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Modeling plant growth and development
 Current Opinion in Plant Biology 7
, 2004
"... Computational plant models or 'virtual plants' are increasingly seen as a useful tool for comprehending complex relationships between gene function, plant physiology, plant development, and the resulting plant form. The theory of Lsystems, which was introduced by Lindemayer in 1968, has ..."
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Computational plant models or 'virtual plants' are increasingly seen as a useful tool for comprehending complex relationships between gene function, plant physiology, plant development, and the resulting plant form. The theory of Lsystems, which was introduced by Lindemayer in 1968, has led to a wellestablished methodology for simulating the branching architecture of plants. Many current architectural models provide insights into the mechanisms of plant development by incorporating physiological processes, such as the transport and allocation of carbon. Other models aim at elucidating the geometry of plant organs, including flower petals and apical meristems, and are beginning to address the relationship between patterns of gene expression and the resulting plant form. Introduction The term 'model' has many meanings in biology. Representative organisms are commonly referred to as model organisms, qualitative hypotheses are referred to as models, and the statistical analysis of experimental data is referred to as modeling. In this review, we consider mathematical models, in which the system under study is described using mathematical formulae. In particular, we look at spatial models of plants, which take plant form into account. Spatial models may treat plant geometry as a continuum (which is particularly justified in the description of individual organs, such as leaves or petals) or as an arrangement of discrete components (also called modules) in space. In the latter case, the definition of components depends on the level of plant organization at which a study is carried out. Frequently used components include individual cells, architectural modules (e.g. internodes, buds, apices, leaves, and flowers), and whole plants in the case of ecological models. The models may be static, capturing plant form at a particular point in time, or developmental, describing the form as a result of growth. Developmental models may in turn be descriptive (or reconstructive), integrating the results of measurements of form over time, or mechanistic, attempting to elucidate the development of form in terms of the underlying biological, chemical, and physical processes. Developmental models are commonly explored using computational or simulation techniques. The simulation software may be generalpurpose, intended to capture a variety of developmental processes depending on the input files, or specialpurpose, intended to capture a specific phenomenon. Input data range from a few parameters in models capturing a fundamental mechanism to thousands of measurements in calibrated descriptive models of specific plants (species or individuals). Standard numerical outputs (i.e. numbers or plots) may be complemented by computergenerated images and animations. There is as yet no consensus regarding the value of computational models in developmental biology. Opinions diverge on the most fundamental issues, such as the role of theory in biological understanding, the usefulness of applying chains of mathematical deductions to biological data, and the appropriateness of transplanting research methodologies from physics to biology Several key benefits have been attributed to the use of computational models. First, they can provide a quantitative understanding of developmental mechanisms when qualitative descriptions are fundamentally inadequate. For example, computational models can assist in the analysis of genetic regulatory mechanisms, characterize phyllotactic patterns, or provide a detailed description of growth dynamics. Second, models might lead to a synthetic (i.e. systemic or integrative) understanding of the interplay between various aspects of development, such as genetic regulation, physiological processes, environmental influences, and the development of the whole plant. And third, the use of computational models can identify of areas of ignorance and guide further empirical research. Adrian Bell, one of the pioneers of plant modeling, summarily characterized these benefits as follows ''The very process of constructing computer simulations to reproduce a particular branching structure can be a useful experience in its own right, even without proceeding to the use of such a simulation to test an hypothesis. Either the mor The models considered in this review have been organized into three classes: models of plant architecture, models of organs and tissues, and models incorporating genetic regulatory networks. The separation between the first two classes reflects the different mathematical structures of the models. This difference is related to the properties of threedimensional space: the arrangement of components of a branching structure brings about problems that are different to those presented by the arrangement of components that extend over areas or volumes (e.g. cells in a tissue must properly fit without gaps or overlaps). From a practical perspective, architectural models have often been motivated by their prospective applications to forestry, agriculture, or horticulture, whereas models of plant organs have been motivated by more fundamental questions of biological development. The emerging class of models that incorporate genetic regulatory networks is treated separately because of yet another modeling methodology and origin, which are related to the modeling of processes within individual cells. Models of plant architecture Models of plant architecture are based on the ecological concept of a plant as a population of semiautonomous modules, and describe a growing plant as an integration of the activities of these modules Modules of the same type may have diverse behaviors due to different states (i.e. values of variables that are associated with the modules) and to signaling between the modules. The convenience of expressing signaling in dynamically changing branching structures (using socalled contextsensitive productions) is an essential feature of Lsystems. An example is the use of context sensitivity in the simulation of the branching pattern and flowering sequence of Mycelis muralis A distinctive feature of Lsystems is that they give rise to a class of programming languages for specifying the models. This makes it possible to construct generic simulation software that is capable of modeling a large variety of plants, plant parts, and processes in plants at the architectural level, given their specifications in an Lsystembased language Although other architectural models are not explicitly expressed using Lsystems, they share the underlying philosophy of describing a growing branching structure in terms of the activities of individual plant modules. Models of organs and tissues Although plant architecture is commonly treated in a modular fashion, the choice between discrete and continuous descriptions of tissues (which form surfaces or volumes) is less obvious. Both approaches, as well as their combinations, are used. These approaches echo two competing views of the relation of cells to an organism: 'cells make an organism' and 'an organism makes cells'. The emphasis on cells as the building block calls for a discrete model. Emphasis on an organism, on the other hand, makes it possible to treat tissues in a continuous fashion, either abstracting from their cellular composition or treating cellular patterns as an effect of higherlevel processes. The concurrent use of both approaches also reflects the fact that models of growing surfaces and volumes are mathematically more complicated than models of linear and branching structures, and definitive modeling methods are yet to emerge. In the discrete approach, the simulation software must manage structural changes that occur in a system described as a growing assembly of modules. For example, when a cell divides, the state variables that characterize the parent cell are no longer part of the description of the whole system and must be removed; the state variables that characterize the daughter cells must be inserted; and the set of equations that relate all of these variables must be updated, taking into account the position of the new cells with respect to their neighbors in the structure. Parametric Lsystems [9] offer a solution to these problems for filamentous and branching structures, but extensions of Lsystems to surfaces In the continuous approach, the tissue is treated as a whole without division into components (at least conceptually; division may be imposed by numerical methods used to implement the models). The problem of dealing with the dynamically changing arrangement of modules is thus avoided. The fundamental notion for describing growth in continuous terms is the strain tensor, a notion defined in the mechanics of continuous media to characterize local expansion or contraction of a material in various directions. Local growth directly affects the local (Gaussian) curvature of surfaces, and causes global changes to the shape of surfaces and volumes. Accordingly, physical experiments, mathematical analyses and computer simulations have demonstrated that the wrinkled shapes of leaves and petals can be produced as emergent phenomena due to differential growth, without direct genetic control Models incorporating genetic regulatory networks Within the bounds of geometric and mechanical constraints, developmental patterns and forms are, in the final account, determined by genes. Pursuing this relationship, Mendoza and AlvarezBuylla [35] integrated numerous experimental data into a regulatory network of 11 genes that control the shoot branching pattern and switch to flowering in Arabidopsis. The network was described as a logic circuit, analogous to those found in computers. Simulations showed that it had several stable states, which could be associated with different cellular fates in flower morphogenesis. Subsequently, the same authors applied a similar formalism to capture the regulatory pathways underlying the differentiation of Arabidopsis root hairs As the methodologies for modeling individual cells Conclusions The methodology for modeling plant development at the architectural level, taking into account diverse physiological processes and ecological interactions between plants, is already well established. Several modeling packages exist, and advanced architectural models are routinely presented in the literature. These models are often created with practical applications to forestry, agriculture and horticulture in mind. The introduction of mathematical modeling and computer simulations as a research methodology in fundamental plant biology is a new phenomenon. A combination of established models that operate at the architectural level with emerging models that operate at the tissue and molecular levels may produce rapid advancements in modeling methodology. The increased availability of detailed data resulting from genomic studies, complemented by the construction of models that incorporate these data, may lead to an indepth understanding of the mechanisms of plant development from genes to phenotypes. In the meantime, computational modeling of plants is becoming a fascinating area of interdisciplinary research.
Robustness: confronting lessons from physics and biology
 Biological Reviews of the Cambridge Philosophical Society
, 2008
"... The term robustness is encountered in very different scientific fields, from engineering and control theory to dynamical systems to biology. The main question addressed herein is whether the notion of robustness and its correlates (stability, resilience, selforganisation) developed in physics are r ..."
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The term robustness is encountered in very different scientific fields, from engineering and control theory to dynamical systems to biology. The main question addressed herein is whether the notion of robustness and its correlates (stability, resilience, selforganisation) developed in physics are relevant to biology, or whether specific extensions and novel frameworks are required to account for the robustness properties of living systems. To clarify this issue, the different meanings covered by this unique term are discussed; it is argued that they crucially depend on the kind of perturbations that a robust system should by definition withstand. Possible mechanisms underlying robust behaviours are examined, either encountered in all natural systems (symmetries, conservation laws, dynamic stability) or specific to biological systems (feedbacks and regulatory networks). Special attention is devoted to the (sometimes counterintuitive) interrelations between robustness and noise. A distinction between dynamic selection and natural selection in the establishment of a robust behaviour is underlined. It is finally argued that nested notions of robustness, relevant to different time scales and different levels of organisation, allow one to reconcile the seemingly contradictory requirements for robustness and adaptability in living systems.
Solving differential equations in developmental models of multicellular structures using Lsystems
 Lecture Notes in Computer Science 3037 (Proceedings of the International Conference in Computational Science ICCS 2004, Krakow
, 2004
"... Mathematical modeling of growing multicellular structures creates the problem of solving systems of equations in which not only the values of variables, but the equations themselves, may change over time. We consider this problem in the framework of Lindenmayer systems, a standard formalism for mode ..."
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Mathematical modeling of growing multicellular structures creates the problem of solving systems of equations in which not only the values of variables, but the equations themselves, may change over time. We consider this problem in the framework of Lindenmayer systems, a standard formalism for modeling plants, and show how parametric context−sensitive L−systems can be used to numerically solve growing systems of coupled differential equations. We illustrate our technique with a developmental model of the multicellular bacterium Anabaena. Reference P. Federl and P. Prusinkiewicz: Solving differential equations in developmental models of multicellular
A Topological Framework for the Specification and the Simulation of Discrete Dynamical Systems
, 2004
"... MGS is an experimental programming language for the modeling and the simulation of discrete dynamical systems. The modeling approach is based on the explicit specification of the interaction structure between the system parts. This interaction structure is adequately described by topological not ..."
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MGS is an experimental programming language for the modeling and the simulation of discrete dynamical systems. The modeling approach is based on the explicit specification of the interaction structure between the system parts. This interaction structure is adequately described by topological notions. The topological approach enables a unified view on several computational mechanisms initially inspired by biological or chemical processes (Gamma and the CHAM, Lindenmayer systems, P systems and cellular automata). The expressivity of the language is illustrated by the modeling of a di#usion limited aggregation process on a wide variety of spatial domain: from cayley graphs to arbitrary manifolds.
Developmental computing
 Unconventional Computation. 8th International Conference, UC 2009, Lecture Notes in Computer Science 5715
"... Abstract. Since their inception over forty years ago, Lsystems have proven to be a useful conceptual and programming framework for modeling the development of plants at different levels of abstraction and different spatial scales. Formally, Lsystems offer a means of defining cell complexes with ch ..."
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Abstract. Since their inception over forty years ago, Lsystems have proven to be a useful conceptual and programming framework for modeling the development of plants at different levels of abstraction and different spatial scales. Formally, Lsystems offer a means of defining cell complexes with changing topology and geometry. Associated with these complexes are selfconfiguring systems of equations that represent functional aspects of the models. The close coupling of topology, geometry and computation constitutes a computing paradigm inspired by nature, termed developmental computing. We analyze distinctive features of this paradigm within and outside the realm of biological models.
Understanding Patchy Landscape Dynamics: Towards a Landscape Language
, 2012
"... Patchy landscapes driven by human decisions and/or natural forces are still a challenge to be understood and modelled. No attempt has been made up to now to describe them by a coherent framework and to formalize landscape changing rules. Overcoming this lacuna was our first objective here, and this ..."
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Patchy landscapes driven by human decisions and/or natural forces are still a challenge to be understood and modelled. No attempt has been made up to now to describe them by a coherent framework and to formalize landscape changing rules. Overcoming this lacuna was our first objective here, and this was largely based on the notion of Rewriting Systems, also called Formal Grammars. We used complicated scenarios of agricultural dynamics to model landscapes and to write their corresponding driving rule equations. Our second objective was to illustrate the relevance of this landscape language concept for landscape modelling through various grassland managements, with the final aim to assess their respective impacts on biological conservation. For this purpose, we made the assumptions that a higher grassland appearance frequency and higher land cover connectivity are favourable to species conservation. Ecological results revealed that dairy and beef livestock production systems are more favourable to wild species than is hog farming, although in different ways. Methodological results allowed us to efficiently model and formalize these landscape dynamics. This study demonstrates the applicability of the Rewriting System framework to the modelling of agricultural landscapes and, hopefully, to other patchy landscapes. The newly defined grammar is able to explain changes that are neither necessarily local nor Markovian, and opens a way to analytical modelling of landscape dynamics.
Molecular Biology
, 1968
"... A protein having similarity with methylmalonylCoA mutase is required for the assimilation of methanol and ethanol by Methylobacterium extorquens AM1 ..."
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A protein having similarity with methylmalonylCoA mutase is required for the assimilation of methanol and ethanol by Methylobacterium extorquens AM1
A.: Simulation of selfassembly processes using abstract reduction systems
 Systems SelfAssembly: Multidisciplinary Snapshots
"... We present in this chapter the use of MGS, a declarative and rulebased language dedicated to the modeling and the simulation of various morphogenetic and developmental processes, like selfassembly processes. The MGS approach relies on the introduction of a topological point of view on various dat ..."
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We present in this chapter the use of MGS, a declarative and rulebased language dedicated to the modeling and the simulation of various morphogenetic and developmental processes, like selfassembly processes. The MGS approach relies on the introduction of a topological point of view on various data structures called topological collections. This topological approach enables a uniform handling of theses data structure by a new kind of rewriting rules called transformations. Using (local) rewriting rules to specify selfassembling processes is particularly adequate because it mimics closely the incremental building mechanism of the real phenomena. The MGS approach is illustrated on the fabrication of a fractal pattern, a Sierpinsky triangle, using two approaches: by accretive growth and by carving. More generally, the notions of topological collections and transformations available in MGS enable the easy and concise modeling of cellular automata on various lattice geometries as well as more arbitrary constructions of multidimensional objects. 1
J.L.: Interactionbased simulations for integrative spatial systems biology
 Understanding the Dynamics of Biological Systems: Lessons Learned from Integrative Systems Biology
, 2011
"... Abstract Systems biology aims at integrating processes at various time and spatial scales into a single and coherent formal description to allow analysis and computer simulation. In this context, we focus on rulebased modeling and its integration in the domainspecific language MGS. Through the not ..."
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Abstract Systems biology aims at integrating processes at various time and spatial scales into a single and coherent formal description to allow analysis and computer simulation. In this context, we focus on rulebased modeling and its integration in the domainspecific language MGS. Through the notions of topological collections and transformations, MGS allows the modeling of biological processes at various levels of description. We validate our approach through the description of various models of a synthetic bacteria designed in the context of the iGEM competition, from a very simple biochemical description of the process to an individualbased model on a Delaunay graph topology. This approach is a first step into providing the requirements for the emerging field of spatial systems biology which integrates spatial properties into systems biology.
SUBMITTED TO THE FACULTY OF GRADUATE STUDIES
, 2006
"... Turing proposed a mathematical model based on chemical reactions and diffusion of substances throughout the xyplane, called reactiondiffusion, to study the emergence of patterns from a homogeneous medium [65]. Hammel and Prusinkiewicz used Lsystems and reactiondiffusion in a onedimensional medi ..."
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Turing proposed a mathematical model based on chemical reactions and diffusion of substances throughout the xyplane, called reactiondiffusion, to study the emergence of patterns from a homogeneous medium [65]. Hammel and Prusinkiewicz used Lsystems and reactiondiffusion in a onedimensional medium to model the formation of sea shell patterns, and in an expanding onedimensional medium to model heterocyst spacing in the cyanobacterium Anabaena catenula [25]. They considered the formation of these two patterns as a continuous deterministic process neglecting the noise which is inherent to such a process. The approach taken in this work is to stochastically model reactiondiffusion in a spatial, and possibly expanding, linear structure using Lsystems. The stochastic simulation method for chemical reaction kinetics which was developed by Gillespie [20] is used to study this model. On the basis of theoretical considerations, the Lsystem modelling language L+C [31] is extended to include a stochastic rewriting strategy based on Gillespie’s algorithm.