Given a weighted, undirected simple graph G = (V, E, d) (where d : E → R + ), the Distance Geometry Problem (DGP) is to determine an embeddingAlthough, in general, the DGP is solved using continuous methods, under certain conditions the search is reduced to a discrete set of points. We give one such condition as a particular order on V . We formalize the decision problem of determining whether such an order exists for a given graph and show that this problem is NP-complete in general and polynomial for fixed dimension K. We present results of computational experiments on a set of protein backbones whose natural atomic order does not satisfy the order requirements and compare our approach with some available continuous space searches.

Motivated by constraint-based CAD software, we introduce a new, very general, rigidity model: the body-and-cad structure, composed of rigid bodies in 3D constrained by pairwise coincidence, angle and distance constraints. We have identified 21 relevant geometric constraints and a new, necessary, but not sufficient, counting condition for minimal rigidity of body-and-cad structures: nested sparsity. We remark that the classical body-and-bar rigidity model can be viewed as a body-and-cad structure that uses only one constraint from this new set of constraints.

We study the rigidity of body-and-cad frameworks which capture the majority of the geometric constraints used in 3D mechanical engineering CAD software. We present a combinatorial characterization of the generic minimal rigidity of a subset of body-and-cad frameworks in which we treat 20 of the 21 body-and-cad constraints, omitting only point-point coincidences. While the handful of classical combinatorial characterizations of rigidity focus on distance constraints between points, this is the first result simultaneously addressing coincidence, angular, and distance constraints. Our result is stated in terms of the partitioning of a graph into edge-disjoint spanning trees. This combinatorial approach provides the theoretical basis for the development of deterministic algorithms (that will not depend on numerical methods) for analyzing the rigidity of body-and-cad frameworks.

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2016-12-23T18:52:10

BackgroundA protein's function is determined by the wide range of motions exhibited by its 3D structure. However, current experimental techniques are not able to reliably provide the level of detail required for elucidating the exact mechanisms of protein motion essential for effective drug screening and design. Computational tools are instrumental in the study of the underlying structure-function relationship. We focus on a special type of proteins called "hinge proteins" which exhibit a motion that can be interpreted as a rotation of one domain relative to another.ResultsThis work proposes a computational approach that uses the geometric structure of a single conformation to predict the feasible motions of the protein and is founded in recent work from rigidity theory, an area of mathematics that studies flexibility properties of general structures. Given a single conformational state, our analysis predicts a relative axis of motion between two specified domains. We analyze a dataset of 19 structures known to exhibit this hinge-like behavior. For 15, the predicted axis is consistent with a motion to a second, known conformation. We present a detailed case study for three proteins whose dynamics have been well-studied in the literature: calmodulin, the LAO binding protein and the Bence-Jones protein.ConclusionsOur results show that incorporating rigidity-theoretic analyses can lead to effective computational methods for understanding hinge motions in macromolecules. This initial investigation is the first step towards a new tool for probing the structure-dynamics relationship in proteins.

Data exchange between different computer-aided design (CAD) systems is a major problem inhibiting information integration in collaborative engineering environments. Existing CAD data format standards such as STEP and IGES enable geometric data exchange. However, they ignore construction history, features, constraints, and other parametric-based CAD data. As a result, they are inadequate for supporting modification, extension and other important higher-level functionality when accessing an imported CAD model from another CAD system. Achieving such higher-level functionality therefore often requires a time-consuming, error-prone, tedious process of manually recreating the model in the target CAD system. Based on techniques adapted from programming language research, this paper presents an approach to exchanging parametric data between CAD systems using formally-defined conversion semantics. We have demonstrated the utility of our approach by developing a prototype implementation that automates the conversion of 2D sketches between two popular CAD systems: Pro/ENGINEER and SolidWorks. We present examples showing that our approach is able to accurately convert parametric CAD data even in cases where models were constructed using operations from the source CAD system that have no direct counterpart in the target CAD system. Although the case study focuses on 2D interoperability, our approach provides formal foundations for supporting 3D and semantic interoperability between CAD systems.

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