Crossing scales,crossing disciplines: collective motion and collective action in the Global Commons

Two conflicting tendencies can be seen throughout the biological world: individuality and collective behaviour. Natural selection operates on differences among individuals, rewarding those who perform better. Nonetheless, even within this milieu, cooperation arises, and the repeated emergence of multicellularity is the most striking example. The same tendencies are played out at higher levels, as individuals cooperate in groups, which compete with other such groups. Many of our environmental and other global problems can be traced to such conflicts, and to the unwillingness of individual agents to take account of the greater good. One of the great challenges in achieving sustainability will be in understanding the basis of cooperation, and in taking multicellularity to yet a higher level, finding the pathways to the level of cooperation that is the only hope for the preservation of the planet.     

Turing pattern with proportion preservation

Although Turing pattern is one of the most universal mechanisms for pattern formation, in its standard model the number of stripes changes with the system size, since the wavelength of the pattern is invariant. It fails to preserve the proportionality of the pattern, i.e. the ratio of the wavelength to the size, that is often required in biological morphogenesis. To get over this problem, we show that the Turing pattern can preserve proportionality by introducing a catalytic chemical whose concentration depends on the system size. Several plausible mechanisms for such size dependence of the concentration are discussed. Following this general discussion, two models are studied in which arising Turing patterns indeed preserve the proportionality. Relevance of the present mechanism to biological morphogenesis is discussed from the viewpoint of its generality, robustness, and evolutionary accessibility.     

Pattern formation in generalized Turing systems

Turing's model of pattern formation has been extensively studied analytically and numerically, and there is recent experimental evidence that it may apply in certain chemical systems. The model is based on the assumption that all reacting species obey the same type of boundary condition pointwise on the boundary. We call these scalar boundary conditions. Here we study mixed or nonscalar boundary conditions, under which different species satisfy different boundary conditions at any point on the boundary, and show that qualitatively new phenomena arise in this case. For example, we show that there may be multiple solutions at arbitrarily small lengths under mixed boundary conditions, whereas the solution is unique under homogeneous scalar boundary conditions. Moreover, even when the same solution exists under scalar and mixed boundary conditions, its stability may be different in the two cases. We also show that mixed boundary conditions can reduce the sensitivity of patterns to domain changes.Supported in part by NIH Grant # GM29123     

Labyrinthine versus straight-striped patterns generated by two-dimensional Turing systems

Striped patterns are often observed on fish skin. Such patterns have been accounted for by reaction-diffusion (RD) Turing-type models, in which two substances can spontaneously form a spatially heterogeneous pattern in a homogeneous field. Among the striped patterns generated by Turing-type models, some are "straight-striped patterns," with many stripes running in parallel, while others are "labyrinthine patterns," in which the stripes often change direction, merge with each other, and frequently branch out. RD models differ in terms of their tendency to generate either labyrinthine or straight-striped patterns. Here, we studied the conditions under which either a labyrinthine or straight-striped pattern would emerge. First, we defined an index for stripe clearness, Sh. Straight-striped patterns (large Sh) are formed if only a narrow range of spatial periods corresponds to an unstable mode. Labyrinthine patterns (small Sh) are formed when a wide range of spatial periods is unstable. More specifically, labyrinthine patterns are formed when the maximum spatial period of unstable modes is more than twice that of the minimum spatial period of unstable modes; otherwise, straight-striped patterns are formed. We then examined RD models with nonlinear reaction terms, including both activator-inhibitor and substrate-depletion models, and we demonstrated that the same conclusions hold with respect to the conditions required for labyrinthine versus straight-striped patterns.     

Coarse analysis of collective motion with different communication mechanisms

We study the effects of a signalling constraint on an individual-based model of self-organizing group formation using a coarse analysis framework. This involves using an automated data-driven technique which defines a diffusion process on the graph of a sample dataset formed from a representative stationary simulation. The eigenvectors of the graph Laplacian are used to construct 'diffusion-map' coordinates which provide a geometrically meaningful low-dimensional representation of the dataset. We show that, for the parameter regime studied, the second principal eigenvector provides a sufficient representation of the dataset and use it as a coarse observable. This allows the computation of coarse bifurcation diagrams, which are used to compare the effects of the signalling constraint on the population-level behavior of the model.     

Turing instability in pioneer/climax species interactions

Systems of pioneer and climax species are used to model interactions of species whose reproductive capacity is sensitive to population density in their shared ecosystem. Intraspecies interaction coefficients can be adjusted so that spatially homogeneous solutions are stable to small perturbations. In a reaction-diffusion pioneer/climax model we will determine the critical value of the diffusion rate of the climax species, below which the equilibrium solution is unstable to non-homogeneous perturbations. For diffusion rates smaller than this critical value, an equilibrium solution remains stable to spatially homogeneous perturbations but is unstable to non-homogeneous perturbations. A Turing (diffusional) bifurcation leads to the formation of spatial patterns in species' densities. Forcing, interpreted as stocking or harvesting of the species, can reverse the bifurcation and establish equilibrium solutions which are stable to small perturbations. The implicit function theorem is used to determine whether stocking or harvesting of one of the species in the model is the appropriate remedy for diffusional instability. The use of stocking or harvesting by a natural resource manager thus influences the long-term dynamics and spatial distribution of species in a pioneer/climax ecosystem.     

Transient dynamics and pattern formation: reactivity is necessary for Turing instabilities.

The theory of spatial pattern formation via Turing bifurcations - wherein an equilibrium of a nonlinear system is asymptotically stable in the absence of dispersal but unstable in the presence of dispersal - plays an important role in biology, chemistry and physics. It is an asymptotic theory, concerned with the long-term behavior of perturbations. In contrast, the concept of reactivity describes the short-term transient behavior of perturbations to an asymptotically stable equilibrium. In this article we show that there is a connection between these two seemingly disparate concepts. In particular, we show that reactivity is necessary for Turing instability in multispecies systems of reaction-diffusion equations, integrodifference equations, coupled map lattices, and systems of ordinary differential equations.     

The limits of convergence in the collective behavior of competing marine taxa

Collective behaviors in biological systems such as coordinated movements have important ecological and evolutionary consequences. While many studies examine within‐species variation in collective behavior, explicit comparisons between functionally similar species from different taxonomic groups are rare. Therefore, a fundamental question remains: how do collective behaviors compare between taxa with morphological and physiological convergence, and how might this relate to functional ecology and niche partitioning? We examined the collective motion of two ecologically similar species from unrelated clades that have competed for pelagic predatory niches for over 500 million years—California market squid, Doryteuthis opalescens (Mollusca) and Pacific sardine, Sardinops sagax (Chordata). We (1) found similarities in how groups of individuals from each species collectively aligned, measured by angular deviation, the difference between individual orientation and average group heading. We also (2) show that conspecific attraction, which we approximated using nearest neighbor distance, was greater in sardine than squid. Finally, we (3) found that individuals of each species explicitly matched the orientation of groupmates, but that these matching responses were less rapid in squid than sardine. Based on these results, we hypothesize that information sharing is a comparably important function of social grouping for both taxa. On the other hand, some capabilities, including hydrodynamically conferred energy savings and defense against predators, could stem from taxon‐specific biology.     

A Turing test of a timing theory

A quantitative theory of timing or conditioning can be evaluated with a Turing test in which the behavioral results of an experiment can be compared with the predicted results from the theory. An example is described based upon an experiment in which 12 rats were trained on three fixed-interval schedules of reinforcement, and a simulation of the predicted results from a packet theory of timing. An objective classification rule was used to determine whether a sample from the data or a sample from the theory was more similar to another sample from the theory. With an ideal theory, the expected probability of a correct classification would be 0.5. The observed probability of a correct classification was 0.6, which was slightly, but reliably, greater than 0.5. A Turing test provides a graded metric for the evaluation of a quantitative theory.     

The evolution of patterning during movement in a large-scale citizen science game

The motion dazzle hypothesis posits that high contrast geometric patterns can cause difficulties in tracking a moving target and has been argued to explain the patterning of animals such as zebras. Research to date has only tested a small number of patterns, offering equivocal support for the hypothesis. Here, we take a genetic programming approach to allow patterns to evolve based on their fitness (time taken to capture) and thus find the optimal strategy for providing protection when moving. Our ‘Dazzle Bug’ citizen science game tested over 1.5 million targets in a touch screen game at a popular visitor attraction. Surprisingly, we found that targets lost pattern elements during evolution and became closely background matching. Modelling results suggested that targets with lower motion energy were harder to catch. Our results indicate that low contrast, featureless targets offer the greatest protection against capture when in motion, challenging the motion dazzle hypothesis.     

Propagation of Turing patterns in a plankton model

The paper is devoted to a reaction–diffusion system of equations describing phytoplankton and zooplankton distributions. Linear stability analysis of the model is carried out. Turing and Hopf stability boundaries are found. Emergence of two-dimensional spatial structures is illustrated by numerical simulations. Travelling waves between various stationary solutions are investigated. Transitions between homogeneous in space stationary solutions and Turing structures are studied.     

Chemical morphogenesis: Turing patterns in an experimental chemical system

Patterns resulting from the sole interplay between reaction and diffusion are probably involved in certain stages of morphogenesis in biological systems, as initially proposed by Alan Turing. Self-organization phenomena of this type can only develop in nonlinear systems (i.e. involving positive and negative feedback loops) maintained far from equilibrium. We present Turing patterns experimentally observed in a chemical system. An oscillating chemical reaction, the CIMA reaction, is operated in an open spatial reactor designed in order to obtain a pure reaction-diffusion system. The two types of Turing patterns observed, hexagonal arrays of spots and parallel stripes, are characterized by an intrinsic wavelength. We identify the origin of the necessary difference of diffusivity between activator and inhibitor. We also describe a pattern growth mechanism by spot splitting that recalls cell division.     

Hinge-bending motion in citrate synthase arising from normal mode calculations

A normal mode analysis of the closed form of dimeric citrate synthase has been performed. The largest-amplitude collective motion predicted by this method compares well with the crystallographically observed hinge-bending motion. Such a result supports those obtained previously in the case of hinge-bending motions of smaller systems, such as lysozyme or hexokinase. Taken together, all these results suggest that low-frequency normal modes may become useful for determining a first approximation of the conformational path between the closed and open forms of these proteins. © 1995 Wiley-Liss, Inc.     

Synthetic Turing protocells: vesicle self-reproduction through symmetry-breaking instabilities

The reproduction of a living cell requires a repeatable set of chemical events to be properly coordinated. Such events define a replication cycle, coupling the growth and shape change of the cell membrane with internal metabolic reactions. Although the logic of such process is determined by potentially simple physico-chemical laws, modelling of a full, self-maintained cell cycle is not trivial. Here we present a novel approach to the problem that makes use of so-called symmetry breaking instabilities as the engine of cell growth and division. It is shown that the process occurs as a consequence of the breaking of spatial symmetry and provides a reliable mechanism of vesicle growth and reproduction. Our model opens the possibility of a synthetic protocell lacking information but displaying self-reproduction under a very simple set of chemical reactions.     

Stripes,spots, or reversed spots in two-dimensional Turing systems

Two-dimensional Turing models can generate stationary striped patterns or spotted patterns, and are used to explain the body pattern formation of animals. We studied the effects of the choice of reaction terms on pattern selection, i.e., which pattern is likely to be formed. We examined in detail a model with linear reaction terms and additional constraint terms that confine two variables within a finite range. In the one-dimensional model, a periodic stationary pattern can be formed only when the activator level is constrained both from below and from above. In the two-dimensional model, the relative distance of the equilibrium level of the activator between the upper and lower limitations determines the pattern selection. Striped patterns are produced when the equilibrium is equally distant from the upper and the lower limitations, but spotted patterns are produced when the equilibrium is clearly closer to one than to the other of two limitations. We then examined models with nonlinear reaction terms, including both activator-inhibitor and activator-depletion substrate type models; we attempted to explain the pattern selection of these nonlinear models based on the results of linear models with constraints. The distribution of the activator level is skewed positively and negatively for spotted patterns and reversed spotted patterns, respectively. In contrast, the skew of the distribution of the activator level was close to zero in the case of striped patterns. This observation provides a heuristic argument of how the location of the equilibrium between the constraints leads to pattern selection.     

Unravelling the Turing bifurcation using spatially varying diffusion coefficients

. The Turing bifurcation is the basic bifurcation generating spatial pattern, and lies at the heart of almost all mathematical models for patterning in biology and chemistry. In this paper the authors determine the structure of this bifurcation for two coupled reaction diffusion equations on a two-dimensional square spatial domain when the diffusion coefficients have a small explicit variation in space across the domain. In the case of homogeneous diffusivities, the Turing bifurcation is highly degenerate. Using a two variable perturbation method, the authors show that the small explicit spatial inhomogeneity splits the bifurcation into two separate primary and two separate secondary bifurcations, with all solution branches distinct. This splitting of the bifurcation is more effective than that given by making the domain slightly rectangular, and shows clearly the structure of the Turing bifurcation and the way in which the! var ious solution branches collapse together as the spatial variation is reduced. The authors determine the stability of the solution branches, which indicates that several new phenomena are introduced by the spatial variation, including stable subcritical striped patterns, and the possibility that stable stripes lose stability supercritically to give stable spotted patterns.. Received: 10 January 1996/Revised version: 3 July 1996     

Stability analysis of Turing patterns generated by the Schnakenberg model

We consider the following Schnakenberg model on the interval (-1,1): where We rigorously show that the stability of symmetric N-peaked steady-states can be reduced to computing two matrices in terms of the diffusion coefficients D₁, D₂ and the number N of peaks. These matrices and their spectra are calculated explicitly and sharp conditions for linear stability are derived. The results are verified by some numerical simulations.Mathamatics Subject Classification (2000): Primary 35B40, 35B45; Secondary 35J55, 92C15, 92C40Revised version: 10 October 2003     

Gravity is a pull! Exploring forces and motion with English learners

English learners (ELs) benefit from inquiry-based science instruction that includes explicit attention to language learning goals. The purpose of this article is to share a third-grade unit on forces and motion which integrates science inquiry and writing in science notebooks with the goal of developing ELs' engagement in science, conceptual understanding, and academic language and literacy skills. We demonstrate how to engage diverse students' background knowledge and use classroom activities and discussion to create bridges between everyday and academic language. We utilize excerpts from Peter, Lucia, and Andrea's science notebooks to explore and highlight how teachers can use this resource as a means of communicating science, during instruction. Through these EL students' journals, we discuss the importance of developing language goals at the word, sentence, and discourse level while promoting and supporting ELs' use of the language of science.