Narrow Results

We discuss the problem of assembling complex physical systems that are structured from the nanoscale up through the macroscale, and argue that embryological morphogenesis provides a good model of how this can be accomplished. Morphogenesis (whether natural or artificial) is an example of embodied computation, which exploits physical processes for computational ends, or performs computations for their physical effects. Examples of embodied computation in natural morphogenesis can be found at many levels, from allosteric proteins, which perform simple embodied computations, up through cells, which act to create tissues with specific patterns, compositions, and forms. We outline a notation for describing morphogenetic programs and illustrate its use with two examples: simple diffusion and the assembly of a simple spine with attachment points for legs. While much research remains to be done — at the simulation level before we attempt physical implementations — our results to date show how we may implement the fundamental processes of morphogenesis as a practical application of embodied computation at the nano- and microscale.

Replication and differentiation of spots in a class of reaction–diffusion equations are studied by extending the Gray–Scott model with self-replicating spots so that it includes many chemical species. By examining many possible reaction networks, the behavior of this model is categorized into three types: replication of homogeneous fixed spots, replication of oscillatory spots, and differentiation from "multipotent spots". These multipotent spots either replicate or differentiate into other types of spots with different fixed-point dynamics, and as a result, an inhomogeneous pattern of spots is formed. This differentiation process of spots is analyzed in terms of the loss of chemical diversity and decrease of the local Kolmogorov–Sinai entropy. Initial condition dependence and robustness of a pattern against macroscopic perturbation are also analyzed. Relevance of the results to developmental cell biology is also discussed.

Many scientists have struggled to uncover the elusive origin of "complexity", and its many equivalent jargons, such as emergence, self-organization, synergetics, collective behaviors, nonequilibrium phenomena, etc. They have provided some qualitative, but not quantitative, characterizations of numerous fascinating examples from many disciplines. For example, Schrödinger had identified "the exchange of energy" from open systems as a necessary condition for complexity. Prigogine has argued for the need to introduce a new principle of nature which he dubbed "the instability of the homogeneous". Turing had proposed "symmetry breaking" as an origin of morphogenesis. Smale had asked what "axiomatic" properties must a reaction–diffusion system possess to make the Turing interacting system oscillate.

The purpose of this paper is to show that all the jargons and issues cited above are mere manifestations of a new fundamental principle called local activity, which is mathematically precise and testable. The local activity theorem provides the quantitative characterization of Prigogine's "instability of the homogeneous" and Smale's quest for an axiomatic principle on Turing instability.

Among other things, a mathematical proof is given which shows none of the complexity-related jargons cited above is possible without local activity. Explicit mathematical criteria are given to identify a relatively small subset of the locally-active parameter region, called the edge of chaos, where most complex phenomena emerge.

A reaction–diffusion Gierer–Meinhardt model of morphogenesis subject to Dirichlet fixed boundary condition in the one-dimensional spatial domain is considered. We perform a detailed Hopf bifurcation analysis and steady state bifurcation analysis to the system. Our results suggest the existence of spatially nonhomogenous periodic orbits and nonconstant positive steady state solutions, which imply the possibility of complex spatiotemporal patterns of the system. Numerical simulations are carried out to support our theoretical analysis.

In this paper we study some qualitative properties of the solutions of a nonlinear flux-limited equation arising in the transport of morphogens in biological systems. Questions related to the existence of steady states, the finite speed of propagating fronts or the regularization in the interior of the support are studied from analytical and numerical points of view.

The simulation of the creation and evolution of biological forms requires the development of computational methods that are capable of resolving their hierarchical, spatial and temporal complexity. Computations based on interacting particles, provide a unique computational tool for discrete and continuous descriptions of morphogenesis of systems ranging from the molecular to the organismal level. The capabilities of particle methods hinge on the simplicity of their formulation which enables the formulation of a unifying computational framework encompassing deterministic and stochastic models. In this paper, we discuss recent advances in particle methods for the simulation of biological systems at the mesoscopic and the macroscale level. We present results from applications of particle methods including reaction–diffusion on deforming surfaces, deterministic and stochastic descriptions of tumor growth and angiogenesis and discuss successes and challenges of this approach.

In the following the reader will find a short description of some issues related to the modeling, analysis and simulation of large populations of living systems, a research field which is currently deserving a considerable interest, and that has been explored during the first 10 editions of the BIOMAT Summer School at Granada.

The inflorescence of Symplocarpus foetidus constitutes good material to analyse the biological processes and physical constraints involved in the development of plants. During the development of the inflorescence, two morphogenetic periods can be distinguished (i) before and (ii) during and after the development of floral parts. In the first period, when the floral primordia appear, the phyllotactic system could be explained by global processes at the inflorescence level. In the second period, the development of floral parts produces patterns which can be explained by local processes at the floral level. In this analysis the author defines the concepts of open system and closed system in phyllotaxis. In a closed system (e.g. spadix) the elements are arranged on a continuous and closed surface. In an open system (e.g. shoot apex) the elements appear on a surface periodically renewed and are removed from each other by the intercalary growth.

A minimum cellular automaton, carrying precise biophysical significance in each rule, is presented to model pigmentation patterns on molluscan shells. We find the following types of modes: self-organisation into stationary (Turing) structures, travelling waves, chaos, and so-called class IV behaviour. The latter consists of a disordered spatio-temporal distribution of periodic and chaotic patches; it differs from chaos in that it has no well-defined error propagation rate. The calculations of the modes agree well with observations in natural shells. In particular, our results suggest evidence in nature for class IV behaviour, a mode that had so far been reported only as the result of simulations. Moreover, we show that patchiness results in a class IV mode from the same algorithm that renders chaos and periodicity; thus, there is no need to invoke two competing pattern generators, as in previous approaches.

A mathematical model for the mechanism of periodic pattern formation in the process of somitogenesis is proposed. It is assumed that metameric arrangement first appears before somite formation at the stage of transition of mesodermal into presomitic cells. It is assumed that the transition occurs in a certain phase of the mitotic cycle and that it can be suppressed due to excretion of some transition inhibitor by presomitic cells. The model demonstrates that periodicity can appear as a result of interaction of the wave of somitogenic cell determination with the mitotic cycles of mesodermal cells. It is shown that the model naturally explains synchrony in the somite formation and the results of heat shock experiments.

A multi-part theorem is presented concerning the morphogenesis of high-symmetry structures made of three-dimensional morphological units (MU's) free to move on the surface of a sphere. All parts of each MU interact non-specifically with the remainder of the structure, via an isotropic function of distance. Summing all interactions gives a net figure of merit, ℐ, that depends upon MU positions and orientations. The structure evolves via gradient dynamics, each MU moving down the local gradient of ℐ. The analysis is reresented with generality in Fourier space, which eases the expression of symmetry.

Structures near symmetry, but far from a local minimum of ℐ, are analyzed. For each, a symmetrical configuration can be found, for which ℐ is an extremum with respect to symmetry-breaking perturbations. Under gradient dynamics, a quadratic measure of such deviations from symmetry decreases monotonically, anywhere in the large basin of attraction of a local minimum. Thus: high symmetry is an attractor.

Application is made to icosahedral virus capsids. The Symmetrization Theorem shows that a stable capsid, maintained by non-specific interactions among its capsomeres, could arise generically in a "bottom-up" process. For animated evolutions that self-assemble into high symmetry, visit

The mathematical modeling of biological morphogenesis processes is considered. Emphasis is placed throughout on the lessons of experience in modeling three-dimensional forms that evolve in time. The qualitative requirements of a model, the general components of a dynamic system, and the products of a morphogenesis modeling program are discussed. Examples are drawn frequently from phyllotaxis.

A theorem is presented concerning the morphogenesis of high-symmetry structures made of three-dimensional morphological units (MU's) free to move in three dimensions or constrained to a surface. All parts of each MU interact non-specifically with the rest of the structure, via an isotropic function of distance. Summing all interactions gives a net figure of merit, ℐ, that depends upon MU positions and orientations. A structure evolves via gradient dynamics, each MU moving down the local gradient of ℐ. The analysis is represented with generality in Fourier space.

A "warping" from a configuration of MU's is a set of MU displacements and/or rotations that slightly perturb nearest neighbor relations; deviations can accrue across the structure, producing large global distortions. A warping behaves qualitatively like a small perturbation, so a warping from a stable equilibrium decays under gradient dynamics. Connection to the Symmetrization Theorem greatly extends the basin of attraction of stable symmetrical configurations. Warped configurations are equivalent as precursors of structure, which helps to understand assembly by accretion.

Numerical illustrations are given in cylindrical geometry, for application to phyllotaxis; and in spherical geometry, for virus capsid structure. For animations of numerical evolutions that find high symmetry via unwarping, see

The self-organization of cells into a living organism is a very intricate process. Under the surface of orchestrating regulatory networks there are physical processes which make the information processing possible, that is required to organize such a multitude of individual entities. We use a quantitative information theoretic approach to assess self-organization of a collective system. In particular, we consider an interacting particle system, that roughly mimics biological cells by exhibiting differential adhesion behavior. Employing techniques related to shape analysis, we show that these systems in most cases exhibit self-organization. Moreover, we consider spatial constraints of interactions, and additionaly show that particle systems can self-organize without the emergence of pattern-like structures. However, we will see that regular pattern-like structures help to overcome limitations of self-organization that are imposed by the spatial structure of interactions.

Chemically dissipative or Turing processes, have been predicted by theoreticians as a way by which an initially homogenous solution of chemicals or biochemicals can spontaneously self-organise and give rise to a macroscopic pattern by way of a combination of reaction and diffusion. They have been advanced as a possible underlying process for biological self-organisation and pattern formation. Until now, there have been no examples of in vitro biological substances showing this type of behaviour. Evidence is presented that microtubule solutions in vitro self-organise in this manner and that similar processes may occur in vivo during embryogenesis.

Morphogenesis is a result of complex biological, chemical, and physical processes in which differential growth in biological systems is a common phenomenon, especially notable in plant organs such as petals and leaves. Mechanisms of these biologic structures have been studied in recent years with a growing focus from the mechanics point of view. However, understanding differential-growth-induced shape formation quantitatively in plant organs remains largely unknown. In this study, we conduct quantitative experimental measurement, theoretical analysis, and sufficient finite element analysis of constrained differential growth of a thin membrane-like structure. By deriving the corresponding strain energy expression of a buckled growing sample, we can calculate the shape function of such membrane structures explicitly. The results of this work demonstrate the effect of growth function, geometry characteristics, and material property. Our research points to potential approaches to novel geometrical design and inspirations on the microscale and the macroscale for items such as soft robotics and flexible electronics.

Recent advances in stem cell biology provide the conceptual framework for the development of cell-based therapies for life-threatening diseases affecting many organs, including the lung. Because of its complexity and structure, cell-based therapy for the lung faces significant technical challenges. Therapeutic goals span a spectra of expectations that might include: (1) regeneration of functional lung tissue, (2) replacement of specific cells affected by inherited or acquired diseases with genetically altered progenitor cells, (3) provision of cells capable of enhancing repair or influencing oncogenesis directly or indirectly, and (4) introduction of cells capable of expressing therapeutic molecules for local or systemic delivery. The technical hurdles required for accomplishing each of these goals are distinct and of various heights. None are trivial. Knowledge of the cellular and molecular basis for specification and differentiation of stem/progenitor cells will be required for the successful application of cell-based therapies for the lung. This chapter reviews concepts derived from study of lung morphogenesis and repair as well as stem cell biology that will be relevant to the development of novel therapies for pulmonary diseases in the future.

In this paper we describe supercomputer simulations for the self reorganization of tissue which has been separated into endoderm, mesoderm, and ectoderm cells.

The five earlier chapters introduced the basic foundation (co-existence of order and complexity, sensitive dependence on initial conditions, complexity <=> presence of in-determinism and unpredictability, necessities to change, the intelligence paradigmatic shift, structural reform, complex adaptive systems and dynamic, and some other fundamental and critical properties/characteristics involved) of the complexity theory and intelligent organization (IO) theory. This chapter introduces the global/holistic complexity-intelligence strategy (with two macro-paths) of the IO theory (although, some sub-strategies/models have been mentioned or partially analyzed in earlier chapters). The new strategy attempts to provide more comprehensive linkages and coverage on some specialized aspects indicating that human organizations must be led and managed differently in the current context because of high complexity density. In this chapter, three sub-strategies of the holistic complexity-intelligence strategy that is vital for nurturing highly intelligent human organizations (iCAS) are examined. They are namely, organizing around intelligence, nurturing an intelligent biotic macro-structure, and the integrated deliberate and emergent strategy.

The intelligence/conscious-centricity aspect begins with a deeper analysis on human level intelligence and consciousness, complexity, collective intelligence, org-consciousness and their associated dynamics relative to that of some other biological species (swarm intelligence), as well as other physical complex adaptive systems (CAS) characteristics. It has been observed that human interconnectivity, interdependency, selforganizing communications, truthful engagement, complex networks, collective intelligence, orgmindfulness, orgmind, and emergence can be significantly dissimilar. The four different perspectives of organizing around intelligence are examined.

Next, the intelligent biotic macro-structure (introduced earlier in Chapters 3 to 4) that resembles a highly intelligent biological being, and is more effective at exploiting information processing and knowledge accumulation, and a smarter evolver are more deeply scrutinized. There exists a high synchrony between organizing around intelligence and the presence of a biotic macro-structure. Thus, the advantages and significance for intelligent human organizations to possess such an inherent biotic macro-structure to better exploit certain biological and complexity associated characteristics and dynamics (including intelligence-intelligence linkage, complexity-centricity, complexity-intelligence linkage, more efficient natural decision-making node, information processing capability, learning and adaptation, knowledge acquisition and creation, organizational neural network, artificial node, and structural and dynamical coherency) to compete more effectively and efficiently in the current ‘raplexity’ context is also illustrated. In addition, the uniqueness and roles of artificial information systems (artificial nodes) is further examined.

Finally, the integrated deliberate and emergent strategy is scrutinized with respect to its significant association with the co-existence of order (deliberate planning, determinism, completeness and predictability) and complexity (continuous nurturing processes, in-determinism, unpredictability, unknown unknowns, risk management, new opportunity, crisis management, self-transcending constructions, futuristic and emergence) — in particular, highlighting the criticality of the deliberate and emergent auto-switch (better ambidexterity). Currently, the holistic integrated smarter evolver and emergent strategist approach is absent in most human organizations.