Tessellations and Pattern Formation in Plant Growth and Development

The shoot apical meristem (SAM) is a dome-shaped collection of cells at the apex of growing plants from which all above-ground tissue ultimately derives. In Arabidopsis thaliana (thale cress), a small flowering weed of the Brassicaceae family (related to mustard and cabbage), the SAM typically contains some three to five hundred cells that range from five to ten microns in diameter. These cells are organized into several distinct zones that maintain their topological and functional relationships throughout the life of the plant. As the plant grows, organs (primordia) form on its surface flanks in a phyllotactic pattern that develop into new shoots, leaves, and flowers. Cross-sections through the meristem reveal a pattern of polygonal tessellation that is suggestive of Voronoi diagrams derived from the centroids of cellular nuclei. In this chapter we explore some of the properties of these patterns within the meristem and explore the applicability of simple, standard mathematical models of their geometry.Comment: Originally presented at: "The World is a Jigsaw: Tessellations in the Sciences," Lorentz Center, Leiden, The Netherlands, March 200

Modelling Visual Search with the Selective Attention for Identification Model (VS-SAIM): A Novel Explanation for Visual Search Asymmetries

In earlier work, we developed the Selective Attention for Identification Model (SAIM [16]). SAIM models the human ability to perform translation-invariant object identification in multiple object scenes. SAIM suggests that central for this ability is an interaction between parallel competitive processes in a selection stage and a object identification stage. In this paper, we applied the model to visual search experiments involving simple lines and letters. We presented successful simulation results for asymmetric and symmetric searches and for the influence of background line orientations. Search asymmetry refers to changes in search performance when the roles of target item and non-target item (distractor) are swapped. In line with other models of visual search, the results suggest that a large part of the empirical evidence can be explained by competitive processes in the brain, which are modulated by the similarity between target and distractor. The simulations also suggest that another important factor is the feature properties of distractors. Finally, the simulations indicate that search asymmetries can be the outcome of interactions between top-down (knowledge about search items) and bottom-up (feature of search items) processing. This interaction in VS-SAIM is dominated by a novel mechanism, the knowledge-based on-centre-off-surround receptive field. This receptive field is reminiscent of the classical receptive fields but the exact shape is modulated by both, top-down and bottom-up processes. The paper discusses supporting evidence for the existence of this novel concept

Evolving Synaptic Plasticity with an Evolutionary Cellular Development Model

Since synaptic plasticity is regarded as a potential mechanism for memory formation and learning, there is growing interest in the study of its underlying mechanisms. Recently several evolutionary models of cellular development have been presented, but none have been shown to be able to evolve a range of biological synaptic plasticity regimes. In this paper we present a biologically plausible evolutionary cellular development model and test its ability to evolve different biological synaptic plasticity regimes. The core of the model is a genomic and proteomic regulation network which controls cells and their neurites in a 2D environment. The model has previously been shown to successfully evolve behaving organisms, enable gene related phenomena, and produce biological neural mechanisms such as temporal representations. Several experiments are described in which the model evolves different synaptic plasticity regimes using a direct fitness function. Other experiments examine the ability of the model to evolve simple plasticity regimes in a task -based fitness function environment. These results suggest that such evolutionary cellular development models have the potential to be used as a research tool for investigating the evolutionary aspects of synaptic plasticity and at the same time can serve as the basis for novel artificial computational systems

The Drosophila Gap Gene Network Is Composed of Two Parallel Toggle Switches

Drosophila “gap” genes provide the first response to maternal gradients in the early fly embryo. Gap genes are expressed in a series of broad bands across the embryo during first hours of development. The gene network controlling the gap gene expression patterns includes inputs from maternal gradients and mutual repression between the gap genes themselves. In this study we propose a modular design for the gap gene network, involving two relatively independent network domains. The core of each network domain includes a toggle switch corresponding to a pair of mutually repressive gap genes, operated in space by maternal inputs. The toggle switches present in the gap network are evocative of the phage lambda switch, but they are operated positionally (in space) by the maternal gradients, so the synthesis rates for the competing components change along the embryo anterior-posterior axis. Dynamic model, constructed based on the proposed principle, with elements of fractional site occupancy, required 5–7 parameters to fit quantitative spatial expression data for gap gradients. The identified model solutions (parameter combinations) reproduced major dynamic features of the gap gradient system and explained gap expression in a variety of segmentation mutants

Gene Circuit Analysis of the Terminal Gap Gene huckebein

The early embryo of Drosophila melanogaster provides a powerful model system to study the role of genes in pattern formation. The gap gene network constitutes the first zygotic regulatory tier in the hierarchy of the segmentation genes involved in specifying the position of body segments. Here, we use an integrative, systems-level approach to investigate the regulatory effect of the terminal gap gene huckebein (hkb) on gap gene expression. We present quantitative expression data for the Hkb protein, which enable us to include hkb in gap gene circuit models. Gap gene circuits are mathematical models of gene networks used as computational tools to extract regulatory information from spatial expression data. This is achieved by fitting the model to gap gene expression patterns, in order to obtain estimates for regulatory parameters which predict a specific network topology. We show how considering variability in the data combined with analysis of parameter determinability significantly improves the biological relevance and consistency of the approach. Our models are in agreement with earlier results, which they extend in two important respects: First, we show that Hkb is involved in the regulation of the posterior hunchback (hb) domain, but does not have any other essential function. Specifically, Hkb is required for the anterior shift in the posterior border of this domain, which is now reproduced correctly in our models. Second, gap gene circuits presented here are able to reproduce mutants of terminal gap genes, while previously published models were unable to reproduce any null mutants correctly. As a consequence, our models now capture the expression dynamics of all posterior gap genes and some variational properties of the system correctly. This is an important step towards a better, quantitative understanding of the developmental and evolutionary dynamics of the gap gene network

Specialization Can Drive the Evolution of Modularity

Organismal development and many cell biological processes are organized in a modular fashion, where regulatory molecules form groups with many interactions within a group and few interactions between groups. Thus, the activity of elements within a module depends little on elements outside of it. Modularity facilitates the production of heritable variation and of evolutionary innovations. There is no consensus on how modularity might evolve, especially for modules in development. We show that modularity can increase in gene regulatory networks as a byproduct of specialization in gene activity. Such specialization occurs after gene regulatory networks are selected to produce new gene activity patterns that appear in a specific body structure or under a specific environmental condition. Modules that arise after specialization in gene activity comprise genes that show concerted changes in gene activities. This and other observations suggest that modularity evolves because it decreases interference between different groups of genes. Our work can explain the appearance and maintenance of modularity through a mechanism that is not contingent on environmental change. We also show how modularity can facilitate co-option, the utilization of existing gene activity to build new gene activity patterns, a frequent feature of evolutionary innovations

Neutrality and Robustness in Evo-Devo: Emergence of Lateral Inhibition

Embryonic development is defined by the hierarchical dynamical process that translates genetic information (genotype) into a spatial gene expression pattern (phenotype) providing the positional information for the correct unfolding of the organism. The nature and evolutionary implications of genotype–phenotype mapping still remain key topics in evolutionary developmental biology (evo-devo). We have explored here issues of neutrality, robustness, and diversity in evo-devo by means of a simple model of gene regulatory networks. The small size of the system allowed an exhaustive analysis of the entire fitness landscape and the extent of its neutrality. This analysis shows that evolution leads to a class of robust genetic networks with an expression pattern characteristic of lateral inhibition. This class is a repertoire of distinct implementations of this key developmental process, the diversity of which provides valuable clues about its underlying causal principles

Mathematics and biology: a Kantian view on the history of pattern formation theory

Driesch’s statement, made around 1900, that the physics and chemistry of his day were unable to explain self-regulation during embryogenesis was correct and could be extended until the year 1972. The emergence of theories of self-organisation required progress in several areas including chemistry, physics, computing and cybernetics. Two parallel lines of development can be distinguished which both culminated in the early 1970s. Firstly, physicochemical theories of self-organisation arose from theoretical (Lotka 1910–1920) and experimental work (Bray 1920; Belousov 1951) on chemical oscillations. However, this research area gained broader acceptance only after thermodynamics was extended to systems far from equilibrium (1922–1967) and the mechanism of the prime example for a chemical oscillator, the Belousov–Zhabotinski reaction, was deciphered in the early 1970s. Secondly, biological theories of self-organisation were rooted in the intellectual environment of artificial intelligence and cybernetics. Turing wrote his The chemical basis of morphogenesis (1952) after working on the construction of one of the first electronic computers. Likewise, Gierer and Meinhardt’s theory of local activation and lateral inhibition (1972) was influenced by ideas from cybernetics. The Gierer–Meinhardt theory provided an explanation for the first time of both spontaneous formation of spatial order and of self-regulation that proved to be extremely successful in elucidating a wide range of patterning processes. With the advent of developmental genetics in the 1980s, detailed molecular and functional data became available for complex developmental processes, allowing a new generation of data-driven theoretical approaches. Three examples of such approaches will be discussed. The successes and limitations of mathematical pattern formation theory throughout its history suggest a picture of the organism, which has structural similarity to views of the organic world held by the philosopher Immanuel Kant at the end of the eighteenth century