 OK, welcome back. This is the third and last part of my lectures on the biophysics of tissues. So we started the first lecture with the mechanics of networks, vertex models, and compartment boundaries. In the next second lecture, I discussed chemical patterns and signals in particular. Like this? Better? Let's hope so, yeah? Not sure if I can regulate the volume, but OK. And today I will talk about how tissue shape emerges. And I will discuss again the example of the fly wing development. And in particular, I will discuss how many cellular events and processes in this tissue contribute together to change the shape and to deform the tissue. And so here you see the characteristic adult fly wing, which we've been showing already a few times. You've seen it also in other talks. It has this characteristic shape, a characteristic wing pattern. And for comparison, I show you below the shape of the wing of a fly that is mutant in the single gene called dumpy. And this mutation causes an abnormally misinformed wing, a misinformed wing. There are different versions of this dumpy mutant. The knockout of this gene is lethal. And mutations of variance degree give gradually stronger misformations of the wing. So one of the points of this lecture will be to understand what is the difference between two flies and how do they come out to look different in terms of their wing shape. I already showed you that we study the tissue morphogenesis during pupil stages in the living system. One can watch the tissue by opening up the pupil case and directly looking at it through the microscope inside the pupil. And I also already showed you this time-lapse movie, 17 hours of pupil wing development. This is the tissue that will give rise to the wing. The e-cut here in staining reveals as a fluorescent label the outlines of cells. And so you also maybe see roughly that this image, this movie, is stitched together from many individual images. You see these lines which come from stitching. So there are many high-resolution microscope images stitched together to create a large-scale image. And each frame that is generated is separated from the next by about five minutes. Now we go through about 17 hours of development. And we see these dramatic dynamics. Many cellular events, cell division, cell rearrangements, giving rise to cell flows, structures form, the morphology of veins emerges. And you can also see the boundary of the tissue moves. It is reshaped and undergoes shear deformations. And at the end, you see already the morphology of an adult fly wing emerging in this process. So what I'm going to discuss today is some of the aspects of this process. In particular, I'd like to ask how large-scale digit deformations emerge from these many individual cellular events. I'd also like to get some understanding of what is the mechanics that analyze these flows and this tissue remodeling. What are the force balances involved? What drives this? And having answers to these questions, we'd like to get some insight in how do these dumpy mutants have a deformed shape? And let's first, before going on to this, give you a large-scale view, because it's helpful to think about the problem. I showed this already yesterday. So here, we color the region of the tissue which will become the wing hinge and the region of the tissue which will become the wing blade differently. And you see here that there is an overall contraction of the wing hinge, which is an important driver of these deformations, of poles on the wing blade. And this happens in the context of this tissue being inside a cuticle. The cuticle is a material that was secreted earlier by these cells. It solidifies. And the tissue can hold on to the cuticle while it undergoes this remodeling. So therefore, I schematically showed this here as attachments of the cuticle. In fact, they are all around. They do exist. And the tissue is sort of being stretched while being attached to a rigid scaffold. So the simple picture is that the hinge contracts while at the outside the tissue is attached. And so the tissue can stretch itself and therefore pull on it, even though the boundary doesn't move by contracting the hinge area. It's a mechanical event I wish the tissue stretches. There are external forces pulling, but they are generated inside the tissue by contracting it while it holds on to the boundary. And as a result, the tissue, particularly I will focus on the wing blade, is undergoing a shear deformation. And from the point of view of the wing blade, it looks like as if there was a force pulling on it from the outside. Now the fact that there are these attachments and that they are important for this tissue morphogenesis, one can show, and it's revealed in experiments where one uses laser ablation at the tissue margins without sort of affecting the tissue itself but cutting material that connects the tissue with the cuticle. And here you see, that's an experiment, an image taking at 32 hours after preparing information when earlier, at the time point, 22 hours after preparing information, a laser cut was performed along this blue line. And as a result, these connections are lost and the tissue is pulled in. Similarly, cutting on one side, at 22 hours the tissue moves in this side. Here the connections are still there and the tissue does not move in. So one can see evidence for these connections. And we have to take these into account when we want to understand this process. OK, so this is a large scale picture just to keep you sort of this as a background when we discuss now what's going on inside the tissue. Now to understand the process is taking place inside the tissue, we are interested in the cellular events, cell shapes, cell shape changes, cell rearrangements. And to study that between the tissue and the cuticle, it's an extra cellular matrix. And the dumpy protein is there as well. So we use automated image analysis to segment these images and to determine the cell outlines as polygonal contours for all the cells in all the frames. And here you see the segmentation for the initial frame. And this is sort of the time dependence of the segmented network of cells. So this shows we have a database with all the cells, their shapes, and we track them over time. We track divisions, track extrusions, and we can follow cell rearrangements. I mentioned before that the images are of very high resolution. And I can just highlight that by zooming into one of these images. And you see here the fluorescence intensity and the polygons that are determined from the segmentation. And we can, for each cell, choose a unique label to follow cells in time. And these numbers are these cell labels. So each cell has a unique identifier. And if one now goes from one frame to the next, one can track these cells. And we can follow cellular events taking place. For example, many cell divisions. For example, this cell becomes larger here. And it is about to divide in two daughter cells. So between this frame and the next frame, this division has happened. And we can now assign two new cell labels to these two daughter cells. And we can, of course, remember which cell they originated from. So we have a database with all this information and all the polygonal vertex positions and their network connections. Now to do an analysis of deformations that are generated by all these events, we start from this polygonal network measured in this tissue. And we construct a conjugate triangular network. And it will become clear in a moment why that is useful. So we have a set of polygons. We can define cell centers as a geometric centers of each polygon. And now we define, using these cell centers, for each vertex, we can construct a triangle shown here. And this triangular network that also fills space, it contains the same topological and shape information as the original polygonal network, as the conjugate network. For example, you see that for each cell bond of the original polygon, each edge of a polygon becomes an edge of a triangle, but perpendicular to the original polygonal network. Each vertex in the original polygonal network corresponds to a triangle in the new network. And each polygon corresponds to a new vertex in the triangle vector. So in the view of what I discussed in my first lecture about the Euler characteristics, the vertices are exchanged by faces. And we get a corresponding network, which of course has the same number of E's the same. The vertices are turned into faces. And the faces are turned into vertices. And we have a conjugate net as the same Euler characteristic. Now we have two networks, polygons and triangles. I will mainly now discuss the triangular network. But one can use many of the quantities I'm discussing. We can equally well define for polygons. And let's start with a definition of how we want to quantify deformations in these networks. And the definition of a tissue deformation tensor, it's a tensorial object, is motivated by continuum mechanics, where deformations are defined as displacement gradients. So I can define for each cell a displacement field of this vector, which then forms a displacement field in this material. This is a vector. Alpha is a vector index. I'm a plane, so everything's two-dimensional. Alpha equals x or y. But displacement is not a deformation. Because if you just have constant displacement, you just move the tissue, the material around. Displacement happens when distances change. And that happens whenever the displacement varies in space. And that's why a deformation is a displacement gradient. And since the displacement is a vector, the displacement gradient is a tensor. It's a matrix if you represent it. Now in order to define this idea for a mesh work of polygons or triangles, we use a very general approach. If you have a continuum displacement gradient defined and you average this over an arbitrary region in space, arbitrary patch of area, so an integrated over area and divide by the total area to average it, then this can be exactly expressed because this integral over derivative depends only on the boundary in a boundary integral around this patch. And I have now a boundary integral around the boundary line of this patch. DL is a line element. N is the normal vector to the boundary line. And U is now the displacement vector on this boundary. So if I integrate the displacement vector, multiply it with a normal vector, generating a tensorial object around the boundary, I can calculate this average deformation. And this is sort of a coarse-grained deformation tensor corresponding to this patch. And now I can apply this definition to arbitrary patches. I can apply it to a region of the tissue, but I can also apply it to a single polygon or to a single triangle. So if I take, for example, a single triangle and I move these vertices, then I will move this boundary line. And then I can perform this integral around the edges of the triangle using the deformation of the boundary line and defining the deformation tensor for this triangle. If I do that for each triangle individually, of course, in order to get a displacement, I have to take one frame of my movie and I look for the same triangle in the subsequent frame. And as these triangles change, this defines me this object and defines me the tensor describing triangle deformation. If I do that for every triangle in the network, I can also do this for polygons. Here I don't need to distinguish between triangles or general polygons. Then this has the nice property that if I average over triangles in a certain patch of the tissue, this integral becomes exactly the sum over the quantities of the triangles if I weigh it with the areas of triangles. So the quantity calculated for a large patch or even for the whole wing, for the whole tissue, only using deformations of the boundary are exactly given by the sum over properties of deformations of individual triangles. That's a very powerful property of this definition. And this definition has the property that it corresponds exactly to the continuum mechanical concept of displacement gradient tensor. That's important to understand to see what we are doing here. Now, this allows us not to calculate U alpha beta for all triangles and also for all groups of triangles, including the whole tissue in our movies. Now U alpha beta is a matrix, as I mentioned. It has four components. And this matrix can be uniquely decomposed into contributions that have a precise physical meaning. And this is schematically shown here. One can decompose this matrix. This is a general non-symmetric matrix. It's a gradient of a vector. In a matrix that is symmetric and traceless, traceless means that the sum of the diagonal elements is zero. And a symmetric traceless matrix of a deformation gradient is a pure shear deformation. If one only looks at the diagonal elements in their sum, that's the so-called trace of the matrix, this defines area changes, corresponds to area changes, expansion or compression of the area. And finally, one can extract an anti-symmetric part of the matrix. That's sort of what if I take the symmetric part here, what remains is the anti-symmetric part. That is associated with pure rotations. We maybe just use a blackboard to illustrate this a little bit because the notation will be used later again. So when I write U alpha beta, I mean by that a matrix which has two dimensions, four components. Alpha and beta can both be x or y. So x, x, u, x, y, u, y, x, u, y, y. Now first, I can decompose this matrix in a symmetric part and an anti-symmetric part. Now the diagonal elements, symmetric means that this is the same as this. The diagonal is also symmetric. So what I do is I write U, this doesn't change. And I symmetrize it here. U, x, y, plus U, y, x, over 2. U, x, y, plus U, y, x, over 2, U, y, y. And so this is not the same. I have to add something. And what I have to add is purely anti-symmetric. So this is now 0, U, x, y, minus U, y, x, over 2. And U, y, x, minus U, x, y, over 2, 0. And this is minus this. So it's anti-symmetric. And now this I can write as a number. U, x, y, minus U, y, x, over 2 times a purely anti-symmetric matrix, plus 1, minus 1, 0. And this matrix I will always call epsilon, alpha, beta, in my talk. And this symmetric matrix I can also decompose. So I call this matrix, this is the symmetric part of U, alpha, beta, I call it U, s. And this I can decompose U, s, x, x, U, s, x, y. Because of symmetric, these two elements are the same. I give this the same name, U, s, x, y, U, s, y, y. This I can decompose into a traceless part and part only corresponding to the trace. The traceless part has the form U, x, x, symmetric, minus U, sorry, y, y, symmetric. Then I have here U, x, y, x, y. So this is just symmetric. It's a traceless symmetric matrix. And here I have U, y, y, symmetric, minus U, x, x, symmetric, over 2. And I have to add the part coming from the trace. That is U, x, x, symmetric, plus U, y, y. Remember, the sum of the diagonal elements, U, x, x, and U, y, y is the trace. And I multiply this with one unit matrix. And this, in my notation, I will always call delta alpha beta. So that's the name of this matrix. That's the name of this matrix. And now you can see all these things happening. Yes? I think it's fixed. So maybe I have to move it. I have to write it up. So you can't see it, yeah? So I said delta alpha beta is the matrix 1, 1, 0, 0, which I use here. And this matrix here, you don't see it, but this is what I call utilde. This is a traceless symmetric part of U. So this is now divided in, so this is sub-directed, this 2. This is utilde that you see here. Then this part here, the trace, is this term here with a delta alpha beta corresponding to this. And this piece here is exactly this term, which is epsilon times the number, which I call psi. And so here I just perform exactly this decomposition. And I call utilde the traceless symmetric part, which is this one. This turns out to be the divergence of the displacement field because, let's write this out for you, because the displacement field is u alpha. If I take d beta u alpha and take the trace of that, then this is the gamma u gamma by sum over gamma. And this is the divergence of u, the vector u. And this is just uxx plus uyy. Let's give this term. And psi is equal to uxy minus uyx over 2. And that's the angle of rotation that corresponds to this deformation. Now this one can do now for each frame in the movie, for each polygon, for each triangle. And defines now local shear deformations, local compression expansion, local rotations. I like to stress that the tensor utilde alpha beta, the traceless symmetric matrix, defines an anisotropy axis. There's no area change. There's no rotation. There's an axis. And this axis defines the axis along which the tissue, the object, the polygon, the triangle gets stretched. Yes? This equation again? So imagine you have a continuous displacement field. And you can define the continuous displacement gradient. Then you can, this is just a standard integration. If you integrate that derivative, you get the boundary term. Now this is exactly equal to that. And so you can calculate this average over your displacement gradient in the arbitrary area by just integrating the displacement around the boundary. Does this answer the question? Is this the question of whether we can track cells or not? And there are lots of issues which will come up later. Let's not worry about the complications now. This is just the concepts. This is all well-defined. Of course, in practice, there will be some issues which we'll have to discuss. Now we do that for two subsequent frames and get the full deformation field, displacement field. And of course, we know the time between them. So we can translate this into velocity fields and deformation rates. So by defining a u divided by a delta t gives us a velocity. I think I showed the velocity field already yesterday. Now again, we can do the same thing. Because of course, this is completely equivalent. We can decompose the velocity gradient, which is the deformation rate of the deformation by the deformation rate. Again, into the similar contributions, there will be a shear rate tensor, which is traceless symmetric. There will be a divergence of the flow field, which gives me the other tropic part, which tells me about compressibility and expansion or compression and expansion. And now there's an angle which depends on time. It's the rotation rate, the vorticity of a flow. And this is just sort of the symmetric part of the tensor. It's just symmetrized, as I explained. The vorticity is the anti-symmetric part, and so on. OK, so that's a discussion of deformations. And for these deformations and deformation rates, we need to at least have two subsequent movie frames to define and measure them. So they talk about changes from one frame to the next. The next important quantity that I want to introduce, and that's the point where the triangulation becomes relevant, is our state quantities. These are quantities which describe shapes. And they have the property that one can define them for a given shape, for a given movie frame. So for each movie, we can define shapes of all the triangles, and also, similar to of the polygons. And that's a state quantity. It's very important to remember for what comes later. And the idea is the following. For each triangle, I want to find a measure of its shape, which is unique, that if I know this measure, I know exactly what the shape of the triangle is. And I do that by using, formally abstractly, a reference triangle, and ask, how do I have to transform the reference triangle to get my specific shape? And I associate the shape measure this transformation. And the transformation is a matrix, S alpha beta. And the reference triangle is sort of the most unbiased triangle I can come up with. It is an equilateral triangle. And for the moment, I'm only talking about shape, not about size. So let's not worry about how big these triangles are. Let's take triangles which keep their area constant in the transformation. And let me just catch this on the blackboard. So I start from a reference triangle, which is equilateral, and which has these side vectors, which I call R1 and R2. And then there's also a vector R3, which is essentially given by the other two, which is R1 minus R2. And now I have an arbitrary triangle, which is completely any possible triangle I can choose, as now vector R2 prime and R1 prime. And the matrix is defined such that the vector Ri can be written as S matrix. So this is now the coordinate independent notation of vectors and matrices. This is prime, and this is Ri. So for all the three sides, I have a unique matrix which transforms these vectors into those vectors. And of course, if it works for R1 and R2, it trivially also works for R3. And this can be done uniquely only for triangles. That's why I'm using a triangulation. If you have more sides than three, there's not a unique solution to that. It will not work. But for triangles, I can define for each arbitrary triangle exactly one matrix S. There's one small issue here that is this is up to rotations. So if I want to have the same triangle, but not only rotated, I have to rotate my reference triangle to get that. And what it means in the end is that I need two things. I need a quantity, which I call triangle elongation. That's a measure of the shape of the triangle. Only shape, not position, not rotation. But also an angled theta, which tells me how this shape is now oriented in space. And this is done the following way. I can uniquely again decompose this matrix S, which I identify just from the geometry of triangles, in a product of two matrices. A pure rotation matrix. It has nothing else than rotate things. And an exponential of a matrix Q. Now, since both matrices shouldn't change the area, they have to have determinant 1. This is obviously true for rotation matrix, because rotations don't change areas. They have determinant 1. It is only true for the exponent of Q if Q is traceless. Because for the exponent of a matrix is defined by the theta expansion 1 plus Q plus Q squared over 2 and higher orders. And we have the determinant of e to the Q is e to the trace of Q. So if the trace is 0, the determinant is 1. So that's why we have here a traceless symmetric matrix. Similar to what we had in the shear deformation tensor. There's some vague similarity, but it's not the same thing. That's why we have to distinguish it. This is a traceless symmetric tensor. And I can uniquely decompose S in these two things. So for each triangle, I can now calculate Q defining an axis and a stretch that have to apply to an equilateral reference to generate my triangle. And an orientation. That's the orientation of the original reference triangle to which I apply the stretching. That's why a first-year rotation, then the stretching. Now, of course, you can, yes, yes. You can count, if you do that in practice, a simple exercise, you can show that it's unique. It's not hard to see. You count degrees of freedoms as the exact same measures. For the moment, I'm discussing only shape changes, not area changes. Next slide is area changes. So now I have this. Now I can add area changes to it. And of course, since this doesn't change area, I put all the area changes in one factor. Very easy. So A0 is the reference area. And A is the actual area of a triangle. And I just have to add this factor. The areas will be sort of trivial in the whole discussion. So now I have, for my triangulation, I can define this now for every individual triangle. It's a perfectly unique definition. It gives me a field of Q tensors and a field of orientation angles. And I'd like to stick to the triangle representation because everything is exact here. I don't have to, because this works so well here. And I can still go to the cell level by, for example, defining the tensor Q, describing the shape of a cell, by simply averaging the Q's of the triangles that are shared by the cell. There's not exactly, but I think there's a perfectly valid definition of the Q tensor for this polygon. Because for polygons, this is not a unique definition of the sense to do it. So let's use this definition as a definition for the polygons. There's no Voronoi diagram here. It's not Voronoi. It's not very far, very different from Voronoi. It's not the same thing. I don't want to talk about Voronoi diagrams here. So I can now define this quantity for each cell. And it is a measure of how unazotropic the shape of the cell is. I can identify this, it's a trace asymmetric tensor which, as I said, defines an axis and strength of an axis. So I can use a red bar. If it burns long, there's more, another trophy. If it's short, there's less, another trophy. And the axis is defined by the orientation of the bar. Now to show what happens in the tissue in terms of this quantity, I average this Q tensor over small patches of cell. So I'll co-drain it and show you the pattern of these red bars in the tissue during this movie. So here we start. They are all quite small. The cells are almost isotropic. They are oriented in all sorts of directions. There's no order. But as the tissue now undergoes these dramatic rearrangements and it's being put under external stress, there is a pattern emerging of a large-scale coordination of cell elongation. It grows up, becomes very strong, and then it relaxes again, which will be quite remarkable. Let's show this again. So that is how the cell shapes averaged over regions evolved in time and it exhibits this extraordinary large-scale order which is revealed by this analysis. The number of cells changes a lot because they're not something to divide. I think I said that before, yes. But there's no growth. The cells divide but they get smaller. You're talking about the veins? The veins are determined by genetic patterns and the cells in the veins have a slightly different behavior from the cells outside. I will not talk about this here. You're still working on that. But the cells essentially become smaller and the rearrangements in the veins are slightly different than outside. In my talk today, I will just average over the whole thing without worrying about the internal structure so much. And you see also that this elongation pattern is not really perturbed by the veins. The veins go at an angle through it. Okay, now comes another important concept where I would like to spend a little bit of time and go into some of the mathematics because it's so elementary and sort of very general for I think for all tissues. There is now a fundamental relationship between the state variables that I introduced. This defining anisotropy of shape. I call this cell elongation. That's a terminology that is sometimes confusing. It means is a measure of the anisotropy of cell shape. So the fact that the shape has an elongated shape is implied by this tensor. It doesn't imply an actual elongation at which it is deformed. So that we have to distinguish state variables, which is the shape, q, and orientation theta, and deformation variables, which I introduced as u, which describes how I go from one triangle to the next triangle into from one frame to the next. So I can have an initial frame. I know what the shape of the triangles is. I go to the second frame. They have new shapes, the same triangles have new shapes. Now I can ask, if I know what the initial and final state variables are, do I know what deformation is that happened in between? And these are different quantities. So I like to relate now. Given initial and final state properties, which I've measured in my frames, without knowing what the next frame will be, can I determine the change, the deformation from these state variables? So the idea is I have a reference triangle for all states. I use my transformation with q and theta to define the initial triangle. I use new variables q prime and theta prime to define the late triangle. I have difference in q. Delta q is q prime minus q. I have difference in orientation angle of the reference, theta prime minus theta. I have difference in area, a prime minus a, and now what is the relationship between these changes and the actual deformation, delta u, that I defined in my first slide, which can be decomposed in pure shear, in pure rotation, and pure area change. Now the area change is simple. What I have here is just delta a over a. That's easy to show. What is harder to do is to understand what these two things are. And this is just an exercise in calculating the shape, the formation of a triangle. It's not nothing special, but it's surprisingly subtle what happens there. So I show you what the result is if you do this calculation. So I calculate first the traceless symmetric part of this deformation, tensor. And I express this in terms of the initial q matrix and its change. So here's the q matrix, here's its change. Also in terms of the change of the angle theta of the reference triangle orientation. And I parametrize the q tensor as a magnitude q times a matrix that has only orientation. And theta defines the angle of the axis along which q describes an elongated shape. So theta is a property of the tensor q, an angle. q is the magnitude of this entropy, which one can calculate from the trace of q squared. This is the matrix multiplied by itself when I'm taking the diagonal sum. And so this matrix can uniquely be written as this. And now if this angle phi changes, I get the contribution here. Now this epsilon here is my anti-symmetric matrix which I introduce on the blackboard. This is the original tensor. And this is a complicated relationship. It has become simple in certain cases. It becomes simple when there are no rotations involved. So if the initial reference triangle and the final reference triangle are not rotated. And if under this change of q, the axis of q does not rotate. That becomes very simple because then delta theta is zero and delta phi is zero, no rotation. Now this whole part disappears. And in this special case, we see that the shear deformation is exactly given by the change in q. And that's why the introduction of this quantity q is so useful and so important. q is a state variable. And if I only know it's change, I know the actual deformation that happens. So changes of q are actual deformations if there are no rotations involved. If there are rotations, it gets more complicated. And you can also immediately understand why it gets more complicated if there are rotations. Because if you take a triangle and you simply rotate it, there is no deformation. But my q tends to change. Because first it had an elongation and stretched shape along one axis and later has a stretched shape along another axis. So if we have a pure rotation of a triangle, delta u is zero and cannot be the same as delta q. And this term corrects for that. So if we have a pure rotation, delta theta equals delta phi. The q rotates exactly the same way as the reference triangle. The g's here drop out, which are the complicated function of sinus hyperbolic of this magnitude q. They drop out, g minus g, so one remains and this is just a co-rotational term which exactly corrects and delta q takes out the rotation. And this is zero. Now in general, this is more complicated. So if you have a stretch without rotation, it's simple. If you have a pure rotation, it's simple. If we have something in between, it's complicated. The second thing we can discuss is the rotation. So that psi is the actual rotational component of the displacement gradient. And this psi can also be calculated in terms of my quantities. If we have a pure rotation, then we see that psi is just the rotation angle because those two are the same and it's dropped out. If we don't have a pure rotation, that's a complicated case, then delta psi is different from the rotation angle. And now usually I'm trying to talk about not about deformations, but about deformations rates because things evolve in time. And then I have to develop this quantity and I'll not look at infinitesimal changes, you. I will divide it by the delta t that corresponds and I will define a velocity gradient. So a deformation rate. And then this becomes, in fact, the shear rate of the system. v tilde alpha beta is the velocity gradient, symmetric, traceless, it's a pure shear deformation rate. This becomes the time depends of the q tensor, dq dt. And this here becomes a correction term that has all the time derivatives in it. And such a time derivative, such an expression is in general terms called a corrotational time derivative because this is a pure time derivative and this takes out rotations. And in fact, this v, it can be written as a time derivative of q taken in a reference frame that rotates with the system at a particular rate and a particular rate is complicated. So therefore, express v as a time derivative and the capital D denotes a corrotational, a sort of time derivative with a complicated definition, a corrotational time derivative. And one way to write it is shown here. I used it in terms of my continuum variable descriptions. This is now with sort of continuum theory. I have a velocity gradient, which can be written as a corrotational time derivative of the q field in the tissue. This corrotational time derivative is defined as the simple derivative. In fact, since I'm tracking triangles in the continuum theory language, it corresponds to a convected time derivative. And this is a corrotational term. Omega is the actual vorticity in the tissue, which I defined before. And d phi dt has to do with the actual change of orientation angle of the tensor q. Now this looks a bit unintuitive. This is just geometry of triangles and nothing there here. But it's surprisingly subtle and it's important to understand material deformations. So to give you sort of a hint at what this equation actually describes, I show you a simple example, where rather than looking at a triangle, which is difficult to see what's going on, I show you a pattern. You can imagine there's some sort of a very fine-drained triangulation of the plane and some of the triangle bonds are black and some of them are white, so that you see a pattern. And now I let this pattern undergo deformations. And of course I cannot use, and so the equations that we are trying to understand are those here. So the actual vorticity of the deformation gradient can be related to reference changes, rotations of reference triangles, rotations of the elongation tensor, and the triangle shear can be composed in terms of changes of states of triangles. Now I start from a circular structure, where I have a pattern on it that you see its deformation and orientation. That's the Minerva, that's the symbol of the Max Planck society. So you can use the direction which it looks as a reference angle of the triangles. So you have equilateral reference triangles in the circular state and if you now stretch the circle to an ellipse, the triangles will be elongated. This elongation corresponds now to the shape of this ellipse, so it'll outline. Now what I will do is I will apply only pure shear deformations to the system so that there's never in the deformation a rotational component. So delta psi is always zero. And I always have pure shear. First I stretch it with pure shear to generate an ellipse shape. Now given this ellipse ellytal shape, I stretch it in a direction at an angle to the original shape so that the axis of the elongation of the ellipse actually rotates effectively, but only because I apply a new stretching at an angle. And by doing that, I create a sequence of deformations. And what happens in this process, and I'll show you this in a moment, is you create actually a rotation of triangles. So the triangle reference orientation changes, C table change. So first I stretch it. Now I deform it along this axis to stretch it at a slightly different angle. Then I continuously change the angles and while I do that, I never impose any rotation to the system. And as a result, the triangles do rotate. The reference states of the triangles do rotate. And that's just geometry. But it's a non-trivial aspect of combining stretching and rotations. And that's what these equations describe. But it's something which in tissue can happen. Cells could do this and therefore the equations have to describe this. Okay, so that was some mathematics. Now we have built some concepts. We have tools. We can measure lots of very interesting quantities. We understand how they're geometrically related. And now we can use that to analyze what happens in a tissue. And that's what I want to do in the remainder of this talk now. So what the important thing one has to memorize from what I just told you in some detail is just that we measure in each frame quantities, tensorial quantities Q that characterize shapes of cells and that the actual shear of these triangles can be expressed as a change of this Q tensor if it is evaluated in a co-rotating reference frame. If there are no rotations, nothing that happens. You just, it's just the same thing. And therefore we can write it as a co-rotational time derivative. Now all of this is based on single triangles and this becomes true for larger tissue patches if there are no neighbor exchanges. So if the mesh work of this triangulation doesn't change its topology. If the triangles stay all the same all the time. The next thing is to understand what happens to these relations if now we have cellular processes that change the neighborship. Cell divisions, T1 transitions. Before we get there, let me first show you the data for the fly wing. So what I plot here is an average over the full wing blade. Everything is aligned along the proximal distal axis. There are no rotations if average over the whole wing. There are local rotations but not global ones. And therefore I can project everything on the X axis and the only component that really matters is the XX component of the tensor. So I can make single plots rather than plotting tensors which is not so easy. So I plot here QXX, characterizing the shape of cells and you tilt the XX characterizing the deformation of the tissue as a function of time, average for the tissue. And you see the blue curve for UXX implies the tissue is undergoing steady shear extending along the X axis. And the slope of this curve, the blue curve is the shear rate that's VXX. So VXX is DUXXBT, this is slope. So you see there is a shear rate with more or less constant which decreases slightly during these seven in hours. What you also see is that the cell shapes behave differently. The cell shapes start out more or less isotropic and then they start to become elongated, also stretched along this horizontal axis. Their allegation reaches the maximum and then they relax the elongation. Now if this triangular network would not undergo any neighbor exchanges, divisions and so on, these two quantities would have to be the same because of this. So the difference between those two comes from all these cell events. So it's striking that this is not the same as DQDT. What is also very striking that will come to this later is that the slope of the green curve is steeper than the slope of the blue curve. That is very revealing because it means that even if the cells would not rearrange, they would have to have the same slope and usually rearrangements lead to a case where the cells would elongate less than the overall tissue. If you have some relaxation processes. But here the tissue is stretched, so I said roughly as if it was pulled on it from the outside but it's coming from the contraction of the hinge. But while the tissue is being stretched, the cells do even more of that, they stretch more. This cannot come from outside, it comes from inside. So there's a very interesting information in this data. So let's discuss this. What it has so far was that cell shape changes can translate to deformation and if the tissue doesn't keep sort of the neighborship relations almost the same and can directly relate tissue shape changes and tissue deformations in a similar relation as cell shape changes and shape deformation. So we can then write for the sheer rate of the patch of tissue, the rate at which it changes the shape of the patch as DQDT because we just average the equation over many cells. Here DQDT is a co-rotational change of averaged elongation. Now, however, it can also change the shape of a patch of tissue without changing the shape of cells. That's only possible if cells change their neighborships. And of course the key example is the T1 transition. I can start from a patch of cells which has a certain shape that's for simplicity take isotropic cell shapes here, one, two, three, four and now I can go to this patch of cells which has a different shape but the cells didn't change their shape but one and three were not neighbors before our neighbors now two and four are no longer neighbors but they were before. So this is possible because we went through a T1 process one bond shrank, we go through a four-fold junction we rope, we open it. And this process can contribute to a shape change that is independent of the tissue that is independent on the shape change of the cell. And what it means is we can decompose the overall shear rate of the tissue in the part that I explained in some detail that comes from shape changes of cells plus a contribution that comes from rearrangements of the network. And I call this, this is a shear rate it's a symmetric traceless tensor and it is the contribution to shear that comes from cell rearrangements. T1 transitions but all other events that change the topology of network can contribute to that. And so extrusion cell divisions it's a one. So now I can plot this extra term. So we have total shear rate of the tissue which is the blue curve can be decomposed in the rate of change of this Q and the shear rate due to cell rearrangements if I integrate this over time I get this cumulative curves and the blue curve is the green curve plus the red curve. The red curve is a contribution to shear standing from internal mesh work rearrangements. It's not so easy to do when one can cut it off but it's quite brutal. Unfortunately we don't know enough about the signals that actually contract their hinge to do the several things but that's of course interesting to find out not to do. The shapes here, I'll come to that. So the idea is now first if I have two curves I can calculate the red curve just by the fact that they have to obey this sum but in fact one can calculate the red curve independently by looking at the individual changes of the mesh work and that's related I think to your question. In fact we can define contributions to the red curve from the individual types of events. We can decompose this red curve, the rate of shear due to cell rearrangements in the part that comes purely from T1 transitions in the part that comes purely from cell divisions and the part that comes purely from cell extrusions and all of these change the mesh work. In the case of a cell division I add a new bond and sort of this leads to a retriangulation. A cell extrusion leads to a retriangulation and this leads to a retriangulation and how to do that I want to illustrate in the example of T1 transitions. So it is epithelium as it evolves cells do extrude. In the vertex model which I showed in my first lecture these extrusions happen in the sense that triangles can collapse and this happens while you have a growth simulation I counted the number of them per 100 divisions and I think in the tissue they extrude. Now the question is in the vertex model do they extrude for mechanical reasons? And usually, so I think they're both possible. Cells can be targeted, change their properties which they less to the extrusion and then they undergo apoptosis usually when they're outside but they can also just be extruded for mechanical reasons and then have to decide to do apoptosis because they're no longer in the right environment. But here this is just about concepts for the moment, yeah? Yeah, I'm not sure what the purpose is and you formulate as it was a purpose but the model generates it. So the mechanics can drive it and I'm not sure if that's what you mean but it can be mechanical triggering of extrusion because of the stress conditions. Yes? It was calculated independently and now here I explain how it is done. Yeah. The point is this is an exact decomposition so by calculating independently it has to add up exactly and that's what it does. So the way to calculate this following so we start from this configuration we end up with this one that's a T1 transition in between. This is the point when it happens. The T1 transition happens at one instant. It's sort of one moment but the network has to be changed. And before the T1 transition we have these two triangles defining sort of this bond here and after the T1 transition we have these two triangles defining the new bond. Now the important point is that this is a quadrilateral. The shape of this quadrilateral doesn't change. The only thing that changes is this thing is flipped. We flip this bond to that bond. That's what the T1 transition does to the triangulation. It just flips one bond while the quadrilateral does not change. Now before we are here, we have triangle aggregations. We can average those two if you want. You call Q minus just prior to this transition and after that we have Q plus after the transition. Triangles change their shape. The quadrilateral doesn't change the shape but the triangles change their shape. And this shape change of triangles associated with the T1 transition we have to take out of this equation because DQDT counts this abrupt topologically induced triangle shape change in and so we have to subtract it with this term R. So what it means, R has to be defined as the change in triangle elongation which is discontinuous instantaneously at a time point T and when this happens. And if you add this correction term here then we can still express V in terms of DQDT. Except that we had to take care that was this sudden rearrangements which change reference conditions and which meant that I don't have a smooth change of triangle shape. So after this correction and therefore the DQDT is sort of not the right measure for V because it contains abrupt changes. So the formations are smooth but triangles behave discontinuously if there is a topological change and that's what this correction term captures and formally if you do it exactly you get delta function contribution at each moment in time when you have a remeshing and you should now do averages over time intervals which you have to do in experiment like going from one frame to the next you have to smoothen that then this sort of becomes smeared out in time which takes in sort of a coarse-grained version of it but this can be done on a computer simulation this can be done exactly instantaneously. And now by this method one can now decompose this tissue shear in several different contributions which are shown here as colored curves. And I didn't show the extrusions here because in the wing, in the pupil wing they're essentially zero, they're boring to look at but if we sum them all up we get exactly the tissue shear and the tissue shear can be calculated independently just by looking at how the boundary line of the tissue changes with time as I mentioned in the beginning and this is exactly the same thing so we check in the surprising how well this works if you put these curves on top of each other. So here I have the oval tissue shear, the Q which I described before I also have not a pure contribution of only T1 transitions I separate out a contribution coming from cell divisions because I can distinguish these events in my database. I didn't plot extrusions but there's one contribution which I have to explain now which we call a correlation contribution and that's a significant contribution in the pupil fly wing which we found quite astonishing. And as I'll show you in the next slide it is related, a term called D it has to do with coarse graining because here I plot in curves not for individual triangles but for an average over the whole blade. An average of the whole blade creates contributions that come from nonlinearities in the average and in fact the dominant contribution to this correlation term is an average of changes of cell shape correlated with the local rotations and this has a significant contribution now where does this come from? Just to explain on this slide so I showed you that for individual triangle the mathematics gives us an exact precise relation between the shear rate and the rate of change of Q the triangle elongation tensor and make the index n to identify this as individual triangle property. Now we are looking at a larger tissue, a network the average this quantity over the area and with the area weight that I introduced before and then we define the coarse grained shear rate and I told you also at the beginning that this is an exact way to calculate the coarse grained shear rate the definitions of my shear rate is as such that this average creates exactly the correct coarse grained shear rate and this is now exactly the same as an average of DQDT of triangles. Now DQDT as I showed you is a very complicated object it is a time derivative that has to be taken in the co-rotating reference frame and therefore it is intrinsic non-linear it's not a linear operator as usual derivatives are is a non-linear object which means that the time derivative of the average of Q is not the same as the average of the time derivative of Q so in order to express this now in terms of my coarse grained variables Q which I want to use in a coarse grained large scale description there's a correction and this correction is nothing else than the difference between those two the difference between DQDT averaged or DDT of the average and that's only due to correlations due to fluctuations if the system if all the triangles do the same thing in this tissue then these two things are the same it can only be different if they are inhomogeneities if there's toccasticity if different triangles are slightly different things and then if you now express this difference using the definition of this co-rotational derivative you get correlation functions appearing here and in particular what is important in this particular experiment is the correlation between local rotations and Q this comes because it's a co-rotational term the co-rotational term multiplies a rotation rate with the Q itself with this epsilon matrix to create a greater rotation it's a non-linear term and we can calculate this term of course from our data and we need to calculate it to sum up all the contribution to tissue shear so this contributes to tissue shear in this continuum code square in description and we were wondering whether these correlations that contribute here there must be a correlation between variables that has a significant contribution to overall tissue shear we were wondering whether this corresponds to interesting processes that take place in this tissue and I should also say not all tissues do this this is the property of the tissue that you have such correlations and we're still not completely clear about what exactly comes from but by looking for example at patterns of rotation so here I draw little red and blue circles on each triangle where blue means clockwise rotation and red means counter-clockwise rotation and if you zoom into these things you see there comes out in lines it has probably to do with another property you see also in the cell elongations on average there is no rotation but there are local rotations clockwise and counter-clockwise which come in lines and these parallel lines they actually correspond to situations where this correlation exists so we think that these patterns of rotation together with cell shape that occur in the tissue generate effectively such a correlation and this correlation has a consequence for the shear of the tissue expressed in terms of coarse-grained variables okay so that's about many details of how the cellular processes generate tissue deformations in the remaining time I now want to make use of this information to discuss the mechanics of this tissue and I will since we don't have enough information about the details of stresses of force generation in this tissue I will take a very coarse-grained point of view so I will go move away from any of these details and just take averages of a larger regions of the tissue and I will also this average is no longer really distinguished between these subtle distinctions of contributions such as shear due to cell division or shear due to T1 transitions anyways the T1 transitions contribute dominantly to the shear due to cell rearrangements and so I will just sum all these up including the correlation term into the quantity R and for now I'm on look at these three curves but you should keep in mind that this red curve has all these interesting features in it and subtleties and distinctions of different types of cellular processes now let's move to the physics of this problem so what I built so far was just building up tools to understand the geometry all of these deformations rearrangements that I discussed so far where essentially geometry a little bit of topology these are just tools which allow me to define precise quantities and to allow which I can measure and allow me also to understand how they are related now I use this so I start from this information and let me first discuss what happens at early times to give you a rough picture now what happens at early times I already mentioned it a region of tissue is sheared and undergoes an elongation on the long axis in this process cells also change their shape and become more elongated on the long axis that's why both the blue and the green curve go up, have a slope however the green curve goes up most deeply and this corresponds to the red line having a negative slope what this means is that somehow the T1 transitions contribute oppositely to shear than all the other things which is a bit first of the surprising but what it actually means is that the T1 transitions change sort of bonds in a way that has to work against the externally driven shear for example it means that negative slope here means that actually while the tissue elongates along this axis and cells stretch bonds parallel to this axis have a larger probability to actually shrink than perpendicular and as these bonds shrink they can drive T1 transitions after they open up perpendicularly which now brings the cell 2 and 4 which were apart closer together and you see in this effect if both the tissue stretches and cells are pulled together by an active contraction as required here to generate this T1 transition which corresponds to this negative slope then the cells are stretched even more strongly than the tissue you see this this cell stretches much more than a patch and that's what this red curve describes so you see from this argument because I have to invoke forces somehow to make this happen that these T1 transitions have to work they have to perform mechanical work and they have to have some energy supply this is an active process this is not a passive process so at the beginning we see the tissue is stretched not really from the outside but with the help of hinge contraction but for some reason the cells are driven to stretch more by active T1 transitions this happens inside the tissue the tissue drives this so cells are stretched more than a tissue and this happens up to this enormous cell shape change has been built up more than the tissue shape change and now we come to later times this is the second part and here now thinks a reverse sign you see now the slope of the red curve has changed sign the T1's have changed the orientation and at the same time the cell shapes to relax what this means is again shown here now the T1 doesn't go the other way now the bonds contract perpendicularly and open here and therefore cells while the whole tissue the tissue patch elongates even more there's still a continuous shear going on the cells can now relax their shape because the cells that are all together here are now can move apart and relax a little bit of this stress and that's why the elongation can go down here while the tissue still continues to shear further that works together with this positive slope now this process can be passive in principle it can be active but it can be passive because here now the T1 transitions follow sort of the stresses that are in the tissue they don't work against these stresses in the tissue they follow the stresses in the tissue so now with this picture that sort of gives you to help understand what's going on we can now build a model a physical model of this process and we have to take into account now the stresses in the tissue and the force balance is in the tissue and the basic idea is to consider cells themselves as elastic bodies that's the easiest starting point so I define the local stress and this is only the shear stress the tildes in my case are always the traceless symmetric parts of the tensor the shear stress is proportional to the tensor Q which defines the cell shape that's just an elastic description if you go from an isotropic to an elongated shape you need to deform the object and there is an elastic response that's the elastic property now in general and because these are active systems there can be also an extra contribution which we call an active stress so if cells generate internal stresses they can deform spontaneously that would be an interpretation of this that stresses generate even if Q is zero alternatively one can also think of the idea that a cell has an intrinsic tendency to be elongated and the stress is not zero when Q is zero and this is sort of what counts for it so there's an active and a passive version of the same equation but since cells are always active systems I will always consider this to be an active term now this is in tensor which is traceless symmetric so it defines an axis so one needs some sort of cell another tropy to have this but of course the clear candidates are the PCP systems which I introduced yesterday so cells are another tropic and so they can do this so this is my equations describing stress two dimensions then of course I will have to use my geometry which I introduced to you in some length I can define the shear rate of this tissue using the fact that there's a contribution from changes in cell shape which then corresponds also to changes in stress and shear due to cell rearrangements now in principle this I could do on a polygonal or a triangular model in a vertex model but for simplicity I will just take a coarse-grained point of view and make a continuum model out of it because we can solve it more easily and also since I missed the continuum theories in my first lecture I can now bring one here in the third lecture so from now on I will think of these as continuum fields in the plane these are also continuum variables in the plane and then the only thing that is really difficult to do correctly because it's completely not understood and completely unknown is to understand what equation to write for R because we have we need to have a constitutive material equation that tells us how R behaves given the state of the tissue in order to have equations that have a unique solution and that's an equation for R now in general this is difficult to do difficult to guess what it might be R is a very complicated term because it involves the shear due to cell rearrangements we don't really know when and how cell rearrangements actually happen probably stress plays a role in driving them but active poses as well so let's start in a simple-minded way let's not worry about non-linearities let's think in terms of a linear theory but since cells are complex objects things are slow things take time I will allow for memory effects so it's not that things just are instantaneous cells can respond with some time because the internal processes it takes some time to respond and there's one unique way to write a linear theory of memory and that's written here what it involves is so that the state is characterized by q because it has a memory it depends on a history of q and because linear I can just superimpose it and this defines a so-called memory kernel that's a function chi that somehow captures all the complicated dynamics that goes in the cell and that determines how the history of what the cell experience in terms of stress or shape changes now allows the continuous period I shouldn't say not a cell but a patch of tissue or patch of tissue with the history of q and the complex process that takes place in the tissue will give rise to a remodeling and one goal is to understand what this function chi looks like can we measure it but writing this down also I will also add the possibility that t1 transitions are actively driven of course I saw in the data that it must be the case so in addition to this part I add a term which similar to this term here comes from active processes requires some local chemical energy entropy like a pcp system to exist will consume ATP to happen and I can also add it to this now this defines me a very simple model which I can solve as a continuum model to discuss how this tissue remodels the only thing that is missing to now solve this to solve this is a force balance equation now in continuum mechanics force balance is equivalent to saying that the stress tensor is divergence free so the stress tensor is has to do with momentum conservation and momentum conservation involves divergence free stress now the stress tensor has the shear part sigma tilde it also has the isotropic part which in the case of stress in two dimensions in three dimensions would be a pressure in two dimensions sort of an area type of pressure or some area tension I still call it p that's the isotropic part of the stress tensor that is the traceless part of the stress tensor in this system stress has no anti-symmetric terms and would be a separate lecture to discuss possible anti-symmetric parts of the stress but which I will not use here now what can we say about so you're using this model now what can we say about this big unknown namely this memory kernel to learn something about it what we what we what we're doing is we make a plot of r versus q taking our data and then trying to analyze what what is chi whether chi exists as it describes this r and q we have measured we know they can just for example plot r x x versus q x x for the whole blade each data point corresponds to the blade at a different time and times are color coded so we are early here and late here another question is can this equation account for this data and it can and it can by the black curve is an example so just fitting this equation to this data and we get this black curve by using a very simple memory kernel which is shown here so the usually the memory kernels can be can be expressed as a superposition of many relaxation processes in the simplest case you can think of a model which has many relaxation processes that are exponential and then you have to superimpose them this defines this memory kernel here we only need the single exponential to account for this data which means there is a some relaxation process that dominates this property to generate shear due to cell rearrangements there is a single relaxation time tau d characterizing the the relaxation the pre-factor of this kernel you can look at the units here has a square of time so there are two time scales here I can identify the tau d again and the tau is a second time scale that is the classical Maxwell time so if there was no memory if the cell rearrangement would be instantaneous given a change in stress this would generate what is called in in in material science a Maxwell model a viscoelastic material and this is the Maxwell relaxation time now this material is much more complex doesn't look like a Maxwell model it's a generalization of it with more relaxation features it has a second time scale and a second relaxation process which is characterized by this memory kernel so we can measure all of this in the data so we find that this relaxation time of this memory effect is four hours this Maxwell relaxation time is two hours we also in this fit get this constant or this number lambda which is in fact reflecting that T1 transitions must be partly active at least in the early stages and the fact that the sign is negative corresponds to that this red curve I showed you had a negative slope so this must be an active term driving this opposite direction but because of the units of this equation this lambda has units of time so we have three timescales here in this single analysis now what does this mean we can turn this knowledge into an understanding of the rheology of the material rheology is the science where you try to understand how a material responds when you expose it to stresses or how the stress changes if you sheer it that's rheology and simple example would be an elastic system which has an elastic stress if you deform it but such materials are much more complex that they don't just respond elasticly they also show flow like behavior and they have memory effects and that's then characterized by rheology now to understand the rheology of this material we can use this equation for the sheer rate of the material combine it with this equation for the sheer due to cell rearrangements eliminate from the discussion R and only have a relationship between Q and V and the relationship between Q and V is essentially rheology because Q is directly a proxy of the stress because I use an elastic model for cells so if the cell is elongated there's elastic stress there so Q is almost synonymous for stress and V is the sheer rate so I can calculate V as a function of Q that's the rheology now let's look at this model which has quite surprising properties I look at the situation where I impose a step change in sheer rate so I start from the system with sheer rate zero and at a particular time I suddenly begin to sheer it at a constant rate and that's this blue slope here so I have zero sheer rate finite sheer rate and now I can calculate what is the response the stress response of the system or in my language here for today it will be the cell elongation response of the system and since it's linear it's very straightforward to calculate this there is an exponential response to a step change there is a characteristic number S in the exponent if you do the calculation this S can become complex so it has a real part and has an imaginary part it becomes complex exactly when this time tau d is bigger than the time tau which it was in the experiment it was four hours it was two hours otherwise it would be real it would be simply relaxation but because this is a complex number with an imaginary part this exponential has not just an relaxation part but it has also an oscillating part and therefore you get a damped oscillation so the response of this tissue to a step change in sheer is a damped oscillation response in this rheological model and of course this is reminiscent to what we see in the tissue we have a sheer which starts here and we get a damped oscillation at least we see one period of the oscillation in the data however an important difference between my very simple argument here and the tissue is that here the slopes are the same at the beginning because at early times there's a purely elastic response in my model and they are parallel and then when the salary arrangement is set in with some delay then we have a difference of two curves however I did not put in here the active T1 transitions so if I now add the active T1 decisions to make them negative then I do indeed generate this negative term and then get qualitatively the type of behavior I see in the actual tissue even though this is a very crude situation where just with a step change of sheer rate so what we see here is that this is we have now material properties of this tissue we have a model for its continuum mechanics it has a very complex rheology it has a surprising rheology which I've never seen in any soft meta system that if you put a change in sheer rate it has an oscillatory response this I think requires an active system you cannot get this in a passive material and that's what the wing does now in order to better understand and what happens in solving this equation putting this number this is a simple exercise you can do it yourself it's very easy it's just a linear model so in order to discuss now the real tissue and in order to get some sort of quantitative comparison we have to solve this continuum mechanical model in 2D and we have to impose boundary conditions boundary conditions are completely totally important for this we cannot discuss this without boundary conditions so what I want to do the remaining couple of minutes and that's what we did in order to analyze these experiments is use a very coarse-grained simplified version of this model to be able to compare to experiments we are lacking a lot of information of patterns spatial patterns of stress generation of these these active terms that I showed you in the tissue so we know that a hinge on a blade behave completely differently but we don't really know what are the differences within one of these tissues so let's just average of the blade and over the hinge separately give the hinge and the blade different material properties and we want to find out what material properties these are we connect everything to an outer frame which is rigid that's the cuticle with some elastic linkers and there can also be some friction when the tissue moves relative to the cuticle and then we see whether such a model by solving this these our equations now for a homogeneous system for a rectangular shape for the hinge in the blade whether we can account for the general features of basic features of this wing and the answers we can we can use that also to determine parameters for the blade and for the hinge so the hinge we have next I didn't discuss here didn't have the time there's of course also stress equation described in the pressure in the tissue and then we have the shear we have the force balances and to putting everything in the hinge there's an active contraction that we need to make it shrink in that sense the contractile part of the material has different properties in hinge and blade and the other material properties can also be slightly different and here you see now comparison of the measured curves cell elongation, tissue shear and shear due to cell rearrangements and together with the solutions of our equations for parameters that have been chosen that this matters well and we think this gives a very clear picture of the basic force balances the basic process to take place so here just overlay our rectangle model to the video of the actual wing the rectangle model describes shape changes of these rectangles which is essentially captures the overall shear deformation of the whole tissue it captures the shape changes of cells here symbolized by a deformed hexagon it accounts for the correct shear due to cell rearrangements and cell flows and sort of shows how the system can by an interplay of boundary conditions hinge contraction and activity one transitions undergo this morphogenetic change now my time is essentially up but I still want to briefly talk about the dumpy mutant which was the whole teaser for the for the whole talk because now we have sort of an approach to study these shape changes we have a basic idea of how this operates in the wild type wing and we can of course look at the same approach at mutants and the dumping mutant is particularly exciting so let's look at this so I mentioned as a mutant with a single gene mutation has a perturbation in the fly wing we can watch this fly in the pupa here you see the wild type tissue in the pupa at early times and here you see the mutant tissue at early times and you cannot really distinguish the two even though at the end they're very different so up to the pupil stage this dumpy mutant has not had did not have really dramatic problems the problems arise now exactly the time when we can watch it that's why it's so nice and we can see what happens to this wing in the same 17 hours when we look at the wild type and it behaves very differently so the same system with just one gene mutation has a completely different dynamics and the most striking difference is that the wing margin rapidly moves in and this hints strongly at a change in boundary conditions so this is a mechanical process active forces generate inside contractions everything attached to a boundary and if these boundary attachments are sort of altered the whole process is dramatically changed and that's what the dumpy mutant does it controls boundary conditions so dumpy is a protein that is part of the extracellular matrix that the tissue secretes and that is the material that was asked before that is also linking the tissue to the cuticle and probably dumpy provides such links maybe between the tissue and the extracellular matrix we don't know exactly what it links but it is a key structural element that connects the tissue at a margin to the extracellular matrix if one looks at in YFP distribution one finds it along the whole margin one finds it also on the surface of the tissue so it also contributes to friction between the tissue and the cuticle and the plane which you also need to understand these mechanics as a parameter which is the friction coefficient and if you look at dumpy mutant you see that somehow it can detach from the margin sort of reminiscent of what happened when we did the laser ablation and from that we proposed that what dumpy mutant actually does or the effect it has is if it's mutated it changes the connections the mechanical connections of the boundary and therefore it changes the mechanical boundary conditions what is a very complex active process that I described so to apply our approach the idea then is to say we would expect that the material properties of the tissue of the cells themselves aren't changed dumpy some are outside so we keep the same parameters that we determined for the wild type for all the tissue parameters but we allow now all the parameters that describe the effects of linkers to the outside to change the stiffness of these springs and also the friction coefficients when the system slides and essentially we change that in the blade region but we also change values in the hint region and this is enough to sort of quite nicely account for the dumpy data and we can do a sort of the same comparison experiment theory that I did for the wild type okay maybe sort of to finish I guess my time is up but do we have still some time probably not so since we want to go for lunch let's maybe have one more slide because the whole thing which I just explained to you about shear and shear deformations one can of course also do for the isotropic part of the deformation and this is much easier I didn't show it because more simple but it's also interesting to see and that is sort of the tissue area dynamics so we can also and it's exactly the same idea it's just easier to do than for shear we can decompose area changes you can do that either for single cells or for patches of tissue he has written for a patch of tissue the rate of change of tissue area can be decomposed in the contribution that comes simply from the fact that cells grow a shrink in the area but their corrections coming from cell divisions and cell extrusions and so we can do the same type of decomposition of areas as we do for shear and here we show what happened in the wild type the blue curve is the area change you see here the wing blade almost doesn't change the area even though there are lots of cell divisions cells become smaller the area it oscillates a little bit this is in fact this is the same oscillation that I showed you before which comes from this complex rheology so there are stresses and there are pressure changes and it changes a little bit the area but there's almost no area change then there are contributions this blue curve can be decomposed and free contributions cell divisions cell area change and extrusions and so if there are no extrusions and if there's if divisions perfectly subdivide the tissue then each division halves the size the area of cells and then this orange curve and this green curve exactly cancel so this comes from this is negative because cells get smaller this is positive because cells divide so if you have division and cells keep the area then the area would grow according to an orange curve if cells get smaller at the same time this compensates and extrusions also tend to make it smaller and here you see the dumpy there are many more extrusions in dumpy and dumpy the area actually shrinks and that has to do with the fact that extrusions are stress dependent as a subject which I cannot go into today because time is up and I'd like to finish here I'd like to thank you for your attention and for your patience being a bit over time today