 The third thing that I want to show you is to again deploy a similar approach in the case of hemoglobin. So, hemoglobin of course, you all know is a protein in your blood it is a tetrameric protein and it has 4 oxygen binding sites. So, this is one particular oxygen binding site and if once it zooms out hopefully you will see that this whole hemoglobin protein has 4 oxygen binding sites. So, this whole tetrameric structure is one hemoglobin protein. And again you can ask similar questions that what is the probability that you have a bound oxygen that this sites on this hemoglobin are bound as a function of let us say oxygen concentration. And again we will use this sort of a discrete state formalism counting. Before we do this hemoglobin we will do a sort of toy protein which is called the dimoglobin, where instead of having 4 oxygen binding sites I will assume it has 2 oxygen binding sites. So, this dimoglobin is a sort of hypothetical protein which has 2 oxygen binding sites. And the basic idea that I want to show is that how does this probability that an oxygen comes and binds with this dimoglobin or ultimately hemoglobin change depending on whether these binding events are cooperative or not. So, if I have this hemoglobin protein and I come and I an oxygen comes and binds in one of these 4 centers does it affect the probability for oxygen to come and bind to any of these other centers ok. And you could imagine that it does because whenever you have some sort of a binding you will produce some sort of a structural change that structural change might make it easier or more difficult for another ligand to come and bind right. So, in general what I want to sort of explore with this is this idea of cooperativity. When you have these multi ligand binding the ligand in this case being oxygen then what role does cooperativity play in this binding process ok. So, here is sorry. So, here is my dimoglobin protein. So, this protein has sort of 2 binding sites 1 and 2 right. So, I will again use 2 state variables sigma 1 and sigma 2 to characterize the internal states right. Sigma 1 is going to be 1 if there is oxygen in this site and 0 otherwise sigma 2 similarly will be 1 if there is an oxygen bound to this site or 0 otherwise. So, the machinery is again extremely similar ok. So, here is the idea that I have this model dimoglobin protein it has 2 oxygen binding sites and I label it using 2 state variables sigma 1 and sigma 2 right, but I need to write down the energy function. So, let me say that I write down an energy like this that if I have a single oxygen that is bound either at site 1 or at site 2 that has some energy advantage of epsilon ok. On the other hand if I have both oxygens bound then the cooperative sort of energy comes into the picture and I have this interaction term which is j j times sigma 1 sigma 2. So, this will only come into play when both sigma 1 and sigma 2 are 1 right. So, therefore, both sites are occupied by an oxygen if not then the energy is simply if you have only an oxygen here or here the energy is simply going to be epsilon ok. So, this is an example of how to introduce cooperativity in a model like this when you have multiple ligands that are binding. So, again I can write down the Boltzmann factors for each of these states I have again 4 states 2 into 2. So, here nothing is bound therefore, the energy is 0 the weight is 1 here it is e to the power of minus beta epsilon minus mu. So, ok let me just what is the grand canonical partition function by the way z to the power of into into ok. So, let me write z small z to the power of n into the canonical partition function of n and then summed over all possible values in that right. Since there is too many z let me just write it as e to the power of beta mu right e to the power of beta mu is the fugacity mu is the chemical potential right. This you are all familiar with the grand partition function. So, here is something. So, where the number of oxygen particles that will bind is variable right you can have 0 you can have 1 could have 2. So, I will use this sort of a grand partition function for malism in order to calculate the probabilities. So, when you have so, when you have just one binding then you will have an e to the power of beta mu into the picture here and here e to the power of beta mu and the canonical partition function for that is simply e to the power of minus beta epsilon. Here when you have both bind when n is capital 2 then they will have e to the power of 2 beta mu and the energies we have written down forgot energies epsilon sigma 1 plus sigma 2 plus 2 j sigma 1 sigma 2. So, the energy of this is minus beta 2 epsilon not 2 j they write 2 j or j 2 epsilon plus j right yes j. So, this is what this term is 2 epsilon plus j minus 2 mu ok. So, that is the weight of that term. If you sum up all of this that gives you the grand partition function right. So, the grand partition function is then e to the power of beta mu n then this z n t. So, in this case we have worked out what is z 0, what is z 1 and what is z 2 and therefore, you can write down what is the grand partition function right. So, this is the total grand partition function. So, now, you can ask that well therefore, what is for example, so now that I have the grand partition function I could ask what is the average occupancy of this dimo-globin molecule right. That is on an average how many oxygen molecules will be bound to this dimo-globin. How do I calculate the average number given the partition function grand partition function? Yes, that is of course true, but what is the short form formula that is in terms of the probabilities which is of course correct. But if you know the partition function then del del log q by del mu right and something k B t. So, you can calculate or if you want as Shubhanic says you can just write down it is a summation. Since this is any way a very small discrete system you can explicitly count and write down what are the probabilities. So, you can write down what is the what is the average number of oxygen molecules that are going to be bound the average occupancy and this you can write. So, this is just this as a function of this oxygen chemical potential. So, what tells you how much oxygen is there that is the chemical potential that signifies the strength of the how much oxygen is there in the environment and that will determine how much oxygen what is the average oxygen occupancy of this dimo-globin ok. All right. So, instead of using the chemical potential you can also express the chemical potential as a function of concentration I have not really shown that I will show that next class. So, we will derive how to write the chemical potential as a function of the concentration and that is simply written as mu is some mu naught k B t log c by c naught. So, if you do not know this you can take this as a given I will derive this next class I will show how to get this. So, the chemical put in the larger the concentration the larger the chemical potential. So, now you can therefore, so you can if you rewrite this previous formula in terms of the concentration you will get some answer. So, here is the average occupancy in terms of the oxygen concentration ok and you will see that what would you so for example, what would you expect if there was no cooperativity if j was equal to 0 you would expect to recover back the ligand receptor formula that we got last class right where we had a delta E basically difference between the two states a single ligand binding to a single receptor right. So, you would expect to recover twice that rather to be precise because you now have effectively what this would if it there was no cooperativity what it would function was that you could have two ligands coming and binding here one here and one here and these would be independent of each other. Therefore, if I consider this whole thing as a single protein the average occupancy would simply be twice that. But because you have this cooperativity because you have this interaction energy j that changes the sort of average occupancy. So, this is just of course, two times whatever you had earlier. So, if you remember this was the formula we got earlier C y C naught e to the power of minus beta. So, this in the absence of j you just get two times that, but in the presence of j it is in the presence of. So, for example, this is the blue term is the curve in the absence of j there is no interaction it just behaves as two sort of independent receptors interacting with two ligands. As you introduce more and more cooperativity the curve shifts more and more to the left right which means that you will get more oxygen occupancy at a lower concentration of oxygen if you have cooperative binding. So, whenever you have one if that if that makes it more likely that you recruit another oxygen to the second side then the average occupancy will increase at a lower value of this oxygen concentration or oxygen partial pressure. If you plot these probabilities. So, this is the probability for example, that there is no oxygen bound which falls as you increase the oxygen concentration. The probability that there is only one oxygen bound sort of peaks somewhere in the middle and then again falls and for higher value of the pressure it is very likely that you will have both sides that are bound by the where oxygen is bound to both sides of this time. So, there is a sort of quantification of how cooperativity affects this binding curves this occupancy probabilities function of the concentration is a sort of shape. So, you can now. So, this is we did this for this you can also do a model in a similar spirit which is a very famous model, but instead of introducing this interaction energy J. Remember our idea was that I have this protein I have this protein which let us say has two subunits if a ligand comes and binds to one of these subunits it may be causes some conformational change which makes it more likely to recruit a second ligand here which I am representing by this interaction energy J. Alternatively you could also say that there are this protein this dimeric protein could exist in one of two possible conformational states right. One is this one is this circle circle state and maybe another is something like this. And you can reframe this problem in terms of this conformational states and that is actually a classic model in the literature which is called the MWC model or the Monarch Wyman Shangu model. So, again the spirit is the same the sort of language that we use is slightly different. So, what you say is that the protein can exist in two states the tenth state or the relaxed state and again I have a state variable associated with that sigma m is 0 if it is in the tenth state it is 1 if it is in the relaxed state. And let us say in the absence of ligands this r state is energetically unfavorable. So, the tenth state is lower in energy. So, this is my r state this is my t state and it has some energy cost of epsilon. On the other hand the ligand might prefer to oops the ligand might prefer to bind to this relaxed state. So, the ligand has a higher affinity to bind in this r state in this relaxed state. So, you can call this epsilon t as some binding energy when protein is in the tenth state epsilon r is the binding energy when the protein is in the r state. And because this ligand has a higher affinity this epsilon r is smaller than x identity it is more stable when it binds in the relaxed state. So, I am sort of reframing the problem using this sort of tenth state this conformational state idea. And again you can write down this sort of an energy. So, you have two sort of sites one and two each of them has a sigma i. In addition you have this conformational state variable sigma m which can be either 0 or 1 depending on whether it is this tenth state or the relaxed state. So, for example, in the tenth state this thing comes into the picture. So, you have some epsilon t depending on how many ligands are bound if both are bound then you have two epsilon t. If it is in the relaxed state then these terms come into the picture there is a conformational energy of epsilon plus depending on how many ligands are bound 1 epsilon r or 2 epsilon r right. So, this is then my Hamiltonian pole system. And again you can sort of list out what are the various states that are possible and what are the corresponding weights for these states. So, for example, if this is these are my tenth states these are my relaxed states I can calculate what is the energies of each piece right. So, this is one of course, this is e tenths minus again there is a e to the power of beta mu this is the same as that and here it is 2 times epsilon t and 2 beta mu right. In these you have a structural energy for the relaxed state which is in addition to this ligand binding which is epsilon. So, here even when nothing is bound it is unfavorable factor of e to the power of minus beta epsilon and then depending on how many are bound you will have whatever epsilon r plus epsilon minus mu and here you will have 2 epsilon r plus epsilon minus 2. So, once you have listed out all of this you can simply write down the partition function and you can calculate again what is the average occupancy. And then depending on how favorable you make your sort of tenth state in comparison to the relaxed state you will get different curves for this occupancy profile as a function of the concentration of the ligand. So, this is written in terms of some energies of the tenth state and the relaxed state and this difference in energies between the tenths and the relaxed concentrations. So, these are sort of two sort of ways you can think about this from an interaction energy perspective that these ligands sort of interact where it and it has some sort of cooperative with the energy J or you can think of this from this confirmation state perspective that when a ligand comes and binds it is more likely to make it go transition into this different state the relaxed state in the same WC sort of model. So, now that we have done this we can now go forget about this dimo globin and go on to the actual hemoglobin it is exactly the same except that you now have four possible ligands right instead of having two states you now have four states and therefore, the terms get more complicated, but the spirit is exactly the same. So, just to start off with I can do the non-interacting model where I say that well I do not so, I will talk in terms of the interaction energy. So, I will talk in terms of this J. So, the first model that you can so, you can build in complexity layer by layer. So, the zeroth order is when I say that each of these four sites on the hemoglobin they are non-interacting when oxygen comes and binds here it does not affect the energy of each of these any of these others. And then you can simply write down the weights there are four sites. So, therefore, there are four variables sigma 1 sigma 2 sigma 3 sigma 4 each of them can be 1 or 0 depending on whether there is an oxygen bound or not and you can write down these states. So, this there are four possible ways to get one oxygen bound right on this four sites. So, you have a factor of 4 to have two oxygen bounds you have six possibilities 4 C 2 to have three bound you have again four possibilities 4 C 3 and to have all four bound there is a single possibility ok. So, you can write down again the partition function. Once you write down the partition function as expected because this is a non-interacting model is just the single partition single particle single ligand partition function raised to the power of 4 right because these are not interacting. And then you can calculate what is the average occupancy and that is just 4 times the single ligand occupancy. We will compare these results to what the actual binding data for hemoglobin shows. So, this is the single ligand occupancy which is sorry this is the non-interacting model where the answer is simply 4 times the single ligand model. You can now start to put in cooperativity. So, for example, the first level is what is called the Pauling model for hemoglobin where you say that you have an interaction energy J when you have more than one ligands that are bound right. So, for example, in here there J does not kick into the picture here there are three bonds sorry here there is one bond. So, there is one J here for example, there are three bonds between this and this there is an interaction energy between this and this between this and this. So, there are three J terms and here there are how many six J terms right. So, I write down my Hamiltonian in this form that I have epsilon sum over i sigma i plus J by 2 sigma alpha sigma gamma where this alpha gamma run over all pairs right of these sides and J by 2 simply because I am double counting I am counting 1 and 2 and 2 and 1 together. And it is a restricted sum in the sense that I cannot have alpha and gamma is the same ok. So, this prime implies it is a restricted sum where alpha cannot be equal to gamma. So, this is then my Hamiltonian and then I can do whatever I can write down the weights right. So, the again there are four possibilities 6 for this 4 for this 1 for this this has 1 beta J terms this is 3 beta J terms this is 6 beta J terms depending on how many bonds you form with this interaction. And again this is therefore, my whole partition function and again I can calculate what is going to be the average occupancy right what is average n. Again you take you can take a derivative with respect to mu of this log of this partition function that will give you the average occupancy. So, you get some complicated term it depends on what is your mu and what is your depends on what is your epsilon and what is your J, but it is definitely not what we had earlier which was just 4 times the single ligand occupancy. It has changed because you have introduced this sort of a cooperative binding in the model that one ligand sort of makes it more favorable to recruit other ligands. So, this is one level of complexity you can keep on building more complexity for example, in the next level you have what are called what is called the Adair model. And there instead of just having pair wise interactions you can have interactions between triplets or between quadruplets as well ok. So, let me just write down the Hamiltonian. So, this and this are the same as I had in the Pauling model right between these are between pairs. Then I have some interaction energy between triplets if you have three things that are there then I have an additional favorable term which is of the strength k. And if I have all four that are bound I have an additional even more favorable and that interaction term is L for example. And again you can write down what are the weights of these states and you can calculate what is the what is the grand partition function and again that will be a function of now not only J, but J, k and L. So, this is more parameters as opposed to this Pauling model, but it may or may not be better suited to fit the actual binding data for democracy. And again you can calculate what is the average occupancy. So, now that I have done this I just want to show some data for this homoglobin and see that which of these actually fits the data whether it is a non-cooperative model or the Pauling model of that air model ok. So, here for example, this dots are this experimental data for the average occupancy as a function of this oxygen concentration on this side. This blue line is the non-cooperative model. So, it is just four times that of this single ligand which as you can see does not fit the data very well. On the other hand both the Pauling and the air models fit the data to a very high degree of accuracy right. So, what this says is that these even if these three vertex or these four vertex terms are present they are not really playing a very important role. It is enough to consider that you have pair wise interactions between the ligands in order to explain the experimental binding curve for homoglobin ok. So, just this Pauling model is actually good enough to explain the binding curve of homoglobin, but definitely you need some level of cooperativity. If you do not have any cooperative binding you cannot reproduce this behavior. Different ligands and receptors will be fitted by different models. There might be ligands where a non-cooperative model might be the correct model to use. There might be ligands where this Pauling model does not work well, but that air model on the other hand will be the correct model to use. So, just because this works for the Pauling works for homoglobin and the non-cooperative does not is no guarantee that you know in general the non-cooperative model is a wrong model. There might be ligands, receptors and ligands where data is actually better fitted by the non-cooperative model. And you can sort of also think about what the effect of this cooperativity is by looking at these probabilities of finding these difference probability that there is no oxygen versus the probability that there is one oxygen versus two and so on. Probability that there is four oxygen. How do these change in the presence and in the absence of this cooperative binding? So, this is that curve. This is just this formula in the I think this is formula in the Pauling model for certain choice of this J and so on. So, if you do not have if you do not have cooperativity then this is your P 0 ok. It falls, but once it falls first P 1 rises you will have some region where it is likely to have only one oxygen bound. Some region by region I mean some concentration of oxygen where it is more likely to have two oxygens bound, some where it is more likely to have three oxygens bound and then you get to this four. Whereas, once you introduce cooperativity it is more like either you have nothing bound or you have all four that are bound. The other ones this 0, 1 and sorry this 1, 2 and 3 they are suppressed to a much larger extent in relation to this non-cooperative model. The moment. So, what it is saying is that the moment is sort of recruit one because it makes it more and more likely that you will recruit more. You very quickly switch from this no oxygen bound to this four all four oxygen bound without these terms ever becoming a dominant factor as opposed to this non-cooperative model where each of this has a regime where they will play a dominant role ok. So, again the statistical mechanics is fairly simple is just a matter of writing down what are the various states and you know the states are reasonably discrete you know 2, 4, 8 whatever you can actually it is countably few. So, you can write down the corresponding states and weights for each of these states and therefore, get what are these sort of occupancy probabilities for different ligands, different proteins, different modifications, science channels and so on and so forth. So, you can take this whole machinery of this calculating the partition function and from there calculating probabilities and apply it across a variety of protein systems ligand receptors or in systems ok. So, I think I will stop here today.