 Hello, welcome to this lecture of bio mathematics. In the last few lectures, we have been discussing about diffusion, like last couple of lectures. We discussed the diffusion equation and then we discussed, how do we calculate the RMS distance from this equation without really solving the equation? Even though diffusion equation was the second order differential equation, at this moment, we will not discuss how to solve it. It is a little more complex, you need to learn some more mathematics to figure out how to learn this or to understand how do you solve this diffusion equation. But without really solving it, something very useful, some relation that is very useful we got, which is the RMS distance and time, the relation between time and the x square average. So, this we will continue to discuss something about diffusion. Initially, in the first part, we will discuss a bit about another quantity, which is average x average. And then, we will go on to discuss some one, another very important relation as far as the diffusion is concerned. So, so far, we took diffusion coefficient as a constant. Today, we will discuss, what is diffusion coefficient related to, what is the, how is, you know, how is the diffusion coefficient related to temperature, viscosity, etcetera, of the medium. So, this relation was the famous Einstein's relation. So, we will discuss Einstein's relation in today's lecture towards the end of this lecture. So, the topic of, we are, we are continuing to discuss this section on applications of calculus and vector algebra in biology. And under this, we are discussing this diffusion. So, we said that, if you start with, if you take a tube, as you see here, if you take a tube and if you put a few number of particles in this or of protein molecules at the middle of this tube, the concentration is only, at t equal to 0, the concentration is only at the middle of the tube. And as we go along, it will spread, the molecules will diffuse. Even, even at this time, maximum number of molecules, large, bigger number of molecules, the concentration is still more here at the center, but there are still, there are some, still, there are some molecules as we go along the tube to the left or to the right. So, this is the concentration profile and the diffusion equation is, the equation that deals with this, spread of this concentration is the diffusion equation. So, the diffusion equation is d C x by d, d C by d t is equal to d, d del square C by del x square, where C is the concentration as a function of position and the time. And by solving this equation, we expect to get concentration as a function of position for any time, that is what you will get. And we defined two qualities, x average and x square average as x is C, C tilde average integral d, x C tilde d x minus infinity to infinity and x square C tilde minus infinity to infinity as x square average. And we also saw that x square average is 2 d t, this is a very interesting relation, an important relation. The square of the position goes as time or in other words, if you take square root both sides, the square root, the r m s distance goes as the root of time. So, that is what the important relation that we saw, that is x r m s, which is this is proportional to the square root of time. So, this is x r m s is proportional to the square root of time. So, this is the interesting relation that we found and this is, so this square root of time or we can write this t power half. So, this is, this t power half is kind of synonymous to diffusion, one would say like it is like something moving like t power half, something moving like square root of time, that is called diffusive motion. So, this is an important relation as far as diffusion is concerned. Now, we will calculate x average, something which we are going to learn. So, the question is, next question is, what is x average? So, that is the question that we want to address next, what is x average? So, we found that in the previous lecture, that x square average is 2 d t. We discussed this relation, as we said this is an important relation, which says that how does the r m s distance, this is the root of this is the x r m s and how does the x r m s is related to time. So, the x r m s, if you take square root on both sides, the x r m s as we wrote previously, look at here, x r m s is root of x square average, this is proportional to time, square root of time. In other words, x r m s is proportional to square root of time or proportional to t power half. So, now the next question is, what is x average? We will calculate the x average the same way as we calculated the x square average. How did we do that? We took this equation, which is the diffusion equation. We multiply both sides by x, we are multiplying both sides with x and integrating. So, you multiply here with x and integrating. So, this becomes integral x d c d x with del by del t outside and here also, the right hand side also we are multiplying with x and integrating. So, integral x del square c by del x square t x. So, we multiplied both sides of this diffusion equation with x and integrated from minus infinity to infinity and what do we get? So, the left hand side as we defined earlier is integral x c tilde x g x is nothing but x average. So, this is the left hand side. So, what do you get? You get on the left hand side. So, let us look at what we get. So, what we have is del by del t of integral minus infinity to infinity x c of c tilde of x d x is equal to d into integral x del square c by del x square c tilde d x. Now, this part as we said last time is nothing but del by del t of. So, del by del t and this is integral x c tilde x d x is nothing but c average. So, this is c average. This one, this part which is I writing it in a box, this part can be written as c average. So, this is equal to d into integral minus infinity to infinity x del square c tilde by del x square d x. Now, just like we discussed last time, this can be written as integral and you can take x as u and this term you can take as del v by del x or d v by d x. So, this is integral u d v by d x. So, this is you can do this rule in calculus called integration by parts. So, integral as we just discussed sometime ago, integral as we just discussed in the last lecture, integral u d v by d x d x can be written as integral u v sorry can be written as u v in the limits minus integral v del u by del x d x. So, we can use this formula, which is the standard formula in calculus which we in the last class we figured out how this formula is coming. And let us say let us use this formula now. So, what did we said is that we just said have a look at here. So, what did we just said that del by del t of c equal to minus this and we call this as u and this as del v by del x. So, now we can apply this formula there. So, if we apply this formula there, if we apply this formula there what do we get. So, we have x as u and del square c by del x square as del v by del x. So, what you would get essentially is this, if we apply this formula what you would get is that del x average by del t is equal to u is x. So, x and v is del c by del x in the limits minus d into integral v d u by d x. So, now v is del c by del x d u by d x is 1. So, this is what you will get. So, let us this what precisely I have written what you would get is that d into x d c tilde by d x in the limits minus d into del c by del x del c tilde by del x d x. So, this is what you would get. Now, if you apply this limit just by arguing that at plus infinity and minus infinity the derivative is. So, by just arguing that this term at you apply this del c by del x at plus infinity and minus infinity and you calculate this term you will get this equal to 0. So, then what you are remained with us just. So, this limit this is 0. So, what you have is just minus d integral del c by del x d x. So, we have just this term. So, now let us see what is this term what this term gives. So, let us think about that term a bit. So, what you end up essentially is that del by del t of x average is minus d del minus infinity to infinity del c tilde by del x d x. So, what is this? So, this is derivative. So, this is essentially minus d into c at the limits. If you take minus infinity and infinity plus infinity at the limits if you calculate the c c at infinity and minus infinity is 0. So, the answer that this integral is essentially 0 because if you do this derivative integral of a derivative is just the function itself. So, you have c c at infinity and c at minus infinity they are 0. So, essentially this derivative this integral is 0. So, you will end up with this relation that del by del t of x average is 0 which means that x average is 0. So, what we got essentially is that x average is 0. We had found that x square average is 2 d t. So, this is two interesting relations that we get that x average is 0, x square average is 2 d t. Now, what does this mean to say that x average is 0? Physically what does that mean? So, let us think about this. So, let us think about this diffusing in a pipe example that we thought. So, we let us take this pipe and to begin with you have some particles here and let us say there is one blue particle here and one particle which I am circling here which is in blue color. So, let us say there is one blue particle here and all other black particles. Now, if you look at this blue particle and in one experiment you might see that this blue particle is going this way. So, it will diffuse some distance x in this way in one experiment. Let us say you are doing the same experiment. Let us say you are doing the same experiment in a different day or you are repeating this experiment. Now, you have again you start with the same condition. So, you know you have the blue particle you put here, when you put the blue particles becomes here and this time it might move in this way. So, it might move the distance x in this particular way. So, if you just keep repeating this experiment of the experiment that we discussed that is adding some amount of proteins at the middle of the tube and looking at where is it going, which way this is going. So, at one experiment you might see that this basically what this is like imagine that you have just one particle that we can detect. Let us say it is fluorescing or it has some different color. So, you will see that in one experiment this particular particle might be going this way. In another experiment this particle might be going this way. In some other experiment it might be going this way. So, if you do many experiments and average over all this. So, sometime it will go in the plus direction sometime it will go in the minus direction. So, minus direction plus direction finally the average you will get 0. So, the first experiment it might have moved minus 3 centimeter. In the next experiment it might have moved plus 3 centimeter. So, if you just repeat this experiment many, many times on average if you find the average of this you would get 0. That is what it precisely means. It means that if you look. So, in a diffusion experiment if you look the position of one particle over many, many experiments the average over all experiments will give you the average position as 0. On the other hand if you calculate the square average minus 3 square is 9 3 square is also 9. So, there is no way this x square can average out to 0. So, you will get some quantity which is 2 d t. So, the meaningful. Since x average is 0 the meaningful quantity is x square average. This is the meaningful quantity that we can that we should know or that that that is useful. Physically useful quantity is x square average in other words the RMS distance. So, the important formula as far as the diffusion motion is concerned it is x square average is 2 d t or x RMS is equal to root of 2 d t. So, this is the important formula. This is the root mean square distance that a particle would travel in a time t. If you do average over many, many experiments this is the root mean square average that things would travel in a time t. Now, what is this d? So, that is the question what is d? We have been discussing we have been having this thing called d for a long time. So, from this we found that d has a unit if you dimension of d is length square by time length square by time. So, it will have a unit meter square per second. So, now what is this d? d is the diffusion coefficient. What does that mean diffusion coefficient mean? Diffusion coefficient essentially that contains the property of the medium that in which you are putting this protein. If you are doing it in water it contains the property of the water like viscosity of the water. It also contains the temperature. So, you can imagine that if the temperature is very large things will diffuse out very fast higher temperature higher diffusion. So, the information about the temperature, viscosity all the property of the medium is put into this one quantity called d. So, the d contains the property of the medium. So, now how do you find out? How does the d depends on the property of the medium? If the viscosity is more how does the d change? If the temperature is more how does the d change? How do you find it out? So, just by learning just by knowing what you learned in mathematics so far and with some intuition with some simple thinking one can figure it out. Actually this was discovered by none other than Albert Einstein in 1905. So, this is what we are going to discuss. So, the relation of the d. So, the relation between the relation between the relation between d temperature and viscosity let me call this eta as the viscosity. So, this relation is called Einstein's relation. So, that is what we are going to discuss now. We will discuss this relation between diffusion coefficient temperature and viscosity as and this relation is known as the Einstein's relation. This was discovered by Albert Einstein in 1905. Albert Einstein for his P S D he was studying about Brownian motion of particles and he discovered this relation. I will tell you in a simple way how do we calculate how do we derive roughly what Einstein did about 100 plus years ago. Take care about 100 and 106 years ago. Einstein derived this relations. This as you might have also heard this 1905 is a very famous year for Einstein. Like he wrote three very famous papers. One paper is related to this Einstein's relation which became very famous and this relation became one of the most popular relations like very highly cited relations in science because this is application on biology, in chemical engineering, in chemistry, in physics, in all sorts of field. Einstein's relation related to diffusion is used. In environmental sciences in you can think of any field virtually any field and this relation will be or is this relation is being used and then he discovered the he explained the photoelectric effect and he also explained he also he also had his famous paper on relativity. So, this three papers made him world famous like all these papers. So, one of the papers even got him Nobel Prize. So, this is a miraculous year as for Einstein's concern and the world of science is concerned. So, we will discuss one of his contribution in that year 1905. It is interesting that just by understanding this simple mathematics we can derive this relation. It is very similar to what we did for Nernst equation. So, we will go in the same line as we did when for to understand Nernst equation. So, let us go ahead and think about Einstein's relation. So, Einstein thought about the following examples. So, he thought there are some particles in water in a beaker and this particle is subjected to some external field. So, let us say there is gravity downwards. So, if there is gravity all this particles with some mass they will be forced to come down because of the external force gravity. For example, it could be either gravity or it could be if they are charged particle it would be even electric field. So, you could think of this is electrophoresis if you wish. So, basically this are you could even think of this as charged particles and then there is some force exerted on this charged particle due to electric field. This could be like some protein molecules under electric field. Now, let this force be minus g times x. Mathematically this force is minus g times x where x is the distance from bottom to top. So, x is the distance starting from the bottom to the top. So, x cap has this particular direction and the force has this particular direction. So, force is acting downwards and the distance is going upwards. So, the f and x are have opposite direction. So, that is why this minus sign. So, g is the amount of force. So, g could be the amount of electric field, electric force due to electric field could be amount of force due to gravity whatever you wish. But g is some force and the magnitude of the force is g and f is the force of the vector force. So, let f is equal to minus g x. So, we can say that the energy. So, for every particle if you want to this particle to go up at distance x it has to spend an energy f dot x. So, if you have a force f look at here, you have a force f. So, which is let us say it is g x cap and let us say that the energy. So, if you have such particles and each of this particle is experiencing a force. So, if this particle wants to go up a distance if it wants to reach a distance x from the bottom, it has to spend an energy f with f dot x. So, which is nothing but f is minus g x cap dot x. So, this is minus g x. So, this has to spend this much energy. So, it is not favorable. So, the magnitude of energy is g x essentially. So, the energy is g x. So, it is not favorable to go up here because the force is in this way. So, it has to spend an energy. So, most of the particle you will find at the bottom because there is a force acting and there is an energy cost to go here. So, if you look at the concentration, if you think intuitively the concentration will be more at the bottom and less at the top. You can think of any particle if you put something into water, you could think it of a sedimentation. Something will fall down to the bottom of the beaker. If you put something which is some objects into water they will fall down because the gravity is attracting it down. So, the concentration of anything will be more at the bottom and less at the top. So, if you plot, if you wish, if you plot the concentration as a function of the distance from the bottom, you will see some exponential relation. There is some reason why it is exponential, but let us intuitively assume that c is, c is proportional to e power minus g x by k b t. So, g x is the energy and it has to be divided by another energy k b t to make it dimensionless. So, the concentration decreases as you go along x. This is what it means. So, if you know this relation, we can derive the Einstein's relation. So, this is the one ingredient that we need to know that the concentration will exponentially decrease as we go to the bottom. How do we get this relation? That we will discuss later, but for the moment just take this relation for granted, which is intuitively clear to you that the concentration will decrease as we go along x and knowing this, we will derive Einstein's relation. So, let us say the concentration will be more here and concentration will be less here and this has this particular functional form. Once we know this, we will follow roughly what we did for deriving Nernst equation. We said that in the case of Nernst equation, there is a current due to electric field or the force. Similarly, here there is a current due to this force, this gravitational force or electric force, which is pulling down this particles downwards. They will want this particle to flow down. So, the flowing down happens with a velocity v and this v is related to the current or the flow j. As we discussed previously in the case of Nernst equation, j f is c concentration times v, which is velocity. Now, any particle moving in water will have a velocity, which is given by f by 6 pi eta a, where f is a force acting on that particle, pi is a constant, eta is the viscosity of the water or the medium and a is the size of the particle. So, there are quantities, which you should remember, eta is the viscosity, a is the size of the particle and f is the force acting on this particle. If you know this much, the velocity is this and the flow is proportional to the velocity. The more the velocity, the more the flow is. So, the flow downwards is c times v. So, this can be written in a different way. The c times v and c, v is force. This can be written as minus g x cap, where x cap is this direction, so minus g x cap. So, substituting this f as minus g x cap, you get j f, the flow as minus c g x cap by 6 pi eta a. So, this is the current that is making. So, this is the flow due to this, this force. It could be gravitational attraction downwards. It could be the attraction due to electrostatic forces or electric field. It could be the electric field of applying downwards, forcing the proteins to move in this particular direction. So, this could be any force, whatever the force you wish, but that force will, will lead to a current or a flow given by this particular formula. As we saw in Nernst equation, we had similar, similar flow. Now, this flow leads to our interesting thing that it makes the concentration more here and the concentration less here. If the concentration is more here and the concentration is less here, diffusion can happen, because diffusion is a flow from higher concentration to lower concentration. So, in principle, things can diffuse back from here to here. It can diffuse back from lower concentration to a higher concentration. So, here it is, sorry, it can diffuse back from higher concentration to a lower concentration. So, here it is higher concentration. Here it is lower concentration. So, from here to here, you can think of, you can think, imagine that there can be some flow due to diffusion or flow due to concentration change. You could think of some kind of diffusional or diffusive flow. So, how much is that flow? So, we said that due to diffusion, there can be a current or a flow and that current J D is related proportional to the derivative of the concentration as we saw previously. J D is proportional to del C by del x and as we go along the X, C decreases with del C by del x is negative. So, with this minus sign, the flow is actually along the X cap direction, which is in this direction shown by this blue arrow. So, we have a diffusion, which is basically taking this in this particular way. We have a flow, which is in this particular way given by J D equal to minus D del C by del x. Now, what is C? We just saw that C is proportional to e power minus G. So, we just said that C is proportional to e power minus G x by k B T. So, that means, C is some constant A, it will be some constant, times e power minus G x by k B T. Now, we also said that J is minus D del C by del x. Now, what is del C by del x? Del C by del x of this will be, so let us find the derivative of this. So, what is the derivative of this? So, del C by del x will be A is there, derivative of E is e power minus G this itself, times the derivative of this, which is minus G by k B T. So, we said that derivative of e power k x is k e power k x. So, we had a k, which is minus G by k B T. So, that is the k, which is coming here. So, by using this relation that we learned, the derivative of e power k x is k e power k x, where our k here, in our k is the k was minus G by k B T. We have del C by del x is A e power minus G x by k B T into minus G by k B T. Now, look at here, what is this A e power minus G x by k B T? What is this part? This part is C itself. Look at here, this is C is A e power minus G T x. So, del C A, C is A e power minus G x by k B T. So, this is C itself. So, del C by del x, del C by del x is nothing but C itself times minus G by k B T, minus C G by k B T. So, what does this mean? This implies that we had J, which is D del C by del x is minus D times C times G by k B T. J is minus D C G by k B T. So, that is what we have here. So, J D is G del C by del x and substituting for D, known that C is proportional to e power minus G x by k B T. And, substituting this, in this, we get J D is D C. There is a plus sign here. I made a mistake and there is a typo in this. So, if you just substitute this minus, there is a minus sign here. By taking this minus sign into a cone, we will get a plus sign here. So, essentially you get this, what is shown in this square, in this rectangle here. What is marked here? The current upwards is D C G by k B T along the x cap direction and we had J F downwards and J D upwards. So, we had current downwards and the current upwards. So, when you have currents in opposite direction, we said that when both these currents balance, that the current upwards and the current downwards when they are, when they balance, we reach equilibrium. We call it equilibrium. For example, if you look at this particular point, there will be some current upward, there will be some current downwards and when these currents balance, we have net current 0 and we reach equilibrium. This is exactly the argument that we discussed in the case of Nernst equation. So, what does that mean? Equilibrium means net current 0, net current is nil. What does it mean? The total current J D plus J E is 0. In other words, J D is equal to minus J E. There is a, what I meant here is J F. This is a typo here, should be J F here. So, what I meant here is that J D plus J F is 0. In other words, J D is equal to minus J F. So, the current due to the force is equivalent opposite. Equal to the opposite sign, that of currents due to the diffusion. So, this is what, this is, this is the condition for equilibrium. So, what do we, how do we, let us try doing this. So, what we have is J D as d C G by K B T and J F as minus G X by 6 by eta A and we assume that J D is equal to minus J F and that is, that means, d C G by K B T is equal to C G by 6 by eta A. So, which implies I can take everything, D alone keep this side and take everything else to the other side and what you would get is that, D is equal to K B T by 6 by eta A. So, this is, says that diffusion coefficient is equal to K B T by 6 by eta A. So, this is the famous Einstein's relation, that this relates the diffusion coefficient to Boltzmann constant, temperature, viscosity and the size of the particle. What does it say? The more the temperature, the more the diffusion coefficient, the more the viscosity, this is inversely proportional. So, if viscosity is very large, the diffusion will be less, which is intuitively clear. Something might diffuse better at water and much less in honey, where honey has higher viscosity compared to water. So, the viscosity of something highly viscous, let us say, take example of tar or honey, you will, it will be very difficult for things to diffuse in tar or a very highly viscous medium and if the size of the particle is very large, look, if you look at the size, A is the size of the particle, if you have proteins that are very huge, very large, they will not diffuse, the diffusion coefficient of those objects will also be very small. So, this is the famous relation called Einstein's relation, which relates the diffusion coefficient to these quantities. And the relation, the, we derived this relation by, in the way we derived Nernst equation by arguing that if you have an object, if you have a beaker and particles under a force F, there will be two currents, the concentration will be more here, because of this force is attracting it downwards and very few particles up. So, since the concentration is less here and concentration is more here, there will be diffusion this way, diffusive current and the current due to the force and when they are equal and opposite, we get, by arguing that they are equal and opposite and equating them, we get this relation, which is D is equal to K B T by 6 by eta A. So, now, we learn two, three things, we learn that X average is 0 in today's lecture. In the previous lecture, we learn that X square average is 2 D T and we also learnt that D is K B T by 6 pi eta A. This is true for a spherical particles typically, if this is not a spherical particles, this 6 pi eta A could be something else, but how are, let us, let us strict limit ourselves to a case of spherical particles, proteins could be thought of as spherical particles. So, this is the, this is another relation that we learnt today and this are the three important relation as well as diffusion is concerned and using simple arguments from calculus and vector errors, we could derive this relations and this has high, very high significance as far as, as far as diffusion is concerned. Now, so, one more interesting thing I just want to share with you is that, so in the beginning, we graphically represented the diffusional profile, the profile of the concentration in this particular manner. Now, what is this mathematical function that can represent this concentration? So, it turns out that the mathematical function C of X is in the case of pure diffusion, that diffusion with no force, this is not the case. So, in the case of diffusion with force, we found that this is e power minus f x by k B T. So, this is diffusion under external field, but when there is no external field, that is, diffusion under no external field or free diffusion, C of X will be having some foam, which is A e power minus some B x square. So, this function is called Gaussian function, this has a bell shaped curve, this is the, this is the bell, this is the bell shaped curve, if you plot this. So, this is called a Gaussian function. So, this is diffusion, this is free diffusion, there is no external, this is free diffusion and the free diffusion is, now if you plot this, if you plot this, we saw that this particular foam and if you plot this, it will have some foam, which is symmetric like this. I am not properly drawing this, just go and see how the Gaussian, plot this function, if you wish like putting some values for A and B, plot it yourself and see how does this look like, this looks like a bell shaped curve, symmetric both sides to the x axis, nicely symmetric around the peak. Now, the free diffusion is governed by this equation del C by del T is equal to D del square C by del x square. It turns out that the diffusion under some external force is governed by the following equation. So, we have this equation plus some effect due to the external force. So, let us say the particle, when they have an external force. So, let us, if you have just no force, this will be the equation, but let us say, each of this particle is experiencing some force downwards or in a particular direction, then each of this particle will get some velocity due to this external force, then the equation will be this. So, this is the diffusion on equation under an external force and this is free diffusion where there is no external force. So, these two are two equations and this equation, this equation has a solution, which has of the form this and this equation has a solution of the form this. So, we will see how do we get to the solution of this later, if we may see this, how do we get this relation, but for this, at this moment, we would not go to solve this equation. We will just understand that there are such differential equations. When it comes to probability, etcetera, we might revisit these equations in a different context, but at this moment, it is, it suffices to say that these are very interesting equations in biology and there are two important relations that as far as this equations, from this equations, from this mathematics, essentially you get two importance, three importance relations, which you should all remember, that is the average distance that a particle moves. If you just mark a particle and ask the question, if you do many, many experiments by marking a particle and ask the question on average where it will go, sometime it will go to the 1 x minus x, some other time it will go to the plus x. So, on an average, it will not go anywhere x average will be 0, that is what this means, but the x square average is 2 g t and the diffusion coefficient is k b t by 6 pi eta a. So, by knowing this relations, with this relations, we will summarize today's lecture. So, the importance relations are x square average is equal to 2 d t and d is equal to k b t by 6 pi eta a. You will come across this relations many, many times in your life and with this we will stop today's lecture. Remember this relations and with this we will kind of completing the section on diffusion. We will go ahead and learn new things in the coming lectures with this we are stopping today's lecture. Thank you.