Robustness and modular design of the Drosophila segment polarity network

Biomolecular networks have to perform their functions robustly. A robust function may have preferences in the topological structures of the underlying network. We carried out an exhaustive computational analysis on network topologies in relation to a patterning function in Drosophila embryogenesis. We found that while the vast majority of topologies can either not perform the required function or only do so very fragilely, a small fraction of topologies emerges as particularly robust for the function. The topology adopted by Drosophila, that of the segment polarity network, is a top ranking one among all topologies with no direct autoregulation. Furthermore, we found that all robust topologies are modular--each being a combination of three kinds of modules. These modules can be traced back to three sub-functions of the patterning function and their combinations provide a combinatorial variability for the robust topologies. Our results suggest that the requirement of functional robustness drastically reduces the choices of viable topology to a limited set of modular combinations among which nature optimizes its choice under evolutionary and other biological constraints.Comment: Supplementary Information and Synopsis available at http://www.ucsf.edu/tanglab

Failure of the Regge approach in two dimensional quantum gravity

Regge's method for regularizing euclidean quantum gravity is applied to two dimensional gravity. We use two different strategies to simulate the Regge path integral at a fixed value of the total area: A standard Metropolis simulation combined with a histogramming method and a direct simulation using a Hybrid Monte Carlo algorithm. Using topologies with genus zero and two and a scale invariant integration measure, we show that the Regge method does not reproduce the value of the string susceptibility of the continuum model. We show that the string susceptibility depends strongly on the choice of the measure in the path integral. We argue that the failure of the Regge approach is due to spurious contributions of reparametrization degrees of freedom to the path integral.Comment: 27 pages, LaTex + uuencoded figure files (13 postscript files

Geodesic grassfire for computing mixed-dimensional skeletons

Skeleton descriptors are commonly used to represent, understand and process shapes. While existing methods produce skeletons at a fixed dimension, such as surface or curve skeletons for a 3D object, often times objects are better described using skeleton geometry at a mixture of dimensions. In this paper we present a novel algorithm for computing mixed-dimensional skeletons. Our method is guided by a continuous analogue that extends the classical grassfire erosion. This analogue allows us to identify medial geometry at multiple dimensions, and to formulate a measure that captures how well an object part is described by medial geometry at a particular dimension. Guided by this analogue, we devise a discrete algorithm that computes a topology-preserving skeleton by iterative thinning. The algorithm is simple to implement, and produces robust skeletons that naturally capture shape components. Under Revie

Skeletons of stable maps II: Superabundant geometries

We implement new techniques involving Artin fans to study the realizability of tropical stable maps in superabundant combinatorial types. Our approach is to understand the skeleton of a fundamental object in logarithmic Gromov--Witten theory -- the stack of prestable maps to the Artin fan. This is used to examine the structure of the locus of realizable tropical curves and derive 3 principal consequences. First, we prove a realizability theorem for limits of families of tropical stable maps. Second, we extend the sufficiency of Speyer's well-spacedness condition to the case of curves with good reduction. Finally, we demonstrate the existence of liftable genus 1 superabundant tropical curves that violate the well-spacedness condition.Comment: 17 pages, 1 figure. v2 fixes a minor gap in the proof of Theorem D and adds details to the construction of the skeleton of a toroidal Artin stack. Minor clarifications throughout. To appear in Research in the Mathematical Science