Point clouds are usually represented either globally, as surfaces, or locally, as sets of points ... more Point clouds are usually represented either globally, as surfaces, or locally, as sets of points with small neighbourhoods. We propose an intermediate representation, called ribbons, which is obtained by partitioning a point cloud into one-dimensional strips. This representation is well suited to the placement of strokes in non-photorealistic rendering, and can be visualized efficiently using quad strips. Methods for performing hatching, cross hatching, and silhouette renderings are presented. Ribbons also allow for the application of curvebased operations to the point cloud.
We propose Partition of Unity Parametrics (PUPs), a natural extension of NURBS that maintains aff... more We propose Partition of Unity Parametrics (PUPs), a natural extension of NURBS that maintains affine invariance. PUPs replace the weighted basis functions of NURBS with arbitrary weight-functions (WFs). By choosing appropriate WFs, PUPs yield a comprehensive geometric modeling framework, accounting for a variety of beneficial properties, such as local-support, specified smoothness, arbitrary sharp features and approximating or interpolating curves. Additionally, we consider interactive specification of WFs to fine-tune the character of curves and generate nontrivial effects. This serves as a basis for a system where users model the tools used for modeling, here weightfunctions, in tandem with the model itself, which we dub a meta-modeling system. PUP curves and surfaces are considered in detail. Curves illustrate basic concepts that apply directly to surfaces. For surfaces, the advantages of PUPs are more pronounced; permitting nontensor WFs and direct parameter space manipulations. These features allow us to address two difficult geometric modeling problems (sketching features onto surfaces and converting planar meshes into parametric surfaces) in a conceptually and computationally simple way.
The plant hormone auxin plays an essential role in the patterning of plant structures. Biological... more The plant hormone auxin plays an essential role in the patterning of plant structures. Biological hypotheses supported by computational models suggest that auxin may fulfil this role by regulating its own transport, but the plausibility of previously proposed models has been questioned. We applied the notion of unidirectional fluxes and the formalism of Petri nets to show that the key modes of auxin-driven patterning-the formation of convergence points and the formation of canals-can be implemented by biochemically plausible networks, with the fluxes measured by dedicated tally molecules or by efflux and influx carriers themselves. Common elements of these networks include a positive feedback of auxin efflux on the allocation of membrane-bound auxin efflux carriers (PIN proteins), and a modulation of this allocation by auxin in the extracellular space. Auxin concentration in the extracellular space is the only information exchanged by the cells. Canalization patterns are produced when auxin efflux and influx act antagonistically: an increase in auxin influx or concentration in the extracellular space decreases the abundance of efflux carriers in the adjacent segment of the membrane. In contrast, convergence points emerge in networks in which auxin efflux and influx act synergistically. A change in a single reaction rate may result in a dynamic switch between these modes, suggesting plausible molecular implementations of coordinated patterning of organ initials and vascular strands predicted by the dual polarization theory.
Auxin plays a key regulatory role in plant development. According to our current understanding, t... more Auxin plays a key regulatory role in plant development. According to our current understanding, the morphogenetic action of auxin relies on its polar transport and the feedback between this transport and the localization of auxin transporters. Computational models complement experimental data in studies of auxin-driven development: they help understand the self-organizing aspects of auxin patterning, reveal whether hypothetical mechanisms inferred from experiments are plausible, and highlight differences between competing hypotheses that can be used to direct further experimental studies. In this chapter we present the state of the art in the computational modeling of auxin patterning and auxin-driven development in plants. We first discuss the methodological foundations of model construction: computational representations of tissues, cells, and molecular components of the studied systems. On this basis, we present mathematical models of auxin transport and the essential properties of pattern formation mechanisms involving auxin. We then review some of the key areas that have been investigated with the use of models: phyllotactic patterning of lateral organs in the shoot apical meristem, determination of leaf shape and vasculature, long-distance signaling and apical control of development, and auxin patterning in the root. The chapter is concluded with a brief review of current open problems.
Bark patterns are a visually important characteristic of trees, typically attributed to fractures... more Bark patterns are a visually important characteristic of trees, typically attributed to fractures occurring during secondary growth of the trunk and branches. An understanding of bark pattern formation has been hampered by insufficient information regarding the biomechanical properties of bark and the corresponding difficulties in faithfully modelling bark fractures using continuum mechanics. This study focuses on the genus Xanthorrhoea (grasstrees), which have an unusual bark-like structure composed of distinct leaf bases connected by sticky resin. Due to its discrete character, this structure is well suited for computational studies. A dynamic computational model of grasstree development was created. The model captures both the phyllotactic pattern of leaf bases during primary growth and the changes in the trunk's width during secondary growth. A biomechanical representation based on a system of masses connected by springs is used for the surface of the trunk, permitting the e...
The hormone auxin plays a crucial role in plant morphogenesis. In the shoot apical meristem, the ... more The hormone auxin plays a crucial role in plant morphogenesis. In the shoot apical meristem, the PIN-FORMED1 (PIN1) efflux carrier concentrates auxin into local maxima in the epidermis, which position incipient leaf or floral primordia. From these maxima, PIN1 transports auxin into internal tissues along emergent paths that pattern leaf and stem vasculature. In Arabidopsis thaliana, these functions are attributed to a single PIN1 protein. Using phylogenetic and gene synteny analysis we identified an angiosperm PIN clade sister to PIN1, here termed Sister-of-PIN1 (SoPIN1), which is present in all sampled angiosperms except for Brassicaceae, including Arabidopsis. Additionally, we identified a conserved duplication of PIN1 in the grasses: PIN1a and PIN1b. In Brachypodium distachyon, SoPIN1 is highly expressed in the epidermis and is consistently polarized toward regions of high expression of the DR5 auxin-signaling reporter, which suggests that SoPIN1 functions in the localization of new primordia. In contrast, PIN1a and PIN1b are highly expressed in internal tissues, suggesting a role in vascular patterning. PIN1b is expressed in broad regions spanning the space between new primordia and previously formed vasculature, suggesting a role in connecting new organs to auxin sinks in the older tissues. Within these regions, PIN1a forms narrow canals that likely pattern future veins. Using a computer model, we reproduced the observed spatio-temporal expression and localization patterns of these proteins by assuming that SoPIN1 is polarized up the auxin gradient, and PIN1a and PIN1b are polarized to different degrees with the auxin flux. Our results suggest that examination and modeling of PIN dynamics in plants outside of Brassicaceae will offer insights into auxin-driven patterning obscured by the loss of the SoPIN1 clade in Brassicaceae.
Like many natural phenomenon, clouds are often modeled using procedural methods, which may be dif... more Like many natural phenomenon, clouds are often modeled using procedural methods, which may be difficult for an artist to control. In this paper, a freehand sketching system is proposed to control the modeling of volumetric clouds. Input sketches are used to generate a closed mesh, defining the initial cloud volume. Sketch analysis as well as the elevation at which the cloud is drawn is used to identify the cloud type and then generate a mesh with the appropriate characteristics for the determined cloud type. The cloud volume can then be edited using Boolean operations that allow for addition and removal of material from existing clouds. The proposed modeling system provides an intuitive framework for generating individual clouds and entire cloud fields, while maintaining the interactive rates necessitated by the sketch-based paradigm.
Point clouds are usually represented either globally, as surfaces, or locally, as sets of points ... more Point clouds are usually represented either globally, as surfaces, or locally, as sets of points with small neighbourhoods. We propose an intermediate representation, called ribbons, which is obtained by partitioning a point cloud into one-dimensional strips. This representation is well suited to the placement of strokes in non-photorealistic rendering, and can be visualized efficiently using quad strips. Methods for performing hatching, cross hatching, and silhouette renderings are presented. Ribbons also allow for the application of curvebased operations to the point cloud.
We propose Partition of Unity Parametrics (PUPs), a natural extension of NURBS that maintains aff... more We propose Partition of Unity Parametrics (PUPs), a natural extension of NURBS that maintains affine invariance. PUPs replace the weighted basis functions of NURBS with arbitrary weight-functions (WFs). By choosing appropriate WFs, PUPs yield a comprehensive geometric modeling framework, accounting for a variety of beneficial properties, such as local-support, specified smoothness, arbitrary sharp features and approximating or interpolating curves. Additionally, we consider interactive specification of WFs to fine-tune the character of curves and generate nontrivial effects. This serves as a basis for a system where users model the tools used for modeling, here weightfunctions, in tandem with the model itself, which we dub a meta-modeling system. PUP curves and surfaces are considered in detail. Curves illustrate basic concepts that apply directly to surfaces. For surfaces, the advantages of PUPs are more pronounced; permitting nontensor WFs and direct parameter space manipulations. These features allow us to address two difficult geometric modeling problems (sketching features onto surfaces and converting planar meshes into parametric surfaces) in a conceptually and computationally simple way.
Proceedings of the National Academy of Sciences, 2011
Biological shapes are often produced by the iterative generation of repeated units. The mechanist... more Biological shapes are often produced by the iterative generation of repeated units. The mechanistic basis of such iteration is an area of intense investigation. Leaf development in the model plant Arabidopsis is one such example where the repeated generation of leaf margin protrusions, termed serrations, is a key feature of final shape. However, the regulatory logic underlying this process is unclear. Here, we use a combination of developmental genetics and computational modeling to show that serration development is the morphological read-out of a spatially distributed regulatory mechanism, which creates interspersed activity peaks of the growth-promoting hormone auxin and the CUP-SHAPED COTYLEDON2 (CUC2) transcription factor. This mechanism operates at the growing leaf margin via a regulatory module consisting of two feedback loops working in concert. The first loop relates the transport of auxin to its own distribution, via polar membrane localization of the PIN-FORMED1 (PIN1) efflux transporter. This loop captures the potential of auxin to generate self-organizing patterns in diverse developmental contexts. In the second loop, CUC2 promotes the generation of PIN1-dependent auxin activity maxima while auxin represses CUC2 expression. This CUC2-dependent loop regulates activity of the conserved auxin efflux module in leaf margins to generate stable serration patterns. Conceptualizing leaf margin development via this mechanism also helps to explain how other developmental regulators influence leaf shape.
The use of computational techniques increasingly permeates developmental biology, from the acquis... more The use of computational techniques increasingly permeates developmental biology, from the acquisition, processing and analysis of experimental data to the construction of models of organisms. Specifically, models help to untangle the non-intuitive relations between local morphogenetic processes and global patterns and forms. We survey the modeling techniques and selected models that are designed to elucidate plant development in mechanistic terms, with an emphasis on: the history of mathematical and computational approaches to developmental plant biology; the key objectives and methodological aspects of model construction; the diverse mathematical and computational methods related to plant modeling; and the essence of two classes of models, which approach plant morphogenesis from the geometric and molecular perspectives. In the geometric domain, we review models of cell division patterns, phyllotaxis, the form and vascular patterns of leaves, and branching patterns. In the molecular-level domain, we focus on the currently most extensively developed theme: the role of auxin in plant morphogenesis. The review is addressed to both biologists and computational modelers.
The practical value of this approach is demonstrated by modeling the architecture of trees and va... more The practical value of this approach is demonstrated by modeling the architecture of trees and vasculature in plants. Trees are modeled using markers of empty space to mediate competition between branches. When vascular patterns are modeled, the markers of empty space ...
Point clouds are usually represented either globally, as surfaces, or locally, as sets of points ... more Point clouds are usually represented either globally, as surfaces, or locally, as sets of points with small neighbourhoods. We propose an intermediate representation, called ribbons, which is obtained by partitioning a point cloud into one-dimensional strips. This representation is well suited to the placement of strokes in non-photorealistic rendering, and can be visualized efficiently using quad strips. Methods for performing hatching, cross hatching, and silhouette renderings are presented. Ribbons also allow for the application of curvebased operations to the point cloud.
We propose Partition of Unity Parametrics (PUPs), a natural extension of NURBS that maintains aff... more We propose Partition of Unity Parametrics (PUPs), a natural extension of NURBS that maintains affine invariance. PUPs replace the weighted basis functions of NURBS with arbitrary weight-functions (WFs). By choosing appropriate WFs, PUPs yield a comprehensive geometric modeling framework, accounting for a variety of beneficial properties, such as local-support, specified smoothness, arbitrary sharp features and approximating or interpolating curves. Additionally, we consider interactive specification of WFs to fine-tune the character of curves and generate nontrivial effects. This serves as a basis for a system where users model the tools used for modeling, here weightfunctions, in tandem with the model itself, which we dub a meta-modeling system. PUP curves and surfaces are considered in detail. Curves illustrate basic concepts that apply directly to surfaces. For surfaces, the advantages of PUPs are more pronounced; permitting nontensor WFs and direct parameter space manipulations. These features allow us to address two difficult geometric modeling problems (sketching features onto surfaces and converting planar meshes into parametric surfaces) in a conceptually and computationally simple way.
The plant hormone auxin plays an essential role in the patterning of plant structures. Biological... more The plant hormone auxin plays an essential role in the patterning of plant structures. Biological hypotheses supported by computational models suggest that auxin may fulfil this role by regulating its own transport, but the plausibility of previously proposed models has been questioned. We applied the notion of unidirectional fluxes and the formalism of Petri nets to show that the key modes of auxin-driven patterning-the formation of convergence points and the formation of canals-can be implemented by biochemically plausible networks, with the fluxes measured by dedicated tally molecules or by efflux and influx carriers themselves. Common elements of these networks include a positive feedback of auxin efflux on the allocation of membrane-bound auxin efflux carriers (PIN proteins), and a modulation of this allocation by auxin in the extracellular space. Auxin concentration in the extracellular space is the only information exchanged by the cells. Canalization patterns are produced when auxin efflux and influx act antagonistically: an increase in auxin influx or concentration in the extracellular space decreases the abundance of efflux carriers in the adjacent segment of the membrane. In contrast, convergence points emerge in networks in which auxin efflux and influx act synergistically. A change in a single reaction rate may result in a dynamic switch between these modes, suggesting plausible molecular implementations of coordinated patterning of organ initials and vascular strands predicted by the dual polarization theory.
Auxin plays a key regulatory role in plant development. According to our current understanding, t... more Auxin plays a key regulatory role in plant development. According to our current understanding, the morphogenetic action of auxin relies on its polar transport and the feedback between this transport and the localization of auxin transporters. Computational models complement experimental data in studies of auxin-driven development: they help understand the self-organizing aspects of auxin patterning, reveal whether hypothetical mechanisms inferred from experiments are plausible, and highlight differences between competing hypotheses that can be used to direct further experimental studies. In this chapter we present the state of the art in the computational modeling of auxin patterning and auxin-driven development in plants. We first discuss the methodological foundations of model construction: computational representations of tissues, cells, and molecular components of the studied systems. On this basis, we present mathematical models of auxin transport and the essential properties of pattern formation mechanisms involving auxin. We then review some of the key areas that have been investigated with the use of models: phyllotactic patterning of lateral organs in the shoot apical meristem, determination of leaf shape and vasculature, long-distance signaling and apical control of development, and auxin patterning in the root. The chapter is concluded with a brief review of current open problems.
Bark patterns are a visually important characteristic of trees, typically attributed to fractures... more Bark patterns are a visually important characteristic of trees, typically attributed to fractures occurring during secondary growth of the trunk and branches. An understanding of bark pattern formation has been hampered by insufficient information regarding the biomechanical properties of bark and the corresponding difficulties in faithfully modelling bark fractures using continuum mechanics. This study focuses on the genus Xanthorrhoea (grasstrees), which have an unusual bark-like structure composed of distinct leaf bases connected by sticky resin. Due to its discrete character, this structure is well suited for computational studies. A dynamic computational model of grasstree development was created. The model captures both the phyllotactic pattern of leaf bases during primary growth and the changes in the trunk's width during secondary growth. A biomechanical representation based on a system of masses connected by springs is used for the surface of the trunk, permitting the e...
The hormone auxin plays a crucial role in plant morphogenesis. In the shoot apical meristem, the ... more The hormone auxin plays a crucial role in plant morphogenesis. In the shoot apical meristem, the PIN-FORMED1 (PIN1) efflux carrier concentrates auxin into local maxima in the epidermis, which position incipient leaf or floral primordia. From these maxima, PIN1 transports auxin into internal tissues along emergent paths that pattern leaf and stem vasculature. In Arabidopsis thaliana, these functions are attributed to a single PIN1 protein. Using phylogenetic and gene synteny analysis we identified an angiosperm PIN clade sister to PIN1, here termed Sister-of-PIN1 (SoPIN1), which is present in all sampled angiosperms except for Brassicaceae, including Arabidopsis. Additionally, we identified a conserved duplication of PIN1 in the grasses: PIN1a and PIN1b. In Brachypodium distachyon, SoPIN1 is highly expressed in the epidermis and is consistently polarized toward regions of high expression of the DR5 auxin-signaling reporter, which suggests that SoPIN1 functions in the localization of new primordia. In contrast, PIN1a and PIN1b are highly expressed in internal tissues, suggesting a role in vascular patterning. PIN1b is expressed in broad regions spanning the space between new primordia and previously formed vasculature, suggesting a role in connecting new organs to auxin sinks in the older tissues. Within these regions, PIN1a forms narrow canals that likely pattern future veins. Using a computer model, we reproduced the observed spatio-temporal expression and localization patterns of these proteins by assuming that SoPIN1 is polarized up the auxin gradient, and PIN1a and PIN1b are polarized to different degrees with the auxin flux. Our results suggest that examination and modeling of PIN dynamics in plants outside of Brassicaceae will offer insights into auxin-driven patterning obscured by the loss of the SoPIN1 clade in Brassicaceae.
Like many natural phenomenon, clouds are often modeled using procedural methods, which may be dif... more Like many natural phenomenon, clouds are often modeled using procedural methods, which may be difficult for an artist to control. In this paper, a freehand sketching system is proposed to control the modeling of volumetric clouds. Input sketches are used to generate a closed mesh, defining the initial cloud volume. Sketch analysis as well as the elevation at which the cloud is drawn is used to identify the cloud type and then generate a mesh with the appropriate characteristics for the determined cloud type. The cloud volume can then be edited using Boolean operations that allow for addition and removal of material from existing clouds. The proposed modeling system provides an intuitive framework for generating individual clouds and entire cloud fields, while maintaining the interactive rates necessitated by the sketch-based paradigm.
Point clouds are usually represented either globally, as surfaces, or locally, as sets of points ... more Point clouds are usually represented either globally, as surfaces, or locally, as sets of points with small neighbourhoods. We propose an intermediate representation, called ribbons, which is obtained by partitioning a point cloud into one-dimensional strips. This representation is well suited to the placement of strokes in non-photorealistic rendering, and can be visualized efficiently using quad strips. Methods for performing hatching, cross hatching, and silhouette renderings are presented. Ribbons also allow for the application of curvebased operations to the point cloud.
We propose Partition of Unity Parametrics (PUPs), a natural extension of NURBS that maintains aff... more We propose Partition of Unity Parametrics (PUPs), a natural extension of NURBS that maintains affine invariance. PUPs replace the weighted basis functions of NURBS with arbitrary weight-functions (WFs). By choosing appropriate WFs, PUPs yield a comprehensive geometric modeling framework, accounting for a variety of beneficial properties, such as local-support, specified smoothness, arbitrary sharp features and approximating or interpolating curves. Additionally, we consider interactive specification of WFs to fine-tune the character of curves and generate nontrivial effects. This serves as a basis for a system where users model the tools used for modeling, here weightfunctions, in tandem with the model itself, which we dub a meta-modeling system. PUP curves and surfaces are considered in detail. Curves illustrate basic concepts that apply directly to surfaces. For surfaces, the advantages of PUPs are more pronounced; permitting nontensor WFs and direct parameter space manipulations. These features allow us to address two difficult geometric modeling problems (sketching features onto surfaces and converting planar meshes into parametric surfaces) in a conceptually and computationally simple way.
Proceedings of the National Academy of Sciences, 2011
Biological shapes are often produced by the iterative generation of repeated units. The mechanist... more Biological shapes are often produced by the iterative generation of repeated units. The mechanistic basis of such iteration is an area of intense investigation. Leaf development in the model plant Arabidopsis is one such example where the repeated generation of leaf margin protrusions, termed serrations, is a key feature of final shape. However, the regulatory logic underlying this process is unclear. Here, we use a combination of developmental genetics and computational modeling to show that serration development is the morphological read-out of a spatially distributed regulatory mechanism, which creates interspersed activity peaks of the growth-promoting hormone auxin and the CUP-SHAPED COTYLEDON2 (CUC2) transcription factor. This mechanism operates at the growing leaf margin via a regulatory module consisting of two feedback loops working in concert. The first loop relates the transport of auxin to its own distribution, via polar membrane localization of the PIN-FORMED1 (PIN1) efflux transporter. This loop captures the potential of auxin to generate self-organizing patterns in diverse developmental contexts. In the second loop, CUC2 promotes the generation of PIN1-dependent auxin activity maxima while auxin represses CUC2 expression. This CUC2-dependent loop regulates activity of the conserved auxin efflux module in leaf margins to generate stable serration patterns. Conceptualizing leaf margin development via this mechanism also helps to explain how other developmental regulators influence leaf shape.
The use of computational techniques increasingly permeates developmental biology, from the acquis... more The use of computational techniques increasingly permeates developmental biology, from the acquisition, processing and analysis of experimental data to the construction of models of organisms. Specifically, models help to untangle the non-intuitive relations between local morphogenetic processes and global patterns and forms. We survey the modeling techniques and selected models that are designed to elucidate plant development in mechanistic terms, with an emphasis on: the history of mathematical and computational approaches to developmental plant biology; the key objectives and methodological aspects of model construction; the diverse mathematical and computational methods related to plant modeling; and the essence of two classes of models, which approach plant morphogenesis from the geometric and molecular perspectives. In the geometric domain, we review models of cell division patterns, phyllotaxis, the form and vascular patterns of leaves, and branching patterns. In the molecular-level domain, we focus on the currently most extensively developed theme: the role of auxin in plant morphogenesis. The review is addressed to both biologists and computational modelers.
The practical value of this approach is demonstrated by modeling the architecture of trees and va... more The practical value of this approach is demonstrated by modeling the architecture of trees and vasculature in plants. Trees are modeled using markers of empty space to mediate competition between branches. When vascular patterns are modeled, the markers of empty space ...
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Papers by Adam Runions