 So, mostly when we have been doing these models we have been although in the initial part when we did numbers and scales we said that the cell is a very crowded environment and we calculated some of the numbers and numbers of proteins and so on. We have mostly been disregarding that while doing the actual calculations right. So, what I will try to do today is at least in some cases try to show that what are what would be the effect of if we actually took into account that the cell lives in this extremely crowded environment what sort of things would change. For example, when we have been doing diffusion through random walks we said that you know molecules take a step whatever and they trace out a trajectory like bacteria or when we were talking about macromolecules like polymers and so on. Again, we said that sometimes we can talk about macromolecules is freely jointed polymers where they can take steps anywhere they want, but in reality often these sites might be blocked by things that are occupied by other proteins or other organelles within the cell and that will have some effect on the statistical properties that we have been calculated. So, that is what we will try to see. So, where I start just to remind ourselves again this so this is again some slides from the first lecture or second lecture. So, inside of the cell for example, the E. coli looks something like this it is extremely crowded the white things were the DNA, the black things were everything else. There is not a lot of free space inside the cell right and in fact, we had calculated numbers basically so for example, we said that the number of proteins sort of estimate of the order of magnitude estimate of the number of proteins gave us some 3 million proteins or so of which there was some 20,000 ribosomes and these take up a large volume of the cell right. Simply one type of protein the ribosome took up about 10 percent of the cellular volume and this was just proteins. So, if you looked at other things smaller molecules like water there was some 10 to the power of 10 molecules of water, there was some 60 million molecule small ions that were floating around in the cell and whenever an object is doing a random walk or diffusion or whatever it is going to come in contact with these obstacles. In fact, if you we had also calculated typically given some protein of the order of nanometers what would be the inter protein separation and that turned out to be of the order of the protein size itself right. So, these were extremely extremely dense objects. So, that is what we will try to see that in this sort of a dense background where you have millions of molecules or billions in the case of water, what sort of an effect will that have on physical properties on dynamics and so on. So, this was that slide we ended up with which listed all the numbers for all these other things. Even the membrane in particular is also a very crowded thing. So, proteins which were diffusing on the membrane are also going to see a lot of other proteins in its part. So, these were sort of the two assumptions that we have at the back of our mind when we have been doing. One is the systems are ideal in the sense that they are sufficiently dilute that we can treat it by a non-interacting approximation. That this molecule does not see any when it is doing a random walk, it does not see any of the other molecules. Other molecules are so far away that on an average it does not really see that which is why we treat it using a non-interacting approximation. And secondly, that the environment is sort of homogeneous everywhere looks roughly the same. And both of these are not true. I will try to show you that the inside of the cell is definitely not homogeneous. The molecules which are traveling or doing random walks inside the cell definitely meet other molecules as they do this. So, just to give you a sense of the density this is some pictures of the site of skeleton which means actins, microtubules, intermediate filaments in different cells. So, for example, if I can read the this is an epithelial cell which has these bundles of actin. And then there is a filamentous mesh of actin inside the membrane. This one if I remember correct is I think the axon somewhere know be this one is the axon of a neuronal cell. And again this has this long filamentous assemblies along which your motors like kinesins and dyneins they travel along these filamentous assemblies they transport cargo from one end to other to another. Then C is I think some collagen fibers and that. So, these are aligned collagen bundles and so on. So, these are extremely dense objects they are definitely not homogeneous. You have these bundler bundle like structures that you often see inside of cells. And so, if you are looking for transport in this sort of a medium you need to take into account the background against this against which this sort of a transport is happening. And these span a range of scales from hundreds of nanometers up until the microns depending on what sort of cells and what sort of structures we are looking for. This is another picture of this site of skeleton and again this is just inside the cell membrane you have this very very dense bunch of actin filaments that are criss-crossing and everywhere so they are spread around everywhere. And it is against this backdrop that things are performing diffusion or motion ok. So, remember that so, let us look at let us start off with a simple case let us look at diffusion. And remember how we had quantified this diffusion coefficient was to say that through these frappe experiments the fluorescence recovery after photo bleaching where we took a cell you bleach out an area which means that you kill all the fluorophores in this region so that it becomes dark. And then so, this region you bleach out this region and then as objects sort of diffuse back into this region from outside you slowly see that there is a recovery of fluorescence. So, if you plotted the fluorescent intensity of this bleached region as a function of time it was some intensity when you bleach it it drops to 0 and then it slowly recovers back to its background intensity. And by studying the kinetics of this recovery curve so, this depended on the diffusion coefficient. So, this was a function of the diffusion coefficient and then by fitting whatever theoretical analysis we did to the actual recovery curves you can get an estimate of the diffusion coefficient itself right. Now, if I look if I take a same if I take one molecule if I take some molecule I do not care what I take some molecule and I found out it is diffusion coefficient in water let us say so, I find out the diffusion coefficient in water versus the diffusion coefficient inside the cell. So, let us call it d-site cytoplasm which one would be higher naively hm water would be higher. So, I can say that right what else can I say so, let us say that the size is whatever r it has a typical size r and if I roughly assume that the cytoplasm is 3 times as viscous as water how much would this cytoplasmic diffusion coefficient would I expect it to decrease yes. D remember was what k B T by 6 pi eta r right. So, I would expect that the ratios of these diffusion coefficients would be simply the inverse ratios of this viscosity and therefore, I might expect that the d-siteoplasm is roughly so, the cytoplasmic diffusion coefficient is roughly one third of the water diffusion coefficient. So, we can try to see whether that sort of matches up . So, we can see how this diffusion coefficient behaves inside the cell as compared to water. So, we have said that it will decrease and naively I would say that it will decrease by this amount it would be one third the water value. Can I say something about how small molecules would fare relative to large molecules? So, for example, if I took a small molecule which is very small versus a molecule which is very big in which case would I sort of roughly expect this sort of relation to hold true . So, in for which molecule would if the cell is extremely crowded if this is diffusing in a very crowded environment both of these in which case would I expect the environment to have more of an effect for the larger ones right. So, if I am going to see deviations from something like this I would expect the deviation to show up more for large molecules as compared to small molecules . So, now, we can just look at some data . So, here are diffusion coefficient the ratio of the cytoplasmic diffusion coefficient to the diffusion coefficient in water for a variety of different molecules. These over here are small molecules this is the green fluorescent protein GFP. These are molecules these are polymers basically dextran or DNA which you can change the molecular weight depending on how long a DNA molecule you have or how big of a dextran molecule you have. And what I see is that for these smaller molecules so, when this molecular weight is sort of small roughly the ratio is around let us say 0.25 ok which is close enough to 0.33 it is somewhat smaller than that, but at least it is close enough to 0.33 ok. So, most of the decrease is sort of accounted by by the change in viscosity, but as you go to larger and larger molecules this sort of relation breaks down you can be smaller by almost 2 orders of magnitude even right. And that effect is due to crowding . So, when you have these other substances that are crowded throughout this and it is trying to of this big molecule is trying to perform diffusion in the background of these smaller molecules you will see more of an effect than would be predicted by a simple stokes Einstein relation and that is what you see for these sort of curves . This one for this very large DNA very large DNA it almost drops to 0 that does not really diffuse at all. You would you would see ok. So, let us see ah let us say that I have a diffusion coefficient inside the cytoplasm let us say I let us say I am plotting this the variance of distance that has been travelled inside the cell and I get something 2 d cytoplase and this is this d cytoplasm that has been plotted . What sort of other effects could you see how else could I modify this relation any idea ok time. So, in this I could also say that this exponent instead of being t to the power of 1 becomes some t to the power of alpha where alpha is less than 1 right and that would be the class of walks which I would broadly call as sub diffusive walks . So, I will be talking mostly of this I will assume that this linear relationship with time holds true and I will try to quantify how this cytoplasmic how the cytoplasmic diffusion coefficient changes, but you should be aware that ah in fact, the more common outcome is actually that this alpha itself changes the walks are no longer purely diffusive mostly inside cells you see sub diffusive walks where this time does not really grow as t to the power of 1. In fact, if you want to see that so for example, this is a movie of a protein which is performing a two dimensional random walk on the cell membrane ok. So, I have this cell membrane and there is a protein which is performing a diffusive walk on this cell membrane. So, these are actual particles which are performing diffusion and you can analyze their trajectories and if you see that the trajectories actually look fairly heterogeneous. There are some trajectories which just by looking at it without quantifying it or whatever without just by looking at it I would say well this one looks something like I would expect my random walk to look and I can then calculate the diffusion coefficient. On the other hand there are trajectories which do not really move at all ok. So, this one for example or this one for example, these just stay stuck there. So, it reflects the heterogeneity of the environment in which this protein is performing the random walk and in cases like this often the more appropriate description is to use instead of trying to think in the context of diffusion you try to think of it in the context of this sub diffusive walks. Now, actually trying to calculate this powers and so on is a complicated business and I will not try I will not venture into that rather what I will do is mostly I will just sketch a 0th order calculation as it were and just see how well that does ok. So, let us say so, I will go back to this sort of a random walk picture and again I will try to think of it as a one dimensional random walk. So, I have this lattice in which this I have this lattice in which this protein or whatever object is doing a 1D random walk. So, it can hop to this side or that side except now I have the caveat that some of these sites are occupied. So, some sites are occupied by some other objects and the volume and the fraction of sites that are occupied is 5. So, I call this 5 as this fraction of sites that are occupied. The non average if I am trying to hop to the right or to the left let us say with equal probability. The probability that I this move is going to be allowed and the move remember a move is going to be allowed only if this site is free if it is already occupied by something then that is not going to be allowed. So, the probability is like half into 1 minus 5 to the right and half into 1 minus 5 to the left and a probability 5 that it just stays there simply because it cannot hop and every time it does this it either hops by plus a minus a or c. So, I simply think of this sort of a random walk where this density of crowders is sort of captured in this volume fraction or this fraction of sites occupied 5. So, if I consider this sort of a walk and I did the usual business of writing down the master equation and then taking the continuum limit what would the diffusion coefficient come out to be relative to the empty site relative to d naught d naught into 1 minus 5 simply. So, this very 0th order calculation sort of tells me how this diffusion coefficient is going to vary as a function of this number of crowders. It is 0th order because the picture is very simple it is 0th order for example, an easy thing I could do if I wanted to sort of incorporate this size dependence that we saw in the experiments is to say that well the object that is diffusing actually occupies multiple sites it is a big molecule and it is diffusing in the background of smaller molecules. So, in order for it to hop it has to have a consecutive whatever of sites it occupies there is to have that many sites consecutively free only then can it hop and then I can come up with an improved version of this model that will take into account the size of this object that is trying to hop. But I am not really trying to go into that I can put it in the assignment, but let us say this is my 0th order approximation and then I can see how this so d naught into 1 minus 5 and then I can see how well this does for actual real data. So, these are different molecules for which this diffusion coefficient has been calculated as a function of this concentration of crowders. And for small enough concentrations I guess you could say it is falling roughly linearly and then it becomes non-linear, but I think on the other hand I do not expect this formula to work very well, but it captures I guess to a certain extent this initial fall of this diffusion coefficient as a function of crowders. If you want to explain this better this curve better and people have done various improvements to this 0th order model in taking into account more and more physically relevant factors you can come up with certain models which will give you a better and better fit to these data. So, this is just one very simple way in which crowding this crowding inside the cell is going to affect the quantities that we are calculating in particular in this case the diffusion coefficient. Moving on so I want to do this this is very interesting at least interesting to me manifestation of this crowding and that is called the depletion interaction. So, let us look at this manifestation of this macromolecular crowding which I will call depletion interactions and the origin is very simple, but the manifestation is everything else.