 So, what I thought we will start is a slightly different topic which is polymers in biology. So, today what I will try to do is try to start with very simple models of polymers and how to write statistical descriptions of polymers and then next couple of classes will go more into examples of biological situations where such calculations would be relevant. So, what are biopolymers for example? So, the most common one is of course, DNA. A polymer is of course, as you know it is something which is made up of repeating subunits, a long chain made up of some repeating subunit, monomers and when you connect many of these monomers together what you get is some sort of a continuous chain which is my polymer. So, DNA is of course, one such polymer and then proteins are of also polymers for example, slightly different kind and as you will see the sort of description that we will use for proteins is slightly different from what we use for DNA, but nevertheless these are also polymers. So, what I want to look at is how to describe sort of structures of these polymeric objects, but structures in a very specific sense structures of polymers, but structures in the sense of statistical structures. So, for example, for a protein you know you have in a amino acid sequence that comprises the protein and then if you do some x-ray crystallography you can get some sort of a three dimensional structure for the protein and some sort of a structure and that is what you think of as the structure of this particular protein let us say. But these are often somewhat static descriptions in reality inside cells in vivo conditions the structure is not often not one fixed thing it can go from one conformation to another depending on what environmental conditions pH salt and so on that you are changing. Similarly, if you were to think of DNA for example, a DNA which is coiled inside a nucleus. So, let us say this is my nucleus and here is my DNA chain that is lying inside this often it is not that let us say whatever the 100th monomer the 100 base pair of this DNA is at some location are 100 and the 201 is somewhere else are 200 and so on. It is not a fixed structure in that sense things sort of fluctuate because of thermal energy and you know often you will find this monomer here or maybe at some later instance of time you will find that same monomer somewhere else and so on. So, what we are interested what we are going to look at is somewhat statistical properties that will characterize for us in a very coarse grain sense things about the structure of this polymer such as basically things like what is the end to end distance of this polymer things like what is the end to end distance or something like what is the radius of gyration of this polymer and I will define all of these terms. So, for example, if I say that well I take a bacterial chromosome let us say which is some n 1 number of base pairs and I take a human chromosome which has some different n 2 number of base pairs. How large will this object be when I put it in solution what is going to be a characteristic size scale that describes this object versus something that would describe that object how does that vary with the number of monomers at the number of base pairs that you have. So, questions like that is roughly what we will be asking in this section or how likely is it for example, for two monomers which are some distance apart on the backbone how likely is it for them to be close to one another something that is called the contact probability. So, we will look at statistical sort of descriptions ensemble average descriptions of these structural quantities. So, at a very basic level what we will start off with is actually something that we have done which is the random walk model. We have done the random walk model in the case of transport. So, diffusion we started from a random walk and we went to the diffusion equation and that gave us time trajectories of particles right. So, if I consider a bacteria is doing random walk it looks something like this in time the trajectory, but if I were to think in terms of sort of space then this could equivalently represent the conformation of a polymer right. So, the underlying mathematics is very similar what was time in my earlier case R square grew with time will be replaced by the number of monomers in your polymer in this case ok. So, that is what we will start off with. There is a most basic model in some sense of a polymer freely jointed chain random walk polymer or Gaussian polymer and then we will try to see how to build in more and more complexity as you go along ok. So, let us start with this random walk polymer. So, let us say I have and again let me start off with 1 d and I will generalize later. So, here is let us say my lattice on which I will do my random walk these are my lattice sides right. I start off from somewhere let us say here and I place 1 link which is like 1 monomer either to the right or to the left. So, let us say I placed 1 link to the right then I placed another one to the right and then maybe I place 1 back to the left and then maybe 3 to the right. So, this is what a polymer in 1 d would look like. If you think in 3d or let us say 2d then of course, you can go up but top bottom left right. So, if you keep placing links your polymer could look something like this. So, you take each step let us say a step of length a which represents the size of the monomers of this polymer that comprise this polymer and then you place them along a lattice and then if you step far back and you look at it it will. So, if you if these step sizes are much smaller compared to the size of the polymer. So, it will look something like a continuous shape right. So, I will work it out in 1d first. So, let us say that this polymer in 1d can take steps to the right or to the left and let us say it takes steps with equal probabilities. So, pr equal to pl is equal to half and let us say this is a polymer with n subunits. So, n monomers. So, this is a n number of monomers. So, the number of configurations of that are possible of this polymer is what it is 2 to the power of n right. So, it is 2 to the power of n. So, this is the total number of confirmations is the number of confirmations is 2 to the power of n. And like we start off with the mind. So, these each such each of these 2 to the power of n confirmations would be one microstate of this system right. And if I start from the micro canonical ensemble, then each of these microstates a priori are equally likely to occur ok. So, then I could ask that well I let us say I am interested in something like the end to end distance ok of this polymer. So, I construct I define my end to end distance r r sorry which is the sum over all these steps x i, i going from 1 to n. I have n monomers. So, I take n such steps I do the left or the right and then I have some. So, here for example, I have taken 1 2 3 4 5 6 steps and this is my end to end distance 1 2 3 4 right. So, the end of that I will get some end to end distance and I can ask what is the expectation value of that and so on. So, for example, in a walk like this in a random walk polymer what would be the mean value of this end to end distance? It would be 0 right because you have equal probability of going to the right or to the left. On the other hand what would be the variance of this random walk polymer would be n square into my step let us say the steps are a half n into step length square right. So, if you look about think about the correspondence this n is like my time in my space diffusive walks this was it r square grew as time here that time is replaced by the number of monomer links that I am space keeping or pleasing. So, r square would grow as the number of number of monomers that you have in your polymer. So, this would be my r square would be n times a square ok. So, this is exactly similar to what we do in a in diffusion. But of course, again we can do better than this we can do better than calculating just the first few moments we can calculate the full distribution right. So, let us say I want to want to derive what is the probability distribution that the n to n distance for a polymer which is n monomers n subunits is r ok. What would that be for is so, I have a polymer made up of n subunits or n monomers and at the end I want to calculate the probability that my n to n distances are. So, if you think in terms of this polymer is a 1 d polymer remember. So, it can take steps to the right or to the left. So, let us say it has taken n r steps to the right and n l steps to the left right. Then how can I express this n r and n l in terms of r and n I can right. So, n r plus n l would be the total number of steps which would be the number of monomers. So, this would be n and what would be r? r would be let us say n r minus n l right times of course, my step length it has to have the dimensions of length. So, n r minus n l into a ok. So, now it just sort of becomes easier that I have this 1 d random work of n steps out of which n r are to the right and n l are to the left and I know what is n r and n l in terms of r and n from these two equations. So, then I could so, I can write that what is the for example, the number of ways omega in which I can have n r steps to the right and n l steps to the left such that my total number of steps is n and what is that this is n c n r or n c n l whichever let me keep n. These are not independent coordinates n l is just n minus n r. So, the n c n r right which is n factorial by n r factorial into n minus n r factorial. So, n l factorial there is a total number of ways in which I can take n r steps to the right given that the total number of steps is capital N. So, then what would be the probability of a configuration like this? What would be the probability of a configuration where I have taken n r steps to the right out of a total of capital N steps yes divided by 2 to the power of capital N that is the total number of configurations that are possible n c n r right. So, you can now just write it in terms of r and n since that is those are the sort of physical quantities that we are interested in. So, the probability of having an end to end distance of r in a polymer which is n steps is then n c n r. So, n factorial n r is what. So, n plus n by 2 plus r by 2 a factorial and then n by 2 minus r by 2 right. I just solve for n r and n l from those two equations and I write it in terms of n. So, this is the probability distribution to have an end to end distance r given that the total number of monomers is n this is just a simple binomial distribution and of course, I have forgotten this 1 by 2 to the power. Now, let us make an approximation. So, I can treat this as a sort of polymer as this continuous sort of an object when the length of the polymer is much larger than these individual step lengths that you are taking right. So, for example, in the DNA each individual step is like a base pair length and remember a base pair is around one third of a nanometer 0.33 nanometers and the total length of the DNA is billions of base pairs right. So, that is my l. So, l is like the total number into the step length and this. So, let me now work in the limit that this length is going to be much larger than the end to end distance that I have. So, if I work in this limit that na much much larger than r then I can massage this equation and get it to a slightly better form. So, this is in the limit basically when the number of steps of this polymer is very large yes is r assigned quantity. So, it is in principle a vector quantity. So, ideally I should write P of r comma n and in fact, when we write 3D polymers that is what we will do. So, if because I was writing in 1D, but even there there is a plus or a minus. So, so that if you take more steps to the left you will get a minus sign for the length ok. So, if I do this sort of an approximation that is a very long polymer this contour this l is called the contour length of the polymer the contour length is much larger than this end to end distance that you have then I can go ahead and sort of few sterlings formula right. So, let me just do that. So, log of P log of P r comma n is going to be n log n n log n minus n minus n by 2 plus r by 2 a log n by 2 plus r by 2 a minus n by 2 minus r by 2 a log n by 2 minus r by 2 a and minus n log 2. So, that is the log of the probability. These n terms say anyway cancel hopefully I have missed out something right. I have missed out these things n log n minus n. So, I have missed out minus minus plus n by 2 plus r by 2 a and 1 from here plus n by 2 minus r by 2 right n log n minus. So, then this n will cancel out with this n by 2 and this n by 2 and we can think of what these logs will look like actually this r by 2 a anyway cancels out that r by 2 a. So, I do not know that. So, now, let us look at what this logs look like. So, I have terms which are like log of n by 2 plus or minus r by 2. So, f terms which are like log of n by 2 plus or minus r by 2 a right. So, let me take that n by 2 common. So, this is let us say log of n by 2 1 plus minus r by n a ok. So, this is log of n by 2 and then plus log of 1 plus or minus r by n a and remember that I am working in the limit that it is a very long polymer. So, r is much much smaller than n a. So, then I know what is the expansion of log 1 1 plus x or 1 minus x. So, I can substitute that. So, let me say what I get is log of n by 2 plus or minus r by n a minus half r by n a whole square plus higher order terms right. So, then let me substitute that here. Let me keep this so that I do not forget. So, I have an n log n over here and I have a minus n log 2. So, that is log of p of r comma n is n log n by 2 if I combine those two terms. Let me get that out into a minus n by 2 plus r by 2 a into log of that. So, into log of n by 2 plus r by n a minus half r by n a whole square that is this term and then I have another minus n by 2 minus r by 2 a log n by minus r by n a minus half. So, then what cancels now? This n by 2 log n by 2 and this one add up to cancel this first one which is an n log n by 2. So, this cancels out this cancels out what else remain anything else cancels out obviously. So, what do I get? Minus minus r by 2 a minus r by 2 a then minus minus plus plus 1 by 4 r square by n a square minus r square by 2 n a square and then here plus hopefully they will cancel out at some point. And then from here I get a minus minus plus r by 2 a then another plus 1 by 4 r square by n a square then a plus and a minus. So, minus r square by 2 n a square and a plus and a minus. So, minus 1 by 2. So, hopefully if I have done it correctly this cancels this cancels. So, what am I left with minus r square by 2 n a square right. So, this gives me roughly log of p r comma n is equal to minus r square by 2 n a square. Right. If you add up these four terms that is what you get. So, basically then in this in this continuum limit what I get is an expression for the probability to the probability to have an end to end distance r for a polymer of length n goes is e to the power of minus r square by 2 n a square. And of course, you can work out the work out what is this factor in front by just demanding normalization that the probability must be normalized. And then you will get the full probability distribution p r of n for this one-dimensional polymer. So, you will get the p of r comma n is 1 by square root 2 pi n a square e to the power of minus r square by 2. Could you have written down this distribution without doing this algebra? What theorem tells us what this distribution will look like? Hmm? Central limit theorem right. If you take independent and identically distributed random variables then in the limit that the number of steps go to infinity you should end up with a Gaussian distribution and that is what we have done. We ended up with a Gaussian distribution. My remember my variance is n a square. So, this is r square by 2 sigma square right. So, this is like the standard Gaussian distribution. So, you could have got. So, this is like consequence of the central limit theorem. You can now generalize this to this was artificial polymer in the sense that this was doing random walk in 1D. You can, but you can generalize this to polymer let us say in 3 dimensions. And again so, for a polymer so, for a random walk polymer in 3D random walk polymer let us say in 3 dimensions. Because it is a random walk polymer again my r is going to be 0 and my r square is going to be some n a square right which means that my probability distribution to have this n to n vector r given some number of steps n will again go as some e to the power of minus something times r should not write a something else times r square with an appropriate normalize with an appropriate normalization factor. So, if you substitute the normalization and you substitute the variance you can work out what this k is what this kappa is what this normalization factor is and then this what should you get. So, what is the probability distribution in 3 dimensions you get e to the power of minus 3 r square by 2 n a square over here and here you will get 3 by 2 pi n a square 2 pi n a square to the power of 3 halfs. So, it is like each component r x r y r z is doing their own random work. So, for each of them you will get minus r x square by n a square and so on and if you so rather if you just put in this variance and this normalization you should get this full probability distribution for the random walk in 3 dimensions ok. So, this is one thing. So, let me just write this is p 1 d and let me keep this p 3 d as well of the n to n vector is 3 by 2 pi n a square to the power of 3 halfs e to the power of minus 3 r square this is n 3.