Combined in silico/in vivo analysis of mechanisms providing for root apical meristem self-organization and maintenance.

Background and aimsThe root apical meristem (RAM) is the plant stem cell niche which provides for the formation and continuous development of the root. Auxin is the main regulator of RAM functioning, and auxin maxima coincide with the sites of RAM initiation and maintenance. Auxin gradients are formed due to local auxin biosynthesis and polar auxin transport. The PIN family of auxin transporters plays a critical role in polar auxin transport, and two mechanisms of auxin maximum formation in the RAM based on PIN-mediated auxin transport have been proposed to date: the reverse fountain and the reflected flow mechanisms.MethodsThe two mechanisms are combined here in in silico studies of auxin distribution in intact roots and roots cut into two pieces in the proximal meristem region. In parallel, corresponding experiments were performed in vivo using DR5::GFP Arabidopsis plants.Key resultsThe reverse fountain and the reflected flow mechanism naturally cooperate for RAM patterning and maintenance in intact root. Regeneration of the RAM in decapitated roots is provided by the reflected flow mechanism. In the excised root tips local auxin biosynthesis either alone or in cooperation with the reverse fountain enables RAM maintenance.ConclusionsThe efficiency of a dual-mechanism model in guiding biological experiments on RAM regeneration and maintenance is demonstrated. The model also allows estimation of the concentrations of auxin and PINs in root cells during development and under various treatments. The dual-mechanism model proposed here can be a powerful tool for the study of several different aspects of auxin function in root

Analysis of cell division patterns in the Arabidopsis shoot apical meristem

The stereotypic pattern of cell shapes in the Arabidopsis shoot apical meristem (SAM) suggests that strict rules govern the placement of new walls during cell division. When a cell in the SAM divides, a new wall is built that connects existing walls and divides the cytoplasm of the daughter cells. Because features that are determined by the placement of new walls such as cell size, shape, and number of neighbors are highly regular, rules must exist for maintaining such order. Here we present a quantitative model of these rules that incorporates different observed features of cell division. Each feature is incorporated into a “potential function” that contributes a single term to a total analog of potential energy. New cell walls are predicted to occur at locations where the potential function is minimized. Quantitative terms that represent the well-known historical rules of plant cell division, such as those given by Hofmeister, Errera, and Sachs are developed and evaluated against observed cell divisions in the epidermal layer (L1) of Arabidopsis thaliana SAM. The method is general enough to allow additional terms for nongeometric properties such as internal concentration gradients and mechanical tensile forces

Tessellations and Pattern Formation in Plant Growth and Development

The shoot apical meristem (SAM) is a dome-shaped collection of cells at the apex of growing plants from which all above-ground tissue ultimately derives. In Arabidopsis thaliana (thale cress), a small flowering weed of the Brassicaceae family (related to mustard and cabbage), the SAM typically contains some three to five hundred cells that range from five to ten microns in diameter. These cells are organized into several distinct zones that maintain their topological and functional relationships throughout the life of the plant. As the plant grows, organs (primordia) form on its surface flanks in a phyllotactic pattern that develop into new shoots, leaves, and flowers. Cross-sections through the meristem reveal a pattern of polygonal tessellation that is suggestive of Voronoi diagrams derived from the centroids of cellular nuclei. In this chapter we explore some of the properties of these patterns within the meristem and explore the applicability of simple, standard mathematical models of their geometry.Comment: Originally presented at: "The World is a Jigsaw: Tessellations in the Sciences," Lorentz Center, Leiden, The Netherlands, March 200

Using cellzilla for plant growth simulations at the cellular level

Cellzilla is a two-dimensional tissue simulation platform for plant modeling utilizing Cellerator arrows. Cellerator describes biochemical interactions with a simplified arrow-based notation; all interactions are input as reactions and are automatically translated to the appropriate differential equations using a computer algebra system. Cells are represented by a polygonal mesh of well-mixed compartments. Cell constituents can interact intercellularly via Cellerator reactions utilizing diffusion, transport, and action at a distance, as well as amongst themselves within a cell. The mesh data structure consists of vertices, edges (vertex pairs), and cells (and optional intercellular wall compartments) as ordered collections of edges. Simulations may be either static, in which cell constituents change with time but cell size and shape remain fixed; or dynamic, where cells can also grow. Growth is controlled by Hookean springs associated with each mesh edge and an outward pointing pressure force. Spring rest length grows at a rate proportional to the extension beyond equilibrium. Cell division occurs when a specified constituent (or cell mass) passes a (random, normally distributed) threshold. The orientation of new cell walls is determined either by Errera's rule, or by a potential model that weighs contributions due to equalizing daughter areas, minimizing wall length, alignment perpendicular to cell extension, and alignment perpendicular to actual growth direction

Modeling the organization of the WUSCHEL expression domain in the shoot apical meristem

Motivation: The above-ground tissues of higher plants are generated from a small region of cells situated at the plant apex called the shoot apical meristem. An important genetic control circuit modulating the size of the Arabidopsis thaliana meristem is a feed-back network between the CLAVATA3 and WUSCHEL genes. Although the expression patterns for these genes do not overlap, WUSCHEL activity is both necessary and sufficient (when expressed ectopically) for the induction of CLAVATA3 expression. However, upregulation of CLAVATA3 in conjunction with the receptor kinase CLAVATA1 results in the downregulation of WUSCHEL. Despite much work, experimental data for this network are incomplete and additional hypotheses are needed to explain the spatial locations and dynamics of these expression domains. Predictive mathematical models describing the system should provide a useful tool for investigating and discriminating among possible hypotheses, by determining which hypotheses best explain observed gene expression dynamics. Results: We are developing a method using in vivo live confocal microscopy to capture quantitative gene expression data and create templates for computational models. We present two models accounting for the organization of the WUSCHEL expression domain. Our preferred model uses a reaction-diffusion mechanism in which an activator induces WUSCHEL expression. This model is able to organize the WUSCHEL expression domain. In addition, the model predicts the dynamical reorganization seen in experiments where cells, including the WUSCHEL domain, are ablated, and it also predicts the spatial expansion of the WUSCHEL domain resulting from removal of the CLAVATA3 signal

A mathematical and computational framework for quantitative comparison and integration of large-scale gene expression data

Analysis of large-scale gene expression studies usually begins with gene clustering. A ubiquitous problem is that different algorithms applied to the same data inevitably give different results, and the differences are often substantial, involving a quarter or more of the genes analyzed. This raises a series of important but nettlesome questions: How are different clustering results related to each other and to the underlying data structure? Is one clustering objectively superior to another? Which differences, if any, are likely candidates to be biologically important? A systematic and quantitative way to address these questions is needed, together with an effective way to integrate and leverage expression results with other kinds of large-scale data and annotations. We developed a mathematical and computational framework to help quantify, compare, visualize and interactively mine clusterings. We show that by coupling confusion matrices with appropriate metrics (linear assignment and normalized mutual information scores), one can quantify and map differences between clusterings. A version of receiver operator characteristic analysis proved effective for quantifying and visualizing cluster quality and overlap. These methods, plus a flexible library of clustering algorithms, can be called from a new expandable set of software tools called CompClust 1.0 (). CompClust also makes it possible to relate expression clustering patterns to DNA sequence motif occurrences, protein–DNA interaction measurements and various kinds of functional annotations. Test analyses used yeast cell cycle data and revealed data structure not obvious under all algorithms. These results were then integrated with transcription motif and global protein–DNA interaction data to identify G(1) regulatory modules

A Pycellerator Tutorial.

We present a tutorial on using Pycellerator for biomolecular simulations. Models are described in human readable (and editable) text files (UTF8 or ASCII) containing collections of reactions, assignments, initial conditions, function definitions, and rate constants. These models are then converted into a Python program that can optionally solve the system, e.g., as a system of differential equations using ODEINT, or be run by another program. The input language implements an extended version of the Cellerator arrow notation, including mass action, Hill functions, S-Systems, MWC, and reactions with user-defined kinetic laws. Simple flux balance analysis is also implemented. We will demonstrate the implementation and analysis of progressively more complex models, starting from simple mass action through indexed cascades. Pycellerator can be used as a library that is integrated into other programs, run as a command line program, or in iPython notebooks. It is implemented in Python 2.7 and available under an open source GPL license

Evolution of Robustness to Noise and Mutation in Gene Expression Dynamics

Phenotype of biological systems needs to be robust against mutation in order to sustain themselves between generations. On the other hand, phenotype of an individual also needs to be robust against fluctuations of both internal and external origins that are encountered during growth and development. Is there a relationship between these two types of robustness, one during a single generation and the other during evolution? Could stochasticity in gene expression have any relevance to the evolution of these robustness? Robustness can be defined by the sharpness of the distribution of phenotype; the variance of phenotype distribution due to genetic variation gives a measure of `genetic robustness' while that of isogenic individuals gives a measure of `developmental robustness'. Through simulations of a simple stochastic gene expression network that undergoes mutation and selection, we show that in order for the network to acquire both types of robustness, the phenotypic variance induced by mutations must be smaller than that observed in an isogenic population. As the latter originates from noise in gene expression, this signifies that the genetic robustness evolves only when the noise strength in gene expression is larger than some threshold. In such a case, the two variances decrease throughout the evolutionary time course, indicating increase in robustness. The results reveal how noise that cells encounter during growth and development shapes networks' robustness to stochasticity in gene expression, which in turn shapes networks' robustness to mutation. The condition for evolution of robustness as well as relationship between genetic and developmental robustness is derived through the variance of phenotypic fluctuations, which are measurable experimentally.Comment: 25 page

Mechanisms of gap gene expression canalization in the Drosophila blastoderm

<p>Abstract</p> <p>Background</p> <p>Extensive variation in early gap gene expression in the <it>Drosophila </it>blastoderm is reduced over time because of gap gene cross regulation. This phenomenon is a manifestation of canalization, the ability of an organism to produce a consistent phenotype despite variations in genotype or environment. The canalization of gap gene expression can be understood as arising from the actions of attractors in the gap gene dynamical system.</p> <p>Results</p> <p>In order to better understand the processes of developmental robustness and canalization in the early <it>Drosophila </it>embryo, we investigated the dynamical effects of varying spatial profiles of Bicoid protein concentration on the formation of the expression border of the gap gene <it>hunchback</it>. At several positions on the anterior-posterior axis of the embryo, we analyzed attractors and their basins of attraction in a dynamical model describing expression of four gap genes with the Bicoid concentration profile accounted as a given input in the model equations. This model was tested against a family of Bicoid gradients obtained from individual embryos. These gradients were normalized by two independent methods, which are based on distinct biological hypotheses and provide different magnitudes for Bicoid spatial variability. We showed how the border formation is dictated by the biological initial conditions (the concentration gradient of maternal Hunchback protein) being attracted to specific attracting sets in a local vicinity of the border. Different types of these attracting sets (point attractors or one dimensional attracting manifolds) define several possible mechanisms of border formation. The <it>hunchback </it>border formation is associated with intersection of the spatial gradient of the maternal Hunchback protein and a boundary between the attraction basins of two different point attractors. We demonstrated how the positional variability for <it>hunchback </it>is related to the corresponding variability of the basin boundaries. The observed reduction in variability of the <it>hunchback </it>gene expression can be accounted for by specific geometrical properties of the basin boundaries.</p> <p>Conclusion</p> <p>We clarified the mechanisms of gap gene expression canalization in early <it>Drosophila </it>embryos. These mechanisms were specified in the case of <it>hunchback </it>in well defined terms of the dynamical system theory.</p