 All right So last time I Actually gave a real chalk talk, but this time it's going to be a fake chalk talk Well, I know I guess physicists like chalk Another big difference between physicists and biologists is that physicists and the habit of Acknowledging that I collaborate is first thing our biologists always do at the end of the talk, but I don't really know what's going to happen at the end of the talk. So I Will acknowledge my my friends and collaborators right away. So all the original bits that I will talk about were done together with so current and former students Nick and Kevin and and a terrific group of postdocs that Came through KTP over the years Madhav and It's a it's a was already mentioned here by Ariya. He went on to work with Ariya at Price and And then of course Our experimental collaborators Eric and his student Matt among them was prominently and of course Sebastian Straykhan who is a postdoc at KTP still and Has contributed enormously to it, but all of that Will come in due time. So I'll start with Somewhat philosophical introduction just Set the general agenda trying to understand that how one goes from genes to geometry and So the well understanding geometry is a complicated business. So and Kind of little steps for little people We will start with a simpler kind of a quantitative phenotype if you like It's morphogenetic flow that I'll explain as we get going and After that introduction now basically a launch into into the construction of Mechanical model of epithelial tissue, which hopefully will give us some insight into this morphogenetic flow and again little steps for little people we are going to Make all sorts of crazy assumptions As we built this little theoretical model, but We'll be afraid to go too far away from the experiment. So we'll be kind of going so tacking in and out so we'll take a little crazy theoretical jump and Then try to check experimentally if There is something to it so I think that that's the general philosophy and We'll see where it takes us so the Inspiration for all of this in many ways comes From Darcy Thompson and Darcy Thompson. I'm sure you you know Was this great perhaps the last of 19th century polymaths, you know working at the turn of 20th century and Now he was thinking about his beautiful shapes in nature and was trying to describe them mathematically and In fact, I think Marianne already showed you this Transformation theory of his you know in this particular way deforming fish into one another, but but the amusing thing is he really thought of morphogenesis as as a physics problem and He was way beyond you know before his time according to this timeline of developmental biology right published in nature I Development by developmental biology didn't even start cracking until 1924 with discovery of the Spelman mangled organizer and again area already mentioned it and You know, I really like this timeline because it actually mentions Not quite the physicist but the mathematician Alan Turing in 52 with his famous paper and chemical basis of morphogenesis and amusingly this article says that Well, not everybody thinks that this was a serious contribution to developmental biology But the editors of nature Actually think that it was so therefore it was But anyway clearly We've learned the amazing amount of Biology in this intervening 100 years and a lot of this insight came from studying So very strong. You might say binary phenotypes, right? So amazingly An ectopic expression of a single gene will transform this antenna on Flys head into a leg, right? So you can switch so one appendage into another and You know exactly The gene that's going to act as the toggle switch but the question of just exactly how you Define this very different shape of a leg compared to an antenna is an entirely different Question, right? It's not a binary question. It's a quantitative question and So geometry is a quantitative phenotype. So you need some other Bag of tricks in order to to deal with that so Wouldn't it be fun to revisit Darcy Thompson's agenda, which was really all about How controlled growth Generates different shapes our controlled growth generates form wouldn't be fun to come back to that and Try to pursue it again and Well, I already confessed to being a physicist and You know, you may wonder just exactly what physics has to contribute to morphogenesis and I guess as a physicist they also quite often give talks in physics departments and then your physics friends Want to know actually is there anything that morphogenesis can contribute to physics? well, anyway, I don't I don't know the answers, but I'll kind of keep these questions in mind so and again physicists like simple models And here I was talking about shapes and complicated shapes But we're going to start really easy so You already heard a lot about the Fly development. So there you go the embryo Larva pupa and this adult very complicated fly and We of course want to understand how That shape Ultimately emerges from what I was going on here, right and we're going to start here and that's that shape is just about complicated enough For for us, right? This is sort of a spherical approximation to the fly and again You already have the necessary background here and but it is a bit out of sequence It would have been better if if I was talking out there, but So you heard about the patterns of Gene expression in the embryo there is this Morphogen gradient in fact there are several morphogen gradients that set up If you like the coordinate system provide positional information Which drives the pattern of gene expression and again, we're going to hear a lot more About that today, right? So there is this Pattern of gene expression which setups set excuse me with sets up the body plan of Fly right and When you look at these static pictures that of course is the striking thing that you see There is this pattern of gene expression. Well, you know, I guess if you know how to visualize this pattern That's what you see you see the stripe pattern, but the moment you can make a movie and And We already seen a movie yesterday the one that Eric showed us oops Okay, I'll just play and so here you just have a Different visualization, right? We are looking at the nuclei Florescently labeled his stones in red and now I'm going to play it again and Grab a hold of it and Do it more slowly. Okay So you see individual nuclei? I'm going back and so that's red and yellow is a Reporter for one particular developmental gene even skipped and it first comes up as this one single stripe which then right then additional stripes come in there you have this nice static pattern and right in the meantime and I'm sure Eric is going to explain this tonight Nuclei cell arise and the moment This Solidization occurs and so the proper epithelial tissue forms cells start moving, right? So now You see this stripe pattern in motion actually the first thing that occurs is Formation of a ventral pharaoh and Gastrulation, so that's the topological change Which Lewis Wolpert calls the most important event in your life, right the topological transition gestulation and The next thing that happens is you start seeing You're going to see elongation along this anterior posterior axis neglected to tell you that Also, what are you looking at? other projections of sort of a proper Sort of 3d image and So obtained by a light sheet microscopy where we basically have an embryo and You illuminate it with a sheet of light light and You move the embryo through the light sheet and Your image what scatters? Or rather you image fluorescent light that is emitted and Here you you you you you see the same embryo from the dorsal side and This region is called posterior mid gut and it is It so started at the pole Question I can't really do two things at the same time. Can I? All right So it starts at the pole and it swings over to the back as the whole thing extends anyway Like Eric was saying yesterday We can watch these movies all day But I'm going to press on and what I'm going to do is also Find another way of displaying what's going on. So This is something that Sebastian and it's a Put quite an effort into so the problem with this light sheet microscopy it generates some wonderful 4d images and These are basically files, you know close to a terabyte and Then the question is how do you even begin to process it? So you immediately have to Find ways of extracting information out of those crazy movies and But you Here you can take advantage of the fact that all the interesting stuff is happening on the surface of this ellipsoid. So They basically put together a little software package that unrolls their ellipsoid Onto a square, right? so there So serious distortions in this map because this whole line is one pole, right? and that line is one pole, but on the other hand you get to see both Ventral and the dorsal side at the same time and It's all but easier to understand what's going on. So what's going on? Now I'll play it back So now you're going to see this formation of the ventral furrow All right, imagination going there and the next thing You see posterior mid-gut this Kind of very hydrodynamic looking plume moving it. So that's that's the story and of course Remember in the previous movie you're very clearly so nuclei. So this is all its cellular resolution and one can keep Analyzing this and Actually the simplest thing to do is just construct the flow field, right? So you do this I guess engineers call it optical flow analysis physicists call it the PIV particle image velocimetry you essentially Take an image in one frame and correlate it with an image in the next frame and to optimize The overlap by fitting Sort of by shifting one relative to the other right and that's a local you do it in small patches and that's your local measurement of Velocity so in this way you can reconstruct the velocity field and indeed you see If we look at this stage of the flow we recognize a Very simple structure here so there is The flow is hyperbolic. There is a hyperbolic fixed point on the dorsal side and then of course if you have a hyperbolic fixed point on One side of the sphere. Let's say in the top of the sphere then you also have to have another hyperbolic fixed point somewhere else and that one happens on the belly and of course this map sort of slices the ventral side so that half of this hyperbolic fixed point shows up here another half over there and Then you realize well If you have these two hyperbolic fixed points on the sphere They also have to be some elliptic fixed points and indeed there are these sort of circular circulate teri flow patterns on on the side and Right, so You can get a velocity map at each point of time but then of course you can put it all together and You know track the flow as a function of time effectively following the motion of cells so following trajectories, right you like doing what So following Lagrangian coordinates in Fluid mechanics speak is the same thing as doing cell tracking so we can do all that and Clearly there's a lot going on and one thing we can do is say well, let's look in some region of the flow and Watch closely what the cells are doing so Let's take a look at what the cells are doing in this lateral region and What they're doing right so that that's part of what I described is hyperbolic flow so in biology literature It's known as a convergent extension and why is it convergent extension because if you look at little rectangle here and Look at it again at the later time it's going to contract in One direction and extend in another direction and that if you then zoom in and look at what the cells are doing You will see the following little dance so In order to Deform like this these cells will have to rearrange they'll have to rearrange by changing their neighbors so I hope you can see that these green cells Actually touching each other here, but later on they don't right so the cells rearrange in this manner They exchange neighbors in whether they Intercalate here, right? So this is known as a t1 process, right? And I guess I Will be mentioning t1 processes later. So you will Remember what it looks like Now of course you can zoom in even further and ask what exactly this interface between the cells looks like and Well, it's complicated and what you find in this interface the coherent molecules which You know link the two cells together right so these are proteins or transmembrane proteins with large extracellular domains which in the presence of calcium kind of zipper up and Provide cell cell adhesion now on the Other side on the inside of the cell these same molecules bind through a bunch of intermediate to Actimize in the cytoskeleton, and then of course there are other important molecules doing signaling And other components so of adhesion machinery and so on so And already we see sort of three scales on which you can think about what's going on here. I Studied out by presenting you with this very global picture of the flow Then it's interesting to understand what's happening on Kind of mesoscale the scale of cells and There's important stuff going on inside the cell in the cytoskeleton but But then if you zoom in and so talk about what's going on on the scale of cells right in these different regions Then you still have a challenge of putting it all together right we can Wonder how Let's say the motion of this posterior mid-gut that boom going on to the dorsal side how Does it depend on what was happening in the ventral pharaoh or In this convergent extension of lateral acutor so We really would like to Elevate our mesoscale description of the process on the scale of cells into a global understanding Okay So maybe that's that's a good moment to ask for questions. So this was just philosophy So now we're going to dive into a Little bit more serious stuff So any questions or any philosophical questions, yes, yeah Well, so I think Eric is going to Tell us a few things about the coupling of what's going on on the surface and what's happening in in the yoke right You know, they're definitely sort of active mechanical processes Going on in the yoke especially so early on when So, you know before we get into What I called more for genetic flow here, but There is a sense in which The flow is dominated by what's what's happening in the cytoskeleton and I guess if I haven't made it clear before I Should make it clear now that there's just a monolayer of nuclei, right and the monolayer of cells Right and the cytoskeleton is really sort of a two-dimensional structure, right? You can think of this as sort of trans cellular mechanical network essentially two-dimensional But on the other hand that you don't have to believe any of us, right allergy is complicated, right? So ultimately we will have to Prove that you can understand the flow just in entirely in terms of this two-dimensional picture The cells that remain on the surface get stretched, except the overall volume Certains they are distorted, but it stays the same So over that 40-50 minutes That's there looks like over a longer time That's not going to complicate Early things Yeah, yeah, well, you know later on yeah You noticed As I was showing in the movie the movie then went on but I kind of stopped talking about it so more complicated things are going to start happening and those will be basically Fundamentally three-dimensional Right, but again little steps for all people will be happy to describe what's happening just with a two-dimensional flow in the beginning So I think that's the that's the logic. Well anyway, but our question is great, so We have this flow and it looks very much like hydrodynamics, but it's unusual hydrodynamics, right? It's driven by internal forces not external forces. These forces are generated within the tissue and And what is this tissue anyway? Is it the fluid? Well, it's flowing, but What kind of a fluid is it? How do we describe? So let's try to build a little model, right? So We're going to zoom in on these cells zoom in a little more and Kind of start building a little model and Of course oversimplifying all the way here, right? So here we have these cells. They're approximately polygonal and There is this atomizing cortex Which forms, you know just below apical surface forms kind of a belt So these are Acton filaments So cross-linked Transiently cross-linked by my myosin and myosin of course is a molecular motor. So it runs along these actin ramps and These bundles Cross-linked by cut here so Complicated But of course the reality is even more complicated. I just told you about one pool of myosin this If you like Cortical or junctional myosin, but there's myosin associated with apical surface of Cells and there's also myosin associated with the basal surface of the cells and so there are other pools of myosin and at some point we will have to we'll be forced to remember about this but for now, let's forget about this and just worry about what's happening in In this in this cortical belt belts. So If you accept this picture then All the mechanics is then confined to the surface of of cells. So let me just replace it by this polygonal structure and So you you of course start recognizing this vertex models that You know Frank talked about last week and I talked a little bit about this week and so We keep track of cells here by specifying coordinates of the vertices and We have a notion that there is some tension associated with Edges and they have some pressure of some sort associated with the innards of of the cells and Going to write down this elastic energy particular simple form will just treat these edges as Springs, so this is just a hook-in spring and then we'll say that there is some pressure which Maintains certain area of the cell okay, and We'll say that dynamics is simply relaxation So dynamics just tries to find the mechanical equilibrium of this network And I'm going to right So in this of a simple model, of course tension is just proportional to right It's just hook-in tech attention there just proportional to deformation of these little springs This is simple But it's not simple enough I'm going to make it even simpler I'll try to convince you that We don't have to worry about pressure differential the pressure differentials between cells that approximately The cells approximately have the same tension Inside, excuse me, not tension pressure or at least differentials are small compared to attention and that basically means that mechanical equilibrium just corresponds to Force balance at every vertex and the forces are just determined by tensions and the directions of these edges Right, so it's just that the all the forces all the tensions Impinging on each and every vertex have to add up to zero So that that's what we call a tension net right, so there are tensions in all edges and I just hope that one of you will ask so with all these tensions why doesn't the whole network collapse and That's actually where where pressure matters so one way of thinking about this uniform pressure It's basically Lagrange multiply on the total area so ultimately The total area covered by this vertex model Tension net is fixed So let's think through the consequences of this mechanical equilibrium business. So At each vertex tensions have to add up to zero so How can we represent the fact that three vectors add up to zero? Well, we can make these three vectors and make a little triangle out of them, right and the way I'm going to deal with this is Right this side corresponds to tension in here, but I have rotated it by 90 degrees right and Then when we look at the force balance at the next vertex that has to make its own triangle, but two vertices Two neighboring vertices share a tension that means that these two triangles share an edge So and so I go around the placate here and I keep balancing tensions and adding triangles and I add triangles and I come to this particular situation that I You know added one triangle here and But it's not obvious that this triangle is going to close that and there is a condition for The closure of this triangulation and this condition is Something has special has to oops something special has to happen with this angle here for us to reach mechanical balance of the whole Placate, okay, it's really a very simple condition. It follows the fact. It's just follows from the sign theorem Sign you know sign law sign law applied to all of these triangles so I defined a Couple of angles for every vertex here and write the product of The ratios of these the signs of these angles has to be equal to one So it's a geometric condition a purely geometric condition. We don't need to know anything about the tensions all we need to know is These angles just the geometry of the network tells us Whether it can possibly conceivably be in tension equilibrium Yeah, so we have these one two three Right, we have three vectors Adding to zero so I'm just going to draw them like this, right? so they add to zero and Now I'm just going to rotate them right so I'll draw it like this and like this And like that, so I guess I didn't correctly represent them here, right? So that that's all that has happened. Yeah, don't worry about the center, right? These triangles live in a different space All we know is I didn't have to draw it that way, right? I could have just drawn it drawn it on Different side, right? It's not important Right. We're triangulating here. Not the real space. We're triangulating the tension space And there's some special condition when they actually become related, but We don't need to get into that. Okay. The important thing is We got something that we can immediately start testing so we just go back to some nice images that we can get for different issues for example this these are the cells on So on the ventral side of the embryo As seen just before they started forming this ventral pharaoh And this is some pretty epithelial tissue from A pupil stage of the fly and Some other ones, right? And what we're going to do is Just digitize the whole bloody image measure all these angles Calculate for each cell that particular product of the ratios of signs and And see what the distribution looks like, right? So in fact what I'm plotting here is All right, so I had sky which was a product of The ratio of signs and now I'm going to take a logarithm of it, right? And this was supposed to be one right for every cell so I take a logarithm it's supposed to be zero and We're now looking at the histogram of these guys and Well, it's not just equal to zero, right? Of course, there's noise the image Is not a perfect Polygonal tiling etc etc so we want to look at the distribution of these guys and Compare to what you might expect in a random network or perhaps not completely random network, but at least the So the network with so the same statistics of angles but scrambled right unconstrained local and That's the red stuff here, right and what we see is that The distribution is piled up Near zero more than a random distribution, right? So it is closer. So real tissue is closer to satisfying this geometric constraint then random Right and because you have so many cells even though There's a lot of noise This is obviously very strongly statistically significant, right? This is real signal. So this is some evidence that You know in favor of Actually two things right not only We're assumed that the tissue is in tension balance We also assume that it's close to mechanical equilibrium, right? So we're testing both In one two and just to convince you that this is not some trivial Fact that that's always true There are tissues where which don't seem to obey this Equilibrium constraint for example Epithelium in the imaginal disc that I was talking about Well, maybe there is some statistical significance there, but I wouldn't Fight for it Well, you just want to construct a knoll, right? So you have to come up with some way of In fact, it's at some point She had it was having dinner in the in the restaurant and There was a polygonal tiling basically Stones on on the wall, right? So he took a picture of that, right? So that's sort of a random polygonal tiling, right? So One way of constructing this random polygonal tiling is by scrambling angles and and that that's what we do so But this is actually a very valid question. So, you know, what represents the relevant knoll? so for example, you can take a vertex model and add lots of Pressure differentials into it, right? Just give every cell its own private pressure. So You no longer have tension balance. It is still a polygonal tiling right and Then you can use that Correct. So that's why when we construct the knoll we preserve the same distribution Yeah, how many When we figured out when you know after the talk, I'm sorry, what well Okay, so you have somewhat non trivial question Let me deal with the it's basically the same question. So let's go over it after but you know There are many ways of doing it. This is this is a good question Yeah Well, you know sign can be zero, right? So, you know, it can be pretty large, right? So You know what actually a little later Will go over sort of a better way of comparing distributions, you know, you're right to asking these questions, right? This was a very very very indirect test Right, we would really like to have a more direct test wouldn't be nice to actually measure tensions Right. So the only problem is that we don't have a good way of measuring tensions so so the standard approach is to Right use laser ablation where you basically is zapped one of those interfaces and then it retracts and you try to measure the velocity of retraction and then say that Attention of that zapped interface Must have been proportional to the observed velocity well, so that that's what I call a non not non destructive method right and so then we all dream of So a nice way of measuring tension perhaps by some frets sensor where there is a little calibrated spring and you know to fluorophores and on the tension the Resonant transfer between the two fluorophores is modulated and you would be able to read it out. Well, I Guess we're still dreaming About sensors like this, but it turns out that it's actually Very non trivial both to make and to interpret and fact measure as well. So What can a theorist do while waiting for experimentalists to come Come up with a better way of measuring tension well Can we figure out what the stresses are just by looking at the picture again, right? So That's what we seem to be doing. We just look at pictures and Trying to learn from them. So we can take this picture and then see If we believe in Our little vertex model Can we assign vertex model parameters in such a way that it will reproduce the geometry? Right. So what are these vertex models parameters? Well Tensions and again in general, there'll be some pressures associated with with the cells, right? So Can we look at the picture and try to infer what these parameters would have to be? So first we have to do a little counting argument So we have One tension per edge and one pressure per cell. So that's Number of unknowns Mechanical equilibrium is a force-balance constraint at each vertex, right? So It's a victorial constraint so they're really twice the number of vertices as constraints and In fact, I'm already redoing this little counting calculation that I Did whenever it was a Monday or Tuesday So we want to find out how this number compares to to that number and Remember we went over Oil is famous theorem which relates the number of vertices edges and and cells and the bottom line of that is Oiler tells us that this number is pretty much equal to that number pretty much because we're not quite Worrying about there are also boundary conditions that we have to worry about so There actually seems to be enough information in these mechanical constraints to infer the parameters But we're actually going to do even better. So This is kind of marginal right there's just enough constraints to infer parameters Right, but if we were in addition to assume that all the pressures were constant right, so certainly we don't have to worry about Any of these parameters and then we're really over constrained So it becomes a rather simple fitting exercise. So Let's do that. So let's take some nice picture so for example a frame from of this movie which comes from tamale quiz lab and They're looking at the lateral ectoderm. So that lateral view of the embryo and My zen is Fluorescently labeled so it is in green here and there's also coherent channel here in red, right and We're going to take some snapshots from here and Do a bit of image analysis so well, that's actually where a lot of hard work goes to actually getting a good segmented images But that's a little technical the bottom line is Once you know exactly what geometry you're looking at you can Infer the tensions and now I'm showing your tension map here. That's a heat map Labeling tensions and I guess maybe one thing that I should have drawn your attention to When we were looking at this picture it was the formation of Mise and filaments right so I think Here you can so start seeing these lines of For the straight lines spanning length of No, two three four cells with High my is Yeah, so that's so that's going to be much of what where I'm going So the idea is that the mechanical equilibrium is fast, right so So the tissue is almost in mechanical equilibrium, right, but then this There is internal remodeling going on and this equilibrium is changing so The flow that you see is really an adiabatic deformation of this mechanical equilibrium, right? So now think of it this way, right, so I can just you know drop my hand like this So it's out of equilibrium. It's just falling on the gravity right I can drop it like this it's in Essentially mechanical equilibrium, but I'm doing something there Right, but it looks like it is falling right so and Excellent question. Thank you very much. So that's exactly how we want to think about this flow not as your sort of hydrodynamic flow External force driving viscous fluid but rather An active material Which is remodeling itself from the inside hopefully I'll get there because I'm going to slowly Fine, you know, so I'm telling you some story we did some whatever least square fitting and I have some color map and I'm going to tell you that you know, perhaps you are seeing some Mies and fibers here right the you know high tension fibers But how do we know If this is true or false again, we still don't have a direct way of measuring tension but So what we're going to do is Well, we don't know tension, but we can visualize my son It's not completely ridiculous to think that tension is proportional to my son Right. So what we're going to do is we'll try to correlate Predict the tension to observe level of my son, right and again, what we're going to do is Compare so correlate cell by cell right so for each cell we know what the tensions are and I'm sorry, we don't know what the tensions are We know what our predicted tensions are Okay, and we can correlate predicted tensions with observed my son and There is some correlation Coefficient there's some correlation number For each cell and then we have lots of cells Okay, so what we're going to do is plot a cumulant distribution function for this whole collection of cells so suppose you have some random variable let's call it x and So this is a probability distribution function. So a cumulant distribution function Also known as the CDM is Basically the integral of the probability distribution. So I will start somewhere here. Ah, we have an inflection point and We'll converge to one some of you Probably know this but if you don't know this if I only Teach you one thing Let it be this thing. So how does one compare distributions? So there is a very good way So if this is a real distribution But you only have a Sample some finite sample drawn from that distribution, right? So, you know, you will want to bin it somehow and you'll have some distribution and you want to know Is what you see empirically consistent? With your model with what you expect Right, so how do you compare? Well? So this is the CDF expected in the model you construct The CDF corresponding to the empirical distribution right and that thing is going to go in steps you look at How far these things deviate from each other? So this is Some measure of distance you look for a maximal deviation between these two curves and Come on Gorov and Smyrnov Then give you the p-value which of course depends on how many samples you have Basically, how many points you've measured? Right, and this is actually completely rigorous and it immediately gives you an idea of whether your empirical distribution is Compatible with with your model. No, the issue here is well, you can generate any It's not just about a number it's a number with a meaning right and here the meaningful number is the probability of This sample being generated by that distribution. That's what you want to know right and Come on Gorov Smyrnov gives it to you right so what we care about is not this distance Right is this little formula that I'm not giving you Right, that's what come on Gorov Smyrnov did right anybody can define this distance but It takes come agarov to interpret it as as a probability So anyway No, no, so another thing. Yeah, no, no, so another beautiful aspect of Coma Gorov construction, you don't need to be you just you order your samples on the line and every time you go Sorry more questions Okay, anyway, so what that that's what we're doing here and The bottom line is that again if you scramble So the mice and measurements and Not sorry if you write if you scramble edges within the cell and Construct the distribution of these correlation coefficients, right? that's our control and This is what happens in the data and you see that the median here corresponds to correlation coefficient about point four and you may say that Point four is not such a great correlation coefficient right but the statistical significance on that is Basically astronomical because you have so many examples, so I think Somebody Along was talking about Making noisy measurements in the lots and lots of cells Right, so that's that's the situation. We'll be looking at smallish correlation coefficients With enormous statistical significance, right? And these correlation coefficients are small because the data is so incredibly noisy right, you know, just think of how much image analysis has gone into this and As was pointed out It's not even obvious. Well, it's obvious that it is not Exactly an equilibrium. They're all the dynamical fluctuation, right? There are measurement fluctuations, but there's the honest dynamical fluctuations going on right but Still remarkably there are correlations. So now we're very happy with that and Then we get really ambitious well Tension is proportional to my zen, but you know, of course We know that there are other things contributing for example could hear in right is an Asian molecule So if what he's in is a little bit like wetting, right? It should be contributing negative Tension, so how about that? Well? Great, so in that particular movie we had another channel the could hear in channel so we can redo this correlation analysis With this little model right and Then examine so the statistical significance now. I'm not plotting the whole CDF. We're just looking at statistical significance as a function of this parameter Okay, and what you see is The statistical significance that is right the logarithm of of the p-value is Dipping at you know some particular value of of alpha and The more data you throw at this right and here's the number of samples the more significant the dip is right so Right remarkably even though this correlation coefficient is not terribly High you can if you have a parametric model You can measure the parameter So you can so think of this as super resolution with the Komogorov Smirnov So in fact, this is exactly the same principle super Resolution microscopy is used as a model Right that you understand something about the diffraction right and if you can get a lot of photons from one molecule you can actually average over a Gaussian and and and achieve this greater resolution so So all these tests are indirect, but we're kind of beginning to to to believe that this mechanical inverse is actually getting you something and well once you get ambitious you can push it further you can and Try to Right with these inferred tensions you can reconstruct the coarse-grained mechanical stress Now on the scale of a few cells and what you observe again, just inferred from Cell geometry there is Yeah, so where we're looking at these different components of stress and in particular we'll see that There's a higher tension Right to one side here, which basically means that you expect The flow responding to this large scale tension imbalance To to go this way Well microscopic picture is just that so they they make a zipper But to go from that to negative tension Well, you know just think of this thermodynamically It's like wetting right, so if there is a free energy gain from You know forming these bonds at the interface will want to be long That's all being said so that that we can't right because so right now. This is all parametric, right? This is basically saying do we have support for this model, right? So we'll need to do something else if we want to Demonstrate to you that with that model we can account for the flow It will come good question. So Yeah, so the question is what's what's fixed, right? So You know the number of coherence is fixed or if you like the chemical potential for coherent is fixed I'm sorry we can say with what? Yeah Right, so still the same thing We're separating time scales right, we're basically saying that the On short time scale the tissue is approximately in equilibrium. It's not quite an equilibrium So we can also push that right we can basically Maybe because of the shortage of time I don't want to One can go a little further and and deal with that So now it let me let me press on because I'm way way way too slow Okay, so so far we were just talking about Right these equilibrium Geometries right and I told you that for a given cell array there is a Transulation of tension space, but what if we go in the opposite direction is this mapping unique? Well, it obviously isn't because if we can construct You know one polygonal tiling that You know a base equilibrium condition then we can construct many more just by redrawing it while preserving all the angles, so There are these angle preserving modes which Related to so the discrete conformal transformation right Which preserve tension balance? Tension balance actually does not define a unique Geometry of the cell array which means that there are easy directions for cells to to To deform right so cells can deform without restoring force without disturbing mechanical equilibrium So if so we expect that this should be visible well remember I argued that We can understand what's going on in the approximation where pressure differentials are negligible and We're going to stick to that so we're going to take a look at Again on the in fly embryo on the ventral side Just before the onset of gastrulation. There's something going on here right the cells are Moving in and coming together and Eric is going to tell us a lot more about this And I'm going to Not play the whole movie right but just sort of cycle through a brief moment So early on in this process right just when the cells are coming in and Take a look at what's going on here. You see that the So some of these cells you know they start more or less comparable area than some of these cells Really get compressed seemingly at least the apical surface gets compressed a lot a lot a lot right Well not at all it is changing Right, but Right, but let's just try to understand what's going on. Yeah Don't worry. I mean water is still incompressible. I'm only looking at the apical surface Right, so if the whole cell did that we would have to worry right So Eric and his troops looked at this process rather rather closely and Demonstrated that this Contraction of apical area of cells is Driven by this other pool of my zen that Is not an our model right that's his my zen on the apical surface of the cells and They observed these transient flashes of my zen and they describe this process as as a ratchet so these apical my zen flashes on and The cell crumples a little bit and then it doesn't go back right. That's the ratchet and Then another flash and it gets a little smaller again and we would like to think of this in terms of Our isogonal modes, so we'll just say well suppose there is a transient perturbation of this mechanical equilibrium right as a transient perturbation but Let me just finish the sentence sorry But this mechanical equilibrium state is not unique. There's a whole manifold So if you get off that manifold, you're not guaranteed that you're going to come back the same place So Still a question. Yeah, can you please speak up? Well, eventually the cells are going to go inside right, but we're just looking at the very first moment Right, and we're only looking at the apical surface right and Did I answer a question and so what is on the same plane? Yeah, where we're just looking at the one plane, right? We're looking close to the surface So they still haven't gone too far off the surface, but then of course they are deforming in 3d So our game is going to be the following So the prediction basically is that we should be able to look closely at the deformation during this process and See to what extent it is isogonal to what extent do the cells preserve the angles as as they shrink and And actually it turns out that we can explain So more than 80% of the variance right In terms of these isogonal mouths. Yeah, so the same question got that as before right what is incompressible? right, so The cell is going to change shape to accommodate Change in volume, you know on other hand, of course, you know the water can come out of Of the cell also, right, but so I will not claim that the cell volume is completely conserved right but The cells and you will Yeah, so the cell certainly deforms like the balloon right fine, so There we go, so we have this notion that This activity of this other pool of myosin of apical myosin just transiently Takes us out of this mechanical equilibrium manifold and then puts us back, but in a different place and and There is an assumption That went into this right that the cortical myosin itself did not change along the way Right that we stayed in the same mechanical equilibrium, so we can go back and actually look at What cortical myosin, you know this green myosin is doing while the cells are shrinking Right, so there's apical myosin that comes on and off the Cell is changing shape, but as far as we can see the average myosin concentration on So associated with the junctional cortex is more or less constant in any case It's not varying nearly as much as as the length so I have a Dilemma here I can go a little longer Or I can just Stop here now and Every start tomorrow, what do they do? Five minutes Okay, I have five minutes fine. Let's rescue the five minutes. Okay, so let's I see what Okay, I can stretch it to the top fine so so all that and you know and I still just been talking about the mechanical equilibrium of this network, but of course it can do a lot more so Like we said that there are little springs and they have some length, but but these are really myosin acta myosin filaments and The intrinsic length is itself a dynamical variable, right? So and we know what the Myosins do myosins walk so we expect that as myosin as myosins walk they will drag The filaments with them and the length of the whole filament will shrink right So the only question is as they walk, but they can only walk If they are not carrying too much load, right? And we know this actually from nice measurements on single molecules that Okay, myosin will move with some velocity, but this velocity depends on Mechanical load and There is some stall force at which they will stop and Beyond that stall force. They will start sliding back. They'll just be dragged back by By the external force, right? So So we expect that this is what's going to happen to these filaments these right if this filament is not under external tension it will contract but if there is a lot of tension it will elongate and Then there is a fixed point here. So corresponding to the steady state of that and What's the question? Oh, I'm sorry. Am I J? Myosin density Sorry. Yeah, that's myosin density, right? So And You notice something here that When this right-hand side is equal to zero, right? Tension is proportional to myosin, right? That in fact is our microscopic basis for the intuition that we already relied on that tension in the filament is proportional to the myosin, right and that basically comes from the stall load that myosin can and Now you have this effectively viscous viscoelastic fluid and at this stage I was going to switch into a little tutorial mode and Remind you what the viscoelastic fluid? Yes, but And I have maybe five minutes three minutes And I'm just going to do this. So let's think for a moment about the creepy Sprint So we have a spring it has some length It has some intrinsic length. This is what it wants to be in unstretched state But it happens to be stretched. So what's it gonna do? Sorry dots stand for time derivatives Well, it will Move under the action of a hook-in Force right which is a mismatch between physical length and the intrinsic length Right and this force will act against the friction. Let's say. All right, but then in addition Let's say that The internal length of the spring is not fixed That it is plastic so if you keep it for too long at Given physical length the internal length will try to adjust to the physical length and that we're going to do simply by saying that this is R minus L and One over tau creep here. Yeah This is our dot Okay, our dot is time derivative of R we're describing the dynamics of so the sliding of Of the spring and speaking of force to make things a little bit more interesting. Let's force it externally So now what's going to happen? So suppose the creep is slow. So what will happen is? As a function of time Let's make things a little more interesting. So we'll say that the will apply a Step in force and then the force will Go back to a zero And here I'm going to plot K R minus L, right? That's so defamation of of the spring so we start on the form and We apply the force and The spring will deform as rapidly as Friction lets it and it will try to Approach the new equilibrium the name and then we'll let the force go and It relaxes back and the characteristic time For this to happen is determined by the balance of spring Constant and the friction. So that's basically new over K That's a power elastic in the meantime What's happening to L here if this time is short and the creep is very slow You know not much is going to happen While there is all this elastic action, but if the force stays on for much longer then we'll start seeing creep and And it's very simple what's going on because we're basically going to see in here This balance is that so RL is Going as is approximately f over K and we got the tau here So time derivative of L is equal to the constant So even I can integrate that standing in the flag board So so this is what we're going to see L as a function of time is Going to ramp up and then of course the force is turned off So this goes back to zero L didn't start at zero L started that whatever it was the beginning and here We got this plastic deformation and I think I'm going to stop here. So that's a little little introduction into creepy springs and Next time I'll so start by building a little bit more comprehensive view of So relating this to this good this elasticity