Cells, walls, and endless forms

International audienceA key question in biology is how the endless diversity of forms found in nature evolved. Understanding the cellular basis of this diversity has been aided by advances in non-model experimental systems, quantitative image analysis tools, and modeling approaches. Recent work in plants highlights the importance of cell wall and cuticle modifications for the emergence of diverse forms and functions. For example, explosive seed dispersal in Cardamine hirsuta depends on the asymmetric localization of lignified cell wall thickenings in the fruit valve. Similarly, the iridescence of Hibiscus trionum petals relies on regular striations formed by cuticular folds. Moreover, NAC transcription factors regulate the differentiation of lignified xylem vessels but also the water-conducting cells of moss that lack a lignified secondary cell wall, pointing to the origin of vascular systems. Other novel forms are associated with modified cell growth patterns, including oriented cell expansion or division, found in the long petal spurs of Aquilegia flowers, and the Sarracenia purpurea pitcher leaf, respectively. Another good example is the regulation of dissected leaf shape in C. hirsuta via local growth repression, controlled by the REDUCED COMPLEXITY HD-ZIP class I transcription factor. These studies in non-model species often reveal as much about fundamental processes of development as they do about the evolution of form. Addres

The genetic architecture of petal number in Cardamine hirsuta

International audienceInvariant petal number is a characteristic of most flowers and is generally robust to genetic and environmental variation. We took advantage of the natural variation found in Cardamine hirsuta petal number to investigate the genetic basis of this trait in a case where robustness was lost during evolution. We used quantitative trait locus (QTL) analysis to characterize the genetic architecture of petal number. Αverage petal number showed transgressive variation from zero to four petals in five C. hirsuta mapping populations, and this variation was highly heritable. We detected 15 QTL at which allelic variation affected petal number. The effects of these QTL were relatively small in comparison with alleles induced by mutagenesis, suggesting that natural selection may act to maintain petal number within its variable range below four. Petal number showed a temporal trend during plant ageing, as did sepal trichome number, and multi-trait QTL analysis revealed that these age-dependent traits share a common genetic basis. Our results demonstrate that petal number is determined by many genes of small effect, some of which are age-dependent, and suggests a mechanism of trait evolution via the release of cryptic variation

Stochastic variation in Cardamine hirsuta petal number

Background and Aims Floral development is remarkably robust in terms of the identity and number of floral organs in each whorl, whereas vegetative development can be quite plastic. This canalization of flower development prevents the phenotypic expression of cryptic genetic variation, even in fluctuating environments. A cruciform perianth with four petals is a hallmark of the Brassicaceae family, typified in the model species Arabidopsis thaliana. However, variable petal loss is found in Cardamine hirsuta, a genetically tractable relative of A. thaliana. Cardamine hirsuta petal number varies in response to stochastic, genetic and environmental perturbations, which makes it an interesting model to study mechanisms of decanalization and the expression of cryptic variation. Methods Multitrait quantitative trait locus (QTL) analysis in recombinant inbred lines (RILs) was used to identify whether the stochastic variation found in C. hirsuta petal number had a genetic basis. Key Results Stochastic variation (standard error of the average petal number) was found to be a heritable phenotype, and four QTL that influenced this trait were identified. The sensitivity to detect these QTL effects was increased by accounting for the effect of ageing on petal number variation. All QTL had significant effects on both average petal number and its standard error, indicating that these two traits share a common genetic basis. However, for some QTL, a degree of independence was found between the age of the flowers where allelic effects were significant for each trait. Conclusions Stochastic variation in C. hirsuta petal number has a genetic basis, and common QTL influence both average petal number and its standard error. Allelic variation at these QTL can, therefore, modify petal number in an age-specific manner via effects on the phenotypic mean and stochastic variation. These results are discussed in the context of trait evolution via a loss of robustness

A Survey of Satisfiability Modulo Theory

Satisfiability modulo theory (SMT) consists in testing the satisfiability of first-order formulas over linear integer or real arithmetic, or other theories. In this survey, we explain the combination of propositional satisfiability and decision procedures for conjunctions known as DPLL(T), and the alternative "natural domain" approaches. We also cover quantifiers, Craig interpolants, polynomial arithmetic, and how SMT solvers are used in automated software analysis.Comment: Computer Algebra in Scientific Computing, Sep 2016, Bucharest, Romania. 201

Algorithmic Analysis of Qualitative and Quantitative Termination Problems for Affine Probabilistic Programs

In this paper, we consider termination of probabilistic programs with real-valued variables. The questions concerned are: 1. qualitative ones that ask (i) whether the program terminates with probability 1 (almost-sure termination) and (ii) whether the expected termination time is finite (finite termination); 2. quantitative ones that ask (i) to approximate the expected termination time (expectation problem) and (ii) to compute a bound B such that the probability to terminate after B steps decreases exponentially (concentration problem). To solve these questions, we utilize the notion of ranking supermartingales which is a powerful approach for proving termination of probabilistic programs. In detail, we focus on algorithmic synthesis of linear ranking-supermartingales over affine probabilistic programs (APP's) with both angelic and demonic non-determinism. An important subclass of APP's is LRAPP which is defined as the class of all APP's over which a linear ranking-supermartingale exists. Our main contributions are as follows. Firstly, we show that the membership problem of LRAPP (i) can be decided in polynomial time for APP's with at most demonic non-determinism, and (ii) is NP-hard and in PSPACE for APP's with angelic non-determinism; moreover, the NP-hardness result holds already for APP's without probability and demonic non-determinism. Secondly, we show that the concentration problem over LRAPP can be solved in the same complexity as for the membership problem of LRAPP. Finally, we show that the expectation problem over LRAPP can be solved in 2EXPTIME and is PSPACE-hard even for APP's without probability and non-determinism (i.e., deterministic programs). Our experimental results demonstrate the effectiveness of our approach to answer the qualitative and quantitative questions over APP's with at most demonic non-determinism.Comment: 24 pages, full version to the conference paper on POPL 201

Scaling Bounded Model Checking By Transforming Programs With Arrays

Bounded Model Checking is one the most successful techniques for finding bugs in program. However, model checkers are resource hungry and are often unable to verify programs with loops iterating over large arrays.We present a transformation that enables bounded model checkers to verify a certain class of array properties. Our technique transforms an array-manipulating (ANSI-C) program to an array-free and loop-free (ANSI-C) program thereby reducing the resource requirements of a model checker significantly. Model checking of the transformed program using an off-the-shelf bounded model checker simulates the loop iterations efficiently. Thus, our transformed program is a sound abstraction of the original program and is also precise in a large number of cases - we formally characterize the class of programs for which it is guaranteed to be precise. We demonstrate the applicability and usefulness of our technique on both industry code as well as academic benchmarks

Stochastic Invariants for Probabilistic Termination

Termination is one of the basic liveness properties, and we study the termination problem for probabilistic programs with real-valued variables. Previous works focused on the qualitative problem that asks whether an input program terminates with probability~1 (almost-sure termination). A powerful approach for this qualitative problem is the notion of ranking supermartingales with respect to a given set of invariants. The quantitative problem (probabilistic termination) asks for bounds on the termination probability. A fundamental and conceptual drawback of the existing approaches to address probabilistic termination is that even though the supermartingales consider the probabilistic behavior of the programs, the invariants are obtained completely ignoring the probabilistic aspect. In this work we address the probabilistic termination problem for linear-arithmetic probabilistic programs with nondeterminism. We define the notion of {\em stochastic invariants}, which are constraints along with a probability bound that the constraints hold. We introduce a concept of {\em repulsing supermartingales}. First, we show that repulsing supermartingales can be used to obtain bounds on the probability of the stochastic invariants. Second, we show the effectiveness of repulsing supermartingales in the following three ways: (1)~With a combination of ranking and repulsing supermartingales we can compute lower bounds on the probability of termination; (2)~repulsing supermartingales provide witnesses for refutation of almost-sure termination; and (3)~with a combination of ranking and repulsing supermartingales we can establish persistence properties of probabilistic programs. We also present results on related computational problems and an experimental evaluation of our approach on academic examples.Comment: Full version of a paper published at POPL 2017. 20 page

Certification of Bounds of Non-linear Functions: the Templates Method

The aim of this work is to certify lower bounds for real-valued multivariate functions, defined by semialgebraic or transcendental expressions. The certificate must be, eventually, formally provable in a proof system such as Coq. The application range for such a tool is widespread; for instance Hales' proof of Kepler's conjecture yields thousands of inequalities. We introduce an approximation algorithm, which combines ideas of the max-plus basis method (in optimal control) and of the linear templates method developed by Manna et al. (in static analysis). This algorithm consists in bounding some of the constituents of the function by suprema of quadratic forms with a well chosen curvature. This leads to semialgebraic optimization problems, solved by sum-of-squares relaxations. Templates limit the blow up of these relaxations at the price of coarsening the approximation. We illustrate the efficiency of our framework with various examples from the literature and discuss the interfacing with Coq.Comment: 16 pages, 3 figures, 2 table

Seasonal regulation of petal number

International audienceFour petals characterize the flowers of most species in the Brassicaceae family, and this phenotype is generally robust to genetic and environmental variation. A variable petal number distinguishes the flowers of Cardamine hirsuta from those of its close relative Arabidopsis (Arabidopsis thaliana), and allelic variation at many loci contribute to this trait. However, it is less clear whether C. hirsuta petal number varies in response to seasonal changes in environment. To address this question, we assessed whether petal number responds to a suite of environmental and endogenous cues that regulate flowering time in C. hirsuta. We found that petal number showed seasonal variation in C. hirsuta, such that spring flowering plants developed more petals than those flowering in summer. Conditions associated with spring flowering, including cool ambient temperature, short photoperiod, and vernalization, all increased petal number in C. hirsuta. Cool temperature caused the strongest increase in petal number and lengthened the time interval over which floral meristems matured. We performed live imaging of early flower development and showed that floral buds developed more slowly at 15°C versus 20°C. This extended phase of floral meristem formation, coupled with slower growth of sepals at 15°C, produced larger intersepal regions with more space available for petal initiation. In summary, the growth and maturation of floral buds is associated with variable petal number in C. hirsuta and responds to seasonal changes in ambient temperature