Systems and methods for modeling morphogenesis

Abstract

Systems, methods, and computer readable medium for modeling a morphogenetic process. In some embodiments, the methods include: constructing a neural cellular automata (NCA), in which the NCA comprises a plurality of cells and each cell in the plurality of cells has an assigned state and obeys local update rules; initializing the NCA to reach a target pattern by selecting a set of parameters; characterizing an evolutionary process of the initialized NCA developing to reach the target pattern based on the local update rules and the set of parameters; and outputting to a user a report of the characterized evolutionary process of the NCA, in which the report allows the user to relate the evolutionary process to morphogenetic pattern formation.

Description

T002814 166118.01533 SYSTEMS AND METHODS FOR MODELING MORPHOGENESIS CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to U.S. Provisional Application No. 63 / 574,743 filed on April 4, 2024, which is incorporated herein by reference in its entirety for all purposes. BACKGROUND

[0002] There exists a need for understanding morphogenetic processes. Using evolutionary algorithms on neural cellular automata is a promising method to study morphogenesis. SUMMARY

[0003] Disclosed herein are systems, methods, and computer readable medium for modeling a morphogenetic process. In some embodiments, the methods include: constructing a neural cellular automata (NCA), in which the NCA comprises a plurality of cells and each cell in the plurality of cells has an assigned state and obeys local update rules; initializing the NCA to reach a target pattern by selecting a set of parameters; characterizing an evolutionary process of the initialized NCA developing to reach the target pattern based on the local update rules and the set of parameters; and outputting to a user a report of the characterized evolutionary process of the NCA, in which the report allows the user to relate the evolutionary process to morphogenetic pattern formation. BRIEF DESCRIPTION OF THE DRAWINGS

[0004] The patent or application file contains at least one drawing executed in color. Copies of this patent or patent application publication with color drawing(s) will be provided by the Office upon request and payment of the necessary fee.

[0005] FIG. 1A shows an illustration of directing encodings of a phenotype of a two- dimensional (2D) smiley-face tissue made of single cells (the smiley face is used because it mimics the electric face pattern that guides craniofacial morphogenesis in vertebrates, like the 1 QB\166118.01533\95647317.2T002814 166118.01533 frog embryo). Each gene encodes a specific phenotypic trait, here of each specific cell type of the tissue, colored blue, pink, and white.

[0006] FIG. 1B shows an illustration of indirect encodings of a phenotype of a 2D smiley-face tissue made of single cells. A deterministic mapping between the genome and different phenotypic traits, here again of each cell type (shown for completeness, but not investigated here due to reasons discussed in Discussion).

[0007] FIG. 1C shows an illustration of multi-scale competency architecture encodings of a 2D smiley-face tissue made of single cells. Encoding of functional parameters of the uni- cellular agents which self-assemble a target pattern via successive local perception-action cycles1 (see FIG. 1D). In FIGS. 1A-1C, the genome, the respective encoding mechanism, and the corresponding phenotype are illustrated left to right; colors indicate cell types, and arrows indicate the flow of information and environmental noise, affecting each cell during the developmental process.

[0008] FIG. 1D shows detailed information-flow-chart of the perception-action cycle of a particular single cell agent, labeled i, in a Neural Cellular Automaton (NCA)-based multi-scale competency architecture (compare to FIG. 1C and Methods, NCA): Starting from a multicellular phenotype configuration at time tk(left smiley-face panel), and following the thick orange arrows, each cell i perceives cell state information about its respective local neighborhood of the surrounding tissue (respectively labeled). This input is passed through an artificial neural network (ANN), substituting the internal decision-making machinery of a single cell, until an action output is proposed that induces a (noisy) cell state update in the next developmental step at time tk+1(details on labeled internal ANN operation and ANN architectures are introduced later in Methods, NCA and Detailed Methods, Artificial Neural Networks).

[0009] FIG. 1E shows a schematic illustration of the evolution of a morphogenesis process with a multi-scale competency architecture acting as the developmental layer between genotypes and phenotypes (see Methods): The genotype (top) encodes the structural (initial cell states) and functional parts (decision-making machinery) of a uni-cellular phenotype (center). The cell’s decision-making machinery is represented as a potentially recurrent ANN (yellow / orange graph) with an adjustable competency level (red knob). Through repeated local 2 QB\166118.01533\95647317.2T002814 166118.01533 interactions (perception-action cycles; detailed in panel D)), the multi-cellular collective self- orchestrates the iterative process of morphogenesis and forms a final target pattern, i.e., a system-level phenotype after a fixed number of developmental steps (bottom left to right) while being subjected to noisy cell state updates at each step (red arrows). The evolutionary process solely selects at the level of the system-level phenoctypes (labeled Final State at the bottom right). Based on a phenotypic fitness criterion, the corresponding genotypes - composed of the initial cell states (bottom left) and the functional ANN parameters (top right) - are subject to evolutionary reproduction - recombination and mutation operations - to form the next generation of cellular phenotypes that successively “compute” the corresponding system-level phenotypes via morphogenesis, etc.

[0010] FIG. 2 shows Typical fitness trajectory over several generations of CMA-ES103 of an NCA-based 8x8 Czech-flag morphogenesis task without (top) and with competency (bottom), corresponding to (i) direct and (ii) an multi-scale competency encoding of the target pattern as discussed here, representative for related experiments at similar system parameters (see FIG.3). We present the historic- (blue) and current (light blue) best fitness value per generation, the current structural fitness (purple), and the mean (black) and variance (gray) of the fitness of the entire population; in the top panel, the structural and phenotypical fitness are equivalent, thus only the latter is shown. The task is solved when a final fitness score of Fj= 64 is reached (marked by the green dashed line), e.g., when 8x8 = 64 cell types are correctly assumed after tD= 25 developmental steps. The cartoon insets represent the perception-action cycle of the NCA, assembling an initial (random) arrangement of cell types into the target pattern; for the direct case (top panel), the NCA’s ANN is disabled, which is illustrated by masking the agential parts in the cartoon.

[0011] FIG. 3A shows the average fitness per generation of the best-performing individual in a population of 65 independent evolutionary processes of the 8x8 Czech flag task, evaluated from left to right at different noise levels and color-coded by the decision-making probabilities; solid lines mark average fitness values, the shaded area marks the standard deviation (to lower values only), and dashed lines indicate when an average fitness threshold of 64 is crossed, solving the problem. 3 QB\166118.01533\95647317.2T002814 166118.01533

[0012] FIG. 3B shows the average fitness per generation of the best-performing individual in a population of 65 independent evolutionary processes of the 8x8 Czech flag task, evaluated from left to right at different decision making probabilities and color-coded by the noise levels; solid lines mark average fitness values, the shaded area marks the standard deviation (to lower values only), and dashed lines indicate when an average fitness threshold of 64 is crossed, solving the problem.

[0013] FIG. 3C shows a heatmap of the average generation number when the fitness threshold of 64 is crossed at particular combinations of the decision-making probability and noise level as detailed in FIG. 3A, FIG.3B; green and red arrows respectively indicate directions along PDof increasing and decreasing values of the avg. fitness at fixed noise values.

[0014] FIG. 3D shows the same as FIG. 3C, partitioned by the respective FF-agent or RGRN-agent architectures used in the respective CMA-ES runs.

[0015] FIG. 4A shows the evolved decision-making probability PDfor different noise levels when a fitness threshold of 64 for the 8x8 Czech flag task is reached; each symbol represents an independent lineage with a color-coding that indicates the number of generations it took for that particular lineage to cross the specified fitness threshold. The green / orange / red dashed lines indicate at which value of PDthe evolutionary process crossed the fitness threshold the fastest / on average / the slowest (i.e., in the least, average, or largest number of generations) for each noise level. The red / green / blue frames emphasize the noise level = 0, 0.125 and 0.25 corresponding to panels FIGS. 3C-3E, respectively.

[0016] FIG. 4B shows the same as FIG. 4A, but the threshold is 70. The red / green / blue frames emphasize the noise level = 0, 0.125 and 0.25 corresponding to panels FIGS. 4C-4E, respectively.

[0017] FIGS. 4C-4E show the latter show the evolution of the decision-making probability / fitness (top / bottom left panel) and the value of the decision-making probability as a function of the corresponding fitness during the evolutionary process of each lineage (right panel) for all lineages at the specified noise level. Results are shown for an RGRN-agent architecture with redundancy R = 1, and are qualitatively similar to an FF-agent architecture. 4 QB\166118.01533\95647317.2T002814 166118.01533

[0018] FIG. 5A shows the average fitness score of 100 independent evaluations of selected NCA results utilized at noise conditions, which have not been experienced during training for an increased total lifetime of 100 time steps. The respective NCAs have been evolved at zero-noise without competency, with a fixed number of tD= 25 developmental steps; the results are based on RGRN-agent architecture with the training conditions given by titles and dashed lines.

[0019] FIG. 5B shows the average fitness score of 100 independent evaluations of selected NCA results utilized at noise conditions, which have not been experienced during training for an increased total lifetime of 100 time steps. The respective NCAs have been evolved at zero-noise with evolvable competency, with a fixed number of tD= 25 developmental steps.

[0020] FIG. 5C shows the average fitness score of 100 independent evaluations of selected NCA results utilized at noise conditions, which have not been experienced during training for an increased total lifetime of 100 time steps. The respective NCAs have been evolved at noise level = 0.25 and competency level of PD=0.5, and deployed under different noise conditions andmaking probabilities. The data presented are based on the same NCA solution as FIG. 5E (indicated by the dashed frames), while the noise level is varied in FIG. 5C and FIG. 5D at a fixed competency level of PD=0.5.

[0021] FIG. 5D shows the average fitness score of 100 independent evaluations of selected NCA results utilized at noise conditions, which have not been experienced during training for an increased total lifetime of 100 time steps. The respective NCAs have been evolved at noise level = 0.5 and competency level of PD=0.5, and deployed under different noise conditions andmaking probabilities. The data presented are based on the same NCA solution as FIG. 5F (indicated by the dashed frames), while the noise level is varied in FIG. 5C and FIG. 5D at a fixed competency level of PD=0.5.

[0022] FIG. 5E shows the average fitness score of 100 independent evaluations of selected NCA results utilized at competency-level conditions, which have not been experienced during training for an increased total lifetime of 100 time steps. The respective NCAs have been evolved noise level = 0.25 and competency level of PD=0.5, and deployed under different5 QB\166118.01533\95647317.2T002814 166118.01533 noise conditions and decision-making probabilities. The data presented are based on the same NCA solution as FIG. 5C (indicated by the dashed frames), while the competency is varied in FIG. 5E at a fixed noise level of =0.25.

[0023] FIG. 5F shows the average fitness score of 100 independent evaluations of selected NCA results utilized at competency-level conditions, which have not been experienced during training for an increased total lifetime of 100 time steps. The respective NCAs have been evolved at noise level = 0.5 and competency level of PD=0.5, and deployed under different noise conditions and making probabilities. The data presented are based on the sameNCA solution as (indicated by the dashed frames), while the competency is varied in FIG. 5F at a fixed noise level of =0.5.

[0024] FIG. 6 showsnumber of generations it takes for the CMA-ES to adapt a pre-evolved NCA solution that can solve the 8x8 Czech-flag morphogenesis task to adapt, respectively, to the 8x8 blue-, white-, red-, and Viennese-, blue\white-, and blue / red-flag morphogenesis tasks instead (see panel insets) and reach a correctness fitness score of 64. We specifically adapted Czech-flag NCA solutions that have been pre- evolved at a noise level of ξc= 0.25, but with corresponding competency levels according to the horizontal axis in3C, and deploy CMA-ES for 1000 generations at the corresponding noise / competency-levels depicted here on the vertical / horizontal axis, and average over multiple CMA-ES runs and corresponding redundancy numbers, R = 1, 2, 4, 8, 16.

[0025] FIG. 7 shows the developmental process of the 8x8-Czech flag task (vertical axis) of selected generations over evolutionary time-scales (horizontal axis) for an NCA evolved with system parameters = 0.25, PD= 50%, and R = 4. Each pixel corresponds to a cell of the NCA, at a given developmental step and generation, in an RGB notation corresponding to the numerical values of the first three cell states, scaled to values between [0, 1]. The top panel shows the current fitness of the respective generations (blue), and the structural fitness at tk= 0 (purple); the green vertical dashed line marks the generation crossing the fitness threshold of Fj= 64 where we consider the problem solved.

[0026] FIG. 8 shows the same as FIG. 7, but for an NCA without competency (e.g., PD= 0). 6 QB\166118.01533\95647317.2T002814 166118.01533

[0027] FIG. 9 shows the same as FIG. 3C, but for a target pattern of a 9x9 smiley face (inset in left panel). Moreover, we here aggregate over R ≥ 4.

[0028] FIG. 10 shows the same as FIG. 7, but for the 9x9 smiley face task, with a fitness threshold of Fj= 81 (see green dashed line on top panel).

[0029] FIG. 11A shows the average fitness per generation of the best-performing individual in a population of 65 independent evolutionary processes of the 8x8 Czech flag task, evaluated from left to right at different noise levels and color-coded by redundancy number R; solid lines mark average fitness values, the shaded area marks the standard deviation (to lower values only), and dashed lines indicate when an average fitness threshold of 64 is crossed, solving the problem.

[0030] FIG. 11B shows the average fitness per generation of the best-performing individual in a population of 65 independent evolutionary processes of the 8x8 Czech flag task, evaluated from left to right at different redundancy numbers and color-coded by the noise levels; solid lines mark average fitness values, the shaded area marks the standard deviation (to lower values only), and dashed lines indicate when an average fitness threshold of 64 is crossed, solving the problem.

[0031] FIG. 11C shows a heatmap of the average generation number when the fitness threshold of 64 is crossed at particular combinations of the redundancy number R and noise level as detailed in FIG. 11A, FIG.11B.

[0032] FIG. 11D shows the same as FIG. 11C, partitioned by the respective FF-agent or RGRN-agent architectures used in the respective CMA-ES runs.

[0033] FIG. 12 shows an example process for modeling morphogenesis.

[0034] FIG. 13 shows an example hardware system.

[0035] FIG. 14 (top) shows a reevaluation of the fitness scores of the best-performing individual per generation of the lineage depicted in FIG. 2 with modified genomes of systematically reduced floating-point precision by first type-casting from 32-bit (single-) to 16- bit (half-)precision and further reducing the number of significant bits (mantissa) in the genes’ 7 QB\166118.01533\95647317.2T002814 166118.01533 binary representation from 10 to 1 (color-coded). Thus, for the 16-bit half-precision genes, thenumber of bits used to encode each gene is given by 6+ 6+S, with S significance (or mantissa)bits, 5 exponent bits, and 1 sign bit (following the IEEE 754-2008 standard). (Bottom) Similar to the top panel but showing the numerical difference of the historically best fitness score (blue curve in top panel) and the re-evaluated fitness scores (thin lines); thick lines on the bottom panel show the fitness deviation smoothed over several generations.

[0036] FIG. 15 (top) shows the same experimental data as shown in the bottom panel of FIG. 2, additionally contrasting the structural fitness of the historically best-performing individual (dashed blue) with the current structural fitness (purple) that corresponds to the best- performing individual from the current generation. (Bottom) The numerical improvement in the maximum fitness score for every generation with respect to the historically best fitness score until the respective prior generation is drawn in a sym-log representation (linear scale between 0–10-2, logarithmic above).

[0037] FIG. 16 (top) shows reevaluated fitness scores of the lineage depicted in the bottom panel of FIG. 2 (and in FIG. 15) at noise levels >0.25, different from those originally experienced in the corresponding evolutionary process; the historically best fitness score (blue) is presented as reference. (Bottom) the numerical difference between the historically best fitness score and the fitness score of the best-performing individual of a particular generation (thin lines); thick lines in the bottom panel show the fitness deviation smoothed over several generations. DETAILED DESCRIPTION

[0038] In accordance with some embodiments of the disclosed subject matter, mechanisms (which can include, for example, systems, computer readable media, and methods) for modeling morphogenesis or a morphogenetic process are provided. Morphogenesis refers to processes which entail developing anatomical or histological order (e.g., shape, form, or pattern) in a system. Morphogenesis is a critical process for cell, tissue, and organism development, regenerative, and aging in biology. A “morphogenetic process” is not confined to biological processes; it may refer to any process of developing order in an initially disordered system, such as active matter, chemical self-organizing reactions, and self-assembling engineered agents. A 8 QB\166118.01533\95647317.2T002814 166118.01533 morphogenetic process can be used to describe some robotics processes, such as swarm robotics which needs to assemble in a specific pattern in order to complete a task, self-organizing patterns in excitable media, and chimeric constructs such as biobots and hybrots. Multi-scale competency architecture (MCA) may refer to a system after undergoing a morphogenetic process. An MCA includes nested layers of active homeostatic agents, each forming the self-orchestrated substrate for the layer above, and, in turn, relying on the structural and functional plasticity of the layer(s) below.

[0039] It is important to understand how MCA can naturally arise from a system. Simulating morphogenesis with neural cellular automata (NCA) using an evolutionary algorithm can provide insights to how complex architectures can arise from disordered systems.

[0040] System

[0041] Neural cellular automata (NCA) can be viewed as systems made up of cells or elements. An NCA may have a target pattern or form. In some embodiments, an image, such as the Czech flag may be used as a target form of an NCA. This simplified model can be used to investigate mechanisms of morphogenesis.

[0042] A “cell” or “element” refers to a specific component of the NCA at a specific location. In some embodiments, an NCA can be viewed as a grid (e.g., a 2D grid or a 3D grid). A cell has a specific location in the grid. Each cell may have one or more “adjacent” or “neighboring” cells. Adjacent refers to cells that are directly next to one another (e.g., in a 1D example, cells at position 5 and 6 are adjacent to one another; cells at position 5 and 7 are not adjacent to one another). In some embodiments, a cell is dictated by a sensory part, a neighbor- wise sensory embedding component, and a controller component.

[0043] Each cell may have an assigned state. The terms “state” and “type” may be used interchangeably herein. A state refers to a descriptor of a cell. A state can be a binary value, integer, real, or even vector-valued. In the example of using a Czech flag as an NCA, a state may be a color corresponding to colors in the flag.

[0044] A cell’s state may evolve during an evolutionary or morphogenetic process. In the example of using a Czech flag as an NCA, the color of a cell may evolve to be the correct color in the flag. A cell may obey local update rules. Local update rules dictate mechanisms in which a 9 QB\166118.01533\95647317.2T002814 166118.01533 cell’s state may change. Update rules may be influenced by sensing behavior (e.g., the ability to sense the state of a neighboring cell) and communicating behavior (e.g., the ability to communicate updates with neighboring cells). In some embodiments, a state of a cell is determined by a previous state of the cell, and another previous state of one or more adjacent cells.

[0045] A cell also possesses an internal recurrent state (akin to transcriptomic expression in GRNs) which is not visible to neighboring cells. This internal state is also constantly updated across developmental time steps, as every cell processes information at every time step.

[0046] In some embodiments, a cell’s behavior is chosen to mimic a minimal set of rules relevant to biological cells. In some embodiments, cells can move on the NCA‘s grid (into neighboring void cells which are not populated by other cells). In some embodiments, a cell may ‘choose’ to die (similar to programmed cell death), leaving voids in the grid of the NCA. In some embodiments, cells can divide into a neighboring void (which is not populated with other cells) and row a target tissue by self-reproduction (e.g., the new cell may copy the original cell’s state, or get a random state assigned, depending on the algorithm’s configuration). In some embodiments, a cell may do nothing. In some embodiments, a cell may grow into a particular shape including a rod, square, circle, or an asymmetric shape such as an arrow (e.g., on square grids or hexagonal grids).

[0047] “Selected parameters” can refer to selected parameters that encode behaviors of individual cells, or encode system-level behaviors. In some embodiments, a “decision making probability” (sometimes displayed as PD) is a selected parameter. PDquantifies the probability at which a proposed update of each individual cell is executed in an environment. Altering the decision-making probability can vary the behavior of an NCA from direct encoding to multi- scale competency.

[0048] An “evolutionary process” refers to how an NCA changes over time, with respect to individual cells and the system as a whole. An evolutionary process may also be referred to a morphogenetic process or a developmental process. In some embodiments, an NCA is evolved using an evolutionary algorithm. The way in which the NCA changes while implementing the evolutionary algorithm is a morphogenetic process. 10 QB\166118.01533\95647317.2T002814 166118.01533

[0049] An evolutionary process may be characterized, using several metrics. In some embodiments, a fitness score may be determined and analyzed over development. Additionally or alternatively, the competency level of a network may be used.

[0050] A “genotype” of a network refers to the set of NCA parameters that govern a system. A “phenotype” of a network refers to a description of the network after development. For instance, a “system-level phenotype” may describe how close to a target is to a target pattern. In some embodiments, a “genotypic fitness” and a “phenotypic fitness” are quantified and recorded over the course of an evolutionary process.

[0051] Additional metrics that can be used to characterize the process of evolution are the quality of solutions found, the efficiency with which these solutions are found, the quality of exploration of the solution space, the lack of tendency to get stuck in dead ends, and the degree of maintenance of diversity in the population. In some embodiments, “transferability,” the efficiency with which a current solution to an existing problem can adapt to novel environmental conditions, may also be quantified.

[0052] Applications and Advantages

[0053] NCAs are not bound to certain architectures. In contrast to similar machine learning paradigms, such as diffusion models or NCAs trained with back propagation in time, the methods described herein are not restricted by gradient learning (vanishing gradients, etc.). Thus, a cell specific neural network may have a recurrent or transformer architecture. Moreover, the described methods offer precise control over neighborhood relations and grid specifications of the NCA. This offers an improvement over other methods, in which NCAs are purely convolutional on square / cubic grids.

[0054] In some embodiments, the systems and methods described herein may be used to study a system which evolves an intrinsic signaling mechanism to perform a task of interest. The systems and methods can be used to analyze the underlying signaling mechanisms, and the ability of the system to complete the task of interest.

[0055] In some embodiments, the methods and systems described herein can be used to model morphogenesis in biological system. This can be used to understand how morphogenesis is enacted in a variety of levels (cellular level, tissue level, organism level). Furthermore, the 11 QB\166118.01533\95647317.2T002814 166118.01533 systems and methods can be used to investigate how biological systems are able to undergo morphogenetic process in the presence of challenges (e.g., damage to the network, damage to cells, cells that obey different update rules, etc.).

[0056] One advantage of the disclosed systems and methods arises from splitting the genome into a structural and functional component. Doing so may serve as a method for open- ended evolution, development, or learning. These two sub-systems are concerned with learning the same objective (e.g., they are redundant to some extent). There is a shift of the competency from the cells’ functionality towards the structural component of the genome (the structural genome eventually covers / solves the problem via the Baldwin effect). Thus, the cells’ functionality becomes increasingly irrelevant in the system, and can be used for something else, so long as it does not destroy the target pattern. Thus, such collaborative redundant subsystems can undergo shifts of competency (from one to the other), allowing the former to deal with novel problems if necessary.

[0057] Another advantage of the disclosed systems and methods arises from the concept that biological evolution contains a degree of intelligence and is not merely a trial-and-error process. The system may be used as a way to model agential (intelligent) systems such as situations in which the materials itself is intelligent. This has application is situations in which there is a population of solutions and a need to efficiently arrive at a final solution without going through all possible permutations (e.g., all possible networks); populations of solutions evolve / improve solutions.

[0058] This system can be used to study embryos (and detect / address birth defects), metamorphosis (to understand maturation and shape change in insect and amphibian populations in ecological contexts), organ regeneration (used for developing applications in regenerative medicine), aging (to develop interventions for longevity), and cancer (to develop detection and normalization strategies in oncology).

[0059] In both biological and engineering applications, this method enables better prediction of self-assembly of form and function, and better methods to design complex multi- scaled functional systems (biobots, swarm robots, healthy normal bodies, robotics, and various kinds of hybrids). 12 QB\166118.01533\95647317.2T002814 166118.01533

[0060] In addition, NCAs can be used to model gene regulatory networks. The NCA’s cell states may correspond to the physiological states of biological cells, while the internal RGRN state of NCAs correspond to transcriptomic expressions of a biological cell. This allows investigation into causal pathways from the bottom up and top down, e.g., how to interact / control the GRN level based on high-level interventions at the tissue level. Moreover, in this scenario, the NCA’s neural network would represent a functional “generative genome.”

[0061] In some embodiments, NCAs may be evolved into small logical circuits (processing information from input cells across the tissue to output cells), and then using such NCA circuits to construct a self-assembling computer. This could be a model for the Physarum computer, exploiting exploratory behavior of the Physarum slime-mold for computation, among many other things.

[0062] In some embodiments, the disclosed systems and methods can be used to study and implement using a chimeric NCA (e.g., using cells of different NCAs on the same grid, and studying their self-assembling scenarios). This allows studying the emergence of symbiotic of adversarial strategies in multiscale systems.

[0063] The disclosed systems and methods can be used to study long-range connections between cells. This can enable a system to transition from purely localized to hybrid long-range communication akin to the central nervous system in human bodies.

[0064] Examples

[0065] The following examples are meant to be illustrative and should not be seen as limiting.

[0066] Example 1: Evolutionary Implications of Multi-Scale Intelligence

[0067] Overview

[0068] In recent years, the scientific community has increasingly recognized the complex multi-scale competency architecture (MCA) of biology, comprising nested layers of active homeostatic agents, each forming the self-orchestrated substrate for the layer above, and, in turn, relying on the structural and functional plasticity of the layer(s) below. The question of how natural selection could give rise to this MCA has been the focus of intense research. Here, we instead investigate the effects of such decision-making competencies of an MCA’s agential 13 QB\166118.01533\95647317.2T002814 166118.01533 components on the process of evolution itself, using in silico neuroevolution experiments of simulated, minimal developmental biology. We specifically model the process of morphogenesis with neural cellular automata (NCAs) and utilize an evolutionary algorithm to optimize the corresponding model parameters with the objective of collectively self-assembling a two- dimensional spatial target pattern (reliable morphogenesis). Furthermore, we systematically vary the accuracy with which an NCA’s uni-cellular agents can regulate their cell states (simulating stochastic processes and noise during development). This allowed us to continuously scale the agents’ competency levels from a direct encoding scheme (no competency) to an MCA (with perfect reliability in cell decision executions). We demonstrate that an evolutionary process proceeds much more rapidly when evolving the functional parameters of an MCA compared to evolving the target pattern directly. More- over, the evolved MCAs generalize well toward system parameter changes and even modified objective functions of the evolutionary process. Thus, the adaptive problem-solving competencies of the agential parts in our NCA-based in- silico morphogenesis model strongly affect the evolutionary process, suggesting significant functional implications of the near-ubiquitous competency seen in living matter.

[0069] Introduction

[0070] Biological systems are organized in an exquisite architecture of layers, including molecular networks, organelles, cells, tissues, organs, organisms, swarms, and ecosystems. It is well-recognized that life exhibits complexity at every scale. Increasingly realized however is the fact that those layers are not merely complex, but actually active “agential matter” which has agendas and competencies of its own. Elsewhere, we have discussed examples of problem- solving in unconventional spaces, including transcriptional, physiological, metabolic, and anatomical space.

[0071] Especially interesting is the ability of these ubiquitous biological agents to deal with novel situations on the fly, which is not limited to brainy animals navigating 3D space, but also occurs with respect to injury, mutations, and other kinds of external and internal perturbations. One example of such problem-solving capabilities are the regenerative properties of some species that can regrow limbs, organs, or entire parts of their bodies when amputated, and - remarkably - stop when the precisely correct target morphology is complete. This can be understood as cellular collectives navigating morphospace, until the desired target shape or goal 14 QB\166118.01533\95647317.2T002814 166118.01533 is reached again. Other examples include the ability of scrambled tadpole faces to reorganize in novel ways to result in normal frog faces, and the normal shape and size of structures in amphibia despite drastic changes in cell number and cell size, which are handled by exploiting different molecular mechanisms to reach correct target morphologies despite novel changes in internal components. Behavioral and morphological plasticity intersect, in cases such as tadpoles made with eyes on their tails, which nevertheless can see and learn in visual assays without needing rounds of evolutionary adaptation.

[0072] The ability to navigate transcriptional and anatomical spaces, using perception- action loops and homeostatic setpoints, is now being increasingly targeted by biomedical and bioengineering efforts. A fascinating body of work exists around the question of how neural and non-neural problem-solving capacities evolved, and how neuro-behavioral intelligence affects evolution. However, we and others have previously suggested that somatic competency pre-dates neural intelligence and has a bi-directional interaction with the evolutionary and developmental process. Thus, here we investigate how evolutionary processes are affected by the competency of the material. Especially important is the inclusion of the middle layer between genotype and phenotype. Mutation operates on genomes, and selection operates on phenotypic performance, but in most organisms, the connection between them is not linear or shallow; instead, developmental physiology provides a deep reservoir of dynamics that strongly alter the process. As a contribution to the study of evolvability and developmental mechanisms potentiating it, we established a virtual embryogeny system focused on anatomical morphogenesis by cells. In this minimal model of morphogenesis, we were able to study the effects of different degrees of cellular competency on the evolutionary process.

[0073] The standard understanding of evolution is schematized in FIG.1A. The genome of an organism encodes aspects of the organism’s cellular hardware, which together define phenotypic traits. Given a competitive environment, natural selection then favors organisms with advantageous traits, and thus, on average, the corresponding genes tend to get passed on to the next generations more frequently. Random mutations may occur, consequently changing traits in the offspring phenotype. This affects the offspring’s reproductive success during the selection stage and, in that way, good traits prevail, and bad ones perish over time. 15 QB\166118.01533\95647317.2T002814 166118.01533

[0074] Important open questions concern ways in which the properties of development – the layer between the mutated genotype and the selected phenotype – are evolved and in turn affect the evolutionary process. Specifically, significant work has been done at the interface of evolution and learning. Significant progress has been made on the question of how evolution produces agents with behavioral competency in diverse problem spaces. We have focused on navigating anatomical morphospace. More specifically, we here investigate in silico the evolutionary implications of the self-orchestrated process of morphogenesis, where local actions of single cells need to be aligned with a global policy of a multi-cellular collective to guide the formation of a large-scale tissue, in turn affecting the underlying evolutionary process. Work on developmental plasticity, chimeras, synthetic biobots, and the ability to overcome novel stressors has highlighted ways in which evolution seems to give rise to problem-solving machines, not fixed solutions to specific environments.

[0075] Thus, the problem-solving capacities of development, regeneration, and remodeling ensure that in many (perhaps most) kinds of organisms, the mapping from genotype to phenotype is not merely complex and indirect (see FIG. 1B), but actually enables evolution to search the space of behavior-shaping signals, not microstates, and exploit modularity and triggers of complex downstream responses (see FIGS. 1C-1D). We have previously argued that both evolution and human bioengineers face a range of unique problems and opportunities when dealing with the agential material of life – not passive or even just active matter, but a substrate that has problem-solving competencies and agendas at many scales. What selection sees is not the actual quality of the genome, but the quality of the form and function of the flexible physiological “software” that runs on the genomically-specified molecular hardware, as schematically illustrated in FIG.1E. This in turn suggests that the actual progress of evolution should be significantly impacted by the degree and kind of competency in the developmental architecture. Prior work has suggested a powerful feedback loop between the evolution of morphogenetic problem-solving and the effects of these competencies on the ability of evolutionary search to produce adaptive complexity. Here, we construct and analyze a new model of evolving morphogenesis, to study how different competency architectures within and among cells impact evolutionary metrics such as rate, robustness to noise, and transferability to new environmental challenges. 16 QB\166118.01533\95647317.2T002814 166118.01533

[0076] To quantitatively study the effects different levels of competency of the decision- making centers in a multi-scale competency architecture have on the process of evolution, we here rely on tools from the research field of Artificial Life which furthers computational and cybernetic models that mimic life-like behavior based on ideas taken from biology; a simple example is cellular automata (CAs). In such CAs, the (numerical) states of localized cells, organized on a discrete, spatial grid, change in time via local update rules. Although typically rather simple “hardcoded” update rules are employed, CAs often display complex dynamics but are not known to exhibit homeostatic (closed-loop) activity. An extension of CAs, termed neural cellular automata (NCAs), utilize artificial neural networks (ANNs) as more flexible trainable update rules, aiming to model the internal decision-making machinery of biological cells. Employing machine learning methods, such NCAs have been trained to perform self- orchestrated pattern-formation (notably, of images from a single “seed” cell) and even the co- evolution of a rigid robot’s morphology and its controller has been demonstrated with such NCAs.

[0077] NCAs exhibit a striking resemblance with the genome-based multi-scale competency architecture of biological life, as illustrated in FIGS. 1C-1E. An organism’s entire building plan is encoded in its genome (corresponding to the NCA’s parameters), while its cells collectively run the self-orchestrated developmental program of morphogenesis (realized by the NCA’s layout and ANN architecture) via perception-action cycles at the uni-cellular level (cell state updates in the NCA, FIG.1D). Starting from an initial cell state configuration of the NCA, the details of a virtual organism are then, step-by-step, “refined” in a collective self-organizing growth phase on the cellular level, and maintained against cell state errors later on in the virtual organism’s lifetime. Thus, a single (trainable) NCA guides the growth and integrity of a virtual organism’s tissue via intracellular information processing and intercellular communication, imitating in silico the multi-scale competency-based process of morphogenesis and morphostasis.

[0078] Here, we deploy a swarm of virtual uni-cellular agents on the spatial grid of an NCA. As illustrated in FIG. 1D, each uni-cellular agent’s internal decision-making machinery is modeled by an ANN that allows each agent to independently perceive the cell states of its adjacent neighbors on the grid and propose cell state update actions to regulate its own cell state 17 QB\166118.01533\95647317.2T002814 166118.01533 over time. The collective of cells thereby forms a spatial pattern or tissue of cell states on the NCA via local communication rules.

[0079] We utilize evolutionary algorithms (EAs) as simulated evolutionary process to optimize the parameters of such NCAs, so the uni-cellular agents evolve to collectively self- assemble a predefined target pattern of cell states in a fixed number of developmental steps; see FIG. 1E for a flow-chart of the evolutionary process. We explicitly separate the NCA parameters into a structural and a functional part. The structural parameters describe the initial cell state, and the functional parameters the weights and biases of the ANN of each agent, as illustrated by the “Genome” in FIG. 1E. Both the structural and functional part of the genome are compiled into a swarm of uni-cellular phenotypes on the grid of the NCA. Thus, starting from an initial cell state configuration, given by the structural part of the genome, the NCA’s uni-cellular agents run the developmental program of morphogenesis via successive perception-action cycles (see FIG. 1D) to self-assemble in successive developmental steps a system-level phenotype, e.g., a two- dimensional pattern of cell states on the NCA. The deviation of these final cell state configurations from a desired target pattern (e.g., here, a Czech flag or smiley-face pattern reminiscent to that of the amphibian craniofacial prepattern) defines the phenotypic fitness score of a particular NCA realization. Based on an entire population of NCAs, and on the corresponding fitness scores, the EA successively samples the genomes of the next generation of NCAs which, over time, evolve to reliably self-assemble the target pattern.

[0080] The conceptually simple process of cell state updates of NCAs and the ANN- based modelling of the uni-cellular decision-making allow us to interfere with (I) the reliability of the cell state update executions, and (II) with the computational capacity of the ANNs that guide each cell’s decision-making. To vary the former (I), we introduce a decision-making probability, PD, that specifies the probability at which a proposed update of each individual cell is executed in the environment (or omitted otherwise). Thus, by tuning the decision- making probability from PD= 0 to PD= 1, we can continuously vary the behavior of the NCA from a direct-encoding scheme without competency to a multi-scale competency architecture with perfect reliability in cell state update executions.

[0081] To systematically vary the computational capacity of the involved ANNs (II), we introduce independent copies of a particular sub-module of the uni-cellular agents’ ANNs, i.e., of 18 QB\166118.01533\95647317.2T002814 166118.01533 the policy module illustrated in FIG.1D. This increases the number of trainable parameters of the ANNs which are responsible for performing the same operation, namely interpreting the cell’s local environment and proposing a cell state update action. Thus, by taking the average output of all redundant policy-modules of a single agent, a cell’s decision-making can be biased by the several redundant paths through which signals are transmitted in the ANN, inspired by error-correcting codes. We explicitly define a redundancy number, R, that specifies how many redundant copies of the policy module are used in the ANNs of an NCA’s cells.

[0082] The decision-making probability (I) and the redundancy number (II) represent two levers of competency in our system (schematically illustrated by the red know in FIGS. 1A-1E), which we can scale continuously (I) or discretely (II) to systematically tune the behavior of an NCA. Throughout the manuscript, we refer to these two parameters as “competency levels”, but we would like to stress that many more options would have been possible to vary the competency in our system. For instance, the particular ANN architecture can have large effects on the competency of the uni-cellular agents; a systematic investigation thereof is out of the scope of this work. Here, we utilize two particular ANN architectures, one based on a feed forward (FF) and one based on a recurrent ANN architecture that is inspired by gene regulatory networks (GRNs), which we thus term recurrent gene regulatory network (RGRN), see Detailed Methods, Artificial Neural Networks for details.

[0083] To study the effects of different competency levels of the decision-making centers in a multi-scale competency architecture on the underlying evolutionary process of a morphogenesis task, we systematically vary in large-scale simulations the decision-making probability (I) and the redundancy number (II) of NCAs with FF and RGRN ANN architectures. Furthermore, we expose the corresponding NCAs to different noise conditions during cell state updates (III) and perform several statistically independent evolutionary searches at each parameter combination (I-III) to investigate the performance of the evolutionary process of finding solutions to such noisy pattern formation tasks.

[0084] Methods

[0085] Neural Cellular Automaton: A Multi-Agent Model for Morphogenesis

[0086] Cellular Automata (CAs) have been introduced by von Neumann to study self- replicating machines and are simple models for Artificial Life. In CAs, a discrete spatial grid of 19 QB\166118.01533\95647317.2T002814 166118.01533 cells is maintained over time, each cell i being attributed a binary, integer, real, or even vector- valued state, , at each step in time The cell states evolve over time via local updated rules, , as a function of its own, , and the numerical states, of its on the grid, that we collect in the matrix. Although typically rather simple “hardcoded” (e.g., predefined) CAs often display complex dynamics and can even be utilizedexample, Conway’s Game of Life or Wolfram’s rule).

[0087] Neural cellular automata (NCAs) extend CAs by replacing the local update rule with more flexible artificial neural networks (ANNs), , where θ denotes the set of trainable parameters of the ANN (see Detailed Methods, Artificial Neural Networks for details). Employing Machine Learning, such NCAs have been trained to perform self-orchestrated pattern-formation (notably, of RGB-images from a single “seed” pixel) and even the co- evolution of a rigid robot’s morphology and its controller has been demonstrated recently with NCAs in silico. Such self-orchestrated pattern-formation is reminiscent of the self-regulated development of a biological organism, from a single fertilized egg cell to a complex anatomical form. Thus, NCAs have been proposed as toy models for morphogenesis.

[0088] An NCA basically represents a grid of cells that are equipped with identical ANNs, each perceiving the numerical cell states of its host’s local environment, , and proposing actions, , to regulate its own cell state:

[0089] , Eq. 1

[0090] and, in turn, the cell states of its neighbors, where we also account for potential noise in the environment during the process of morphogenesis. Thus, each cellular agent can only the numerical states of its direct neighbors, , at an instant of time, , and, in turn, communicate with these neighbors via cell state updates, following a policy that is approximated by an ANN with parameters θ. Through the lens of Reinforcement Learning, an NCA can thus be understood as a trainable, locally- communicating multi-agent system that can be utilized such that the collective of cells achieves a target system-level outcome (see Detailed Methods, Reinforcement Learning Agent’s Perception- Action Cycle). 20 QB\166118.01533\95647317.2T002814 166118.01533

[0091] In contrast to previous contributions of in silico morphogenesis experiments in NCAs, we here do not use standard convolutional filters in our ANN architectures but utilize permutation invariant ANNs with respect to a cell’s neighbors, (see FIG. 1C for an illustration). Inspired by Y. Tang and D. Ha, Advances in Neural Information ProcessingSystems, 2021, this is achieved by partitioning a cell’s ANN a sensory part, , preprocessing the state of each neighboring cell separately (i.e., its own,into a respective sensor embedding,wise sensor embeddings are (ii) dimension) into acontext vector s, is then used as the input of (iii) a recurrent feedback connections, that eventuallyoutputs the cell’s action, ; for details we refer to Detailed Methods, Artificial Neural

[0092] Due to the mean-aggregation of a cell’s sensory embedding, each cell completely loses its ability to spatially distinguish between neighboring (and even its own) state inputs and thus fully integrates into the tissue locally. We would like to stress the close relation of our approach to the concept of breaking down the computational boundaries of a cell’s “Self” via forgetting and to the scaling of goals from a single agent’s objective to a system-level objective.

[0093] To model the developmental process of morphogenesis, we here employ NCAs on a two-dimensional square grid with the objective that all cells of the grid assume their correct, predefined target cell type, , after a fixed number of tDdevelopmental time steps, starting from an initial cell state configuration . We attribute a number of NGelements of the NC-dimensional cell state of an NCA as indicators for one of 1, . . . , NGdiscrete cell types, such that ; the remaining elements of the cell state represent hidden states of a cell that can be utilized by the NCA for intercellular communication. We explicitly define each cell’s type, , as the argument (i.e., the index) of the maximum element of the NG- dimensional indicator vector: 21 QB\166118.01533\95647317.2T002814 166118.01533

[0094] Eq. 2 pattern (realized by a set oftarget cell types for the entire grid) thus boils down to finding a suitable set of NCA parameters (e.g., “Genotype” in FIG. 1E) that minimizes the deviation of each cell i’s type from after developmental time steps, e.g., after the developmental stage of the virtual organism (see “System-level Phenotype” in FIGS. 1A-1E from left to right, and details below). Here, we are interested in the evolutionary implications of biologically inspired multi- scale competency architectures, the latter being modeled by our morphogenetic NCA implementation. We thus introduce in Methods, Neuroevolution of NCAs, and utilize in Results, evolutionary algorithms to evolve suitable sets of NCA parameters that maximize a fitness score based on comparing the “final” cell types of the NCA, , with the predefined target cell types .

[0096] Neuroevolution of NCAs: an Evolutionary Algorithm Approach to Morphogenesis

[0097] Evolutionary algorithms (EAs) are heuristic optimization algorithms that maintain and optimize a set, i.e., a population, , of parameters , also termed individuals, over successive generations to maximize an objective function, or a fitness score . Inspired by the ideas of natural selection and the DNA-based reproductionbiological life, EAs (i) predominantly select high-fitness individuals of a given population for reproduction, and utilize (ii) crossover and (iii) mutation operations to generate new offsprings by (ii) merging the genomic material of two high-quality individuals from the current population, , and (iii) occasionally mutating the offspring genomes by adding typically Gaussian) noise to the parameters; the symbol indicates a genuine merging operation of two genomes, which may depend on the particular EA implementation. In that way, a population of individuals is guided towards high fitness regions in the parameter space , typically over many generations of successive selection and reproduction cycles (i)- (iii). 22 QB\166118.01533\95647317.2T002814 166118.01533

[0098] In contrast to biological life, many use-cases of EAs do not require a distinction between individuals in the parameter space, i.e., genotypes , and the corresponding organisms in their natural environment, i.e., phenotypes, : while the genetic crossover and mutation operations of biological reproduction rely on biomolecular mechanisms on the level of RNA and DNA, (e.g., are performed in the genotype space), selection typically happens at the much more abstract level of an organism’s natural environment (e.g., in the phenotype space). Carrying this through computationally can be resource-demanding, depending on the complexity of a simulated environment. Nevertheless, to address the asymmetry between genotypes and phenotypes in multi-scale competency architectures, it is essential to evaluate the EA’s fitness score in the phenotype space instead of the genotype space, .

[0099] We explicitly separate the genotype and phenotype representation of individuals by introducing a biologically inspired developmental layer 1 in between genotypes and phenotypes, , as illustrated in FIGS. 1A-1E. More precisely, we follow Methods, NCA andprocess of morphogenesis in silico by utilizing NCAs. We treat an NCA j’s parameters, such as the set of initial cell states and the corresponding ANN parameters , as theorganism’s genome,

[0100] , Eq. 3(S) and a functional (F) part, as indicated by the superscripts. We then perform a fixed number of tD developmental steps employing Eq.1, and interpret the corresponding set of “final” cell types of the entire NCA, see Eq. 2, as the mature phenotype,

[0102] , Eq. 4

[0103] Representing a two-dimensional tissue of cells.

[0104] In an effort to evolve the parameters, xjof an NCA j to achieve morphogenesis of a two-dimensional spatial pattern of cell types, pj, that resembles a pattern of predefined target cell types , of a total of Njcells on an square grid (see Methods, NCA), we define the phenotype-based fitness score r(pj) as 23 QB\166118.01533\95647317.2T002814 166118.01533

[0105] , Eq.5after tDdevelopmental steps, (ii) is the number of time steps at which the entire target cell type pattern is correctly assumed, i.e., whenever for all I, and (iii) is the number of successive time steps, and , where all cell types stagnate, i.e., where for all i. With Eq. 5, we thus reward the entire NCA j by counting all correctly assumed cell developmental steps (while discounting all incorrect cell types), we reward maintaining the target pattern over time with a factor of and discount a stagnation of a suboptimal pattern over time by a factor of . We consider the problem solved if a final fitness score of is reached. Notably, there is no explicit fitness or reward feedback on the level of the uni-cellular agents in our system; the fitness score is solely used as the selection criterion for sampling the next evolutionary generations, so the cellular collective needs to evolve an intrinsic signaling mechanism to successfully perform the requested morphogenesis task.

[0107] The here proposed setting of genotypes, xj, corresponding phenotypes, pj, and associated fitness scores, , given by Eqs. 3-5, respectively, can be used in combination with any black-boxgenetic algorithm. We rely on the well established Covariance Matrix Adaptation of Evolutionary Strategy (CMA-ES) to simultaneously evolve the set of initial cell state configurations (i.e., structural genes, ), and the set of corresponding ANN parameters of an NCA (i.e., functional genes ) with the objective of a purely self- orchestrated formation of a 2D spatial tissue as illustrated by FIGS.1A-1E and described by Eq. 1.

[0108] Results

[0109] The System: An Agential Substrate Evolves to Self-Assemble the Czech Flag

[0110] Evolution works with an active rather than a passive substrate, i.e., with biological cells with agendas of their own. Thus, at every stage of development during morphogenesis collective decisions are made at vastly different length- and time scales within an organism, guiding the formation of the mature phenotype. We aim to model exactly this process via Neural 24 QB\166118.01533\95647317.2T002814 166118.01533 Cellular Automata (NCAs) and employ evolutionary algorithms (EAs) to evolve the parameters of such NCAs, so the latter perform well on a target morphogenesis task (see Methods). Without loss of generality, we consider a Nxx Ny= 8x8 Czech flag pattern (as a more complex version of the classic French Flag problem of morphogenesis) as the target pattern for our in silico morphogenesis experiments, with a fixed number of Nj= 64 cells in total, NG= 3 distinct cell types (colored blue, white, and red, respectively) and NH= 1 hidden state, which renders the dimension of the NCA’s cell state as NC= 4. We use a square grid of cells with N = 8 neighbors per cell and with fixed boundary conditions (see Detailed Methods, Fixed Boundary Condition Handling of the Neural Cellular Automaton).

[0111] Starting from a genotype xjdefined in Eq. 3, we perform a number of tD= 25 developmental steps per morphogenesis experiment to “grow” a phenotype pj, described by Eq. 4, based on which the fitness score r(pj) is evaluated following Eq. 5 (see FIGS. 1A-1E for an illustration of this process). During this entire process, we limit the magnitude of the numerical cell state values ci(tk) at all time steps tkto the interval lc= [ 3, 3], and, analogously, limit the magnitude of the proposed actions ai(t) of each uni-cellular agent to the interval la= [ 1, 1]. This is achieved by clipping the numerical values of ci(tk+1) after a cell state update described by Eq.1, and the ANN outputs ai(t) to the respective limits lcand la. The noise level ξcdefined in Eq.1 is counted in units of the action limits, max(la), and is thus sampled from a Gaussian distribution with zero mean and standard deviation ξcindependently for each of the NC= 4 cell state elements, thus affecting the cell state updates during development; the actual numerical values for the hyperparameters above turned out to be well suited for the problem at hand, especially to reasonably compare and discuss simulation results for the means of this contribution, but are not crucial for the more general aspects on the evolutionary implications of multi-scale intelligence discussed here.

[0112] To study the effects of different types of decision-making machinery within a cell, we utilize two different architectures for the NCA’s artificial neural networks (ANNs), a Feed Forward (FF) and a recurrent ANN inspired by gene regulatory networks (RGRNs). Notably, the RGRN-agent architecture augments cells with an internal memory that is independent of their states in the NCA and can thus not be accessed by the cells’ neighbors. To balance the length of the structural genome and functional genome defined in Eq. 3,25 QB\166118.01533\95647317.2T002814 166118.01533 the two ANN architectures, FF and RGRN, are chosen such that the number of parameters and is roughly the same as the number of initial cell states . Thus, the ANNs utilized here (and detailed in Table 1 of Detailed Methods, Artificial Neural Networks) are tiny compared to Mordvintsev, A., et al. Distill 5 (2020).

[0113] For each experiment of evolving an NCA’s parameters, i.e., for each independent run of the EA, we typically utilize a population, , of individuals and a maximum number of generations. As the EAs ultimate fitness criterion, we consider the average of statistically independent fitness scores of simulations starting from an individual j’s xjandresulting at a phenotype pjafter tDdevelopmental steps; program described via Eq. 1 is imperfect due to the developmental noise applied to the cell state updates and can thus lead to different, noise-induced phenotypic realizations. Typical values used here for the corresponding reward factors defined in Eq. 5 are and . We consider the problem solved if a fitness of is reached, but since we reward individuals to maintain the target (via ). The maximum possiblefitness score after developmental time steps is in this example. Further details about the hyper-parameters of the EAresources can be found in Detailed Methods, Covariance Matrix Adaptation Evolution Strategy (CMA-ES).

[0114] Direct vs. Multi-scale Encoding: Cellular Competencies affect System Level Evolvability

[0115] We aim in this contribution to investigate the evolutionary implications of biologically inspired multi-scale competency architectures. Thus, we compare two qualitatively different evolutionary processes both with the objective of morphogenetic pattern formation but whose genomes either (i) directly encode phenotypic features of a two-dimensional target pattern (see FIG.1A) or (ii) encode cellular competencies of a multi-scale architecture that gives rise to the same phenotypic features (see FIG. 1C). Notably, different definitions of direct and indirect encodings in multi-agent systems have been used in the literature. Here, we specifically distinguish between structural parameters, , in the search space that directly 26 QB\166118.01533\95647317.2T002814 166118.01533 encode features of the phenotype, i.e., specific initial cell types , and functional parameters , that indirectly, or rather functionally, encode the target pattern by parametrizing the intercellular communication and intracellular information processing competencies of the NCA that facilitate the self-orchestrated pattern formation process.

[0116] If no ANN at all were present in our model, i.e., , and in the absence of noise, ξc= 0, we would re-establish a direct mapping between genotype and phenotype, as , and thus a direct encoding of the target cell type pattern can be achieved, . However, by default, we allow each cell to successively regulate its own cell state a target homeostatic value via an iterative perception-action cycle defined by Eq.1 and, moreover, to communicate in that way with neighboring cells. More specifically, each cell updates its cell state solely based on its own and the states of its adjacent neighbors which, in turn, update their states based on their respective local environment. We explicitly avoid direct environmental feedback to the cells’ perception (such as an individual or collective reward signal) but fully restrict the NCA to intercellular communication (via cell state updates) and intracellular information processing. These uni-cellular agents thus exhibit a certain level of problem-solving competencies that can be utilized for the challenge at hand, in our case for a collective system-level objective of forming a specific two-dimensional target pattern.

[0117] With the explicit partitioning of the genome into a structural part, i.e., , and a functional part, i.e., , we can study the effect of direct vs. multi-scale, or competency-driven encoding of phenotypic traits in the process of evolution, and, moreover, quantitatively tackle the question whether competent parts affect the process of evolution and evolvability. In any case, the initial cell state pattern is given by the structural part of the genome. Thus, in the absence of noise and without any active functional part in the genome, the set of initial cell states directly represents the final pattern, while otherwise cell states can either be modified passively by noise in the system or actively through actions by the cells during the developmental stage. Thus, we employ CMA-ES to either evolve the (i) structural, or both (ii) the structural and functional part of the genome of an NCA simultaneously with the shared objective of self-assembling an Czech flag pattern in developmental time steps in the presence of noise, ξc= 0.25 (see Methods for details). More explicitly, in case (i) we restrict the evolutionaryto search 27 QB\166118.01533\95647317.2T002814 166118.01533 only the space of direct phenotypic encodings, while in case (ii) we allow evolution to evolve both a structural and a functional part of the genome, thus giving it the opportunity to prioritize one over the other (the results of this experiment are presented in FIG. 2).

[0118] We can see in FIG. 2 that both the evolution of the (i) direct and (ii) multi-scale encoding schemes of the target pattern can be achieved with the presented framework and a fitness threshold of is reached after ≈300-600 generations, thus solving the problem. However, depending on the encoding scheme (i) or (ii), we can identify clear qualitative differences in the strategy and the “efficiency” of the evolutionary process i.e., how many generations it takes to reach a certain fitness threshold and eventually converge (see top and bottom panel of FIG. 2, respectively): The respective fitness score of the direct case (i) grows steadily and almost monotonically over successive generations until the threshold of is reached in 428 generations, and the EA converges after 679 generations (although at a lower maximum fitness of after 942 generations. In contrast, the evolutionary process of the multi-scale case (ii) undergoes significant leaps as reflected by the corresponding fitness score which can increase rapidly if a suitable innovation, e.g., a favorable crossover or mutation event in the functional parameters θj, occurs; the initial standard deviation of the fitness of the entire population is significantly larger compared to the direct case (i), yet the threshold fitness of is reached in 428 generations, and the EA converges after 679 generations (although at a lower maximum fitness of ).

[0119] The results provided here are based on selected evolutionary optimization runs, that are representative of related experiments with similar parameterizations. However, one should keep in mind that such results are always susceptible to chance in initial conditions or mutations in the EA, but also to the developmental noise; moreover, hyperparameters of the evolutionary search or even the specific ANN architectures can influence the evolvability of such NCA systems. Thus, we present in Results, Evolution Exploits Competency over Direct Encoding below a more statistically significant analysis of the evolutionary implications of direct and multi-scale encodings under various conditions of the cellular agents’ competency levels and the developmental noise. 28 QB\166118.01533\95647317.2T002814 166118.01533

[0120] Our separation of the genotype xjinto a structural , and into a functional part , moreover allows us to extract the structural (or genotypic) fitness along an entire evolutionary history. We define the structural fitness as the fitness score of a phenotype with evolved structural genes , but with disabled agency . Notably, in the direct case (i) we have , which is illustrated in FIG. 1A and reflected in the top panel of FIG. 2; the structural fitness of the multi-scale case (ii) is explicitly visualized in the bottom panel of FIG. 2. In the latter case, the structural fitness remains essentially detached from the phenotypic fitness, during the entire evolutionary history (which also explains the convergence to a suboptimal maximal fitness level of in this particular NCA solution, as the final Czech flag pattern first needs from thecorresponding imperfect initial cell configurations, . This all suggests that, in contrast to (i), the EA in (ii) can make the most use of exploring the functional part of the genome, i.e., the space of behavior-shaping signaling and information processing, and, in turn, that the mere presence of competent parts drastically changes the search space accessible to evolution; to illustrate this explicitly, we present in Detailed Methods, Direct vs. Multi-scale Encoding: Morphogenetic Development Over Evolutionary Time-Scales and illustration of the evolution of the morphogenesis space.

[0121] Interestingly, we still observe a slow but steady increase of the structural fitness in the long term in case (ii) owed to the small additional reward signal, rT, reinforcing the cellular agents to maintain the target pattern over time. This can most efficiently be achieved if the agent starts from a perfect set of initial cell types, representing a particular sub-space in the parameter space that might not necessarily be easily accessible to the EA at all stages during the evolutionary search. However, we would like to stress that such a slow transfer of problem- specific competencies from an agential, highly adaptive functional part, , to a rather rigid structural part, , of the genome could be a manifestation of theeffect. Through a computational lens, such a competency transfer would also allow, as soon as the structural part of the genome is reliable enough, to re-purpose the system’s competency to adapt to other, independent tasks, and thus may facilitate the in biology ubiquitous effect of polycomputing in related systems. 29 QB\166118.01533\95647317.2T002814 166118.01533

[0122] While remaining neutral with respect to the system level fitness score, this competency transfer seems to affect the entire population presented in FIG. 2 as reflected by the successively increasing population-averaged structural fitness score. Notably, and as detailed in Detailed Methods, Direct vs. Multi-scale Encoding: Neutral Transfer of Hierarchical Competencies Affects Uni-Cellular Robustness, we identified an associated decrease in the robustness against increasingly noisy cell state updates of the corresponding solutions with larger structural fitness. This suggests a reduction in uni-cellular competencies and might relate to the “paradox of robustness.” Through a computational lens, such a competency transfer would also allow, as soon as the structural part of the genome is reliable enough, to repurpose the system’s competency to adapt to other independent tasks, and thus may facilitate the, in biology, ubiquitous effects of adaptability and polycomputing in related systems.

[0123] This all illustrates that an agential material, or more precisely, a substrate composed of competent parts, can have significant effects on the process of evolution and evolvability, especially for morphogenesis tasks. We thus conclude that, if competent parts are available, evolution prefers exploiting competency over direct encoding - if the environment requires competency at all (see below). This leads to the conclusion that “competency at the lowest level greatly affects evolution and evolvability at the system level.”

[0124] Evolution Exploits Competency Over Direct Encoding, If Necessary

[0125] Here, we investigate the effects of varying different levels of competency at the cellular level of a multi-scale competency architecture on the evolutionary process of morphogenesis. More specifically, we introduce the decision-making probability (I), PD, which constrains the ability of each cell individually to perform cell state updates in the environment: PDdefines the probability at which a proposed cell state update of each individual cell in the NCA is executed (or otherwise omitted). Thus, varying the decision-making probability from PD= 0 to PD= 1 smoothly transitions the system’s behavior from a direct encoding scheme without competency to an increasingly reliable multi-scale competency architecture (see FIG. 2).

[0126] Another, somewhat hidden level of competency we already discussed in Methods is each cell’s ANN architecture: An RGRN-agent with internal memory can acquire and execute tasks differently than a simpler FF-agent without any feedback connections except for its cell 30 QB\166118.01533\95647317.2T002814 166118.01533 state . Comparing the evolutionary implications of such functionally different ANN architectures is, however, not trivial, and is thus kept to a minimum here.

[0127] However, we parameterize both FF and RGRN agents such that their controller part of the ANNs (see FIG.1C, Methods, and Detailed Methods, Artificial Neural Networks) are (II) stacks of R redundant copies of the same controller ANN, each copy with its own set of parameters, which take the same pre-processed aggregated sensor embedding as input, and whose individual outputs are averaged into a single action-output of a cell. Inspired by redundancy in error-correcting codes, we thus allow cells with higher values of this redundancy numbers, R, e.g., with many alternative routes through the controller part of the ANN, to integrate environmental signals more generally compared to R = 1, thus affecting the cells competency.

[0128] While scaling from PD= 0 to PD= 1 smoothly increases a cell’s competency to reliably regulate its cell state, increasing R enhances the computational capacities of the uni- cellular agents. Henceforward, we interpret (I) PDand (II) R as two competency levels in our system which we can vary (I) continuously and (II) discretely.

[0129] Analogous to Results, The System and Direct vs. Multiscale Encoding, we thus utilize CMA-ES to evolve the genotypic parameters of an NCA to self-assemble the 8 ×8 Czech flag pattern under different conditions (I-II), and expose the cells to different noise levels (III), ξc, during cell state updates defined in Eq. 1.

[0130] In FIGS. 3A-3B, we present the corresponding fitness scores of a maximum of 2000 generations of CMA-ES for different noise levels , averaged over different values of the decision-making probability for both FF- agents and RGRN-agents. Moreover, for each realization of and PDwe utilize experiments with different redundancy numbers and employ 15 statistically independent EA runs for each parameter combination ξc, PDand R, and thus arrive at 75 statistically (and functionally, with respect to an agent’s ANN architecture) independent fitness trajectories per (PD, ξc)-combination. See Results and Detailed Methods, Covariance Matrix Adaptation Evolution Strategy for more details on the EA parameters. In FIG.3C, we present the average number of generations it takes to solve the problem (to reach a fitness threshold of Fj= 64) for each combination of PDand , aggregated over the agents’ ANN architectures, FF or RGRN, 31 QB\166118.01533\95647317.2T002814 166118.01533 and the respective redundancy numbers R for 15 statistically independent EA-runs each; in FIG. 3D, we present the data from FIG. 3C but separately for both ANN architectures.

[0131] We observe in FIGS. 3A-3D that, depending on these two parameters, PDand ξc, for no or very low noise levels, , the evolutionary search is most efficient, e.g., finds the solution in the least number of generations on average, for low values of the competency level PD≈ 0. Thus, in these situations, direct encoding (achieved via PD= 0) seems to be preferable to competency-driven encodings with PD> 0 (as indicated by the bottom red arrow in FIG. 3C); this is partly owed to the specific definition of the cell types given by Eq. 2, making a noise-less search very simple for the EA. However, for more realistic, noise conditions > 0, the situation changes drastically. With increasing noise level, the evolutionary efficiency of NCA’s with higher competency levels is significantly larger compared to low competency levels and, especially, to the direct encoding scheme (as indicated by the green arrows in FIG. 3C); for noise levels of = 0.375 and 0.5, the EA does not even find solutions for the direct encoding case with PD = 0 in 2000 generations as cell state updates become increasingly necessary to counteract the noise in the system. There is a clear trend of increasing the evolutionary efficiency in our in silico morphogenesis experiments by increasing the competency level for increasingly difficult environments with high noise levels.

[0132] Thus, we conclude that scaling competency has a strong effect on the process of evolution, and in realistic situations (with moderate to high noise) competency may greatly improve the evolutionary efficiency and evolvability of collective self-regulative systems.

[0133] It might be noteworthy, that for evolving the Czech flag pattern, essentially no qualitative difference in the evolutionary efficiency between FF-agents and RGRN-agents with the given number of parameters was observed. Also, the evolutionary implications of utilizing a number of R > 1 redundant copies within the controller ANNs of an NCA’s cells is much less pronounced, compared to the results depicted in FIGS. 3A-3D, as can be seen in FIG. 11A-11D of Detailed Methods. However, for more advanced problems such as assembling a smiley-face pattern (see Detailed Methods), RGRN-agents seems to outperform a simpler FF-agent significantly in terms of evolutionary efficiency. Moreover, a larger redundancy number of is required by the evolutionary process to more efficiently evolve an NCA’s 32 QB\166118.01533\95647317.2T002814 166118.01533 functional parameters compared to a direct encoding scheme, hinting at a capacity bottleneck of the deployed ANNs.

[0134] There is a Trade-off Between Competency and Direct Encoding Depending on Developmental Noise

[0135] A careful analysis of the results shown in FIGS. 3A-3D reveals, that the largest competency level of PD= 1 does not result in the highest evolutionary efficiency for any presented noise level. On the contrary, populations with slightly lower competency levels of PD= 0.5 or even PD= 0.25 perform best at noise levels and 0.125, respectively (as indicated by the green and red arrow-ends in . In fact, cells with an initially randomgenome (comprising the ANN and initial cell parameters) that are forced to make “uninformed”, i.e., initially random, decisions at every time step can interfere with the performance of the EA, as even initially perfect cell state configurations will be destroyed during such a randomized developmental stage. We suspect that this leads to corresponding delays in the evolutionary search compared to situations where populations can better rely on the structural part of the genome. Indeed, populations with “overconfident” actions can be trapped in local optima for many generations at all stages of the EA, which, in our system, may only be resolved by very specific but random mutations of the functional part of the genome (as we show later through FIGS. 4A-4E in Results, Trade Off Between Competency and Direct Encoding). This is reflected in FIGS. 3A-3B by the large deviations in the average fitness trajectories for large PDvalues.

[0136] The insights from above lead to the questions, of whether there is a “natural”, or optimal competency level, with respect to the decision-making probability PD, or whether a mutable competency level can be utilized by the evolutionary process to improve the efficiency of guiding a population towards high fitness regions in the parameter space. Thus, we include the decision-making probability as an additional competency gene, , into the NCA’s genome , see Eq. 3, and we perform in silico morphogenesis evolution experiments of the Czech flag pattern with different noise levels, , analogous to Results, Evolution Exploits Competency over Direct Encoding. We analogously limit the numerical range of the competency gene to the interval [-3, 3], and extract the corresponding decision- 33 QB\166118.01533\95647317.2T002814 166118.01533 making probability via . Notably, for the experiments shown in this sub-section, we use an L2- the genotypic parametersthrough subtracting , fitness score defined in Eq. 5, with .

[0137] In we present the evolved competency level for different noiselevels after a fitness threshold of 64 and 70 is crossed, respectively, for 10 independent lineages per noise level for an RGRN-architecture. The problem is considered solved at a fitness of 64, but since we reward the NCAs to maintain the target pattern over time via r(T)in Eq. 5, a higher maximal fitness score of 70.25 can be reached after tDdevelopmental steps for sufficiently long evolution. Thus, we here relate FIG. 4A to the evolutionary stage of having achieved the process of morphogenesis, and FIG. 4B of having achieved morphostasis. For both cases, we essentially see two strategies emerging (see also FIGS. 4C-4E): (i) one, where competency is maximized very early during the evolutionary process that then remains near the maximally possible value of PD= 1, and (ii) a hybrid strategy where a significantly lower competency level is assumed that still allows to solve the problem.

[0138] Notably, strategy (i) is predominantly pursued at high noise levels where large cell state fluctuations in the environment favor informed actions by the cellular agents. In contrast, the second strategy (ii) emerges more frequently in lineages evolved at low noise levels where, especially at very low noise levels , most of the evolutionary processes result in solutions that avoid competency altogethera direct encoding scheme (PD=0) is evolved. Intermediate competency levels evolve in the corresponding intermediate noise regime. Following the trend of evolving morphogenesis (by crossing a fitness score of 64) to morphostasis (by converging to the maximal fitness value of ≈ 70), in FIGS. 4A-4B, we see that the two strategies (i) and (ii), “sharpen” during the course of the evolutionary process, such that PDpredominantly converges to the minimally or maximally possible values of 0 and 1, depending on the environmental conditions.

[0139] We also illustrate the evolved competency level of the particular lineage at all noise levels in FIGS. 4A-4B at which the respective fitness threshold is crossed in the least-, and maximum number of generations (and on average) amongst all 10 independent lineages per noise level. This clearly reveals that evolutionary processes that follow a more direct encoding strategy (ii) can evolve the problem at hand efficiently - if this is permitted by the developmental noise. 34 QB\166118.01533\95647317.2T002814 166118.01533 However, when increasing the noise level, the evolutionary process can afford to evolve - or put differently, increasingly relies on evolving - the multi-cellular intelligence of the NCA to perform morphogenesis and morphostasis, thus following a third strategy (iii) that integrates both strategies (i) and (ii) in a non-trivial way. We observe in FIG.4A that the most efficient strategy for evolving morphogenesis seems indeed to be such a hybrid approach (iii), where a minimally necessary competency level is utilized at a specific noise level such that the corresponding evolutionary process can, again, be very efficient in solving the task.

[0140] Moreover, this also holds for the stage where morphostasis is reached, (see FIG. 4B). Lineages that efficiently evolved to solve morphogenesis in our experiments also (typically) evolve to solve morphostasis efficiently. To emphasize this, we present in FIGS. 4C-4E the “temporal dynamics” of the population-wise highest fitness and the corresponding competency level per generation for all lineages at selected noise levels ; we also present for all corresponding lineages that have been evolved at selected noise levels in thegenotypic competency level against the corresponding phenotypic fitness scores rj, and we find an apparent yet non-trivial relation between these two quantities. Typically, an initial rise in fitness rjin early generations is associated with a decline in which is more pronounced at lower noise levels. For intermediate noise levels we find that often assumed a minimum (i.e., a minimally required yet finite competency level) when the evolutionary process reaches a fitness level of ≈ 64. We suspect that this allows the evolving morphogenetic process to establish good starting configurations based on changes in the structural genome, which can most efficiently be done at a minimal(ly necessary) competency level given a certain developmental noise level in the environment. However, the competency is then quickly pulled towards a maximum level of when the EA converges at a maximum fitness score of 70, at the morphostasis stage. For large noise levels, e.g., as depicted in Results, the competency level rises with the corresponding fitnessa much more monotonic way, emphasizing the necessity of the corresponding NCAs to utilize the cellular competency to solve the problem already at an early stage of the evolutionary process.

[0141] Curiously, we also see lineages that settle at the highest possible competency levels throughout their evolutionary history, even in conditions without noise, as can be seen in Results. Here, an initial “frozen accident” may cause an entire lineage to maintain high 35 QB\166118.01533\95647317.2T002814 166118.01533 competency levels due to a lack of diversity in the corresponding gene, although this is not even necessary to solve the task. However, these high competency levels early on during the evolutionary process can cause the population to stagnate at sub-optimal regions in the parameter space for many generations if the corresponding policy of the cells is sub-optimal but rigid to strategy changes via small mutations in the genome. The population seems “trapped”, until a favorable mutation or crossover event occurs in the functional part of the genome of an individual, that guides the entire population towards higher fitness scores, eventually solving the problem. We suspect that this is also the reason for the lower evolutionary efficiency of the “most competent” configurations (with PD= 1) compared to slightly less competent cases (with PD= 0.5) of the experiments depicted in FIGS. 3A-3D.

[0142] Thus we conclude, that if the evolutionary process can afford to evolve its own competency level, there seems to be a trade-off - during the entire course of the evolutionary process - between “going direct” or “going competent”, depending on the developmental noise. Moreover, randomly initialized starting conditions may favor either direct or multi-scale encoding strategies, which may not only affect the “final” competency level the evolutionary process converges to, but can also greatly influence the efficiency of the evolutionary process itself. In general, the most efficient strategy for evolving morphogenesis tasks seems to be a non- trivial tradeoff between finding a suitable initial cell state configuration that then allows the competency-based self-assembly of the target pattern to “kick in” and solve the task efficiently.

[0143] Competency Can Lead to Generalization

[0144] We are ultimately interested in the question of whether a substrate of competent parts shows abilities to generalize to environmental conditions that have never been experienced by its evolutionary predecessors, and hence would allow the evolutionary process to adapt an organism to changing environmental conditions more efficiently compared to a direct encoding scheme. Thus, we systematically vary in FIGS. 5A-5F the system parameters, specifically the noise level and the decision-making probability competency level, for selected NCA solutions of the Czech flag problem that have been trained with certain sets of the system parameters above.

[0145] For instance, we utilize NCA solutions that have been evolved to solve the Czech flag problems in tD=25 developmental steps (see above) under zero-noise conditions without and with evolvable competency. Here, we utilize such solutions for larger noise levels of 36 QB\166118.01533\95647317.2T002814 166118.01533 and for lifetimes of 100 time-steps and present the average fitness values of 100 statistically independent simulations at each particular noise level in FIGS. 5A-5B, respectively, without any further evolutionary optimization. Analogously, we expose NCA solutions that have evolved with a competency level of PD=0.5 and noise levels of =0.25 and 0.5, respectively, to vastly different noise levels of compared to the conditions during their respective evolutionary processes, and resent the results in FIGS. 5C-5D. Eventually, we again deploy the latter NCA solutions but vary the competency level . Instead, at respectively fixed noise levels of = 0.25 and 0.5, with results in FIGS. 5E-5F. Notably, we only consider the of the fitness score, i.e., the first term in Eq. 5 by setting rT=0 and rS=0.

[0146] The results in FIGS. 5A-5F demonstrate that the performance of the here evolved NCAs, optimized with evolutionary methods to assemble and maintain a target morphology over time at particular system parameters, differs greatly between NCA solutions that follow the direct- or multi-scale encoding paradigms when subjected to novel environmental conditions. The typical fitness over the lifetime of an NCA without competency that encodes the target phenotype pattern directly (see FIG. 5A) is constantly affected by random fluctuations and thus decreases in fewer time steps with increasing noise levels in a diffusive process; the duration of how long the corresponding maximum fitness score of 64 can be maintained, and the speed at which the fitness eventually decays during the lifetime of the here discussed Czech flag NCA depends on the particular noise-level and on the values of the initial cell states, which are limited numerically to the interval [ 3, 3] for each cell. In contrast, NCA solutions with larger competency levels that have been evolved at finite noise-level conditions still perform well - and can maintain the target pattern for exceptionally long times - also when changing the system parameters dramatically, (see FIGS. 5B-5F). Note the noise-level axis of ξc= 0 to 1, compared to maximum noise-levels of ξc= 0.5 during training.

[0147] Thein panel FIG. 5B are especially curious, as the corresponding NCA has been trained to evolve its decision-making probability alongside the structural and functional parts of the genome at zero noise conditions. While no competency at all would have been required to solve this task, the presented NCA solution evolved to afford a maximum competency of PD= 1 (see FIG. 4C). Strikingly, this particular NCA is capable of resisting much larger noise levels of ξc≈ 0.25 while maintaining the pattern perfectly for at least tD= 25 steps,37 QB\166118.01533\95647317.2T002814 166118.01533 and the average fitness score of 100 independent solutions does still not drop below a certain threshold of ≈ 40-50 for even higher noise levels and for 100 time steps. Notably, there appears to be a bifurcation of the long-term behavior of these NCA solutions (not shown here) where the NCA maintains the target pattern perfectly for long times, while in other independent runs, the fitness drops quickly.

[0148] We thus show that NCAs that have evolved to assembly and maintain a target pattern within a relatively short developmental stage are capable of maintaining the corresponding target pattern over much longer time scale without any further optimization, and thus show great signs of functional, morphostatic generalizability. Moreover, the here-discussed in silico morphogenesis and morphostasis model systems are capable of handling - essentially on the fly system-parameter combinations neither they, nor their evolutionary ancestors ever experienced before. Thus, we conclude that such multi-scale competency architectures, whose substrate is composed of competent rather than passive parts, can be more than capable of generalizing to changes in their environment (within reasonable boundaries) by allocating robust problem-solving competencies at many scales.

[0149] Competency Can Augment Transferability to New Problems

[0150] Deducing from the discussion in Results, Competency Can Lead to Generalization about the generalizability of multi-scale competency architectures1 towards changing environmental conditions, such systems should also exhibit increased evolvability and transferability properties to new problems: if such multi-scale competency architectures are capable of adapting their behavior towards changing environmental conditions on the fly during a single lifetime (see FIGS. 5A-5F), this has great consequences for the evolutionary process when environmental conditions change.

[0151] Thus, we utilized the NCA solution discussed in FIG.4C and FIG. 5D and performed subsequent CMA-ES on the Czech flag problem at changed environmental conditions, e.g., at higher noise levels. Between a single or at most a handful of generations are necessary for soling the task even at intermediate and high noise levels of ξc=0.25 and 0.5 (not shown here).

[0152] To emphasize the potential of transferability of multi-scale competency architectures, we here investigate the adaptation-capability of pre-evolved NCAs when their 38 QB\166118.01533\95647317.2T002814 166118.01533 objective function is suddenly changed e.g., when the environment starts selecting for different target patterns than the one they have originally been evolved for. More specifically, we utilize NCA solutions and discussed through FIGS. 3A-3D, which successfully solve the Czech- flag task, and additionally perform 1000 evolutionary cycles of CMA-ES on a related blue-, white-, red-, and Viennese-, blue / white, and blue / red- flag morphogenesis task for various noise and competency levels. We allow changes to both the structural and functional parts of the genomes of the pre-evolved NCA.

[0153] In FIG. 6, we present the corresponding number of generations it takes for 1- 60 CMA- ES runs on average to adapt a pre-evolved, i.e., “informed”, NCA solution that can solve the Czech-flag morphogenesis task to then solve the respective new morphogenesis task under different environmental conditions. We see a clear advantage in terms of evolvability and adaptability of pre-evolved individuals at high-competency levels (in contrast to individuals with lower competency levels) so that adaptation can happen in as few as ≈10 generations. While the Czech→blue-, white-, and red-flag tasks are trivial (see top panels in FIG. 6), computationally the Czech → Viennese-, blue / white, and blue / red=flag adaptation tasks (bottom panels in FIG. 6) are more complicated. Still, the latter can be solved in as few as ≈20 generations compared to 100 generations of evolving a corresponding randomly initialized NCA to solve the Czech-flag problem from scratch, as shown above.

[0154] Thus, we conclude that pre-evolved (or “informed”) competency at subordinate scales of a multi-scale competency architecture greatly enhances a collective system’s capability of adaptation. Thus, a competent and informed substrate has great effects on a multi-scale competency architecture’s evolvability towards changing environmental conditions and on the transferability of already acquired (evolved) solutions to new problems.

[0155] Conclusion

[0156] We have investigated the evolutionary implications of multi-scale intelligence on the example of in silico morphogenesis of a two-dimensional tissue of locally interacting cells that are equipped with tunable decision-making machinery. More specifically, we have utilized evolutionary algorithms (EAs) to evolve the parameters of Neural Cellular Automata (NCAs) on morphogenesis tasks under various conditions of the competency level of the uni-cellular agents and the developmental noise in the system. 39 QB\166118.01533\95647317.2T002814 166118.01533

[0157] In this model of a multi-scale competency architecture1, a two-dimensional grid of locally interacting cells is tasked to self-assemble and maintain a global spatial target pattern of predefined cell types, here primarily of a two-dimensional, Czech flag pattern. Each uni-cellular agent’s internal decision-making machinery is modeled by an artificial neural network (ANN), allowing these cells to independently perceive the cell states of their adjacent neighbors on the grid and propose actions to regulate their own cell state over time, thereby communicating with their neighbors. Both the ANN parameters and the initial cell states of all permanent cells represent the parameters of the NCA and are optimized by EAs for a specific in silico morphogenesis task at hand, thus forming the functional and structural part of the system’s genome, respectively.

[0158] To investigate the effects of competency in a multi-scale competency architecture on the underlying evolutionary process, we vary (I) the reliability an NCA’s uni-cellular agents can independently regulate their cell types during a noisy developmental stage. We thus specifically define a “competency level” parameter in our system as the decision-making probability at which proposed actions of uni-cellular agents are considered in the NCA’s corresponding cell state updates (or omitted otherwise). This allows us to continuously scale the NCA’s competency level from a direct encoding scheme of the target pattern (no competency) to a multi-scale competency architecture that self-assembles the pattern with perfect reliability in cell decision executions. Furthermore, we introduce (II) a variable number of redundant sub- modules in the NCA’s ANN, each with an independent set of functional parameters, which we can control in our system as another “axis” of competency based on redundancy and computational capacity of the cells’ decision-making machinery.

[0159] In large-scale simulations, we systematically vary these two competency levels (I, II), expose the corresponding NCA to different noise conditions (III), and perform several statistically independent evolutionary searches at each parameter combination (I-III). In that way, we demonstrate that an evolutionary process proceeds significantly more rapidly (on average) on noisy pattern formation tasks when evolving the parameters of a multiscale competency architecture compared to evolving the target pattern directly (with no competency involved).

[0160] Our multi-scale competency architecture model and the corresponding evolutionary optimization process comprise several scales. At the smallest scale (1), each 40 QB\166118.01533\95647317.2T002814 166118.01533 structural and functional gene is represented by a floating point number. The functional genes parameterize the behavior of artificial neurons (2), our atomic decision-making centers, which are then hierarchically arranged into layers of artificial neurons (3), sub-modules of interconnected layers (4), to an ANN with a predefined architecture (5). Thus, even the uni- cellular phenotypes (6) in our system - ANN-based agents that maintain a particular internal cell state - are composites of smaller (proto-competent) decision-making centers down the hierarchical ladder. The composite uni-cellular agents perceive the cell states of their grid neighbors (7) on the NCA, perform potentially several cycles of internal calculations, and eventually update their own cell state in a single developmental step. In that way, clusters of different tissue types (8) may be formed in successive developmental steps. A fixed number of developmental steps comprise the lifetime of a single NCA, giving rise to a self-assembled phenotypic tissue of cell types on the entire grid of the NCA (9), e.g., as in our case, to the Czech flag pattern. The quality of each individual in an evolutionary population of NCAs (10) is evaluated via a phenotypic fitness score, quantifying the deviation of the assumed cell types from a target pattern. Based on the fitness scores of a particular generation of NCAs, the genotypes of potentially better- adapted successor generations are successively sampled by the EA, closing the loop (1) and forming the largest scale in our system, an evolutionary lineage (11). Eventually, on a meta-scale (12), we compare the efficiency of the evolutionary process at different system parameters (I-III), i.e., at different competency- and noise levels, by analyzing the fitness trajectories of statistically independent lineages evaluated at the same system parameters.

[0161] We demonstrate that especially in the presence of developmental noise, affecting cell state updates during morphogenesis, the evolutionary process favors a multi-scale competency-based realization over a direct encoding scheme of the target pattern. Moreover, when the competency level itself was left as an evolvable parameter to the EA, there appeared to be a non-trivial dynamical tradeoff in the evolutionary process’ efficiency between exploiting the competency level of its components or the direct, pre-patterning-like encoding of the target pattern. We thus report that under realistic conditions (e.g., at moderate noise levels), an evolutionary process can be significantly more efficient when working with an agential- rather than a passive material.

[0162] Notably, we explicitly omit a reward or fitness feedback from the environment to the NCAs’ uni-cellular agents’ perception, restricting the cells’ decision-making solely to local 41 QB\166118.01533\95647317.2T002814 166118.01533 communication of cell state updates between grid neighbors. Thus, the cells need to figure out their own communication protocol such that their single-agent decisions align with the global (multi-agent) system-level objectives of assembling the correct target pattern. These uni-cellular competencies are acquired over evolutionary time scales and can be understood as emergent behavior-shaping signaling.

[0163] On a more technical note, we specifically employ permutation invariant ANNs as trainable update functions of the NCAs and successfully evolve the corresponding models to perform the here studied pattern formation tasks. We thus show that, contrary to previous assumptions, a perfect spatial resolution of neighboring cell states in an NCA is not necessary but that a mean-aggregated neighboring cell state can be sufficient for single cells to reliably contribute to the objective of a larger scale collective. Strikingly, we show that such uni-cellular agents do not even need to distinguish between their own states and the states of their neighbors to achieve this task, thus fully integrating into the tissue locally and essentially losing their individuality.

[0164] Also in contrast other research, we do not start our morphogenesis experiments from a single “alive” cell but instead evolve the initial cell states of all permanent cells on the grid of an NCA, while the uni-cellular agents are constantly challenged to correct their state from developmental noise (notably, a process reminiscent to the de-noising steps of Diffusion Models). This allows us to explicitly distinguish between the evolutionary implications of (i) direct and (ii) multi-scale competency-based encodings of the target pattern, where we either constrain the evolutionary process to (i) only evolve the structural part of the genome, or to (ii) evolve both the structural and functional part simultaneously. Admittedly, the choice of the structural part of the genome limits the scalability of the approach, as the size of the structural genome will grow correspondingly with the number of cells in the system. However, as occurs with biomechanical, biochemical and bioelectric pre-patterning, the initial states of an NCA of moderate size could be seen as a coarse-grained scaffold, based on which an NCA of potentially much higher resolution can run its multi-scale competency-based developmental program to self- assemble a high-resolution target pattern. Alternatively, we suggest utilizing a Compositional Pattern Producing Network (CPPN) to indirectly encode the initial states of all cells on the grid of an NCA, allowing such a hybrid approach to perform in-silico morphogenesis at scale. Unfortunately, it has been proven difficult, if not unfeasible, to exactly reproduce predefined 42 QB\166118.01533\95647317.2T002814 166118.01533 target patterns reliably with neuroevolution of CPPNs alone, which is why we here refrained from this approach; we emphasize, however, that gradient-based methods such as Neural Radiance Fields (NeRF) to train CPPN-like architectures might be an interesting workaround.

[0165] We find that fully evolved NCA solutions, capable of performing the morphogenesis tasks discussed above, show great signs of generalizability toward changing the system parameters, and can, without any further evolutionary optimization or training, handle noise and competency levels that are vastly different from the training conditions. Consequently, this leads to increased evolvability of such competency-based models to changing environmental conditions: a subsequent evolutionary process can adapt a pre-evolved solution to altered environmental conditions within a handful or sometimes even a single generation. Moreover, we demonstrate that such pre-evolved NCA solutions can even quickly adapt to new, yet related problems. Specifically, we modified the objective function of our evolutionary process from the Czech flag task to self-assemble a blue-, red-, white-, Viennese-, diagonal blue\white and blue / red flag instead, respectively. In most of these situations, an adaptation of an existing NCA solution to the new problem can be done in significantly fewer generations than evolving the initial Czech flag task from a randomly initialized configuration. Typically, these adaptations happen the faster the larger the competency level of the NCA, while for the direct encoding scheme (or in situations with low competency) the structural part of the genome is too dominant to allow quick adaptations by the EA. This suggests that multi-scale competency architectures allow the underlying evolutionary process to not over-train on priors, thus augmenting adaptability through a competent substrate.

[0166] We conclude that not only can evolutionary processes efficiently utilize and bring forth the intriguing multi-scale problem-solving machines of biological life, but that the efficiency of such evolutionary processes, as well as the generalization abilities, evolvability, and transferability of the corresponding phenotypic outcomes, are strongly affected by the level of competency of the underlying agential material. An intriguing open question is whether this implies a positive feedback loop that enhances that quality over time. Judging from the considerable effects of scaling the competency in the here studied still shallow multi-scale system on a rather simple in silico evolutionary process (e.g., CMA-ES), it becomes increasingly evident that the vastly more complex multi-scale competency architecture of biological life cycles back and thus affects the process of evolution itself. 43 QB\166118.01533\95647317.2T002814 166118.01533

[0167] For future directions, our multi-scale competency framework is easily extendable to simulate tissue growth via cell migration or division actions proposed by the NCA’s underlying ANNs. More specifically, our framework allows for a minimal set of biologically relevant uni-cellular actions, such as a cell state update, cell division, migration, cell death, and an identity operation, only constrained by the NCA’s spatial grid. Furthermore, the framework is capable of handling flexible ANN architectures, potentially allowing us to investigate intriguing competencies such as active inference through utilizing world model architectures in a (neuro)evolutionary context. Our system, so far, has a fixed hierarchical architecture that deviates from the scale-free competency architecture of biological life with open-ended functional adaptation (where any abstraction layer becomes the basis for the next one). Thus, in future work, we aim to model precisely this behavior by introducing multiple layers of horizontal communication pathways in an NCA that the ANN-based agents can dynamically traverse in the vertical direction. Moreover, by choosing a proper fitness function related to measuring scale- invariant pattern formation, critical dynamics, or applying the free-energy principle, we are confident to achieve a biologically more accurate model of the scale-free dynamics and open- ended evolution of life. Such computational models could thus further quantitative studies of the communication strategies and boundaries of individual- and groups of cells in an agential, potentially adversarial Umwelt, with possible applications in individual- and collective aging (as morphostasis defects) or cancer research.

[0168] Detailed Methods

[0169] Artificial Neural Networks

[0170] Inspired by biological neural circuits, an Artificial Neural Network (ANN) is an inter-connected network of artificial neurons (AN). Each such AN maps a set of inputs , onto a single number , usually through a non-linear filter, : The output of an AN can be defined as a parameterized function , with weights and bias .

[0171] Commonly organized in layers of Ans, a Feed Forward ANN represents a parameterized non-linear function , transforming an input , over consecutive hidden layers of ANs, , to an output vector . More specifically, the output of layer I, defined by 44 QB\166118.01533\95647317.2T002814 166118.01533

[0172] ) Eq. A1

[0173] Becomes the input , to the next deeper layer, i+1, through layer-wise filtered dot-products with the and bias vectors.

[0174] Training an ANN thus boils down to optimizing a set of parameters, , i.e., the entire network’s weights and biases, such that an input is mapped (with minimal deviation) to a desired output. In this manuscript, we utilize ANNs as the trainable update function of a neural cellular automaton (NCA) and optimize the corresponding ANN parameters via evolutionary algorithms to study the evolutionary implications of multi-scale intelligence on the example of morphogenesis.

[0175] Notably, in contrast to previous contributions of NCA-based morphogenesis, we do not rely on predefined convolutional filters in our ANN architectures to preprocess a cell’s local environment. Based on its own cell state, , and the states of its direct neighbors , as a given time step tk, which we formally collect in , to evaluate a sensor embedding, . The latter is averaged alongdimension to form aof fixed size s that is permutation invariant with respect to the cell’s neighborhood on the NCA (also see Methods, NCA: a Multi-agent model for morphogenesis). This context vector represents the cell’s internal representation of its local environment on the cellular grid of the NCA.

[0176] Each cell i independently proposes an update, , to its own state , potentially at every time step tkfollowing Eq. 1. This update is computed by a controller ANN, , based on the cell-specific context vector . Thus, the set of ANN comprises the sensory and controller network parameters andwe have not specified a particular architecture for either or . Although the presented approach is agnostic to the particularly chosen ANN architecture, we 45 QB\166118.01533\95647317.2T002814 166118.01533 here rely on rather simple implementations of ANNs: For the sensory ANN, , we utilize a Feed Forward architecture with hyperbolic tangent activation function , with 4 inputs, 8 neurons in a single hidden layer, and 8 output neurons, defined the (s=8)-dimensional context vector), resulting in a total of 112 parameters. For the controller ANN we utilize two different architectures, a Feed Forward (see FF-agent in Results, The System and Eq. A1) and a recurrent ANN that is inspired by both, Recurrent ANNs (RNNs) and Gene Regulatory Networks (see RGRN-agent in Results, The System and Eqs. A2 and A3 below).

[0178] The Feed Forward controller architecture consists of 8 input units (i.e., the context vector from the sensory ANN), a single hidden layer with 6 neurons and a hyperbolic tangent activation function, and 4 output neurons without activation function, resulting in 82 parameters in total; thus the genuine FF-agent architecture in Example 1 comprises a total number of NFF= 194 parameters (see Results, the System). The RGRN controller architecture (see details below) consists of 8 input units, a single self-regulated recurrent state with 3 neurons (with an internal hyperbolic tangent activation), and 4 output neurons (without an activation function), resulting in 52 parameters in total; thus the genuine RGRN-agent architecture in Example 1 comprises a total number of NRGRN = 164 parameters (see Results, The System). In Table I we summarize the FF-agent and RGRN-agent’s architectures and parameter counts. Sensory ANN (Num. Param.) Controller ANN (Num. Param.) Total Num. Param. FF-agent Feed Forward (112) Feed Forward (82) 112 + R 82 RGRN-agent Feed Forward (112) RGRN (52) 112 + R 52

[0179] Table I: Architecture and number of parameters in sensory and controller ANNs of the two different agent architectures used in this contribution. The total number of parameters depends on the redundancy number R of the controller ANN (see Results, The System).

[0180] Finally, we define the RGRN architecture, , that relies on both an instantaneous input, , and athe previous iteration of the network to generate an output, . First, we define the self- regulated recurrent state h(tk) as

[0181] , Eq. A246 QB\166118.01533\95647317.2T002814 166118.01533

[0182] which thus is maintained over time by a factor of and updated by a factor of via integrating the external stimuli x(tk), and recurrent memory, h(tk-1), through the trainable matrices , and bias vectors bU, bV, respectively. Second, we evaluate the network’s output, , based on the RGRN’s recurrent state h(tk), following

[0183] , Eq. A3

[0184] Having introduced the weight matrix and bias vector , and a non-linear activation function .

[0185] Following ideas from Hiscock, BMC Bioinformatics, 20, 214 (2019), we thus utilize with Eq. A2 an ANN that maintains a self-regulated (or “gene regulated”) state, h(tk). However - and dropping the bias vectors for convenience below - the second term in Eq. A2, i.e., , is reminiscent to the kernel of an RNN, thus allowing the RGRN to integrate new information (i.e., external stimuli) into its regulatory behavior. Thus, the state update of h(tk) corresponds to regulating the network’s recurrent state (or “gene expression”) conditional to external stimuli. Furthermore, explicitly separating the self-regulated recurrent state from the RGRN’s output allows us to utilize the RGRN as a controller, e.g., to use its output, y(tk), for updating the cell state of an NCA in Eq. 1.

[0186] Here, we set τ1= 0.75, τ2= 0.25 and chose as the identity transformation (i.e., no, or linear activation of the RGRN’s output), and we apply Eq. A23 times (updating h(tk) in every cycle) before forwarding the final value of h(tk) to Eq. A3 to generate the RGRN’s output.

[0187] For all ANN implementation, we here relied on PyTorch.

[0188] Reinforcement Learning Agent’s Perception-Action Cycle

[0189] We utilize a Neural Cellular Automaton (NCA) for morphogenesis tasks of 2D target patterns. In such a setting, each cell of the NCA represents an autonomous agent that perceives details about its local environment (e.g., the cell states of its direct neighbors on the NCA’s spatial grid) and proposes actions to update its own state. Here, we summarize the terminology behind this perception-action cycle of an agent in an arbitrary environment of a Reinforcement Learning (RL) setting (see FIG. 1B for an illustration). 47 QB\166118.01533\95647317.2T002814 166118.01533

[0190] Based on an agent’s perception of the environment, i.e., a state skmeasured at time step tk, the agent’s goal is to manipulate the environment by taking an action ak- resulting in a state update sk+1in the next time step - to collect as much reward, (provided by the environment), as possible.

[0191] Formally, an agent picks its actions according to a policy , i.e., a typically complicated function which might be parameterized via hidden variables θ. Artificial Neural Networks (ANNs) as universal function approximators are promising candidates that can be trained to fit an agent’s optimal policy (mapping states skto optimal actions ak). Here, we thus utilize ANNs as a trainable update function of an NCA and deploy evolutionary algorithms (see Methods, Neuroevolution of NCAs) to find the optimal policy (here, of morphogenesis tasks), via optimizing . This enables an agent to choose actions aiming at maximizing the expected cumulative reward (or maximum fitness, in our terms).

[0192] The particular functional choice of the reward signal defines the agent’s task via positive (or negative) reinforcement. In our case, the cumulative reward Rαof all Nαagents (i.e., of all cells on the grid) after tDtime steps is summed up to the fitness of the entire NCA. There is no general procedure for creating effective reward

[0193] Crucially, we here do not provide the cellular agents with environmental reward feedback directly, but only use the cumulative reward, Rα, as a fitness criterion for the evolutionary algorithm. Thus, the collective of cells needs to evolve a signaling strategy to communicate desirable or prohibitive cell state updates during the corresponding developmental stage.

[0194] Fixed Boundary Condition Handling of the Neural Cellular Automaton

[0195] We employ Neural Cellular Automata (NCAs) with fixed boundary conditions on a two-dimensional square grid (see Methods, Neuroevolution of NCAs). Each cell I is associated with integer grid-coordinates (xi, yi) on the NCA’s grid of size with and .

[0196] The neighborhood of cell i is defined by all directly adjacent cells , i.e., that share a border or a corner with cell i. Since we consider a square grid in this contribution, 48 QB\166118.01533\95647317.2T002814 166118.01533 the grid coordinates of all N-9 neighbor cells are given by permutations of , with , excluding the identity of .

[0197] For cells at the boundaries of the grid, some neighbors with coordinates or , respectively, will be out of bounds. Thus, we clip all neighbor coordinate to the intervals [1, Nx] and [1, Ny], respectively via , and , and replace the index ivwith the correspondingly indexi’vof the clipped coordinates . Collecting the numerical state values of theneighborhood of cell i thus yields . The matrix then represents the input network (seeMethods, Neuroevolution of .

[0198] Covariance Matrix Adaptation Evolution Strategy (CMA-ES)

[0199] Covariance Matrix Adaptation Evolutionary Strategy (CMA-ES) is a popular evolutionary algorithm: a multivariant normal distribution is utilized to model the (genotypic-) distribution of a set or a population of parameters that are evaluated against an objective function. Roughly speaking, this evaluated fitness score of an individual is associated with its probability of survival, and thus for participating in the reproduction of the next generation. The parameters of the multivariant normal distribution, i.e., the mean and covariance matrix, are successively updated based on selecting the best individuals from a given population (or, more precisely, by weighting the relative importance of an individual by its fitness score) such that high-fitness individuals are generated with high likelihood by the Gaussian model. Thus, iteratively sampling “offspring” generations and adapting the model covariance matrix (and its mean) based on the population’s fitness scores guides the evolutionary population toward high fitness regions in the parameter space over successive generations. Typically, also the numerical step size of the parameter update is adapted according to some inter- and intra-generation fitness measures. In a nutshell:

[0200] 1. CMA-ES typically starts with a standard (or parameterized) multi-variant normal distribution with the dimension given by the number of parameters (or genes).

[0201] 2. At each evolutionary cycle, a new population of a fixed number of individuals is sampled from the model. 49 QB\166118.01533\95647317.2T002814 166118.01533

[0202] 3. Each individual is evaluated against a fitness function, which quantifies the corresponding individual’s probability of being selected for reproduction to form the next generation.

[0203] 4. The mean and covariance matrix of the normal distribution, and a step-size parameter, are updated such that high-quality individuals are generated with high likelihood by the generative model.

[0204] 5. The process (2-4) is repeated until a convergence criterion is met.

[0205] In the CMA-ES experiments presented in this contribution, we used an initial normal distribution with zero mean, μ0= 0, and a standard deviation of typically σ0= 2−4, and we disable step-size adaptation.

[0206] We specifically utilized the open-source pycma Python implementation of CMA- ES from Hansen et al. Cma-es / pycma: r3.3.0 (2023).

[0207] Direct vs. Multi-scale Encoding: Morphogenetic Development Over Evolutionary Time-Scales

[0208] In FIG. 7, we explicitly illustrate the developmental process of the Czech flag task over evolutionary time-scales for an NCA evolved at noise-level of = 0.25, decision- making probability PD= 50%, and redundancy number R = 4; in FIG. 8, weillustrate the same developmental process for an NCA without competency, i.e., with PD= 0.

[0209] FIG. 7 and FIG.8 illustrate how the evolutionary process learns how to construct the target pattern over generations, depending on the competency of the underlying substrate: either driven by intercellular communication-based self-assembly of the target pattern that continuously corrects potential errors of the developmental program, or via directly encoding the target pattern into the structural part of the genome to resist developmental noise for as long as possible. Moreover, while the target pattern for the competent first case is quickly (and robustly) self-assembled and maintained over time potentially much longer as the tD= 25 developmental time steps the phenotypes have been selected for - in the direct case the initial cell state eventually gets destroyed by the noise during the developmental process.

[0210] Thus, in the former case, illustrated in FIG. 7, the structural fitness of the initial cell states (at tk= 0) remains decoupled and rather low compared to the highest fitness of the 50 QB\166118.01533\95647317.2T002814 166118.01533 population of the phenotypes, even long after the problem is solved. In contrast, in the latter case, illustrated in FIG.8, the initial cell state needs to evolve towards the target pattern directly, resulting in high structural fitness values at tk= 0, which are then progressively decreased by the noise during the developmental process, resulting in correspondingly lower phenotypic fitness values.

[0211] Direct vs. Multi-scale Encoding: Neutral Transfer of Hierarchical Competencies Affects Uni-Cellular Robustness

[0212] Above, we compare an evolutionary process operating on (i) the direct encoding of the structural traits of an 8×8 Czech flag pattern with the evolution of (ii) the parameters of an NCA-based multi-scale encoding of the same pattern. Here, we specifically investigate case (ii) more closely, relating the slow but steady long-term improvement in the structural fitness, long after the problem is effectively converged, to the Baldwin effect, and neutral evolution and the “paradox of robustness.”

[0213] In the top panel of FIG.15, we again show the entire evolutionary lineage presented in the bottom panel of FIG. 2, but we specifically contrast the structural fitness of the historically best-performing individual with the structural fitness of the best-performing individual of every generation. We see that after generation 860, the historically best structural fitness (which we explicitly track for numerical reasons) deviates from the structural fitness of the still-evolving population.

[0214] We interpret this slow but steady improvement in the structural fitness as a manifestation of the Baldwin effect, where (evolutionarily) acquired uni-cellular competencies (here, to assemble a target pattern) are shifted to hard-wired phenotypical traits. Especially between generations 860 and 1823, this transfer of competency is caused by corresponding successive adaptations to the structural genomes of the entire population as reflected by the increasing mean of the entire population’s structural fitness depicted in FIG. 15. However, this process happens without significant changes to the entire system’s fitness score between successive generations as illustrated in the bottom panel of FIG. 15, where we explicitly present incremental numerical improvements to the lineage’s fitness score. Thus, these adaptations to the structural genomes remain neutral with respect to the fitness of the entire system. 51 QB\166118.01533\95647317.2T002814 166118.01533

[0215] Eventually, a random event in generation 1823 causes a slight improvement in the so-far historically best fitness score of generation 860 (from =69.781 to 69.828). However, this newly assigned historically best individual is now qualitatively different from the previous one (cf. purple and blue-dashed curves in FIG. 15), as its policy is shifted from a rather competency- based NCA (generations 860–1822) to a solution that relies more heavily on directly encoded phenotypic traits (generations ≥1823). This, in turn, affects the new solution’s robustness against increasingly noisy cell state updates as illustrated in FIG. 16: we re-evaluate the fitness score of the entire genetic lineage depicted in FIG. 15 (which we tracked in the original experiment) at modified noise levels larger than experienced during the respective evolutionary process >0.25. While the best-performing individuals for generations 750–860 can generalize well to different noise levels, we observe successively decreasing performance of the preceding generations, especially at higher noise levels.

[0216] Thus, this could be a manifestation of the paradox of robustness, which states that in the presence of a higher-level control mechanism, the system’s lower-level (agential) components may become increasingly unreliable. In our case, the higher-level mechanism would be represented by an increasingly accurate initial cell state configuration, which is a global system-level encoding of the target pattern. The lower-level components are represented by the ANN-based uni-cellular agents of the NCA, performing local error correction to assemble and maintain the target pattern through cell state updates. Notably, and depending on the noise level, the former may be more difficult to acquire by an evolutionary process, starting from a randomly initialized population, but the latter might become increasingly specialized the better the structural genome is adapted.

[0217] Since for a particular noise level, a precise enough initial cell state configuration is sufficient to solve the problem (FIGS. 5A-5F), the NCA competencies may in such cases become successively unnecessary in the long run, and thus lose their relevance to the evolutionary process. Consequently, such a shift in competency renders a system less robust against perturbations and changing environmental conditions as depicted in FIG. 16 (and also reflected in FIG. 14). Notably, such a combination of the Baldwin effect (shift in competencies), neutral evolution (not affecting the system-level fitness scores), and the paradox of robustness (less competent components) might even explain the reduced capabilities of NCAs pre-evolved 52 QB\166118.01533\95647317.2T002814 166118.01533 at lower rather than higher noise levels to adapt to modified problems of self-assembling different target patterns as observed in FIG. 6.

[0218] We also conjecture that this mechanism of heterogeneous agents’ competencies cooperating on a common system-level objective might be important for adaptability and evolvability in biological systems. Notably, robustness mechanisms might be only partially active. For example, chaperones that prevent proteins from misfolding allow for more exploration of mutations while destabilizing the fold, giving the system more time to find a restabilizing mutation. In such cases, the "protected layers" are still intermittently exposed to selection and unlikely to deteriorate. However, if the specialized competencies of certain agential components can be replaced by a different mechanism in the same organism more cheaply, these increasingly redundant components might be repurposed to fulfill different tasks, thus potentially facilitating open-ended evolution.

[0219] Direct vs. Multi-scale Encoding: Evolution and Morphogenesis of a Smiley Face Pattern

[0220] Above, we primarily investigate the evolutionary implications of multi-scale intelligence on the example of morphogenesis of an Czech flag pattern. To test, whether our findings in Results, Evolution Exploits Competency over Direct Encoding generalize to different target patterns, we here present results for a much more involved task, namely a smiley face pattern (see FIGS. 1A-1E) which has several internal boundaries of (i) the face, (ii), the eyes, and (iii) the mouth; all other parameters are the same as for the Czech flag task.

[0221] We thus perform an analogous study to Results, Evolution Exploits Competency over Direct Encoding and present the results in FIG. 9 (reminiscent to FIG. 3C), but for redundancy numbers (as we found that smaller controller networks perform systematically worse on the task, suggesting a capacity bottleneck of ANNs with R < 4 in this case). Analogously to the much simpler Czech flag task, we can learn from FIG. 9 that, while in the low noise regime, direct encoding can lead to a more efficient evolutionary process, in situations with increasing developmental noise higher competency levels (here again realized via the decision-making probability) can significantly enhance the efficiency of the evolutionary process of a morphogenesis task. Notably, and due to computational reasons, we evaluated only two to three independent evolutionary processes for every combination of the system parameters 53 QB\166118.01533\95647317.2T002814 166118.01533 (noise-level, decision-making probability, and redundancy number) for the results depicted in FIG. 9. However, the overall trend of the evolutionary efficiency of (i) directly encoding the target pattern and (ii) encoding the functional parameters of a multi-scale competency architecture is consistent with our previous results discussed in Results, Evolution Exploits Competency over Direct Encoding. Due to the increased complexity of the smiley face task, the critical noise level that separates the evolutionary efficiency of (i) and (ii) is correspondingly shifted to larger values of here ≥ 0.25 (see FIG. 3C).

[0222] Analogous to FIG.7, wein FIG. 10 the developmental process of the 9 9-smiley face task over evolutionary time-scales for an NCA evolved at noise-level of = 0.25, decision-making probability PD= 50%, and redundancy number of R = 4 in an RGBattributing the numerical values of the first three cell states, scaled to values 1], respectively (see Methods, NCA).

[0223] Evolution Exploits Redundancy at the Cost of a More Complex Search Space

[0224] Analogous to Results, Evolution Exploits Competency over Direct Encoding, we here present the evolutionary efficiency of the same morphogenesis experiments of the Czech flag problem depicted in FIGS. 3A-3D, but present as a function of the redundancy number R - instead of the decision-making probability PD- and the noise level ; for a given combination of R and , we additionally utilized different values for PD= 0.25, 0.5, 1.0 and performed 15 statistically independent runs of the EA for each parameter combination, resulting in 45 independent evolutionary runs per (R, )-tuple.

[0225] Although there appears to be an effect of R on the evolutionary efficiency (see FIGS. 11A, 11C, and 11D), the results are less pronounced compared to FIGS. 11A-11D. Despite considerable uncertainty in the evolutionary efficiency, as shown in FIGS. 11A-B, we can learn from the heatmaps, FIGS. 11C-11D, that at low noise levels of = 0 or 0.125, large R values appear favorable over lower ones, whereas, at larger noiseof = 0.5, populations with lower values of R perform better on average. For intermediatelevels of = 0.25 and 0.375, we observe an “optimal” redundancy number of 4, in this particular54 QB\166118.01533\95647317.2T002814 166118.01533

[0226] This suggests a trade-off in the evolutionary efficiency of redundancy - as an affordance of competency - and the corresponding increase in the number of overall parameters of the functional genome.

[0227] Morphogenesis at Scale With a Hybrid Compositional Pattern-Producing Network – Neural Cellular Automata Model

[0228] The particular choice of especially the structural part of the genome x(S) in Eq. 3 limits the scalability of our multi-scale competency approach of morphogenesis to significantly larger systems, as the size of the structural genome will grow correspondingly with the number of cells in the system. However, by utilizing Compositional Pattern Producing Networks (CPPNs) the parameters, θH, of a hyper-network , could replace the structural genes in Eq. 3 such that the initial states of each cell i are indirectly encoded by the hyper-network based on their relative spatial positions (xi / Nx, yi / Ny) on the Neural Cellular Automaton’s (NCA;s) grid via .

[0229] However, it has proven to be difficult, if not numerically infeasible, to reliably and exactly reproduce a two-dimensional target pattern using CPPNs. Thus, we here propose a hybrid approach for morphogenesis at scale of a CPPN indirectly encoding the initial cell states of an NCA, whose uni-cellular agents are then challenged to self-assemble the desired target pattern in a morphogenetic developmental stage. This would allow for scaling the target pattern arbitrarily either during training or during deployment since the number of cells on the NCA’s grid does not affect the size of the (structural part of) genome.

[0230] References for Example 1

[0231] M. Levin, “Darwin’s agential materials: evolutionary implications of multiscale competency in developmental biology,” Cellular and Molecular Life Sciences 80, 142 (2023). C. Fields and M. Levin, “Regulative development as a model for origin of life and artificial life studies,” Biosystems 229, 104927 (2023).

[0232] C. Fields and M. Levin, “Competency in navigating arbitrary spaces as an invariant for analyzing cognition in diverse embodiments,” Entropy 24, 819 (2022).

[0233] M. Levin, “Collective Intelligence of Morphogenesis as a Teleonomic Process,” in Evolution "On Purpose": Teleonomy in Living Systems (The MIT Press, 2023). 55 QB\166118.01533\95647317.2T002814 166118.01533

[0234] K. D. Birnbaum and A. S. Alvarado, “Slicing across kingdoms: Regeneration in plants and animals,” Cell 132, 697–710 (2008).

[0235] A. K. Harris, “The need for a concept of shape homeostasis,” Biosystems 173, 65–72 (2018).

[0236] M. Levin, A. M. Pietak, and J. Bischof, “Planarian regeneration as a model of anatomical homeostasis: Recent progress in biophysical and computational approaches,” Seminars in Cell & Developmental Biology 87, 125–144 (2019).

[0237] L. N. Vandenberg, D. S. Adams, and M. Levin, “Normalized shape and location of perturbed craniofacial structures in the xenopus tadpole reveal an innate ability to achieve correct morphology,” Developmental Dynamics 241, 863–878 (2012).

[0238] J. Cooke, “Scale of body pattern adjusts to available cell number in amphibian embryos,” Nature 290, 775–778 (1981).

[0239] G. Fankhauser, “Maintenance of normal structure in heteroploid salamander larvae, through compensation of changes in cell size by adjustment of cell number and cell shape,” Journal of Experimental Zoology 100, 445–455 (1945).

[0240] D. J. Blackiston and M. Levin, “Ectopic eyes outside the head in Xenopus tadpoles provide sensory data for light-mediated learning,” Journal of Experimental Biology 216, 1031–1040 (2013).

[0241] G. Pezzulo and M. Levin, “Remembering the body: applications of computational neuroscience to the top-down control of regeneration of limbs and other complex organs,” Integrative Biology 7, 1487–1517 (2015).

[0242] J. Davies and M. Levin, “Synthetic morphology with agential materials,” Nature Reviews Bioengineering 1, 46–59 (2023).

[0243] J. M. Baldwin, “A new factor in evolution,” The American Naturalist 30, 441–451 (1896).

[0244] W. H. Thorpe, “Animal learning and evolution,” Nature 156, 46–46 (1945).

[0245] G. G. Simpson, “The baldwin effect,” Evolution 7, 110 (1953). 56 QB\166118.01533\95647317.2T002814 166118.01533

[0246] G. E. Hinton, S. J. Nowlan, et al., “How learning can guide evolution,” Complex Systems 1, 495–502 (1987).

[0247] R. K. Belew, “When both individuals and populations search: Adding simple learning to the genetic algorithm,” in Proceedings of the 3rd International Conference on Genetic Algorithms (Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 1989) p. 34–41.

[0248] D. Ackley and M. Littman, “Interactions between learning and evolution,” in Proceedings of the Second Conference on Artificial Life, Vol. 10, edited by C. G. Langton, C. Taylor, J. D. Farmer, and S. Rasmussen (Addison-Wesley, Santa Fe, NM, USA, 1991) pp. 487– 509.

[0249] G. Mayley, “Landscapes, Learning Costs, and Genetic Assimilation,” Evolutionary Computation 4, 213–234 (1996).

[0250] L. Bull, “On the Baldwin Effect,” Artificial Life 5, 241–246 (1999).

[0251] S. Nolfi and D. Floreano, “Learning and evolution,” Autonomous Robots 7, 89– 1131999).

[0252] H. Dopazo, “A model for the interaction of learning and evolution,” Bulletin of Mathematical Biology 63, 117–134 (2001).

[0253] B. H. Weber and D. J. Depew, Evolution and Learning: The Baldwin Effect Reconsidered (The MIT Press, 2003).

[0254] F. Mery and T. J. Kawecki, “A fitness cost of learning ability in Drosophila melanogaster,” Proceedings of the Royal Society of London. Series B: Biological Sciences 270, 2465–2469 (2003).

[0255] E. Crispo, “The Baldwin effect and genetic assimilation: revisiting two mechanisms of evolutionary change mediated by phenotypic plasticity,” Evolution 61, 2469– 2479 (2007).

[0256] T. J. Kawecki, “Evolutionary ecology of learning: insights from fruit flies,” Population Ecology 52, 15–25 (2009).

[0257] I. Paenke, T. J. Kawecki, and B. Sendhoff, “The influence of learning on evolution: A mathematical framework,” Artificial Life 15, 227–245 (2009). 57 QB\166118.01533\95647317.2T002814 166118.01533

[0258] K. M. Hoedjes, H. M. Kruidhof, M. E. Huigens, M. Dicke, L. E. M. Vet, and H. M. Smid, “Natural variation in learning rate and memory dynamics in parasitoid wasps: opportunities for converging ecology and neuroscience,” Proceedings of the Royal Society B: Biological Sciences 278, 889–897 (2010).

[0259] R. A. Watson and E. Szathmáry, “How can evolution learn?” Trends in Ecology & Evolution 31, 147–157 (2016).

[0260] A. Livnat and C. Papadimitriou, “Evolution and learning: Used together, fused together. a response to watson and szathmáry,” Trends in Ecology & Evolution 31, 894–896

[0261] C. Fields, J. Bischof, and M. Levin, “Morphological coordination: A common ancestral function unifying neural and non-neural signaling,” Physiology 35, 16–30 (2020), pMID: 31799909.

[0262] F. Baluška and S. Mancuso, “Deep evolutionary origins of neurobiology: Turning the essence of ’neural’ upside-down,” Communicative & Integrative Biology 2, 60–65 (2009), pMID: 19513267.

[0263] F. Baluška, W. B. Miller, and A. S. Reber, “Cellular and evolutionary perspectives on organismal cognition: from unicellular to multicellular organisms,” Biological Journal of the Linnean Society 139, 503–513 (2022).

[0264] L. Shreesha and M. Levin, “Cellular competency during development alters evolutionary dynamics in an artificial embryogeny model,” Entropy 25 (2023), 10.3390 / e25010131.

[0265] V. Liard, D. P. Parsons, J. Rouzaud-Cornabas, and G. Beslon, “The Complexity Ratchet: Stronger than Selection, Stronger than Evolvability, Weaker than Robustness,” Artificial Life 26, 38–57 (2020).

[0266] J. Huizinga, K. O. Stanley, and J. Clune, “The Emergence of Canalization and Evolvability in an Open-Ended, Interactive Evolutionary System,” Artificial Life 24, 157–181 (2018).

[0267] W. C. Ratcliff, J. D. Fankhauser, D. W. Rogers, D. Greig, and M. Travisano, “Origins of multicellular evolvability in snowflake yeast,” Nature Communications 6 (2015), 10.1038 / ncomms7102. 58 QB\166118.01533\95647317.2T002814 166118.01533

[0268] J. L. Payne, J. H. Moore, and A. Wagner, “Robustness, Evolvability, and the Logic of Genetic Regulation,” Artificial Life 20, 111–126 (2014).

[0269] K. Raman and A. Wagner, “The evolvability of programmable hardware,” Journal of The Royal Society Interface 8, 269–281 (2011).

[0270] C. L. Nehaniv, “Evolvability,” Biosystems 69, 77–81 (2003).

[0271] H. Hoenigsberg, “Cell biology, molecular embryology, lamarckian and darwinian selection as evolvability,” Genet Mol Res 2, 7–28 (2003).

[0272] T. F. Hansen, “Is modularity necessary for evolvability?” Biosystems 69, 83–94 (2003).

[0273] M. A. Bedau and N. H. Packard, “Evolution of evolvability via adaptation of mutation rates,” Biosystems 69, 143–162 (2003).

[0274] M. Kirschner and J. Gerhart, “Evolvability,” Proceedings of the National Academy of Sciences 95, 8420–8427 (1998).

[0275] G. P. Wagner and L. Altenberg, “Perspective: Complex adaptations and the evolution of evolvability,” Evolution 50, 967–976 (1996).

[0276] R. Raff, “Developmental mechanisms in the evolution of animal form: origins and evolvability of body plans,” Early life on earth 84, 489–500 (1994).

[0277] R. Raff, The Shape of Life: Genes, Development, and the Evolution of Animal Form, The Shape of Life: Genes, Development, and the Evolution of Animal Form (University of Chicago Press, 1996).

[0278] P. Alberch, “From genes to phenotype: dynamical systems and evolvability,” Genetica 84, 5–11 (1991).

[0279] S. A. Frank, “Natural selection maximizes Fisher information,” Journal of Evolutionary Biology 22, 231–244 (2009).

[0280] S. A. Frank, “Maladaptation and the paradox of robustness in evolution,” PLoS ONE 2, e1021 (2007). 59 QB\166118.01533\95647317.2T002814 166118.01533

[0281] S. A. Frank, “Developmental selection and self-organization,” Biosystems 40, 237–243 (1997).

[0282] S. A. Frank, “The design of adaptive systems: Optimal parameters for variation and selection in learning and development,” Journal of Theoretical Biology 184, 31–39 (1997). K. O. Stanley and R. Miikkulainen, “A Taxonomy for Artificial Embryogeny,” Artificial Life 9, 93–130 (2003).

[0283] C. Waddington, “The strategy of the genes,” L.: Allen and Unwin (1957).

[0284] C. H. Waddington, The strategy of the genes (Routledge, 2014).

[0285] D. Noble, “Modern physiology vindicates darwin’s dream,” Experimental Physiology 107, 1015–1028 (2022).

[0286] J. A. Shapiro, “Engines of innovation: biological origins of genome evolution,” Biological Journal of the Linnean Society 139, 441–456 (2022).

[0287] A. Szilágyi, P. Szabó, M. Santos, and E. Szathmáry, “Phenotypes to remember: Evolutionary developmental memory capacity and robustness,” PLOS Computational Biology 16, e1008425 (2020).

[0288] E. Szathmáry, “Toward major evolutionary transitions theory 2.0,” Proceedings of the National Academy of Sciences 112, 10104–10111 (2015).

[0289] C. L. Buckley and R. A. Watson, (2022, manuscript in preperation).

[0290] K. Kouvaris, J. Clune, L. Kounios, M. Brede, and R. A. Watson, “How evolution learns to generalise: Using the principles of learning theory to understand the evolution of developmental organisation,” PLOS Computational Biology 13, e1005358 (2017).

[0291] E. Fox Keller, “Elusive locus of control in biological development: Genetic versus developmental programs,” Journal of Experimental Zoology 285, 283–290 (1999).

[0292] E. Jablonka, “The evolutionary implications of epigenetic inheritance,” Interface Focus 7, 20160135 (2017).

[0293] K. Laland, T. Uller, M. Feldman, K. Sterelny, G. B. Müller, A. Moczek, E. Jablonka, 60 QB\166118.01533\95647317.2T002814 166118.01533

[0294] J. Odling-Smee, G. A. Wray, H. E. Hoekstra, D. J. Futuyma, R. E. Lenski, T. F. C. Mackay, D. Schluter, and J. E. Strassmann, “Does evolutionary theory need a rethink?” Nature 514, 161–164 (2014).

[0295] M. Elgart, O. Snir, and Y. Soen, “Stress-mediated tuning of developmental robustness and plasticity in flies,” Biochimica et Biophysica Acta (BBA) - Gene Regulatory Mechanisms 1849, 462–466 (2015).

[0296] O. T. Eldakar and D. S. Wilson, “Eight Criticisms not to make about Group Selection,” Evolution 65, 1523–1526 (2011).

[0297] S. Karve and A. Wagner, “Environmental complexity is more important than mutation in driving the evolution of latent novel traits in e. coli,” Nature Communications 13, 5904 (2022).

[0298] S. Karve and A. Wagner, “Multiple novel traits without immediate benefits originate in bacteria evolving on single antibiotics,” Molecular Biology and Evolution 39, msab341 (2022).

[0299] A. Wagner, “The molecular origins of evolutionary innovations,” Trends in Genetics 27, 397–410 (2011).

[0300] A. Wagner and W. Rosen, “Spaces of the possible: universal darwinism and the wall between technological and biological innovation,” Journal of The Royal Society Interface 11, 20131190 (2014).

[0301] D. S. Wilson, “A theory of group selection.” Proceedings of the National Academy of Sciences 72, 143–146 (1975).

[0302] R. Calabretta, A. D. Ferdinando, G. P. Wagner, and D. Parisi, “What does it take to evolve behaviorally complex organisms?” Biosystems 69, 245–262 (2003).

[0303] G. Schlosser and G. P. Wagner, eds., Modularity in Development and Evolution (University of Chicago Press, Chicago, IL, 2004).

[0304] B. M. Stadler, P. F. Stadler, G. P. Wagner, and W. Fontana, “The topology of the possible: Formal spaces underlying patterns of evolutionary change,” Journal of Theoretical Biology 213, 241–274 (2001). 61 QB\166118.01533\95647317.2T002814 166118.01533

[0305] K. H. Ten Tusscher and P. Hogeweg, “Evolution of networks for body plan patterning; interplay of modularity, robustness and evolvability,” PLoS Computational Biology 7, e1002208 (2011).

[0306] G. P. Wagner, M. Pavlicev, and J. M. Cheverud, “The road to modularity,” Nature Reviews Genetics 8, 921–931 (2007).

[0307] G. P. WAGNER and P. F. STADLER, “Quasi-independence, homology and the unity of type: A topological theory of characters,” Journal of Theoretical Biology 220, 505–527 (2003).

[0308] D. A. Power, R. A. Watson, E. Szathmáry, R. Mills, S. T. Powers, C. P. Doncaster, and

[0309] B. Czapp, “What can ecosystems learn? expanding evolutionary ecology with learning theory,” Biology Direct 10 (2015), 10.1186 / s13062-015-0094-1.

[0310] H. I. Schreier, Y. Soen, and N. Brenner, “Exploratory adaptation in large random networks,” Nature Communications 8, 14826 (2017).

[0311] Y. Soen, M. Knafo, and M. Elgart, “A principle of organization which facilitates broad lamarckian-like adaptations by improvisation,” Biology Direct 10, 68 (2015).

[0312] T. Uller, A. P. Moczek, R. A. Watson, P. M. Brakefield, and K. N. Laland, “Developmental Bias and Evolution: A Regulatory Network Perspective,” Genetics 209, 949– 966 (2018).

[0313] R. A. Watson, R. Mills, and C. L. Buckley, “Global Adaptation in Networks of Selfish Components: Emergent Associative Memory at the System Scale,” Artificial Life 17, 147–166 (2011).

[0314] R. A. Watson, R. Mills, C. L. Buckley, K. Kouvaris, A. Jackson, S. T. Powers, C. Cox,

[0315] S. Tudge, A. Davies, L. Kounios, and D. Power, “Evolutionary connectionism: Algorithmic principles underlying the evolution of biological organisation in evo-devo, evo-eco and evolutionary transitions,” Evolutionary Biology 43, 553–581 (2016). 62 QB\166118.01533\95647317.2T002814 166118.01533

[0316] R. A. Watson, G. P. Wagner, M. Pavlicev, D. M. Weinreich, and R. Mills, “The evolution of phenotypic correlations and "developmental memory",” Evolution 68, 1124–1138 (2014).

[0317] J. Bongard, “Morphological change in machines accelerates the evolution of robust behavior,” Proceedings of the National Academy of Sciences 108, 1234–1239 (2011).

[0318] J. Bongard, C. Laschi, H. Lipson, N. Cheney, and F. Corucci, “Material properties affect evolution’s ability to exploit morphological computation in growing soft-bodied creatures,” in Artificial Life Conference Proceedings (MIT press One Rogers Street,

[0319] Cambridge, MA 02142-1209, USA, 2016) pp. 234–241.

[0320] S. Kriegman, N. Cheney, and J. Bongard, “How morphological development can guide evolution,” Scientific Reports 8, 13934 (2018).

[0321] M. Levin, “Bioelectric networks: the cognitive glue enabling evolutionary scaling from physiology to mind,” Animal Cognition 26, 1865–1891 (2023).

[0322] G. Pezzulo and M. Levin, “Top-down models in biology: explanation and control of complex living systems above the molecular level,” Journal of The Royal Society Interface 13, 20160555 (2016).

[0323] V. Nanos and M. Levin, “Multi-scale chimerism: An experimental window on the algorithms of anatomical control,” Cells & Development 169, 203764 (2022).

[0324] D. Lobo, M. Solano, G. A. Bubenik, and M. Levin, “A linear-encoding model explains the variability of the target morphology in regeneration,” Journal of The Royal Society Interface 11, 20130918 (2014).

[0325] M. Levin, “The Computational Boundary of a “Self”: Developmental Bioelectricity Drives Multicellularity and Scale-Free Cognition,” Frontiers in Psychology 10 (2019), 10.3389 / fpsyg.2019.02688.

[0326] M. Levin, “Technological approach to mind everywhere: An experimentally- grounded framework for understanding diverse bodies and minds,” Frontiers in Systems Neuroscience 16 (2022), 10.3389 / fnsys.2022.768201. 63 QB\166118.01533\95647317.2T002814 166118.01533

[0327] L. Pio-Lopez, J. Bischof, J. V. LaPalme, and M. Levin, “The scaling of goals from cellular to anatomical homeostasis: an evolutionary simulation, experiment and analysis,” Interface Focus 13, 20220072 (2023).

[0328] C. G. Langton, Artificial life: An overview (Mit Press, 1997).

[0329] J. Von Neumann, A. W. Burks, et al., “Theory of self-reproducing automata,” IEEE Transactions on Neural Networks 5, 3–14 (1966).

[0330] M. Games, “The fantastic combinations of John Conway’s new solitaire game “life” by Martin Gardner,” Sci. Am. 223, 120–123 (1970).

[0331] B. W.-C. Chan, “Lenia: Biology of artificial life,” Complex Systems 28, 251–286 (2019).

[0332] X. Li and A. G.-O. Yeh, “Neural-network-based cellular automata for simulating multiple land use changes using gis,” Int. J. Geogr. Inf. Sci. 16, 323–343 (2002).

[0333] A. Mordvintsev, E. Randazzo, E. Niklasson, and M. Levin, “Growing neural cellular automata,” Distill 5 (2020), 10.23915 / distill.00023.

[0334] S. Pontes-Filho, K. Walker, E. Najarro, S. Nichele, and S. Risi, “A unified substrate for body-brain co-evolution,” in From Cells to Societies: Collective Learning across Scales (2022).

[0335] E. Hansen and G. W. Walster, Global Optimization using Interval Analysis: Revised and Expanded, 2nd ed. (CRC Press, 2004).

[0336] L. N. Vandenberg, R. D. Morrie, and D. S. Adams, “V-atpase-dependent ectodermal voltage and pH regionalization are required for craniofacial morphogenesis,” Developmental Dynamics 240, 1889–1904 (2011).

[0337] R. W. Hamming, “Error detecting and error correcting codes,” The Bell System Technical Journal 29, 147–160 (1950).

[0338] C. Wang, D. Sklar, and D. Johnson, “Forward error-correction coding,” Crosslink 3, 26–29 (2001).

[0339] Y. Zhang, M. C. Fontaine, V. Bhatt, S. Nikolaidis, and J. Li, “Arbitrarily scalable environment generators via neural cellular automata,” arXiv preprint arXiv:2310.18622 (2023). 64 QB\166118.01533\95647317.2T002814 166118.01533

[0340] D. E. Rumelhart, G. E. Hinton, and R. J. Williams, “Learning internal representations by error propagation,” in Parallel Distributed Processing: Explorations in the Microstructure of Cognition, Vol. 1: Foundations (MIT Press, Cambridge, MA, USA, 1986) p. 318–362.

[0341] T. W. Hiscock, “Adapting machine-learning algorithms to design gene circuits,” BMC Bioinformatics 20, 214 (2019).

[0342] S. Wolfram, A New Kind of Science (Wolfram Media, 2002).

[0343] M. Cook, “Universality in elementary cellular automata,” Complex Syst. 15 (2004).

[0344] K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural networks 2, 359–366 (1989).

[0345] R. S. Sutton and A. G. Barto, Reinforcement Learning: An Introduction (The MIT Press, 2018).

[0346] Y. Tang and D. Ha, “The sensory neuron as a transformer: Permutation-invariant neural networks for reinforcement learning,” in Advances in Neural Information Processing Systems, edited by A. Beygelzimer, Y. Dauphin, P. Liang, and J. W. Vaughan (2021).

[0347] L. Wolpert, “Chapter 6 positional information and pattern formation,” in Current Topics in Developmental Biology Volume 6 (Elsevier, 1971) p. 183–224.

[0348] J. Sharpe, “Wolpert’s French Flag: what’s the problem?” Development 146, dev185967 (2019).

[0349] M. Gabalda-Sagarra, L. B. Carey, and J. Garcia-Ojalvo, “Recurrence-based information processing in gene regulatory networks,” Chaos: An Interdisciplinary Journal of Nonlinear Science 28, 106313 (2018).

[0350] S. Biswas, S. Manicka, E. Hoel, and M. Levin, “Gene regulatory networks exhibit several kinds of memory: Quantification of memory in biological and random transcriptional networks,” iScience 24, 102131 (2021). 65 QB\166118.01533\95647317.2T002814 166118.01533

[0351] S. Biswas, W. Clawson, and M. Levin, “Learning in transcriptional network models: Computational discovery of pathway-level memory and effective interventions,” International Journal of Molecular Sciences 24, 285 (2022).

[0352] M. Etcheverry, C. Moulin-Frier, P.-Y. Oudeyer, and M. Levin, “Ai-driven automated discovery tools reveal diverse behavioral competencies of biological networks,” (2023), 10.31219 / osf.io / s6thq.

[0353] K. O. Stanley and R. Miikkulainen, “A taxonomy for artificial embryogeny,” Artificial life 9, 93–130 (2003).

[0354] J. Bongard and M. Levin, “There’s plenty of room right here: Biological systems as evolved, overloaded, multi-scale machines,” Biomimetics 8, 110 (2023).

[0355] C. Gershenson and D. Helbing, “When slower is faster,” Complexity 21, 9–15 (2015).

[0356] A. Mordvintsev, E. Randazzo, and C. Fouts, “Growing isotropic neural cellular automata,” in The 2022 Conference on Artificial Life, ALIFE 2022 (MIT Press, 2022).

[0357] J. Sohl-Dickstein, E. Weiss, N. Maheswaranathan, and S. Ganguli, “Deep unsupervised learning using nonequilibrium thermodynamics,” in Proceedings of the 32nd International Conference on Machine Learning, Proceedings of Machine Learning Research, Vol. 37, edited by F. Bach and D. Blei (PMLR, Lille, France, 2015) pp. 2256– 2265.

[0358] J. Ho, A. Jain, and P. Abbeel, “Denoising diffusion probabilistic models,” Advances in neural information processing systems 33, 6840–6851 (2020).

[0359] A. Q. Nichol and P. Dhariwal, “Improved denoising diffusion probabilistic models,” in Proceedings of the 38th International Conference on Machine Learning, Proceedings of Machine Learning Research, Vol. 139, edited by M. Meila and T. Zhang (PMLR, 2021) pp. 8162–8171.

[0360] P. Dhariwal and A. Nichol, “Diffusion models beat gans on image synthesis,” in Advances in Neural Information Processing Systems, Vol. 34, edited by M. Ranzato,

[0361] A. Beygelzimer, Y. Dauphin, P. Liang, and J. W. Vaughan (Curran Associates, Inc., 2021) pp. 8780–8794. 66 QB\166118.01533\95647317.2T002814 166118.01533

[0362] J. Song, C. Meng, and S. Ermon, “Denoising diffusion implicit models,” (2022), arXiv:2010.02502 [cs.LG].

[0363] D. Savic, B. Belintzev, L. Beloussov, and A. Zaraisky, “Morphogenetic activity prepattern in embryonic epithelia,” Progress in clinical and biological research 217A, 101—104 (1986).

[0364] P. Hunt and R. Krumlauf, “Deciphering the hox code: Clues to patterning branchial regions of the head,” Cell 66, 1075–1078 (1991).

[0365] H. L. Ashe and J. Briscoe, “The interpretation of morphogen gradients,” Development 133, 385–394 (2006).

[0366] M. Levin and C. J. Martyniuk, “The bioelectric code: An ancient computational medium for dynamic control of growth and form,” Biosystems 164, 76–93 (2018).

[0367] R. B. Palm, M. G. Duque, S. Sudhakaran, and S. Risi, “Variational neural cellular automata,” in International Conference on Learning Representations (2022).

[0368] K. O. Stanley, “Compositional pattern producing networks: A novel abstraction of development,” Genetic Programming and Evolvable Machines 8, 131–162 (2007).

[0369] B. G. Woolley and K. O. Stanley, “On the deleterious effects of a priori objectives on evolution and representation,” in Proceedings of the 13th annual conference on Genetic and evolutionary computation, GECCO ’11 (ACM, 2011).

[0370] B. Mildenhall, P. P. Srinivasan, M. Tancik, J. T. Barron, R. Ramamoorthi, and R. Ng, “Nerf: representing scenes as neural radiance fields for view synthesis,” Commun. ACM 65, 99–106 (2021).

[0371] T. Parr, G. Pezzulo, and K. J. Friston, Active Inference: The Free Energy Principle in Mind, Brain, and Behavior (The MIT Press, 2022).

[0372] D. Ha and J. Schmidhuber, “Recurrent world models facilitate policy evolution,” in Advances in Neural Information Processing Systems, Vol. 31, edited by S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett (Curran Associates, Inc., 2018). 67 QB\166118.01533\95647317.2T002814 166118.01533

[0373] S. Pontes-Filho, P. G. Lind, and S. Nichele, “Assessing the robustness of critical behavior in stochastic cellular automata,” Physica D: Nonlinear Phenomena 441, 133507 (2022).

[0374] J. M. Beggs, “The criticality hypothesis: how local cortical networks might optimize information processing,” Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 366, 329–343 (2008).

[0375] W. L. Shew and D. Plenz, “The functional benefits of criticality in the cortex,” The Neuroscientist 19, 88–100 (2012).

[0376] W. L. Shew, W. P. Clawson, J. Pobst, Y. Karimipanah, N. C. Wright, and R. Wessel, “Adaptation to sensory input tunes visual cortex to criticality,” Nature Physics 11, 659– 663 (2015).

[0377] W. P. Clawson, N. C. Wright, R. Wessel, and W. L. Shew, “Adaptation towards scale- free dynamics improves cortical stimulus discrimination at the cost of reduced detection,” PLOS Computational Biology 13, e1005574 (2017).

[0378] F. Habibollahi, B. J. Kagan, A. N. Burkitt, and C. French, “Critical dynamics arise during structured information presentation within embodied in vitro neuronal networks,” Nature Communications 14 (2023), 10.1038 / s41467-023-41020-3.

[0379] S. A. Jones, J. H. Barfield, V. K. Norman, and W. L. Shew, “Scale-free behavioral dynamics directly linked with scale-free cortical dynamics,” eLife 12, e79950 (2023).

[0380] A. D. Wissner-Gross and C. E. Freer, “Causal entropic forces,” Phys. Rev. Lett. 110, 168702 (2013).

[0381] L. Pio-Lopez and M. Levin, “Aging as a morphostasis defect: a developmental bioelectricity perspective,” (2023), 10.31219 / osf.io / wkhx4.

[0382] W. B. Miller Jr, F. Baluška, A. S. Reber, and P. Slijepcˇevic´, “Why death and aging ? all memories are imperfect,” Progress in Biophysics and Molecular Biology (2024), https: / / doi.org / 10.1016 / j.pbiomolbio.2024.02.001.

[0383] W. S. McCulloch and W. Pitts, “A logical calculus of the ideas immanent in nervous activity,” The Bulletin of Mathematical Biophysics 5, 115–133 (1943). 68 QB\166118.01533\95647317.2T002814 166118.01533

[0384] M. L. Minsky, Theory of neural-analog reinforcement systems and its application to the brain-model problem (Princeton University, 1954).

[0385] M. Minsky, “Steps toward artificial intelligence,” Proceedings of the IRE 49, 8– 30 (1961).

[0386] F. Rosenblatt et al., Principles of neurodynamics: Perceptrons and the theory of brain mechanisms, Vol. 55 (Spartan books Washington, DC, 1962).

[0387] M. Minsky and S. Papert, “Perceptron: an introduction to computational geometry,” The MIT Press, Cambridge, expanded edition 19, 2 (1969).

[0388] D. E. Rumelhart, G. E. Hinton, and R. J. Williams, “Learning representations by back-propagating errors,” nature 323, 533–536 (1986).

[0389] Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521, 436–444 (2015).

[0390] I. Goodfellow, Y. Bengio, and A. Courville, Deep Learning (MIT Press, 2016).

[0391] A. Paszke, S. Gross, F. Massa, A. Lerer, J. Bradbury, G. Chanan, T. Killeen, Z. Lin, N. Gimelshein, L. Antiga, A. Desmaison, A. Kopf, E. Yang, Z. DeVito, M. Raison, A. Tejani, S. Chilamkurthy, B. Steiner, L. Fang, J. Bai, and S. Chintala, “PyTorch: An imperative style, high-performance deep learning library,” in Advances in Neural Information Processing Systems 32 (Curran Associates, Inc., 2019) pp. 8024–8035.

[0392] N. Hansen, yoshihikoueno, ARF1, G. Kadlecová, K. Nozawa, L. Rolshoven, M. Chan,Y. Akimoto, brieglhostis, and D. Brockhoff, “Cma-es / pycma: r3.3.0,” (2023).

[0393] Example 2: Illustrative Embodiments of Methods and Systems Described Herein

[0394] FIG. 12 shows a flow chart of an exemplary method of modeling morphogenesis. At step 1202, a neural cellular automata (NCA) is constructed. The NCA may include a plurality of cells and each cell in the plurality of cells has an assigned state and obeys local update rules. At step 1204, the NCA may be initialized to reach a target pattern by selecting a set of parameters. At step 1206, the method may characterize the evolutionary process of the initialized NCA. In some embodiments, the evolutionary process may occur while the NCA is developing 69 QB\166118.01533\95647317.2T002814 166118.01533 to reach a target pattern based on the local update rules and the set of parameters. At step 1208, the method may output a report of the characterized evolutionary process to a user. The report may allow the user to relate the evolutionary process to morphogenetic pattern formation.

[0395] In FIG. 13, an example 1300 of a system (e.g., a data processing system) for characterizing a protein in accordance with some embodiments of the disclosed subject matter is shown.

[0396] In some embodiments, computing device 1304 and / or server 1316 can be any suitable computing device or combination of devices, such as a desktop computer, a laptop computer, a smartphone, a tablet computer, a wearable computer, a server computer, a virtual machine being executed by a physical computing device, etc. As described herein, system 1000 can present information about the characterized protein to a user (e.g., a researcher and / or a physician).

[0397] In some embodiments, communication network 1302 can be any suitable communication network or combination of communication networks. In some embodiments, communication network 1302 can be any suitable communication network or combination of communication networks. For example, communication network 1302 can include a Wi-Fi network (which can include one or more wireless routers, one or more switches, etc.), a peer-to- peer network (e.g., a Bluetooth network), a cellular network (e.g., a 4G network, a 5G network, etc., complying with any suitable standard, such as CDMA, GSM, LTE, LTE Advanced, WiMAX, etc.), a wired network, etc. In some embodiments, communication network 1002 can be a local area network, a wide area network, a public network (e.g., the Internet), a private or semi-private network (e.g., a corporate or university intranet), any other suitable type of network, or any suitable combination of networks. Communications links shown in FIG.13 can each be any suitable communications link or combination of communications links, such as wired links, fiber optic links, Wi-Fi links, Bluetooth links, cellular links, etc.

[0398] FIG. 13 additionally shows an example of hardware that can be used to implement computing device 1304 and server 1316 in accordance with some embodiments of the disclosed subject matter. In some embodiments, computing device 1304 can be used to execute one or more set of instructions to identify a behavioral catalog. In other embodiments, computing device 1304 can be used to identify therapeutic interventions. In still other embodiments, 70 QB\166118.01533\95647317.2T002814 166118.01533 computing device 1304 can be used to identify a configuration of parameter of a gene regulatory network to perform a desired function.

[0399] As shown in FIG. 13, computing device 1304 can include one or more hardware processor 1306, one or more displays 1308, one or more inputs 1310, one or more communications 1312, and / or memory 1314. In some embodiments, processor 1306 can be any suitable hardware processor or combination of processors, such as central processing unit, a graphics processing unit, etc. In some embodiments, display 1308 can include any suitable display devices, such as a computer monitor, a touchscreen, a television, etc. In some embodiments, inputs 1310 can include any suitable input device and / or sensors that can be used to receive user input, such as a keyboard, a mouse, a touchscreen, a microphone, etc.

[0400] In some embodiments, communication systems 1312 can include any suitable hardware, firmware, and / or software for communicating information over communication network 1302 and / or any other suitable communication networks. For example, communications systems 1312 can include one or more transceivers, one or more communication chips and / or chip sets, etc. In a more particular example, communications systems 1312 can include hardware, firmware and / or software that can be used to establish a Wi-Fi connection, a Bluetooth connection, a cellular connection, an Ethernet connection, etc.

[0401] In some embodiments, memory 1314 can include any suitable storage device or devices that can be used to store instructions, values, etc., that can be used, for example, by processor 1306 to present content using display 1308, to communicate with server 1316 via communications system(s) 1312, etc.

[0402] Memory 1314 can include any suitable volatile memory, non-volatile memory, storage, or any suitable combination thereof. For example, memory 1314 can include RAM, ROM, EEPROM, one or more flash drives, one or more hard disks, one or more solid state drives, one or more optical drives, etc. In some embodiments, memory 1314 can have encoded thereon a computer program for controlling operation of computing device 1304. In such embodiments, processor 1306 can execute at least a portion of the computer program to present content (e.g., images, user interfaces, graphics, tables, etc.), receive content from server 1316, transmit information to server 1316, etc. 71 QB\166118.01533\95647317.2T002814 166118.01533

[0403] In some embodiments, server 1316 can include a processor 1318, a display 1320, one or more inputs 1322, one or more communications systems 1324, and / or memory 1326. In some embodiments, processor 1318 can be any suitable hardware processor or combination of processors, such as a central processing unit, a graphics processing unit, etc. In some embodiments, display 1320 can include any suitable display devices, such as a computer monitor, a touchscreen, a television, etc. In some embodiments, inputs 1322 can include any suitable input devices and / or sensors that can be used to receive user input, such as a keyboard, a mouse, a touchscreen, a microphone, etc.

[0404] In some embodiments, communications systems 1324 can include any suitable hardware, firmware, and / or software for communicating information over communication network 1302 and / or any other suitable communication networks. For example, communications systems 1324 can include one or more transceivers, one or more communication chips and / or chip sets, etc. In a more particular example, communications systems 1324 can include hardware, firmware and / or software that can be used to establish a Wi-Fi connection, a Bluetooth connection, a cellular connection, an Ethernet connection, etc.

[0405] In some embodiments, memory 1326 can include any suitable storage device or devices that can be used to store instructions, values, etc., that can be used, for example, by processor 1318 to present content using display 1320, to communicate with one or more computing devices 1304, etc. Memory 1326 can include any suitable volatile memory, non- volatile memory, storage, or any suitable combination thereof. For example, memory 1326 can include RAM, ROM, EEPROM, one or more flash drives, one or more hard disks, one or more solid state drives, one or more optical drives, etc. In some embodiments, memory 1326 can have encoded thereon a server program for controlling operation of server 1316. In such embodiments, processor 1318 can execute at least a portion of the server program to transmit information and / or content (e.g., results of a tissue identification and / or classification, a user interface, etc.) to one or more computing devices 1304, receive information and / or content from one or more computing devices 1304, receive instructions from one or more devices (e.g., a personal computer, a laptop computer, a tablet computer, a smartphone, etc.), etc.

[0406] In some embodiments, any suitable computer readable media can be used for storing instructions for performing the functions and / or processes described herein. For example, 72 QB\166118.01533\95647317.2T002814 166118.01533 in some embodiments, computer readable media can be transitory or non-transitory. For example, non-transitory computer readable media can include media such as magnetic media (such as hard disks, floppy disks, etc.), optical media (such as compact discs, digital video discs, Blu-ray discs, etc.), semiconductor media (such as RAM, Flash memory, electrically programmable read only memory (EPROM), electrically erasable programmable read only memory (EEPROM), etc.), any suitable media that is not fleeting or devoid of any semblance of permanence during transmission, and / or any suitable tangible media. As another example, transitory computer readable media can include signals on networks, in wires, conductors, optical fibers, circuits, or any suitable media that is fleeting and devoid of any semblance of permanence during transmission, and / or any suitable intangible media.

[0407] A number of references to patent and non-patent documents are made throughout the publication, each of which is herein incorporated by reference in its entirety.

[0408] While the invention has been described above in connection with particular embodiments and examples, the invention is not necessarily so limited, and that numerous other embodiments, examples, uses, modifications and departures from the embodiments, examples and uses are intended to be encompassed by the claims attached hereto. 73 QB\166118.01533\95647317.2

Claims

T002814 166118.01533 CLAIMS What is claimed is:

1. A method of modeling a morphogenetic process, comprising: constructing a neural cellular automata (NCA), wherein the NCA comprises a plurality of cells and each cell in the plurality of cells has an assigned state and obeys local update rules; initializing the NCA to reach a target pattern by selecting a set of parameters; characterizing an evolutionary process of the initialized NCA developing to reach the target pattern based on the local update rules and the set of parameters; and outputting to a user a report of the characterized evolutionary process of the NCA, wherein the report allows the user to relate the evolutionary process to morphogenetic pattern formation.

2. The method of claim 1, wherein the target pattern comprises a desired position for each cell in the plurality of cells based on the cell’s assigned state.

3. The method of claim 1 or 2, wherein the local update rules are controlled by artificial neural networks, wherein the artificial neural networks comprise partitions of sensory components, neighbor-wise sensory embedding components, and controller components.

4. The method of any one of the preceding claims, wherein a state of a given cell in the plurality of cells is determined by a previous state of the given cell, and another previous state of one or more adjacent cells.

5. The method of any one of the preceding claims, wherein the evolutionary process is implemented by an evolutionary algorithm. 74 QB\166118.01533\95647317.2T002814 166118.01533 6. The method of any one of the preceding claims, wherein the selected set of parameters directly encodes phenotypic features of a target pattern.

7. The method of any one of the preceding claims, wherein the selected set of parameters encode cellular competencies.

8. The method of any one of the preceding claims, wherein the NCA evolves an intrinsic signaling mechanism to perform a task of interest.

9. A system for modeling a morphogenetic process, comprising: a non-transitory computer-readable medium having stored thereon instructions that, when executed by a processor, cause the processor to: construct a neural cellular automata (NCA), wherein the NCA comprises a plurality of cells and each cell in the plurality of cells has an assigned state and obeys local update rules; initialize the NCA to reach a target pattern by selecting a set of parameters; characterize an evolutionary process of the initialized NCA developing to reach the target pattern based on the local update rules and the set of parameters; and output to a user a report of the characterized evolutionary process of the NCA, wherein the report allows the user to relate the evolutionary process to morphogenetic pattern formation.

10. The system of claim 9, wherein the target pattern comprises a desired position for each cell in the plurality of cells based on the cell’s assigned state. 75 QB\166118.01533\95647317.2T002814 166118.01533 11. The system of claim 9 or 10, wherein the local update rules are controlled by artificial neural networks, wherein the artificial neural networks comprise partitions of sensory components, neighbor-wise sensory embedding components, and controller components.

12. The system of any one of claims 9-11, wherein a state of a given cell in the plurality of cells is determined by a previous state of the given cell, and another previous state of one or more adjacent cells.

13. The system of any one of claims 9-12, wherein the evolutionary process is implemented by an evolutionary algorithm.

14. The system of any one of claims 9-13, wherein the selected set of parameters directly encodes phenotypic features of a target pattern.

15. The system of any one of claims 9-14, wherein the selected set of parameters encode cellular competencies.

16. The system of any one of claims 9-15, wherein the NCA evolves an intrinsic signaling mechanism to perform a task of interest.

17. A non-transitory computer-readable medium having stored thereon instructions that, when executed by a processor, cause the processor to: construct a neural cellular automata (NCA), wherein the NCA comprises a plurality of cells and each cell in the plurality of cells has an assigned state and obeys local update rules; initialize the NCA to reach a target pattern by selecting a set of parameters; 76 QB\166118.01533\95647317.2T002814 166118.01533 characterize an evolutionary process of the initialized NCA developing to reach the target pattern based on the local update rules and the set of parameters; and output to a user a report of the characterized evolutionary process of the NCA, wherein the report allows the user to relate the evolutionary process to morphogenetic pattern formation.

18. The non-transitory computer-readable medium of claim 17, wherein the target pattern comprises a desired position for each cell in the plurality of cells based on the cell’s assigned state.

19. The non-transitory computer-readable medium of claim 17 or 18, wherein the local update rules are controlled by artificial neural networks, wherein the artificial neural networks comprise partitions of sensory components, neighbor-wise sensory embedding components, and controller components.

20. The non-transitory computer-readable medium of any one of claims 17-19, wherein a state of a given cell in the plurality of cells is determined by a previous state of the given cell, and another previous state of one or more adjacent cells.

21. The non-transitory computer-readable medium of any one of claims 17-20, wherein the evolutionary process is implemented by an evolutionary algorithm.

22. The non-transitory computer-readable medium of any one of claims 17-21, wherein the selected set of parameters directly encodes phenotypic features of a target pattern.

23. The non-transitory computer-readable medium of any one of claims 17-122, wherein the selected set of parameters encode cellular competencies. 77 QB\166118.01533\95647317.2T002814 166118.01533 24. The non-transitory computer-readable medium of any one of claims 17-23, wherein the NCA evolves an intrinsic signaling mechanism to perform a task of interest. 78 QB\166118.01533\95647317.2

Images