Evolution of the Morphology and Patterning of Artificial Embryos: Scaling the Tricolour Problem to the Third Dimension

Transcription

1 Evolution of the Morphology and Patterning of Artificial Embryos: Scaling the Tricolour Problem to the Third Dimension Michał Joachimczak and Borys Wróbel Computational Biology Group, Department of Genetics and Marine Biotechnology Institute of Oceanology, Polish Academy of Sciences Powstańców Warszawy 55, Sopot, Poland Abstract. We present a model of three-dimensional artificial embryogenesis in which a multicellular embryo develops controlled by a continuous regulatory network encoded in a linear genome. Development takes place in a continuous space, with spherical cells of variable size, and is controlled by simulated physics. We apply a genetic algorithm to the problem of the simultaneous evolution of morphology and patterning into colour stripes and demonstrate how the system achieves the task by exploiting physical forces and using self-generated morphogen gradients. We observe a high degree of robustness to damage in evolved individuals and explore the limits of the system using more complex variations of the problem. We find that the system remains highly evolvable despite the increased complexity of three-dimensional space and the flexible coding of the genome requiring from evolution to invent all necessary morphogens and transcription factors. Keywords: artificial embryogeny, gene regulatory network, morphogenesis, embryo patterning, positional gradients, French flag model, cellular differentiation. 1 Introduction The field of artificial embryogenesis investigates how the compact genomes of living organisms are able to encode the structure of extremely complex multicellular organisms using a limited number of genes. As a rule, multicellular biological organisms start their development from a single cell and form through cell division. Before each division, a copy of the genome is made. Symmetry-breaking mechanisms cause the differential expression of the genes in the descendant cells. However, the embryo is shaped not only by the fact that different gene products are present at different concentrations in different cells. Physical interactions between the cell components, as well as between the cells and their environment result in emergent phenomena that allow bridging the overwhelming discrepancy between the amount of information encoded in the genome and the complex structure of the organism. Furthermore, living developing systems display a remarkable robustness to perturbations. On the one hand, this demands better understanding, while on the other, it raises hopes of employing developmental models G. Kampis, I. Karsai, and E. Szathmáry (Eds.): ECAL 2009, Part I, LNCS 5777, pp , c Springer-Verlag Berlin Heidelberg 2011

2 36 M. Joachimczak and B. Wróbel to solve various engineering problems and to overcome the limitations of evolvability present in direct encoding schemes ([1,2]). Many approaches to generating morphologies of artificial organisms have been proposed, and have frequently been evaluated using the so-called French flag problem, proposed more than 40 years ago by Lewis Wolpert ([3]), which consist of developing a patterned embryo with three colours (blue, white, red) in three areas. Several recent papers present biologically-inspired models of cellular development in which each cell is controlled by a gene regulatory network. In some, the development takes place on a 2D lattice with cells occupying fixed locations (e.g., [4,5,6]). In others, for example those using the Cellular Potts Model, the cells occupy multiple locations on a lattice, and their shapes are determined by the simulated physical interactions of their membranes ([7,8]). Other approaches include elastic interactions between the cells ([9] and recently [10]). Three-dimensional simulations ([11,9,12,13]) remain scarce, partly because it is not always necessary to investigate underlying phenomena, and partly because more degrees of freedom make 3D multicellular development a much more difficult task. In this work, we attempt to scale the French tricolour problem to three dimensions by evolving genomes which code gene networks regulating the development of multicellular ellipsoidal bodies with multicolour patterning. Patterning (cell differentiation) is understood as expressing particular gene products in different areas of the embryo. The cells in our model form bodies through elastic interactions, are suspended in continuous space (no grid is used), and produce spatiotemporal gradients of morphogens, perceived by other cells. 2 The Model We employ an extended version of the system introduced in [12], where it was used to evolve three-dimensional morphologies of multicellular organisms. The conceptual framework remains unchanged, but many features that were found to be superfluous were removed, facilitating a better understanding of and control over the whole system. Furthermore, biological relevance was improved by modelling product accumulation and degradation. An overview of essential features and concepts is provided, for additional details please refer to [12]. As the main interest of our work is the emergent properties of evolved regulatory networks, our system is designed to minimize limitations imposed on topologies that can evolve. In our model, linear genomes can contain any number of regulatory units. Regulatory units are composed of one or several regulatory elements and one of several genes (genetic elements that code products, which include transcription factors and morphogens). Any number of connections between regulatory units can exist. 2.1 Genome and Genetic Elements The genome in our model is a list of genetic elements that fall into three classes: elements that code products (called genes); regulatory elements (called promoters); special elements (that code the external inputs and outputs of the regulatory network). The genome is parsed sequentially, and regulatory units are formed whenever a series of

3 Evolution of the Morphology and Patterning of 3D Artificial Embryos 37 G reg. unit #1 reg. unit #2 reg. unit #3 S G P P G P G S G P G P P type sign 0,1,2,3,4 or 5-1 or 1 a special element: external factor (0) or effector (1) co-regulated genes a promoter: additive (2) or multiplicative (3) a gene: internal product (4) or morphogen (5) x 1 x 2 x n position in N-dimensional space Fig. 1. The genome and the structure of a single genetic element. Each element consists of a type field, a sign field, and a sequence of N real values used to determine affinity to other elements (N =2wasusedinthispaper). promoter elements is followed by a series of genes, with special elements assigned to input and output nodes at a later stage. By computing affinities between all gene products and all promoters, connections between regulatory units are formed, and, thus, a gene regulatory network (GRN) emerges. Fig. 1 provides an overview of the process. In our system, promoters come in two types: additive and multiplicative (see below). The products can be either internal or external; only external products (morphogens) can diffuse from one cell to another. Each genetic element in our system encodes a point in N-dimensional space (Fig. 1). Affinity between products and promoters is determined as a function of Euclidean distances between associated points in this space. The lower the Euclidean distance is between these points, the higher the affinity is between gene products and promoters. A cut-off distance is used to prevent full connectivity in the network. The product of sign fields of the two elements determines the sign of the connection (activatory or inhibitory). One can imagine (or actually visualize if N<3) thatas genomesevolve (and the element coordinates change), points in N-dimensional sequence space that correspond to the elements approach one another or move away. Note, however, that this space is used solely as a mechanism for determining connectivity and bears no relation to the 3D space in which multicellular development takes place. The activation of each promoter in a regulatory unit is the sum of the concentration of all binding products weighted by their affinities. The sum of activations of all additive promoters is multiplied by the product of activation of all multiplicative promoters. 2 The result (A) is fed to a sigmoid function f A (A) =. This is interpreted 1+e (A 1) as the production rate (positive or negative) of all products in a given regulatory unit, i.e., dl dt = f A (A) L, wherel is the current concentration. This results either in an increase of concentration of a product (if the synthesis is higher than degradation) or in increased degradation. The presence of a multiplicative promoter in a regulatory unit results in a strict requirement for the presence of a binding product, otherwise the unit is not expressed. 2.2 Development Cells occupy continuous positions in 3D space and are modelled as elastic spheres (of various sizes). If two cells overlap, a repulsive force proportional to the amount of overlap is applied. In particular, after cell division the daughter cell overlaps with the mother and will slowly move away. To maintain the coherent structure of the embryo, adhesive forces are simulated: cells stick together whenever they are at a close distance.

4 M. Joachimczak and B. Wróbel Fig. 2. Mechanics of the developmental model. Left: each cell has an internal orientation vector H and two perpendicular helper vectors that can be rotated by three effectors (outputs of the network) - see p. 19 of [14] for explanation of such a system. Right: simplified diffusion is modelled without the use of a grid, by using past values of morphogen expressions. An outburst of morphogen production in the embryo followed by quick degradation produces a wave travelling through the system. Furthermore, fluid drag is simulated to prevent erratic movements. Morphogen diffusion (Fig. 2, right) is simulated by calculating the perceived levels of diffusive substances in a given point in space as a function of distance and the historic values of production from all the sources (the farther the source, the older value is used). Special elements code either for the outputs of the network (effectors) or inputs. Effectors correspond to specific cell action (division, death, size increase of the daughter cell, and rotation of the cell orientation vector; cell orientation determines where the new cell is placed after division, similarly to 3D L-systems [14]). In addition, the composite of the activation of colour effectors defines the cell colour. We use either two (red, blue) or three (red, green, blue) colour effectors, so for example, cells that express both red and blue are pink. Division or death of a cell occurs when the activation of the corresponding effector is above a set threshold. Similarly, in some experiments cells were coloured only when the activation of the colour effector was above a set threshold. In all the experiments in this work, the GRNs are given as the input an external factor that was present in all cells at the same concentration (set at 1) throughout development. This input was necessary to initiate any activity, with the exception of experiments where two maternal morphogen gradients were deployed in the environment and could serve the same purpose. Direct regulatory connections between external factors and effectors were not allowed to prevent trivial solutions. At the time of division, the state of GRN is copied to the new cell, which can differ from the mother only by its rotated orientation vector and size. As these properties cannot directly influence the state of GRN, an external symmetry breaking mechanism is necessary. Development is allowed to take 300 time steps, in each time step both the position of the cells and the state of GRN in each cell is updated. 2.3 Genetic Algorithm and the Fitness Function Genetic operators in our system act on the level of element fields (changing element type, sign bit, or disturbing the coordinates of an associated point in space). Duplications and deletions of single elements or multiple elements are allowed. The results shown in this work were obtained using a fairly standard genetic algorithm with a

5 Evolution of the Morphology and Patterning of 3D Artificial Embryos 39 population size of 300, elitism, tournament selection, and multipoint crossover for sexual reproduction (for 30% of the individuals in each generation). Evolutionary runs were initiated with individuals consisting of 5 randomly created regulatory units, usually requiring a few thousand tries before a single individual capable of starting division would appear (as this requires random emergence of connection from external factor 1 to some regulatory unit and then to the division effector). Evolutionary runs where terminated after no improvement over 500 generations was detected, resulting in runs lasting for about generations. To assess the fitness of individuals, evolved morphology was compared to a target shape at the voxel level. Each voxel outside the target and inside some cell decreased the fitness. Each voxel inside the target and inside a cell increased fitness proportionally to the correctness of colours expressed by this cell (i.e., 1 for perfect match, 0 for completely reversed pattern expression). Thus, both the correct morphology and patterning were rewarded. To reward the emergence of multiple colours at the very early stages of evolution, fitness was divided by the number of colour effectors minus the number of colours present in the individual (as in [8]). We found this improved evolvability greatly. 3 Results and Discussion We investigated the evolvability and scalability of the system by performing a series of experiments, each repeated 10 times. In the first series of experiments, two colour effectors were used, allowing the cells to be coloured red or blue if their activation (a real value between 0 and 1) was above a threshold of 0.5, resulting in four possible colours - with pink for expression of both effectors and white for no expression (rather than black, so that actual French tricolour could be obtained). The target shape seen in Fig. 3 (left) was used. Development was terminated after 300 time steps, with a hard limit on the number of cells (i.e., cells could not divide after the limit was hit). Obtaining correct morphology turned out to be a fairly trivial task for the GA, owing to the exploitation of physics. Embryos would apply minor variations to the orientation of division vectors and the repulsion of cells would take it from there, creating elongated morphologies (Fig. 3, right). The only input to the network in these experiments was a static external factor, perceived in each cell at the constant level of 1. The symmetry of the embryo development is broken because, after division, cells shift in space and perceive slightly different concentrations of morphogens. This results in self-emerging positional information (explored earlier by Knabe et. al. in [8]). Fig. 3. Left: default target shape and colour pattern,right: small variations to orientations of divisions generate an elongated shape by exploiting the physics of the system (same individual as in Fig. 4)

6 40 M. Joachimczak and B. Wróbel (a) (b) (c) Fig. 4. Development of the best individual with 2 colour effectors (a); below, self-generated gradients of positional information employing two different morphogens (b,c): left - production level of a morphogen in each cell (blue to red colour map), right - normalized morphogen density maps in the space surrounding the embryo. Fig. 4a presents snapshots from the development 1 of the best individual obtained in 10 runs, with a 200 cell limit. Typically, runs would result in 30-40% yield of threecoloured individuals, the others would remain stuck in local suboptima. A more detailed analysis of the individual shown in Fig. 4a allows locating the clear asymmetric production of two distinct morphogens centred in the posterior and anterior of the embryo and reveals gradients generated in space (Fig. 4bc). Only two gradients are shown in the figure, even though two more morphogens with similar patterns were also present. Although this is not the only possible solution for generating anterior/posterior axis (single morphogen at one extreme of the embryo would suffice in principle), all analysed individuals developed using the differential production of at least two morphogens. One of the problems identified in early experiments was that when evaluated by their similarity with target pattern after 300 steps, the patterns were rarely stable. Typically, they would sweep through the embryo (driven by diffusing waves of morphogens) or oscillate. We have managed to partially alleviate this by calculating overall fitness as an average of similarity values taken every 5 simulation steps in the last 50. Individuals that were largely stable through this period would then be obtained. However, in most cases, the pattern degraded if development was allowed to continue beyond its default lifetime of 300 steps. We note, however, that it is (sadly) a common feature of living systems to degrade if their lifespan is extended beyond what they were selected for by evolution. In the next series of experiments, we investigated the stabilizing role of the gradients of substances present in the environment of the developing embryo. Two external factors diffused from sources external to the embryo at its two extremes (similarly to Bicoid and Nanos, which determine anterior/posterior axis in Drosophila embryo development; see, e.g., [15]). This allowed for stable embryo patterning, but only if the cells were additionally prevented from producing their own morphogens (which makes 1 Supplementary materials, including videos of development and gradient formation, are available at

7 Evolution of the Morphology and Patterning of 3D Artificial Embryos 41 (a) (b) (c) Fig. 5. Robustness to damage: the effects of removing a single cell from the embryo at the 2- (a), 4- (b), or 8-cell stages (c) of development; in (c) only half of possible cases is shown. Removing cells at further stages has a notable, albeit diminishing, effect. The same individual as in Fig. 4 is shown. (a) (b) (c) (d) Fig. 6. More difficult evolutionary targets presented to the system (top); two best individuals out of 10 runs for each target are shown below. Note that for (a) and (b) the lack of expression of colour effectors is drawn as black. those maternal factors the only inducers of differential expression). The dominant role of self-produced morphogens can be explained by the fact that, in our system, morphogens produced by the embryo can reach much higher concentrations than the diffusive substances present in the environment. Robustness to perturbations such as mutations or damage is one of the essential features observed in developmental systems (see e.g. [2,4,16]). Although all the individuals presented in this paper evolved in the absence of any developmental stochasticity, they were found to respond extremely well to the removal of a single cell, even at the very early stages of development (Fig. 5). This suggests that fault tolerance developed as an effect of mutational robustness and can be considered to be an indication of good evolvability ([17]). To assess the scalability of the model, different approaches to patterning were evaluated. Fig. 6a shows the use of non-thresholded colour effectors. This turned out to be a harder problem, which is understandable if one considers that it was now not only necessary to reach a certain threshold of colour expression, but also to maximize (or repress) it in a given section of the embryo. Rather surprisingly, evolutionary runs (terminated after 500 generations without improvement) took much longer, taking even up to generations (compared to about in the experiments with thresholding). This indicates that the fitness landscape of this problem is actually much less rugged, and provides many more opportunities to fine-tune individuals.

8 42 M. Joachimczak and B. Wróbel The task can also be made harder by introducing the third colour effector (Fig. 6b), making it necessary to both express a certain colour and repress the expression of two others in each area. As expected, lower fitness was reached in general, but some tricoloured individuals were obtained. Fig. 6cd show solutions to other variations of the patterning problem - the emergence of four areas or multiple stripes. Both tasks were solved using two colour effectors. These problems are still within range but approaching the limits of evolvability of the current setup. 4 Conclusions We present a system capable of evolving solutions to the French flag problem using simulated physics and self-generated gradients of morphogens dynamically diffusing in the environment. The computational cost of adding a third dimension and implementing simple physics was negligible compared to the cost of the simulation of a regulatory network in every cell. Considering the availability of sophisticated physics engines, it would be very interesting to see how this developmental system (and other comparable models) would be able to exploit more complex simulated physics, such as flexible cells. Thus, our further work will focus on maintaining the biologically-relevant features of the network while introducing more elements to simulate the physical aspects of multicellular development in our system. Acknowledgements. The computational resources used in this work were obtained through the support of the Polish Ministry of Science and Education (project N N and N N ), the Tri-city Academic Computer Centre (TASK) and the Interdisciplinary Centre for Molecular and Mathematical Modelling (ICM, University of Warsaw; project G33-8). References 1. Bentley, P., Kumar, S.: Three ways to grow designs: A comparison of embryogenies for an evolutionary design problem. In: Proceedings of the Genetic and Evolutionary Computation Conference, Orlando, Florida, USA, vol. 1, pp Morgan Kaufmann, San Francisco (1999) 2. Stanley, K., Miikkulainen, R.: A taxonomy for artificial embryogeny. Artificial Life 9(2), (2003) 3. Wolpert, L.: Positional information and the spatial pattern of cellular differentiation. J. Theor. Biol. 25(1), 1 47 (1969) 4. Miller, J.F.: Evolving a self-repairing, self-regulating, french flag organism. In: Deb, K., et al. (eds.) GECCO LNCS, vol. 3102, pp Springer, Heidelberg (2004) 5. Beurier, G., Michel, F., Ferber, J.: A morphogenesis model for multiagent embryogeny. In: Proceedings of ALIFE X, pp MIT Press, Cambridge (2006) 6. Chavoya, A., Duthen, Y.: A cell pattern generation model based on an extended artificial regulatory network. Biosystems 94(1-2), (2008) 7. Hogeweg, P.: Evolving mechanisms of morphogenesis: on the interplay between differential adhesion and cell differentiation. J. Theor. Biol. 203(4), (2000)

9 Evolution of the Morphology and Patterning of 3D Artificial Embryos Knabe, J.F., Nehaniv, C.L., Schilstra, M.J.: Evolution and morphogenesis of differentiated multicellular organisms: autonomously generated diffusion gradients for positional information. In: Proceedings of ALIFE XI, pp MIT Press, Cambridge (2008) 9. Hotz, P.E.: Genome-physics interaction as a new concept to reduce the number of genetic parameters in artificial evolution. In: Congress on Evolutionary Computation, CEC 2003., vol. 1, pp (2003) 10. Doursat, R.: Programmable architectures that are complex and self-organized: From morphogenesis to engineering. In: Proceedings of Artificial XI, pp MIT Press, Cambridge (2008) 11. Kumar, S.: Investigating Computational Models of Development for the Construction of Shape and Form. PhD thesis, Dept. of Computer Science, University College London (2004) 12. Joachimczak, M., Wróbel, B.: Evo-devo in silico: a model of a gene network regulating multicellular development in 3D space with artificial physics. In: Proceedings of Artificial XI, pp MIT Press, Cambridge (2008) 13. Haddow, P.C., Hoye, J.: Achieving a simple development model for 3D shapes: are chemicals necessary? In: Proceedings of the 9th Annual Conference on Genetic and Evolutionary Computation (GECCO 2007), pp ACM, New York (2007) 14. Prusinkiewicz, P., Lindenmayer, A.: The algorithmic beauty of plants. Springer, New York (1996) 15. Carroll, S.B., Grenier, J.K., Weatherbee, S.D.: From DNA to Diversity: Molecular Genetics and the Evolution of Animal Design. Blackwell Publishers, Malden (2004) 16. Eggenberger, P.: Exploring regenerative mechanisms found in flatworms by artificial evolutionary techniques using genetic regulatory networks. In: The Congress on Evolutionary Computation, CEC 2003, vol. 3, pp (2003) 17. Federici, D., Ziemke, T.: Why are evolved developing organisms also fault-tolerant? In: Nolfi, S., Baldassarre, G., Calabretta, R., Hallam, J.C.T., Marocco, D., Meyer, J.-A., Miglino, O., Parisi, D. (eds.) SAB LNCS (LNAI), vol. 4095, pp Springer, Heidelberg (2006)