Pattern Formation in Biology, Vision and Dynamics

The following sections are included:

• PREFACE

• CONTENTS

The variety of natural patterns makes it difficult to analyze and compare them in a systematic manner. We address this problem by focusing on the computational aspects of pattern formation processes. They are characterized in terms of the number of morphogenetic agents, the computing capability of each agent, and the forms of information transfer between the agents and their environment. This computational analysis can be applied to a wide range of patterns. It highlights the fundamental, algorithmic similarities between processes that may be implemented using different physical or physiological mechanisms in nature. It also confirms earlier observations that similar or identical patterns can be created by fundamentally different processes. The tradeoffs between computational characteristics of these processes lend themselves to a formal analysis, which could lead to the formulation of a “computational theory of morphogenesis” on the basis of the theory of algorithms.

In this paper the impact of hydrodynamics on a nutrient driven growth process is studied by using a simulation model. A very simple model is used to simulate the growth process, growth is modelled as an aggregation process in three dimensions, in which new sites are added with a certain probability to the aggregate. The growth probability is determined by the local amount of absorped simulated nutrient. The nutrient distribution is modelled by using the lattice Boltzmann method in combination with a tracer step, and where tracer (the simulated nutrient) is released from certain source sites and being absorbed at the boundary nodes of the aggregates. With this method it is possible to study the dispersion of nutrients caused by a combination of flow and diffusion, an increasing influence of hydrodynamics can be expressed by a relatively higher Péclet number, a Péclet number of zero indicates that nutrients are only dispersed by diffusion. Growth of the aggregates is studied under bi-directional flow conditions. In the simulations it is found that the degree of compactness gradually increases with an increasing influence of hydrodynamics. Furthermore it is observed that the aggregates (for all Péclet numbers) develop a roughly radiate symmetry under bi-directional flow conditions. Both observations correspond to qualitative observations made in marine sessile organisms, as for example sponges and corals, where growth forms tend to change gradually from compact shapes under conditions where the organism is exposed to water movement, and into thin-branching forms under sheltered conditions. The growth forms of these organisms have either a roughly radiate symmetry or a flattened morphology. Furthermore it is was found that the highest degree of nutrient absorption takes place at the upstream side of the simulated objects, which corresponds to experimental work on food particle absorption in octocorals and stony corals under low flow velocity conditions.

Pseudo-random number generators, stochastic evolutionary growth models and texture images are treated in a unified way in terms of dynamic systems with nonlinear feedback and their evolutionary behavior.

We use crystal growth as a simplified model for biological cell division, and we employ discrete approximations to Riemannian geometry as rough measures of the shapes of the organisms thus created. The first two sections are nontechnical, expository, and speculative. The first section creates a fantastical crystalline growth model of a tree trunk whose shape oscillates wildly over the lifetime of the tree, even though each cell of the trunk senses, responds, and reproduces according to simple local laws. The second section speculates on the underlying geometry in the beautiful microscopic view of rat tissue and suggests that the discrete geometry of tissue plays a role in evolutionary value. The remainder of the paper develops the notion of mathematical crystal growth and discrete geometry in more depth, with the main result being a mathematical demonstration of the experimental fact that, asymptotically, growth crystals generically exhibit the structure of self-similarity fractals.

Wang tiles are square tiles with colored edges. The plane is tiled with Wang tiles by placing them edge-to-edge in such a way that colors on contiguous edges are identical. A tiling is called periodic if there is a translation of the plane that keeps the tiling unchanged, otherwise the tiling is non-periodic. A set of tiles is called aperiodic if it only admits non-periodic tilings. This article discusses new results and applications of aperiodic Wang tilings. We present recently discovered small aperiodic tile sets. Instead of geometric arguments the construction is based on elementary arithmetics on the tiles. Then we discuss special types of tile sets that have the property that each tile is completely determined by two of its neighbors. Such tiles provide examples of one-dimensional cellular automata that exhibit unexpected behaviour.

Self-enhancement coupled with one or more antagonistic reactions is the crucial element in pattern forming reactions. Depending on the parameter, this can lead to patterns in space and/or in time, they can be either extremely robust and reproducible or highly variable. Complex patterns result from a linkage of many pattern forming reactions, one pattern generates the prerequisites for the next. The support these models have obtained recently by molecular-genetic observations give rise to the hope that in the future an interplay between theory and experiment will lead to a still better understanding of this central issue.

Free of functional constraints, the diversity of patterns of on shells of mollusk provide a rich source to study the properties of dynamic systems in general. Everyday we are confronted with systems that have an inherent tendency to change. The weather, the stock market, or the economic situation are examples in which self-enhancing and antagonistic processes play also a decisive role. The shell patterns are sufficiently complex to be a challenge but also sufficiently simple to be accessible to modelling. Their one-dimensional character and the preservation of the history of their formation provide unusual help for deciphering these patterns. They illustrate the range of behavior that can be generated by modifications of a basic mechanism. They can be regarded as a natural exercise book to study dynamic systems.

Tropical molluscs exhibit very complex pigmentation patterns on their shells. In this paper we intend to show that some of those patterns can be understood as the instability of limit cycles in spatially extended dynamical systems.

Sunflowers do not inspire only painters; for botanists, mathematicians, physicists, there is a specific beauty in the spiral order of their inflorescences. They are the most complex manifestation of a quasi crystalline order which is ubiquitous in plants. The study of these structures has developed into a whole domain called phyllotaxis. It is historically remarkable that one of the pioneering works in this field is due to A. and L. Bravais (1835, 1837). This investigation preceded A. Bravais's famous work on crystallography where he gave the classification of the 14 possible periodic lattices in three dimensional space. In this latter work he used the same type of geometrical arguments for lattices of point-like elements that he had established in his botanical work. Very probably, for Bravais the order of plants served as a starting point to imagine the order in crystals.

The following sections are included:

• Introduction

• Cell patterning from the perspective of the cell body

• What lies behind analyses of cell patterning?

• A simple autoregenerating cell pattern related to organogenesis

• Simulation of cell patterns and the uncovery of division rules
  • The Simulation System for the Psilotum Apex
  • Identification of cell types
  • Merophyte formation by apical divisions
  • Wall-offsetting leads to refinement of the wall production system
  • The distribution of apical triangular cells and the problem of dichotomous axis branching

• Possible regulatory factors underlying cell patterning and its simulation at the psilotum apex
  • A System's View of Division Wall Site Selection
  • Cytological View of Division Wall Site Selection

• Was there a precursor of the psilotum pattern?

• Prospects

• Acknowledgments

• References

Cellworks are L-systems generating 3D networks representative of the network of cell walls in plant tissues. They provide a scaffold for the development of plant forms. The model is a topological, deterministic and interactionless approach to plant morphogenesis with a secondary geometrical interpretation.

By means of cellworks, and using double wall labelling, it is possible to search algorithmically for patterns of cell division activity that lead to particular patterns of cellular arrangements. Within meristems, which are responsible for growth and form of plants, it is shown how the autoreproduction of the shape of apical cells maintains the specific form of plant structure. Furtheron, the way patterns are formed permits the construction of a general classification of plant meristems, internal, apical or in layers, based on a progressive complexification of the shape of apical cells and of the set of cell production rules.

Special attention is given to the geometrical interpretation of the topological cellular arrangements, important for the plant morphologist.

The following sections are included:

• Introduction

• Patterns of the shoot apical meristem
  • Developmental variation of cellular patterns
  • The geometry of a growing apex
  • Phyllotaxis

• Cellular patterns of vascular cambium
  • Storied pattern
  • The pattern of growth activity
  • Structural wavy patterns

• Conclusions

• Glossary

• References

The following sections are included:

• Introduction

• Stress anisotropy

• Stress distribution

• Conclusions

• Glossary

• References

Plants, unlike animals, grow throughout their lifetimes. The plant body enlarges by the continuous activity of a special localized regions called meristems which are centers of cell division and cell expansion. A plant grows in length by its apical meristems, of the stem and root, and in width by a sheath of lateral meristems. Organs such as leaves, flowers, and fruits, however, rarely have sharply localized meristems but enlarge throughout much of their extent, as an animal body does, until growth ceases. When they reach maturity all their tissues stop growing and there is no meristematic region set apart by which further growth may be accomplished. Meristems are of much interest to workers of plant morphogenesis. They provide a “continuous embryology” (Sinnott 1960) for the plant and offer an important point for studies of problems of plant development. This paper deals with the meristem of the shoot apex. It generates cells, which subsequently elongate and differentiate into the primary tissues of the stem and leaves. The apical part of the shoot apex has the form of a regular dome situated, mostly, above the level of the youngest leaf primordia. The dome is the ultimate source of all the cells of the stem. I have worked out a simulation-based model for growth and cell divisions in the shoot apical dome described, mathematically, by the growth tensor (Hejnowicz and Romberger, 1984). Here, the model is considered in two cellular layers, one related to the axial section and the other related to the surface of the dome. In both, cell patterns coming from the apex of spruce seedlings will be developed.

DNA nanotechnology entails the exploitation of DNA molecules for non-biological purposes. The properties that make DNA such a successful molecule in living systems are also very powerful when applied in chemical systems; DNA nanotechnology has resulted in the assembly of polyhedra, topological figures, nanomechanical devices and crystals. Unusual motifs of DNA, particularly branched molecules and structures derived from them, can be combined with sticky-ended association to direct the assembly of multiply-connected species. Branched junctions have been combined in this fashion to create molecules whose helix axes have the connectivities of a cube and of a truncated octahedron. A solid support-based methodology has been developed to facilitate the construction of DNA objects. Owing to the plectonemic nature of the DNA double helix, DNA polyhedra are complex catenanes of cyclic single strands. A half-turn of double helical DNA has a structure that permits it to behave as a unit tangle in topological construction. In addition to catenanes, it is straightforward to prepare specific DNA knots; Borromean rings have also been constructed from DNA. The goal of directing the assembly of periodic matter from DNA components requires a rigid species, for which the DNA double crossover molecule has been employed. Predictable patterns result when the double crossover motif is incorporated into triangular molecules which are ligated in one dimension. Two-dimensional crystals of double crossover molecules have been assembled and shown to possess specific programmed topographic patterns. It is possible to alter the patterns enzymatically and chemically, so that features may be added or subtracted readily. The rigidity of the double crossover molecule has been combined with the B-Z transition, thereby creating a DNA nanomechanical device. Future applications of DNA nanotechnology entail the assembly of threedimensional crystalline arrays, the use of DNA to direct the assembly of other molecules, and the applications of unusual DNA motifs to DNA-based computing.

We propose use of three dimensional graph structures for solving computational problems with DNA molecules. Building blocks of k-armed branched junction molecules are used to form graphs. Algorithms to solve the Hamiltonian Cycle problem and the 3-vertex colorability problems are presented. Both of these algorithms require constant number of laboratory steps regardless of the size of the graph. Two mathematical problems that raise from this approach are initiated.

An introduction to DNA computing for non-specialists is given in a novel fashion. Wet lab solutions using circular double stranded DNA molecules are sketched for several classical algorithmic problems. Laboratory tests of these suggestions are in the initial stages.

One of the most important properties of DNA molecules is their double stranded structure, based on the Watson—Crick complementarity of nucleotides. This new data structure and the operation of producing double stranded sequences by sticking together complementary single stranded strings can be used as basic ingredients in building two types of computability models, the so-called sticker systems and Watson—Crick automata. The former are language generating mechanisms, the latter are automata-like devices. Under certain conditions, both of them are proved to be computationally complete, that is, they have the same power as Turing machines. This is essentially due to the structure of DNA molecules, which is closely related to the twin-shuffle language, known to be a generator of the family of recursively enumerable languages.

The first insightful approach to visual shape perception was provided in the 1910's by the Gestalt school, with the formulation of rules according to which local visual features are perceptually grouped into larger units forming separate shapes (proximity, closeness, good continuation, etc.). Similarly oriented textural elements are perceptually grouped, and used to support 3D interpretations, suggesting that some of the tools used in the perception of static shapes are derived from tools in motion perception and optic flow analysis.

Another approach, rooted in differential geometry, was developed in the 1970's. Here, shape perception is conceived as mainly concerned with the computation of local surface curvature. Features like shading gradients, surface reflection spots, internal contour terminators would be clues from which local surface curvature would be inferred, in a consistent manner, over the whole surface.

The nature of the shape-generators used by the brain to make sense of the visual inputs is still mysterious. It is speculated here that in 3D, the shape generators would be rather crude ones, and would make use of rules such as “the height associated to a point inside a contour (assumed to be horizontal) increases with its remoteness from the contour”. The 2D shape generators seem to be economical and accurate. As a metaphor, one might suggest the equipotential curves that are produced in electrostatics by judiciously positioning a few charges.

The ability of the human visual system to rapidly categorize complex natural scenes presents a major challenge to our understanding of pattern recognition mechanisms. In particular, recent experiments that have measured the speed with which the visual system processes information impose very severe constraints on the coding mechanisms that are used. Indeed, the classical view, which holds that the information is transmitted in the form of a firing rate code, appears to be too slow and inefficient to be compatible. We argue that alternative coding schemes that make use of the pattern of spikes across a population of cells, and more specifically the order in which they fire could be more plausible hypothesis.

A method for compressed representation of photo realistic images by means of geometric models (vectorization) is described. This method is based on a general approach, developed in the framework of Singularity Theory. Applications to compressed representation and processing of three-dimensional scenes and animations are given. No specific knowledge in Singularity Theory or in Image Processing is assumed.