# Geometric aspects of epithelial tissue mechanics – Michael Moshe

the organizer for inviting me and I will tell you about several tissue mechanics and specifically I will tell you today about a bazillion tissue this is the work done with mark pawuk and Cristina Marchetti from Syracuse University but I need to start and tell you about what are the serial tissue so these are just two dimensional layers composed of biological cells adhere to each other forming a surface like tissue there are many types of tissue over epithelial tissue which differ mostly by the shape of the quantitating cells here are this really simplest example there is the squamous cuboidal and columnar epithelial tissue and famous examples are the skin and pancreas in uterus as biological objects these are complex systems they have so it is a complex system with with cell can divide and they can they can be mobile so in principle the behavior of Sorority issue is complicated but the reason I'm telling you about the systems because sometimes if despite of the complication there are unique behavior that can be explained based based on very simple rules and there are many examples today I will tell you about one example and then I will show you about a new way to describe such materials so the system I will tell you about to motivate my talk is from a paper published three or three years ago by park at all about bronchial epithelial tissue which are just the layer that lining your lungs in their study they they were interested in the differences between epithelial tissue bronchial tissue from automatic and non automatic donors so what you see here are this is a tissue that was cultured on a petri dish with cells taken from a non automatic donor and this is from a somatic donor and what are you going to see are videos that show the motion of these tissues so each cell is going to push its friends and to try to crawl and so what you can see is that in the non elastic case there is a clear motion cells are able to push their friends and to propagate so there is a some kind of flow so it is a fluid like phase after eight days these motion stops and there are evidences that supported self still try to crawl but there is some solidification with the automatic donors even after 14 days you can still find that cells are able to propagate and there is still emotion now as I said this is a complicated system but in this case you can completely forget about the origin for the motion and why cells push their friends and what determines their geometric properties of each cell but you can just ask simple question like what is the relation between the local geometry of each cell and the global behavior and this is what they did in this work they study the relation with the parameter and the shape parameter so for each cell you can measure the perimeter and area and define this dimensionless number there is a lower bound on it for the circle a circle has 3.5 4 5 and any other shape has a larger one and if you have large shaping a parameter it means that you the shape is elongated or have arms and what they found by measuring averages from different donors and different cultures from the same donor here is what they found this is the NHANES multi case after six days the shape parameter reaches a critical value approximately three point eight one and in that matter case it does not reach this value now you might be worried about this law very large error bars but using statistical metals of bootstraps they were able to show that there is a very probability of ninety-five percent to be on a much much smaller window here so you can believe these results but you should still be suspicious I will tell you later why but this invites hypothesis it is possible that what determined this mechanics is the shape parameter so we will not ask what is the biological origin of this shape parameter but we can ask given this shape parameter what is going to be the mechanics and to explain this the authors of this work what they did was to go to a very famous model known as the vertex model and to study the mechanic so now I'm going to describe it the vertex model originally used to study soap bubbles soap films and but later by Honda and Fryman and new liquor adapted to study mechanics of several tissue so what is the vertex model here is an example of piece from a cellular tissue as you see it really it is a three-dimensional object but we will be interested now in planar tissue so we are if there are remain planar and the thickness is much smaller than the system size we can describe it using the projection and in this vertex model there is a an approximation for shape cells as polygons although they don't must have be polygons but they don't have to be polygons but it's just an approximation M and now we can say that there is a mechanics associated with this network so the mechanic says that once you get you have a configuration of your cellular tissue characterized by the vertices of the polygons you can associate with it an energy and now I will explain you each term here so this term in the energy is vanishes for area discrepancies from some target area so this is the actual area of cell number I it has some target area what it describes actually is three-dimensional deformation if the thickness is changed there is a manager to cut it assume that there is some preferred and if you assume a conservation of volume something that is not but you can argue about it but it's just the simplest model then you have an cost associated with thickness so it is effectively like energy associated with area discrepancies there is another term the perimeter for each cell that comes from cell contractility at the top of each cell you have a contractile ring it contracts and if you have a different perimeter you pay energy another term comes from effective line tension that comes additive due to some coherent molecule sitting between the cells and making an effective adhesion now you can rearrange this functional to a simpler form by completing the square and you have two main terms wanted punishes for error discrepancies and wanted punches for perimeter discrepancies now you can characterize this functional using two dimensionless numbers M which are the ratio between the rigidities and the shape parameter but now this shape parameter in this year is not the one that measured in experiment it is the target one you have target perimeter and target area okay so it can be a field that is the characterize the local preferred state of each cell now what they did in the work they studied this model they they set the following argument we observe a fluid fluid like behavior fluid like behavior corresponds to a rearrangement of cells locally so here is how it looks like you have for example four neighboring cells a rearrangement corresponds to shrinking this edge to a point and then open it in the transverse direction so you have local rearrangement and flow is collective behavior of such many many local rearrangement so the question would be what is the energy barrier for such a local rearrangement is function of the shape parameter M so here is the numerical result they studied it for different rigidity ratios and they found always that adds function of the shape parameter below some critical critical values 3.81 you have finite energy barrier and above it you have a very small almost zero energy barrier and comparing with this experimental result everybody should now be very happy we have a very simple model that recovered this kind of and it is really nice but the question is what is this three point one number where it comes from so I talked to is the guy that simulated this vertex model the max B from Northeastern University and what I asked him is what will happen if you take not disorder or network of cells but the perfect hexagonal lattice and what you get apparently is not three point eight one but the different number you get three point seven two and for the disordered case you get three point eight one the origins of this number is very simple it's just the you remember told you that there is a lower bound on the shape parameter but this is for all shapes if your shape is hexagon the lower bound is three point seven two I mean if it is a Pentagon it is three point eight one why Pentagon is the relevant for the disordered case it is not clear now later I might suggest an explanation but this now raises several questions we have a nice model that explain the transition but in the solid regime is it an elastic solid is it something else how to treat this material and we know that there are these local t1 transformations they are also observed in amorphous solids or other similar systems and especially the solid fluid transition is it similar to melting is it just like an jamming there are many different possibilities and in general to study the behavior of seller at issue there exist many phenomenological models that couple between biology and mechanics but there is no continuum theory they start on from the vertex model and we cover this kind of behavior of like the transition and explain the behavior in the different regimes so this is basically the goal of this talk to show you how to derive such a model and two possible predictions that can be tested and say interesting things about this this kind of different material okay so two to show you how to derive this continuum theory I need to give a brief reminder all of you are familiar with the geometric approach to elasticity so I will not expand too much but it's very brief and simple in elasticity usually it is assumed that given a solid there exist a global target configuration that is free from stresses we know that in practice it is not the case now I'm not talking I'm sorry I'm not talking about zero tissue now I'm taking a short break to talk about elasticity and then I'll go back to the cell or tissue so we know that in general given a solid it it is not necessary have global stress free configuration so classic elasticity should should be refined somehow and the Assumption the way to refine it is to assume that although a global target configuration is not exist so maybe at least locally if I cut a small element of the material it can relax all stresses so the way to quantify now the solid is using a metric tensor what is a metric tensor is just a simple way to define inner product so if this is our solid we can choose an arbitrary coordinate system and we can cut out of it a small element you hold this element you can measure these these lengths these lengths and the angle between them so you have three degrees of freedom in two dimension it is sufficient to determine the metric these are this is going to determine the three degrees this the determine the actual Petric and now you can relax you can relax this element then find a new configuration which is the stress-free configuration locally so we have actual metric and a target metric and now these lenses and the angle are going to be to define the values of the target metric and we can define a measure of the formation this is just the difference between the actual metric and the target one given you can think about it like you know in the spring it's like Delta X the displacement now given a displacement you can define an energy exactly like in the spring you have 1/2 K Delta X square so now it is the tensorial form you have the elastic tensile time times the you have the strain tensor squirt basically and this is what we want to describe except that it is not an elastic solid and as you can see it is this picture from the experiment that I showed you is quite amorphous and also in elasticity if you want to describe amorphous solid you better be able to first describe ordered solids like crystals so my first step would be to describe a to give you a continuum description of cellular tissue and ordered one that is completely uniform is no any complicated structure and now some of you may see that it is not really perfectly ordered the picture that I put here there are some defects topological dislocations but this even just justify the need in understanding first the perfect exact analysis because then we can ask question about how defects looks like in this kind of material and later go to the amorphous structure okay so what I would really want to have if this is our cellular tissue I would like to have the continuum theory which means that a small element which contains many cells will be described using a continuum theory of this form with some actual metric and some target perimeter an area where I know that this perimeter term should be the analog of this term and the area should be done a log of data how to do it it's in fact it's quite simple we know the the continuum limit is of this form we have the area at the perimeter and that's it we only need to be able to write area and perimeter in terms of metric and area in terms of metric is simple its determinant the perimeter is a bit more complicated because given a metric at the point it is you know if you want if you want to know what is the perimeter you need to know what is the shape so there is an additional structure the shape of the underlying network so if it is hexagons hexagonal lattice we have these six vectors and this is the perimeter it is just the sum of the lengths so now we have an energy functional written in terms of metric for area and perimeter discrepancies and this is nice but it is far from being the end of the story because this is not the form of elasticity theory that were familiar with we are used to work with strains and stress and writing equilibrium equations etc so let me show you now why this is a problem to get this kind of elasticity theory and how to solve it and for this I want for the moment to forget about the area term let's assume that we have only perimeter term later I will put it back into the game so if you have only perimeter term we want this energy to be of this familiar form with some target metric the problem is that there is no target metric and I will show you now with an example of square lattice of cells assume that this configuration is of uniform square cells with perimeter equal exactly to their target value here is another configuration with exactly the same actual perimeter you have two different structures now assume that I stretch this isotropically so I have a different configuration and I can stretch also this one now if I want to associate strain with this configuration with which configuration should I compare it with this one or that one and the same with this one intuitively it looks like it makes sense to compare this with this one because this closer compared with this one it looks like it is highly deformed so this is only a very rough intuitive argument to convince you that what you should do is to define you have a set of possible target configuration not only one and in terms of metric you have a set of possible target geometries and the question is what is this set assuming we know what is desert the only thing you need to do is to choose the metric that is closest your actual configuration given on configuration choose the closest one and then calculate the energy and the way to calculate this set is very simple you have in your matrix three degrees of freedom you have now a constraint you this is constraint for the target matrix you have a target perimeter you have three degrees of freedom for the matrix so you remain with two three degrees of freedom so this set of metric is a two-dimensional surface in target configuration space so the same you can do for the area and you end up with two terms one for perimeter one for the area and the energy functional four for this system is going to be exactly of this form only with the additional minimization with respect to this target geometries now this is the important part of the story which make it non linear or non analytic and give rise to this phase transition in the rigidities and now I guess that it is not very intuitive now with all these symbols so I can now give you a very simple interpretation for this theory what are these G a and g p the sets of possible target geometries so here is the way you can think about is this is only an illustration although the real calculation gives similar pictures so you have the set of all possible target configuration with the same perimeter and the set of all target configuration with the same area and they can be distinct any configuration some point on this in this space is necessarily the deviate either from perimeter in perimeter or in area the relative configuration of these two surfaces depends on the shape parameter you can take for small shape parameter smaller from some critical value there are distinct for some critical value there is a shared point which means that there is only one configuration that can satisfy both area and perimeter and if you further increase it you have a set of possible target configurations that can satisfy both area and perimeter simultaneously and this now give us hope to understand what what's going on in this fluid solid transition if you are below some critical value you are frustrated you cannot satisfy both Hamilton's lis above this value you can satisfy both simultaneously and what are this value is just a simple calculation of these metrics and what you get is that for hexagonal lattice it is three point seven two for pentangle Pentagon's local Pentagon will give you three point eight one and now the question would be ok so what can we do with it so we can first start the uniform networks of cells and we can just stretch them and ask what is the effective Young's modulus so here is the result for this is an energy displacement curve for incompatible critically compatible and compatible tissue and what you get is that for the incompatible case do the energies gapped and you can now estimate the young's modulus based on approximating the curve right behind the minima to a parabola and what you get is this effective Young's modulus about three point seven two the Youngs modulus is very very small so this is consistent with this picture of having a very low resistance to local rearrangement if the Youngs modulus is very small so you can rearrange locally very easily just by pushing your neighboring friends as I said here it is three point eight one for the amorphous case and my guess it is it that it really depends on the distribution of the cells my guess is that in their simulation mode there was mostly Pentagon's heptagons and hexagons but if you have many triangles you can get completely different numbers so my guess is according to this calculation is that it's going to be strongly depends on the distribution of the polygons type that you have so this is the source for the difference but the main message is that the emergence of rigidity here is just a result of geometric incompatibility between these two sets of metrics now I have how much time okay no but I started okay so now two predictions I want to give two predictions one for the compatible regime and one for the incompatible I will start with the incompatible case okay so this is the for the energy functional that we have it is the usual one plus this minimization right but when you have two squares you know you can complete the square it is a standard exercise even in distance oriole form and what you can get is effective target geometry which is function of G bar P and G bar a and the elastic tensors plus a residual term which is not the function of the actual metric it depends only on these target metrics and elastic tensors but we still have this minimization in this case you know you have this to target geometry so you have a preferred effectively some actual metric that minimize this energy so let's say that we found that the without any external load this is the optimal configuration and target geometries for this cellular tissue so we can expand this energy around this minimizer okay so what for small deviation it is just like elasticity but if you really still aware of the existence of target geometries you get something a bit different with decent minimization so we have a regular elasticity with target metric and something that is positively definite for deviations in the target geometries now when I I'm very excited because this is exactly what describes plastic events in amorphous solids the target geometry is a fixed properties but if you change the intrinsic geometry just by local rearrangement it can cost you energy this is like the energy barrier that we had before in the incompatible case suddenly what used to be zero mode now cost some finite energy and this is exactly what we have in amorphous solids if you take an amorphous solid and shear it very slightly it responds elastically but if you allow for plastic arrangement it corresponds to changing the intrinsic metric there is an energy cost and now if you ask what will be this rearrangement you should use to minimize with respect to all possible target geometries we are while taking this cost into account and in a previous work with a tomorrow Katja and George Henschel we showed that if you take such a solid and put some distribution of defects in it of local defect and it strengthened the material strengthen it means that if you will share it there will be a shear band if you add this defect the shipment will form only for higher critical strain this suggests that you see it is just as exactly the same form so it suggests that authorship the question are the shear bands in a more in such a cell or tissue and the answer is that in fact yes and it it is a common trick in simulations if you simulate uniform lattice of hexagons you immediately get shear bands so the trick to avoid this in in vertex models is to introduce some disorder into this shape parameter and this prevent the formation of shear bands and just a second so the first prediction is that they will form a shear band and the critical strain for forming this your band really depends on the variability of the shape parameter very similar to to the to the amorphous solids so this is here mu and nu runs for P and a the perimeter and area and you have Delta G bar wa a the deviation of the target metric from its optimal value which was found in the absence of external load so this is like changing the intrinsic geometry inside the space of zero mode but because of the incompatibility it cost you some finite energy yeah yeah yeah there is a contraction in this menu yeah yeah yeah yeah yeah and the important part is that this term is positively definite so this is a finite cost for local rearrangement in this zero mode field it's something that you calculate from the G zero in the previous calculation no no just a second I'm going to complete it I need one minute is it okay so the last thing I will not show you how I calculate it but in the compatible regime of to target matrix and you can ask the following question if I take this surface and I want to impose it on a curved surface in elasticity we know that stretching must be part of the story and you can never do it isometrically but now in the compatible region since you have so many degrees of freedom the only thing that you should satisfy is satisfying the target area target perimeter and require that this should be descend to describe the surface and these are only five constraints so it appears that a simple calculation shows that you can take this model of epithelial tissue impose it on a curved surface without deviating so it is a new kind of isometric embedding of curved surface of several tissue two curved surfaces and this is important if you want to understand the bending of the pasilla I would only just want to conclude there are possible applications now because if you understand defects in this geometry or or the phase transition can be repeted not as they unjam but really as a fluid exotic face transition transition and since I'm over time I will leave it here and thank you and we'll think mark and Christina for this [Applause]