 So, this is just a sort of example for this ligand receptor binding. You can use a similar approach for different problems. For example, if you think of this let me come back to this RNA polymerase process. So, this is the whole transcription and translation process we are just interested in this RNA polymerase. And again you can calculate this P bound right as a we calculated the entropy, but now if you said that when this RNA polymerase ok. So, let me just say this little clearly. So, let us now consider a similar problem for this RNA polymerase binding. So, I have this one I have this 1B DNA which has these sites right. I forget about the physical 3D structure of the DNA and I consider that this RNA polymerase can come unbind and it can sort of hop around this. Maybe this site is a specific site. So, this is the site where it wants to go unbind. So, it has some energy for going and sitting on this on this specific site let me call it specific, but it can also bind non-specifically to this sequence ok. So, it can bind weakly to this other other sequences over here and let me call the energy for that as epsilon non-specific ok. Again it is an approximation in that energy will depend on the exact sequence that you have, but let me say that all sequences as long as it is non-specific binding will have some energy epsilon NS. When it goes and sits on the site that it wants to so that it can start feeding the gene that energy is lower and that I will call as my epsilon S. And then again we will do if you do the same business that you have NP number of polymerases, you have N number of available sites and so on. You can calculate what is the probability that an RNA polymerase is going to be bound specifically to this site and that is again going to look very similar. So, that is again going to look very similar to this whatever expression we had. So, again if you calculate what is P binding. So, this I will leave because it is exactly the same sort of a calculation P bound is going to be P which is the let me call it NP by N e to the power of minus beta delta E by 1 plus NP by N e to the power of minus beta delta. So, exactly what we had, but now delta E is the difference between the specific binding energy and the non-specific binding energy. So, epsilon specific minus epsilon. And again if you go back and look at experimental data for different sort of proteins binding on this DNA. So, this is lack protein in E coli, this red curve, this is T 7 I think is a bacteriophage. So, this is another polymerase which is the blue one is a polymerase that is found in the bacteriophage T 7. So, again these are experimental measurements of the probability that this polymerase molecule is bound in two different organisms as a function of the number of RNA polymerase molecules. And again you see that the curves look exactly as would be predicted by this sort of Langmuir absorption isotherm. You can read off what would be the values of this delta E's, what is the difference in the specific versus the non-specific binding energies for these different polymerases in different organisms. So, it is true that these are very simple calculations, but even so you can often look at experimental data and get some information out about the underlying biological system using these very simple models like this. If one can of course, build more and more complicated models, but what I wanted to show at least for today was that even these simple models often can be used to interpret real experimental data. So, we will continue with this binding with a little more maybe little more biologically complicated models where you can have multiple ligands and so on and so forth different complications which we look at this is nice. So, I will just take two minutes to describe this. So, you know that inside cells hydrophobic interactions are often a large or hydrophobic forces are a large or often a very important factor which determines for example, even structure of protein folding and so on. This lipid bilayer formation where you keep the hydrophobic tails inside and the hydrophilic parts outside and so on. So, what is the origin of this hydrophobic forces you can think of them in terms of a sort of entropic picture very hand waving picture, but nonetheless. So, here is water right H 2 O the orange thing is the oxygen the white ones are the hydrogens. The oxygen forms the hydrogen bond with a hydrogen molecule of a neighboring water right. So, here is one oxygen here is the hydrogen of a neighboring water molecule it forms a hydrogen bonding between them and similarly. So, whenever you have an oxygen next to a hydrogen you can form a hydrogen bonding often you will find this sort of a tetrahedral structure tetrahedral symmetry. So, this sits at the center of the tetrahedron and these are the four vertices of the tetrahedron due to this oxygen binding sorry due to this hydrogen bonding. So, now you can think of this that let us say you have a C of such water molecules which are arranged in this sort of a tetragonal structure. So, if I think of this water molecule at the center it can form. So, this form sits at the center of the tetrahedron and it can form these oxygen molecules depending on which way they are oriented can form hydrogen bonding with waters at the other vertices of the tetrahedron right. So, these are the six possible confirmations depending on which way your oxygen molecule oxygen bonds point. So, here they are pointing along this axis there one is pointing along this one is pointing along that and so on ok. So, these are the microstates in some sense corresponding to this hydrogen bonding network or this hydrogen bonding tetrahedral ok. For this one water molecule which sits at the center of this tetrahedron ok. So, you can say that the number of microstates that are possible. So, each of these confirmations is a microstate for that water molecule right. So, I can say that the number of microstates in this case is going to be 6 ok depending on which way this water molecule is oriented is that clear. So, now let us say that you place you replace a water molecule and you place a molecule which does not like to form a hydrogen bond right. So, let us say you replace at any one vertex. So, let us say you replace at this vertex instead of a water molecule you place a polar molecule which should not like to form a hydrogen bond. Then what could happen if there is no water molecule on this vertex then this configuration is no longer allowed because it cannot form a hydrogen bond there is no water here. This configuration is not allowed that configuration is not allowed is that clear. This out the central oxygen cannot form a hydrogen bond with the water here because there is no water here it cannot form a hydrogen bond with the new molecule here these ones are still fine because here this oxygen is forming a hydrogen bond with the water which is present there that I have not replaced. So, if I replace the water molecule by a new molecule which does not form a hydrogen bond at any one of these vertices ok, then I reduce the number of available conformations from 6 to 3 right. So, in this new case you can call it omega nu I have only 3 available conformations. So, by introducing a molecule which does not like to form a hydrogen bonding I have reduced the number of available conformations from 6 to 3 again this is very simple to take it with a pinch of salt, but still the basic principle is this ok. So, if I now say that therefore, what is the change in entropy right what is this delta s that delta s is k b log 6 minus k b minus k b log 3 right which is therefore, minus k b log 2 per molecule of water that I have replaced ok. So, this is k b log 2 per molecule of water that I have replaced by this new molecule which does not like to form a hydrogen bond. Therefore, the total if I want it the total free energy cost given that I had replaced n such molecules would go roughly as n k b t log 2 ok. This n is in some sense ok. So, if you are faced with a mixture of hydrophobic and hydrophobic molecules and water what it would like to do is that it. So, this delta g is something like n k b t log 2 where n is the number of molecules which share an interface with water right. So, what it would like to do then is to minimize this number of n right if it wants to to minimize its free energy it would like to minimize the number of contact points of this molecule with water which is what this hydrophobic interaction does right. So, if I think of lipid membranes or whatever or vesicles it reduces the interaction of these hydrophobic parts with water. So, that this n this effective n which is the number of hydrophobic molecules which come into contact with water that gets reduced and therefore, that stabilizes the structure ok. So, the heart of it is this sort of you can think of it is this again of course, a very simple way, but you can think of it as this reduction in the number of available microstates when you place a hydrophobic molecule over here ok. You want to reduce it reduces the number of available confirmations and therefore, that has a change in the free energy. So, even this hydrophobic forces which are this fundamental forces you can think in terms of this changes in free energy of changes in entropy that are associated with the change levels ok. I think that is all I have yes that is all I have. So, again these are the reference chapters from Nelson and from Phillips and of course, please if you have forgotten please read up the stat mech part. We will be using a little more stat mech as we go along today was very simple ok.