nanoHUB-U Principles of Nanobiosensors/Lecture 2.1: Settling Time - Shape of a surface
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[Slide 1] Welcome back. This is Lecture 4. And I would first briefly remind you of the topics that we have been thinking about, about biosensors. And the topic of today's discussion will be the shape of a surface.
[Slide 2] So I'll recap Lecture 3, remind you what we had been talking about. And then we'll talk about the shape of an object described by fractals. Now, the word fractals may frighten you. Don't worry, I'll explain it very simply. And then I'll talk about regular fractals followed by when a surface is randomly structured but still defined by a single number. I'll end with the outline of the course because then we'll really get started from the next lecture onward.
[Slide 3] You see you may remember from the last lecture that although there are wide variety of sensors, but once you sort of arrange them in some sort of shape according to their shape the planar sensors are far less sensitive compared to nanowire or cylindrical sensors. Other sensors having complicated shape, for example here is a collection of nanowires or a parallel collection of array of nanowires that has response somewhere in between. And we wanted to know what is it in the geometry of this structure that gives rise to this extraordinary sensitivity of nanobiosensors.
[Slide 4] Now, of course, we tried immediately to see whether the surface to volume ratio, that argument that molecules surrounding a nanowire are giving extraordinary sensitivity, that fell flat. Because although we were able to explain why the sensitivity goes up by a certain amount, let's say a factor of five or so, when you make the nanowire diameter small. So that's good, the factor of five is nothing to sneeze about. But that really didn't explain the extraordinary sensitivity gain of nanobiosensors. And I told you that the real secret lies in how a jellyfish catches its food, so and, it uses the same aspect of geometry, that's the shape of the geometry in order to catch the food. And that is what we are going to discuss next. And we'll call it geometry of diffusion.
[Slide 5] Why I use this term you will know in a few minutes. Now, if you have taken a geometry class in high school then your teacher may have
[inaudible] to tell you a surface is two dimensional, a line is one dimensional, and a point is a zero dimensional object. Now, of course, you may not have questioned him or her because it seems so obvious. But think about it for a second. If you had instead a collection of disks, nanodots of some sort randomly scattered on a surface, or a collection of sticks, of nanowires, for example, in that case what dimension would the surface actually be? It's neither a planar surface because the whole thing isn't covered, nor is it a single line. So, therefore, is the surface definition of the surface completely undefined in this case? It turns out that's really not the case.
[Slide 6] You can, let me show you how it works. Let's take a same surface and put it in a set in a grid. Now, you see when you have a surface that are arranged as a planar surface if you divide it by the factor of two then in a surface all four cells will be occupied. In a line only two of the four will be occupied. And for a dot, this green dot, only one of them occupied. Divide it one more time, all occupied on the surface, only four occupied in the line, and still the single one occupied in this for a dot. And, therefore, if you keep doing it you will soon get the idea that the number of -- as you keep dividing the number of cells occupied goes as h squared. h squared is the division. It goes linearly with h for a line, and it goes h to the positive which is, by the way, I hope you remember that's one, because it's always one cell that gets occupied. So that goes as h to the positive zero. So now you can immediately see that if you plot the log of h as function of log over one over h in this you'll immediately see that this will be fractal, -- the slope would be, the slope of this line will give you the fractal dimension for a surface, that for a line and that for a dot. So because the dot doesn't change, the occupation doesn't change. In the blue line it changes linearly for the surface, it changes as a square. As a result you can immediately see why your surface should be two dimensional, a line one dimensional, and a dot a zero dimensional object. Now, this recipe immediately tells us how to think about a random collection of sticks or nanowires or nanodots, for example. But before I get there let me tell you how to think about a surface which is slightly more ordered.
[Slide 7] We'll call that a regular fractal. Consider a line as shown here in the red. And then let's take out one third of it. Then of the remaining two pieces, there are eight pieces, let's divide it into two, three, and then take out the middle piece here and here. And let's keep doing it. If you keep doing it the residual surface or the residual line is not a full line, nor is it a full dot. It's not a single dot. And, therefore, the fractal dimension of the surface would be somewhere between zero and one. So let's calculate again. So if you divide it into one third then N will be two because only two are taken the one is thrown out. If you divide it into nine pieces then four will be occupied, twenty-seven, eight will be occupied. And in general if h goes as one over t to the power N the number of cells occupied will be two to the power of N. Again, you can calculate the fractal dimensions simply by log, taking the log of these quantities. And once you have done that you will see that dimension of this fractured line is only .63. It is not one, not zero, but somewhere in between. Now, in general you get the rule, the rule is divide things in N pieces and keep m pieces. And if you keep doing it over and over again the surface you will create is, will be the fractal dimension of the corresponding surface. The log of the ratios, the ratio of the logs will give you the fractal dimension of the surface. Now, this is a line. No sensor is actually just a single line. Rather most sensors are surfaces. So what about fractal dimension of a surface?
[Slide 8] In order to know the fractal -- before I get there you see in order to create this fractal surface this surface is you don't have to throw out predetermined pieces. For example, if you divide it into four you can throw away the middle two and keep throwing away the middle two of the residual pieces. That will create one surface. That will be fine. That will have a certain fractal dimension, log of four in the denominator, log of two in the numerator. But you could also have done something quite different. You could have taken the four pieces, but instead of throwing away just the middle two you could throw away the first and third and keep doing this, first and third, first and third, and by doing that you can also create a surface that follows the same algorithm and, therefore, has the same fractal dimension. But they don't look the same at all. Now, these two examples we'll call regular fractal where you're following the same rule over and over again. But, of course, you could have followed a random rule that in the first iteration you can throw away the middle two, in the next iteration you can throw away the first and third, first and third. And, therefore, keep mixing up the rules but so long you keep the same number of pieces. And out of the total number of pieces that you originally divided the line with you'll be good. So, therefore, these surfaces will be called a random fractal, and two before that I showed will be called irregular fractal. And so therefore fractal surfaces will not be unique, and they can be either regular or irregular.
[Slide 9] Now, let's talk about surfaces. Same rules once again. Let's say you divide the whole surface by stripes. Initially you divide the whole surface into groups of three. Throw away the middle piece, then of the remaining piece you throw away the middle piece and keep the remaining ones and iterate the process. What is the fractal dimension of this surface? The fractal dimension of this surface you can again divide it into one third and count how many red lines or red stripes you have in this structure. So in the beginning when you divided it into only three you will have six, one, two, three, four, five, six. Six boxes will be occupied. Then if you divide it into nine pieces on either side then you will have 36. Count it up. It should be a good exercise, and you'll be able to see how it works out. And very soon you will see that when h goes as one to the power, three to the power N, that total number will go as three to the power N multiplied by two to the power N. And once you calculate the fractal dimension what you will see the fractal dimension is one because it is uniform in this direction plus the fractal dimension along this line which is, from the previous example was .63.
[Slide 10] So the fractal dimension of this surface is 1.63. In general you don't have to follow exactly the same rules. As I mentioned before fractal, that regular fractal comes in a wide variety of forms. So, for example, this surface is exactly the same as the other surface because now you have sort of striped on the horizontal axis. Previously you did it on the vertical axis. And, in fact, in both cases the fractal dimension would be exactly the same. Now, what about this fractal structure, random collection of rods? Does it have anything to do with these fractals? Can it have exactly the same fractal dimension as the other two?
[Slide 11] Let's find out. So here I have redrawn a set of nanowires, for example, on the surface, which will eventually become nanobiosensors. Biomolecules would be attached to the nanowires themselves, but we'll get to that later on. So, again, let's start by following the rules. Divide it into two, two pieces, and once you do you see all four are actually occupied in this particular case because there's nanowires in each one of those boxes. Now, if you keep doing it, keep dividing it, very soon you will find that there will be boxes which are no longer occupied. And so the box number will not go as h squared but will go at somewhat slower rate because there will be empty boxes. And you could keep doing it. And as you keep doing it then there will be more and more numbers. But the number will not go as h squared but somewhere less than h squared. And in this case you can show that the fractal dimension of this nanotube network is 1.54. It is neither completely two dimensional nor one dimensional but somewhere in between. And this fractal dimension will play a great role in the story that I'm going to tell you.
[Slide 12] Now, in many cases what we'll it find useful, that when you have a random surface like this, instead of handling the random surface directly, what we should do is convert it to a equivalent regular surface. Because you see equivalent regular surface will be much easier to analyze and handle, and we'll be able to do mathematics on it that would have been difficult otherwise. So how do we get from the regular to irregular surface? Very simple. You remember that we have already mentioned for the regular surface with stripes that D fractal dimension of the regular surface is one plus log m plus divided by log n. Let's say this particular surface has a fractal dimension of 1.54. The previous slide we saw that's 1.54. But here let's assume that it's approximately 1.5. Now, let's make m as equal to two. So the number of pieces we are going to keep is two. We can put m in here, D I know is 1.5. So once I solve it I will get n is equal to four. And once I get n equals four I can now generate the equivalent surface. I'll draw a line segment, I'll draw a line segment, cut it up into four pieces, throw away the middle two. Then again of the remaining cut it up into four pieces, throw the remaining two and keep doing this. And once I have done this then the surface I have created has exactly the same surface property as the original irregular surface. And this equivalence will be very important when you see how biomolecules get captured on these surfaces.
[Slide 13] So with this information now let me tell you the outline of the course. Because now we have all the three pieces that we needed to know in place. We know why nanobiosensors are important. We know the types of biomolecules that we are interested in, DNA, protein, glucose and similar type of -- and viruses and bacteria. And we now know that the geometry of the surfaces can be described by fractals, and there are three types of sensors we are interested in potentiometric, amperometric and cantilever. With this basic information now we can check out the outline of the course.
[Slide 14] The first part of the course has to do with how long the biomolecule takes in order to reach the sensor surface. It cannot get there instantaneously because it is diffusing around, randomly walking around. And so there is a time necessity before it can reach the sensor surface. The lower the density of biomolecules longer it will be before a certain number can find its way to the sensor surface. And that information is captured here. Rho naught is the density of the target, ts is the average time needed for the molecules to get to the sensor surface. And DF is a fractal dimension of the sensor itself. And we'll find that these informations are all connected up. Lower the density longer the time one needs in order to sense something. Now, this fundamental relationship will tell us that regardless of how sensitive the sensor is there is a fundamental lower limit, the diffusion limit for which dictates what is the minimum density one can detect by using a specific sensor. Now, this relationship might remind you of Heisenberg's principle where the energy and the time are also related to a constant. If you want to measure something with a high precision then you must allow for longer time. If you want to measure a low density you must allow for longer duration. You must wait for long before you can declare that the analyte is present or not. Now, this is all this technology agnostic. It doesn't matter whether you are using amperometric sensors, potentiometric sensors or even a cantilever sensor. It doesn't matter. Rather, this is a general principle that applies to all biosensors.
[Slide 15] Now, next from lecture 11 to 22 we'll talk about when the biomolecules have finally landed on the sensor surface how does the sensor respond? Now, in this case, of course, amperometric sensors are very different from potentiometric sensors or cantilever based sensors. The physics of the sensor matters. And that we'll spend some time in thinking through the issues clearly. And in that case we'll think about primarily how the sensor responds to the intrinsic properties of the biomolecules, namely the mass of the biomolecule. You may remember bacterias are heavy. And whereas the DNA and glucose are actually very light. And so depending on the intrinsic property of the biomolecule we'll be using different sensors to detect them. Now, we'll also at that point will discuss the noise limit associated with detection. Because in a noisy environment a sensor may not may be able to tell apart between a molecule that is target molecule which has actually landed versus a random fluctuation in the sensor characteristics. Geometry of the sensor will again play a very important role as we will see.
[Slide 16] And, finally, we'll talk about one of the most but often most misunderstood aspect of sensing. And that has to do with selectivity. You see, if two molecule comes in and lands on the sensors, the target molecule and the P for parasitic molecule, meaning some molecule that you really don't want to detect, in that case if both of them trigger the same response on the sensor surface then it really doesn't matter how sensitive the sensor is or how fast it can detect it. In fact, this sensor is completely useless because it's not selective to the molecule that we're interested in. And it turns out that this problem of selectivity can be defined as a signal to noise ratio problem. So this T is the signal, P is the noise. And we'll see that theories of information,the information theory, the various results of the information theory will be directly relevant. Also, will be relevant are the fundamental issues with random sequential absorption, that how these molecules essentially arrange themselves on the surface.
[Slide 17] And that will be lectures 22 through 30. So, therefore, what you'll see a key and repeating or recurring aspect in this course is the essential role geometry plays. Geometry plays a significant role in defining the sensitivity limit. It defines the screening, how the molecules arrange themselves around the sensor. And it defines the selectivity, that some molecules which are the target molecules lets say blue and the green molecules we said is the parasitic molecules how they arrange themselves and what the signal to noise ratio would be is again defined by the geometry of the surfaces as well as geometry of the molecules. And you will see nanobiosensors in particular, this role of geometry, surface geometry is fundamental.
[Slide 18] And we'll find that over and over again in various contexts. So let me conclude then. I tried to tell you in the beginning that the surface to volume ratio is often used to describe a sensor surface or a sensor's response, but it really doesn't work in terms of interpreting the dramatic gain in sensitivity. Instead what we'll see, what we have discussed is that if you characterize the sensor surface with fractal dimension that will turn out to be far more powerful. The second aspect regarding fractals I wanted to mention was that you can have a regular fractal where the elements are generated by a set, set of rules, a given set of rules. Or you can mix it up. And in that case it can have a random surface like a random collection of nanowires that's again described by a fractals dimension. And you can go back and forth between them, and that will ease analysis. And, finally, I mention that the course is arranged in a way that we'll discuss three fundamental topics associated with nanobiosensing. One is settling time. How long do you have to wait before you have a signal? The second was sensitivity. That once the molecule lands how sensitive is the sensor so that it can respond to it? How many molecules on the minimum do you need before you have a good signal to noise ratio? And, finally, one of the most important is selectivity. Can the sensor tell apart between the target and the parasitic molecules? And that is how the course is arranged. So we'll get started in that from the next lecture. So until next time take care.