of density, the settling time is the same. If you increase the number of capture probes, more policemen on the street, then you will see the whole thing will come down. And here, you have put, let's say a picomolar of concentration. So below a picomolar, therefore the time becomes independent of the analyte concentration. And an important thing is the knee. The knee is when the two concentrations, analyte and the target molecules, they become equal. And the point is that in both cases, this becomes independent of the analyte concentration. Because as I mentioned before, that someone or else of the capture probes will capture it, and therefore, it really doesn't matter that what the analyte density itself is.
[Slide 13] So that gives you, this is why throwing more biosensors into the mix, into the solution allows you to detect solutions much faster. And remember, the time is density. That shorter the time for the-- If you can detect something for a shorter period of time, it also equivalently means for a given time, you can detect at a lower concentration. And that's what allows us to push the attomolar concentration as has been demonstrated by many experiments.
[Slide 14] Now, of course, this is not the only way you could fragment the space by throwing things in the solution. One thing you can do is essentially have a sensor, fragmented many sensors, looking for the same molecule. So whatever solution you have which contains the rare analyte, you spread it in and all. And in this case, let's say this one sort of has the molecule and it will immediately be recognized. Because within this volume, effectively, the concentration is very high because you have just limited L, made the L small. The others one will not find anything, that's true. You have wasted a lot of space, but for this molecule in particular, it will be placed very close to the sensor surface, and therefore will be detected almost immediately. Now it has many advantages. That it has bigger area, it has redundancy, and you can do many analytes in parallel, but also because all of this green space you have to sort of data dead spaces. So therefore, it sort of is wasteful and many times it can be very expensive if you want to do it this way. But if you wanted to beat the diffusion limit, the point is, and you don't go worry about money, in this particular case, you can certainly do so by this approach.
[Slide 15] Finally, I will not go into the details of this particular scheme, but this combines two ideas. One is fragmenting the space that I just told you. So assume that you have biomolecules sort of decorating a bid, and the biomolecules are shown here, this is the DNA. And in this case, again you have spread it out into individual wells, so they are very close to the sensor surface. So the diffusion time would be very small because L is small. Further, what is done and I will show you later in the course, that instead of letting this DNA to diffuse, which is a big molecule, rather, here is secondary reaction product. The proton is allowed to diffuse. So proton becomes sort of the proxy for the DNA. Proton as you know is much lighter, right, compared to a DNA. And so, therefore, they can diffuse much faster, and therefore the D is reduced-- D is greatly enhanced as well. L is reduced. D is enhanced, and tau can be very small. As from the simulation we have seen, that within a fraction of a second, in fact, you can detect these biomolecules. Now, this I will get into a much longer detail down the road, so therefore at this point, you don't have to understand all the details. The only thing I ask you to do understand, this idea about space fragmentation is very similar to the biobarcode approach in the solution policemen looking for the analyte. And this idea about using a proxy to enhance the diffusion coefficient, both are very powerful concept that are available in modern technologies, technologies that are available in many labs. So therefore, this would be very important but we need to connect it up with the concepts of the-- of this biobarcode approach.
[Slide 16] So let me end here. So, I try to explain to you the biobarcode approach and how it sort of "beats the diffusion limit." It still has the diffusion limit, it just makes the effective L smaller. The biobarcode itself doesn't sense anything. It's like you capture it, but then you have to bring it to the sensor by using a magnetic field because these are magnetic particles so you can attract them by magnetic field. And then do the sensing at a later state. So this is, the capture process is very fast. Now, if your sensor is not sensitive, that may take a while for-- to detect the sensor, detect the molecule. So, that you can do at a later state. Now, this using multiple sensors. So you could have different molecules making magnetic particles essentially looking for different analytes. And so therefore, you can do these things in parallel. And that's why this approach is called biobarcode approach. And in fact, if you use multiple sensors distributed in space, that is equivalent to doing this biobarcode in solution. And I think in both cases it increases cost, but it certainly beats the diffusion limit. So I will talk about-- continue discussing another approach, which is also very beautiful in the next lecture.