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Morphogenesis is the generation of structural patterns through a dynamic process. The mathematical basis of morphogenesis has long been studied with the key initial work by Alan Turing. This paper explores the consequences of a circuit basis for reaction-diffusion systems on morphogenesis, including the reachability of patterns and the logical basis of pattern stability. We consider how morphological patterns can arise through iterative computation and produce robust forms. Through an exhaustive analysis of reaction-diffusion dynamics in a minimal model of morphogenesis, we show how the stability and reachability of morphologies are influenced by their circuit basis. We show that this model exhibits similar behavior to the recently experimentally observed dynamics not accounted for by Turing's original model. We conclude by presenting an additional class of metastable patterns exhibited in this model.

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... As helpful as the model is in hypothesizing high-level principles, they do not elucidate the underlying biological complexity. Models based on reaction-diffusion or artificial neural networks, on the other hand, capture some of the biological complexity but suffer from the drawback that it is difficult to elucidate the underlying nonlinear causal relationships using conventional analysis techniques [40,41]. ...

What information-processing strategies and general principles are sufficient to enable self-organized morphogenesis in embryogenesis and regeneration? We designed and analyzed a minimal model of self-scaling axial patterning consisting of a cellular network that develops activity patterns within implicitly set bounds. The properties of the cells are determined by internal ‘genetic’ networks with an architecture shared across all cells. We used machine-learning to identify models that enable this virtual mini-embryo to pattern a typical axial gradient while simultaneously sensing the set boundaries within which to develop it from homogeneous conditions—a setting that captures the essence of early embryogenesis. Interestingly, the model revealed several features (such as planar polarity and regenerative re-scaling capacity) for which it was not directly selected, showing how these common biological design principles can emerge as a consequence of simple patterning modes. A novel “causal network” analysis of the best model furthermore revealed that the originally symmetric model dynamically integrates into intercellular causal networks characterized by broken-symmetry, long-range influence and modularity, offering an interpretable macroscale-circuit-based explanation for phenotypic patterning. This work shows how computation could occur in biological development and how machine learning approaches can generate hypotheses and deepen our understanding of how featureless tissues might develop sophisticated patterns—an essential step towards predictive control of morphogenesis in regenerative medicine or synthetic bioengineering contexts. The tools developed here also have the potential to benefit machine learning via new forms of backpropagation and by leveraging the novel distributed self-representation mechanisms to improve robustness and generalization.

... The object that emerges from collective fabrication thus participates in its own development, since it participates passively in the construction (Michener, 1974). Turing (1952) proposed the concept of "morphogen" and integrates, in a mathematical model, in the form of non-linear differential equations, biochemical data, more particularly from enzymology and solution physics (Harrington, 2016;Lesne, 2012;Szalai et al., 2012). Turing demonstrates that, starting from a homogeneous medium, one can arrive at a periodic spatial distribution of the concentration of these morphogens. ...

Additive manufacturing is based on a deterministic principle of spatially localized material transformation. By using spatial energy resolution modes, it is thus possible to produce an object that meets a set of specifications in a step-by-step manner. This paper attempts to take the opposite view by examining whether it is interesting to use knowledge on “morphogenesis” (cf. Alan Turing) and other conceptual ideas of spontaneous self-organization to achieve a desired shape. This new and more systemic form of 3D/4D printing enables the self-assembly and establishment of elements (voxels) in various environments through the use of programmable or smart matter. The necessary convergence of interacting elements requires a related decisive activity, however, to the sign of unknown, indecision, complexity, unpredictable bifurcations, in short of scientific promise that cannot be robustly kept. Beyond this frame of generalized non-decidability—a form of abandonment of self-organization paradigm in favour of deterministic doctrines—, there are paths to partial success where desired spatial transformations can be created through (mastered?) disorder. The paper presents the conceptual bases of 3D self-organization with some illustrative examples. The objective is to highlight the interests of this disruptive vision with its realistic limits.

... The limits of phenomena that can be understood from the perspective of biological circuit engineering have yet to be discovered [65]. The field and future of synthetic biology is well beyond the scope of this review, and so we direct the interested reader to some of the extensive reviews of the field [86,24,55]. ...

In nature, gene regulatory networks are a key mediator between the information stored in the DNA of living organisms (their genotype) and the structural and behavioral expression this finds in their bodies, surviving in the world (their phenotype). They integrate environmental signals, steer development, buffer stochasticity, and allow evolution to proceed. In engineering, modeling and implementations of artificial gene regulatory networks have been an expanding field of research and development over the past few decades. This review discusses the concept of gene regulation, describes the current state of the art in gene regulatory networks, including modeling and simulation, and reviews their use in artificial evolutionary settings. We provide evidence for the benefits of this concept in natural and the engineering domains.

... Louise Magbunduku and Kyle Harrington showed how swarm intelligence can be used to build structures and, ultimately, liveable habitats; this may be further advanced in the architectural morphogenesis and growing building via reaction-diffusion, Figure 1 (Harrington, 2016). Pavlina Vardoulaki showed how units that communicate and collaborate as swarms could in effect efficiently build simple to complicated structures. ...

Imagine evolving swarms of robots interacting and by doing so reshaping and cultivating our habitat. This habitat could be here on Earth, on a distant planet or moon, or within a self-contained spacecraft. What would these robots look like and made of what type of material? What kind of information, hardware or software? What are the architectural necessities? There are many open questions when trying to envision the future of architecture; but, in this particular workshop, the goal was not only to imagine the future but also to create it. With this particular goal in mind, the Living Architecture workshop at European Conference on Artificial Life (ECAL) 2017 brought together practitioners from the sciences and architecture to share ideas and technologies to examine possible paths forward. Living Architecture is a specific substantiation of the broader notion of Living Technology where living systems or artificial systems with life-like properties are developed towards technological applications. In Living Architecture, objects designed in the built environment would contain living systems as part of their functionality (such as bioreactors for energy) or artificial distributed systems with feedback loops (such as neural networks or artificial intelligence). In this way, Living Architecture represents a congruence in functionality and form between living systems, technology and architecture.

Development as it occurs in biological organisms is defined as the process of gene activity that directs a sequence of cellular events in an organism which brings about the profound changes that occur to the organism. Hence, the many chemical and physical processes which translate the vast genetic information gathered over the evolutionary history of an organism, and put it to use to create a fully formed, viable adult organism from a single cell, is subsumed under the term “development”. This also includes properties of development that go way beyond the formation of organisms such as, for instance, mechanisms that maintain the stability and functionality of an organism throughout its lifetime, and properties that make development an adaptive process capable of shaping an organism to match—within certain bounds—the conditions and requirements of a given environment. Considering these capabilities from a computer science or engineering angle quickly leads on to ideas of taking inspiration from biological examples and translating their capabilities, generative construction, resilience and the ability to adapt, to man-made systems. The aim is thereby to create systems that mimic biology sufficiently so that these desired properties are emergent, but not as excessively as to make the construction or operation of a system infeasible as a result of complexity or implementation overheads. Software or hardware processes aiming to achieve this are referred to as artificial developmental models. This chapter therefore focuses on motivating the use of artificial development, provides an overview of existing models and a recipe for creating them, and discusses two example applications of image processing and robot control.

Cistanches Herba (CH) is thought to be a "Yang-invigorating" material in traditional Chinese medicine. We evaluated neuroprotective effects of Cistanches Herba on Alzheimer's disease (AD) patients. Moderate AD participants were divided into 3 groups: Cistanches Herba capsule (CH, n = 10), Donepezil tablet (DON, n = 8), and control group without treatment (n = 6). We assessed efficacy by MMSE and ADAS-cog, and investigated the volume changes of hippocampus by 1.5 T MRI scans. Protein, mRNA levels, and secretions of total-tau (T-tau), tumor necrosis factor-α (TNF-α), and interleukin- (IL) 1β (IL-1β) in cerebrospinal fluid (CSF) were detected by Western blot, RT-PCR, and ELISA. The scores showed statistical difference after 48 weeks of treatment compared to control group. Meanwhile, volume changes of hippocampus were slight in drug treatment groups but distinct in control group; the levels of T-tau, TNF-α, and IL-1β were decreased compared to those in control group. Cistanches Herba could improve cognitive and independent living ability of moderate AD patients, slow down volume changes of hippocampus, and reduce the levels of T-tau, TNF-α, and IL-1β. It suggested that Cistanches Herba had potential neuroprotective effects for moderate AD.

The Belousov-Zhabotinsky (BZ) reaction has become the prototype of nonlinear chemical dynamics. Microfluidic techniques provide a convenient method for emulsifying BZ solutions into monodispersed drops with diameters of tens to hundreds of microns, providing a unique system in which reaction-diffusion theory can be quantitatively tested. In this work, we investigate monolayers of microfluidically generated BZ drops confined in close-packed two-dimensional (2D) arrays through experiments and finite element simulations. We describe the transition from oscillatory to stationary chemical states with increasing coupling strength, controlled by independently varying the reaction chemistry within a drop and diffusive flux between drops. For stationary drops, we studied how the ratio of stationary oxidized to stationary reduced drops varies with coupling strength. In addition, using simulation, we quantified the chemical heterogeneity sufficient to induce mixed stationary and oscillatory patterns.
Graphical abstract

Significance
Turing proposed that intercellular reaction-diffusion of molecules is responsible for morphogenesis. The impact of this paradigm has been profound. We exploit an abiological experimental system of emulsion drops containing the Belousov–Zhabotinsky reactants ideally suited to test Turing’s theory. Our experiments verify Turing’s thesis of the chemical basis of morphogenesis and reveal a pattern, not previously predicted by theory, which we explain by extending Turing’s model to include heterogeneity. Quantitative experimental results obtained using this artificial cellular system establish the strengths and weaknesses of the Turing model, applicable to biology and materials science alike, and pinpoint which directions are required for improvement.

We consider a new cellular automata rule for a synchronous random walk on a two-dimensional square lattice, subject to an exclusion principle. It is found that the macroscopic behavior of our model obeys the telegraphists's equation, with an adjustable diffusion constant. By construction, the dynamics of our model is exactly described by a linear discrete Boltzmann equation which is solved analytically for some boundary conditions. Consequently, the connection between the microscopic and the macroscopic descriptions is obtained exactly and the continuous limit studied rigorously. The typical system size for which a true diffusive behavior is observed may be deduced as a function of the parameters entering into the rule. It is shown that a suitable choice of these parameters allows us to consider quite small systems. In particular, our cellular automata model can simulate the Laplace equation to a precision of the order (/L)6, whereL is the size of the system and the lattice spacing. Implementation of this algorithm on special-purpose machines leads to the fastest way to simulate diffusion on a lattice.

From an engineering point of view, the problem of coordinating a set of autonomous, mobile robots for the purpose of cooperatively performing a task has been studied extensively over the past decade. In contrast, in this paper we aim to understand the fundamental algorithmic limitations on what a set of autonomous mobile robots can or cannot achieve. We therefore study a hard task for a set of weak robots. The task is for the robots in the plane to form any arbitrary pattern that is given in advance. This task is fundamental in the sense that if the robots can form any pattern, they can agree on their respective roles in a subsequent, coordinated action. The robots are weak in several aspects. They are anonymous; they cannot explicitly communicate with each other, but only observe the positions of the others; they cannot remember the past; they operate in a very strong form of asynchronicity.We show that the tasks that such a system of robots can perform depend strongly on their common agreement about their environment, i.e. the readings of their environment sensors. If the robots have no common agreement about their environment, they cannot form an arbitrary pattern. If each robot has a compass needle that indicates North (the robot world is a flat surface, and compass needles are parallel), then any odd number of robots can form an arbitrary pattern, but an even number cannot (in the worst case). If each robot has two independent compass needles, say North and East, then any set of robots can form any pattern.

We study a binary-cell-state eight-cell neighborhood two-dimensional cellular automaton model of a quasi-chemical system with a substrate and a reagent. Reactions are represented by semi-totalistic transitions rules: every cell switches from state 0 to state 1 depending on if the sum of neighbors in state 1 belongs to some specified interval, cell remains in state 1 if the sum of neighbors in state 1 belong to another specified interval. We investigate space-time dynamics of 1296 automata, establish morphology-bases classification of the rules, explore precipitating and excitatory cases and scrutinize collisions between mobile and stationary localizations (gliders, cycle life and still-life compact patterns). We explore reaction–diffusion like patterns produced as a result of collisions between localizations. Also, we propose a set of rules with complex behavior called Life 2c22.

A method for evolving a developmental program inside a cell to create multicellular organisms of arbitrary size and characteristics is described. The cell genotype is evolved so that the organism will organize itself into well defined patterns of differentiated cell types (e.g. the French Flag). In addition the cell genotypes are evolved to respond appropriately to environmental signals that cause metamorphosis of the whole organism. A number of experiments are described that show that the organisms exhibit emergent properties of self-repair and adaptation.

We present a lattice-gas automaton approach to coupled reaction-diffusion equations. This approach provides a microscopic bases for exploring systems which exhibit such interesting features as oscillatory behavior and pattern formation. Two-species systems are analyzed in detail. As an application of the formalism, we construct the microscopic dynamics for a system described by the Maginu equations; simulation results show excellent agreement with the phenomenological predictions. Most important is the result showing that we obtain Turing-type structures by a purely microscopic approach.

A central focus of postgenomic research will be to understand how cellular phenomena arise from the connectivity of genes and proteins. This connectivity generates molecular network diagrams that resemble complex electrical circuits, and a systematic understanding will require the development of a mathematical framework for describing the circuitry. From an engineering perspective, the natural path towards such a framework is the construction and analysis of the underlying submodules that constitute the network. Recent experimental advances in both sequencing and genetic engineering have made this approach feasible through the design and implementation of synthetic gene networks amenable to mathematical modelling and quantitative analysis. These developments have signalled the emergence of a gene circuit discipline, which provides a framework for predicting and evaluating the dynamics of cellular processes. Synthetic gene networks will also lead to new logical forms of cellular control, which could have important applications in functional genomics, nanotechnology, and gene and cell therapy.

It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogeneous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances. Such reaction-diffusion systems are considered in some detail in the case of an isolated ring of cells, a mathematically convenient, though biologically unusual system. The investigation is chiefly concerned with the onset of instability. It is found that there are six essentially different forms which this may take. In the most interesting form stationary waves appear on the ring. It is suggested that this might account, for instance, for the tentacle patterns on Hydra and for whorled leaves. A system of reactions and diffusion on a sphere is also considered. Such a system appears to account for gastrulation. Another reaction system in two dimensions gives rise to patterns reminiscent of dappling. It is also suggested that stationary waves in two dimensions could account for the phenomena of phyllotaxis. The purpose of this paper is to discuss a possible mechanism by which the genes of a zygote may determine the anatomical structure of the resulting organism. The theory does not make any new hypotheses; it merely suggests that certain well-known physical laws are sufficient to account for many of the facts. The full understanding of the paper requires a good knowledge of mathematics, some biology, and some elementary chemistry. Since readers cannot be expected to be experts in all of these subjects, a number of elementary facts are explained, which can be found in text-books, but whose omission would make the paper difficult reading.

Proto-organisms probably were randomly aggregated nets of chemical reactions. The hypothesis that contemporary organisms are also randomly constructed molecular automata is examined by modeling the gene as a binary (on-off) device and studying the behavior of large, randomly constructed nets of these binary "genes". The results suggest that, if each "gene" is directly affected by two or three other "genes", then such random nets: behave with great order and stability; undergo behavior cycles whose length predicts cell replication time as a function of the number of genes per cell; possess different modes of behavior whose number per net predicts roughly the number of cell types in an organism as a function of its number of genes; and under the stimulus of noise are capable of differentiating directly from any mode of behavior to at most a few other modes of behavior. Cellular differentation is modeled as a Markov chain among the modes of behavior of a genetic net. The possibility of a general theory of metabolic behavior is suggested.

Cellular automata are used as simple mathematical models to investigate self-organization in statistical mechanics. A detailed analysis is given of "elementary" cellular automata consisting of a sequence of sites with values 0 or 1 on a line, with each site evolving deterministically in discrete time steps according to definite rules involving the values of its nearest neighbors. With simple initial configurations, the cellular automata either tend to homogeneous states, or generate self-similar patterns with fractal dimensions $\simeq${} 1.59 or $\simeq${} 1.69. With "random" initial configurations, the irreversible character of the cellular automaton evolution leads to several self-organization phenomena. Statistical properties of the structures generated are found to lie in two universality classes, independent of the details of the initial state or the cellular automaton rules. More complicated cellular automata are briefly considered, and connections with dynamical systems theory and the formal theory of computation are discussed.

The novelty of this paper is the study of emergence of diffusion-induced (Turing-like) patterns from a microscopic point of view, namely, in terms of cellular automata. Formally, the cellular automaton model is described in lattice-gas terminology [H. Bussemaker, A. Deutsch, E. Geigant, Phys. Rev. Lett. 78 (1997) 5018–5021]. The automaton rules capture in abstract form the essential ideas of activator–inhibitor interactions of biological systems. In spite of the automaton’s simplicity, self-organised formation of stationary spatial patterns emerging from a randomly perturbed uniform state is observed. Fourier analysis of approximate mean-field kinetic difference equations [J.P. Boon, D. Dab, R. Kapral, A.T. Lawniczak, Phys. Rep. 273 (1996) 55–147] yields a critical wave length and a “Turing condition” for the onset of pattern formation.

In a system in which anonymous mobile robots repeatedly execute a “Look–Compute–Move” cycle, a robot is said to be oblivious if it has no memory to store its observations in the past, and hence its move depends only on the current observation. This paper considers the pattern formation problem in such a system, and shows that oblivious robots can form any pattern that non-oblivious robots can form, except that two oblivious robots cannot form a point while two non-oblivious robots can. Therefore, memory does not help in forming a pattern, except for the case in which two robots attempt to form a point. Related results on the pattern convergence problem are also presented.

To construct sophisticated biochemical circuits from scratch, one needs to understand how simple the building blocks can be
and how robustly such circuits can scale up. Using a simple DNA reaction mechanism based on a reversible strand displacement
process, we experimentally demonstrated several digital logic circuits, culminating in a four-bit square-root circuit that
comprises 130 DNA strands. These multilayer circuits include thresholding and catalysis within every logical operation to
perform digital signal restoration, which enables fast and reliable function in large circuits with roughly constant switching
time and linear signal propagation delays. The design naturally incorporates other crucial elements for large-scale circuitry,
such as general debugging tools, parallel circuit preparation, and an abstraction hierarchy supported by an automated circuit
compiler.

Fas-zillierend: Die durch Brom vermittelte inhibitorische Kopplung zwischen wässrigen Nanolitertröpfchen, die die Belousov-Zhabotinsky(BZ)-Reaktionslösung enthalten, erzeugt gegenphasige Oszillationen und stationäre Turing-Muster (siehe Bild; rot: reduzierter, blau: oxidierter Katalysator). Durch Wahl des Oszillatorsystems und des Abfangreagens sollten Systeme mit kontrollierbarer inhibitorischer oder exzitatorischer Kopplung zugänglich werden.

We propose a mapping to study the qualitative properties of continuous biochemical control networks which are invariant to the parameters used to describe the networks but depend only on the logical structure of the networks. For the networks, we are able to place a lower limit on the number of steady states and strong restrictions on the phase relations between components on cycles and transients. The logical structure and the dynamical behavior for a number of simple systems of biological interest, the feedback (predator-prey) oscillator, the bistable switch, the phase dependent switch, are discussed. We discuss the possibility that these techniques may be extended to study the dynamics of large many component systems.

Proto-organisms probably were randomly aggregated nets of chemical reactions. The hypothesis that contemporary organisms are also randomly constructed molecular automata is examined by modeling the gene as a binary (on-off) device and studying the behavior of large, randomly constructed nets of these binary “genes”. The results suggest that, if each “gene” is directly affected by two or three other “genes”, then such random nets: behave with great order and stability; undergo behavior cycles whose length predicts cell replication time as a function of the number of genes per cell; possess different modes of behavior whose number per net predicts roughly the number of cell types in an organism as a function of its number of genes; and under the stimulus of noise are capable of differentiating directly from any mode of behavior to at most a few other modes of behavior. Cellular differentation is modeled as a Markov chain among the modes of behavior of a genetic net. The possibility of a general theory of metabolic behavior is suggested.

We analyze a class of ordinary differential equations representing a simplified model of a genetic network. In this network, the model genes control the production rates of other genes by a logical function. The dynamics in these equations are represented by a directed graph on an n-dimensional hypercube (n-cube) in which each edge is directed in a unique orientation. The vertices of the n-cube correspond to orthants of state space, and the edges correspond to boundaries between adjacent orthants. The dynamics in these equations can be represented symbolically. Starting from a point on the boundary between neighboring orthants, the equation is integrated until the boundary is crossed for a second time. Each different cycle, corresponding to a different sequence of orthants that are traversed during the integration of the equation always starting on a boundary and ending the first time that same boundary is reached, generates a different letter of the alphabet. A word consists of a sequence of letters corresponding to a possible sequence of orthants that arise from integration of the equation starting and ending on the same boundary. The union of the words defines the language. Letters and words correspond to analytically computable Poincare maps of the equation. This formalism allows us to define bifurcations of chaotic dynamics of the differential equation that correspond to changes in the associated language. Qualitative knowledge about the dynamics found by integrating the equation can be used to help solve the inverse problem of determining the underlying network generating the dynamics. This work places the study of dynamics in genetic networks in a context comprising both nonlinear dynamics and the theory of computation. (c) 2001 American Institute of Physics.