Dynamical modules in metabolism, cell and developmental biology.

Modularity is an essential feature of any adaptive complex system. Phenotypic traits are modules in the sense that they have a distinguishable structure or function, which can vary (quasi-)independently from its context. Since all phenotypic traits are the product of some underlying regulatory dynamics, the generative processes that constitute the genotype-phenotype map must also be functionally modular. Traditionally, modular processes have been identified as structural modules in regulatory networks. However, structure only constrains, but does not determine, the dynamics of a process. Here, we propose an alternative approach that decomposes the behaviour of a complex regulatory system into elementary activity-functions. Modular activities can occur in networks that show no structural modularity, making dynamical modularity more widely applicable than structural decomposition. Furthermore, the behaviour of a regulatory system closely mirrors its functional contribution to the outcome of a process, which makes dynamical modularity particularly suited for functional decomposition. We illustrate our approach with numerous examples from the study of metabolism, cellular processes, as well as development and pattern formation. We argue that dynamical modules provide a shared conceptual foundation for developmental and evolutionary biology, and serve as the foundation for a new account of process homology, which is presented in a separate contribution by DiFrisco and Jaeger to this focus issue.

Auxin influx importers modulate serration along the leaf margin.

Leaf shape in Arabidopsis is modulated by patterning events in the margin that utilize a PIN-based auxin exporter/CUC2 transcription factor system to define regions of promotion and retardation of growth, leading to morphogenesis. In addition to auxin exporters, leaves also express auxin importers, notably members of the AUX1/LAX family. In contrast to their established roles in embryogenesis, lateral root and leaf initiation, the function of these transporters in leaf development is poorly understood. We report that three of these genes (AUX1, LAX1 and LAX2) show specific and dynamic patterns of expression during early leaf development in Arabidopsis, and that loss of expression of all three genes is required for observation of a phenotype in which morphogenesis (serration) is decreased. We used these expression patterns and mutant phenotypes to develop a margin-patterning model that incorporates an AUX1/LAX1/LAX2 auxin import module that influences the extent of leaf serration. Testing of this model by margin-localized expression of axr3-1 (AXR17) provides further insight into the role of auxin in leaf morphogenesis.

Bioattractors: dynamical systems theory and the evolution of regulatory processes.

In this paper, we illustrate how dynamical systems theory can provide a unifying conceptual framework for evolution of biological regulatory systems. Our argument is that the genotype-phenotype map can be characterized by the phase portrait of the underlying regulatory process. The features of this portrait--such as attractors with associated basins and their bifurcations--define the regulatory and evolutionary potential of a system. We show how the geometric analysis of phase space connects Waddington's epigenetic landscape to recent computational approaches for the study of robustness and evolvability in network evolution. We discuss how the geometry of phase space determines the probability of possible phenotypic transitions. Finally, we demonstrate how the active, self-organizing role of the environment in phenotypic evolution can be understood in terms of dynamical systems concepts. This approach yields mechanistic explanations that go beyond insights based on the simulation of evolving regulatory networks alone. Its predictions can now be tested by studying specific, experimentally tractable regulatory systems using the tools of modern systems biology. A systematic exploration of such systems will enable us to understand better the nature and origin of the phenotypic variability, which provides the substrate for evolution by natural selection.

The inheritance of process: a dynamical systems approach.

A central unresolved problem of evolutionary biology concerns the way in which evolution at the genotypic level relates to the evolution of phenotypes. This genotype-phenotype map involves developmental and physiological processes, which are complex and not well understood. These processes co-determine the rate and direction of adaptive change by shaping the distribution of phenotypic variability on which selection can act. In this study, we argue-expanding on earlier ideas by Goodwin, Oster, and Alberch-that an explicit treatment of this map in terms of dynamical systems theory can provide an integrated understanding of evolution and development. We describe a conceptual framework, which demonstrates how development determines the probability of possible phenotypic transitions-and hence the evolvability of a biological system. We use a simple conceptual model to illustrate how the regulatory dynamics of the genotype-phenotype map can be passed on from generation to generation, and how heredity itself can be treated as a dynamic process. Our model yields explanations for punctuated evolutionary dynamics, the difference between micro- and macroevolution, and for the role of the environment in major phenotypic transitions. We propose a quantitative research program in evolutionary developmental systems biology-combining experimental methods with mathematical modeling-which aims at elaborating our conceptual framework by applying it to a wide range of evolving developmental systems. This requires a large and sustained effort, which we believe is justified by the significant potential benefits of an extended evolutionary theory that uses dynamic molecular genetic data to reintegrate development and evolution.

A shift toward smaller cell size via manipulation of cell cycle gene expression acts to smoothen Arabidopsis leaf shape.

Understanding the relationship of the size and shape of an organism to the size, shape, and number of its constituent cells is a basic problem in biology; however, numerous studies indicate that the relationship is complex and often nonintuitive. To investigate this problem, we used a system for the inducible expression of genes involved in the G1/S transition of the plant cell cycle and analyzed the outcome on leaf shape. By combining a careful developmental staging with a quantitative analysis of the temporal and spatial response of cell division pattern and leaf shape to these manipulations, we found that changes in cell division frequency occurred much later than the observed changes in leaf shape. These data indicate that altered cell division frequency cannot be causally involved in the observed change of shape. Rather, a shift to a smaller cell size as a result of the genetic manipulations performed correlated with the formation of a smoother leaf perimeter, i.e. appeared to be the primary cellular driver influencing form. These data are discussed in the context of the relationship of cell division, growth, and leaf size and shape.

LEAFPROCESSOR: a new leaf phenotyping tool using contour bending energy and shape cluster analysis.

*Significant progress has been made in the identification of the genetic factors controlling leaf shape. However, no integrated solution for the quantification and categorization of leaf form has been developed. In particular, the analysis of local changes in margin growth, which define many of the differences in shape, remains problematical. *Here, we report on a software package (LEAFPROCESSOR) which provides a semi-automatic and landmark-free method for the analysis of a range of leaf-shape parameters, combining both single metrics and principal component analysis. In particular, we explore the use of bending energy as a tool for the analysis of global and local leaf perimeter deformation. *As a test case for the implementation of the LEAFPROCESSOR program, we show that this integrated analysis leads to deeper insights into the morphogenic changes underpinning a series of previously identified Arabidopsis leaf-shape mutants. Our analysis reveals that many of these mutants which, at first sight, show similar leaf morphology, can be distinguished via our shape analysis. *The LEAFPROCESSOR program provides a novel integrated tool for the analysis of leaf shape.

Regulative feedback in pattern formation: towards a general relativistic theory of positional information.

Positional specification by morphogen gradients is traditionally viewed as a two-step process. A gradient is formed and then interpreted, providing a spatial metric independent of the target tissue, similar to the concept of space in classical mechanics. However, the formation and interpretation of gradients are coupled, dynamic processes. We introduce a conceptual framework for positional specification in which cellular activity feeds back on positional information encoded by gradients, analogous to the feedback between mass-energy distribution and the geometry of space-time in Einstein's general theory of relativity. We discuss how such general relativistic positional information (GRPI) can guide systems-level approaches to pattern formation.

Development: dissecting the dynamics of segment determination.

A novel combination of high-resolution time-course expression data and computational modelling has provided a remarkably detailed picture of a key stage of Drosophila segment determination, highlighting the dynamic nature of this process.