DevoWorm Posters, 25 Aug 2022

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From YouTube: Mathematics of DevoWorm

Description

Overview of the "Mathematics of DevoWorm" poster: presented at Find Your Inner Modeler V, August 2022.

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Hello, my name is bradley alicea and I'm going to be presenting this poster mathematics at diva worm on behalf of the diva worm group. uh This represents the activities that we've been involved with for about eight years now and we're a participatory, open source, computational developmental biology group.

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So we have our weekly meetings, we participate in conferences, we have special events and we try to incorporate people from around the world with different areas of expertise relevant to what we want to do, and this poster is organized around a series of questions.

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So, if you'd like to actually first of all, if you'd like to know more, you could visit our website here. We have our publications page here. This has all of our publications working papers, posters that we presented and everything we're affiliated with the open worm group, open room foundation and the orthogonal research and education lab.

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So we've organized this poster around a series of questions and each of these questions have sort of we've kind of answered them in different ways, and so I'm going to introduce a number of models here, based on some of them, based on analyzing data and some of them being more purely theoretical, but they do answer a number of questions in developmental biology and approach it from this sort of computational first approach.

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So this is uh the first question: how do we capture the multitude of interactions and state changes which can be characterized as cell identity that occur across embryogenesis, and our answer has been in the form of discrete dynamic, on categorical models.

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So I have a couple of things here. These letters represent. If you go to the bottom of the poster, there are some key references that should demonstrate some of these approaches in practice. So the first approach I will talk about here is category theory, so category theory involves transformations from one category to another of different objects.

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These objects are cells, but they can be other things. We use a number of different. um We use category theory to build a compositional system of a developing embryo, so you can see here what we're trying to capture. Is this embryo transforming from a sphere of a single cell sphere or a two cell sphere to something that's differentiated, that's asymmetrical! That has a lot of cells in it of different categories.

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The second approach is developmental game theory. So this is an example of developmental game theory. This is a first mover approach to the embryo, into growth of different poles of the embryo and different tissues of the embryo. So you can see that in the first move you have a division from one cell to a two cell state. In the second move, you have an asymmetric division of the one cell and not of the two, the second cell. In the third move, you have the second cell dividing, but it's now occupying less space.

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There's an asymmetry that's been introduced in the fourth move. You have another round of cell division, that's asymmetrical and so forth, and you can see that it starts to approximate an embryo that has different tissues, has asymmetrical divisions and has some asymmetrical structure.

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Our second question involves: how do we capture pattern formation and cell migration dynamics as a computational process? So, just now, we've talked about cell division.

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We've talked about cell cell fate cell states and that process both dynamically and capturing the sort of the categorical structure of it in this case we're looking at a different uh problem, which is how do you characterize pattern, formation and cell migration, all those things that you see in videos of embryos, that kind of look like they're moving around or that they're quote unquote living, and this is accomplished through two tools: uh cellular automata and spatial data structures.

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So you can see here that we have our spatial complexity and self-organization theme. This is reference c. This is our uh five tuple uh representation, so we have. This is typically what we use to characterize embryos. We work on the c elegans embryo, but we work on other embryos as well, and you can take things like cell tracking data and you can build this sort of representation.

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Furthermore, you can simulate an embryo and you can also build this representation. So if we walk through the representation, we see that we have a point cloud of cells with centroids with positions in an embryo space and it represents basically five parameters.

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The first three is the volume or the the three dimensional shape, so an x y and z position in the chord in a coordinate system.

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Then you have a time uh parameter, which is of course, the time of division or the time since the fertilization or the single cell was uh born. And then you have this angle, so this theta is actually an angle. You can use this for other things too. There are other parameters that we can introduce into this model that extend it out. So, for example, if you decide you want to transform these coordinates into other types of coordinates, that's possible.

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We start off with a three-dimensional uh point cloud, but we can transform these volumetric parameters into other parameters. We can you know instead of having uh continuous timing and have discrete time and instead of having angle, we can have other features as well. So we can introduce things like gene expression. We can introduce things like uh migration, so we have all these different things we can play around with, but this is the base representation.

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Then we move to cellular automata, which are actually a bit different than this, and this is an example of a two-dimensional cellular automata, where there's random activity and that random activity over time ends up forming these these uh these uh patterns, these uh dotted patterns- and this looks like the um code of a zebra or some or a coat of a tiger or a lion, or something like that.

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So this is actually pattern formation, and so we use cellular automata rules to derive these types of patterns, and we, this is a little bit divorced from this representation, but it actually on a two-dimensional scale. uh You can map between these two representations, so the cellular, automata, outputs things uh as sort of patterns you can classify the patterns, you can define them with rules and we have a tool called morphozoic, which classifies this space at multiple scales and outputs uh discrete patterns with discrete categories here. So this is a these.

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Are a nice set of tools that we've worked on. um This is a von neumann neighborhood. This is related to the cellular automata in that they're neighboring cells to a focal cell and these cells there's a radius of this neighborhood and that's a local set of interactions between cells and you have global sets of interactions across multiple cells. So the local interaction is just a little cluster here, but you have global interactions that occur for a wider spatial range and we can capture all that with these kind of models.

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The next question is: how do we characterize the organization of developmental cells, expanding more morphological structures and emerging connectomes, and this we do this through a number of techniques. We use neural networks, complex networks and trees. So you can see over here for reference d and reference these in the bibliography. As I said this, these all characterize developmental functions. So these are all when things are transforming when complexity is being added to say, like a connectome or you have some behavior that you want to see.

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The output of you can characterize that with a neural network or a lineage tree which shows how cells divide and the lineage that they come from and how that lineage uh changes over time and you can trace it back to a mother cell and it tells you a lot about the structure of the cells in their origin and development.

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So you can see we're really working towards the open worm, vision of a virtual organism- and this is a virtual c elegans and it has almost a thousand cells, the connectom having about 300 cells, and so this is this lineage tree one fold to represent all of these cells and c elegans. You can trace them back quite easily, but we can also use a neural network to characterize the connectome.

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So we can characterize output behaviors using a neural network where we can characterize, you can actually do machine learning techniques and deep learning techniques to pick up some of the features in the cells. So you have microscopy images. You can pick up features, so we use neural networks in different ways, and then we have these uh complex networks which are actually quite different from neural networks or lineage trees and that they're.

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In our case, we use something called embryo networks which are spatially uh explicit networks that have you know, maybe represent cells in a diff in a certain area of the anatomy and the connections between them. So the connections between them might be signaling mechanisms, they might be proximity, they might be actual physical connections with gap junctions and things like that, but this rep this represents some sort of connectivity in between cells and then these networks grow in development.

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So that's a major uh challenge of this approach, but we actually have ways we've published on this, to characterize these networks, to characterize the growth in networks and then to characterize this uh differentiation process within both regular networks and hyper networks. Hypergraphs, and so we, you know we're interested in a lot of that type of work as well.

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So then we go up to this side of the uh poster and we talk we. We do. We've done a lot of work in the past. We don't do maybe so much anymore, but on lineage tree dynamics. So before, as I said, there's a lineage tree. You go from a single cell to multiple cells. These lineage trees tend to differentiate over time, so you get sub trees and analyzing.

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These trees are very important to understand things like define, mutants and their phenotypes, or to understand differences between species where sometimes even in c, elegans, there's a difference between the hermaphrodite and the male, and so you can understand these things.

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By looking at the lineage tree, you can see that at every level of the lineage tree you get uh larger, you can get memory with more cells and more differentiation, and so we've done a lot of different types of analysis of these trees versus random trees versus trees and other species, and we've actually done things with information theory and with uh you know, geometric approaches to give us a signal that we can look at and compare across different conditions, and so this is another. This is simulating lineage trees in this graph.

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This is where we can simulate lineage trees using different uh distributions, so lineage trees. You know we might assume that they're normally distributed that they branch in a very regular interval. So maybe it's some sort of uh gaussian function or some sort of uh normal distribution, but it may actually not be that, and so we can look at different types of distributions for different types of lineage trees and from that we can uh sample and and simulate embryos in different ways.

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Sometimes the you know our simulations may not make any sense biologically, but we can generate these new possibilities by doing this. This sort of mathematical modeling.

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So that brings us to another question: how do we characterize the rate, recombination and conservation of growth and form? And so we do that through lineage modeling, military modeling and differentiation tree modeling, which is actually something that one of our collaborators richard gordon originated and he's written a number of books on it, and the idea is that you have this tree. It's like a lineage tree, except it. It focuses on sort of the next level up which is tissue differentiation in c elegans.

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A linear differentiation tree is essentially trying to understand like what happens when different tissues differentiate.

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But it's not they're the same thing as what you might see in say like a mammalian species or a lizard species, and so you know the the the differentiation tree sometimes is very similar to lineage tree.

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Sometimes it's not so what I've done here is I've had I introduced a figure that shows uh development and differentiation and agent- and this is a computational agent and this just it sort of resembles a uh braidenburg vehicle, and the idea is that we're showing that they're different developmental stages and that these different developmental stages and the tissues that result from these different stages.

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uh You know you can build a tree of this and it characterizes all the different stages of development and maybe some of the variation that you see in development. So this is a three-layered embryo. This is something that approximates something you'd see in nature, uh where you have three germ layers and an embryo. That's your starting point with a so it just branches off early and then later you have different variations of b, which is where these three germ layers differentiate into these sensors and effectors.

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This outer shell and then this inner shell, which will become the connecto and then c, which branches off from b, which becomes more complex, is where the connectome is actually hollowed out and then filled with nervous system, cells or neurons.

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This middle shell is differentiated from the core connectum and then this outer shell exists, and then you have your uh sensors and effectors down here and that's that's, basically the structure of a semi-mature agent, so you can see with agent modeling, and this is done because it's a very easy example, uh you can build this structure of differentiation tree uh and so finally, we also do we've done a lot of work on differentiation, trees and computational development.

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So you know: we've looked in c elegans, we've done some studies of differentiation, trees and and some of the cell tracking data and some of the other data. We've done some things with molecular data and and other types of comparative biology and we've.

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You know built computational models and integrated it with differentiate, studying differentiation, trees, and so that is reference f on the poster, and so then now, if we go down to the bottom, actually we don't have f on this poster, but this is uh topic f, so they're, actually a through c or a through e, which are the references here. So we have a reference from the general neuroinformatics one from biosystems on um the origins of the embryo that features the game. Theory work c. Is this morphozoic?

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So this is the cellular automata work d is the emerging connectome, so this says actually emerging connectomes.

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So this area of the the c elegans, the brain, it's it forms in development, and so this paper kind of focuses on taking data, that's available for the m for embryogenesis and then mapping it to the adult connectome and seeing what the emergence of that connectome looks like in c elegans and then finally, information isometry techniques.

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This is something that refers to this figure up here. That's what paper this is taken from is just looking at these different transformations of lineage trees and differentiation, trees and characterizing them. Statistically so I hope you've enjoyed this poster. It gives you an idea of what we're doing again. This is our website here. This is our publications page here and thank you for your attention.