Richard Kennaway

Generation of Leaf Shape Through Early Patterns of Growth and Tissue Polarity

Shape-Shifting Signals Although orthogonal signaling systems seem to direct various developmental processes, few tissues remain in the same shape as they are at initiation to that of the final form. Arabidopsis leaves are free of the cell migrations that complicate animal development, and thus allowed Kuchen et al. (p. 1092) to track and model the trajectory of leaf growth under a variety of perturbations. Varying the values of parameters in their model produced outputs of different leaf shapes ranging from obcordate, ovate, and oval to elliptic, and offered predictions for genes that regulate the developmental process. The meristem at the growing tip of plants is home to stem cells and is the source of newly differentiating shoots and leaves. New leaves make their first appearance as bulges at the side of the dome-shaped meristem. Although these developmental events are under hormonal control, they also seem to be constrained by the physical properties of the meristem. Kierzkowski et al. (p. 1096) tested physical effects acting on the shoot apical meristem of growing tomato shoots that alter turgor pressure. Again, mathematical modeling combined with observations of plant tissue helped to define the different zones in the meristem that respond to diverse mechanical stimuli. A model for the development of leaf shape describes how it arises through oriented growth and tissue deformation. A major challenge in biology is to understand how buds comprising a few cells can give rise to complex plant and animal appendages like leaves or limbs. We address this problem through a combination of time-lapse imaging, clonal analysis, and computational modeling. We arrive at a model that shows how leaf shape can arise through feedback between early patterns of oriented growth and tissue deformation. Experimental tests through partial leaf ablation support this model and allow reevaluation of previous experimental studies. Our model allows a range of observed leaf shapes to be generated and predicts observed clone patterns in different species. Thus, our experimentally validated model may underlie the development and evolution of diverse organ shapes.

Generation of Diverse Biological Forms through Combinatorial Interactions between Tissue Polarity and Growth

A major problem in biology is to understand how complex tissue shapes may arise through growth. In many cases this process involves preferential growth along particular orientations raising the question of how these orientations are specified. One view is that orientations are specified through stresses in the tissue (axiality-based system). Another possibility is that orientations can be specified independently of stresses through molecular signalling (polarity-based system). The axiality-based system has recently been explored through computational modelling. Here we develop and apply a polarity-based system which we call the Growing Polarised Tissue (GPT) framework. Tissue is treated as a continuous material within which regionally expressed factors under genetic control may interact and propagate. Polarity is established by signals that propagate through the tissue and is anchored in regions termed tissue polarity organisers that are also under genetic control. Rates of growth parallel or perpendicular to the local polarity may then be specified through a regulatory network. The resulting growth depends on how specified growth patterns interact within the constraints of mechanically connected tissue. This constraint leads to the emergence of features such as curvature that were not directly specified by the regulatory networks. Resultant growth feeds back to influence spatial arrangements and local orientations of tissue, allowing complex shapes to emerge from simple rules. Moreover, asymmetries may emerge through interactions between polarity fields. We illustrate the value of the GPT-framework for understanding morphogenesis by applying it to a growing Snapdragon flower and indicate how the underlying hypotheses may be tested by computational simulation. We propose that combinatorial intractions between orientations and rates of growth, which are a key feature of polarity-based systems, have been exploited during evolution to generate a range of observed biological shapes.

On “on graph rewritings”

Abstract In 1984, Raoult has given a description of graph rewriting. His description is operational, despite the similarity which his constructions have to the category-theoretic concept of a pushout. We describe a modification to Raoults description of graph rewriting which allows the reduction of a redex to be described as a pushout, in a category of graphs where morphisms are not required to preserve graph structure. Our description can also handle term rewrite rules whose right-hand sides consist of a variable. Raoult specifically excludes such rules from his treatment. Rules of this form include the K combinator, the identity function, selector functions for extracting components of data structures, and the conditional. We prove the correctness of this implementation of term rewriting.

Graph Rewriting in Some Categories of Partial Morphisms

We present a definition of term graph rewriting as the taking of a pushout in a category of partial morphisms, adapting the rather ad hoc definitions we gave in [Ken87] so as to use a standard category-theoretic concept of partial morphism. This single-pushout construction is shown to coincide with the well-known double-pushout description of graph rewriting whenever the latter is defined. In general, the conditions for the single pushout to exist are weaker than those required for the double pushout. In some categories of graphs, no conditions at all are necessary.

Dactl: An Experimental Graph Rewriting Language

Dactl is an experimental language programming language based on fine grain graph transformations. It was developed in the context of a large parallel reduction machine project. The design of the language is outlined, and examples given of its use both as a compiler target language and as a programming language. Dactl has a formal semantics and stable implementations on a number of platforms.

Experience with and Requirements for a Gesture Description Language for Synthetic Animation

We have created software for automatic synthesis of signing animations from the HamNoSys transcription notation. In this process we have encountered certain shortcomings of the notation. We describe these, and consider how to develop a notation more suited to computer animation.

Evolution of Allometry in Antirrhinum

Correlated variation in shape and size (allometry) is a major component of natural diversity. We examined the evolutionary and genetic basis for allometry using leaves and flower petals of snapdragon species (Antirrhinum). A computational method was developed to capture shape and size variation in both types of organ within the Antirrhinum species group. The results show that the major component of variation between species involves positively correlated changes in leaf and petal size. The correlation was maintained in an F2 population derived from crossing two species with organs of different sizes, suggesting that developmental constraints were involved. Identification of the underlying genes as quantitative trait loci revealed that the larger species carried alleles that increased organ size at all loci. Although this was initially taken as evidence that directional selection has driven diversity in both leaf and petal size, simulations revealed that evolution without consistent directional selection, an undirected walk, could also account for the parental distribution of organ size alleles.

Director strings as combinators

A simple calculus (the Director String Calculus-DSC) for expressing abstractions is introduced, which captures the essence of the “long reach” combinators introduced by Turner. We present abstraction rules that preserve the applicative structure of the original lambda term, and that cannot increase the number of subterms in the translation. A translated lambda term can be reduced according to the evaluation rules of DSC. If this terminates with a DSC normal form, this can be translated into a lambda term using rules presented below. We call this process of abstracting a lambda term, reducing to normal form in the space of DSC terms, and translating back to a lambda term an implementation. We show that our implementation of the lambda calculus is correct: For lambda terms with a normal form that contains no lambdas (ground terms), the implementation is shown to yield a lambda calculus normal form. For lambda terms whose normal forms represent functions, it is shown that the implementation yields lambda terms that are beta-convertible in zero or more steps to the normal form of the original lambda term. In this sense, our implementation involves weak reduction according to Hindley et al. [9].

Comparing curried and uncurried rewriting

Abstract Currying is a transformation of term rewrite systems which may contain symbols of arbitrary arity into systems which contain only nullary symbols, together with a single binary symbol called application. We show that for all term rewrite systems (whether orthogonal or not) the following properties are preserved by this transformation: strong normalization, weak normalization, weak Church-Rosser, completeness, semi-completeness, and the non-convertibility of distinct normal forms. Under the condition of left-linearity we show preservation of the properties NF (if a term is reducible to a normal form,then its reducts are all reducible to the same normal form) and UN→ (a term is reducible to at most one normal form).We exhibit counterexamples to the preservation of NF and UN→ for non-left-linear systems.The results extend to partial currying(where some subset of the symbols are curried),and imply some modularity properties for unions of applicative systems.

Implementing term rewrite languages in DACTL

Dactl is a low-level language of graph rewriting, intended for programming highly parallel machines. The language includes, but is not restricted to, the limited form of graph rewriting which is commonly used to implement functional languages such as Miranda, ML, Hope, and Clean. In contrast to these functional languages, where the order in which subterms are evaluate (the evaluation strategy) is fixed for all programs, in Dactl the evaluation strategy is programmed explicitly. We define a translation of a functional language into Dactl, describe the problems encountered and their solution, and outline a proof that the translation is correct.