TY - JOUR AB - The physical distance between presynaptic Ca2+ channels and the Ca2+ sensors that trigger exocytosis of neurotransmitter-containing vesicles is a key determinant of the signalling properties of synapses in the nervous system. Recent functional analysis indicates that in some fast central synapses, transmitter release is triggered by a small number of Ca2+ channels that are coupled to Ca2+ sensors at the nanometre scale. Molecular analysis suggests that this tight coupling is generated by protein–protein interactions involving Ca2+ channels, Ca2+ sensors and various other synaptic proteins. Nanodomain coupling has several functional advantages, as it increases the efficacy, speed and energy efficiency of synaptic transmission. AU - Eggermann, Emmanuel AU - Bucurenciu, Iancu AU - Goswami, Sarit AU - Jonas, Peter M ID - 3317 IS - 1 JF - Nature Reviews Neuroscience TI - Nanodomain coupling between Ca(2+) channels and sensors of exocytosis at fast mammalian synapses VL - 13 ER - TY - JOUR AB - We introduce two-level discounted and mean-payoff games played by two players on a perfect-information stochastic game graph. The upper level game is a discounted or mean-payoff game and the lower level game is a (undiscounted) reachability game. Two-level games model hierarchical and sequential decision making under uncertainty across different time scales. For both discounted and mean-payoff two-level games, we show the existence of pure memoryless optimal strategies for both players and an ordered field property. We show that if there is only one player (Markov decision processes), then the values can be computed in polynomial time. It follows that whether the value of a player is equal to a given rational constant in two-level discounted or mean-payoff games can be decided in NP ∩ coNP. We also give an alternate strategy improvement algorithm to compute the value. © 2012 World Scientific Publishing Company. AU - Chatterjee, Krishnendu AU - Majumdar, Ritankar ID - 3314 IS - 3 JF - International Journal of Foundations of Computer Science TI - Discounting and averaging in games across time scales VL - 23 ER - TY - JOUR AB - We consider the offset-deconstruction problem: Given a polygonal shape Q with n vertices, can it be expressed, up to a tolerance ε in Hausdorff distance, as the Minkowski sum of another polygonal shape P with a disk of fixed radius? If it does, we also seek a preferably simple-looking solution P; then, P's offset constitutes an accurate, vertex-reduced, and smoothened approximation of Q. We give an O(nlogn)-time exact decision algorithm that handles any polygonal shape, assuming the real-RAM model of computation. A variant of the algorithm, which we have implemented using the cgal library, is based on rational arithmetic and answers the same deconstruction problem up to an uncertainty parameter δ its running time additionally depends on δ. If the input shape is found to be approximable, this algorithm also computes an approximate solution for the problem. It also allows us to solve parameter-optimization problems induced by the offset-deconstruction problem. For convex shapes, the complexity of the exact decision algorithm drops to O(n), which is also the time required to compute a solution P with at most one more vertex than a vertex-minimal one. AU - Berberich, Eric AU - Halperin, Dan AU - Kerber, Michael AU - Pogalnikova, Roza ID - 3115 IS - 4 JF - Discrete & Computational Geometry TI - Deconstructing approximate offsets VL - 48 ER - TY - JOUR AB - Computing the topology of an algebraic plane curve C means computing a combinatorial graph that is isotopic to C and thus represents its topology in R2. We prove that, for a polynomial of degree n with integer coefficients bounded by 2ρ, the topology of the induced curve can be computed with bit operations ( indicates that we omit logarithmic factors). Our analysis improves the previous best known complexity bounds by a factor of n2. The improvement is based on new techniques to compute and refine isolating intervals for the real roots of polynomials, and on the consequent amortized analysis of the critical fibers of the algebraic curve. AU - Kerber, Michael AU - Sagraloff, Michael ID - 3331 IS - 3 JF - Journal of Symbolic Computation TI - A worst case bound for topology computation of algebraic curves VL - 47 ER - TY - JOUR AB - Arrays of vertically aligned ZnO : Cl/TiO2 and ZnO : Cl/ZnxTiOy/TiO2 core–shell nanowires (NWs) were prepared by means of the combination of two solution-growth processes. First, single-crystal ZnO NWs with controlled n-type doping were grown on conducting substrates by a low-cost, high-yield and seed-free electrochemical route. These NWs were covered by a titanium oxide shell of tunable thickness mediating successive adsorption-hydrolysis-condensation steps. Using this atomic-layer growth procedure, titania shells with controlled thickness and the anatase TiO2 phase were obtained after sintering at 450 °C. Higher sintering temperatures resulted in the formation of ZnO : Cl/ZnxTiOy/TiO2 core–shell NWs by the interdiffusion of Zn and Ti ions at the ZnO–TiO2 interface. The performance of ZnO : Cl/TiO2 and ZnO : Cl/ZnxTiOy/TiO2 core–shell NWs towards photoelectrochemical (PEC) water splitting was investigated as a function of the titania shell thickness. Furthermore, the performance of such core–shell NWs as photoelectrodes in dye-sensitized solar cells was also characterized. The TiO2 presence at the ZnO : Cl surface promoted a two-fold increase on the produced photocurrent densities, probing their potential for PEC and optoelectronic applications. Electrochemical impedance spectroscopy was used to corroborate the lower resistance for charge transfer between the NWs and the electrolyte in the presence of the TiO2 shell. AU - Fan, Jiandong AU - Zamani, Reza AU - Fábrega, Cristina AU - Shavel, Alexey AU - Flox, Cristina AU - Ibáñez, Maria AU - Andreu, Teresa AU - López, Amtonio AU - Arbiol, Jordi AU - Morante, Joan AU - Cabot, Andreu ID - 346 IS - 41 JF - Journal of Physics D: Applied Physics TI - Solution-growth and optoelectronic performance of ZnO : Cl/TiO2 and ZnO : Cl/ZnxTiOy/TiO2 core–shell nanowires with tunable shell thickness VL - 45 ER - TY - JOUR AB - The induction of a signaling pathway is characterized by transient complex formation and mutual posttranslational modification of proteins. To faithfully capture this combinatorial process in a mathematical model is an important challenge in systems biology. Exploiting the limited context on which most binding and modification events are conditioned, attempts have been made to reduce the combinatorial complexity by quotienting the reachable set of molecular species into species aggregates while preserving the deterministic semantics of the thermodynamic limit. Recently, we proposed a quotienting that also preserves the stochastic semantics and that is complete in the sense that the semantics of individual species can be recovered from the aggregate semantics. In this paper, we prove that this quotienting yields a sufficient condition for weak lumpability (that is to say that the quotient system is still Markovian for a given set of initial distributions) and that it gives rise to a backward Markov bisimulation between the original and aggregated transition system (which means that the conditional probability of being in a given state in the original system knowing that we are in its equivalence class is an invariant of the system). We illustrate the framework on a case study of the epidermal growth factor (EGF)/insulin receptor crosstalk. AU - Feret, Jérôme AU - Henzinger, Thomas A AU - Koeppl, Heinz AU - Petrov, Tatjana ID - 3168 JF - Theoretical Computer Science TI - Lumpability abstractions of rule based systems VL - 431 ER - TY - JOUR AB - The potential to control the composition and crystal phase at the nanometer scale enable the production of nanocrystalline materials with enhanced functionalities and new applications. In the present work, we detail a novel colloidal synthesis route to prepare nanoparticles of the ternary semiconductor Cu2GeSe3 (CGSe) with nanometer-scale control over their crystal phases. We also demonstrate the structural effect on the thermoelectric properties of bottom-up-prepared CGSe nanomaterials. By careful adjustment of the nucleation and growth temperatures, pure orthorhombic CGSe nanoparticles with cationic order or polytypic CGSe nanoparticles with disordered cation positions can be produced. In this second type of nanoparticle, a high density of twins can be created to periodically change the atomic plane stacking, forming a hexagonal wurtzite CGSe phase. The high yield of the synthetic routes reported here allows the production of single-phase and multiphase CGSe nanoparticles in the gram scale, which permits characterization of the thermoelectric properties of these materials. Reduced thermal conductivities and a related 2.5-fold increase of the thermoelectric figure of merit for multiphase nanomaterials compared to pure-phase CGSe are systematically obtained. These results are discussed in terms of the density and efficiency of phonon scattering centers in both types of materials. AU - Ibáñez, Maria AU - Zamani, Reza AU - Li, Wenhua AU - Cadavid, Doris AU - Gorse, Stéphane AU - Katchoi, Nebll AU - Shavel, Alexey AU - López, Antonioo AU - Morante, Joan AU - Arbiol, Jordi AU - Cabot, Andreu ID - 377 IS - 23 JF - Chemistry of Materials TI - Crystallographic control at the nanoscale to enhance functionality: Polytypic Cu2GeSe3 nanoparticles as thermoelectric materials VL - 24 ER - TY - JOUR AB - We summarize classical and recent results about two-player games played on graphs with ω-regular objectives. These games have applications in the verification and synthesis of reactive systems. Important distinctions are whether a graph game is turn-based or concurrent; deterministic or stochastic; zero-sum or not. We cluster known results and open problems according to these classifications. AU - Chatterjee, Krishnendu AU - Henzinger, Thomas A ID - 3846 IS - 2 JF - Journal of Computer and System Sciences TI - A survey of stochastic ω regular games VL - 78 ER - TY - JOUR AB - In this Letter we present detailed study of the density of states near defects in Bi 2Se 3. In particular, we present data on the commonly found triangular defects in this system. While we do not find any measurable quasiparticle scattering interference effects, we do find localized resonances, which can be well fitted by theory once the potential is taken to be extended to properly account for the observed defects. The data together with the fits confirm that while the local density of states around the Dirac point of the electronic spectrum at the surface is significantly disrupted near the impurity by the creation of low-energy resonance state, the Dirac point is not locally destroyed. We discuss our results in terms of the expected protected surface state of topological insulators. © 2012 American Physical Society. AU - Alpichshev, Zhanybek AU - Biswas, Rudro AU - Balatsky, Alexander AU - Analytis, James AU - Chu, Jiunhaw AU - Fisher, Ian AU - Kapitulnik, Aharon ID - 387 IS - 20 JF - Physical Review Letters TI - STM imaging of impurity resonances on Bi 2Se 3 VL - 108 ER - TY - JOUR AB - The directional transport of the phytohormone auxin depends on the phosphorylation status and polar localization of PIN-FORMED (PIN) auxin efflux proteins. While PINIOD (PID) kinase is directly involved in the phosphorylation of PIN proteins, the phosphatase holoenzyme complexes that dephosphorylate PIN proteins remain elusive. Here, we demonstrate that mutations simultaneously disrupting the function of Arabidopsis thaliana FyPP1 (for Phytochrome-associated serine/threonine protein phosphatase1) and FyPP3, two homologous genes encoding the catalytic subunits of protein phosphatase6 (PP6), cause elevated accumulation of phosphorylated PIN proteins, correlating with a basal-to-apical shift in subcellular PIN localization. The changes in PIN polarity result in increased root basipetal auxin transport and severe defects, including shorter roots, fewer lateral roots, defective columella cells, root meristem collapse, abnormal cotyledons (small, cup-shaped, or fused cotyledons), and altered leaf venation. Our molecular, biochemical, and genetic data support the notion that FyPP1/3, SAL (for SAPS DOMAIN-LIKE), and PP2AA proteins (RCN1 [for ROOTS CURL IN NAPHTHYLPHTHALAMIC ACID1] or PP2AA1, PP2AA2, and PP2AA3) physically interact to form a novel PP6-type heterotrimeric holoenzyme complex. We also show that FyPP1/3, SAL, and PP2AA interact with a subset of PIN proteins and that for SAL the strength of the interaction depends on the PIN phosphorylation status. Thus, an Arabidopsis PP6-type phosphatase holoenzyme acts antagonistically with PID to direct auxin transport polarity and plant development by directly regulating PIN phosphorylation. AU - Dai, Mingqiu AU - Zhang, Chen AU - Urszula Kania AU - Chen, Fang AU - Xue, Qin AU - McCray, Tyra AU - Li, Gang AU - Qin, Genji AU - Wakeley, Michelle AU - Terzaghi, William AU - Wan, Jianmin AU - Zhao, Yunde AU - Xu, Jian AU - Jirí Friml AU - Deng, Xing W AU - Wang, Haiyang ID - 3110 IS - 6 JF - Plant Cell TI - A PP6 type phosphatase holoenzyme directly regulates PIN phosphorylation and auxin efflux in Arabidopsis VL - 24 ER - TY - JOUR AB - A cell membrane can be considered a liquid-phase plane in which lipids and proteins theoretically are free to diffuse. Numerous reports,however, describe retarded diffusion ofmembrane proteins in animal cells. This anomalous diffusion results from a combination of structuring factors including protein-protein interactions, cytoskeleton corralling, and lipid organization into microdomains. In plant cells, plasma-membrane (PM) proteins have been described as relatively immobile, but the control mechanisms that structure the PM have not been studied. Here, we use fluorescence recovery after photobleaching to estimate mobility of a set of minimal PM proteins. These proteins consist only of a PM-anchoring domain fused to a fluorescent protein, but their mobilities remained limited, as is the case for many full-length proteins. Neither the cytoskeleton nor membrane microdomain structure was involved in constraining the diffusion of these proteins. The cell wall, however, was shown to have a crucial role in immobilizing PM proteins. In addition, by single-molecule fluorescence imaging we confirmed that the pattern of cellulose deposition in the cell wall affects the trajectory and speed ofPMprotein diffusion. Regulation ofPMprotein dynamics by the plant cell wall can be interpreted as a mechanism for regulating protein interactions in processes such as trafficking and signal transduction. AU - Martinière, Alexandre AU - Lavagi, Irene AU - Nageswaran, Gayathri AU - Rolfe, Daniel J AU - Maneta-Peyret, Lilly AU - Luu, Doan-Trung AU - Botchway, Stanley W AU - Webb, Stephen E AU - Mongrand, Sebastien AU - Maurel, Christophe AU - Martin-Fernandez, Marisa L AU - Kleine-Vehn, Jürgen AU - Jirí Friml AU - Moreau, Patrick AU - Runions, John ID - 3113 IS - 31 JF - PNAS TI - Cell wall constrains lateral diffusion of plant plasma membrane proteins VL - 109 ER - TY - JOUR AB - Auxin is a key coordinative signal required for many aspects of plant development and its levels are controlled by auxin metabolism and intercellular auxin transport. Here we find that a member of PIN auxin transporter family, PIN8 is expressed in male gametophyte of Arabidopsis thaliana and has a crucial role in pollen development and functionality. Ectopic expression in sporophytic tissues establishes a role of PIN8 in regulating auxin homoeostasis and metabolism. PIN8 co-localizes with PIN5 to the endoplasmic reticulum (ER) where it acts as an auxin transporter. Genetic analyses reveal an antagonistic action of PIN5 and PIN8 in the regulation of intracellular auxin homoeostasis and gametophyte as well as sporophyte development. Our results reveal a role of the auxin transport in male gametophyte development in which the distinct actions of ER-localized PIN transporters regulate cellular auxin homoeostasis and maintain the auxin levels optimal for pollen development and pollen tube growth. AU - Ding, Zhaojun AU - Wang, Bangjun AU - Moreno, Ignacio AU - Dupláková, Nikoleta AU - Sibu Simon AU - Carraro, Nicola AU - Reemmer, Jesica AU - Pěnčík, Aleš AU - Xu Chen AU - Tejos, Ricardo I AU - Skůpa, Petr AU - Pollmann, Stephan AU - Mravec, Jozef AU - Petrášek, Jan AU - Zažímalová, Eva AU - Honys, David AU - Rolčík, Jakub AU - Murphy, Angus S AU - Orellana, Ariel AU - Geisler, Markus AU - Jirí Friml ID - 3114 IS - AN 941 JF - Nature Communications TI - ER-localized auxin transporter PIN8 regulates auxin homeostasis and male gametophyte development in Arabidopsis VL - 3 ER - TY - JOUR AB - PIN-FORMED (PIN) protein-mediated auxin polar transport is critically important for development, pattern formation, and morphogenesis in plants. Auxin has been implicated in the regulation of polar auxin transport by inhibiting PIN endocytosis [1, 2], but how auxin regulates this process is poorly understood. Our genetic screen identified the Arabidopsis SPIKE1 (SPK1) gene whose loss-of-function mutations increased lateral root density and retarded gravitropic responses, as do pin2 knockout mutations [3]. SPK1 belongs to the conserved DHR2-Dock family of Rho guanine nucleotide exchange factors [4-6]. The spk1 mutations induced PIN2 internalization that was not suppressed by auxin, as did the loss-of-function mutations for Rho-like GTPase from Plants 6 (ROP6)-GTPase or its effector RIC1. Furthermore, SPK1 was required for auxin induction of ROP6 activation. Our results have established a Rho GTPase-based auxin signaling pathway that maintains PIN2 polar distribution to the plasma membrane via inhibition of its internalization in Arabidopsis roots. Our findings provide new insights into signaling mechanisms that underlie the regulation of the dynamic trafficking of PINs required for long-distance auxin transport and that link auxin signaling to PIN-mediated pattern formation and morphogenesis. AU - Lin, Deshu AU - Nagawa, Shingo AU - Chen, Jisheng AU - Cao, Lingyan AU - Xu Chen AU - Xu, Tongda AU - Hongjiang Li AU - Dhonukshe, Pankaj AU - Yamamuro, Chizuko AU - Jirí Friml AU - Scheres, Ben AU - Fu, Ying AU - Yang, Zhenbiao ID - 3111 IS - 14 JF - Current Biology TI - A ROP GTPase dependent auxin signaling pathway regulates the subcellular distribution of PIN2 in Arabidopsis roots VL - 22 ER - TY - JOUR AB - The dynamic spatial and temporal distribution of the crucial plant signaling molecule auxin is achieved by feedback coordination of auxin signaling and intercellular auxin transport pathways [1, 2]. Developmental roles of auxin have been attributed predominantly to its effect on transcription; however, an alternative pathway involving AUXIN BINDING PROTEIN1 (ABP1) has been proposed to regulate clathrin-mediated endocytosis in roots and Rho-like GTPase (ROP)-dependent pavement cell interdigitation in leaves [3, 4]. In this study, we show that ROP6 and its downstream effector RIC1 regulate clathrin association with the plasma membrane for clathrin-mediated endocytosis, as well as for its feedback regulation by auxin. Genetic analysis revealed that ROP6/RIC1 acts downstream of ABP1 to regulate endocytosis. This signaling circuit is also involved in the feedback regulation of PIN-FORMED 1 (PIN1) and PIN2 auxin transporters activity (via its constitutive endocytosis) and corresponding auxin transport-mediated processes, including root gravitropism and leave vascular tissue patterning. Our findings suggest that the signaling module auxin-ABP1-ROP6/RIC1-clathrin-PIN1/PIN2 is a shared component of the feedback regulation of auxin transport during both root and aerial development. AU - Xu Chen AU - Naramoto, Satoshi AU - Robert, Stéphanie AU - Tejos, Ricardo AU - Löfke, Christian AU - Lin, Deshu AU - Yang, Zhenbiao AU - Jirí Friml ID - 3112 IS - 14 JF - Current Biology TI - ABP1 and ROP6 GTPase signaling regulate clathrin mediated endocytosis in Arabidopsis roots VL - 22 ER - TY - JOUR AB - We consider two-player zero-sum stochastic games on graphs with ω-regular winning conditions specified as parity objectives. These games have applications in the design and control of reactive systems. We survey the complexity results for the problem of deciding the winner in such games, and in classes of interest obtained as special cases, based on the information and the power of randomization available to the players, on the class of objectives and on the winning mode. On the basis of information, these games can be classified as follows: (a) partial-observation (both players have partial view of the game); (b) one-sided partial-observation (one player has partial-observation and the other player has complete-observation); and (c) complete-observation (both players have complete view of the game). The one-sided partial-observation games have two important subclasses: the one-player games, known as partial-observation Markov decision processes (POMDPs), and the blind one-player games, known as probabilistic automata. On the basis of randomization, (a) the players may not be allowed to use randomization (pure strategies), or (b) they may choose a probability distribution over actions but the actual random choice is external and not visible to the player (actions invisible), or (c) they may use full randomization. Finally, various classes of games are obtained by restricting the parity objective to a reachability, safety, Büchi, or coBüchi condition. We also consider several winning modes, such as sure-winning (i.e., all outcomes of a strategy have to satisfy the winning condition), almost-sure winning (i.e., winning with probability 1), limit-sure winning (i.e., winning with probability arbitrarily close to 1), and value-threshold winning (i.e., winning with probability at least ν, where ν is a given rational). AU - Chatterjee, Krishnendu AU - Doyen, Laurent AU - Henzinger, Thomas A ID - 3128 IS - 2 JF - Formal Methods in System Design TI - A survey of partial-observation stochastic parity games VL - 43 ER - TY - CONF AB - We propose a new learning method to infer a mid-level feature representation that combines the advantage of semantic attribute representations with the higher expressive power of non-semantic features. The idea lies in augmenting an existing attribute-based representation with additional dimensions for which an autoencoder model is coupled with a large-margin principle. This construction allows a smooth transition between the zero-shot regime with no training example, the unsupervised regime with training examples but without class labels, and the supervised regime with training examples and with class labels. The resulting optimization problem can be solved efficiently, because several of the necessity steps have closed-form solutions. Through extensive experiments we show that the augmented representation achieves better results in terms of object categorization accuracy than the semantic representation alone. AU - Sharmanska, Viktoriia AU - Quadrianto, Novi AU - Lampert, Christoph ID - 3125 IS - PART 5 TI - Augmented attribute representations VL - 7576 ER - TY - CONF AB - Let K be a simplicial complex and g the rank of its p-th homology group Hp(K) defined with ℤ2 coefficients. We show that we can compute a basis H of Hp(K) and annotate each p-simplex of K with a binary vector of length g with the following property: the annotations, summed over all p-simplices in any p-cycle z, provide the coordinate vector of the homology class [z] in the basis H. The basis and the annotations for all simplices can be computed in O(n ω ) time, where n is the size of K and ω < 2.376 is a quantity so that two n×n matrices can be multiplied in O(n ω ) time. The precomputed annotations permit answering queries about the independence or the triviality of p-cycles efficiently. Using annotations of edges in 2-complexes, we derive better algorithms for computing optimal basis and optimal homologous cycles in 1 - dimensional homology. Specifically, for computing an optimal basis of H1(K) , we improve the previously known time complexity from O(n 4) to O(n ω  + n 2 g ω − 1). Here n denotes the size of the 2-skeleton of K and g the rank of H1(K) . Computing an optimal cycle homologous to a given 1-cycle is NP-hard even for surfaces and an algorithm taking 2 O(g) nlogn time is known for surfaces. We extend this algorithm to work with arbitrary 2-complexes in O(n ω ) + 2 O(g) n 2logn time using annotations. AU - Busaryev, Oleksiy AU - Cabello, Sergio AU - Chen, Chao AU - Dey, Tamal AU - Wang, Yusu ID - 3129 TI - Annotating simplices with a homology basis and its applications VL - 7357 ER - TY - CONF AB - In this work we propose a new information-theoretic clustering algorithm that infers cluster memberships by direct optimization of a non-parametric mutual information estimate between data distribution and cluster assignment. Although the optimization objective has a solid theoretical foundation it is hard to optimize. We propose an approximate optimization formulation that leads to an efficient algorithm with low runtime complexity. The algorithm has a single free parameter, the number of clusters to find. We demonstrate superior performance on several synthetic and real datasets. AU - Müller, Andreas AU - Nowozin, Sebastian AU - Lampert, Christoph ID - 3126 TI - Information theoretic clustering using minimal spanning trees VL - 7476 ER - TY - CONF AB - We propose synchronous interfaces, a new interface theory for discrete-time systems. We use an application to time-triggered scheduling to drive the design choices for our formalism; in particular, additionally to deriving useful mathematical properties, we focus on providing a syntax which is adapted to natural high-level system modeling. As a result, we develop an interface model that relies on a guarded-command based language and is equipped with shared variables and explicit discrete-time clocks. We define all standard interface operations: compatibility checking, composition, refinement, and shared refinement. Apart from the synchronous interface model, the contribution of this paper is the establishment of a formal relation between interface theories and real-time scheduling, where we demonstrate a fully automatic framework for the incremental computation of time-triggered schedules. AU - Delahaye, Benoît AU - Fahrenberg, Uli AU - Henzinger, Thomas A AU - Legay, Axel AU - Nickovic, Dejan ID - 3155 TI - Synchronous interface theories and time triggered scheduling VL - 7273 ER - TY - JOUR AB - The structure of hierarchical networks in biological and physical systems has long been characterized using the Horton-Strahler ordering scheme. The scheme assigns an integer order to each edge in the network based on the topology of branching such that the order increases from distal parts of the network (e.g., mountain streams or capillaries) to the "root" of the network (e.g., the river outlet or the aorta). However, Horton-Strahler ordering cannot be applied to networks with loops because they they create a contradiction in the edge ordering in terms of which edge precedes another in the hierarchy. Here, we present a generalization of the Horton-Strahler order to weighted planar reticular networks, where weights are assumed to correlate with the importance of network edges, e.g., weights estimated from edge widths may correlate to flow capacity. Our method assigns hierarchical levels not only to edges of the network, but also to its loops, and classifies the edges into reticular edges, which are responsible for loop formation, and tree edges. In addition, we perform a detailed and rigorous theoretical analysis of the sensitivity of the hierarchical levels to weight perturbations. In doing so, we show that the ordering of the reticular edges is more robust to noise in weight estimation than is the ordering of the tree edges. We discuss applications of this generalized Horton-Strahler ordering to the study of leaf venation and other biological networks. AU - Mileyko, Yuriy AU - Edelsbrunner, Herbert AU - Price, Charles AU - Weitz, Joshua ID - 3159 IS - 6 JF - PLoS One TI - Hierarchical ordering of reticular networks VL - 7 ER -