Well, why don't you have just create an environment where as long as they're in their environment, the environment will uplift them and the environment was transformed in one percent a day. And that's what we're seeing. Actually, we wrote a book called ADHD and autism. We did a case study. 26 kids lowest performing in the nation. They were in about bottom 20 percent. They couldn't count to 60. They couldn't memorize the alphabet. That's how bad it was.  Four months later, using our technology and our frequencies and then with a special math education method, they scored in the top 1% in the nation in the math. So this is quantitative. It's not like, oh, they were a little bit more calm. Oh, they were a little bit smarter or more focused. No, they scored from the bottom 20% now up to the top 1% in the nation in their math scores, which is nearly unheard of unprecedented. Okay. What do we do? They had they had educators come to an art into our classrooms, trying to figure out what we're doing. And they're saying, what did you how are you able to help this child is this girl who we couldn't get to talk for two years, right? And now we're looking at her. Is this the same person? She's leading the classroom and teaching the class. Okay. So so these things are happening everywhere that we're using these technologies. And then with the schools, we're seeing it very clearly. You know, in homes, we're seeing so many success stories you can see in our website. So yeah, we're transforming people, young people, old people, even pets and animals. We're transforming environments in ways that I think are very revolutionary and very groundbreaking. And I think we're just getting started. And we do have scientific studies. For example, we have the case study. We have a pilot study with 200 people and then we're doing another double blinded. I don't, you know, all that all that fancy stuff double blind.


So, in the last class we had just begun analyzing the electro-dynamic lumped parameter model of an acoustic transducer, where we left in the last class was that we had identified some break points or resonance points, specifically there was an electrical resonance between R e and L e and that was happening at around 1820 hertz that is what we calculated. Then there was a resonance as a consequence of C m and M m and that was happening at 89 hertz and then an acoustic resonance between because of M a and R a that was occurring at 717 hertz or something like that. The other thing we had done was that we had decomposed this transfer function v naught prime over v naught in terms of u coil over v times v naught prime over u coil. So, what we will do today is we will develop a board plot for the first ratio u coil over v and see how it changes as frequency changes and then we will also develop a board plot for the second transfer function v naught prime over u coil and then once we have these two board plots together then we will add them up and see how the overall transfer function of the overall transducer looks like. As we develop this relation you may want to remember that an ideal transducer should have a very flat transfer function over a range of frequencies. So, the wider that flat area is from an ideal standpoint the better it is from a listening purposes. The other feature of a good transducer would be that not only it should be flat, but it should be as high as possible because what that means is that if I send in a little amount of electrical energy it gets converted into more and more of sound. So, actually the efficiency of very good acoustic transducer does not exceed one or at the most if you are really smart 2 percent. So, that should give you some overall perspective the overall efficiency of sound system or a transducer hardly exceeds 1 or 2 percent. So, we will start developing a relation for u coil over v v naught prime over u coil. So, we will do this in two steps step one is we are going to do v naught prime over u coil. Now, when I look at this circuit my v naught prime is the voltage across this acoustic resistor r a over ac square and u coil is basically just the voltage across the transformer after the electrical circuit and the current in the mechanical circuit is essentially force and the current in electrical circuit is obviously current. So, when I look at this circuit what I infer is that to get this ratio all I have to consider is basically the element m a times ac square and r a over ac square that is the only thing I have to worry about. So, the equivalent circuit is something like this. So, this is m a times ac square this r a over ac square my input voltage will be in this case u coil and what I will be measuring and what I am interested in is v naught prime. So, this just for sake of simplicity is something like this I just simplified the notation right and if I compute impedance if I compute the impedance and I try to find the ratio the relationship which I get is v naught prime over u coil comes to is complex frequency times capacitance which I have noted as m times r over 1 plus s m r. So, now I construct a board plot. So, on my horizontal axis I have logarithm of omega and in my vertical axis I have decibels omega tends to 0 how will the board plot look like as omega or that is s which is j omega as omega goes towards 0 the low frequency asymptote how will it look like. It will go through the origin and it will be a straight line with a positive slope of 20 decibels per decade and once it crosses a certain threshold then as omega goes towards infinity what happens this ratio essentially converges to 1 right at infinity. So, log of infinity log of 1 is 0. So, the high frequency asymptote is something like this and this what is this value 0. 0 decibels what is this break point this point 1 over m r right what was the 717 hertz we had calculated ok. So, this is my board plot for v naught prime over u coil let us call this figure a. So, we have finished developing a board plot for the second portion which is this now we will do for this one. In the second step step 2 what we are going to compute is board plot for u coil over v in the last class we had found that there was a resonance happening at 89 hertz right. So, below 89 hertz u coil over v is this is my value v and this is u coil below 89 hertz what is going to happen is that most of the current is going to pass through C m the mechanical inductance the impedance due to m m r m and all these things will be extremely high for f less than 89 hertz my equivalent circuit is going to look like C m and that is my u coil why did I neglect L e in the circuit I dropped out L e in the circuit on the electrical side I have not included L e why have I done that L e becomes important as we had calculated after 1800 and 20 hertz below 1820 R e is going to dominate right. So, I can neglect the impact of L e for f less than 89 hertz the board plot in this case is going to look like how will it look like it will be a straight line slanting positively and the slope will be 20 dB per decade and this is 89 hertz. So, now I have to go from 89 to what is the next break point 700 and 17 hertz right. So, for f 89 hertz is larger than is smaller than f and that is 700 and 17 hertz. So, my electrical circuit is going to look like. So, now I will have the capacitor m m both these capacitors are going to be because very low current is going to pass through this inductor because I am above 89 hertz and the capacitors will become active. Which one in an asymptotic sense. So, as s becomes infinite as s becomes infinite the numerator is becoming more and more as s tends to infinity numerator and infinite denominator they become more and more equal. So, the ratio goes to 1 regardless of the values of m and r. So, this is my reduced circuit for this frequency range 89 hertz to 717 hertz again this is v and this is u y. So, my board plot in this case and as frequency goes higher and higher the impedance offered by these. So, I get my this was my original line going up to 89 hertz then I have a negative slope and the negative slope is 20 dB per decade and this is going up to 717 hertz. So, again on the horizontal axis I am plotting logarithm of omega and on the vertical axis I have decibels and this is still u coil over v. So, we know response below 89 hertz response above 89 hertz up to 717 hertz. You would ask ourselves a question that what happens at 89 hertz specifically. So, that is what we will once if we know the answer to that we will know this value how many dB is at 89 hertz. So, that is what we will find. So, what happens when f equals 89 hertz. So, at 89 hertz on the other side above the beyond the transformer the circuit will look like there will be a capacitor. There is an inductor which is mm capacitance is yeah. So, I have capacitance of mm inductance of C m and resistance of 1 over R m in a general sense I am just calling them L C r and when L and C resonate and they are in parallel. Their overall impedance does it go to 0 or does it go to infinite when they are parallel. If it goes to infinite we will see that if we just find the impedance of this block Z block is 1 over S L plus S C inverse equals S L 1 plus S square C L. So, at resonance the impedance offered by this entire block is infinite because 1 plus S square C L becomes 0. So, no current is going at resonance when an inductor and capacitance capacitor they are in parallel and they resonate no current flows through them and all the current goes through the resistor. Our original question was that what is happening at 89 hertz what that means is all current is going through the resistance. So, just at 89 hertz my equivalent circuit is u coil for this one if I actually compute the value I get u coil over v equals 1 over 9 times 162 over 162 plus 8. I am just basically plugging in the values which we had recorded earlier. So, this comes to 0.1059 in decibels it becomes if this is w 1 then w 1 becomes 20 dB 0.1059 log minus 20 dB about. So, I go back and modify this graph and I make this peak value minus 20 dB. One note of correction in all these cases the value here is a resistor below 1820 hertz it is going to be a resistor because that is larger than the inductor and both are in series. So, this also needs correction. So, I apologize. So, now we have figured out the board plot from 0 hertz at 89 hertz this value becomes minus 20 decibels beyond 89 hertz again it will start rolling off at a slope of minus 20 dBs per decade. So, our next one is for the range 717 hertz to 1820 hertz. So, again looking at making judgments which components to include which components should not be included my equivalent circuit looks like the only thing I have is m m this is my u coil is everyone clear why this is the case the board basically what this picture tells me is that the board plot will not change its characteristics beyond 717 hertz also the slope will continue to fall at the same rate the value. So, my modified board plot becomes 89 hertz and it keeps on going till 1820 hertz that is my omega this is in decibel and what I am plotting is u coil over v this is up to 1820 hertz. Finally, for f greater than 1820 hertz I get. So, here I have a resistance and a capacitor above 1820 hertz what is the change going to happen inductor becomes active the electrical inductor becomes active and the slope of this line will be what it will have a slope of minus 40 decibels because now I have to earlier we had seen that if I put a capacitor and inductor in series which is the same case here the slope becomes minus 40. So, I will go back and modify it will make a new one clearly this figure is not to scale and this value is minus 20 decibels. So, again this is u coil over v we started with the fairly not a very complex circuit, but the fairly complex circuit, but just by making identifying the break points decomposing our transfer function into small manageable ratios. We have been able to plot the board plots for all the important all the components which are going to play a role. So, now what I am going to do is I am going to find the board plot for the entire function which is essentially a sum of those two board plots. My final board plot will look like something like this. So, this final board plot will be a plot of v naught prime over v what is the slope of the first segment 40 decibels and this is 20 decibels and this guy is minus 40 decibels per degree looking at this picture what conclusions do you draw? That is the first thing that from 89 to 717 hertz you can use it and whatever is going in it will be faithfully reproduced at the same level output response will be same will have a flat response. Now, in this entire analysis I mean we have plotted it up to infinite hertz this range could be as high, but as we were developing the lumped parameter model of the entire system we had made certain assumption and the most central assumption was my size of the transducer or radiating area shall not exceed lambda over 2 power. So, I have not done the calculation, but the validity of this board plot is only good to the extent that whatever let us say lambda over 2 pi comes to 1200 hertz I have not calculated but you can go and calculate what that number is. So, this board plot can be relied upon only up to that particular frequency that is the second thing. The first one is that I have flat frequency response from 89 to 717 hertz if everything else is good then I can rely I can use this in this way. The second thing is that I may have to restrict the usage of this board plot at a cutoff frequency which corresponds to lambda over 2 pi to be hyperbolic. The third thing is lumped parameter model is especially from mechanical standpoint is that I am assuming that the masses are rigid I have a diaphragm and it just moves like this. Now, in reality when you have any membrane or any structure it has its own modes it could be that the diaphragm starts breaking up meaning that it starts exhibiting its own modes at 300 hertz 400 hertz I do not know if that is the case then that is another word of caution that this particular board plot has its validity as long as rigid masses remain rigid. We have assumed that the diaphragm is infinitely stiff and it does not bend and twist as it moves back and forth. So, someone while this analysis is done at a more detailed and a refined level someone has to go into a modal analysis of the diaphragm using some finite element method and figure out what are the modes at which the diaphragm starts exhibiting its own it is no longer rigid. So, that is the third thing. In what sense the lumped parameter model? No lumped parameter model it gives you a guideline if because it is a relatively straight forward way just doing the diaphragm finding modes of diaphragm all it will tell you is that the diaphragm is going to have. So, when you do this is your diaphragm right and then here you have a voids file I am exaggerating can everyone see this or can you see.  So, when you do and then there is also a suspension here. So, when you do modal analysis of this using finite element method you model you may create elements on the spider you create elements on the surround on the phone everywhere you have elements and then you do modal analysis naturally the first mode which is the finite element analysis shows it will fit out will show will be what it will be equivalent to your mechanical resonance k over m right. So, there is a k associated with these springy elements and then there is the overall mass. So, the first mode will be let us say m 1 will be I do not know what we calculate k over m 75 right. So, it will calculate it is 75. So, what that means is that this entire structure and when you draw the mode shifts you will see that finite element will predict that this entire thing moves up and down in the modal analysis that is a pistonic or a rigid body motion that is fine and may be some other mode m 2 what it will do is that it starts instead of moving like that the cone also has a propensity the structure to do this right. I mean it can move like this it can also do this because it is hinged here, but at a higher frequency. So, this was so that so this can do this. So, this is called rocking mode this is going too deep into transducer design. So, we are not developing some systems in this class since you are, but it is still higher what starts happening is may be in the next class I will show you some pictures that this cone it no longer remains rigid. So, it starts doing suppose this is a cone surface it starts doing something like this and then you have to this 3D when you do a 3D picture of this 3D if I look at it from the top side you will see something like this. So, some part is coming up some part is going down and so on and so forth what that means is that the cone is no longer rigid. So, at frequencies above that particular number the value or the faith which I will have in the predictions of my lump parameter model will be limited. They may still be valid in a quantitative, but they will still qualitative, but they will have a limited value. So, but again we start typically some lump parameter model because it gives you an overview gives you an overview how the system is going to behave and it is very easy to tweak specific parameters and adjust this flat range. If I want my flat range from let us say 40 hertz to 1000 hertz suppose the requirement is that I want my transducer particular transducers to start working from 40 hertz to 300 400 hertz then I can use this kind of approach to figure out what are the optimum parameters how much is my moving mass going to be once I know that I can figure out the size and the thickness of my cone in different part and I can design them. I know from this approach what is going to be the stiffness of my system. So, then accordingly once I have that target number I can design the shape of my spider shape of this around and figure out how to get that particular number. It also tells me what is the value of BL going to be which helps me design the magnet. The same model also tells me what is going to be the optimum value of RE which helps me design the voice well. It also tells me what is going to be the optimum value of AC radiating area and that essentially helps me understand how big my transducer has to be. So, there is a very strong value in using this approach in figuring out the overall design at a coarse level and once I have figured out the coarse level important parameters of the design then I go part by part and design them at a more refined level. So, that is the value of this system overall at a system level modeling the entire system. So, what we will do again in next 10, 15 at the most 20 minutes is we will change play with a couple of parameters in this circuit and see how it impacts the overall response of the system. So, that will again give you a close layer the value of this particular way of modeling transducer. What we had assumed in this particular model was that the transducer is mounted in how is it mounted? Infinite. Infinite backfill. So, you have an infinitely large wall it mounted there the wall is rigid it does not move back and forth and the speaker is radiating sound outside in the positive half plane. Now, in a real application you do not have infinite backfill. So, in a lot of conventional systems you may have seen big boxes right where the mount is the speaker. So, what we will try to find out is what does that do to the performance? Transducer in a box or sealed enclosure. So, I have a box this diaphragm is moving in and out and let us say the volume of this box is V naught. Physically we have talked about this earlier what does a volume do? It acts as a spring. So, I have this entire circuit and the stiffness of this spring is V naught over rho c square and if I have to remove that transformer I also put an ac square in the denominator. So, this is acting like a inductor. This inductor is going to be parallel will it be in parallel to the mechanical inductor or in series to the mechanical inductor which is the spring? It will be in parallel if you add two springs in series the overall stiffness does it go up or does it go down? It goes down right if you have the two springs in parallel the overall stiffness goes up. So, what this box is doing is it is going to increase the overall stiffness. Physically I am trying to explain all I have to do here is I have to add one more element. I just erase some things for so that I can put in more stuff I will put one more compliance number and this will be C B compliance of the box. And the compliance of the box is this value to make my life simpler for analysis purpose. I can assume that C B equals C M it makes things simpler. What that does is that my F M mechanical resonance it goes up by how much? Basically if C B equals C M then I can add these two up together right C M and C B I can just make that make the overall thing as two C M. My mechanical resonance will go up by a factor of my overall system response will be this was my original curve 89 hertz. Basically what I will get is I will get another parallel line this is 89 times root 2 89 times root 2 that is this by omega log omega. So, the dark black line solid black line is the original line this dotted line and it corresponds to the performance of the transducer in a box whose stiffness is same as the stiffness of original transducer. So, if that is the case then this gap A A prime is off by half of A and off cave is factor of 2 right half of root 2 times root 2 this drop of 6 decibels. So, basically what you have done is once you put system in a box have reduced the operating back of the transducer. So, that helps you understand why in conventional systems will come in boxes which are very large. So, that the additional stiffness exerted by the box onto the whole system if my system becomes large and large this A prime moves towards point B A if I move towards B. So, if I have to bring it closer to A then I have to keep on increasing the size of the box. So, that is why in a lot you have seen in auditoriums I mean sometime they have boxes as a book as a room it is something I mean height of the room something comparable to the height of the box. So, that is the motivation will do one or two another one is what they call a ported box. We have talked about this ported box earlier in one of the earlier lecture again in context of boxes we have developed. So, I have my original transducer and then I have a port length is L its area is A T what this does is it add following elements to the box. So, it is V naught over rho naught C square A C square this is basically the stiffness of the box and then rho naught L over A T times A square this is the acoustic mass which we had talked about earlier port side act as acoustic masses. So, in this circuit what I am going to do is I am going to add this additional circuit this value the inductance is V naught over rho C square A C square and this is rho naught L over A P times A C square. This is square will be a numerator. It is an LC circuit how I am interested in two parameters one is whatever is the volume value in physical sense sound coming out of the port that is one thing I am interested in the other thing which I am interested in is whatever what is the spectrum of sound as it comes out of the speaker there are two sources which are radiating energy. So, this is the I call it V cap sound is coming out of the speaker that one surface sound is also coming out of the port that is another surface I am interested in both of them and in a very quality sense we will try to see how things change at resonance below and above resonance and so on and so forth. So, the first thing is that this additional circuit has its own there. So, it resonates which is there in blue because the capacitor and inductors are in parallel are in series their overall impedance goes down to 0 when they are parallel their impedance shoots to 0. So, when that happens almost all current goes through this loop. So, at resonance of this LC circuit you will have a peak in terms of d B s field, but it will be in a narrow band around the resonance it is in. So, one at LC resonance all current goes through port I am putting port in parallel this is because I am using mixing current with port. Second think about it physically what happens when frequencies are extremely low when you have very low frequencies meaning below mechanical resonance when the speaker is moving out there will be a vacuum which will get created in the or low pressure condition which will get created in the box and air will come in from the port and it will go in you have a suction condition happening when speaker is moving out. So, the transducer is trying to generate a positive pressure and port is trying to generate just the inverse of it and both these things will almost cancel out each other at low frequencies. So, the output at low frequencies will be negligible point about frequency is important because at higher frequencies the time it will be required the inertia effects will become important enough. So, that the time required for air to come in and compensate for the pressure will not be large enough that high frequencies this will not happen, but at low frequencies this will happen at low frequencies very low output. What this case happen with all all sorts of frequency. All sorts of frequency what is low frequency depends on the system parameters, but it will it is not specific to this particular transducer, but it will happen in any case. Third at mechanical resonance mechanical resonance implying C M and M M when they resonate the overall impedance we had seen was infinite because C M and M M are in parallel. What that means is physically that at that point the piston does not move much even though current is going through there is impedance and will not move much if the impedance is infinite. So, the piston does not move much and all the energy it gets channelized through the port all the energy gets channelized through the port. So, the volume velocity at mechanical resonance of the port is very high. So, you can design the system in such a way that the at resonance the volume at the contribution of port is high at specific frequencies where the transducer itself is not contributing and at other frequencies where port does not do much your transducer plays an important role. So, a lot of so what this system does is its performance is still not as great as that of an infinite baffle, but in reality you cannot have an infinite baffle and if you want to get closer and closer to infinite baffle you need very large boxes which is not practical. So, people have started using in a lot of cases ports even in small sound systems boom boxes which you may have you may see a small tube going right support there is a volume inside and there is support inside that is all physically. The third example will consider is increased mass by mass I physically mean mass of the phone mass of voice well part of the mass of the these springs what happens if I increase them without increasing or changing other engine ring parameters specifically resistance stiffness and all those numbers when you increase your mass the resonance is going to come down. So, that is my original curve when my mass goes up then in theory my curve looks like this what I get here is that at a reduced resonance I get some extra decibels this is about if I increase my mass by if I double it then I get an enhancement of about 3 decibels which is significant my overall broadband response goes down by 6 dB, but my operating width becomes a little larger operating width becomes a little wider. So, this is a good story as well as a bad story good story because my operating width has become a little larger. So, what I can do is that to get to the same sound performance pressure level I can put in more power and I can get more energy more sound energy for a wider band the bad thing is that I have to put in more power this is what I wanted to cover in today's lecture I also wanted to quickly recap on an assignment problem which I had given in one of the earlier assignments. And the question was I think that merits a little bit explanation the question was that if I have a ball of radius r naught this was in assignment 5 and it is pulsating uniformly and it emits an energy such that at radius r 1 the total pressure not pressure the power output is something like this this corresponds to t time period what I am plotting here is power per unit area this value is 3 times r positive number m this value is negative m and one of the questions was that if I have this graph what is the phase difference between pressure output going pressure and velocity what is the phase difference phase angle of the output going to be. So, there were some questions on that particular this particular part of the problem I just want to capture that for starters what is the average power in this case m times 4 pi r square 4 pi r square because this is power per unit area some people miss that we know that power per unit area we had developed this relation is half of real e u star plus half of real e u e 2 j omega t. Now, before I go more into detail I just wanted to just by looking at this picture we can make some inferences that given that there is minus 1 this graph for power is not symmetric if it was perfectly symmetric then the average power will be 0 what that means is that it will not have any resistive element in it this is a combination of some resistive element and it is an inductive or capacitive element in the overall system. And we know that for a point source or a source of finite size as it emit radially into far field there is a real component in its impedance and there is an imaginary component this asymmetric function is a consequence of the presence of real and imaginary components. So, this picture should give us some idea how to figure out what is the phase relation if it was perfectly symmetric then the phase would have been either minus 90 or plus 90 if it was everything was 0 0 or positive then the phase would have been 0 because it is purely resistive. The expectation is to figure out what is the phase difference and finding it in actual number based on the information provided.    So, I 1 over z and then I have e to j omega t 1 over z star is 1 over what is that relation rho naught c minus 1 over j omega r 1 rho naught 1 over z is this entire thing there is no negative sign, but there is a positive sign. So, z star is just basically I put a m. So, I plug this up here I know that power over area is also the other thing is the 3 m 2 m 2 m this is also power over area right some this graph the average power is m which is and the transient component is 2 m cosine 2 pi 2 pi t over t. So, I put all this in this original relation and what I get is m plus 2 m cosine 2 pi t over t equals my first component which is real p u star half.  So, now I am writing the transient component 2 r 1 square now before I expand I just wanted to make a small comment I am assuming when I am developing this relation the transient proportion of I am assuming that z is modulus times e j phi I can assume that.  Say that again p plus times p plus second term does not contain star I am assuming yes. So, as I said I am assuming that p plus is p plus is what I get is. So, I have 2 sides in the equation and now what I will do is I will just equate the steady state part with the steady state part as the first step. So, let us call this equation 1. So, comparing steady state parts I get I am just talking about the first part yes you are right. So, comparing only the steady state parts on both sides I get times 1 over rho naught c.  So, now I use this equivalence put it in transient and then equate the two transient portions sides.


Okay, maybe the first thing I would like to ask is if there's any question on the topics we have discussed together, also general questions over the basis I'm starting from, the biological basis, clear enough or should I spend some more time on discussing those, all clear. What I would like to do now is to start delving in some more details in the framework which I've highlighted in the first part of my lecture. And so the following, I think I will only give two more lectures, so overall what I will discuss is on one hand what's the data and what the data is telling us about our chromosomes are folded and is what I call part one and then the slide. And then in part two of my lectures, I guess tomorrow and then day after, I'll try to enter instead on the use of polymer physics and machine learning to make sense of the data and what could be the physical principle behind the organization that I discussed with you. I will touch a number of topics and the first thing I really want to do is to mention at least some of the main co-workers I have on this and I will discuss approaches done in collaboration with Ana Pombo in Berlin at MDC with whom we have an outstanding collaboration but also Stefan, Max Planck, Jim, Dag, Marieke in Oxford, Josie and Mikael, Colin in Edinburgh, Davideh also in Edinburgh, Hanperhella in Naples and most importantly I really want to stress the driving force behind most of what I'm going to discuss with you which is the team work with me in Naples and in part in Berlin. Okay, so what's the data and what's the picture which is emerging from the analysis of the data? So I told you already that we know that chromosomes are not randomly folded in the nucleus of cells. You recognize the picture, this is the human cell nucleus and those are the different chromosomes and you see they form territories so they form sort of geographical map and they tend to have statistically changing that but say specific pattern so say chromosomes one and three tend to be close to each other five with further apart and so on. As I told you the limitation of microscopy is a resolution limitation so how can we really go to the scale of single genes and single regulators to understand who's contacting. And that has been done in a real major step forward in the field by Jobdecker roughly 10 years ago when he envisaged the high C technology. Anja told me he has already given you a glimpse of what's that about so if I'm getting too boring please stop me otherwise I will try to summarize that for you again. The idea behind high C is also very very nice and very very simple and it's roughly the following we want to measure who's contacting whom genome white and in the high C matter the idea is the following suppose you have two sides the blue and the orange which are in contact in some way with something which holds them together then the idea is simple and is the following suppose you can by using some special enzyme restriction enzymes that's nice you can cut your genome in pieces in fragments so you see those enzymes cut at those specific position genome white and so after the action of those enzymes the genome is is scattered in pieces and if you think about you have two types of pieces you have single fragments those who are not interacting with anybody else and instead you have pairs which are hold together by the fact that they were in contact original and then by some biochemical trick you can ligate the extremities of those fragments and you come up with a circle and the circle includes a portion of DNA which originally was at a given position and another portion the orange one which was at another position and so you sequence those and now it's just a matter of power of sequencing it's just a matter of sequencing because you sequence what you have and when you sequence a fragment like those you change them in something like that I don't want to enter into many days when you one of those fragments you sequence two portions which should be very far away and so you can map where one piece of the fragment is from and where the other is from and you just count how many times the two fragments have been sequenced together and you come up with a map like this this code that I see contact map and the entries here tell you how many times those two fragments the orange and the blue have been sequenced together and so that's a proxy but how frequently the two are together and you can do that genome-wide and in this type of experiments the resolution is only given by the sequencing depth so how much you can sequence your DNA you have and in fact there is another technical thing which is the length and the other technical thing is is how big are the fragments the the average length of a fragment is given by the enzyme's uses and so that's the other limiting thing for the resolution but current developments you can you can go down to say hundreds of bases so very very very short fragments in fact in this technology there is an important bias which is summarized here and it's related to the ligation step so the fact that you want to for the technology to work you have to to bring together the two dangling ends of the fragment you want to sequence so this is the ligation step and the ligation step has a problem if you think about it is it is introducing a strong bias because suppose for instance that you have three segments close one to the other I told you that one gene and average has four regulators but consider the simple case where you only have three things together when you ligate by definition either you ligate one pair or the other pair or the other pair and all the other pre-key combinations you can think of and so when you sequence one of the ligation products by definition you're sequencing only one of them and so although you have three triplets sorry three doublets here because it is one triplet three pairs you only sequence one and so you have a bias of 66% in the estimation of contacts and that's one of the reason why we developed the other technology I mentioned gum but anyway the high C was a real a real major step ahead because it was the first time that we could access with very fine details in a quantitative way the frequencies of interactions of genome segments a genomic scales and so it was the first time we could access a genomic scales who's interacting with whom which genes and which regulators and we could start delving into that absolutely so I don't know if you hear the question the question is is the pattern random or you see non-random patterns and I think if you stare at it you immediately see that there are non-random patterns isn't it so you see that there are blocks blocks within blocks and there are portions which are more strongly interacting than matters oh yeah exactly you're this is precisely type of questions that the community is working on at this moment so I dose you're right let me let me go very slow on this so this is the contact map of an entire chromosome this is in mouth this is human chromosome for no this is mouse chromosome 14 so this is almost the entire chromosome 14 which is 100 megabases roughly and this is telling you uh which portion of it are interacting with itself and I think what you notice is that there are interactions at different length scales you have sort of let me call them more local interactions although on that scale these are distal interactions look at this block say this is 10 mega roughly find that you have a block of interactions this means that within 10 megabases things are are contacting each other in non-random fashion but what I think you notice is that there are all interactions in bigger blocks and if this is 10 this is 20 30 so very distal interactions maybe what you notice is that there are all interactions at much bigger scale at the scale of the entire chromosomes and this is the picture which I think is emerging from the data I'm going to discuss in a second so far from random long-range and this is exciting because you see from a physicist's point of view this is quantitative data it's not just the picture okay the two are contacting but what can we do with that we have frequencies of contact or proxy for with all the biases I mentioned so the gamma technology essentially was an alternative to that to to IC and I told you the idea is to avoid the ligation problem in fact we started earlier than IC it took 10 years though to to make all the steps I discussed before and in a natural I repeat the concept to avoid the ligation step you use the statistical idea so you can't slice this to the nucleus you see who's present in the nucleus by sequencing in the slice by sequencing and by collecting statistics you can come and decide well A and B are in contact or not and the advantage here is that you can do that without the ligation so with no biases with respect to high C and you can do that for very few cells because you see you we work at the single cell level and say in standard approach high C is not single cell so you have a you have to squash a number of cells to have enough DNA to reconstruct the content and you see the patterns as well so the patterns are there independently of the technology you use and now this is what I want to discuss so I'm following up on your question so the this is again the map I showed you same chromosome chromosome 14 roughly 100 mega and I think by staring at it by eye you see the patterns as we already started discussing but an important step ahead was made a few years later the introduction of IC with slightly better data what a group in California and in France discovered that if you zoom along the diagonal notice this is two mega this is 100 so this is two orders of magnitude higher in resolution you're looking within one of those blocks very very high resolution with respect to the previous experiment and what came out is that you see it is organized and sort of diagonal matrix the content matrix and I think you have understood what what that means are blocks of interactions the genome the chromosome is composed of a sequence of of regions where each region has strong interaction with itself and much weaker interactions with the rest of the chromosome because you see if I move along with the chromosome it is this block here which is strongly interacting with itself and then there is another portion which is also strongly interacting with itself but much weaker with neighboring objects and then another and those blocks you see my eye was the scale it's roughly one mega in fact it's half a mega one mega and they've been named very quick name for physicists topologically associated domains pads but anyway that's to mean that along a chromosome you can see a chromosome as a sequence of regions with strong interactions with themselves and very weak interaction with the rest of the chromosome it's in the picture which emerged cartoon how chromosomes fault uh was this in 2012 you have different regions which strongly interacting with each other with themselves and very likely with the others however I think by staring at the data as as you notice you see that it's more complex than that but first let me give you let me express my frustration in the way in which tats are defined and they're defined a very heuristic way so think of I think all the techniques you have studied in the previous lesson in this course and your background as a physicist is enough to go well beyond the state of the heart and the definition of tats they are defined this way at least one of definition now that is a little literature on the topic but say the basic definition of tats is roughly the following suppose you take a site on your chromosome you literally count how many interactions it has on the left and on the right and you make the difference of those two numbers and normalize in some way if you look at that this was named the directionality index di if you plot di along the sequence you see that this is sort of a step function it's very high very positive in one portion then it becomes negative and remain negative then a positive again and and you see and by I by I you see that more or less the blocks of consistent sign correspond to the blocks that you see in the matrix but my frustration is that this is by I in sense that would you call this the passage to a new tad or just a little fluctuations in the data and so that's why there is a literature on the topic how we define exactly that and my impression is that this is going to depend on on the depth of the data so the quality of the data the level of noise etc etc etc but even today practically this is the say heuristic definition of what it had is and of course what came out is that if you have better data then you start discovering structures within the top so it is no longer one mega but it's half a mega and then it's no longer half a mega it's 100k because there are sub-tads and and so on and in fact what we showed is that you have to be careful in this and you have to really take an account of confidence thresholds and and so on but the here the what I would really like to convey to you is that any good idea in this field is good you can improve on this without even me keeping lecturing or just mind I guess but anyway as I told you the impression is that staring at the data that not only you have fundamental blocks of interactions and maybe blocks within blocks but you have also bigger structures I showed you the map of the entire chromosomes and you immediately saw interactions a chromosomal scale so all just magnitude bigger than single times and so we showed that again in a very stupid and heuristic and frustrating way which I try to illustrate just to give you the sense of how this type of the picture we have of our chromosome folders has changed so we run our own experiments and this is again high c data and again in mouse cells and to be precise what we did was an experiments in the modal system where you move from embryonic stem cells to neural precursors and then neurons post-metotic neurons because we wanted to have two types of information on one hand how chromosomes fold and on the other hand how that changes during differentiation is the architecture of a neuron chromosomal architecture of a neuron distinct different from the architecture of a as you asked at the beginning of a nembronics stem cell and so we have a time course three points and we have high c data our own high c data and we have transcription data cage cage data is technology to to produce to have information which gene is being transcribed and I guess you have understood the reason we want to link architectural changes to transcriptional change so what do you see here is a an example of our data this is a by the way this this is a 2015 the original thing it's only three years ago but it's already old in this field the quality of the data this is five mega and I've changed the color scheme rather than using the one I showed you before used by by yob and the workers so just red and white we added shades being the data slightly better we had the shades to give a sense of the scales of interactions and I think here you immediately perceive what what you could already see in the original data that is to say that there are yes blocks of interactions along the diagonal if you want to call them blocks this would be the tads the black numbers the squares with the black numbers but you clearly see the tads do interact one way the other look at one and two you clearly see the day form a higher order structure so you see where I what I have in mind you know what is the higher order going to structure proteins this means that you have a local folding and then folding a bigger scale whereby distal portions come together here is the same you see half a mega blocks come together in a one mega block and they come together in a two me at someone you have I'll tell you better you're right I've not still described that that's why you are confused let me go to that so I was trying to first introduce the the concept and then they'll be into some more so by staring at the data if you define tads the way I showed you before so very heuristically the tads are those blocks but you see by eye that they are interacting this is the point I'm trying to to make you see that one and two are forming a bigger block and four five and six are also forming another block and this block is distinct from that and so on and so a heuristic approach to try to see to identify higher order structure has as he asked is the one that we followed this is a very nice very stupid one and is the following suppose you want to find them most likely candidates forming and higher order structure well you have to look at the pair of tads which share the most interactions and so we start from the list of tads in this cartoon the list of tads and we bring together and we call a higher order domain the pair which is the most strongly interacting pair by definition this is the most likely candidate to be forming an higher order structure and we add that higher order domain we call it a metatad back to the list of the remaining domains and iterate so this is stupid hierarchical clustering really stupid really naive maybe totally wrong but nevertheless the idea is that at each iteration starting from the tads identified as I mentioned before and I don't like how they are identified but whatever tads are we start from them that each iteration we are selecting we are bringing together clustering together the most interacting pair of objects and so by definition that's the most likely candidate and you come up with the sort of a tree as you expect by hierarchical clustering and the tree tells you at each level what's the organization at that scale now this is really naive and the first thing we had to do of course was to check whether the objects defined in this way are really bringing some statistical information so you bring together the most interacting pair but is that interacting above noise or not so of course you can apply that to whatever you want but maybe the objects you are defining the higher order structures are meaningless from a statistical point of view their interactions are comparable to background and so we of course made a number of tests including experiments to show the admitted tads do exist and I want to mention only one for the sake of privacy please absolutely exactly I like this this is the first question comes to the mind of any statistical physicist so is there a scaling variant if you apply Kavanaugh transformations to those matters is how they look I would get to that and it's not that trivial as you can think it's not just a power law let me put it this way it's more complex that's my but anyway let me let me go exactly in that direction step by step at some point we have to stop it's already lemon um very quickly uh one of the tests we we made to uh to show that those objects high order structures are real at least from an architectural point of view as shown here and I try to guide you through this and then we meet again tomorrow so there is the following suppose you have two domains that one and that one whichever they are fundamental tads who were high order meta tads we define their interaction i as littered the number of contacts they share in the high c or gam matrix so you count how many interactions they have overall high of course is interesting it's giving you a measure of how strongly they are interacting but you want to compare that with this certain say in hill hypothesis the background and then we produce a random control model which I want to briefly mention because this is important in the random model is the following you take the high c data and you randomize the matrix but you not fully randomize randomly randomize but you randomize sub diagonal by sub diagonal because you see immediately by either there is a genomic distance effect and so you want you don't want to mix up the trivial genomic distance effects with which you must keep and that you would have also in background system in a random system with real interaction and so if you think about a sub diagonal is the locus of pairs which have the same genomic distance and so we are randomizing only the contacts for all the pairs at that given genomic distance and that's why the randomization bootstrap is done sub diagonal by sub diagonal in this way in the random model we hold the trivial effects just linked to genomic distance you expect that interaction at large genomic distance are weaker than interactions closer to the diagonal you want to keep that this is trivial but then all the other patterns are washed away so that's the control system we have and so i come to intersect and so uh what we measure then is the real interaction i and what we call the control or background interaction that is to say what would be i in the control system i just said and now i'm going to discuss after answering that question but what we find right this is a bootstrap so you resample from the same distribution and so you can produce how many samples you want so i had very good quality a random model so what we find is to cut short a longer story is shown here in this plot you see here the ratio of what is the real interaction by the control interaction so how as a function of the sides of the metatars uh considered and the size of the metatars is counted as the number of fundamental tats included in that metatars and in green you have roughly the expected random background and in blue instead it's the real signal and the expectation we had was well if there are only tats in the system there are no higher order structures this is no scale scaling whatsoever the expectation would be well if tats do not interact with each other or their interaction is comparable to what you expect in a background system the blue signal should rapidly collapse to the green and instead you see that the blue remains statistically above the green at the huge scales comprising hundreds of tats so i told you that a tad is roughly half a mega so 200 tats is an entire chromosome this is showing you that what you noticed by a very same instant i showed the data that there are significant interactions at chromosomal scales and what we found is that this is not only in our mouse model not only in mouse embryonic stem cells not only in mouse precursors neuro precursors not only mouse neurons it is in human cells all the higher mammals where high-speed data are available show that hierarchical organization of chromosomes and that's tested and confirmed by fish experiments so we run experiments by fish microscopy so different technology to show that indeed bigger portion of chromosomes come together at very big distances genomic distances and so this has changed the picture we have of chromosomes which at least the way we think of it is that we have no longer single independent tats but whatever those tats are and i don't like definition tats as you understand they tend to coalesce and form higher order structures exactly as in proteins and this is and there is a hierarchical organization of domains within domains across scales from very tiny scales to very big scales order of magnitude bigger comprising entire chromosomes well i really would stop here so see you tomorrow unless you have questions