 So, these are single molecule experiments here for example, is a protein that I have fixed one end to a tethering surface and then using an AFM tip you pull the other end of the protein right and you can plot a similar curve like this that as a function of how much force you apply what is the extension of the polymer right or you can use something like an optical trap to pull again to again to apply exert forces. What you get is something that looks like this at least in the initial part, but then of course, it deviates because this one we argued only for very small extensions. So, if you look at DNA for example, here is a DNA let us say you pull it using one of these AFM tips or an optical trap or some other set up what you see is a force versus extension curve like this. Initially for very small extensions it sort of grows linearly, but then as we argued that after all you cannot stretch the polymer indefinitely this extension has to be less than equal to the total contour length of the polymer. So, therefore, as you pull it more and more as you apply more and more force the curve will become more and more nonlinear and you cannot go beyond this one on this slide right because that is the maximum possible extension of this maximum possible extension of this curve. Of course, as you keep pulling at some point the polymer will break, but let us say we are not getting into that region, but this is how what the curve will look like. So, force versus extension will look something like this right this being the maximum possible extension. What this does is the and if you do it for so, this response of this polymer depends on the particular polymer that you are pulling. For example, if you use two different DNA sequences and you pull them you will get a slightly different force characteristics ok. So, this force extension curve is sort of what is like a fingerprint of the polymer that you are pulling. So, it is a characteristic of the particular polymer that you are pulling and therefore, it acts like a some sort of fingerprint ok. If you pull a different sequence of DNA you would get a slightly different curve and so on. Now, this part at least we understand that this part is the linear part which we argued last class, but we can try and see if we can derive this force extension curve in the full limit without using this small extension approximation ok. So, that is what I will try to do first and we will see whether we recover a plot like this ok. So, let us see. So, first I will do it for a one dimensional chain and again then I will generalize to a 3D polymer. So, first I will do it for a 1D polymer. Remember that the free energy is like E minus j dot x. I am using the Gibbs free energy in this case because I will be working in ensembles where this j remember is what is a generalized force x is a generalized displacement right. So, like pressure and volume and so on. So, in this case the pair that I am interested in is the force versus the extension. I have written x for the extension. So, stick to that the force and the extension. So, we apply a constant force and the extension sort of responds to this force that I am applying. Of course, you can see that as I pull as I pull this polymer more and more, my entropy sort of decreases because the polymer has very little a configurational entropy that is available to it. If I were to pull it completely, it has nothing to do except being a straight line. If it was if I did not pull it at all it has a lot of entropy among the number of available microstates and you will. So, let me get rid of j and x. So, let me just write F and L. Those are the appropriate things in this case and the force as you again know is given by minus del G del L right. Just like pressure would have been del G del V exactly like that the force is minus del G del L. Remember pressure has an ad hoc extra negative sign because of how the original convention was defined. Every other generalized force and generalized displacement will come with the negatives. So, that is what we will try to do. So, let me formalize. So, let us say I have a polymer, a one-dimensional polymer with the total contour length of n times a right and it is the number of segments that I have is my coon length of the length the size of a monomer and so therefore, that is my L total. This is a 1D polymer. So, I just have two possible options. I can go take a step to the right or to the left and those are given by nr and nl and therefore, the length of the polymer the end to end distance. So, the end to end distance is nothing, but nr minus nl times a how much excess of one over the other that you have. And remember I will what I will be doing is this sort of random works I have a polymer. So, it is an ideal polymer which means that there are no interactions between the monomer segments which means that I do not really have any. In reality of course, there is some bond energy and so on, but at least for this ideal polymer I assume that there are no interactions between the different bonds. So, then I can write down my Gibbs free energy. My Gibbs free energy is minus f times L and minus Ts. So, minus kBT log of the number of microstates the microstates for this given extension L given that the total possible extension the maximum possible extension is L total ok. Now, I know what this omega is right we have done this before. So, my omega is simply how to distribute these n monomers such that nr in are on the right and nl are on the left. So, that is my n factorial by nr factorial nl factorial. So, we have basically everything I have my L in terms of nr and nl I have my omega in terms of my nr and nl. So, I can just put everything over here and calculate what is therefore, the Gibbs free energy. So, let me just write that down. So, the Gibbs free energy is going to be minus f times L. So, nr minus nl a and then from this part we will get minus kBT n log n minus n minus nr log nr plus nr minus nl log nl plus nl. Now, what I need to do to find out the forces basically to minimum to find out this derivative of this free energy with respect to the length or equivalently what I will do is I will take a derivative with respect to nr because these are basically the same things. So, I want to minimize my free energy. So, I want to minimize this free energy this free energy with respect to my length of the polymer L. So, let me just do del G by del nr and that is minus 2. So, this I will just write down you just check that the mathematics is correct it is a simple derivative plus kBT log of nr plus kBT nr by nl minus kBT log n minus nr minus kBT n minus nr by n minus nl till I thank the shares. And if I now set this equal to 0 what I will get is a relation. So, what I will get if I solve for nr is that nr by nl is equal. So, this is basically nothing but nr by n minus nr nl is nothing but n minus nr if you do the maths correctly this should come out to be 2 F A by kBT which means that the extension or let me derive let me define the relative extension rather because that was what it was plotted over there. The relative extension is let me say z is nothing but l by l total. So, this can vary between 0 and 1 and this is nothing but l is nr minus nl l is nr minus nl and l total is nr plus nl the a's cancel each other. And then if I substitute the solution I got for nr and nl what I will get over here is e to the power of 2 F A by kBT minus 1 by e to the power of 2 F A by kBT plus 1 ok. And this is a simple function what function is this there is nothing but the tan hyperbolic right. So, this is nothing but tan hyperbolic F A by kBT what is the tan hyper what is tan hyperbolic x for small x anyone remembers for small x tan hyperbolic x is just x which means that for small forces you will just get that the relative extension z is simply nothing but F A by kBT right which again tells you that the extension is proportional to the force. So, we recover back the result that we got in the last or last to last class that if I were to write l then that is nothing but l max or l total whatever I am writing l total times A by kBT into F right. And then that gave gave me my spring constant is kBT by l total into A which is what we had last class. So, that gave me my spring constant of the polymer was kBT by l total. So, this so that is a consistency check that whatever we have done in the small x limit when the extension is small or the forces are small it should come out like a hook in spring and that it has. But quite generally you can write down the extension if the forces become large then this is no longer a linear relationship it is now given by the stan hyperbolic ok. And if you plot that if you plot this z if you plot this relative extension if you plot this relative extension as a function of this force again you will get this sort of linears saturating to 1. So, this is now the relative extension so it will saturate this is just this curve with the axis split. So, this is precisely the sort of behavior that you see in these experiments for these DNA DNA double strand pulling. So, this is precisely the sort of behavior that you see in these experiments and if you for example, if you know the force and you know the temperature at which you are doing this experiment this tells you precisely what is the Kuhn line for this particular polymer in order for in order to model this polymer as a freely joint chain as a two dimensional random yes. So, for example, in that AFM setup it is one end was fixed and the other end was being pulled with the particular force right. So, for example, if I look at this setup over here this end was fixed and I was pulling on that end. You could do different things you could also pull both ends and so on depending on what experiment you do depending on the setup you might need to alter the calculation a bit, but for a setup like this this would be an appropriate calculation to do. Actually so for example, one thing is I did this for a 1D polymer you can actually do this for a three dimensional polymer do exactly this similar sort of a thing because in reality the DNA that we are pulling is a 3D polymer. So, let us say if I do a 3D polymer and I should get something very similar of course, I will get a different expression then you can see what it is ok. So, let us say I have a polymer again this is of n links. So, the L total is n times a and so on. And again I work for this ideal of polymer chain where there are no interactions between the segments and these there are no correlations between the segments which means that each monomer sort of does its own thing independent of the other right, independent of any others. So, if I want to write down the partition function Zn for independent links like a freely jointed chain what I will have is that this is nothing, but the single particle partition function raised to the power of n right. This is the non-interacting case exactly like it have done in ideal gas. I can calculate this Z1, I can calculate this Z1 this is nothing, but integral let us say e to the power of minus beta h plus beta j dot x. So, f L so on again this is ideal. So, I do not know why I wrote the h there is no h in this case. So, this is nothing, but e to the power of beta f times a cos theta. I should not write L here let me write a because I am working for the single part single link partition function. And this I would then need to integrate over all possible confirmations. So, d phi 0 to 2 pi d theta 0 to pi. If I do this integration what I will get is a 4 pi k Bt by f A sin hyperbolic f A by k Bt. So, that is my single particle partition function Zn is just the Z1 to the power of n and the Gibbs free energy G is nothing, but minus k Bt log of Zn. So, log of minus n k Bt log of Z1 right. And this average L the extension is nothing, but minus del G by del f. So, this is nothing, but n k Bt n k Bt log n k Bt del of log Z1 by and again you can take you can do this differentiation I am not doing it explicitly. I will just write down the results. So, if you do this what you will get is this extension is nothing, but n times a cot hyperbolic f A by k Bt minus k Bt by f A. So, the relative extension Z is nothing, but this term inside the brackets cot hyperbolic of f A by k Bt minus k Bt by f A. You can again check that for consistency in that if f A is very if this argument is very small it should give me back a hooky in spring. So, you can expand cot hyperbolic x for small x and for small x this is cot hyperbolic x is 1 by x plus x by 3. So, if you do this the 1 by x will exactly cancel this term. So, the leading order term will be x by 3. So, in the small x limit what I will get is that this relative extension let me just write is nothing, but x by 3. So, f A by 3 k Bt which means that the spring constant of this 3-dimensional polymer spring constant of this 3-dimensional polymer is nothing, but 3 k Bt by L total times a which is again what we got last class. But now we have the full force extension relationship for a 3-dimensional polymer this is how it would look. And again if you plot this relative extension L by N a as a function of the force you will get a curve which looks exactly like this like you seen in these experiments.