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This computer program calculates the orientational variants, the operators and the composition table of a groupoid.

A computer program called GenOVa, written in Python, calculates the orientational variants, the operators (special types of misorientations between variants) and the composition table associated with a groupoid structure. The variants can be represented by three-dimensional shapes or by pole figures.

In this paper we provide a review of selected mathematical ideas that can help us better understand the boundary between living and non-living systems. We focus on group theory and abstract algebra applied to molecular systems biology. Throughout this paper we briefly describe possible open problems. In connection with the genetic code we propose that it may be possible to use perturbation theory to explore the adjacent possibilities in the 64-dimensional space-time manifold of the evolving genome.

With regards to algebraic graph theory, there are several minor open problems we discuss. In relation to network dynamics and groupoid formalism we suggest that the network graph might not be the main focus for understanding the phenotype but rather the phase space of the network dynamics. We show a simple case of a C6 network and its phase space network. We envision that the molecular network of a cell is actually a complex network of hypercycles and feedback circuits that could be better represented in a higher-dimensional space. We conjecture that targeting nodes in the molecular network that have key roles in the phase space, as revealed by analysis of the automorphism decomposition, might be a better way to drug discovery and treatment of cancer.

In this paper we discuss a symbolic-numeric algorithm for Boolean operations, closed in the algebra of curved polyhedra whose boundary is triangulated with algebraic patches (A-patches). This approach uses a linear polyhedron as a first approximation of both the arguments and the result. On each triangle of a boundary representation of such linear approximation, a piecewise cubic algebraic interpolant is built, using a C1-continuous prism algebraic patch (prism A-patch) that interpolates the three triangle vertices, with given normal vectors. The boundary representation only stores the vertices of the initial triangulation and their external vertex normals. In order to represent also flat and/or sharp local features, the corresponding normal-per-face and/or normal-per-edge may be also given, respectively. The topology is described by storing, for each curved triangle, the two triples of pointers to incident vertices and to adjacent triangles. For each triangle, a scaffolding prism is built, produced by its extreme vertices and normals, which provides a containment volume for the curved interpolating A-patch. When looking for the result of a regularized Boolean operation, the 0-set of a tri-variate polynomial within each such prism is generated, and intersected with the analogous 0-sets of the other curved polyhedron, when two prisms have non-empty intersection. The intersection curves of the boundaries are traced and used to decompose each boundary into the 3 standard classes of subpatches, denoted in, out and on. While tracing the intersection curves, the locally refined triangulation of intersecting patches is produced, and added to the boundary representation.

A computer program has been written to reconstruct the parent grains from EBSD data of phase transition materials.

A computer program called ARPGE written in Python uses the theoretical results generated by the computer program GenOVa to automatically reconstruct the parent grains from electron backscatter diffraction data obtained on phase transition materials with or without residual parent phase. The misorientations between daughter grains are identified with operators, the daughter grains are identified with indexed variants, the orientations of the parent grains are determined, and some statistics on the variants and operators are established. Some examples with martensitic transformations in iron and titanium alloys were treated. Variant selection phenomena were revealed.

In this paper we study the formal algebraic structure underlying the intrinsic classification algorithm, recently introduced in Singer et al. (SIAM J. Imaging Sci. 2011, accepted), for classifying noisy projection images of similar viewing directions in three-dimensional cryo-electron microscopy (cryo-EM). This preliminary classification is of fundamental importance in determining the three-dimensional structure of macromolecules from cryo-EM images. Inspecting this algebraic structure we obtain a conceptual explanation for the admissibility (correctness) of the algorithm and a proof of its numerical stability. The proof relies on studying the spectral properties of an integral operator of geometric origin on the two-dimensional sphere, called the localized parallel transport operator. Along the way, we continue to develop the representation theoretic set-up for three-dimensional cryo-EM that was initiated in Hadani and Singer (Ann. Math. 2010, accepted).

This paper introduces perspex algebra which is being developed as a common representation of geometrical knowledge. A perspex can currently be interpreted in one of four ways. First, the algebraic perspex is a generalization of matrices, it provides the most general representation for all of the interpretations of a perspex. The algebraic perspex can be used to describe arbitrary sets of coordinates. The remaining three interpretations of the perspex are all related to square matrices and operate in a Euclidean model of projective space-time, called perspex space. Perspex space differs from the usual Euclidean model of projective space in that it contains the point at nullity. It is argued that the point at nullity is necessary for a consistent account of perspective in top-down vision. Second, the geometric perspex is a simplex in perspex space. It can be used as a primitive building block for shapes, or as a way of recording landmarks on shapes. Third, the transformational perspex describes linear transformations in perspex space that provide the affine and perspective transformations in space-time. It can be used to match a prototype shape to an image, even in so called 'accidental' views where the depth of an object disappears from view, or an object stays in the same place across time. Fourth, the parametric perspex describes the geometric and transformational perspexes in terms of parameters that are related to everyday English descriptions. The parametric perspex can be used to obtain both continuous and categorical perception of objects. The paper ends with a discussion of issues related to using a perspex to describe logic.

Existing 3-dimensional acoustic target tracking methods that use wired/wireless networked sensor nodes to track targets based on four sensing coverage do not always compute the feasible spatio-temporal information of target objects. To investigate this discrepancy in a formal setting, we propose a geometric model of the target tracking problem alongside its equivalent geometric dual model that is easier to solve. We then study and prove some properties of dual model by exploiting its relationship with algebra. Based on these properties, we propose a four coverage axis line method based on four sensing coverage and prove that four sensing coverage always yields two dual correct answers; usually one of them is infeasible. By showing that the feasible answer can be only sometimes identified by using a simple time test method such as the one proposed by ourselves, we prove that four sensing coverage fails to always yield the feasible spatio-temporal information of a target object. We further prove that five sensing coverage always gives the feasible position of a target object under certain conditions that are discussed in this paper. We propose three extensions to four coverage axis line method, namely, five coverage extent point method, five coverage extended axis lines method, and five coverage redundant axis lines method. Computation and time complexities of all four proposed methods are equal in the worst cases as well as on average being equal to Θ(1) each. Proposed methods and proved facts about capabilities of sensing coverage degree in this paper can be used in all other methods of acoustic target tracking like Bayesian filtering methods.

In this paper we present a package for implementing exact kinetic data structures built on objects which move along polynomial trajectories. We discuss how the package design was influenced by various considerations, including extensibility, support for multiple kinetic data structures, access to existing data structures and algorithms in CGAL, as well as debugging. Due to the similarity between the operations involved, the software can also be used to compute arrangements of polynomial objects using a sweepline approach. The package consists of three main parts, the kinetic data structure framework support code, an algebraic kernel which implements the set of algebraic operations required for kinetic data structure processing, and kinetic data structures for Delaunay triangulations in one and two dimensions, and Delaunay and regular triangulations in three dimensions. The models provided for the algebraic kernel support both exact operations and inexact approximations with heuristics to improve numerical stability.

The effects of crystal-plasticity on the U-Th-Pb system in zircon is studied by quantitative microstructural and microchemical analysis of a large zircon grain collected from pyroxenite of the Lewisian Complex, Scotland. Electron backscatter diffraction (EBSD) mapping reveals a c.18° variation in crystallographic orientation that comprises both a gradual change in orientation and a series of discrete low-angle (<4°) boundaries. These microstructural data are consistent with crystal-plastic deformation of zircon associated with the formation and migration of dislocations. A heterogeneous pattern of dark cathodoluminescence, with the darkest domains coinciding with low-angle boundaries, mimics the deformation microstructure identified by EBSD. Geochemical data collected using the Sensitive High Resolution Ion MicroProbe (SHRIMP) shows a positive correlation between concentrations of the elements U, Th and Pb (ranging from 20–60 ppm, 30–110 ppm, and 14–36 ppm, respectively) and Th/U ratio (1.13 – 1.8) with the deformation microstructure. The highest measured concentrations and Th/U coincide with low-angle boundaries. This enrichment is interpreted to reflect enhanced bulk diffusion of U and Th due to the formation and migration of high-diffusivity dislocations. 207Pb/206Pb ages for individual analyses show no significant variation across the grain, and define a concordant, combined mean age of 2451 ± 14 Ma. This indicates that the grain was deformed shortly after initial crystallization, most probably during retrograde Inverian metamorphism at amphibolite facies conditions. The elevated Th over U and consistent 207Pb/206Pb ages indicates that deformation most likely occurred in the presence of a late-stage magmatic fluid that drove an increase in the Th/U during deformation. The relative enrichment of Th over U implies that Th/U ratio may not always be a robust indicator of crystallization environment. This study provides the first evidence of deformation-related modification of the U-Th system in zircon and has fundamental implications for the application and interpretation of zircon trace element data.

Tensor scale (t-scale) is a parametric representation of local structure morphology that simultaneously describes its orientation, shape and isotropic scale. At any image location, t-scale represents the largest ellipse (an ellipsoid in three dimensions) centered at that location and contained in the same homogeneous region. Here, we present an improved algorithm for t-scale computation and study its application to image interpolation. Specifically, the t-scale computation algorithm is improved by: (1) enhancing the accuracy of identifying local structure boundary and (2) combining both algebraic and geometric approaches in ellipse fitting. In the context of interpolation, a closed form solution is presented to determine the interpolation line at each image location in a gray level image using t-scale information of adjacent slices. At each location on an image slice, the method derives normal vector from its t-scale that yields trans-orientation of the local structure and points to the closest edge point. Normal vectors at the matching two-dimensional locations on two adjacent slices are used to compute the interpolation line using a closed form equation. The method has been applied to BrainWeb data sets and to several other images from clinical applications and its accuracy and response to noise and other image-degrading factors have been examined and compared with those of current state-of-the-art interpolation methods. Experimental results have established the superiority of the new t-scale based interpolation method as compared to existing interpolation algorithms. Also, a quantitative analysis based on the paired t-test of residual errors has ascertained that the improvements observed using the t-scale based interpolation are statistically significant.

The SPARQL LeftJoin abstract operator is not distributive over Union; this limits the algebraic manipulation of graph patterns, which in turn restricts the ability to create query plans for distributed processing or query optimization. In this paper, we present semQA, an algebraic extension for the SPARQL query language for RDF, which overcomes this issue by transforming graph patterns through the use of an idempotent disjunction operator Or as a substitute for Union. This permits the application of a set of equivalences that transform a query into distinct forms. We further present an algorithm to derive the solution set of the original query from the solution set of a query where Union has been substituted by Or. We also analyze the combined complexity of SPARQL, proving it to be NP-complete. It is also shown that the SPARQL query language is not, in the general case, fixed-parameter tractable. Experimental results are presented to validate the query evaluation methodology presented in this paper against the SPARQL standard to corroborate the complexity analysis and to illustrate the gains in processing cost reduction that can be obtained through the application of semQA.

Background

Computational methods for problem solving need to interleave information access and algorithm execution in a problem-specific workflow. The structures of these workflows are defined by a scaffold of syntactic, semantic and algebraic objects capable of representing them. Despite the proliferation of GUIs (Graphic User Interfaces) in bioinformatics, only some of them provide workflow capabilities; surprisingly, no meta-analysis of workflow operators and components in bioinformatics has been reported.

Results

We present a set of syntactic components and algebraic operators capable of representing analytical workflows in bioinformatics. Iteration, recursion, the use of conditional statements, and management of suspend/resume tasks have traditionally been implemented on an ad hoc basis and hard-coded; by having these operators properly defined it is possible to use and parameterize them as generic re-usable components. To illustrate how these operations can be orchestrated, we present GPIPE, a prototype graphic pipeline generator for PISE that allows the definition of a pipeline, parameterization of its component methods, and storage of metadata in XML formats. This implementation goes beyond the macro capacities currently in PISE. As the entire analysis protocol is defined in XML, a complete bioinformatic experiment (linked sets of methods, parameters and results) can be reproduced or shared among users. Availability: (interactive), (download).

Conclusion

From our meta-analysis we have identified syntactic structures and algebraic operators common to many workflows in bioinformatics. The workflow components and algebraic operators can be assimilated into re-usable software components. GPIPE, a prototype implementation of this framework, provides a GUI builder to facilitate the generation of workflows and integration of heterogeneous analytical tools.

The oligonucleotide r(GGACUUCGGUCC) has been observed to adopt a hairpin conformation by solution NMR and a double helical conformation by X-ray diffraction. In order to understand this apparent conflict, we used time-resolved fluorescence depolarization and 19fluorine NMR to follow the secondary structure of this dodecamer as the solution composition was changed stepwise from the NMR experimental conditions to those used for crystallization. Calculation of the dodecamer concentration in the crystal (180 mM strands) and the cation concentration needed for neutrality (>2 M) prompted investigation of a tethered species, in which two dodecamers are connected by a string of 4 nt, geometrically equivalent to approximately 100 mM strands, in 2.5 M NaCl. The RNA tetraloop and its DNA analog maintain a single-strand hairpin conformation in solution, even under the conditions used to grow the crystal. Under high salt conditions, the tethered RNA and DNA analogs of this sequence yield secondary components which could be the double helical conformation. Crystal contacts in addition to solvent changes and high RNA concentrations are needed to obtain the double helix as the predominant species.

Background

Given n strings s1, …, sn each of length ℓ and a nonnegative integer d, the CLOSEST STRING problem asks to find a center string s such that none of the input strings has Hamming distance greater than d from s. Finding a common pattern in many – but not necessarily all – input strings is an important task that plays a role in many applications in bioinformatics.

Results

Although the closest string model is robust to the oversampling of strings in the input, it is severely affected by the existence of outliers. We propose a refined model, the CLOSEST STRING WITH OUTLIERS (CSWO) problem, to overcome this limitation. This new model asks for a center string s that is within Hamming distance d to at least n – k of the n input strings, where k is a parameter describing the maximum number of outliers. A CSWO solution not only provides the center string as a representative for the set of strings but also reveals the outliers of the set.

We provide fixed parameter algorithms for CSWO when d and k are parameters, for both bounded and unbounded alphabets. We also show that when the alphabet is unbounded the problem is W[1]-hard with respect to n – k, ℓ, and d.

Conclusions

Our refined model abstractly models finding common patterns in several but not all input strings. We initialize the study of the computability of this model and show that it is sensitive to different parameterizations. Lastly, we conclude by suggesting several open problems which warrant further investigation.

Background

Comparison of metabolic networks is typically performed based on the organisms' enzyme contents. This approach disregards functional replacements as well as orthologies that are misannotated. Direct comparison of the structure of metabolic networks can circumvent these problems.

Results

Metabolic networks are naturally represented as directed hypergraphs in such a way that metabolites are nodes and enzyme-catalyzed reactions form (hyper)edges. The familiar operations from set algebra (union, intersection, and difference) form a natural basis for both the pairwise comparison of networks and identification of distinct metabolic features of a set of algorithms. We report here on an implementation of this approach and its application to the procaryotes.

Conclusion

We demonstrate that metabolic networks contain valuable phylogenetic information by comparing phylogenies obtained from network comparisons with 16S RNA phylogenies. The algebraic approach to metabolic networks is suitable to study metabolic innovations in two sets of organisms, free living microbes and Pyrococci, as well as obligate intracellular pathogens.

Background

Improving the accuracy and efficiency of motif recognition is an important computational challenge that has application to detecting transcription factor binding sites in genomic data. Closely related to motif recognition is the CONSENSUS STRING decision problem that asks, given a parameter d and a set of ℓ-length strings S = {s1, ..., sn}, whether there exists a consensus string that has Hamming distance at most d from any string in S. A set of strings S is pairwise bounded if the Hamming distance between any pair of strings in S is at most 2d. It is trivial to determine whether a set is pairwise bounded, and a set cannot have a consensus string unless it is pairwise bounded. We use CONSENSUS STRING to determine whether or not a pairwise bounded set has a consensus. Unfortunately, CONSENSUS STRING is NP-complete. The lack of an efficient method to solve the CONSENSUS STRING problem has caused it to become a computational bottleneck in MCL-WMR, a motif recognition program capable of solving difficult motif recognition problem instances.

Results

We focus on the development of a method for solving CONSENSUS STRING quickly with a small probability of error. We apply this heuristic to develop a new motif recognition program, sMCL-WMR, which has impressive accuracy and efficiency. We demonstrate the performance of sMCL-WMR in detecting weak motifs in large data sets and in real genomic data sets, and compare the performance to other leading motif recognition programs. In our preliminary discussion of our CONSENSUS STRING algorithm we give insight into the issue of sampling pairwise bounded sets, and discuss its relevance to motif recognition.

Conclusion

Our novel heuristic gives birth to a state of the art program, sMCL-WMR, that is capable of detecting weak motifs in data sets with a large number of strings. sMCL-WMR is orders of magnitude faster than its predecessor MCL-WMR and is capable of solving previously unsolved synthetic motif recognition problems. Lastly, sMCL-WMR shows impressive accuracy in detecting transcription factor binding sites in the genomic data and used in the assessment of Tompa et al.

A review of published tetartohedrally twinned macromolecular structures is presented, together with details of the recent structure determination of triclinic tetartohedrally twinned crystals of human complement factor I.

Tetartohedral crystal twinning is discussed as a particular case of (pseudo)merohedral twinning when the number of twinned domains is four. Tetartohedrally twinned crystals often possess pseudosymmetry, with the rotational part of the pseudo­symmetry operators coinciding with the twinning operators. Tetartohedrally twinned structures from the literature are reviewed and the recent structure determination of tetartohedrally twinned triclinic crystals of human complement factor I is discussed.

E. coli bacterioferritin was crystallized in a novel crystal form from different conditions and the structure was solved. The crystals belonged to space group P213 and diffracted to a resolution of 2.5 Å.

Escherichia coli bacterioferritin was serendipitously crystallized in a novel cubic crystal form and its structure could be determined to 2.5 Å resolution despite a high degree of merohedral twinning. This is the first report of crystallographic data on ‘as-isolated’ E. coli bacterioferritin. The ferroxidase active site contains positive difference density consistent with two metal ions that had co-purified with the protein. X-ray fluorescence studies suggest that the metal composition is different from that of previous structures and is a mix of zinc and native iron ions. The ferroxidase-centre configuration displays a similar flexibility as previously noted for other bacterioferritins.

The pitfalls of experimental phasing are described.

Developments in protein crystal structure determination by experimental phasing are reviewed, emphasizing the theoretical continuum between experimental phasing, density modification, model building and refinement. Traditional notions of the composition of the substructure and the best coefficients for map generation are discussed. Pitfalls such as determining the enantiomorph, identifying centrosymmetry (or pseudo-symmetry) in the substructure and crystal twinning are discussed in detail. An appendix introduces com­bined real–imaginary log-likelihood gradient map coefficients for SAD phasing and their use for substructure completion as implemented in the software Phaser. Supplementary material includes animated probabilistic Harker diagrams showing how maximum-likelihood-based phasing methods can be used to refine parameters in the case of SIR and MIR; it is hoped that these will be useful for those teaching best practice in experimental phasing methods.

Based on the minimum shear criterion, a direct and simple method is proposed to calculate twinning elements from the experimentally determined twinning plane for Type I twins or the twinning direction for Type II twins. It is generic and applicable to any crystal structure.

The fundamental theory of crystal twinning has been long established, leading to a significant advance in understanding the nature of this physical phenomenon. However, there remains a substantial gap between the elaborate theory and the practical determination of twinning elements. This paper proposes a direct and simple method – valid for any crystal structure and based on the minimum shear criterion – to calculate various twinning elements from the experimentally determined twinning plane for Type I twins or the twinning direction for Type II twins. Without additional efforts, it is generally applicable to identify and predict possible twinning modes occurring in a variety of crystalline solids. Therefore, the present method is a promising tool to characterize twinning elements, especially for those materials with complex crystal structure.

Fractal analysis has been shown to be useful in image processing for characterizing shape and gray-scale complexity. Breast masses present shape and gray-scale characteristics that vary between benign masses and malignant tumors in mammograms. Limited studies have been conducted on the application of fractal analysis specifically for classifying breast masses based on shape. The fractal dimension of the contour of a mass may be computed either directly from the 2-dimensional (2D) contour or from a 1-dimensional (1D) signature derived from the contour. We present a study of four methods to compute the fractal dimension of the contours of breast masses, including the ruler method and the box counting method applied to 1D and 2D representations of the contours. The methods were applied to a data set of 111 contours of breast masses. Receiver operating characteristics (ROC) analysis was performed to assess and compare the performance of fractal dimension and four previously developed shape factors in the classification of breast masses as benign or malignant. Fractal dimension was observed to complement the other shape factors, in particular fractional concavity, in the representation of the complexity of the contours. The combination of fractal dimension with fractional concavity yielded the highest area (Az) under the ROC curve of 0.93; the two measures, on their own, resulted in Az values of 0.89 and 0.88, respectively.

Digital breast tomosynthesis (DBT) mammography is a promising imaging technique that has the potential to improve detection of early-stage breast cancers. DBT mammography can provide quasi three-dimensional information by reconstructing the breast volume from a number of low-dose mammograms acquired over a limited angular range. Previous studies have shown that iterative reconstruction methods such as simultaneous algebraic reconstruction technique (SART) can give satisfactory image quality in DBT. However, due to the finite size of the detector and the limited field of view, DBT reconstruction contains artifacts caused by the truncated projection-view (PV) images. We developed methods that use a local intensity equalization strategy and a geometrical tissue-compensation method to remove two types of truncation artifacts: detector boundary discontinuity and underestimation of the attenuation pathlength. A custom-built breast phantom and a selected DBT patient case were used to evaluate the improvements. A GE prototype DBT system was used to acquire 21 PVs in 3° increments over a ±30° angular range. Experimental results demonstrated that the artifact reduction methods can improve the image quality at the boundaries with enhanced contrast-to-noise ratio and increased background uniformity, recover the obscured breast structural information, and achieve an overall reconstruction quality comparable to that without truncation.

Crystallization in lipidic mesophases (in meso) has been successfully used to obtain a number of high-resolution membrane protein structures including challenging members of the human G protein-coupled receptor (GPCR) family. Crystallogenesis in arguably the most successful mesophase, lipidic cubic phase (LCP), critically depends on the ability of protein to diffuse in the LCP matrix and to form specific protein-protein contacts to support crystal nucleation and growth. The ability of an integral membrane protein to diffuse in LCP is strongly affected by the protein aggregation state, the structural parameters of LCP, and the chemical environment. In order to satisfy both requirements of diffusion and specific interactions, one must balance multiple parameters, such as identity of LCP host lipid, composition of precipitant solution, identity of ligand, and protein modifications. Screening within such multi-dimensional crystallization space presents a significant bottleneck in obtaining initial crystal leads. To reduce this combinatorial challenge, we developed a pre-crystallization screening assay to measure the diffusion characteristics of a protein target in LCP. Utilizing the Fluorescence Recovery After Photobleaching (FRAP) technique in an automated and high throughput manner, we were able to map conditions that support adequate diffusion in LCP using a minimal amount of protein. Data collection and processing protocols were validated using two model GPCR targets: the β2-adrenergic receptor and the A2A adenosine receptor.

With a dedicated breast CT system using a quasi-monochromatic x-ray source and flat-panel digital detector, the 2D and 3D scatter to primary ratios (SPR) of various geometric phantoms having different densities were characterized in detail. Projections were acquired using geometric and anthropomorphic breast phantoms. Each phantom was filled with 700ml of 5 different water-methanol concentrations to simulate effective boundary densities of breast compositions from 100% glandular (1.0g/cm3) to 100% fat (0.79g/cm3). Projections were acquired with and without a beam stop array. For each projection, 2D scatter was determined by cubic spline interpolating the values behind the shadow of each beam stop through the object. Scatter-corrected projections were obtained by subtracting the scatter, and the 2D SPRs were obtained as a ratio of the scatter to scatter-corrected projections. Additionally the (un)corrected data were individually iteratively reconstructed. The (un)corrected 3D volumes were subsequently subtracted, and the 3D SPRs obtained from the ratio of the scatter volume-to-scatter-corrected (or primary) volume. Results show that the 2D SPR values peak in the center of the volumes, and were overall highest for the simulated 100% glandular composition. Consequently, scatter corrected reconstructions have visibly reduced cupping regardless of the phantom geometry, as well as more accurate linear attenuation coefficients. The corresponding 3D SPRs have increased central density, which reduces radially. Not surprisingly, for both 2D and 3D SPRs there was a dependency on both phantom geometry and object density on the measured SPR values, with geometry dominating for 3D SPRs. Overall, these results indicate the need for scatter correction given different geometries and breast densities that will be encountered with 3D cone beam breast CT.

We are developing a database implementation to support temporal data management for the T-HELPER physician workstation, an advice system for protocol-based care of patients who have HIV disease. To understand the requirements for the temporal database, we have analyzed the types of temporal predicates found in clinical-trial protocols. We extend the standard relational data model in three ways to support these querying requirements. First, we incorporate timestamps into the two-dimensional relational table to store the temporal dimension of both instant- and interval-based data. Second, we develop a set of operations on timepoints and intervals to manipulate timestamped data. Third, we modify the relational query language SQL so that its underlying algebra supports the specified operations on timestamps in relational tables. We show that our temporal extension to SQL meets the temporal data-management needs of protocol-directed decision support.