[Show abstract][Hide abstract] ABSTRACT: Gravitropism, the slow reorientation of plant growth in response to gravity, is a major determinant of the form and posture of land plants. Recently a universal model of shoot gravitropism, the AC model, was presented, in which the dynamics of the tropic movement is only determined by the conflicting controls of (1) graviception that tends to curve the plants toward the vertical, and (2) proprioception that tends to keep the stem straight. This model was found to be valid for many species and over two orders of magnitude of organ size. However, the motor of the movement, the elongation, was purposely neglected in the AC model. If growth effects are to be taken into account, it is necessary to consider the material derivative, i.e., the rate of change of curvature bound to expanding and convected organ elements. Here we show that it is possible to rewrite the material equation of curvature in a compact simplified form that directly expresses the curvature variation as a function of the median elongation and of the distribution of the differential growth. By using this extended model, called the ACĖ model, growth is found to have two main destabilizing effects on the tropic movement: (1) passive orientation drift, which occurs when a curved element elongates without differential growth, and (2) fixed curvature, when an element leaves the elongation zone and is no longer able to actively change its curvature. By comparing the AC and ACĖ models to experiments, these two effects are found to be negligible. Our results show that the simplified AC mode can be used to analyze gravitropism and posture control in actively elongating plant organs without significant information loss.
Frontiers in Plant Science 01/2014; 5:136. · 3.60 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Gravitropism, the slow reorientation of plant growth in response to gravity, is a key determinant of the form and posture of land plants. Shoot gravitropism is triggered when statocysts sense the local angle of the growing organ relative to the gravitational field. Lateral transport of the hormone auxin to the lower side is then enhanced, resulting in differential gene expression and cell elongation causing the organ to bend. However, little is known about the dynamics, regulation, and diversity of the entire bending and straightening process. Here, we modeled the bending and straightening of a rod-like organ and compared it with the gravitropism kinematics of different organs from 11 angiosperms. We show that gravitropic straightening shares common traits across species, organs, and orders of magnitude. The minimal dynamic model accounting for these traits is not the widely cited gravisensing law but one that also takes into account the sensing of local curvature, what we describe here as a graviproprioceptive law. In our model, the entire dynamics of the bending/straightening response is described by a single dimensionless "bending number" B that reflects the ratio between graviceptive and proprioceptive sensitivities. The parameter B defines both the final shape of the organ at equilibrium and the timing of curving and straightening. B can be estimated from simple experiments, and the model can then explain most of the diversity observed in experiments. Proprioceptive sensing is thus as important as gravisensing in gravitropic control, and the B ratio can be measured as phenotype in genetic studies.
Proceedings of the National Academy of Sciences 12/2012; · 9.81 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Branching morphogenesis is a widely spread phenomenon in nature. In organogenesis, it results from the inhomogeneous growth of the epithelial sheet, leading to its repeated branching into surrounding mesoderm. Lung morphogenesis is an emblematic example of tree-like organogenesis common to most mammals. The core signalling network is well identified, notably the Fgf10/Shh couple, required to initiate and maintain branching. In a previous study, we showed that the restriction by SHH of Fgf10 expression domain to distal mesenchyme spontaneously induces differential epithelial proliferation leading to branching. A simple Laplacian model qualitatively reproduced FGF10 dynamics in the mesenchyme and the spontaneous self-avoiding branching morphogenesis. However, early lung geometry has several striking features that remain to be addressed. In this paper, we investigate, through simulations and data analysis, if the FGF10-diffusion scenario accounts for the following aspects of lung morphology: size dispersion, asymmetry of branching events, and distal epithelium-mesothelium equilibrium. We report that they emerge spontaneously in the model, and that most of the underlying mechanisms can be understood as dynamical interactions between gradients and shape. This suggests that specific regulation may not be required for the emergence of these striking geometrical features.
[Show abstract][Hide abstract] ABSTRACT: We derive the analytical solutions to the second order generalised
gravi-proprioceptive equation given in our recent paper [Bastien et al. 2012].
These equations show how plants adjust to the surrounding gravitation field and
highlight the fact that the plant must be able to not only sense its local
posture with respect to the gravitational field, but also to sense its own
local curvature. In [Bastien et al. 2012] we obtained explicit analytical
solutions of these equations in terms of (sums of) Bessel functions, and in the
present paper we derive these solutions.
[Show abstract][Hide abstract] ABSTRACT: Singing-sand dunes have attracted curiosity for centuries and are now
the subject of controversy. We address here two aspects of this
controversy: first the possible link between the frequency heard and the
shear rate (for a gravity avalanche on a dune slip-face, scaling as
0.4g/d, with d the ‘mean’ grain diameter), and second, the
assumed necessity of a layered dune structure under the avalanche that
acts as a resonator. Field recordings of singing dunes over the world
reveal that they can present very different spectral characteristics: a
dune with polydisperse grains produces a very broad and noisy spectrum,
while a dune with sorted grains produces a well-defined frequency.
Performing laboratory avalanches on a hard plate with singing-dune sand
shows that there is no need for a dune below the sand avalanche to
produce the singing sound, and a fortiori neither for the dune's layered
structure nor for its particular sound transmission. By sieving the
polydisperse grains, the same well-defined frequency is obtained to that
of the dune with sorted grains, with the same diameter-frequency
relation. The various frequencies heard in the field avalanches match
the shear rates not calculated from the average size, but from the
various peaks of the grain size distributions.
Geophysical Research Letters 10/2012; 39(20):20310-. · 3.98 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: • Premise of the study: How leaf shape is regulated is a long-standing question in botany. For diverse groups of dicotyledon species, lamina folding along the veins and geometry of the space available for the primordia can explain the palmate leaf morphology. Dubbed the kirigami theory, this hypothesis of fold-dependent leaf shape regulation has remained largely theoretical. Using Acer pseudoplatanus, we investigated the mechanisms behind the two key processes of kirigami leaf development. • Methods: Cytological examination and quantitative analyses were used to examine the course of the vein-dependent lamina folding. Surgical ablation and tissue culturing were employed to test the effects of physical constraints on primordia growth. The final morphology of leaves growing without steric constraints were predicted mathematically. • Key results: The cytological examination showed that the lamina's abaxial side along the veins grows substantially more than the adaxial side. The abaxial hypergrowth along the veins and the lamina extension correlated with the lamina folding. When a primordium was released from the physical constraints imposed by the other primordia, it rapidly grew into the newly available space, while maintaining the curvature inward. The morphology of such a leaf was predicted to lack symmetry in the lobe shapes. • Conclusions: The enhanced growth on the abaxial side of the lamina along the veins is likely to drive lamina folding. The surgical ablation provided clear support for the space-filling nature of leaf growth; thus, steric constraints play a role in determination of the shapes of folded leaves and probably also of the final leaf morphology.
American Journal of Botany 08/2012; 99(8):1289-99. · 2.59 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Sand is known to show a variety of uncommon physical features that do
not fit the behavior of liquid or solid state. A good example of the
inherent difficulties encountered when trying to describe collective
grains behavior is the penetration of an intruding object into a
granular medium. Such problems involve large coordination numbers, and
the medium response dramatically depends on the volume fraction. On the
fringe of these studies, we consider here the penetration of a
cylindrical shell (typically an upside down glass) into dry sand, and
report what we called the "blown-air effect". The air initially trapped
escapes when the shell is pushed into sand, flowing through the granular
medium. This flow dilates the sand and considerably eases the
penetration of the shell. This is very different from what happens in
liquids: when pushing a top-closed shell into a liquid, the trapped air
increases the buoyancy and opposes the penetration. We show that the air
flow does not change the general dynamics of penetration, suggesting
that fluidization only involves an effective smaller volume fraction.
Despite its simplicity (only a glass and some sand are needed to observe
the effect), this experiment nicely illustrates the sometimes
counter-intuitive behavior of granular media. Penetration in sand is
also a critical issue in industry, and this work may help improving
burying methods. Ref: Penetration and blown air effect in granular
media R. Clément, S. Courrech du Pont, M. Ould-Hamouda, D.
Duveau, and S. Douady Phys. Rev. Lett. 2011 Science News:
[Show abstract][Hide abstract] ABSTRACT: Diatoms, the major contributors of the global biogenic silica cycle in modern oceans, account for about 40% of global marine primary productivity. They are an important component of the biological pump in the ocean, and their assemblage can be used as useful climate proxies; it is therefore critical to better understand the changes induced by environmental pH on their physiology, silicification capability and morphology. Here, we show that external pH influences cell growth of the ubiquitous diatom Thalassiosira weissflogii, and modifies intracellular silicic acid and biogenic silica contents per cell. Measurements at the single-cell level reveal that extracellular pH modifications lead to intracellular acidosis. To further understand how variations of the acid-base balance affect silicon metabolism and theca formation, we developed novel imaging techniques to measure the dynamics of valve formation. We demonstrate that the kinetics of valve morphogenesis, at least in the early stages, depends on pH. Analytical modeling results suggest that acidic conditions alter the dynamics of the expansion of the vesicles within which silica polymerization occurs, and probably its internal pH. Morphological analysis of valve patterns reveals that acidification also reduces the dimension of the nanometric pores present on the valves, and concurrently overall valve porosity. Variations in the valve silica network seem to be more correlated to the dynamics and the regulation of the morphogenesis process than the silicon incorporation rate. These multiparametric analyses from single-cell to cell-population levels demonstrate that several higher-level processes are sensitive to the acid-base balance in diatoms, and its regulation is a key factor for the control of pattern formation and silicon metabolism.
PLoS ONE 01/2012; 7(10):e46722. · 3.53 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The arborescent architecture of mammalian conductive airways results from the repeated branching of lung endoderm into surrounding mesoderm. Subsequent lung's striking geometrical features have long raised the question of developmental mechanisms involved in morphogenesis. Many molecular actors have been identified, and several studies demonstrated the central role of Fgf10 and Shh in growth and branching. However, the actual branching mechanism and the way branching events are organized at the organ scale to achieve a self-avoiding tree remain to be understood through a model compatible with evidenced signaling. In this paper we show that the mere diffusion of FGF10 from distal mesenchyme involves differential epithelial proliferation that spontaneously leads to branching. Modeling FGF10 diffusion from sub-mesothelial mesenchyme where Fgf10 is known to be expressed and computing epithelial and mesenchymal growth in a coupled manner, we found that the resulting laplacian dynamics precisely accounts for the patterning of FGF10-induced genes, and that it spontaneously involves differential proliferation leading to a self-avoiding and space-filling tree, through mechanisms that we detail. The tree's fine morphological features depend on the epithelial growth response to FGF10, underlain by the lung's complex regulatory network. Notably, our results suggest that no branching information has to be encoded and that no master routine is required to organize branching events at the organ scale. Despite its simplicity, this model identifies key mechanisms of lung development, from branching to organ-scale organization, and could prove relevant to the development of other branched organs relying on similar pathways.
PLoS ONE 01/2012; 7(5):e36925. · 3.53 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Leaves are packed in a bud in different ways, being flat, rolled, or folded, but always filling the whole bud volume. This "filling law" has many consequences, in particular on the shapes of growing folded leaves. This is shown here for different types of folding and packing. The folded volume is roughly a part of an ellipsoid, with the veins on the outside rounded face and the lamina margin on the adaxial plane. The veins on the abaxial side protect the fragile lamina inside. The first general consequence of the folds and the space limitation of the lamina growth is the presence of symmetries on the leaf shape, and the second is the quantitative relationships between the sizes of the lobes and sinuses. For particular geometries, the leaf lamina can be limited by lateral veins, creating spoon-like lobes, or tangent cuts, creating asymmetrical wavy perimeters. Changes in the packing between different cultivars correspond to changes in the mature leaf shapes. Each particular case shows how pervasive the geometrical consequences of the filling law are.
Journal of Theoretical Biology 08/2011; 289:47-64. · 2.35 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We propose a method for quantitative characterization of spatial networklike patterns with loops, such as surface fracture patterns, leaf vein networks, and patterns of urban streets. Such patterns are not well characterized by purely topological estimators: also patterns that both look different and result from different morphogenetic processes can have similar topology. A local geometric cue--the angles formed by the different branches at junctions--can complement topological information and allow the quantification of the large scale spatial coherence of the pattern. For patterns that grow over time, such as fracture lines on the surface of ceramics, the rank assigned by our method to each individual segment of the pattern approximates the order of appearance of that segment. We apply the method to various networklike patterns and find a continuous but sharp dichotomy between two classes of spatial networks: hierarchical and homogeneous. The former class results from a sequential growth process and presents large scale organization, and the latter presents local, but not global, organization.
[Show abstract][Hide abstract] ABSTRACT: The map of a city's streets constitutes a particular case of spatial complex
network. However a city is not limited to its topology: it is above all a
geometrical object whose particularity is to organize into short and long axes
called streets. In this article we present and discuss two algorithms aiming at
recovering the notion of street from a graph representation of a city. Then we
show that the length of the so-called streets scales logarithmically. This
phenomenon leads to assume that a city is shaped into a logic of extension and
division of space.
[Show abstract][Hide abstract] ABSTRACT: Cities are living organisms. They are out of equilibrium, open systems that never stop developing and sometimes die. The local geography can be compared to a shell constraining its development. In brief, a city's current layout is a step in a running morphogenesis process. Thus cities display a huge diversity of shapes and none of the traditional models, from random graphs, complex networks theory, or stochastic geometry, takes into account the geometrical, functional, and dynamical aspects of a city in the same framework. We present here a global mathematical model dedicated to cities that permits describing, manipulating, and explaining cities' overall shape and layout of their street systems. This street-based framework conciliates the topological and geometrical sides of the problem. From the static analysis of several French towns (topology of first and second order, anisotropy, streets scaling) we make the hypothesis that the development of a city follows a logic of division or extension of space. We propose a dynamical model that mimics this logic and that, from simple general rules and a few parameters, succeeds in generating a large diversity of cities and in reproducing the general features the static analysis has pointed out.
[Show abstract][Hide abstract] ABSTRACT: Sand is known to oppose an increasing resistance to penetration with depth. This is different from what happens in liquids since granular media, usually nonthermal systems, oppose solid friction to the motion. We report another striking and "counterintuitive" difference between the penetration dynamics observed in sand and in liquids. When pushing a top-closed shell (e.g., an upside down glass) into a liquid, the trapped air increases the buoyancy and opposes the penetration. It is more difficult to push a top capped cylinder than an opened one vertically into liquids. In contrast, the penetration is considerably easier in dense sand when cylinders are top capped. In this discrete and biphasic medium, the trapped air escapes from the shell, fluidizes the sand, and eases the motion.
[Show abstract][Hide abstract] ABSTRACT: The song of dunes is a natural phenomenon that has arisen travellers' curiosity for a long time, from Marco Polo to R.A. Bagnold. Scientific observations in the XXth century have shown that the sound is emitted during a shear flow of these particular grains, the free surface of the flow having coherent vibrations like a loud speaker. The sound emission is also submitted to a threshold effect with many parameters like humidity, flow speed, surface of the grains. The sound has been reproduced in laboratory avalanche experiments close to the natural phenomenon on field, but set in a channel with a hard bottom and a few centimeters of sand flowing, which contradicts explanations of the sound that involve a sand dune under the avalanche flow. Flow rates measurements also show the presence of a plug region in the flow above the sheared band, with the same characteristic length as the coherence zones of the sound. Finally we show experimentally that the Froude number, once modified to take into account the height of this plug band, is the parameter that sets the amplitude of the sound, and produces a threshold that depends on the grain type.
[Show abstract][Hide abstract] ABSTRACT: We propose a new method for quantitative characterization of spatial
network-like patterns with loops, such as surface fracture patterns, leaf vein
networks and patterns of urban streets. Such patterns are not well
characterized by purely topological estimators: also patterns that both look
different and result from different morphogenetic processes can have similar
topology. A local geometric cue -the angles formed by the different branches at
junctions- can complement topological information and allow to quantify the
large scale spatial coherence of the pattern. For patterns that grow over time,
such as fracture lines on the surface of ceramics, the rank assigned by our
method to each individual segment of the pattern approximates the order of
appearance of that segment. We apply the method to various network-like
patterns and we find a continuous but sharp dichotomy between two classes of
spatial networks: hierarchical and homogeneous. The first class results from a
sequential growth process and presents large scale organization, the latter
presents local, but not global organization.
[Show abstract][Hide abstract] ABSTRACT: Barchans are crescentic dunes which occur in mainly mono-directional winds. Shape, aspect ratios and velocities of these dunes have been studied as if they were in equilibrium. However, following a study of the shape and migration of 11 barchans of different sizes for 18 months in the field on Moroccan Atlantic Sahara, we show that they only appear to be in a stationary state if studied over a long timeframe (at the scale of the year or several years), but are never in equilibrium at the scale of weeks or months. Rather, they are always ‘trying’ to reach a possible equilibrium state but never have enough time to accomplish this. This may be the main reason for the large variation observed in previous measurements, and justifies some caution in what can be deduced from them.
[Show abstract][Hide abstract] ABSTRACT: Firstly introduced in social science, the notion of central-ity has spread to the whole complex network science. A centrality is a measure that quantifies whether an element of a network is well served or not. This article focuses on cities' street network (seen as a communication network) and studies two classical centralities (the closeness and the straightness) plus a new one (the simplest centrality). To this we introduce a mathematical framework which allows considering a city as a geometrical continuum rather than a plain topological graph. The color plotting of the various centralities permits a visual analysis of the city and to diag-nose local malfunctionings. The relevance of our framework and of the centralities is discussed from visual examples of French towns and time complexity.
[Show abstract][Hide abstract] ABSTRACT: Cities can be compared to living organisms. They are out of equilibrium, opened systems that never stop developing and sometimes die. The cityÃ¢ÂÂs growth is guided by needs in local distribution and in communication among its parts. The local geography can be compared to a shell constraining its development. In brief, a cityÃ¢ÂÂs current layout is a step in a running morphogenesis process. Thus cities display a huge diversity of shapes and none of traditional models from random graphs, complex networks theory or stochastic geometry takes into account geometrical, functional and dynamical aspects of a city in the same framework. We present here a global mathematical model dedicated to cities that permits describing, manipulating and explaining citiesÃ¢ÂÂ overall shape and layout of their street systems. This streets-based framework includes an algebraic formalism, a static analysis of citiesÃ¢ÂÂ main features (topology of first and second order, anisotropy, streets scaling) and a dynamical model which from simple general rules can reproduce a large diversity of cities. Comment: 12 pages, 13 figures