 I just want to sort of set up the scene and say what the kind of the outcome of this presentation, at least I would like to be and what kind of the way how we are going to run it. So, first of all, I would rather more focusing on developing practical understanding of knowledge about this subject that I'm going to talk about. Which by the way, that's probably I should have mentioned on the previous slide, it's the, it will be the discussion about the form factors structure factors and the polydispersity. And the, yes, as I said, it's it will be more practical than a rather theoretical. So I won't be going very much into the details into the map, simply because I think it's quite difficult to to do it over the zoom. And not basically seeing how people react on that on all of the equations that you present them. And once you present them, and the, and it's also maybe not particularly useful for the projects that you've been working with because most likely will stand on the shoulder of giants and some of the things that I've been able to contribute from that from what already been developed. And so, so, then, yeah, you can basically use this, this knowledge in the more practical terms, and I put up some questions to stimulate some thinking about you awake after the lunch break. And so I would appreciate if you can actively participate in this presentation. And also feel free to stop me at any time. So this presentation is for you. It's not really for me. I think you already saw this probably twice already today. And as you are well aware, the small angle scattering experiment involves the probing particle that it's interacting with the sample and then we observe the scattering, which is then detected on some the detector. And I also assume that was introduced at the other end result advance lecture, but we have the we define usually the scattering vector called q, which is the difference between the incident vector wave vector and the and the and the q is related to the scattering angle. And what we measure in this small angle scattering experiment is the intensity function. That's just to remind you what you probably heard already. And then when we have the actual instrument, we have the way to produce the neutrons then to go through the all the neutron optics. And interact with the sample and the we are able to detect the image on the detector and the instrumentation part will be covered in them. I think it's the lecture after this one. So you'll definitely hear more about this, but just looking from this part from the from the sample to detector would be obtained is the duty pattern. But it's for the for the isotropic samples. It's also isotropic and therefore it can be radically average in order to get the one this spectrum. And that's what we usually get from small angle scattering. We get the curve that represents that represents intensity in the function of the scattering factor. So now we are coming to the first question, which is sort of trying to get some some idea what what is your intuition about the experiment as I heard. Many of you already have experience with the different techniques and new different scattering techniques so therefore you may also know particular answer but even if you don't know exact terms. Try to think of what components should be included in the model to explain small angle scattering data, not necessarily sounds at this stage. I wrote sounds here but I mean it at this stage it will be also applicable to the x-ray scattering just sort of think in the way that you are one of these pioneers of the small angle scattering technique. You know what your sample looked like and you know what the pattern that you obtain. So now how to sort of couple this in the what kind of theory developed to go from this stage to to this one. So I would appreciate if you open up chat. And then just type whatever comes up to your mind. So someone mentioned scattering land density. There is a wavelength sample geometry normalization. Okay, so if there are no more ideas let's go to this one. So scattering intensity. It can be defined with the with the sort of three different terms. And yeah essentially that will go through the details through more detailed explanation about these things but it can be simply divided into the pre factor form factor and the structure factor. And the and what they basically stand for is so we can by the way everyone mute themselves. It's a I know it's my fault that I asked you to type if if you're typing and they have a microphone nearby that's my fault. No worries no worries at all. Right so I do mentioned the SLD for for this and that's that was very good thought about this because I'm in this pre factor this first term that it's mentioned here it's something that takes into account the contrast. And that's essentially the difference between the scattering land density of the molecule and the background or the solvent that we have. And what also it's taken into consideration is this number of particles. So, yeah, basically how much of the sample do we have so as its mass and the team. We also are not able to measure something very small from this mind that scattering because if the mass of the particle is just very small and it's not, it's diluted, then of course the signal is very low. What here is represented it's this thing related to the scattering land density contrast so that's I think you all rise this idea because what also I think was covered by advance in the previous lecture you the signal change considerably while you modify this factors. So this is something that we can call pre factor this definition by the way is kind of lose I would say there are different ideas of how to how to name it, but just for the simplicity we can divide the overall scattering intensity into these three terms and I call this pre factor. The second term is the is the form factor and that's something that represents the interference of neutrons scattered from the from the different part of the same object so this is the interrupt particle interactions of the molecules that we have the in in our sample. The next term we just called the structure factor that's on the other hand represents the interference between the different objects so that's why the previous one was the intro particle interaction that's interaction between the particles. So now, the second question is, once we learned what are the three basic components. What do you think will we need to define this form factors even the fact they are accounting for the intro particle interactions. And if you can express some thoughts on the chat. Be grateful. And if the question is not clear then also speak up. Okay, it's not clear. So, what I'm trying to kind of sense here is where you have a sample right and in this case it's this spherical sample that we have this very simple illustration of. So, what do you think are the parameters that you need to describe the sphere in order to define the form factor. So I know it's maybe a little bit difficult in terms of the, because we don't really know what the form factor look like, but what kind of the basic geometrical parameters would be in terms of the, in terms of the sphere to to describe it. Yeah. Okay, yeah, exactly that's that's going in the in this direction so the the form factors explaining this in intro particle interactions or in other ways that size and shape of the particle. So in order to define them we essentially need to know that what are the geometrical parameters of it. So what we see here is the are the different geometrical objects, there is the core shell. Okay, there is a sphere and cylinder and some polymer chain, and each of this have different different form factor associated with so there is a different form factor for the sphere cylinder and so on. And in order to define them, we need to sort of basically use the formulation which is based on the on the on the simple geometrical parameters so in terms of sphere. We can define the form factor using the basically radius on the sphere that it's presented here, and the, and the form factor looks like this like presented here on this, on this, on this plot. And just to give you the idea how this is developed so the form factor is essentially the squared scattering amplitude that for the homogeneous volume can be written in this as this integral. And from here, using some mathematical tricks, we can, this is the trick that needs to be used, but we can derive this formula. So, just to give you basically the idea how this can be derived, the form factors required the basic geometrical parameters and can describe and describe shape and size of the particle that we are studying using small angles category. For cylinder, for example, this form factor has this and can be explained using this formula, which involves the use of the best of functions and the, and the angle which is defined as an angle between the cylinder axis and the scattering vector, the shape of this form factor is as it explained here. So the form factors very much depends on the system that you study and as I said, it's one can benefit of what being developed already and there are many fact the form factors that being already developed for the different systems just to give you another idea that's the core shell particle that's one of the presented before. And again, not going very much into the details formula can be as follow. And here it's a little bit of the mixing with this pre factor, because it's not that easy to kind of deconvolute one from the other, but the, the sort of overall idea is that we can define the scattering particle and density and the radius of the core and the outer shell of this core shell particle in order to obtain the form factor. And like, if you go to the complicated systems and that's, for example, this virus particle that I've been working with, you can that can be explained as an empty core RNA part that is order protein part and the, and then the kind of the solid rigid part using the atomic representation. And then one can divide one can design this complicated form factors and then they can be explained with, for example, something like onion model. But as I said, many of this hand has been developed already. And then the sort of the different group of the form factors that is available. It's the one for that that can be calculated directly from the coordinates. So some of you in the introduction mentioned that the working with the proteins. And for this particular systems, you usually calculate the form factors directly from the coordinates. And here is the example for this monoclonal antibody that here is shown the different and different values comparing to the experiment that's maybe not very important. But what I would like to show is this formula, which is called the debate formula, which is the which is explaining the difference between the, which is explaining the form factor or in this case it's already intensity with the respect to these functions that takes into account the distance between the each contributing scattering atom. And the numbers in these terms are atoms so it's calculating the this function over the difference of the different contributors contributions from the from the from the each scattering atom and the and what needs to be kind of considered here is the is that when you have many atoms that's a formula becoming quite cumbersome and therefore there are some ways to either to simplify the calculations or simplify the representations of the of the proteins and when you when you calculate this. What I would like to do now is to go to the to to SAS view and show you how this can be practically calculated so I will stop share this for the moment. And by the way, if you have already started installed, you feel free to to join me with with in this calculations because we will also see if if you can and you can calculate if that works properly. So the question is, do you see the screen now with with SAS view. Yes. Okay, good. Yes. So, so I mentioned the sphere as a as a as a one of the models. And then we can do so what I what I did now I open SAS here and choose fear from the category. And as you can see we have a this is the category sphere so it's not essentially the model, but the I'm choosing the sphere from the sphere category. And what I will do now I will hit this calculate button and then show plot. And that's, we should get something that looks similar to what we already saw on the one of the slides that I showed previously. So now what I just would like to demonstrate and, as I said, if you're also now playing with SAS view can manipulate it yourself. Then what would happen if we change this radius, which is the, which is the this parameter that governs the the form factor for it. If we go to the smaller values, which is the 10 angstroms, then what you see that this sort of characteristic bumps that are occurring been moved to towards a higher cube values. Right, and that's something that you would expect in the sort of looking through the four year glasses because it's kind of the reference, reciprocal space and the, the smaller distance in the, or the smaller distance in the actual case corresponds to the higher distance in the q space. And while if we now go to something like the hundred angstrom 10 nanometers, you can see here that we get this rough curve, which is in principle something that you shouldn't be seeing. I mean, not that rough. So, and the reason for this is that we need to increase the number of points that are this calculations are done. So that's what I just did in this other setting. So, yeah, so when we had them, maybe let's do it one more time. So when we had this at the 50 radius 50 angstrom radius, then and the, and the, we had a sort of first picture while if we increase the the radius, then this first bump moves towards the low q, low q value. And then I mentioned it already already that many of these form factors have been analytically calculated. So just looking at this list. For example, for the cylinder we have quite a few options. Yes, and essentially, you name it, there are roughly 74 factors available from SAS to to to be directly applied to your data. So let's now go back to the presentation. But as I said, sadly, it's not the, sadly covers many of this, but it's not the, the only resource. Many, many form factors, not necessarily being ported into SAS, but if you are looking for some particular system, this might be a good reference. And there is also software called SAS fit, which also provides some additional models. And yes, and such view has quite a few of them. Now, what to do if everything fails, or in the sense that that we don't really know what the, what the analytical formula of the form factor is. However, we, for example, know what are the, how the molecules can be explained in terms of the, in terms of the coordinates. And for this Monte Carlo simulations are actually a good solution. So it's really working well for the structures with many degrees of freedom. And, and allows for the easy something of the different parameters for the either form factor and the structure factor that we just covered in the, in the minute. But essentially the sort of idea for this is that we do this Monte Carlo simulations, if they're relatively sampling from the different parameters, generating some configurations for the p of q and s of q, and then comparing it with the data. And we do this until the algorithm converges. And based on this, we can estimate what the, what the form and tractor factor looks like. This is just the simple illustration for the sphere form factor. And in order to see that we can recover the intensity and also something which is called the per distance distribution function, which is something that I will talk about in the next lecture on Wednesday, if I remember correctly. Okay, so that covers the, the form factor. We'll switch gears a bit now and we'll discuss the structure factor. So now, again, a question, which hopefully it's, it's understandable. But, and it's what should we consider when defining inter particle interactions. So if you have, if you can share any thoughts on this. So now we are not in the regime of the sort of single sphere, but what is the in how to define the interaction between spheres. Please share in the chat if you have any thoughts about this. Okay, I see some questions. Oh, sorry answers coming to this question. That's very good. Yes, I think it's a good intuition that you have about this. So, the, and we can for the, for the purpose of this presentation and in the, in the majority of the cases, say that the structure factor can be calculated using this correlation function this G of R, which is essentially the average of the normalized density of the atoms at the and the given radius or given shell from the calculated from the center of the particle. So it's essentially corresponding to the density and packing of the and the interaction of the of the atoms. And for the diluted system, the G of R functions can be represented as the, as a sort of the what you say, step function. However, for the, for the concentrating systems, it's, it gets shaped that it's with the maximum and then slowly decaying. However, for the order systems that looks more like this with this periodic peaks. And if you've been working with the fraction data, you may be kind of familiar with this formulation because what I think in the fraction is something which is referred usually as a lattice factor that is related to this G of R. Nevertheless, the sort of what we are getting from here is the this when we calculate this structure factors from this different G of R for the diluted system we get the S of Q, the structure factors equals to one, which if you recall this formula here. That means that in the diluted diluted regime, we essentially have to only take into account the pre factor and the and the form factor, we can assume that our S of Q is equals one and therefore we don't need to take it explicitly into consideration. However, for the, for the, for the, for the concentrated systems that is not really the case and therefore it has to be taken into account. It's similar in the for the for the order systems. However, they are not many of the, of the topic of today's presentation because they are this regime is typically not covered by science. So, excuse me, what did you know what was your I missed the. What did you say that this GR what it shows. So, so the, so what I was showing here is the relation between G of R and S of Q right. What is G of R. What is G of R that's the, that's the correlation function which is the representing the average of the normalized density of atoms. Yeah, I guess I don't really have a good illustration of this, but I mean that's essentially one can think about the parking of, of the, of, of this atoms in the, in the, in the shell. So essentially that's, that's kind of related to the interaction potential between the, between the atoms. So then we should calculate that before we put it in this. I will, I will cover this because I mean, the, the, the S of Q calculated from this there are also sort of in SAS view there are four different models to choose from. And the, and you also can kind of apply them directly to the data. If this question. Thank you. Okay, so, so as I said, there are. There are four different ways to in SAS view, not in general because in general there can be more S of Q, but just for the, for the simplicity, let's discuss this for different options. And the, and the sort of the simplest one, it's called the hard sphere, which enables the calculation of the structure factors from the spherical particles in solution. And through their hearts for interaction. So it's essentially assuming that we have the excluded volume. And then we have the, and then we can account for the, for the interaction between them as a rigid objects. So, and that's usually quite a good approximation for the, for the proteins and other nanoparticles, it can also be at two large extent used for the, for the object that are freely rotating. And therefore, they are occupying the volume as sort of defined as the, as the effective values. So, if you, for example, have the cylinder that it's freely rotating that's taking up the volume, which is the larger than the volume of the cylinder itself. But one can also apply this approximations. To our approximation to this cases. What is really kind of important here is that the, when we just calculate this. If you using this hard sphere model that as you can see the intensity in the, in the low Q region is is at minimum, and then then it's, then it's increasing, which is kind of the opposite from what you learned about the form factors which, which always have the intensity, the higher intensity in the, in the low Q region. So when you apply this to the data and then this will see in the, in the minute, then the you should expect the overall intensity to be decreasing in this region. So the sort of extension of this hard sphere model to the quality quality of particles with charge interactions is something which is called the higher MSA model and then I've been trying to get away from the from introducing the closures here, because that's really a way that this structure factors can be derived using something which is called Einstein's earnings equations that I'm not going to do it today, but this MSA says stands for closure and that's the sort of mathematical way of copying together this Einstein's earnings with the G of R function. I guess you, if you are into this you probably know it and know what I'm talking about if, if not then don't worry that's not really important, but the, the, the sort of the and then the purpose of this potential is to enable the screened columbic interaction between the particles. And again the, the pattern of the S of Q calculated for this it has the minimum in the, in the low Q region. The last two can be sort of tackled together as these are for the colloidal particles with narrow attractive and potential and they are both defined using the square wall. And there is a really technical difference I would say when you account either for the square wall or the sticky hearts here, because they are essentially different in the way that you use this closure functions. So now, what I will try to do again, I'll try to. Okay. Open society again, and let's look at our sphere that we generate before. And the, so just let me try to make sure sorry it's on this small screen is becoming quite clunky with the, with this zoom and the, and that's the at the same time but hopefully can manage. That's the, and these are these different structure factors so let's try to take this hard sphere potential and as you can see here the ones this is applied. Then the intensity decrease in the low Q region. And sort of the similar should happen. When we do the, and this higher MSA. So yes, if you are. If you're now playing with such you can take a look basically what is the effect if you apply one or the other coming back to this one. I just want to say that there are different methods to include this structure factors into the calculation. Not going very much into the details of it, but we essentially have three different options. Which the, the one is called the monodized person approximation. And that's for the spherical symmetric interaction potential, and it is independent of the part of the particle size. Then we have a decoupling approximation. And that's for the applicable for the, for the particles with small anisotropies and the local monodized person approximation. And that's the for the particle of certain size. We just surrounded by the by the particle with the same size. Maybe that become a little bit clearer when we talk about actually polydispersity. In the next few slides what I, what I exactly mean in terms of the, in terms of the polydispersity. Just, yes, again, if you, if you really need. Then I just want to point out that there is an option to do this. And in SAS view, we have the option to include the monodized person approximation, which is as a default, but also the decoupling approximation and the way you choose this is the by choosing different terms here. Right. So now we go to the only dispersity. Whoops. That didn't work exactly the way I wanted. Sorry. Hit the wrong button. Yes. Okay. So, in principle, we have three types of polydispersity, we can account for something which is called size polydispersity, when all particles have similar shape, but different size. Then we have something like a shape polydispersity, so they differ in a shape, both in a shape and size. And the, I will have this illustration on the next slide so we can see examples for this. And then something which is referred to the conformational polydispersity, which is really like for the particles that have the identical molecular mass so they kind of the same. Yeah, the same mass essentially, but they can adopt the different conformations. And I didn't hear anyone mentioning the working with the flexible proteins during the introduction, but sorry if I missed it, but that's that's essentially the case. So, now, one more question, but I'm sorry that absolutely last. How to, what is your feeling, how can we account for this polydispersity so so far we learned that we, and that we have the preform and structure factor. They were coupled together through the different terms. And so, like everywhere to add on the top of this, how would you think polydispersity can be accounted for. Yeah, if you have any, any thoughts, please share, if not, I will just go to the next slide and try to explain it. Okay, yes. So the, the, maybe not the answer that I was looking for but the something that the that mathematically can be used for the for the including polydispersity and that's definitely the case with the with the size polydispersity is essentially to introduce the distribution function for this. So, if we have this size polydispersity, which is illustrated here so we have for example the spherical particles that differ with the radius, and therefore we can say that we have a polar dispersity of radius so that we have a distribution of radius of the different particles. And the and therefore the, the polydispersity can account for the average in intensity for the population of the particles not just the single one and, as I said the convenient way to do this is the is to introduce this dysfunction. And again, in such view, that's can be done to introducing the the function with essentially with the number of points. And here we are using just the Gaussian representation, which is not the, not the only option, but with this, you can essentially define this this distribution for the parameter like if we if we have this radius. And so we, that will be a mean radius here and have with here for the for the Gaussian distribution. We can define the something we just call the polydispersity ratio in order to define this distribution. And what we get as a result is the normalized intensity by the average particle volume right because I mean the situation that we have now it's the particles have a different volume each and therefore we have to account for the average particle volume. So that's a size polydispersity. Now when it comes to the shape and dispersity that usually what is being done. It's something different. So if we have this case for the proteins that they can coexist for example in the, in the monomer and the, and the dimer state. And we also have a some fraction or the data reported for the mixture so let's say we have 20% of the monomer and 80% of dimer what one can do one can deconvolute this fractions and and essentially account for the contributions calculated from the, from the combined intensity that it's the, that is the, it's, it's the, it's the weighted sum of the intensity calculated from this debate formula as I presented for the, for the calculated the proteins. So this, this I small I K is in this case given, and then we have the fractions and that's kind of the purpose of this is to get the fractions back in order to say how much contribution we have of the each species. Kind of the similar trick works for the for this conformational ensembles that's the one of the protein systems. showing the flexibility so this part is flexible, this linker and the two domains and again one can kind of do this similar trick when it comes to the to the inferring the fractions. Yeah, the overall the problem of the of the getting a lot of this parameters out of the systems is problematic in the sense that they usually you have quite a one scattering curve and you have to deconvolute all this information so that's a little bit of the problematic when you have to account for the overfitting. Okay, before I will go to summary just very quick illustration again with such view for the polydispersity. So now, let's maybe go for the simpler case we just account for no structure factor here and let's maybe change this. We use the 50 again. So that should look something like this. And now the polydispersity is the is the is the option that you have here, but what it does it's deactivates this. This is this tab here. And what we can do, we can define the sort of the ratio of the polydispersity. And hopefully that that would work. But as you can see, and that's kind of having quite dramatic effect on the resolution of these features. And that we can have compared to the to what happens when we have no polydispersity applied. So that's a kind of conceptually important because the kind of the weekend by not accounting or accounting properly for polydispersity. We can be sort of working in the and we can we can essentially infer the wrong model parameters. So just to sum up form factors, which represent the size and shapes shape of the of the of the objects. So this is the representing the interference from the different parts of the same objects of inter particle interference structure factor represent the difference between different objects and can be accounted for using the different potentials and and there are three different types of the polydispersity. And sometimes there is also something mentioned the angular dispersity, polydispersity, which is something related to the anisotropic systems that probably Elizabeth will talk about when she will introduce the magnetic samples and the and yes, and there are different ways to to account for this polydispersity depending on which type it is. What hasn't been covered in this presentation, as I said from the beginning, there was no this wasn't really meant to provide a very rigorous derivation either from the form and structure factors for the structure factors in particular with the integral equation theory and the Einstein-Zernick equation and that it's sort of underlying basis for the developing this that hasn't been covered. Sorry about that, but that's that's quite difficult to do it as I said, as an online lecture, plus I think if you're interested into this, there is always possibility to learn more. And the, and what I didn't really cover it's also the discussion about background so that was something that I glossed over but if you might have seen in the insights with there there was some background field to fill in so that is also important for the for the data analysis. Resolution smearing that's also not being covered these are visually instrumental effects that that can contribute to the some loss of features in the in the curve. That's something that the to some extent will be covered at the instrumentation and lectures and I also didn't talk about the orientation and or slash magnetic form factors. And that's something that Elizabeth will be talking about tomorrow, I suppose. And, and yeah, and probably some more stuff that I haven't thought of. And just to finish off the take home message from from this one is that the what we've been doing we've been essentially through introducing this pre factor and form factor and the structure factor, in the model to the data. But what is important to remember is that this is the data that contains a low information content. And therefore, it's always one needs to be careful with that with there with the feeding in order to sort of put too much emphasis on the on the model rather than data and therefore not to overfit, but also the optimal experimental design is is is really key to successful data analysis. And you will hear more about the instrumentation and the experimental design in the coming talks. So hopefully that that will also give you some understanding of what can be done with this respect. And with this, I would like to finish and happy to take any questions. And I saw there was one on the chat already. And so Andy. That's from you, I suppose. Yeah, that's me. I'm asking about the structure factor for the elongated particles like like long rows. Right, right. Okay, yeah, so I very briefly mentioned this and sorry that wasn't clear. But in principle, one, as I said, I mean for the at least this heart sphere can, if you assume that these are freely rotating the structure factors. So freely rotating molecules, then you in principle can account for this as introducing the effective values of this molecule, which is like taking the radius of the freely rotating cylinder, and therefore in some cases that that can be used for the for explaining interactions and therefore calculating the structure factor. I'm pretty sure it's just that I'm not aware of that for the cylindrical molecules that also been developed something specific. I don't know if this answers. Yeah, that answer. Thank you. Okay, I will stop sure now.