 So good afternoon. Today and in the next days, I'm going to give free lectures on various aspects of the dynamics and organization of tissues during development. So the theme is biophysics of tissues. I'm working at the Max Planck Institute for the Physics of Complex Systems in Dresden in Germany. That's the institute you can see here on the right. It's an institute of theoretical physics that covers a very broad range of areas in theoretical physics. And for about 15 years, we have been developing together with a second Max Planck Institute in Dresden the MPI of Molecular Cell Biology and Genetics, a joint research program at the interface between physics and biology. And now also, we're bringing computer science. We are currently building a new center for systems biology to develop quantitative approaches to biology and to theoretical biology. Many of the subjects I'm going to present in these lectures have emerged from collaborations in this context and this research program. There were many people involved. And I'd just like to acknowledge right at the beginning, many of the people that have contributed to some of the subjects I'm going to present. I'd like to highlight in particular collaborations with the group of Suzanne Eaton, also with Christian Daman, who is now at TU Dresden. He used to be also at the Max Planck Institute. Also collaborations with colleagues at the Institute Curie in Paris, Jacques Four and Jean-François Journie, and a longstanding collaboration with Marcos Gonzales Gaetan. It started when Marcos had a group in Dresden and he's now a number of years at the University of Geneva. So the organization of these lectures is outlined here. So roughly, this will be the three lectures. I'm not sure exactly how the timing will work. So I'll start today with models for the physics of tissues and materials and the dynamics. And we'll start with cell-based vertex models and move up to continuum theories of tissue dynamics if time permits. And I will illustrate a number of biological processes that can be highlighted and that I understood using such biophysics approaches. In the second lecture, I will move to signaling processes and regulation. I will focus on planar cell polarity. That is sort of an end-as-a-tropic organization in the tissue that gives each cell an orientation and that can regulate and guide morphogenetic processes. And I will discuss gradients of growth factors that regulate growth. And in the last lecture, I will discuss how these systems sort of can give rise to shapes and how in the context of the fly-wing tissue is shaped. So let's get started with basics of tissue biophysics. And most of the systems I'm going to discuss in these lectures are epithelial tissues. So these are sort of sheet-like organizations of cells. And I just sketch here the general idea of epithelia as schemes. So simple epithelia, which are sort of at the center of my talks, are simple singular cell layers. And they can come in different geometries that distinguish as squamous when they're sort of flat, cuboidal, intermediate, and columnar, if these are sort of long cells packed together. And many of the tissues I'm going to show you are actually columnar epithelia. Epithelia can also come in multi-layers and stratified, pseudo-stratified, and sets sort of the general scheme of the type of systems we're going to discuss. Now I will focus in my lectures on many aspects of tissues, but they will all be discussed in the same system as a model system. So I will discuss the fruit fly, drozophila, as a biological model system for experiments and discuss biophysics in the context of fly development. So the basic question that we're interested in is to understand how many cells in such a tissue can organize collectively in space and time, create structures, morphologies, patterns, and thereby, organs and organisms. And the model is the fly. Here just to remind you of how a fly is made, first a fertilized egg with rice to an embryo, which hatches from the egg, becomes a larvae, which undergoes several stages while it grows and feeds itself. Inside the larvae are later-side tissues that, while the larvae is feeding, grow inside the larvae, are patterned inside the larvae. The larvae then at some point becomes a pupa, so it puts itself in some protective shell, pupil case, and inside there is now a dramatic remodeling. Many of the larval cells die, and the tissues that were growing inside are undergoing morphogenetic remodeling and assembly and emerges from this pupa and adult fly. So here is a scheme of that setting. So we have larvae here and adult fly here, and the tissues that will give rise to the adult fly are here shown, they're organized within the larvae and are called imaginal disks, and these different colors correspond to these different organs here. Now what will be discussed during my lectures is in particular the fly wing. That's a very important model system for morphogenesis. The fly wing is extremely well-studied. It is a very simple tissue in the sense that it is a two-dimensional structure. It's easy to observe. Also it may have many advantages for theory to discuss the fly wing. And second tissue which will come up, I think in the second lecture, is the fly eye, which is a completely different adult structure, but it also starts from an imaginal disk that at the beginning has many similarities to also the wing imaginal disk. So here you see pictures of these imaginal disks taken out of a fly larvae. This is a wing imaginal disk. This is an eye imaginal disk. Already you see the sort of structures forming that will become the omatidia of the fly eye. Here you see the adult organ and the adult wing. One important structural element of these imaginal disks is that from the very beginning they are organized in different cell compartments. The first I should say that what is such an imaginal disk that you see here is essentially a single cell layer of the topology of a sphere, which has been flattened. So if you look at the structure, you see essentially two layers of cells and we're mainly focusing on one of them. That is the layer from which the wing actually forms in the end. The second layer below is less important. But this whole tissue is from the very beginning organized in different compartments with cells that can be distinguished by their genetic markers and they are separated by sharp interfaces called compartment boundaries. And cells within these compartments are lineages that don't mix. So the compartment boundary is a boundary across which these cell lineages don't mix. And there are two important compartment boundaries sort of sketched in the scheme and you may have imagined that they exist in this vaginal disk as a so-called dose of ventral compartment boundary. So dose of ventral essentially means the front and the back of the organism. And there's an posterior-posterior compartment boundary. Anterior-posterior corresponds to front and head and tail of the organism. So this little region here, this is called wing pouch will become the fly wing blade. Sort of this region here and that crosses this tissue, there are two compartment boundaries perpendicular, approximately perpendicular to each other. In particular, the anterior-posterior compartment boundary also sets up signaling systems. It serves as an organizer for signaling molecules that will help pattern this growing tissue and to help it to become a fly wing. So this is chemically shown here. So we have a tissue with anterior and posterior cells that can be distinguished. And this distinction is still visible at the adult level. So here you see from Christian Dammann's lab a fly in which a gene is expressed with a fluorescent label, GFP, that is only expressed in the posterior compartment. And you can see this sharp compartment boundary even at the adult fly. And this compartment boundary exists throughout the whole development process. And as I mentioned, it is used to set up signaling systems that we'll discuss in the next lecture. I'll discuss, I'll show you tomorrow the roughness of the line. We'll discuss it then. It's not, yeah, it is quite straight. I have to look it up. I can tell you later. I didn't write it on a slide, so I forgot. Now this imaginal disc grows inside the larvae. It sets, starts out with about 50 cells that are laid aside to become this imaginal disc later the wing. And these 50 cells divide subsequently. There are about 10 rounds of cell division. And because each 10 rounds of cell division, each division gives twice as many cells. We have two to the 10 is about 1,000. We go from 50 to 50,000 cells within five days. And here you see images of wing imaginal discs taking at different times. After this growth process, the tissue has sort of a larger shape. There's still dramatic remodeling going on, but the main growth phase happens during this stage in the larvae. And the larvae has to eat a lot to make all this happen. There's also death. So this is not, yeah, there's also death, but still all of magnitude-wise is still correct. So this is now in movie of a wing imaginal disc that was taken out of a larvae, put in a culture medium, and then observed under the microscope. And here a caterin, GFP, or caterin staining is used to make visible the outlines of cells. So this is not a pouch region, which corresponds roughly to this region here, shown in red, which will become the wing blade of the adult fly. And now a time lapse movie about 14 hours now shows these cell divisions. So you see here, whenever cells become divided, round up, become spherical, and you see them as circular, dark spots. And within 13 hours, so usually the cells divide every eight hours or so doing this growth process. In the culture, they're not totally happy to grow a little bit slower. So roughly each cell divides a bit more than once during this movie, sort of. One and a half rounds of generations as compared to the 10 generations of the full growth process. After the wing imaginal disc has grown for five days and about 10 rounds of cell divisions, the fly moves to the pupil stage and then there are still some dramatic remodeling processes going on. And one can also observe this remodeling of the tissue in the pupa, in vivo. What one does then is one takes the pupa and can cut it open, can cut open the pupil case and then directly observe the tissue under the microscope. This is the advantage that is in vivo, so there's not a medium which is not perfect for the tissue. So this is sort of the normal condition. However, this can be done not at a very early stage because one needs these conditions where opening the pupil case doesn't sort of kill the fly. And if one does it too early, sort of this happens and after about 14, 15 hours after preparing formation, this works very well. And I show you now what this looks like. So that was between sort of the movie I showed you before of the wing imaginal disc. And this process now, there were some morphogenetic events that are not so easy to observe in vivo, but now we have a wing tissue 16 hours after the pupa is formed and we can watch the subsequent 17 hours. Again, using a cation label and you see the many cells. These are now many more cells than before because there were more rounds of cell division, but it's the same tissue, but later in time. And again, we see in extraordinary dynamics, there are still many cell divisions going on. However, no growth anymore. So these cell divisions subdivide cells and make them smaller. And we see a very interesting collective dynamics of cells giving rise to the characteristic morphologies of fly wing. Particularly that is characteristic veins that are formed. We see that these spots here will become sensory organs, mechanocensors that the fly brain uses as inputs to be able to fly. This is what is called a wing blade. What is down here is called a wing hinge. That's the part that's connected to the body. And particularly noticeable are cell flows which reveal that these really cells rearranged. This is a very dynamic material that is remodeled and reformed in this process with the help of many cellular events. Yes. How critical are those in your head? They were in there and it was complicated for them. But cells in different regions go pretty much the same way for this here. I mean, how critical are they initially? Yeah, but what you call initially is just the first moment when we are able to look at it. But of course there were events before, but it's clear that at the beginning you have less in homogeneities than later. So many of these vein structures where cells are smaller, or the wing hinge has cells that become much smaller. So this happens during the process. So many of these cells become smaller in area while we observe this. And this is sort of a tissue contraction that is part of what drives movements and flows here. And we'll come to that in the last lecture. So we were seeing, so the final adult wing is again exactly two layers of cells. So the upper layer of the mini-imensional this folds outward, that becomes two or less identical layers of cells. This folding outward is what we can't observe because it happens sort of in the blind moment between the two movies. And now we have exactly two layers of cells which presumably are very similar. We can observe one of them. The second one is harder to observe because we have to go deeper into the tissue. So we see only one of them. But the second one is presumably doing the same thing. And the adult fly wing consists of exactly two layers of cells. But when there's flies out of hatches out of the pupa, comes out of the pupa, most of the wing cells die. Before they secrete, it's called cuticle, which is an exoskeleton and that forms a material of the actual fly wing in the end. I think doing this process, they also come closer together. So they're the big, but this is something which is hard to observe. So I cannot tell you so much about it. Here I observe just one of these two layers. Yes? What's going between the two layers? We don't really know, but there's some fluid, probably some exoskeleton matrix. The area is more or less constant. I'll discuss in the third lecture. So in the wing, imagine all this is sort of one tissue. It folds out and connects, except for the veins. The veins then are hollow parts where fluid can move to supply mechanocenters, for example. Do you understand the question? For the food, we just cut it since we can drain everywhere and then... What do you mean, just cut it? I'm not sure if this is so easy to do. I show you always only one layer. The other way is very difficult to see. Usually, when you cut typically, you cut both, but I'm not sure. Some fluid, some complex, some matrix, some fluid. Cannot tell you exactly. So from these examples, it becomes clear that this tissue is very dynamic. It undergoes some patterning. It's a process that is guided by chemical signals. They express genes, but it's an inherently active mechanical process. So this material is an active material in which, for example, cell division, cell death, and cellular force generation introduce stresses. So they're internally generated forces and stresses, and then there are chemical patterns and signals that guide this process, and it's a fundamentally mechanochemical, morphogenetic process. So we think of this as integration of force generation in chemical signals, where chemical signals guide and coordinate, activate forces generation, and force generation such as contractions of certain regions, such as active shear stresses in the tissue, then generate flows and movements, which then also feedback on these chemical signals. So a general theme to understand such morphogenesis is to be able to study this integration of active mechanical events guided by chemical signals and the tight integration feedback of these components. Both dipoles. So I'll discuss that further. The point is that whenever a cell divides, it corresponds to a force dipole in the system because there is no net force, there's always force and counter force, and such force dipoles introduce stresses or correspond to active stresses. So cell division introduces a force dipole, but it's something that will give us more. So I want to first discuss some of the basic mechanics and biophysics of such cell packings in two dimensions. This wing tissue, if you look at one layer, is essentially a column epithelium, and which is dynamic, and we'd like to have tools and concepts to discuss how such cell packings can remodel dynamically. In particular, we use one approach that we call a vertex model, that is sort of representing each cell by its geometry, simplified as a polygon. So in that sense, we are sort of not trying to capture details of the bending or wiggles or curvature of the individual edges of cells, cell bonds, but we want to keep track of the essential shape of cells sort of approximated as a polygon. And therefore we discuss such a cell configuration as a network of polygons. And that's, of course, a sort of a projection of the three-dimensional tissue in two dimensions, but it has the advantage that many of the experimental information that we obtain, like the type of movies that I showed you, do such a projection, and directly represent this polygon network. However, we still have to keep in mind that the system is three-dimensional, and you have to think of this as an effectively two-dimensional description of a three-dimensional system based on the information that we obtain from two-dimensional microscopy. Now, I should also highlight that when we look at this tissue in the microscope and we're using this criterion staining, what we see most clearly, what it has is most well-defined network properties that we see here, is the network of so-called adherence junctions. This is not the full sort of cortical plane that corresponds to the lateral surface of the cells, but there are sort of concentrated belt-like junctions that have enriched actin myosin and also adhesion molecules, and they are sort of very clearly and focused, visible. However, the mechanics of this whole network is not only generated by these adherence junctions, but it's also generated by the tension, antigen, cortical tension that was introduced by Carl Philippe in the last talk along the whole surfaces of adjacent cells, and also the elasticity of, let's say, the areas is stemming from the whole, or it also has contributions from the whole bulk elasticity of the cell. However, it's a simplified picture when you can think of this such a polygon network as representing the network of adherence junctions, and that's what we're usually observing in the microscope. Now, when we take this approach, and we're now going to describe the physical properties of such a network of polygons, we have to take a few things into account. First, we have, of course, a very dynamic tissue. You've seen that in the movies, and we have to be aware of the sort of relevant time scales. It's a soft material. We like to understand it's material properties. They are active processes that give rise to forces, stresses, and movements, as I already mentioned. And the first sort of hint at time scales that we have to know comes from a laser ablation experiment when you can observe such a cell packing in the microscope. And during a few minutes, it typically does not change much. And so, when I can think of this as a force balance configuration, and I can ask if we perturb this force balance, how does it relax, and how fast does it relax? This one can do with a laser ablation experiment. So, we observe the tissue and with a UV laser, one can ablate an edge or a bond between two cells, you see here. And then as a result, this system locally relaxes. The vertices at the end of the ablated bond move apart because the bond is on a tension, and this tension is no longer there after ablation. And I can also measure deformations at further distances from this ablation and study the relaxation of this tissue. And here you see a typical relaxation curve. This is the distance, or the change of length of the initial bond, which now, because the vertices move apart, increases in time. And interesting is the time this happens in about one minute. So, we can see that local perturbations relax within one minute, and everything, and this has to not be compared to other timescales. So, if you now look at the dynamics of this tissue, we have the one minute timescale for local perturbations. We can now compare that with the timescale between cell divisions, which certainly remodel the tissue and make it deform. And as we've seen before, cell divisions happen within a few hours. It's much, much longer than this local, this fast relaxation. And then we have another important sort of type of timescale that is, what does it, how long does it take to build something, to create shapes and patterns. This is now tens of hours to days. And when we are interested to these timescales starting from cell divisions to the forming of patterns, we can use sort of the separation of timescales and take the idea that because local perturbations relax within a minute, on the few minutes timescale, this network is in a force balance relax configuration that is slowly evolving in time, sort of in a quasi-static time evolution. That's a picture that's very useful and that we're going to use. So the idea now is to describe these sort of on short times established force balances in this network in a mechanical model, which can then be used to evolve this tissue in time by always staying in a force balance configuration but changing the parameters of all these cells slowly in time, which corresponds to slow changes of the properties of cells with time. And similarly, when a cell division occurs, this corresponds to a small timescale and we're not looking at what happens at these minute timescales. So this force balances can be described by writing, so we want to have a network configuration that is force balance. And in the simple picture where all these force balances stem either from what we call an area elasticity that the cells have a preferred area and whenever the area is changed by external forces, there's work associated with this. In addition, the cells have contractile tensions or sometimes also expansive tensions on these edges or bonds of cells. One can describe force balances by minimizing the potential function. So this is not a thermodynamic potential or a potential, as one would use it in equilibrium physics because this is an inherently non-equilibrium system. It is just a consequence of this geometry of tensions and pressure that allows us to write any work associated with the deformation as a potential function. And the potential, what we call as a work function because it describes the work associated with the deformation that we have put forward is shown here. It depends on, it's a function that depends on the configuration of the polygons such that a force balance state corresponds to a situation where this derives with respect to all the vertex positions is zero. This corresponds to a force balanced force free state. The contributions are first what we call an area elasticity. So alpha is the index for a cell. A alpha is the actual area of this polygon. A zero is a preferred area, an intrinsic property of the cell or of tissues at its position. And K is an area elastic modulus. And now one should think of this as sort of a three dimensional system. So this area elasticity is an effective two dimensional property that comes from the bulk properties of the cell. In some simple picture, you can imagine that these cells have a mechanism to keep their volume constant in three dimensions. And the tissue because of the adhesion properties and the cortical tensions has a preferred height. And the volume together with the height of these columnar cells then defines the area. And whenever the area in 2D is different from this preferred area that is set via the volume and the height, the cell has to undergo a three dimensional shear deformation. And therefore this area elasticity is related to the three dimensional shear modulus of the cell and to the height of the tissue. But in two dimensions, we don't see this three dimensional background. We just have an effective area elasticity modulus that one can try to measure if we have only two dimensional information. The next contribution here is a simple cell bond tension. So LIJ is the length of the bond connecting the vertices INJ and lambda is a line tension. Now in general, this line tension is not necessarily constant. Of course it can in principle vary and can be different on each individual bond if cells locally regulate contractility for example of ectomyazine. So in principle the lambda itself can be different for each bond. I will today only consider simpler cases where the lambdas are property of the tissue and are the same, mostly everywhere with one exception. In addition, one could imagine that in general this tension changes with the cell state and if the cell for example is forced to have a very long big perimeter, it is harder to recruit other material to these bonds and maybe the tension effectively decreases a bit. In general one would expect that one has to write down the Taylor expansion for this tension or for this energy in powers of bond length. And in general this gives a lot of terms. A very simplified way of keeping track of that is to add one quadratic term as a next order which also helps to stabilize things and to simplify it by giving it one coefficient for one cell perimeter. So alpha is the perimeter, the sum of all the bond length of cell alpha and that's sort of a simplified choice to add a second order term that somehow takes into account that if a cell becomes very big, the perimeter becomes very big then the bond tension may effectively increase or decrease depending on the sign of this camera. Yeah, so that's these, I call this cell perimeter elasticity and with this model can now be used to discuss the mechanics and dynamics of such polygonal networks in different biological contexts. And the idea is now to at every given time take a force balance quadrpiguration and allow these coefficients A0, the lambdas, the gammas and the case to evolve slowly in time and thereby the tissue obtains the dynamics and the idea behind this is that we have the separation of time scales and that we don't have to necessarily write an explicitly dynamic equations which essentially would capture the fast relaxation process. Of course, this is a somewhat simplification that is not always the right thing to do and one can also write explicit dynamic equations for these polygons by rather than just minimizing E sort of writing taking the imbalance of forces in a given network as a force and introduce some kinetic coefficients. But this gets quickly tedious and for many situations where the separation of time scales is very clear, this is not necessary. So I will discuss to gay the simple picture where I will describe networks of local minima of E which are force balanced in the sense that there are no forces acting on the vertices. Yes, these are two independent coefficients. You can think of this as a Taylor expansion and then you have linear terms quadratic cubic and I cut it off after the quadratic level and I only collect a few quadratic terms that I find important just to make it simpler. And this coefficient I can for each cell I can have different one if I want and here for each bond I can have a different one in principle but these are independent coefficients which the cell may regulate by biological processes. Now let me first discuss the states and networks for which this function E is an absolute minimum. I will show you that these are not really the configurations that in practice are the most important ones but they are very useful to know because these are important reference states to discuss what happens in a real tissue. Yes, yes, but there's also the linear, effectively has some spring like properties, that's right because I've cut it after the second order, yes. But it could be anharmonic in reality because there could be cubic terms also. The idea that I should not think of it as a hook and spring but it's the simplest version to start. If you cut a Taylor expansion you always get a spring or one could also write it that way but if I have lambda different each bond I can no longer do it. But this can be also written as L minus and not square. Yes, so what I want to discuss now are the states of minimal energy of this. These are certainly force balance states and these are sort of if you want the most relaxed configurations. Yes, yes. A note of the parameter is a property of the cells. It's a property of the cells. You have to measure it. Each cell has an A naught. Similar to for lambda and gamma and K these are things that cells possess as properties, yes. So in order to discuss these energy minima it is very useful to turn this into a dimensionless form for this discussion. And to do that this has units of an energy and I can divide it by an energy and then this has no dimension anymore and the energy I want to use to divide it is K times A zero square and you see that K times A square is an energy. So this is a dimensionless energy and if I write this dimensionless form then this depends on dimensionless parameters. And I take the simple case where all the lambdas all the bonds have the same lambda and all the cells have the same gamma. And then I have two dimensionless variables. These are just numbers. It is gamma divided by K A zero and lambda divided by K A zero over three half and you can easily convince yourself that these are dimensionless. And now I can ask for a set of these two dimensionless numbers. And I only, so I had sort of more parameters here but I now have only two relevant parameters left after making a dimensionless. That's why it's very useful to make a dimensionless. And now I can ask for each pair of these dimensionless numbers what does the lowest E network look like? And I first showed you the result and then I explain you how one obtains it. So I have two axes of this diagram. I have the normalized tension and I have the normalized perimeter elasticity. Now if the perimeter elasticity is set to zero if I have only simple line tensions then these line tensions must be positive for the thing to be stable. And then I have only this axis here and I only allow the positive values down here and for these positive values I always find as a ground set is a perfect hexagonal network. If I also put in now the finite perimeter elasticity this stabilizes the system if it's positive to coefficient even if the line tension is negative and that's why this axis is important to extend the range. So if I now add this perimeter elasticity I get now a whole two-dimensional region of parameter values where the perfect hexagonal lattice is the ground state the most relaxed configuration of this network. However, if I now change the line tension towards smaller values and typically negative values then I can cross a line and in this line the hexagonal lattice is no longer the minimum of the energy and irregular, typically irregular networks become the minimum. And the hexagonal network one can easily show if you make a small deformation of it shear deformation or many different any type of deformation will increase the value of E and therefore this has a shear modulus it's like a solid that requires work to undergo shear while if you go into the soft network region because this line you need no energy to shear it and therefore it is no longer a solid and we call it a soft network and you can think of it as a fluid like state. So already these ground states have a transition between the solid and the fluid which is quite remarkable. You could use the shear modulus if you want or you can use the hexagonal order. You could choose order parameters for that but the transition is a dramatic transition it's a real phase transition in this ground state diagram. Okay I'll also come to that the hatched region there the network becomes unstable towards collapse of the cells. So that's a region that's unphysical and I'll also come to that. So if you push the parameters here so you have a hexagonal network and if you now increase the line tension the cells get compressed more and more by this line tension and then if you cross the line down here suddenly they collapse to zero area. There's a transition to zero area. Yes? Here not always but here always. Here I'm talking about ground states. Only ground states. Dynamics comes later. Okay now let me show you how one can obtain this diagram. It's an interesting mathematical exercise to find these ground states. And I told you about the dimensionless form. And the nice thing is that I can write this dimensionless form for the special case where these parameters are all the same as a sum of a dimensionless energy per cell which only depends on the dimensionless area of the cell and the dimensionless parameter of the cell. And the dimensionless area is simply a divided by a zero. I'm sorry this a zero is the same as this a zero here. And the dimensionless parameter is the parameter of the cell divided by the square root of the preferred area. These are two dimensionless number characterizing the shape and size of a polygon. And this little e can now be expressed very simply in terms of this dimensionless area and the dimensionless parameter. I have now my two parameters the gamma parameter the parameter elasticity the dimensionless one shows up here and the line tension shows up in the preferred parameter. That's what you asked before one can write it as an L minus L zero squared and dimensionless is a P minus P zero squared. So this dimensionless parameter sorry this preferred parameter is just given by the ratio of these dimensionless numbers. And the minus sign implies that I need a negative tension in somehow to get a positive preferred parameter. Okay so in order now to discuss ground states I will go in two steps. I will first ask what is the lowest energy polygon? So what polygonal shape minimizes my dimensionless e? And when I have understood what polygonal shapes have lowest e I will see where they can assemble them to plain filling networks. And then I will get candidates for minima of these networks. So first step is polygons of minimal e. So I have an n-sided polygon of arbitrary shape and I ask which of these shapes minimizes e? First for given n and later we'll vary n. n is the number of edges of the polygon. That question. So now we have only to minimize the function e of a, p for a single polygon. And the first step I'm using is I take my polygon and I rescale it by a number psi, scale factor psi. And I will now want to minimize with respect to psi. So keeping the shape the same trying to find the correct size that minimize the energy. So for this purpose I introduce a function e prime which is the same as e before except that I've rescaled it by the factor chi. So for the area I have to multiply with chi square or for the parameter multiply with chi. Now I calculate d e d chi. I demand that it's variation zero respect to a rescaling and I get a condition for the a's and the p's. So that's one condition the minimum has to fulfill. And from this equation I find three possibilities. I can have a case where p is smaller than p zero and then a must be larger than one because one side is positive the other one is negative. They can add up to zero only with this combination. One condition is a equals one and p equals p zero. That's in fact the absolute minimum of this quadratic potential. That's the absolute minimum. And then as the possibility p larger p zero is smaller than one. Now the first one cannot be a stable minimum because if I take such a state because p is smaller than p zero I find that if I derive e respect to the parameter this is negative which means if I increase the parameter I will reduce e and this is unstable respect to increasing the parameter at fixed area. This cannot work. This is not a solution. So I have two cases left and these two cases will give me most of the phase diagram. Let's first look at the case p equals p zero equals one. That's the simplest case. That's an absolute minimo of e. These are always solutions. And they can exist for any n where n is the number of edges of a polygon. And in general these are polygons that are irregular inside and they have the property that the perimeter is exactly a prescribed value and the area is a reference area. So if you have a polygon that has exactly the preferred area and the perimeter is the prescribed parameter you have a minimum. But this is only possible if the perimeter is bigger than the smallest possible perimeter of an inside polygon which is a regular polygon. So for each end there exists one regular polygon which has the minimal perimeter for this polygon class. And so to have such a solution p which is exactly equal to prescribed p zero must be bigger than p n and p n can calculate for every n easily for regular polygons. So these are solutions. But they occur only under certain conditions. The other possibility is that p is larger than p zero a is smaller than one. This is the case where we have sort of a cell that is compressed by the p larger than p zero means as an effective compression. And therefore a is smaller than one. It's slightly reduced compared to the preferred area. And in this case the edp is positive which means the perimeter is minimized. So in this condition the solution is a regular n-sided polygon. So we have either irregular polygons typically with parameters larger than p n or we have regular polygons with parameter given by p n. And the transition between those two happens exactly at this particular value p n. So from this argument already one can now draw a diagram what type of polygons minimize as individual polygons minimize this energy not yet networks. And this gives us already a diagram which looks like the final one except with some additional pieces because for each polygon class we can draw a line where we go from a perfectly regular polygon is optimal versus when an irregular polygon which increased parameter is optimal. And for n equals 6, 5, 4, 3 I've drawn this line here. So here we have the region where p larger p zero a smaller one where regular polygons are optimal here we have the region where these irregular polygons are optimal. And depending on n we have different lines and these different lines are shown here as a function of n. Now we understand sort of roughly what polygons are optimal. One important point to mention is that if I now compare a three-sided, four-sided, five-sided polygon how do they compare? Clearly as in this third case the parameter for given area is minimized the absolute minimum is the circle and for large n we're getting closer to a circle for small n we are far from a circle so large n have a lower p than small n so 6 is better than 5 is better than 4 is better than 3 but in principle of course circles would be optimal but now we see of course circles we cannot fill the plane so of course we can fill the plane but not the circle so the question is now how to find ground states, arrangements of polygons that fill the plane and for that I have to briefly to use the concept of counting vertices, edges and faces in the so called Euler characteristics which can be calculated for arbitrary polygonal meshes so the Euler characteristics is the number obtained by counting all vertices in the meshwork counting all edges counting all faces which is number of cells calculate v minus e plus f and this number has very universal properties first it's easy to see that this number doesn't change when I have such a network and I add or remove bonds and you can easily convince yourself if you add a new bond somewhere you create typically two new vertices you add an edge you add two new vertices but you also create a new face and therefore the number doesn't change also if you remove a bond you can check that easily even if you take a new bond that hits exactly a vertex also true whenever any way you add or remove a bond the number doesn't change which means you can remove all bonds until you get just one big polygon the number still doesn't change and for a single polygon you can easily see that the number let's say if you have polygon in this case one, two, three, four, five edges we have five vertices we have five edges we have one face and usually one counts in the way that what is outside the polygon is also counted as a face we have two faces I explain why in a moment so in this case, Kai is two the reason why we take two faces is that it's easier to put all of this on a sphere put our pentagon on a sphere and we have one face here and one remaining face outside and everything is closed so for a sphere if you put an arbitrary measure on a sphere you can always define this number two, irrespective of what kind of mesh work you're using more generally, it only depends on the topology of the manifold on which this mesh work is drawn is there a question? I'm not exactly sure yes, so I'm talking I think of a two-dimensional manifold and I'm putting a polygon or mesh work on it it doesn't matter where the plan is it can be curved, it doesn't matter the only thing that matters is the topology curvature doesn't play a role here in this argument and whether it's a sphere or whether an infinite plane is the same thing if you associate infinity with one point on a sphere now if you have a manifold which is two-dimensional you put a mesh work on it and you close it to the topology of a sphere it will always be two and more generally, it depends only on the topology of the manifold on which you draw it so for Kai is always just given by two times one minus g genus g is zero for a sphere it's one for a torus it's two for a bratzel a donut with two holes and so on and so forth now the reason I'm saying that here is that often we are using periodic boundary conditions we're taking always planar mesh works but sometimes it's useful to use periodic boundary conditions where sort of on the rectangle at the boundary one side of the rectangle is identified with the opposite side and this is also true for the other and this has a topology of a torus so if you have periodic boundary conditions and this is associated with this here and this is associated with this here then you can first wrap it on the cylinder and if you now have periodic boundary conditions here you can think in three-dimensional space it will be sort of a torus and for a torus we have g equals one and then Kai equals zero so if you do it like here on a sphere we get the number two and for a torus we get number zero and just an example the topology of a sphere is an example cube or any polyhedron in three-dimensional space has topology of a sphere you get Kai equals two for a cube you have eight vertices twelve edges, six faces and you get two and here you give you an example of a square lattice in the plane with periodic boundary conditions this corresponds to the torus now we have Kai four vertices these are these four points we have eight edges now this edge goes up here and comes up here so this is the same edge as this one this is the same as this one we have one, two, three, four and those four make eight and we have four faces one here and three here and the other three are mirror images of the other here so it's four and zero now this is just a side now we can apply this to our polygonal mesh work now this number for any polygonal mesh work which is let's say with periodic boundary conditions must be zero and I can now so I use periodic timing means periodic boundary conditions now if I take mesh works like this I always have three-fold vertices now the multiplicity of my vertices I call M M is the number of bonds coming out from a vertex and without loss of generality we can often always use three in particular in biological tissues because even if you have a four-fold higher on four-fold vertex so this would be a three-fold vertex now if you have a four-fold vertex and we zoom in here there may be a small detail there's a tiny bond here which cannot be resolved so let's just replace this by a tiny structure like this and go back to three-fold vertices if what it wants to in practice of course this may not be visible or may not exist then one talks about higher order vertices but let's for the purpose of the discussion here focus on M equals three and now we can explicitly write the Euler characteristics so we sum to get these numbers we sum over all cells for each cell we get one to the faces for each edge we have to count the N edges of cell alpha but since one edge has two neighbors so for this cell we count this edge and for this cell we count this edge so we have to divide by two and similarly with the vertices the number of vertices is obtained by first counting all the vertices of a given cell but then this vertex is shared by M cells M is the multiplicity of the vertex there so we have to put N over M and this gives us the Euler characteristic which is zero now the sum over N can be expressed in terms of the average number of neighbors of polygons in the network so the average number is the inverse number of the faces times the sum over all the neighbors of all the cells so therefore we find that the average is now related to M we can calculate the average knowing what M is now you see that for M so M equals two doesn't make sense because for a vertex it has to be at least M equals three so the smallest M we can have is three so if you put in M equals three we find that which is such a network we find that we have six here divided by two is three sorry 12 divided by two is six no sorry six divided by one sorry for that so we have on average six neighbors and that's a mathematically hard constraint there's no way around it whenever you draw with M equals three must have on average six neighbors you can have individually different numbers of neighbors but the whole network is always six and you can also go to larger M if you have systematically all vertices more than three links and essentially these are examples so M equals three corresponds to hexagons M equals four corresponds to rectangles M equals six corresponds to triangles above that we cannot build a lattice with smaller values of M so if we want to build polygonal networks that have polygons of the same polygon class to tile the plane the only ones that we have are these three possibilities now we already said that hexagons are better that pentagons and squares heptagons would be even better they're not used to tile the plane so the natural ground state is a lattice of hexagons now of course one may be able to minimize the energy further of the whole network by putting in a few polygons that have a higher neighbor number and having in addition a few other polygons which have smaller than six neighbors the average has to be six even if it's not a hexagonal network this is the possibility to find more complex solutions that are not made of identical polygons now it turns out that in the large region of this diagram one can show that one cannot decrease E by mixing polygons of different classes with some of them being higher than six some of them being lower than six therefore in all of this green region the perfectly hexagonal lattice is the ground state now since this is the case we can now go back to our lines here now we have now found a ground state in the green region and I come to some subtleties here you see some other colors here it's a side remark but what we're seeing in green essentially in these dotted lines is the boundary beyond which we can no longer prove that a hexagonal ground state is a ground state so here it is now if we meet this line where it says N equals six that means now that the perfectly hexagonal lattice is no longer the ground state but a slightly deformed lattice still consisting of hexagons because between these two lines I only can have non-regular hexagons as the ground state but not yet pentagons only after the second line I will have non-regular hexagons and non-regular pentagons as ground state and now then I have non-regular quadrilaterals and then I have non-regular triangles that's why I have these several lines so after the first line what happens is the shear modulus of the hexagonal lattice goes to zero and I can deform it without increasing E for a finite deformation but I have to stay within hexagons this means I have a soft lattice but I cannot yet rearrange cell without increasing E the next thing that can happen is that I can even deform the meshwork as much that I can get neighbor exchanges and when this happens I'm in a situation where it's not just the shear modulus is zero but I can move cells freely around without changing E for this I need these so-called T1 transitions these are neighbor exchanges this is a situation where I start with a rearrangement of polygons I change neighbors so that two and four were neighbors before and one and three become neighbors afterwards this requires a topological remeshing and to do that I have to go through a fourfold junction so I can shrink one bond between two and four I go to a fourfold junction and then I reopen a new bond so that two and four are now separate and one and three are now together and when I do that in practice I shrink this bond I have to go from a hexagon to a pentagon so this can only happen when pentagons are allowed to be ground states if I discuss about ground states which means back in this diagram the n equals six line is the line where the solid becomes soft the shear modulus vanishes the n equals five line is the line where I can get T1 transitions and neighbor exchanges so below the n equals five line I have a really fluid lattice I'm talking about very simple models not about real systems so what you're asking has to do now with a real system which one would have to discuss in this particular detail these are just concepts to be used to apply to systems and then these questions come up separately here I don't think of a bilayer or of two identical layers two layers do exactly the same thing as it was I should mention that this is a discussion of ground states Lisa Menning has pushed this further and has investigated whether this transition also exists if you have complex morphologies away from ground states and has shown that this pentagon line is a transition line in more general arrangements of cells and you find this line exactly by this argument that I outlined here so there's a general transition between a solid like network when P0 is smaller than this critical value which has a value of 3.81 if you calculate the dimensionless perimeter and you get a fluid like state where you can move cells relative to each other without increasing E and they have more irregular shapes now we come also to this region here which was asked before so one can use the same discussion we know now here we have regular polygons we know the minimum is sort of a net measure of hexagons and we can ask what happens if we move to higher values of lambda and what can happen is the area of this polygons becomes smaller and smaller and approaches zero and this can happen continuously and it happens in fact discontinuously down here and the purple line here is the line at which the hexagonal lattice becomes locally unstable so at the dark black line the hexagonal lattice is the minimum you can go in the mesh region and having a metastable hexagonal lattice and at the purple line it actually jumps to zero area now there's some extra structure down here which I find amusing so we can zoom in so the dashed lines as I mentioned before is a line where our exact proof for the hex signal to be the absolute minimum fails so beyond this line we may find non-hexagonal ground states and Douglas Staple who investigated it some years ago he actually found non-hexagonal ground states and here you see this dashed line where the proof is no longer works this as well and then he finds two regions where non-hexagonal lattices have a lower e than hexagonal ones this little tiny region here there are four eight lattice so squares and eight n equals eight polygons coexist in the lattice with an average n equals six that is even lower than hexagons and here is a region of three twelve lattice solutions again average n is six that have an even lower energy and the purple line is a line at which the hexagon lattice loses stability and the black line is the line where now the collapse states are minimal energy okay so that's a mathematical discussion of the ground states of this model I just used some time to explain how to get them because I think you get some ideas about the physics of these networks but now I want to use this type of model to look at biological systems to come and get sort of the type of questions that was asked now in taking this as a tool to study let's say growing tissues and I said before I use a quasi-static approximation and in the case of a tissue growth everything is in a force balance until a cell divides a new cell bond is introduced by division the new area that is created with a new bond or divides one cell in two smaller areas but the preferred areas are still larger and so there will be now a larger pressure because of this area elasticity term will push the neighbors at the side and we look for the next minimum this force balance state within a minute or so this situation will relax to a new force balance network configuration and this now gives us a quasi-static time evolution driven by cell divisions and I'll show you what this looks like in this model yes we should look in the movies but if you look in the movies it happens quite quickly I cannot tell you exactly this could be slightly longer but it's much shorter than a time between two divisions that's the separation of time scales that I want to use so what I do is start from a perfectly hexagonal lattice the ground state this here is my periodic box so you see whatever sticks out at one side of the box then fits in on the other side of the box so these opposite sides of the rectangle are connected it's of a toroidal topology if you want in order to avoid boundary effects and here already one cell has divided this thing has relaxed there's still a code to show you how many neighbors each cell has and I started from the ground state and now dividing cells we perform work on this network and we're generating a non-ground state state in the same time it grows because as I said before whenever a cell is divided there are two smaller areas but the preferred area is bigger the cells want to grow and therefore there's a pressure locally that pushes the neighbors to the side and such a calculation creates we're using random orientations of the new bonds that are introduced at divisions sort of a non-oriented cell division and we just stochastically pick cells in a completely random fashion for division and this creates now a mesh work that is far from the ground state but you can still ask what material properties it has and you can ask what are the statistics of these shapes and sizes of cells and in fact that this is a meaningful question one can see by just plotting the energy dimensionless per cell as little e before now as a function of time as this network grows this is a level that corresponds to the perfectly hexagonal network that I explained before and because now cell divisions work on this network this thing goes up and it fluctuates a lot because it's a disorder stochastic process but if you create more and more cells so it becomes self averaging and so it converges to a well defined value that becomes stable and defines the state of the system and in this very well defined state we can do statistics we can for example now ask what is the fraction of inside cells in this mesh work and starting from this perfectly regular hexagonal network we've now created a disorder network which has many different neighbor numbers so of course the average neighbor number must be 6 if we triangular 3 fold junctions and here you see the statistics again fluctuates a lot at the beginning and then converges to well defined values for many cells so we have a very we can define the statistical properties of the so thereby generated tissue interestingly here you see the pentagons are the most common ones hexagons are here octagons and some rectangles quadrilaterals and again such states that generated by growth they still exhibit this phase transition that we found in the ground state and they exhibit that in the same line as Lina Menning has studied in some detail and I should also mention that because there is this characteristic value 6.81 that corresponds to the pentagons that are regular that defines this transition line Lisa and her coworkers have suggested that in order to distinguish between these phases one may just by observation look at the value of P measured in an experimentally observed tissue and if P is bigger than its values one would expect it to be fluid and if it's smaller than expected to be solid where this is indeed a case it's not clear how general this is but at least sort of the fact that cells sort of look more like parameters look more like those of perfect pentagons seems to be seems to allow the tissue to more easily go through T1 transitions and flow so you can measure P for cells P is the perimeter divided by the square of the area of the cell as a dimensionless perimeter and if you take a picture of cells and you calculate this for every cell you average it over the region and you ask whether this value is bigger or smaller than this characteristic value of 3.81 that's what they did and that's what works pretty well but it comes from this transition in the phase diagram but this is just geometry there's no, only this two-dimensional physics I explained to you these arguments don't need a further dimension you don't have to know the further dimension now I showed you one example of this growth that gives rise to statistics of cells and the statistics of cell packing that one obtains depends on where one is in this diagram what I just showed you was this point parameter space where this network becomes a solid so the ground state is a hexagon but of course after growth and it grows it's not a hexagon it's disorder but it's still a solid it will be sort of liquid here but I can also show you what happens in 2 and 3 let's just show you these three cases the first growth that I showed you this would be case 2 which is up here and there we see much more variability between areas and case 3 which is now in the soft network has a bit more elongated cells because there we have parameters that are bigger than the regular polygons and you can visually distinguish these different networks so the statistic of the network is characteristic it depends on the physical parameters which guide the remodeling of this network as we go through this simulated growth process and we sort of used measurements in the wing disc to characterize the statistics of cells and to compare with this model so one can segment these images define the vertices of the cell bonds detect sort of the polygons quantify cell morphologies and also what we also did is to use laser ablation and the perturbations resulting from it and comparing that to laser ablation simulated in this vertex model to see how this tissue relaxes and by this comparison we could sort of find a very small region in our parameter space in which we would generate the same behavior as we see in the wing disc and that's sort of very close to this case one which I showed you before sort of one point in this two-dimensional phase diagram and here you see in the growth simulation the distribution of neighbor numbers and measure the wing disc of neighbor numbers and as one subtlety I should point out when you do an experiment and you measure the neighbor number you have a finite resolution and if two vertices are very small very close nearby you cannot distinguish them so there's always a cut-off below which you cannot distinguish two vertices and then you count this as a four-fold vertex and not as two three-fold vertices now to compare experiment and theory we have to use the same cut-off in the simulation analysis the simulation you have all the information there's no cut-off necessary but in order to compare numbers you have to use a cut-off in the data analysis and that's why here the average of n is not exactly six because we have used this cut-off it has to be six if m equals three but if there's a few four-fold vertices then the average of n can change and in here you see that also the area of statistics corresponds to the experiments which is not the case at these sorry at these other parameter values okay I should also mention that in the simulation we also can count the number of t1 and t2 transitions and here I have a statistic of t t2 transitions there was roughly also measurements which we could compare to now we also have the wing-imensional disc in culture where we can try to count t1 transitions which has not yet been done in comparing to such simulations now in the remaining few minutes I want to show you one example also where this vertex model has proven to be very useful and that is the question of what makes these compartment boundaries straight and what are the principles underlying the organization of these tissue compartments in the wing disc and of course we can use our model of growth in this vertex model to see how interfaces between different cell populations behave and if we just take start from again the hexagonal lattice with two groups of cells blue and red a blue cell divides into blue cells a red cell divides into red cells but I have all the mechanical properties the same everywhere so I don't as before all the tensions of bonds all the preferred areas remain all the same in the whole network so the same growth processes you saw before just that I color the cells and then you see that you cannot maintain a sharp interface the growth process makes the interface irregular and you can even get islands so you don't get boundaries and the tissues can mix which is not happening in the wing imaginal disc as it grows and that raises the question of how does this tissue prevent this from happening in such islands and first one suspicion is that something special happens at the interface when two cells of different sort of compartment property meet and this can be started again using laser ablation of such cell bonds so one can in this tissue ablate bonds look at this is the case one parameters which is the ones that work perfectly for to create the morphology of cell packings in the wing imaginal disc and the ground state is hexagonal yes the ground state would be hexagonal does not create a ground state for obvious reasons I should maybe also mention it cannot create the ground state because it would have to go over energy barriers to go to hexagons while in this other phase in the phase diagram where you can actually remodel it even if you grow it it stays in the ground state all the time that's why this phase transition of the ground state diagram remains exactly the correct phase transition line even if you look at grown tissues okay now we're looking at laser ablation in this network and we can do that by ablating bonds a a means in the interior compartment p p means in the posterior department a p means exactly on the compartment boundary a 1 a 2 means next to the compartment boundary on the epical side the first and second cell next to the compartment boundary to see whether there is some gradient or whether it's something localized at the compartment boundary and we see that the a p bonds exactly compartment boundary bonds behave completely different from the other bonds they move faster and further apart when you ablate them and it's very significant all the other curves fall on the same line if you average over many experiments and these guys are very significantly far away and this suggests because they move much more and also initially faster that they're just under higher tension so to to put this in our conceptual framework we now no longer make all the bond tensions the same in the tissue but we now say those bonds where a blue and a red cell meet they somehow have a higher tension that lambda parameter is larger than for cells either in a or in p either red or blue and we give the red and blue cells the same bond tension because there's no reason to suspect that they're different because they fall on the same curve here and then we ask can we account for the for these experiments and also for the morphology of the compartment boundary by changing this tension and here you see lambda equals one means lambda is the multiplication factor we use to increase the tension along the compartment boundary bonds so here this is the old case they have the same tension as in the bulk and here they have a four times increased tension and first this four times increased tension can account for the can smoothen the boundary and it avoids all these islands and it also counts for this laser explanation and ablation experiments and this was sort of more extensive work to show that there's really contractility regulated by cells upregulated probably by factor of three to four at the anterior posterior compartment boundary to ensure that these cell populations don't mix and the boundary straight. Now we can quantify this further by measuring the roughness of these boundaries so how far they deviate from a straight line and one way to do that is sort of to find the average straight line or by connecting for example the first and the last point and then calculate the variance of the perpendicular displacement so we call this height variable relative to the straight line which can be positive or negative we subtract the mean for each local cell we can define this height we subtract the mean and average the square so this gives us a variance of this height and we do that as a function of a length along the boundary where one can measure this variance so you want to see how this variance of the perpendicular deviations this height variable increases as the lateral length that we use increase and this is showing this plot so this is a distance along the boundary over which we measure the variance as a standard measure to define roughness of interfaces also in condensed metaphysics and this is the quantity this variance normalized with cell size as a length scale and you see here two values of experiments and three curves of these growth simulations so the purple one corresponds to lambda equals 4, this lambda equals 2.5 lambda equals 1 and so these larger values of lambda produce the roughness that corresponds to the experimental measures so this work has suggested that component boundaries are maintained with the help of a locally increased probably actualizing driven contractility only along the bonds between the components and this is not only true for the anterior posterior component boundary more recently we've also shown that variant of this is also true at the dorsal ventral component boundary very briefly I show you here again the same types of plots for the dorsal ventral component boundary exactly on the dv boundary we have a significantly increased response to ablation which we interpret as an increased tension all the other bonds behave differently this looks exactly as in the ap boundary other subtleties which make it different for example we see that this roughness as a function of time reduces with time and this increased tension along the dorsal ventral component boundary is increased with time and we've speculated about several effects that together allow this dorsal ventral component boundary to smoothen as the tissue grows so I think one and a half hours are sort of up so this is a good moment to stop for today and then let's continue tomorrow with the next subjects thank you very much for your attention