DevoWorm Network Science, 11 Jun 2021

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From YouTube: Euler Cycles for “Life”

Description

Recorded presentation (10 minutes) to the TopoNets Satellite session of the Networks 2021 conference. Slides are available here: https://figshare.com/articles/presentation/Euler_Cycles_for_Life_developing_biological_structure_using_multi-cell_networks/14773362

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Hello, my name is bradley alicea, there's my twitter handle and I'm from the open room foundation and the divawarm group. Today, I'm going to be talking about oiler cycles for life, so this talk is going to focus on euler paths and euler paths are ways to evaluate a network by looking at the edges instead of the nodes, and so euler paths are sort of come from this idea of the seven bridges of koningsberg, which is shown there, and you can apply this method to a number of different types of single cell multicellular organisms.

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So, for example, you have diatom cells, which are single cells which have a lot of internal structure or volvox colonies, which are groups of cells that exist in a colony, and so we can take these. uh What we call multi-cell networks- and I want to differentiate between the multi-cell idea and the application to biological cells. Each internal region of a network, the area between the edges is considered a cell and biological cells, by contrast, can be analyzed using this method, so you can have multiple network cells in a single cell or a network cell.

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That combines a number of cells within it or it can represent a single cell- it's pretty flexible in that way, but why are cell boundaries important? So if we're using this analogy between network edges and cell boundaries, why is that important in biology? It's very important for a number of reasons.

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One reason is because it serves as a conduit between cells, so you have things exchanged like mrna molecules or ions or other types of things like uh blood flow or liquid flow or oxygen. Things like that that can be exchanged within in between cells and along the boundaries, and so this is a very important thing to understand and we'll get into some of the reasons why that might lead to multicellularity later, but we're going to evaluate something called an euler circuit.

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So an euler circuit is a set of edges that exist in a network and it's the idea that we're going to cross every independent edge of us in a multi-cell network once and only once so. We can start off by crossing each edge, but we can only cross each edge once and only once if we cross an edge twice or if we failed across an edge within our network, then that network is considered to be an incomplete, euler circuit.

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If we can meet these criterion, then it's a complete euler circuit and there are a number of consequences to that. uh There are a number of sides of a number of sides that exist. They have to equal zero.

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So there are a number of uh shapes that lend themselves to this, but that doesn't mean that we can't evolve these uh networks or these euler circuits to be very complex and include multiple shapes and of course it depends on the biological system, which is why it's not just an exercise in uh seeing if we can fit the model to the data but to see what we can do with this technique, and so even sided shapes like squares and rectangles and hexagons uh are written in an even number of copies that usually results in an euler circuit of zero uh or even an odd uh number of replications for hexagons.

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So let me show you how you evaluate an euler path, so you can use something simple like this graph online interface and you can build a network, draw it out and then find the shortest path in that network. So you can define the nodes and edges and then define the path, and so this graph has an eulerian path, that's complete and it shows the path through this network.

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In this case, though, this network does not have a complete euler path, and so this is shown in the interface as being an incomplete path. It doesn't tell you how many, uh how many steps it missed or how many steps it exceeded it by, but we'll just say that it doesn't have an eulerian solution, and so this this shape. I want to take this and actually get to another issue, which is that we can take one of these networks and deform them.

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So now that we know whether it's eulerian or not, then we can actually distort the shape. So that has rough. The network has roughly the same number of edges and configuration of edges, but then we can create even more interesting shapes with them.

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So we can do things like rotate and scale and shear and make perspective uh projections of these networks, and so this is an example of a graph with a complete eulerian path, but instead of the original shape, we've deformed it deformed it so that one of the edges is stretched out, and so this is reminiscent of conformal mappings, where you have a number of points in a morphology and then the morphology shifts- and you can, you can see that as a sort of a deformation of that basic form.

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So a number of tests that we can do to show how this technique works in terms of evolving phenotypes and evolving multicellular systems comes from testing out. Our triangle motif so in this case we're going to build a lineage tray of these multi cells, we're going to start with a triangle and then we're going to add another triangle on on the side. So we're going to have a simple replication rule, but we find that that simple replication yields an incomplete oil or path. So now we have a choice.

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We can either add in a secondary shape, which is this hexagon and that actually restores our original euler path, but it introduces something called modularity, which means that now it has a separate, it has a new sort of piece to it. That's going to replicate and it's going to be a differentiated morphology or we can add another triangle and we can still end up with an incomplete euler path and then, of course, we can add additional.

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We can have a simple replication rule which gets us further away from the oil complete euler path, which we can. Then you know remedy by introducing these new shapes in the middle and having this new modularity that emerges, and you can do the same thing here in this path, where you add a shape in a different way.

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So then you have modularity as well. So basically what happens is when you have an incomplete euler path in these lineage trees. You add something to see what recovers the euler completeness and then that usually leads to something called modularity or to have like a differentiated phenotype with two or three different modules in it, and so this is kind of what we see with diatoms, where they have these triangular shapes or other shapes, and then they grow. They accrete their body and you know they're.

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They do this in different ways, and this is an interesting exercise in seeing how single cells and multicellular systems kind of emerge and emerge complexity.

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And so then we can do the same thing, though, with a rectangular shape and a simple replication rule in our transformation rule. So now in this tree we have a simple rectangle. We have a replication rule which preserves that euler completeness. We have another duplication rule which preserves that oil or completeness but an asymmetric transformation. Even that actually uh keeps that oiler completeness.

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So this is actually a pretty robust shape, but then, finally, we get to a point where we do this h asymmetric transformation twice and we end up with something: that's not oil or complete. So this lineage is not going to be oil or complete. However, this one here you keep going and you add in you know, maybe you add in another a new shape and you end up losing your oil or modular or your oil or completeness, and then eventually you get to this modularity, where you recover it.

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In this case, we have in this branch of the tree. We have fractal reproduction. So this is the case I talked about earlier, where we have replication rules that occur recursively and we have this fractal structure. So it's it's, it emerges at different. You know numbers of cells, it always retains that uh euler completeness and you have things that have maybe very little internal structure, but you know grow to be very large.

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So this is our tree here and then finally, we have this hexagonal motif, where we just have a replication rule and a duplication rule, and we can retrain euler completeness across the smoke.

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This motif, and so again, hexagons are very good at this, and so this might explain some of the things in nature where you have cells that just keep replicating and they don't somehow form like things that are hugely complex. They just kind of grow and grow and grow, and so that you know there are ways we want to continue with this type of work. To see what kinds of shapes we can find, what kind of roles we can find and so forth. So I wanted to put a plug in from growth and form.

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So this is the direction we want to go in to sort of look at on growth and form in a new way and put numbers on it and think about it in terms of network theory, thank you for your attention and join us in our groups, our dvoram group and our openworm foundation. You can join our slack with that link or you can visit us on the web at that link.