 Great, so welcome everybody, so it's a pleasure to have you here connected remotely Alma Dalco. Alma studied physics in Padova, in Torino, and also studied piano in Venice. She got her PhD in ETH, in the group of Martin Ackerman, and moved recently to Harvard as a post-doctor. And today is going to talk about work in experimental microbiology. So please Alma, you're welcome to start. Okay, thank you very much for the introduction and for having me. I will discuss today some of my interests that are quite broad. And today I'm going to talk about the spatial organization of organism functions in microbial communities. This is what I mostly did during my PhD at the ETH in Zurich. I'm afraid there's going to be a little delay between one slide and another, unfortunately, because of the connection, I hope you will still be able to follow me well. Okay, so what is interesting about microbes is that they are widespread all over Earth and they perform very important processes. For example, microbial communities in the soil cycle the elements while microbial communities in our guts help us with our health by, for example, synthesizing vitamins. And most of the processes that microbial communities perform are collective processes in the sense that they arise from interactions between different species. What you see here in the picture is a consortium where methanoxidines in archaea and sulfur-reducing bacteria perform collectively metabolic process. Particularly, they are able to break down methane and in this way they reduce methane emission in the atmosphere. And this specific metabolic process is something that no single organism in isolation can do, but only this consortium of two interacting species is able to perform. So most bacteria live in communities like the one shown in the picture. And bacteria in these communities secrete compounds and take up compounds such as sugars or amino acids. In this way, individuals exchange metabolites. And this metabolic exchange is done between cells that are within a certain spatial distance. So typically individuals interact within a certain neighborhood. This creates a network of interactions between species. And if we are able to know the spatial arrangement of the different species and the range at which they interact, the spatial range at which they interact, we are able to reconstruct the spatial network of interactions that occur inside the community. All of these interactions together, so the spatial interaction network, collectively, that reminds what processes the community has a hold can perform. So it's very important to understand the spatial organization of organisms within a communities and they range at interactions between species to understand what processes the community has a hold can perform. Now the question is, we might be able to observe the relative location of organisms, but how can we measure the interaction range between different species. So this was the core of my interest when I started working in microbial communities. And the spatial distance at which organisms interact is especially important when there are metabolic dependencies between organisms. So metabolic dependencies are quite widespread in nature. Often, organisms cannot produce all compounds such as amino acids that are needed for growth, and they might exchange those compounds with another species. So here we see two interdependent cell types that exchange some beneficial compounds. As they grow, they will tend to segregate and the question is, are they still able, depending on the range at which they interact, to exchange those metabolic compounds. So if they segregate too much, they might not be able to exchange those compounds. So learning about the interaction range between species will tell us which interactions might or might not occur inside a community. We specifically hypothesized that in such a system where two different types need to exchange beneficial compounds. If they would interact at a very small spatial scale, at a small range, they might not be able to grow well because they cannot receive far away from the partner the compound that they need to grow. So we asked what is the range of interaction in a real system. We studied this question in a simple setup. So we wanted to study a very simplified microbial communities where we could really be quantitative in our measurements. For that, we developed a synthetic consortium of two Escherichia coli cell types. These two cell types were unable to produce an amino acid each. One was unable to produce the amino acid proline and the other one was unable to produce the amino acid triptide. And these two cell types were also carrying a fluorescent marker so that we would be able to distinguish them under fluorescent microscopy because one cell type would be green and the other cell type would be red. These two cell types are able to grow together despite their inability to produce an amino acid because they can exchange this amino acid with the partner. So when growing in isolation, they're not able to grow, but when mixed together, they're able to exchange amino acids and grow. Now we needed a setup where we could measure for long periods of time growth of single cells within a complex community. So we needed to observe hundreds or thousands of cells all together and understand how they interact with their partner. To do so, we developed this experimental setup based on a microfluidic technique. So in this setup, we can feed through some feeding channels, a specific medium, for example, some medium-containing sugars or a mixture of amino acids. And the environment can be tightly controlled by the experimenters, by me in this case. And how the environment looks like from the perspective of the bacteria, I will show it in a minute. Now we play a video where we are entering inside the microfluidic device and we see that there is a main channel where the media flows. And on these main channels, some side chambers face. And these side chambers are chambers that host bacteria in a monolayer. So what we do is that we load bacteria in these side chambers and as time passes, the bacteria will fill the chambers and create a monolayer community. The size of the chambers allow about 1500 cells to grow together. The type of images we get out are time-lapse movies of the growth of these two cell types together. So we are able to observe not only the spatial organization of the two cell types in time, but also if we zoom in, we are able to observe the single cells elongation, so the growth rate of single cells. This data allows us to ask the question, how does the cell grow depending on the neighbors that it has around? So at which spatial scales do the two cell types interact? The naive expectation is that when cell type A is surrounded by cell type B, it will grow faster because it can receive more of the amino acids. The question is at which neighborhood does the cell receive amino acid from? So this is really a good experimental data set to ask, how large is a cell's interaction range? Or to be more specific, how we phrase the question is, what is the neighborhood that predicts a cell's growth rate, so that influences a cell's growth rate? To address this question, we had to develop an analysis pipeline that allowed us first to identify the two cell types automatically and then to track them in time so that we could measure for the same individual the length and the growth rate in time. So the movie I'm playing shows how this process is done, and at the end what we have is a full spatial information about the growth rate of cells that is now plotted as a heat map, so lighter colors indicate faster growth rate. And we know how the individual cells growth in different locations in the communities, and we can correlate the speed of growth with the identity of the neighbors around the cell. So what is the neighborhood that predicts a cell's growth rate? So what we do now is that we select single cells, we measure the growth rate, and we measure the fraction of the other type around the cells, the focal cell we are measuring. If we plot many single cell growth rates, we obtain a scatter plot, where there's a positive correlation between growth of a focal cell and fraction of the other time. Now we have analyzed a specific size of neighborhood, but this might not be the one that gives the highest correlation between growth and fraction of the other type. So we can change and iterate this process with different interaction ranges with different neighborhoods analyzed, let's say, and we obtain different scatter plots. And every time we will obtain a different correlation score between growth of the cell in the middle and fraction of the other type within these specific neighborhoods we are analyzing. So when we iterate this process, we obtain a curve of the correlation, which is the Sperman correlation, so it's a ranking correlation, we obtain a curve that has a peak at a certain distance. And for the purple cell type, this distance is 11 micrometers, which is about three times the cell length. So it's about three micrometers of the cell length. And this means that when we analyze on an 11 micrometer neighborhood around the cell, this neighborhood gives the best prediction of the cell's growth rate. We can repeat the same analysis for the other cell type, and we find that the peak of the correlation is shifted. So this means that the other cell type interacts at a much shorter landscape, which is about one third of the other. So both cell types are found to interact very locally within a few micrometers from themselves, but the two interaction ranges are different. So when we found those results, which are consistent across 10 different replicates, we really wonder what sets the interaction. So it was surprising for me to think that the world that these bacteria see, it's very limited around oneself. I like to think that cells live in a small world. This is how a cell perceives the world around itself. It's a very limited horizon that the cell sees. And the question is, when do cells live in a small world? So what are the parameters that set the interaction range around the cell? And to address this question, we decided to build a biophysical model of the amino acid exchange. So we know that two cell types are exchanging amino acids, and this exchange affects the individual's growth rate. So what are the mechanisms that drive this amino acid exchange? So we know that cells have an internal concentration of amino acids, and this internal concentration can be reduced, can be increased by uptake of amino acids from the environment. So here, I will plot the terms of the equations, and I stands for internal concentration of amino acids, while E stands for external concentration of amino acids. So there is a positive term that is due to uptake of amino acids from the environment into the cells. There is a negative term that is due to the leakage, which is passive leakage of amino acids into the environment, and the environment allows for diffusion. So we can fully write those equations, and then there is an additional assumption that we have to make, which is that, yes, that cells grow. But what we assume is that the illusion due to growth is always perfectly balanced by the production of the amino acid each type can produce. In other words, this means that the two cell types are limited in their growth only by the amino acids they cannot produce, while the other one is always produced enough, let's say. By applying this assumption, we have to write two differential equations for the two cell types. So what we obtain is a system of equations where for cell type A, we will have a constant concentration of one of the two amino acids, which is, as I said, always maintained at the right level. While for the other amino acid, we will have the balance of uptake leakage and degradation due to growth. And the same will be valid for the other cell type, but with the other amino acid swap, let's say. Okay, so those are the equations that we have to solve. And if we are able to solve those equations, we can know the internal concentration of amino acids. And from those, we can calculate the growth rate of cells. So we can apply this model on a grid and know at any point in the grid, what is the internal concentration of amino acid external concentration and the growth rate of cells, depending on the location of all the other cells and their identity. So what we did is that we took this model and we applied it to our real spatial configuration that we measured in the lab. By simplifying a bit these configurations, we can really solve these equations at different location in the chamber in the communities. And we can predict with this model, the growth rate at different locations and how it depends on the neighbors. So what we did is that we took parameters that are known in literature for the uptake leakage and diffusion of our two amino acids, triptopan and Brolin, and we applied those equations to estimate growth rate of cells. And if we do this analysis that we did also with the, with the experimental growth rate, if we do this analysis on the predicted growth rate, we find the same correlation. So we can see that the model is able to recapitulate the correlation analysis and the presence of an interaction range between the two cell types. So both the experimental and the predicted growth rates suggest that the two cell types interact at the finite range and that these two ranges are different. This means that we can explain the range of interaction with these few mechanisms that we included in the model, which are the uptake leakage and diffusion of amino acids. So what of those parameters, the uptake, the leakage and the diffusion, which of those parameters set exactly the interaction range of these two cell types and why is one cell type interacting at a shorter range compared to the other. So I will tell you the answer and I will show you also a little simulation that makes this answer more concrete. So we found that the two different ranges are due to different uptakes of the amino acids. So that the uptake rate of amino acids really sets the range of interaction between the two different species, while the other parameters like the leakage and the diffusion have a minor effect on this range. So I want to show you a small simulation where we are placing one cell type on the left and the other cell type on the right, and we are calculating the growth rate away from the interface. So we will assume that one of the two cell types has high uptake rate of the amino acid while the other one has low uptake rate of the amino acid it needs. And what we find is that the growth profile is very asymmetric. So for the type that has a high uptake of the needed amino acid, the growth rate is really confined around the interface. And this is because cells close to the interface take up amino acid and leave no amino acid for the cells behind them. So a high uptake rate of amino acid corresponds to a small growth range. And what we demonstrated, and I'm not going into detail is that the growth range as I defined here the range at which you can grow around the interface. It's proportional to the interaction range. And the growth range can be really calculated analytically. It can be calculated and what we find out is that the two parameters that mostly affect the interaction range are the uptake of amino acids and the diffusion in the media, while leakage has a much minor influence on the range. And what is interesting in general is that for molecules like amino acid, the diffusion constant doesn't vary much among one molecule and the other. Well, the uptake rate varies a lot. So for example, in our case, proline and tryptophan have a difference in the diffusion constant of about 0.75 one of another. While the uptake rate can be 16 times higher for tryptophan than from proline. So tryptophan has a much shorter interaction range or leads to much shorter interaction range is because it is taken up at very high rates. So we kind of uncovered that the interaction range is small whenever uptake is high, and also I didn't enter into detail so much but when the density of cells is high. And this because the density of cell reduces the diffusion of amino acids, and therefore shortens the range of interactions between cells. What we can do with this model is also try to understand what would happen if you would artificially change the landscape of interaction, which is easily done by changing the uptake rates in the model of the two amino acids. So the question is do cells grow faster if they interact on a large range. So this is what we did we tried and we changed parameters and applied the same model with different parameters on the on the data on the spatial configurations that we what we that we got in the lab. So we found out that if we change the interaction range of the two cell types, the growth rate increases. So in this plot that I'm showing whatever is beyond one means that it gives an higher growth rate than what is actually measured in our data. So if we increase the interaction range artificially in the model, we predict an increased growth rate in our cells. And this is because the two amino acids are more efficiently exchanged at this larger ranges. The other test we did was to ask whether and then an increased mixing would increase the growth rate of ourselves. What we did is that we took a real arrangement and we randomized these arrangements and we found out that cells are predicted to grow faster if they increase their mixing. And again, this is due to the fact that they interact on a small range. So they need to have the partner close by in order to receive the amino acids and grow. And if they're more mixed, the whole community has a whole can grow can grow faster. So far we have been talking mostly about the molecular exchanges and the interaction rules of cells, but what we are also interested is understanding how we can predict global properties of communities, such as the fraction of the two cell types, or their mixing from the local interaction rules, so that we can fully scale up from the molecules to the local interactions to the global properties of the communities and connect these different scales. So what I mostly talk so far is about data in a biophysical model that connects the molecular scale to the individual, individual scale. We found out that there are few biophysical parameters that set the interaction range and the and the growth of these cells. What we want to do now is to have an understanding and a way to predict the global properties from those local interaction rules. So what would be to have a full understanding and a model, let's say to map the molecular properties up to the global properties of a community. So now the second part of my talk will focus on how we can scale up the behavior of single cells and their interaction to predict global properties of communities such as community composition community growth rate, or sorting of cells. And to do this, we decided to go for a mathematical framework that we call a pair approximation in which we take a spatial system and we approximate it with a simplified description of the links between two cell types. So we can describe fully the spatial system by specifying the probability of finding links between here, I will have two cell types, the green and red. So I have, I need just four probabilities to describe the whole spatial system. And I want to convince you that this information is enough to have the properties we want about the community, because these four probabilities allow us to calculate, for example, the mixing of the two cell types, their clustering and also the global fraction of the two of the two of the two cell types. Specifically, the probability of finding a link between G and G, this will be the clustering of the cell type G green. So our goal is to fully specify these probabilities to be able to understand not only the global composition of two cell types, but also their spatial arrangement. Now to find these four quantities, we will have to first write some dynamical equations for for these quantities for these probabilities. And I will now guide you through this process. So as a first thing, we can reduce our, the number of equations, because we know that only three independent variable existing in the, in the, in the system has the sum of these four probabilities has to add to one. So we will have to track only three of those four probabilities. Now to write the dynamical equations, we need to do three things. We need to identify the processes that change the pairwise probabilities. So for example, agreeing could replace a red in this case. We have to calculate the rate at which each of these process or events happen. And then we have to calculate the change in pairwise probability during this event. So if you are able to do these three steps, we are able to write down the different dynamical equations for this for probabilities. So I want to discuss a little bit about the assumptions of this formalism. That will allow us to write these equations. So the first assumption we have is that cells live on a grid and that there are two directed two graphs that we have to take into account. And this is due to the fact that the two cell types have different interaction ranges, which means that if a cell is in the neighborhood, if I read cells in the neighborhood of another cell, this might not be reversible in the sense that there are neighbors of, yeah, it's a bit hard by word, but from the graph you can visualize that some cells might be, let's say, linked to a cell, but the reverse is not true. So we have to consider two superpose graphs of interactions. The other thing we have to, we assume is that according to what we see in the data, there is a frequency dependent fitness, which means that cells grow faster when they're surrounded by the partner type and this is because they receive more of the and the cells that affect a focal cell's growth rate are only those present within its interaction range. So we have two different interaction ranges for the two different cell types and we have two different linear function, fitness functions for the two different cell types. The last assumption we have is that cells reproduce according to a birth best process. So they will reproduce according to their fitness function and so the birth rate is proportional to their fitness. When a cell is ready to reproduce, it will replace a random neighbor. And yes, so they will replace only one of the cells close by in space. So with these three assumptions, we can write down our differential equations. The important thing to note is that because we have two different interaction ranges, the links are not symmetric is what I was mentioning before. You can be friend of someone, but this someone is not friend to you. So let's try to understand how we can parameterize this mathematical framework, even the information we have built so far. So for this mathematical framework, we need a definition of a fitness mattress. And here you can see the W, the fitness can be calculated from from this fitness fitness matrix by counting the neighbors of the two of the two types, either the red or the green type within a certain neighbor. So for example, in the example I'm showing here, we have a red cell that interacts with four neighbors, and two of these neighbors are green. So that we can calculate from this information, the fitness of the focal cell in the center. In our specific case, the fitness mattress is very simple. We have measured it. And we know that this is just just depends linearly on the fraction of the partner type within a certain interaction range. So we can really take the parameters that we find in the data and and plug those parameters in our per persummation model. We can also do, instead of utilizing the data, we can also take the biophysical model and estimate those parameters directly. In fact, we have from literature, the uptake, the leak, the leakage rate and the diffusion of the twin masses. And this is enough to estimate the number of neighbors that are needed in the per persummation and the and the fitness functions. So we have either a direct mapping between molecular parameters and our per approximation, or we can plug in measurements from the data. Anyways, we can parameterize the per approximation in our specific system. So let's now go back to the dynamical equations and let's try to write them now. So we will have to, let's say, work with the transform the probability of having a link between two particles. We will have to write it as a function of the number of links between the different particles, as I write down now. So the goal would be to write the differential equations for the number of links, which then can serve immediately to calculate the probabilities we're interested in. Okay, so let's see what is needed to write down these equations for the number of links and how they change in time. So we will have to do a sum over all the possible events that can change the number of links in this case from red to green. So we will have to have a sum over all possible events. We have to calculate the rates at which these events happen. And then we will have to multiply it by the change in the number of rings that each event causes. Okay, so we're going to break down this process by first identifying which events change the pairwise probability and calculate the rate and then calculate the change during this event, the change in the pairwise probability during this event. Okay, so let's see first how we can take into account all the possible events. I'm taking now this example, and I will show you that there are two events we have to, we have to account for. And one is the green particle replacing a red particle, and the other one is the opposite event. So those are all events that can change the number of links between red and green particles. And then we have to calculate the rate and we break down this rate as, as, as you see. So we, we will calculate the number of green cells that exist around the cell, the probability that this green cell reproduce and the average fitness of this green cell. I hope that with the delay you can still follow this part. Maybe I can go a bit more quickly. So let's say we are left with one last step which is calculating the difference in the, in the link before and after the event. And here I'm showing how we do this. Also without going in detail, I hope I convinced you that we can fully write down, write down these dynamical equations for the number of links. And from those, we can calculate the probabilities of having links between green, green, green, red, and all of these pairwise interactions. Now, with these dynamical equations, we have a system that can be sold at steady state, and by solving a steady state, we have the steady state description of the, of the spatial system. And the steady state description tells us what is the equilibrium frequency between the two cell types, what is the relative clustering of each cell type, and, and their mixing. So let me show you the results that the model predicts and, and the experimental data. So our experimental data. Here you see 60 replicates so 60 different communities evolving in time. And what we see is that the 60 communities in average. Is a composition of about 20% of the type that is called that that cannot produce trip on what I call Delta T. So the community is quite skewed at steady state. There's only one every five cell that is that is Delta T type. And this is can be predicted from our pair approximation model, quite accurate. And I want to show you that also without deriving this equation, but we can calculate this quantity the probability of having Delta T, and this one this this probability in the limit of large interaction ranges so large neighborhoods around both down to what is the the well mixed scenario so the scenario where everyone interacts with everyone. And what the let's say the limit equation sell us is that the type that has the highest maximum growth rate so where the fitness function reaches the highest maximum is the one that is more. So in general, this model suggests that the community composition depends mostly on the maximum growth rates, but it depends also on the interaction ranges, but to a minor extent. The consequence that this model find of the interaction range is that because cells place offspring close to themselves. This means that the fraction of the partner around the cell is reduced compared to the average presence of the partner so the local composition around sell is different from the global composition of the whole community. We can also calculate the difference, and you can see it here. In this equation, these equations, especially is specifically saying that the local composition. The local fraction of a partner is lower than is low global fraction. And this is because cells tend to have their kids so they're similar around themselves rather than partner. Here I'm plotting how the the ratio between local and global composition changes has a function of the interaction range. So what we see is that has the interaction range become very large than the local versus global composition. Attains one so they are the same. Well, as the interaction range decreases the local composition of the of the partner type is lower so so cells tend to be surrounded by their own kids rather than the other part. And this relates to our finding that cells would grow faster if they would be surrounded by the by the by the other type, but yet they tend to create classes of their own cell type. So we from our pair approximation model, we can also directly estimate the growth rate of cells because there's a simple linear dependence between the average composition to the clustering of cells and their growth rate. And the clustering of cell is something we can calculate from our pair wise probabilities. So from those probabilities we can estimate the growth rate of the communities and indeed, while our data suggests that randomizing arrangement so let's say in a well mixed scenario, the growth rate would be about 10% higher. The pair of presumation predicts that the growth rate will be about 7% higher so it kind of agree with the data in saying that short range interaction reduce the growth of cell types because cells interact only with themselves from which they cannot retrieve the seminar so that mixing is important in those communities to maintain interaction between two cell types. Here I want to show you a last consequence of this short range interaction, which I find interesting as a general generalization to other types of communities. So what we're looking at here is a plot that tells us how does the community growth rate in a spatial setting compare to the well mixed scenario where interaction ranges let's say are infinite so everyone interacts with everyone. So what this plot shows is that the community growth rate is reduced so you see darker buttons let's say blue is reduced when the two cell types have very asymmetric maximum growth rate. So the maximum the fitness function is very different for the two cell types. And moreover, you can see that some of these asymmetric communities so where the maximum growth of the two cell types is very different. These asymmetric communities might even collapse so not be able to grow in spatial settings while being able to grow in well mixed scenarios. So it seems that short ranging interactions might be especially dangerous for communities that that display very asymmetric growth rate which is in general what happens in natural communities. So, let's say to summarize we find that cells grow slower when the interaction ranges are small, and that communities the one that are very asymmetric in the growth rate might even collapse in these spatial settings under short interaction ranges. Okay, so to summarize this part of the pair approximation. What we did was to specify the interaction rules by taking these rules either directly from the data, or from the biophysical models that maps molecular events uptake leakage and diffusion to this individual based individual rules. So we can specify the rules. And from those rules, we can predict the spatial patterns the frequency of types and their average growth rate by using this pair approximation framework. So this allows us to go from local interactions to global predictions for communities. Again, the global predictions that we found are that the community composition mostly depends on the growth rates on the relative growth rates between the two cell types, but it also depends on the relative interaction ranges. Moreover, we find that in general, when interaction ranges are small, the local composition of a neighborhood is different from the global composition of the community. And that in general cells grow slower when they interact at short ranges. Okay, so to conclude what I discussed is let's say framework to go from biophysical from from biophysical mechanics of interactions. So the uptake and release of compounds to the local rule of interactions between cells and from there to the global properties of a community. And in general, I believe that this type of bottom up approach can really reveal what are some basic principles that are relevant in natural communities. So here I applied this, this model and analysis to a very simple communities composed of two cell types. And I showed that we can develop some microfluidic devices to quantitatively measure single cell growth rates and relate these growth rates to the local composition of the community. And in general this approach can be applied to more complex communities as long as they they can be measured precisely and for longer periods of time. So this in this simple communities, I believe, can tell us and help us understand how natural communities behave and and can help us, let's say, disentangle those more those complex cases by using this bottom up approach. Now I want to thank the people that helped me in this process and this in these projects. And I want to thank specifically done key fit Susan Schlegel, Martin Ackerman, Simon van Vliet, Christophe Fuert and Michael Brenner, who helped me in different ways, either with the model, or inspired me, especially in the case of Michael Brenner to expand this research beyond the microbial world, towards more, let's say, different horizons, and which is now what I'm doing during my postdoc is, is let's say, not well beyond the microbial world. Okay, I'm happy to take questions. And I hope it was easy to follow despite the delay with the slides. Yes. Thank you. Let's thank Alma. So we have time for questions. So if you want to ask a question, you can unmute and start talking. Let's see if we are able to sell for guys. Maybe the word to someone or I have a question. Yeah, there is a question from Jean. Okay, wondering why do you have asymmetry in your interaction. I couldn't get that point. Why you have sorry asymmetry in the interaction in the model. Why did you have asymmetry in the direction. I mean the biophysical model so we did not impose any asymmetry. Sorry, am I still sharing my screen. Am I still sharing my screen. Maybe I can put myself on that me. Okay, you should see me. So in the, in the model we did not impose any asymmetry. We actually took biophysical parameters that were known on the uptake and the fusion of these amino acids. And we are trying to like the probabilities of switching from G to red. Yeah. So we assume the most general case where we assume that the two cell types might interact on different graphs so one one cell types has many more neighbors with whom it interacts and the others. Is that the question. So we just allow it you can in principle have that they interact at the same range but our data suggests that this is generally not true. So you have to have a more complex setup where you are taking into account links of two different colors, let's say the links of the green are towards many more neighbors than they doing something of the red. Yeah, I don't know if I answered the question I hope. Hi, thanks for the. So yeah, my question is more about the. So you talk about the specific setup where you have a monolayer of cells, right. And what do you have you have you had an idea of what would happen if you consider like a different setup where you have multiple layers of the same community. In this case the interaction range I guess it would be longer so you and maybe in this case also diffusion could play a more important role. Because I mean let's let's think at the layer three there is a cell that is leaking some amino acid this actually can go down in the in the layers now can diffuse. So basically the ranges should be bigger. So what we actually do in the biophysical model is that we assume a three dimensional space. So we model diffusion has in 3D. But then, practically, there's one dimension that is homogeneous because it's so shallow it's just one micrometer high, right. So, effectively the problem reduces to a two dimensional problem. But my idea, I'm not sure but I don't think that the interaction ranges would be longer in 3D. I think actually what matters mostly is the density of cells. So if you have cells that are far away diffusion goes faster because diffusion must be corrected by the density of cells so cell packing reduces diffusion. And I believe that with the same cell packing are same two dimensional system is kind of the same as a three dimensional system. So you think this is relevant even in a natural community where basically there are there are no mono layers or it's very unlikely and you have like different type of physical structures like in three dimensional. Yeah, so you will have set up having the walls and the and the ceiling of the chamber in a natural system you would have another layer of cells doing exactly the same as this one you're studying. I think. And then my idea is that the interaction range will be isotropic in the other dimensions as what we measure in 3D. Okay, but I'm not sure. Yeah. Yeah, because I guess the layers somehow they mix with each other they can also you know cells now in this case can grow not only laterally but also up and down. So like a cluster they go can go down or up or left or right or whatever. So in this case like also many other factors that mix the community more. And actually I think they change the interaction range in like more locally you can have cluster that makes more and then you know other part of the community maybe it's more fixed. Yeah, I don't know. I don't know. Anyways, there are in national communities there are many other factors that we don't account for, for example cells might move, which makes obviously the story very different. And, yeah, in general, other factors or flow, for example, that we don't, we don't take account. But thank you. Thank you. Yes. Thankings but you need to unmute yourself. I should be unmuted now. Can you hear me. Yeah. In fact, my question was precisely what you moved on to right at the end of your comment there I was just about to ask in fact that you model the communications through just a diffusion process and that was precisely what I was going to ask actually if if you actually had motions and flow so you would have an a direction as well. How much that would actually change the picture because you would essentially have like a non local transport by the flow. Would that make sense in terms of the differences between the two types. Once you actually have a much longer range transport term in there. Yeah, I think yes. And I can tell you that, let's say in our experiments, trying more complex setup where there was flow, for example, all in all didn't allow the two cell types to grow together. And that's because they really need, yeah, they really need sufficient amount of these amino acids. So if you have, if you have like a system where amino acids are dragged away by flow. They would just never engage in this holistic relationship. So they, they need to have, let's say, they need to be, for example, packed enough, packed enough, and there needs to be a little dispersion away so that they can really engage in this interaction. Let's say, in theory, it's an interesting question in practice. It actually is something you cannot almost observe. That's my, that's my take on the experiments. Okay, thank you very much. I really enjoyed it. Thank you.