nanoHUB-U Principles of Nanobiosensors/Lecture 2.2: Settling time Classical Sensors I
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[Slide 1] Welcome back. Now we're getting into the real heart of this course. As I mentioned that there will be three big segments for this five week course. The first one is about settling time. How long does it take for a biomolecule to find the sensor because after all if it doesn't find the sensor then there's no point in having a high-sensitivity sensor lying around. And let me get started on that set of lectures that will discuss this issue about settling time.
[Slide 2] So I'll begin with a brief introduction and a reminder. The main point we are trying to address is there are so many sensors in the literature. How do we classify them, how do we understand them about what their ultimate limits are and if there is any fundamental limit to begin with. And as we will soon see that the geometry of diffusion actually defines the response time that if a sensor has a specific geometry, then it will respond differently compared to some other sensors that has a slightly different geometry regardless of what the sensing mechanism itself is. And we will do some simple arithmetic in order to understand this response time. But I'll also show you a slightly more detailed version of it called diffusion capacitance that will also give us this fundamental diffusion limit and then I will conclude.
[Slide 3] You may remember this picture from the previous lectures that we have these various sensors with different sensitivity millimolar all the way to attomolar, 10 to the power minus 18 molar and divide it into 3 orders of magnitude changing the enhancement in the sensitivity. And as I said, planar sensors are somehow less sensitive. The sensors that we had in 1970's are somehow less sensitive compared to the nano biosensors, nanowires and nanotube base sensors that are used in recent literature. In fact, there is something called a bio barcode sensor which responds even faster, so we'd like to understand where does this extraordinary sensitivity come from. And one thing you immediately notice that it really doesn't matter what type of senses you have in here because some may be potentiometric, some may be cantilever based and some maybe amperometric, regardless it looks like that nano biosensor provides the extraordinary sensitivity gain.
[Slide 4] So let's think about a sensor in the simplest way. Let's consider a sensor's immersed in a fluid volume, having a random shape, and the y shaped molecule are the receptors which will catch the biomolecule from the solution. This could be for example antibody for proteins. It could be the DNA, the receptor DNA, if the red molecules are the DNA fragment themselves so they can catch each other. We'll not think about those details right now, we'll think about these red molecules as random particles moving around, diffusing around in solution. And therefore, described by the diffusion equation d is a diffusion coefficient. And if you have a given distribution of analyte molecules red this will evolve as a function of time following the diffusion equation. And once the molecule actually reaches the sensor surface there is a probability that depending on how many sites are empty, receptor sites are empty, then a fraction of them will be captured by the sensors. And we want to know how long will it take before a certain number of molecules are captured by the sensor surface, so the sensor is ready to respond with a given signal-to-noise ratio. So this is a general diffusion capture problem. You can see this is nothing to do so much with sensors per se, but it is a completely general problem. It could be a jellyfish waiting for the food to come to it by diffusion and catching it. It could be a cell while the protein is capturing other proteins from the surrounding, completely general problem. Sensor just happens to be a particularly good illustrative example for such a diffusion capture problem. Now one thing, unfortunately these problems in 3-d with this type of capture dynamics is very difficult to solve and you may need actually big computers. But big computers, the results from big computers may be impressive, but they may not be informative. We will be using something far more simpler to see how this system behaves.
[Slide 5] And it turns out the result is actually very, very simple. It turns out the number of molecules captured when they're going through this complex diffusion process, getting captured by complex sensors with very different geometries is simply given by the initial analyte density rho naught and it's a
[inaudible] function of the fractal dimension of the surface and you may recall that the fractal dimensions are defined by the shape of the surface rather than the surface to volume ratio for example. Now this simplification or this beautifully elegant result, this simple result is only strictly true if the sensor can capture all the molecules that it comes in contact with which is perfectly sticky and doesn't let anybody go. And generally as a result anything that comes in is captured. Now this is an idealization, but we're looking for limits, we are asking that even if you could make the best sensors which catches every molecule that comes in contact with, how long will it take before it catches a certain number of molecule Nt?
[Slide 6] Now you can immediately see before we get to the results directly that it does depend on the analyte density rho naught. Why, for example, if you had a micromolar solution. Do you remember that you have million copies of this red molecules in 100 micron by 100 micron by 100 micron box? In that case as soon as you insert the sensor a certain fraction of the molecule they are close by and they will be captured. However, if you are in the picomolar far fewer molecules are swimming around, the first one may get captured accidentally quickly, but in general others will take longer compared to the previous case. And if you have really low density like a femtomolar. You have a single molecule within that box it will take forever for the molecule to actually show up and if you stop before that time, you may actually say that the molecule doesn't exist. So the sensitivity limit if it's defined by the time it takes for how long you are willing to wait, this ts but equally important it depends on the analyte density which is rho naught. Now what about this shape of the fractal surface?
[Slide 7] So I'll begin with the answer and then I will show you how we get to the answer. And the answer simply is the lowest limit any sensor can detect is that number of analyte is given by the number of molecules that you need to activate the sensor. And it is defined by how long you are willing to wait and the fractal dimension of the surface itself. So, therefore, all these three sensors actually even if they had the same mechanism of transduction, which is potentiometric sensors and all three cases their minimum detection limit will be defined by the shape of the sensors and nothing else. You don't need to know anymore physics than that. Let me show you how this result comes out. I'll start simple and gradually we'll build up.
[Slide 8] Consider a planar sensor and these two green things are lets say source and drain and the surface is planar. The capture molecule, y shaped capture molecules are there and the red molecules are getting captured. This is a sticky sensor surface every time a red molecule lands on the sensor surface it is immediately caught. Now here is a side view all the molecules are shown in here and the sensor surface is on the bottom. What will happen as a function of time? Well as a function of time the molecules will be absorbed on the surface and like a blotting paper this will be captured on the sensor surface. So the question you want to ask how many will be captured in a given time t because we need at least an s of them in order for the sensor to have a given signal to noise ratio.
[Slide 9] Now fortunately before we do the proximal solution there is an exact solution to this problem. For example, if you took the same diffusion equation for the red molecules and simply asked how many molecules will be captured by the sensor surface. In that case, the answer is no and the answer is a horrendous thing that is proportional to the analyte density which is understandable. If you have more density is more then it will capture more at a given time and it's given by some error function. And with the square root of D t if it diffuses slow of course, the end number of molecules will be slightly different.
[Slide 10] For the time being, let's not worry about it because I'm going to show you a simpler way of doing it. I'm just showing you exactly how you could solve the problem if you wanted to. And so what will happen and that as a function of time assume the sensor is on the left and the analyte is filling the same infinite space on the right. For a small amount of time if you looked at dynamic profile you would see the blue line, then the red and then the green line. And these molecules which are in here they are lost from the solution because they have been captured by the sensor surface. So you could easily calculate how many molecules have been captured from the sensor surface by simply asking this question, how many did you have originally. You originally this whole region was filled with biomolecules. And what do you have at a given time t and integrate throughout this space and if you do so, you will see the number of particles captured is proportional to the density of biomolecules and square root of D t multiplied by the factor of the order of 1.
[Slide 11] Now what did just happen? You see, first of all let me tell you what happened is sort of something that you have seen before. Do you remember that if you had a diffusion equation I showed it in the second lecture or so that if you inject a drop of ink in water for example then it diffuses? And we saw that the diffusion distance as a function of time is given by square root of D t. On average of course, every molecule is randomly moving around, but on average the square root of D t is how far they move. Now think about the inverse problem let's assume that the whole region is filled with uniform analyte density, let's say it's completely inky. And in that case you put a single strip of absorbing material, let's say blotting paper right at x equals 0. In this case there will be an inverse problem that gradually this analyte will be depleted because that blotting paper will capture those particles. The interesting thing is that even in that case this is a middle symmetry of this problem that the amount that gets depleted is also given by square root of D t because a molecule which starts from here within square root of D t can now walk around and get captured. If somebody started from here there is no probability that on average it will be able to be captured by this. So, therefore, the depleted volume goes also as the square root of D t. With this basic information in place you see you don't have to do complicated math anymore just a simple geometrical argument will do.
[Slide 12] So let's think about how it does it, how it can be done. Remember the exact profile with the error functions and all these complicated results and the result is here the number of particles is given by analyte density and square root of D t. You see you could have gotten the same result much more simply. Remember the amount that gets depleted, the amount of biomolecules that get caught by this sensor is given by square root of D t for a given time t. And the height of this analyte density, the bulk density is rho naught. And only the molecules which start from within the square root of D t can be captured by the biosensor. And so, therefore, you could simply say the number of molecules captured is equal to the area of the lost triangle. And the lost triangle is half height of the triangle and the base of the triangle. Compare these two results. These two results, this came from solution of differential equation. Here I did nothing simply calculated the area of the triangle and in fact, we did a factor of 2, both results are essentially identical.
[Slide 13] Now in fact, you could be even a little bit even more sloppy and still get the results that is pretty good. For example, instead of thinking about it as a triangle you could simply think about as the whole region has gotten depleted. Yes, you will be off by your factor of 2, but we will not be thinking about factor of 2 here, we are after this concentration will change orders of magnitude from millimolar to femtomolar. So therefore, we are simply happy with the approximately the factor of 2 or so and that should be completely all right. Here this is the same picture as this one except that we have assumed that the entire region has gotten depleted. So the lost rectangle is given by rho naught square root of D t. that is how many molecules it has captured on the sensor surface. Remember in one dimension is the planar surface analytes are getting deposited on the sensor itself.
[Slide 14] Now, therefore, if you are with me up to this point, then you should also agree that the number of molecules captured N sub t of a is rho naught multiplied by the distance it got depleted, square root of D t as I just showed you in the last slide. And so, therefore, if you need a certain number of molecules in order for the sensor to get activated, let's make that number N s, it could be 100, it could be 500 or it could be 10 depending on the sensor. Put that N s, solve this equation the square root of D t goes to the other side and the sensing time or the settling time now becomes inversely proportional to the square of the analyte density. What it simply means that at a low analyte density it takes forever for the sensor to catch the molecule. At high analyte density of course, you can catch things relatively quickly. Slower the diffusion the more viscous the media the longer is the settling time before you can be clear whether a molecule is present or not. Now this dependence of square root of D t is called a diffusion slow down because farther out you go the molecules through random work has more and more difficulty of reaching the sensor surface.
[Slide 15] Now what about you can immediately see now that this idea is so powerful that you can easily solve a wide variety of problems. For example, consider the nanowire sensor. The nanowire sensor here is sort of a cross-sectional view of the nanowire, the analytes are all around, the red analytes, with a density of rho naught. After a certain time t a fraction of those analytes will be caught by the nanowire itself. You can easily calculate how many will be captured, N t is the number of molecule that got captured on this nanowire surface 2 pi N R is a sensor surface. Of course, you can multiply with l for the length, but we'll not think about the length per se by unit length let's say. Now what is that number? Well it's very simple, rho naught is the original density pi R squared. R squared is the region it got depleted minus N R squared is the original nanowire volume. If you assume this is small and R is simply square root of D t, then you can see this is the area of the circle. The bigger circle it got depleted and from that if you set N is equal to N s the number you need in order for detection -- in order to detect, then you can immediately solve this equation to find the settling time is inversely proportional to the analyte density. Not as a square, but is directly proportional and the power of rho naught is equal to the dimension of the sensor itself. Now in this case you see that total number of particle that gets captured goes lineally with t. It looks like as if there has been no diffusion slowdown. And the reason is yes, there is diffusion slowdown, but it is catching from a larger and larger area. So although, far from a given distance the capture is less efficient, but because it is capturing from a larger area that compensates. So it appears as if diffusion has forgotten to play its role.
[Slide 16] Now you may not have understood how these two simple things -- simple formula can change our prospective about nano biosensing dramatically, let me explain and show you how it works. What I have plotted on the x-axis is the settling time t s, the length of the time you have to wait. And the x-axis I start from micromolar to femtomolar, low densities. Now you may remember that for a planar sensor I have the settling time going as 1 over rho squared. If you take log on both sides you will see the slope will be minus 2. If you have higher density of course, your settling time will be low you will be able to detect something within a second if the density is high. On the other hand, if the density is low this car will shoot up and, therefore, it may take typical 9 seconds is like 10 years or so. And so it may take in order to get a picomolar you will have to wait 10 years before you get a signal. Now by that time the patient may be dead, so you shouldn't try that. On the other hand if you took a nanowire sensor, remember the nano biosensors often use nanoscale devices. In that case the t s goes as 1 over rho naught, 1 over rho naught on a log log plot is given by a line with a slope of minus 1. At high density there's not much difference, there are so many molecules both can capture reasonably well. At low density, however, this takes far lower time than compared to your planar sensors because molecules can sort of come from each side in a surround media rather than a planar sensor, which it can only come from one side and limited by diffusion. Experiments essentially support these ideas and I'll show you more experiments later. But the important point I want to make is that if you're only willing to wait for let's say a couple of hours, let's say a thousand seconds, then nanowire sensors will be 5 to 6 orders of magnitude more sensitive compared to a planar sensor. Hence, the extraordinary sensitivity gain of nanowire sensors. So this immediately tells us that the geometry essence of the geometry in nano biosensing especially for diffusion limited system.
[Slide 17] Now I'll take a break at this point and what we'll do is that I explained that why a nanowire sensor is far more sensitive to a planar sensor. In the next part of the lecture I will explain how the nanosphere doesn't really give you much. And then we'll about how these nanostructure sensors how do they respond, but that will be down the road. Let's take a five-minute break and then we'll come back.