 So, in the same spirit I can do this other thing which is this two spheres now and again so ok let me just sketch the calculation is this clear ok. So, let me do this case of when I have these two large spheres. So, I have these two large spheres with their own excluded zones. And if I have if these approach close to one another there will be a common region the overlap volume which I need to calculate. So, let us say that these spheres are of radius r or both of these the small particles are of radius small r and let us say that the center to center distance between these two spheres let me write that as capital D. So, in what range of capital D will there be a force? So, there will be a force if D is of course, has to be larger than 2 r when the two spheres are touching and smaller than 2 r plus. And what will be the excluded volume? The volume that is excluded by these two large spheres is simply the fourth third y the excluded volume of each of these spheres r plus r whole cube there are two such spheres. So, 2 into 4 third pi r plus r whole cube minus whatever is the overlap volume minus whatever is the overlap volume. If the spheres are far apart then there is an overlap volume and this is simply whatever is excluded if they are closed then you will have some non-zero value overlap. And the overlap volume is what the overlap. So, if I draw it like this the overlap volume is simply two times the volume of the spherical cap right and you can use the standard things for the spherical cap. So, I will not derive the spherical cap volume that is a standard geometric k. So, you can calculate with this overlap volume goes as 2 pi by 3 r plus r minus d by 2 whole square into 2 r plus 2 times small r plus d by 2. So, this if you just look up or if you derive the volume of the spherical cap and it is two times that volume you should be able to do that. Then remember that what I want to calculate is this force due to depletion which I had written as minus del G excluded volume del D in this case the separation of the spheres and G was just V excluded by V box. So, this was minus n k B T by V of the box whatever box that you have del V excluded by del D and the V excluded is just minus of V overlap. So, this would just become V overlap and then if you just put in this del del D of this overlap volume. So, there is a D over here what you will get is this expression if you simplify a little bit and again n is small n is nothing but this n by V box is the density of these crowders again. And again you will see that there is an attractive interaction simply because of entropy smaller crowders want to maximize their space and that is going to lead to an attractive interaction of this form. See another way of saying sum up is that if I have many many crowders the increase in entropy because of the crowders is of order n whereas, the decrease in entropy because of this bigger particles is of order 1 right whatever it is it is of order 1. Given that you have a thermodynamically large number of crowders the system will always try to go in the direction that maximizes the entropy of the crowders themselves ok which is why even if I did not take into account explicitly the large particle entropy over there in that original calculation this conclusion still holds true. If this if the number of this large particles becomes comparable to the number of these crowders then of course, this calculation will no longer hold. But that is an implicit assumption that you have simply because of volume restrictions you have a few of these large particles and many many many of these smaller crowders. So, you can put in some numbers actually typical numbers of protein sizes and so on just to get an estimate of what ranges this forces line. So, if I put let us say that this small bigger particles are a micron in size and these smaller particles are of some nanometers and I have some concentration of this smaller particles which is again let us say millimolars. If you put it together in this sort of a formula what this will give you is that this depletion force if I put in exactly this numbers it comes out something like 15 peak on units which is an appreciable force in the cellular context. So, for realistic densities and realistic sort of sizes you get an appreciable amount of depletion force on these larger particles. So, it is not so small that it will not have any effect. In fact, it is large enough that in many cases one definitely must take this sort of depletion forces into account. So, what what sort of cases would one see a depletion force like this? So, here are some ok. So, firstly before I do that here are experiments experimental results. So, this is a setup where I have two large particles which have been constrained to move only along one axis through the use of optical traps. And you vary the concentration of these smaller crowders. So, this is 5 microgram per millimol 10 microgram 25 microgram. So, these are three experiments and you can calculate the free energy. How do you calculate the free energy? You do this experiment many times you. So, basically the only variable I have here is this D the separation between two spheres because I have constrained them to move along a line. You can calculate the probability distribution of how at what separation you see these large particles P of D and that is going to be something like e to the power of minus beta g. So, you observe this probability distribution experimentally. From there you can infer what is going to be this free energy and you can then plot this free energy as a function of the separation of the spheres. And this red line if I am not mistaken is the result of this calculation. The simple sort of calculation two spheres the depletion force between two spheres. The spheres were 1.25 this large spheres were of radius 1.25 microns and you see that this sort of decreases until you reach this 1.25 microns. Beyond that it will shoot up because you cannot interpret the two spheres. And you cannot see over here, but maybe if you look at this the value of the minima the height of the minima in that sense that sort of increases as you increase your prouder concentration which is basically this n by v box. So, that is also consistent with what you see over here ok. So, you can directly do experiments and sort of verify this sort of very simple calculation, but nevertheless it captures most of the essential business ok. So, why would it be useful? So, this is actually a very nice review paper by Martin Duzzo et al in 2006 if I am not mistaken. It argues perhaps a little ambitiously that a lot of cellular organization is actually driven by this depletion attraction, but what is definitely true is regardless of how important it is. It is definitely true that in many cases you need to take into account. It may not be the driving force, but it will definitely be an important component in order to understand these structures. So, if you have small if you have spheres they are going to be driven together in the presence of crowders, if you have long stiff macromolecules they are going to be oriented because of crowders. In fact, if you have a somewhat flexible polymer you can show that you can drive a sort of helix formation again simply because of the presence of crowders themselves and that is actually a follow up paper by Randy Cammion. So, you simply argue that you have a cylinder which has some radius t in the presence of many many small crowders of sizes r ok. Now, this has some overlap volume of course, if you keep it in the straight conformation. If you drop it if you configure it into a helix because of this overlap volumes the available entropy to this smaller particles is going to increase and therefore, there is going to be some force driving it towards this helical state ok. And if if you do this calculation you can show that the pitch and the radius of the helix that you form will depend on the crowder radius. So, this is a non dimensionalized crowder radius r by this t the radius of the cylinder and you can show that for very small crowders for tiny crowders you approach a pitch to radius ratio of around 2.5 roughly approximately. And if you look at alpha helices of proteins the pitch to radius ratio is around 2 ok. It is not evidence of anything it is not saying that alpha helices in proteins are driven by this sort of a crowding, but it is simply saying this could be one of the factors that drive alpha helix formation or in general any helix formation helices are extremely common in biology we saw those helical tails of bacterium and so on which was a protein. DNA of course, is a helix proteins of these alpha helices. So, it turns out simply by this force due to this depletion interaction you can drive a rod like object into forming a helix. They even go further and say that well it is not that this rod I can fold it any way I like this rod might have some bending rigidity I might have to pay some energy cost if I wanted to bend this rod. And we have seen in polymers that this bending rigidity is characterized by this persistence length right the length scale over which this tangent tangent correlations will decay. And so, you can plot which of these structures will be the free energy minimum depending on this ratio of this persistence length versus the crowded radius. And in some cases you will the helix structure will be favored in some cases this tube structure will be favored. If your persistence in length is very large your rod is very stiff it costs a lot of energy to bend it. Then of course, you would like to be in this stretch tube because this is inverse LP. On the other hand if it is very floppy if you can bend it very easily you are more likely to go and fall into this helix combination. So, this is also an interesting calculation to look at actually how you can calculate the pitch and radius as a function of this crowded size. They go even further and they say that well this can also be behind again these are simply conjectures, but it is interesting conjectures nonetheless. For example, we saw that when chromosomes organize one of the motives one of the common motives for this formation of these large loops right. And they argue that when these genes need to become active and so on. So, if I have a stretch of DNA if I have some stretch of DNA something like this you might have promoter complexes. So, large protein complexes coming and sitting upstream of the gene start site and similarly another promoter complex coming and sitting here. And then if these are sufficiently large you would have some sort of depletion interaction between these two which would drive the formation of a loop. So, which would bring this bring this coil close together and that could initiate transcription or repress transcription. And for certain other scenarios as well in the context of chromatons such as heterochromatin and so on I am not going into the details. But the idea is that if you calculate the magnitude of these depletion interactions they are large enough that they might be able to drive some of the features that we discussed during this chromosome packing lecture. There is another paper for example. So, these are disparate papers looking at disparate biological systems all arguing for a common conclusion that this crowding forces need to be taken into account in a diverse in various diverse scenarios. So, this is the paper that looks at compartments inside the nucleus. So, you have your cell and so I have my cell and inside the cell I have my nucleus and inside the nucleus will often find large aggregates for example, some are called nucleoli or nucleolus if it is singular then there are other objects which are called PML bodies and so on. So, this is a set of experiments where they actually take this nucleus. So, this is these are nuclei inside this nuclei you will see these compartments which are the nucleoli and the PML bodies in this case. What they do is that by lowering the salt concentration they increase the volume of the nucleus and then by introducing various crowders they decrease again the volume. So, when they increase the volume you change this N by V box basically. Once you change this N by V box what they see is that you disrupt the formation of these aggregates inside. So, these aggregate this greens are the nucleoli and these reds are these PML bodies in this case. So, when you have increased the volume of the nucleoli of the nucleus sorry not the nucleolus. So, when you increase the volume of the nucleus you change this concentration of crowders and that is sufficient to actually disrupt the formation of the that is sufficient to disrupt these macromolecular assemblies the nucleolus and the PML bodies in this you will see that these green spots have disappeared in this the red spots have disappeared. If you bring the nucleus back to its original size these again form back. So, what they are arguing is that these macromolecular assemblies that we see inside the nucleus a major driving force behind their organization is simply this crowding effects due to various small particles that are present inside the nucleus itself. So, it is this depletion interaction that causes the formation of these bodies. This is again a different paper this is somewhat in vitro. So, they use mixtures of rods and spheres the rods in this case being actually filamentous viruses and they show that this depletion interaction can lead to very nice micro phase separations where you have these rod like viruses interspersed with these spheres and then again viruses a lamellar structure self-organized they have done nothing to organize it this way it is a self-organized structure arising purely out again out of this depletion interactions. So, there is a variety of proof experimental proof that shows that this depletion interaction is actually something that is the magnitude of the depletion interactions tens of piconutens is actually large enough that you need to take into account when you are talking about self-assembly or macromolecular assembly in the cellular context. It is something that is not often done, but it is something that is important enough that needs to be taken into account ok. So, you might ask that well given that you know the cell is very crowded you have millions and millions of objects why do not I see everything just clumping up together. If I have argued that these are large enough forces what prevents aggregation. So, what prevents aggregation? One thing is that of course, we have done a very ideal calculation I have not when I wrote down the free energy I simply wrote down this minus Ts part minus Ts part in principle there is of course, an e part as well most importantly electrostatic interactions right often these proteins are charged and so on. So, you will have electrostatic interact. So, you will have this energetic part in the free energy and that will change the equilibrium from what we have argued. Secondly, this there are dynamic structures so, cytoskeletons and vesicles. So, it is not the level of crowding is not uniform throughout the cycle of the cell you can have aggregates that are disrupted simply because the structures the underlying structures are dynamic. Then there are a lot of active processes the cell actively spends energy in trying to prevent aggregation. Further there is constrained motion due to the cytoskeleton and so on. So, for example, if you think about this very dense actin assemblies that we saw in the first few slides maybe there are large proteins which would want to aggregate, but they are prevented because of this filamentous actin that is present in the background. So, there are a variety of reasons why not everything is clumped together, but that does not mean that you should neglect these forces that arise due to the issue. Let me stop you can also show I do not have time for that. So, I will just give it as an assignment maybe you can also show that. So, this case was a case where I said that I have these large spheres and if I put them in a background of these small small crowders then they are going to drive assembly they are going to feel an attractive force towards each other ok. So, this is an attraction this is an attraction attractive interaction due to crowding, but if you do the opposite limit somewhat related to what Samarthi were asking that if I have now many many of let us forget about the smallest spheres themselves. If I have many many of these large objects right which have their own excluded volumes then this is going to lead to a this is going to lead to an effective repulsion again. So, again due to crowding, but due to self crowding in some sense. So, if I have many many of these large objects then effectively because they cannot occupy the same volume if you do this similar sort of calculation you will see that there is a force, but that force is now positive which means it is a repulsive force. And so, if you do this calculation and maybe if you do this if you link these large objects then what you have got is now self avoiding polymer right and you can calculate this entropy this free energy of this entropy due to this self avoiding beads and hence calculate. So, and hence calculate how this r square is going to go as a function of the number of these crowders now the crowders then being the monomers themselves in this case. And you can show that for self avoiding beads it is not half, but it is whatever 3 by the dimensionality plus 2. So, I might do it on Tuesday or I can give it in the sign ok. So, let me stop here for the time being.