 Okay, let's get started with the second lecture. I'll come back. So yesterday I started with the first lecture on the biophysics tissues, using a vertex model. It did not really have the time to go also into continuum theories, but I think I will have time also to talk a little bit about it in the next lecture. So I will directly go to the second subjects and discuss signals and chemical patterns in tissues that sort of interplay with mechanical events and the biophysics I described yesterday. I will today first start with the morphogen gradients and growth control and then depending on how much time there is we'll also talk about playing a cell polarity. So the theme is now that we discussed sort of the tissue as a mechanical network of cells in which we have forces, tensions that can drive rearrangements, cell divisions are active processes that is an active material. And this active material is now guided with a hub of chemical signals and this is a tight integration. And the first subject today will be to discuss self-organized growth, sort of this interplay within signals and force generation in a growth process that gives rise to an organization of tissue growth with the help of chemical feedbacks. The second part I will also discuss cell polarity and as a chemical and as a piece of cells which can guide spatially and as a tropic processes in tissues. But let's talk about growth first. So now I'll go back to my favorite system the developing imaginal wing disc in the lava of the fly, which then the pupa is remodeled to become the adult fly wing. And as mentioned yesterday, this is a small tissue which grows in about 10 rounds of cell division. And one can study that by taking out these discs at different stages of lava development and sort of just measuring the area. You can also look at the cell number and see how the system grows in time. So here you see outlines of such discs at different times. This is a wild type wing disc and here we have wing disc shapes of discs of a fly that has a GFP labeled growth factor on morphogen DPP, which will become important in this lecture. Usually this GFP is there in order to see this growth factor distributed in the tissue. But it also has a small phenotype in growth. There's an overexpression of DPP in the system and these have slightly different shapes. Just for you to notice that. And this is what the data looks like. So we also quantified an anthropies of growth that you don't want to go into today. The area of the tissue is a function of time. And this is a single logarithmic plot. So the logarithm of the area in micrometer squared is a function of time in hours. And a straight line would be an exponential growth in this plot. So if I take this experimental data, the red data points are wild type wing discs. The blue are GFP DPP wing discs. If I take a smooth curve as a fit and it's tangent to this curve is a growth rate. And therefore you see that the growth starts fast and gradually slows down. And maybe it eventually stops and goes to zero. Now I define the area growth rate G, units inverse time, which is the time derivative of area divided by area. And the solid lines which are used as a fit here correspond to a growth process where this area growth rate is not constant but decreases systematically exponentially with time. So there is a growth rate G zero at time T zero. Tor is a characteristic sort of relaxation time in a physical language over which some process relaxes to zero. And so this growth rate relaxes to zero. If I take now this relation for the growth law and I integrate this to determine how the area evolves in time, I get this expression. This is somewhat complicated form. The area is the exponential of something one minus the exponential of time. And this function is the fit function used here. And from this fit one can determine the important parameter torque, which is the characteristic time over which the growth rate seems to decrease in this growth process. And this is for these discs about 30 hours. It's not this is sort of a new time that comes in the kinetics of how growth changes with time. That's now sort of the phenomenon that we're trying to understand. We have a group of cells that grows and that stops growing. That stops growing at the right size. So this has to do with how does the system can determine its size? And also the question of how can many cells together grow in a coordinated and controlled way? That doesn't, of course you could say each cell has its own cell division rate and they just grow exponentially and the thing is completely uncontrolled until it dies. But here this has to be very controlled and slowed down gradually as a collection to find the right size. And the right size is the final size, which is well-defined here. If you put T goes to infinity here, this exponential is zero, and you get a well-defined size, A zero, E to the G zero torque. That's the final size of this growth process. And the cells have to do that somehow. Of course there are some hormones coming from the larvae to tell this what to do in particular when transitioning from the growth phase here to the pupil phase. This is done by hormones that come from outside. But the growth phase itself must be coordinated between cells and I think of this as a self-organized growth process where the decision of cells, whether or not to divide and to grow comes from chemical signals. And I'd like to discuss now how this can be done and propose a mechanism that actually can produce exactly this behavior. Now one important question is, is this growth process sort of homogeneously distributed in the tissue? Does every piece of tissue grow at the same rate or are the pieces of tissue we grow faster than other pieces of tissue that would be in homogenous growth? And with Albert Wattlich some years ago, we looked at this by using phosphor histone staining to see mitotic cells in fixed wing discs at different stages. And here you see wing discs at different stages and you see all the little spot dots are cells that are just doing mitosis. These are spotty patterns so it's a stochastic type of process which cells divide. But more or less it's spread evenly through the disc. And here's a quantification of that. So this is sort of consistent with the idea that the growth rate and the cell division rates are essentially constant everywhere and the whole tissue grows equally everywhere. So we have to understand homogenous growth and we have to understand the growth rate that decays with time and understand how this can be regulated by chemical signals. Also more recently we now have these wing disc and culture medium where we can actually see cell divisions happen. We haven't really analyzed this in very much detail but I would say roughly this is consistent with this homogenous growth. You see cell divisions everywhere that during this movie every cell divides about once and you cannot see regions where there's more cell division than others. It's to first order is a homogenous growth process. So I introduced yesterday compartment boundaries and already mentioned that they are important organizers for the chemical signals and patterns that form in the imaginal discs as they grow. And this is also a key organizer for growth regulation. So I'll focus here on the AP boundary. And I explained yesterday how it can be maintained and established with the help of mechanical tension. What is now important is to realize that there are chemical differences on both sides. The posterior tissue expresses a molecule that spreads a morphogen called hedgehog. Very important signaling molecule. But the posterior tissue doesn't possess the receptor for hedgehog and the posterior tissue does not activate the signaling pathway of hedgehog. The anterior tissue has the receptor and it's sensitive to hedgehog ligand that spreads from the posterior side and activates hedgehog signaling in a region close to the compartment boundary where the cells first encounter the hedgehog molecules that come from the posterior side. So there's a narrow stripe of cells near the compartment boundary but on the anterior side which are different from the others and in this narrow group of cells one important gene is expressed, DPP or decapentaplegic which is expressed in a narrow row of cells is secreted and which also spreads in the tissue. And DPP is considered to be a morphogen. The definition of a morphogen is that it is locally produced and secreted in certain regions and has signaling effects after moving through the tissue to cells at a distance from where it was secreted. So DPP has a maximum near this anterior posterior about the boundary, it spreads in the tissue and it forms a graded concentration profile in the tissue. And it's a very important signaling molecule so the name already hints at the fact that if it is perturbed or absent there's a dramatic phenotype and the fly will not make it. DPP is involved in tissue patterning so many genes expressed in the disk are expressed in patterns, for example to create later the vein patterns, they're in gene expression patterns established early and such patterns depend on the DPP signal and there are genes which are expressed in a region of high DPP levels and not in regions of low DPP there's a whole zoology of patterns and genes that are known here and DPP is involved in the tissue patterning but secondly DPP is an important growth factor it's known to be involved in growth control and that's what I will focus on here today. So here you see a wing imaginal this is a pouch region which has a GFP DPP expressed and the GFP signal is very strong in this region near the compartment boundary where this monolithic is actually produced but you find it everywhere in the tissue where no expression takes place but you find the DPP that is emerging from this source region and one can quantify the concentration gradient or less the fluorescence gradient using fluorescent microscopy of DPP as a function of the distance from the source and this can be well done because if you're going to the... on this side here the source is in the anterior tissue and if you're looking at the posterior tissue we can use the compartment boundary as a zero coordinate and then we measure distance from the zero coordinate and plot the intensity as a function of this position x and the red beta is the DPP signal after subtracting background. Question and here I also use a fit to this profile which is measured which is an exponential function and one can usually fit this profile very well with simple exponentials you can argue whether this is a good fit or a bad fit but one can fit them and by fitting that there is one parameter that is determined that is a characteristic length the distance over which this profile typically decays and the red data is DPP GFP I also show in blue a different molecule wingless which we also quantified now how do we can... how can we discuss such a graded concentration profile and I will take some visibility here sort of continuum picture not cell based which we could also do and think of this as a molecule that is secreted locally as a source and then it can spread which essentially is an unbiased diffusion like motion but one should not think in terms of simple Brownian motion this is really an effective diffusion that involves cellular processes can involve cellular traffic and processes and then it is binding to receptors on the cell surface it can be internalized to cells and typically when it's inside the cells and endosomes it can trigger signaling activity can have consequence and target genes and then it's taken out of the pool of molecules that's spread in the tissue so this looks like an effective loss or of course it's also at the end the molecules in the cell are degraded so effectively there's a lost term in this equation so we have a time dependence of this level is governed by a source which is only non-zero in the small stripe where it's produced there's a diffusion term because it can spread if the tissue itself moves and there can be self flows because of growth then we also have a convection term or there can also be dilution effects and we have a degradation term corresponding to molecules being taken out from this mobile pool and now if we look at the concrete case of DPP we find the growth turns out to be quite slow as compared to the dynamics of the establishment of these profiles and therefore we typically neglect these terms first it sort of relaxes quickly to a quasi-steady state and convection is not very important but of course that's just a simplification for the discussion in general one has to take these terms into account now by neglecting this and thinking of it as a quasi-steady state governed by diffusion and degradation this is now a differential equation for the profile which can be solved and this has exponential solutions that's also a reason why the exponential fit is a useful one because it connects to this limit and the decay length that one can measure here is in this model related to the ratio of the diffusion to coefficient and the degradation rate so this is an effective diffusion coefficient tissue diffusion coefficient one has to be careful with the definition what it exactly means and this is sort of an effective degradation rate which one can define in more complex cell-based models which have a lot of trafficking process involved and I don't want to go into these details here today yes? DPP is not a transcription factor but DPP triggers a signaling a DPP signaling pathway and the DPP sits in endosomes can be recycled back to the cell surface or it can go to degradation pathway but in these endosomes it can activate the DPP signaling pathway which then brings transcription factors to the napliers yeah? yeah? so effective diffusion is something which one can measure and we have values for it and it is what it is but the point is it is not the same as Brownian diffusion of a molecule in the cell or even an outsider cell it's the apparent diffusion coefficient if you follow the molecule of a long distance over many cells yes? that do respond yes the sense in the anterior and in the endosomes yeah so I'm not 100% sure about exactly how the hedgehog system works here but I suspect it's similar to what I'm doing here it would be a diffusion and degradation term you get a graded profile and then you have a threshold and it gives you a region where it is active as I say if you take such a model seriously then you could also have a gradient of hedgehog and then you have a threshold and it sits at a finite size but it's not a subject of my talk and I don't know as much about this system and I don't know about this system but that's at least if you look at hedgehog you see it being graded and you see only a stripe activity but what exactly the mechanisms are why it is in a stripe I don't want to discuss here because that's not a subject of my talk today you see okay so we have these concepts of this effective diffusion coefficient and effective degradation rate and I could give a separate talk only about the subject but that's not the subject I don't want to go into more details today now I want to talk about growth control and now the important sort of striking issue here is we have a system that grows self-organized homogeneously in space as you see here in a chemically coordinated manner with the help of growth factors but these growth factors are not at all spread homogeneously in the tissue they're totally graded there's a huge concentration here and then they're graded so what how can we make sense of a situation where growth factors are completely inhomogeneous very strongly inhomogeneous while growth is homogeneous and can we use all these ingredients to build a system that does the right thing and that behaves as we see in the tissue that's what I want to do now and so the idea is the question is what are these mechanisms of self-organized growth control and just to put it on context there are a number of ideas in this context that we should discuss so the early suggestions were that it's not the DPP levels that somehow stimulate growth because there's an inhomogeneous but that is the slope of the profile and if you take a simple picture of a simple source and a simple sink somewhere else you get a linear gradient profile and then the slope is constant everywhere so it's a simple picture that slopes may do the job then another important idea is by Lars Hofnagel and Boris Schreimann that mechanical stresses play an important role to homogenize growth so the picture is that if you have locally self-studying to divide more or grow more quickly than elsewhere you sort of build up local stress and pressure and growing against pressure is suppressed physically maybe also by mechanosensing processes and this sort of balances sort of the increase in growth by a mechanical feedback one difficulty with this idea as a general organiser of growth in the whole issue is that it sort of needs growth regulation which is sort of or growth that is sort of radially organized in the center there is more growth than in the periphery and therefore you build up more stress in the center well if you have a line that organizes growth you would not build such stresses if along the line things are constant and only in the second dimension you get in homogeneities so this is certainly an important component but I want to propose a different idea and of course these ideas can also work together they're not they're not necessarily incompatible with each other but one which is really based on using chemical signals alone they can already create homogenous growth and then you can think of such mechanical effects as an additional feedback to help smoothen fluctuations so I want to propose a temporal rule rather than a spatial rule temporal rule that is sort of the time dependence of the DPP signal which regulates growth and stimulates growth and this can do the job without the need of mechanical feedback as a question and then how fast do you grow but then how fast do the cells you can try to make a model I would like to discuss it with you but you have to control it if you regulate it what you're describing is not really regulation so now to get to develop this idea let me first start with the quantification of the DPP profiles of a time as this imaginal disc grows inside the larvae so let's work that was done by Otto Gvaltlik in the lab of Markus Gonzalez Geithan and here you see profiles of DPP in the wiggy vaginal disc at different times you see at times here as a function of distance from the source and then they can all be fit by exponential functions and there are two important things to notice so one is at the beginning the amplitude the levels are small at the end they are high high levels furthermore of course the discs at the beginning are short that's why they stop early and later the discs are larger so they reach out further but also this decay length the lambda that we use as a fit parameter here increases with time so the small discs have a small lambda the large discs have a big lambda that's why I call it say here this lambda of T so as the tissue grows somehow this decay length of the DPP profile increases and as I mentioned that this deeper layer as we always think of it being related to diffusion coefficient and degradation rate this implies that these things are dynamic quantities from this data we find a remarkable property of the profiles that we call scaling they have two different scaling properties if we look at how they change as the tissue becomes larger I mentioned already they increase in amplitude and their increase in reach but both of these changes follow simple laws if we look at how does the decay length the length lambda increase with the system size we see there is a clear linear correlation essentially proportional as the tissue grows the range over which this profile decays follows the growth secondly the amplitude increases as the tissue size increases here plot area of the tissue versus tissue size and this is now a double logarithmic plot and in the double logarithmic plot I can fit it with a straight line and a straight line in the double logarithmic plot is a power law this means that the maximum level of the amplitude of my profile increases with the area to some power beta and this power beta is about zero point six in this tissue and this is also what we call a scaling this profile scales whatever that means I should also stress that a very simple minded way to understand such a scaling would be that the tissue grows and that the profile is just stretched with a growing tissue that would be one mechanism to get such a linear relationship between decay length and size however in such a scenario if that was the case the concentration would go down because you have to keep the same molecules in your distribution and as the system locally grows the same number of molecules is now distributed over a larger area so the concentration is diluted and goes down we see the contrary the concentration goes up so there's a non-trivial process here taking place and it goes up in a well-defined power law for the moment it's just a phenomenological observation it's not a growth rate, no it's just means that if you double the area and doing growth C0 goes up by 2 to the beta that's what it means increases as the size of the tissue increases as the tissue grows so the maximal length of which the greatest measure increases with time if that was your question this we don't know yet, we'll discuss it later for the moment I'll just tell you what we see not yet how it's interpreted fixed experiments averaged over many disks do you see? because life, yeah it's only very recent that we can culture these disks and it's this work was redone and fixed at that time the only way to do it was to take them out but also these current cultural experiments they behave differently than in vivo so it's better to use real disks that grew in a real larvae okay so we have the scaling behavior now by using sort of these exponential fits I may have biased the analysis I want to show you now that this scaling is independent of whether my fit function is a good fit function or not and for that we can just normalize these profiles so I define a relative position relative to the length L up to the margin of the tissue R which goes from 0 to 1 and I normalize also the amplitude with the concentration at x equals 0 and then I can superimpose at different sizes these profiles and I get sort of a characteristic average profile another way to do that is to superimpose all these curves in bind them according to how often a curve passes through a little grid element and then make a histogram of this accumulation of curves and you see they're superimposed on one master curve even though they belong to a very different system sizes and in this case we now define a function f of R which starts from 1 which goes to which starts with f equals 1 at 0 and goes to r equals 1 and is characterizing the invariant shape of the profile as this tissue grows and then the actual profile can be sort of determined by multiplying this with a time-dependent amplitude by taking into account that the tissue grows so we have the idea that profile does not change with the increasing tissue size the amplitude changes and it's stretched with growth now we come to the question of what is going on here I mentioned before that this lambda we think of it as a ratio of an effective diffusion coefficient and an effective degradation rate both of which are properties one can measure but which result from complex cell biological processes involving intercellular trafficking don't want to go into this but the cell can regulate and change those quantities and one can try to see what changes when lambda changes during growth and here we estimate using Thrap experiments this effective degradation rate is a function of tissue area we've also found from our work that the effective diffusion coefficient seems to vary little while the effective degradation rate varies a lot and the suggestion here is that the effective degradation rate decreases as the tissue becomes larger therefore as the tissue is larger molecules have a longer lifetime and even if you have a constant source secretion rate you still increase the levels that would explain both things, it could explain why the overall amplitude increases because the degradation rate is decreased as the tissue becomes larger and it explains why the lambda goes up and if k behaves like one of l squared as this data suggests lambda is proportional to l so our interpretation here is that somehow the degradation rate of DPP varies with time it's probably regulated itself in order to achieve scaling of this profile and therefore we get this phenomena which I showed you now this now raises the big question of what regulates the degradation rate and how and why yes here I don't have a beta on this slide the increase I didn't say it explains the exponent it's probably exponently a full self-controlled for need of a full system, everything together but now this raises the big question of what regulates and how is this degradation rate regulated and that's the subject of the scaling mechanism which I don't really want to go into in my talk today it's also a subject which is not yet fully resolved there are some very important ideas I just want to highlight Nama Barkhai's work so she has proposed a system of regulation of the degradation rate by a molecule that she called an expander and she called this whole process an expansion repression feedback the second molecule is sort of not known exactly what it is it spreads in the tissue and in her expansion repression feedback model the expander has a second equation similar to the first one for the DPP the second equation with a source and a diffusion and degradation it's considerably typically long-lived it is produced at regions where DPP is low sort of when a gradient dies out too quickly there's a large region where this expander is produced it diffuses in the tissue and the degradation rate is suppressed by the expander and then you get a feedback system which has the features that it can do something that looks very much like scaling so that system is a proposal for how the scaling could work independently of this work in collaboration with Markos we also proposed a regulation mechanism that can achieve scaling we also introduced a molecule that regulates degradation let's call it expander in our original version it was a long-lived molecule that was present and didn't need to be secreted all the time and essentially as the tissue grew it was just diluted and therefore the concentration decreases, the concentration decreases like 1 over L squared if the tissue has an area L squared and if now this degradation rate is proportional to the expander molecule, imagine that each expander molecule is needed to degrade a DPP molecule and you get such a law and then you get exactly what you need, K goes like 1 over L squared and this thing scales beautifully so these are two possibilities, I don't want to I think both of them don't really explain what's actually going on but these are good ideas to follow up Nama Baike also pointed out that a molecule called pentagon might be an interesting candidate for this expander even though I'm not sure this is really definite but it's an important idea so pentagon is a molecule that somehow controls DPP signaling, is spread and secreted the tissue has many of the properties that an expander molecule should have the name comes from the fact that in a pentagon mutant you're lacking one vein which is a bit surprising phenotype in the context of this discussion but it clearly affects the DPP profile and can regulate the behavior of DPP in the tissue and DPP signaling but I should say we still don't really know how this expander system works, I don't want to go deeper into the subject let's from now on just suppose there is a beautiful expander system at work which makes this gradient scale as the tissue grows and take the consequences of that now what is the implication or purpose of scaling? first it's somehow an obvious sort of idea when you see that these profiles scale as the tissue grows and they scale in a non-trivial way as I said, it cannot be explained simply by the growth itself there must be something on top that regulates this as the tissue grows this hints at this being a sort of a part of this general system that is able to generate scalable patterns this is a system that in for example many different fly species is at work and creates wing patterns and wings and structures at different sizes so this developmental systems must intrinsically be able to create scalable patterns and of course having scalable morphogen signals could be important for that secondly I'd like to show you that the scaling naturally provides a simple growth control system so if the system scales as I'll show you it can be used for growth control in a very simple way and to get at that let's first look at one interesting quantity which will become important so if I defined a DPP level at position r at time t I can also look at how does it depend on time when the system grows and evolved in time now because the system and the profile is just rescaled during growth the time dependence essentially only comes from the pre-factor so the time dependence that we find in the local region of the tissue if we move with the growing tissue so the constant r is coming from the change in amplitude of course it's modulated by the shape of the profile but if rather than looking at the absolute time dependence we're looking at a relative rate of change if we divide if you if you ask what is the relative rate of change given the levels that we're having what is the percentage change of the concentration per unit time this ratio is independent of position it's everywhere the salmones in the tissue and that's a signal that can be detected everywhere in the signal in the tissue and it's the same everywhere so this could be used control homogeneous growth even though the signal itself has a spatial profile now what does the signal look like if we now take our measurement of the amplitude C0 as a function of area we can ask how does this signal behave in effect if I have this power law which I introduced before and I now calculated C dot divided by C the relative rate of change using this power law I find that it's the same as beta times A dot over A beta is the slope of this curve and A dot over A is the growth rate now this is a correlation this shows us that this signal which is available everywhere in the tissue and has the same value everywhere even though the signal carrying molecule comes in a gradient it's exactly proportion to the growth rate and that's the thing that we want to control so from this correlation from this observation came to the suggestion as a hypothesis that the causality may be such that C dot over C actually controls growth the correlation comes from the fact that it has growth control at this level implemented what it means is that the DPP signaling system is activated it's a very complex signaling process involving gene expression coupling to many systems in itself effectively measures the relative rate of change of the signal and as an output after this complex black box has done its work generates cell growth that is proportional to C dot over C and the provision beta then describes a property of this whole growth control process in the cell that I don't understand in detail it's a black box it measures the relative rate of change as an input it produces growth at the output and it has this property beta is the parameter describing the system and this is at this level hypothesis this can explain this curve but we have to find out whether this is actually true whether this is the mechanism that works let's first use theory and try to understand whether it can work that's if we implement that does it do the right thing first I'd like to comment on why would a signaling system measure C dot over C that sounds at first a little bit surprising or unusual but the opposite is true it's the most natural thing a signal system would measure and that's what I'd like to explain to you in fact measuring C dot over C essentially means that you're having an adaptive sensor a system that does not measure absolute levels but adapts its sensitivity to the actual levels that's for example what all signaling systems have to do sorry all sensory systems have to do your eyes your ears all sensors are adaptive sensors they detect not absolute values but relative changes and therefore they can operate over large ranges if you have a very dim signal you may have to be much more sensitive you need your eyes become more sensitive to in the dark because you want to see very small differences if you have sunlight your eyes get very insensitive the same thing is also true in chemotaxis you see there clearly only works if you if you measure chemicals in a sensory system which is adaptive the olfactory system is adaptive it's also a chemical sense and we propose that all these these sensory systems that cells possess to detect incoming signals from their neighbors during development must also be adaptive systems in order to be able to operate over large ranges of concentration and the idea that systems detect relative changes was also stressed in other papers in particular Uli Uli Alon has termed fault change detection as the term that describes the measurement of such a relative change also in the context of chemotaxis Barca and Leibler have done seminal work to discuss how such adaptive systems can be constructed and can do that very easily with simple feedbacks one can make can build a sensor that is adaptive and has the property that is output measures essentially C dot C of course cells don't take real time derivatives and it also doesn't make a real sense what in practice happens the sensor has an internal measurement time let's call it TOR and then it measures the relative change during a time TOR and this is then a coarse-grained version of the time derivative which essentially does the same thing okay so with this I have established that is the most natural thing for a cell to detect relative rates of change that's the best thing it can do if it does it and if it now uses that information to regulate growth then we are in this system that I'm describing that is supported by this data now there's just a couple of comments of course this law cannot work for arbitrary large signals that there's a maximal growth rate the cell cannot go faster at so in practice the growth rate will be cut off that has a maximal rate so for small signals there may be a linear increase and at some point it has to has to saturate and now if we have now such a law we can think of what's happening so we have a system that grows because it grows the gradient has to be rescaled of DPP because the gradient is rescaled by a reduction of degradation rate molecules have a longer lifetime the DPP profile builds up because the source brings in more molecules and this increase of the DPP profile in particular you see not going up now triggers a wave of growth and this growth now completes this circle and now we have a system that self-organized that's the level of organized growth each wave of growth triggers a new wave of growth of course in a continuous manner and in the end it has to happen in a way where the growth rate slowly decays until the system has reached its its correct size if each wave of growth increases a bigger wave of growth then you're in trouble the system will blow up has to be in the right operating range now let's try out these ideas with the type of concepts I also introduced yesterday let's build a system as a model that has all the ingredients only local interaction rules very simple rules and let's see if the system does the right thing so we take a vertex model where I've shown you yesterday that it can grow but now rather than what I did yesterday where the growth was completely stochastic I now want to control it 100% with molecular concentrations with DPP so each cell has now a DPP concentration I is a cell index this DPP concentration can have a source term if the cell happens to be in the compartment boundary it has a degradation term the diffusion is described by exchange between cell neighbors so now discrete diffusion model if there's a level difference between neighboring cells there's a diffusion flux and it defines a diffusion coefficient we also build in our model the hedgehog system we start from two compartments you also saw the simulation yesterday with two compartments with the compartment boundary that is maintained with extra attention now we have a hedgehog's level yes thank you there's a typo here the hedgehog system has sourced in the whole compartment where it's produced there's also degradation to give to the gradient that I talked about that we mentioned earlier so one can speculate a model for the hedgehog system here then whenever the hedgehog levels are beyond some threshold in the right compartment the source is switched on we need a gradient of scales that's important so we need to control the degradation rate and we do that with an expander that is not produced because it's there from the beginning it's just diluted that's the dilution process so it diffuses and is diluted whenever the cell divides and then the only remaining ingredient is the things that should grow so we say the cells grow the preferred area of the cell that I introduced yesterday is time-dependent and the growth rate is proportional to C dot over C by this growth law and C dot over C we have available in our calculation as well because we know what C dot is, we know what C is now whenever the cell has grown to twice its initial size then we divide it and we give it a random cell division orientation and let's try of course all these ideas that I presented should work but we're not sure if it really works we have fluctuations and mechanics and so on so let's try it out so here we should first show you the cell divisions we start from a small piece of tissue there's a compartment boundary here we have a growth rate, a growth law we have cell divisions when the cell area is doubled we have it expanded it was in scaling and let's look at what happens it actually grows in a beautiful way homogeneous growth growth cell divisions are stochastic for interesting reasons even though the whole process is deterministic the way this is set up but it has to do with the irregularities of the of the network you see the compartment boundary growth slows down with time the red cells are the ones that divide and growth until it stops that's what a system does I can now show you the underlying signals that produced what would you just saw here in the simulation here you now see the two compartments posterior and anterior there are some, as I showed you yesterday that's taken care of by my energy function yes there's no contradiction because it's an active system it uses a food to be able to perform work yeah that's what's described by the mechanical model but of course it's a non-equilibrium active system which can produce forces it wasn't my whole point at the beginning of the lecture we can discuss it later so we have a posterior department we have an anterior department compartment we have a compartment boundary we have hedgehog and blue we have dvp in in green so here you see first the system viewed by its compartment boundaries then we switch to show real that hedgehog is only secreted here then we have a gradient because of degradation and the dots here are those cells which are above the hedgehog threshold and which will secrete dpp and now you see the dpp signal dpp is produced where these dots are and it spreads on both sides and this with the expander it does scale therefore this whole profile stretches as it grows and if we now take this model and we plot the area as a function of time we have this red curve and the black dots are the growth curves actually the same data I showed you at the beginning for the wing growth so we can by just putting beta equals 0.6 in our growth rule we can fully account for this growth curve and this was very promising and we found this quite exciting next thing we did we also looked at a second organ that's called a halter and the halter grows from what is called a halter disk it is some sort of a little winglet it doesn't have the role of a real wing but it's used in fly for stabilization by the fly it's much much smaller than the real wing but it grows by the same principles and the same is a very similar disk here you see the growth of the halter disk and it can be described by the same model but slightly changed initial conditions which depend on the fact that the halter disk starts out slightly differently so this model can account for the growth process fully that we observe now so far i've only shown you that we have a model that can explain the data and if you put sort of at work in a simulation actually works, it's stable and works it doesn't prove that the correlations we see in the data follow from the causality of this model now one next step because we were sort of intrigued by this but we were not sure that whether this is actually the right mechanism we were wondering whether there are tests that are very strong where we have a situation where the same components are at work but the timing and the time dependence is completely different and whether we can still use the same arguments and for that purpose we looked at the I-imaginal disk which is the disk which is important for the fly eye and the I-imaginal disk also has a boundary between a material and posterior region but rather than being stationary as in the wing disk it moves it propagates through the tissue and it leaves in its wake a tissue which then generates this hexagonal lattice of a material and it moves at a certain velocity so as a consequence as there is also a DPP source it is furrow that moves through the tissue we have now a moving gradient and this is a completely different spatial temporal process so let's see whether in this system our idea still works that's a very strong test of the idea so here the sketch again we have the posterior anterior compartment we have a furrow morphogenetic furrow which moves at a velocity vs in its wake there's the formation of a mentity I will not talk about this at all I'm interested in the tissue here in front the anterior tissue into which this furrow moves this is the tissue which actually grows and where cell divisions take place and we want to understand those so here you see a cultured eye disk this is the furrow these are the forming of a didger this is the side of the tissue we're interested in the DPP source is along this furrow this furrow moves at about three micrometers per hour this is now the movie it runs and in front of this moving furrow we have lots of cell divisions here now the question is is this growth and division process in this interior tissue driven by DPP and if yes is it driven by the same rules that we identified in the wing there are some division of left side they're not DPP driven and they're not contributing to growth that's subdividing in the wing under the conditions of the wing I'll explain later then we can talk about it because using the same model now here changes many things and the question is does it change things in the way they actually have it let's first look at what goes on so we have a we can measure the anterior distance LA and the posterior distance LP and here you see the time depends as a function of so posterior is shown in black now as this thing moves a constant velocity through the tissue and there's essentially no growth here the posterior size just increases linearly with time and the slope of this curve is the measured velocity of the for all that move through the tissue the posterior tissue is just increased linearly as the for all that's the slope here the anterior tissue is more complex because the for all moves it's often eaten up by the incoming for all and decreases but at the same time there's a lot of growth there which makes an increase so initially growth wins because it's quite large as other growth wins and finally the for all wins and as a region where the size of approximately constant which is also useful region to look at because I think that's simpler to study so the LP DT the posterior is just V S times T and the interior with is now this combination of growth and the moving for all so what matters here is the growth along the X axis integrating the growth rate gives us a overall length gain and we have to subtract the for all understanding places now let's discuss wing versus versus I in the winged is we have uh... fixed source and we have a profile which scales as the tissue growth because of size-scaling now with this growth law we get a profile that in a rescaled system it creates an amplitude and you get homogenous growth ever been a tissue we now apply the same idea to the ideas let's say for simplicity the case the size of this issue is more than constant that we have to worry about scaling the simplest picture is that this thing is just translated for the tissue space because of the velocity v s and so this is common genius growth now from this configuration we go to a shifted configuration which means now that in front of the for all levels go up with the maximum further way and then go down and in the back of the for all they go down so if the system is stimulates growth within relative rate of change increasing the level growth will be driven in the front but not in the back and it will be driven in a spatially inhomogeneous way that has a maximum of certain distance from from from the origin now the question is is that's what's happening and can we account for what's happening that's the prediction of the of the model and as we can quantify and we can test it and see what it works so I'm not talking about the back I'm not talking about the back I'm just saying this model explains why there's no DPP dependent division happening but other pathways can still regular division and but they're out of my game but I'm talking about the front but as a big asymmetry being back in front that's what I would emphasize you can the DPP effect can be switched off in the back while it's completely active in the front with this mechanism that we propose which explains that this thing can do its own game completely separate from this now we can sort of collect information we have a interesting what a cell receives a local level C cell of T of DPP and now the profile for simplicity I say it's just translated so is that there's a time independent profile C of X but I have to look at it at position Excel and Excel is measured relative to the position of the furrow because where the furrow is is my maximum of my profile now C cell of T now depends on the dynamics of this tissue C cell of T comes from the growth induced contribution to the velocity and to the furrow induced contribution I showed that here there's these two contributions to the tissue dynamics and sort of we have to plug that in here and now we can calculate what is the local C dot over C signal that the cell receives in the spirit of what I explained before and you see if I take the derivative with respect to time of C cell of T I have to first take the space derivative DCDX and then multiply with DXDT so DCD DCDX is this gradient here and DXDT is this factor and if I now divide by C cell this just gives me C so this is now a relationship calculating the local C dot over C now we propose that this signal drives local growth by our law that G the area growth rate is beta minus one C dot C cell over C cell now first I like to mention it's related to what I just said before the local growth rate changes creates a gradient of this growth velocity the relationship in the growth velocity VG and the local growth rate in X direction is just this gradient furthermore the full growth rate which we generate here is the sum of growth rate in X and in Y direction and we introduce here an growth anisotropy parameter which tells us how strong the Y growth is compared to the X growth that's something we can measure epsilon is this growth anisotropy and if epsilon is one it's isotropic epsilon zero is completely anisotropic and is near one if you do the measurement so G is two times GX now we can plug this together yeah we put this expression for GX in here we put this expression for G in here and then we can equal those two things and we can we first write C dot cell over C cell is now a factor combining beta and one plus epsilon with a gradient of VG we then use a new coefficient gamma which is our beta times one plus epsilon which now depends on the growth anisotropy and now we can this must be the same as that so we get a equation which turns out to be differential equation in space for the growth pattern so given the profile of DPP which is time independent but moves through the tissue given the velocity at which this this profile moves through the tissue we can now calculate the spatial dependence of growth if you know what VG of X is we can we can calculate G of X also we have everything this allows us to calculate the position dependent growth profile as a solution of a differential equation and this differential equation can be solved which is quite nice and compared to experiments so that's the solution to the differential equation there is a growth rate that can that is minus VS that's the velocity of the furrow and then the space derivative of the normalized DPP profile to a power gamma where gamma is this combination of our growth control coefficient and the growth anisotropy now this can be tested experimentally so in this i-disc we measure a proxy for C of X it turns out to be most convenient with many experiments to measure the DPP signaling profile using in this case it's a PMAT that's probably the matter of the transcription factor that's triggered by the DPP signal and that's here shown in black so that's something similar to the DPP growth curves that I showed you before it's a different proxy for the same type of information and we fit a smooth curve to it in order to be able to plug it into this into this function and calculate GX and so we take the experimental data we fit a smooth function we calculate this and we know everything on this equation we know what gamma is we know what VS is we calculate G of X and what we get is the dashed blue line and we can also measure G of X in the experiment by counting cell divisions and looking at growth and we get the red data points now you see the two are quite similar remarkably similar the main difference is a little shift in time if we slightly shift the blue curve to the left and we actually can account completely for the data and what this means is there's a time delay between the signal and the event of growth and we can measure this time delay by using this shift as a fit parameter this essentially means it takes one hour between the C dot C signal has arrived at the cells and the growth to be outcome to be effective that's the shift and now we can account for this curve I should say we did the fit only up to here and we didn't we didn't take seriously this end piece but if you look at this this piece of data which deviates from the smooth function is consistent with the deviation down here it even seems to work in these places here even though we didn't actually do it because the data is not reliable there now this also works after we manipulate things and I don't want to go in all details but we can manipulate the the system and we all and as long as we're going to break it completely this works for example here I'm looking at a pentagon mutant the pentagon mutant as I showed you before has a it has a scaling phenotype in the wing disc it also has one in the eye as I'll show you in a moment it changes the profile of the PMAT as it would also do in the wing disc it changes the gradient of TPP and if we use this change profile of PMAT now as an input use our same model we can fully account for the experimentally observed growth profile in this mutant if we adjust the delay from 1 hour to 1.4 hours I should mention that we can also study scaling in the eye disc there is scaling in the wild type so if we in the wild type if we plot um different profiles on top of each other and we scale them um taking into account sort of the varying size LA which doesn't vary very much but it's enough to reveal that when we in the wild type they all overlap and in the pentagon mutant they don't overlap anymore the pentagon mutant has a scaling mutant similar to the wing as well and then maybe one other example that I don't want to put too many but you find them in the paper um we can even go to extremes and expressing DPP sort of brutally essentially everywhere um in this compartment and it still works now naively you won't think by expressing everywhere against a completely flat profile it's not really true when it gets a profile that has still a maximum here but it doesn't go down it stays finite because there's expression now everywhere of DPP this model predicts the blue or this red curve the red data points are the data does account for this dramatic quantitatively can account for this dramatic perturbation of the system and there's now significantly high delay for reason we don't understand but this model can explain all of this and it was constructed in conditions of the fly wing which were totally different so for us this is extremely strong evidence that this is actually what controls growth in these disks so by the way you mean growth you mean that you said growth or you mean I mean I have the right to say and so what do you think so in the model I showed you before the model was that the C.C. drives cell growth and division happens when the cell size goes to a threshold in our experiments we can measure cell sizes and we can measure division rates and we can measure tissue lengths and we use those to deduce the growth profile the area growth profile so you think the delay is the time to change the sub-cycle? no the delay we sort of I treated all of the complex signaling process from the DPP pathway to the growth output to the fact that somehow material is produced to make more proteins in the cell and ribosomes activated and what is needed to actually grow this whole system we treat as a black box is characterized by one parameter beta which describes how C.C. drives and by one delay parameter which tells how long it takes to get from input to the output we describe the whole system by two parameters without explaining exactly how these two parameters are created in this very complex system the next step would be to really understand this whole signaling system and see what times internal times give rise to these deltas and betas we're just saying we need that two important properties of this whole huge signaling process from DPP to growth that can be described by beta which tells me how sensitive growth response to changes in relative changes in DPP level and how long it takes for this response to to appear after the input it's unimportant but it's just a choice for our vertex model simulation in our vertex model we use different versions completely unimportant we first started with models where the timing of cell division was regulated by DPP not the growth rate and the growth was always set by the division rate these details are not resolved by what I'm explaining you could implement what I'm saying in small variants and it still follows the same principles and it demands a lot of data to exactly figure out which variant does it work but the point is this is a very general principle that is rigorous and robust and can work in a number of variants but the principle would be the same no I don't assume anything of this type in one version of the vertex model which I use for illustrative purposes I use the idea that C dot C gripe cell growth and cell division follows when the cell reaches a certain size on the general level I only say tissue grows the area of the tissue is linked to C dot C how it's implemented in detail I don't discuss here whether cell division drives growth or the growth size division I don't comment on here it's not necessary to know this for the principle I'm discussing okay so that's there's a few minutes left but I wanted to discuss a few things so first example was how can we use a feedback loop between growth factors and cell division and cell growth to self-organize growth of the tissue and the second example I want to give in this lecture is to discuss a little bit plane of cell polarity there are 15 minutes or so left is it right? 20 so I will use there are many minutes to give you a brief introduction to cell polarity as a second important system that we've been very interested in to discuss how chemical signals organize the tissue a bit an epithelial tissue and we're staying with the fly wing going back from the eye to the wing and another very far distance and I introduced you this epithelium which has an network of adherence junctions close to the epical surface and this is related to the fact that individual cells are polarized perpendicular to the epithelial plane there's an epical side and there's a basal side which just means that the tissue has two different sides what I now want to talk about is a different polarity a polarity in addition to the simple epical basal polarity namely an asymmetry of cell sides in the plane of the tissue that's called planar cell polarity and that's a very important concept and system to guide morphogenic processes in particular we have to understand not only how tissue size arises which I was sort of alluding to in my last part of my growth control but also how shape arises and shape has to do with anisotropy and about doing things different along one axis and along another axis and for that one is planar cell polarity and one obvious sort of signature of planar cell polarity in the fly wing is the fact that on the adult fly wing they have so-called wing hairs so if you zoom into this wing we see each cell grows a little hair and these hairs form a beautifully aligned pattern they come from the point that they move in a certain direction and hairs from neighboring cells are almost aligned and overall there's a beautiful pattern of hair alignments and here we quantified the local directions of these hairs in the whole adult fly wing there's a beautiful long range pattern with anisotropic and defines the direction in the plane of the tissue and this wing hair polarity is guided by a system of proteins that are called core planar cell polarity proteins I will look at two different types of polarity systems and this one I call core PCP sense of planar cell polarity and this system guides orientation of features in many cases for example the fly wing hairs you just mentioned and if you have mutants of such proteins that are related to this PCP system you can get phenotypes of these orientations the same system also guides the fur pattern for mammals you get a mutant of such proteins in the mouse and you get fur pattern mutant and it also guides for example the orientation of mechanosensory hair cells in the ear which are nicely oriented and get misoriented the same type of proteins are mutated and here you see the list of proteins there are lots of them which some work together the blue and red form together one system and I distinguish them in blue and red because some of them fall on the proximal side of the cell and some form of distal side of the cell and this is chemically shown here these proteins assemble near the cell membrane near all these junctions and the blue ones assemble towards the proximal direction the red one towards the distal direction but they interact across these membranes between neighboring cells so they form actually so it's chemically chemically shown here this flamingo exists on both sides these are sort of the two membranes and you see some of these they form links across from one cell to the other and then there are others which are grouped together they form complex of many molecules together which have a typical orientation from one cell to the other this defines polarity and then the picture is that if you look at two neighboring cells the blue and the red groups of proteins they would bind to each other across the membrane and form big complexes well within a cell they would stay as far as possible from each other that's why they sort of segregate within a cell and bind across the membrane all the feedback this is maybe not a good symbol here if you think of it as a real feedback, it's not a feedback this is just a repulsion it just means they segregate I'm just describing something here in words there's no mathematics here I'm saying that these molecules accumulate on the membrane they bind across here and they somehow stay apart within a cell that's the observation and there are lots of ideas and models about how this could work I don't want to go into here into the cellar details I want to go into how this can be used to organize tissues I should mention but this will not be I will not have much time to go into this there is a second important PCP system that we've also investigated while we called this the core PCP system this one is based on two proteins called FAT-ADUXES and this we can call it FAT-DUXES PCP system FAT-ADUXES are two cation type molecules which are unconventional they form adhesive links between cells which can be polar, which are different can polarize and here's another scheme so one can one can have a situation where these links are typically oriented and this is thought to be guided also by gradients of both of DUXES and by another molecule called full-jointed but I don't want to go into details here the idea is that this is a polarity system which may be guided by concentration gradients by expression gradients but I don't think I come today to the point where this becomes relevant just to tell you that two different systems I will focus now on the core system and the first thing that we were interested in is to measure the patterns of this polarity during development to understand how it emerges in the tissue during the dimensional disc growth and in the pupil phase in order to at the end give rise to a pattern that guides the wing hairs in the adult to do that we're using GFP or YFP labeled versions of individual PCP proteins that's done in Suzanne Eaton's lab and you see here for example the image where the fluorescence intensity can be seen in the cell contours not as nice image as the usual e-catering that we usually used it's sort of a little bit more spotty image but it can be analyzed in terms of another tropies now the difficulty is the system is polar it creates a vectorial axis but because these molecules sit very close to each other on cell bonds the fluorescence signal doesn't distinguish whether we're on the side of the left cell or we're on the bond of the right cell so what we actually see is intensity that loses the vectorial information but we can still pick out an axis we don't know where the vector points to but we can define an axis and such an axis is what is called liquid crystal and physics and emetic it's just an axis it's not a vector but an emetic it's a vector without axis and it can be in two dimensions described by a two by two matrix where the sum of the diagonal elements is zero and it's symmetric so a matrix which is symmetric and has some of the zero diagonals is characterized by two numbers a vector two-dimensional would also be two components so it has also two numbers but it's as a matrix it defines not a vector with a pointing end but just an axis and to define now these components one can sort of look at another tropies around the cell by integrating the intensity that is measured over the angle one can define these two numbers and this now defines an axis which we can represent by a little yellow bar and the length of the bar tells you how another tropic this region is and you see that even though by eye you wouldn't pick out the planar polarity axis in this fluorescence image by doing this quantitative analysis you see there is a there's an axis here global order in this PCP distribution at least order within this little image now we can also get information about the vector where it points to that is a little bit more tricky what one does here is to express clones in the tissue which lack the GFP which don't express the GFP protein and at the boundaries of these clones one can actually know that the fluorescence comes from one side and not the other side and so by integrating around the clone boundary the intensities we can extract a vector but this is not a the polarity vector for the whole clone it's not a single cell property is a clone property but of course then the idea is getting both pieces of information the individual pneumatics the axis for the individual cells which we can also average in small groups and getting the information about the vectors of clones we can construct the polarity picture in the tissue and this we can do at different times in the wing imaginal disk and we can watch how polarity emerges and then how it is organized later in time and here we see a very early pattern this is a very early disk and we have lots of little arrows these are little vector errors that we can measure and they are quite disordered but there is local cell polarity in the plane already established but it's not yet organized in a pattern it's disordered and then at later times we get a beautifully organized cell polarity pattern now we also can also discuss how can such order emerge if we start from a disordered system going to an ordered system and for that we can go back here again you see the scheme of what these order looks like at later times and this is some data the pneumatics and the vectors but let's now discuss some simple physical ideas about how such order can be established and again I use the vertex models that I've introduced yesterday now we start with the vertex model and we complement in the vertex model additional variables that capture PCP levels on cell bonds so each cell bond now has two new variables it has a sigma i alpha variable and a sigma i beta variable so i is the bond index and alpha beta is the cell index so each bond has now two variables for the two cells and these variables change dynamically in time with rules they ensure that within a cell the overall level is constant so conservation of the number blue and red repel each other and blue and red attract each other when they are sitting on the same bond and this can be done in a simple way defining a function for some energy and dynamics which someone tries to minimize energy we don't think of this as a real energy is just a simple way to set up a model that follows these rules we also allow in general a coupling between these PCP molecules and cell elongation which essentially means that the properties of these molecules also depend on how long the bonds are and now one can ask would one get order in such a system if we first build a tissue then initialize these variables sort of in an uncoordinated way and let the system evolve in time that's an example of what you see here and I don't draw here the bond variables I only draw vectors calculated for each cell that is taken from the bond variables the system is based on bond variables and then you see you get a local coupling they tend to align locally but because of the initial disorder you don't get a globally ordered pattern you get these swirls and defects and the fundamental question is how can this tissue prevent this from happening how do we get a tissue which has not such defects even when you start out as you saw from a disordered organization of polarity and what we propose here is this can actually work orders established first in a small system which then grows maintaining the order and propagating it to larger scales that's an interesting idea because this is in physical system when discusses often how such order can be achieved but usually there's no growth involved so the idea that you can get order at large scales by growing a small system making large is usually not discussed in physics so here we start from it now here I show you both the bond variables and the arrows which has a different version of the movie we start from a small system we first let this quickly relax growth is slow and as it grows it can first sort of tend to align in a small patch the patch is smaller than a typical distance between the defects you saw before therefore it can create a nice pattern there and as it grows it can maintain this order carried to large scales so that's important steps we have now given arguments of how to start from an initial condition that is more of this order than small to build up nice order and large scales and the next question then is how does the system go from the wing imaginal this to the final pupil stage and since time is short let me just say a few things here we measure in the pupil wing so in the same method I showed you yesterday watching inside the pupil the pattern and this is the polarity pattern at the early times of the pupil stage this is the polarity pattern at the late times of the pupil stage during the dynamics of the wing in the pupil that I showed you yesterday there is a remodeling reorientation of the venous polarity and so you see how the vectors turn sometimes the order is strong it rotates while it rotates the order gets weaker in between and becomes strong again and here I show you again the movie of yesterday with flow fields of cells what we proposed here is that the actual flows of cells in this process coupled to the orientation of polarity and remodels it and in particular interesting if you have such a flow field one can first calculate deformation rates by taking the spatial gradients of the velocity so the velocity is a vector the gradient of the vector of velocity is a tensor it's a matrix this velocity gradients can be decomposed in different parts if one takes a symmetric part of this matrix one has the shear rate if one takes the anti-symmetric part of this of this matrix one gets the vorticity the rotation rate and both of them are important for this remodeling first I can show you okay this again a picture about how this whole pattern is generated the wing hinge some of contracts and then pulls on this tissue which then stretched and flows we'll talk about this more tomorrow if we now just look at the vorticity of the flow we see that there are clockwise and counterclockwise rotations of the tissue and those will certainly locally rotate tissue patches and they will rotate polarity and this is one contribution to the remodeling tissue polarity in the stage but if you look at the numbers it accounts only for a part of the actual rotation of cell polarity in the tissue and the second contribution we argue comes from the shear that is the deformation the shear deformation part of the flow and here I show you a vertex model simulation with cell polarity at an angle and now we shear this whole this whole tissue in this case using anisotropic division and this rotates the axis of PCP it's a very general very generic phenomenon that if you have a shear flow and you have anisotropic objects they will rotate in a shear flow one can also bring this up to a continuum description of initial and final polarities so the general picture is that if you have a flow that the shear flow converges here and diverges here and you put a let's say a rot-like object in it it will rotate in the flow and this is characterized by such an equation when u is a coefficient that depends on the object and if mu is negative one will align with the direction at which the flow expands and if mu is positive it will align with the direction where the flow converges and using this picture we can analyze what's going on in this fly wing we can measure mu we find that mu is negative it would be a simulation of this process time is up maybe just as a final word we can also use this to understand mutants without going into any details just to show you these two slides so these are these are different mutants these are the regular calculation of patterns which we can calculate using such approaches and we can essentially understand a quite large set of certain self-polarity mutants using this idea of coupling of flows to self-polarity and with that I should stop here to allow you to get some lunch thank you very much