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Rational design of DNA sequences for nanotechnology, microarrays and molecular computers using Eulerian graphs.

Nucleic acids are molecules of choice for both established and emerging nanoscale technologies. These technologies benefit from large functional densities of 'DNA processing elements' that can be readily manufactured. To achieve the desired functionality, polynucleotide sequences are currently designed by a process that involves tedious and laborious filtering of potential candidates against a series of requirements and parameters. Here, we present a complete novel methodology for the rapid rational design of large sets of DNA sequences. This method allows for the direct implementation of very complex and detailed requirements for the generated sequences, thus avoiding 'brute force' filtering. At the same time, these sequences have narrow distributions of melting temperatures. The molecular part of the design process can be done without computer assistance, using an efficient 'human engineering' approach by drawing a single blueprint graph that represents all generated sequences. Moreover, the method eliminates the necessity for extensive thermodynamic calculations. Melting temperature can be calculated only once (or not at all). In addition, the isostability of the sequences is independent of the selection of a particular set of thermodynamic parameters. Applications are presented for DNA sequence designs for microarrays, universal microarray zip sequences and electron transfer experiments.

Algorithms↗

Group graph of the genetic code.

The genetic code doublets can be divided into two octets of completely degenerate and ambiguous coding dinucleotides. These two octets have the algebraic property of lying on continuously connected planes on the group graph (a tesseract) of the Cartesian product of two Klein 4-groups of nucleotide exchange operators. The K X K group can also be broken into four cosets, one of which has completely degenerate coding elements, and another that has completely ambiguous coding elements. The two octets of coding doublets have the further algebraic property that the product of their internal exchange operators naturally divide into two exactly equivalent sets. These properties of the genetic code are relevant to unraveling error-detecting and error-correcting (proof-reading) aspects of the genetic code and may be helpful in understanding the context-sensitive grammar of genetic language.

Genetic Code↗

Structural resemblance between the families of bacterial signal-transduction proteins and of G proteins revealed by graph theoretical techniques.

The first application of a novel technique for the identification of common folding motifs in proteins is presented. Using techniques derived from graph theory, developed in order to compare secondary structure motifs in proteins, we have established that there is a striking resemblance in the tertiary fold of the Salmonella typhimurium Che Y chemotaxis protein and that of the GDP-binding domain of Escherichia coli elongation factor Tu (EF Tu). These two protein structures are representatives of two major macromolecular classes: CheY is a signal-transduction protein with sequence homologies to a wide range of bacterial proteins involved in regulation of chemotaxis, membrane synthesis and sporulation; whilst EF Tu is one of a family of guanosine-nucleotide-binding proteins which include the ras oncogene proteins and signal-transducing G proteins. The similarity we have found extends far beyond the previously recognized resemblances of each protein's fold to that of a generic nucleotide-binding domain. The lack of significant sequence homology between the two classes of proteins may mean that the common fold of the two proteins constitutes a particularly stable folding motif. However, an alternative possibility is that the strong three-dimensional structural resemblance may be indicative of a remote shared common ancestry between the bacterial signal-transduction proteins and the GDP-binding proteins.

Algorithms↗

Three-dimensional vector graphing of the H reflex.

Comparison of latencies of the H reflex from side-to-side is considered a valid indicator of pathology, but amplitudes are too variable to allow similar comparison. Three pairs of electrodes were placed circumferentially and longitudinally along the calf to record the H reflex in three axes simultaneously. An H reflex with the greatest amplitude was produced by incrementally increasing submaximal stimulation to the tibial nerve. Three-dimensional vector graphs were constructed at three levels in each calf in ten normal individuals. Best fit curve Procrustes statistical analysis showed an average of 82.2 to 90.6% agreement of 3-D shape left-to-right with greater agreement at distal levels. Standard deviation ranged from 11.3% proximally to 8.2% distally. This represents much closer agreement than established norms for amplitude, which can vary from two to four times side-to-side. Three-dimensional vector analysis holds promise to further understanding of peripheral and central electrophysiologic phenomena.

Adult↗

A dynamic graph for documentation of gestational age.

A graphic format is presented for the display and storage of data relating to gestational age. The graph permits rapid retrieval and synthesis of often confusion information and is thereby useful in the management of complicated pregnancies.

Female↗

The conceptual basis of mathematics in cardiology: (I) algebra, functions and graphs.

This is the first in a series of four articles developed for the readers of. Without language ideas cannot be articulated. What may not be so immediately obvious is that they cannot be formulated either. One of the essential languages of cardiology is mathematics. Unfortunately, medical education does not emphasize, and in fact, often neglects empowering physicians to think mathematically. Reference to statistics, conditional probability, multicompartmental modeling, algebra, calculus and transforms is common but often without provision of genuine conceptual understanding. At the University of Vermont College of Medicine, Professor Bates developed a course designed to address these deficiencies. The course covered mathematical principles pertinent to clinical cardiovascular and pulmonary medicine and research. It focused on fundamental concepts to facilitate formulation and grasp of ideas. This series of four articles was developed to make the material available for a wider audience. The articles will be published sequentially in Coronary Artery Disease. Beginning with fundamental axioms and basic algebraic manipulations they address algebra, function and graph theory, real and complex numbers, calculus and differential equations, mathematical modeling, linear system theory and integral transforms and statistical theory. The principles and concepts they address provide the foundation needed for in-depth study of any of these topics. Perhaps of even more importance, they should empower cardiologists and cardiovascular researchers to utilize the language of mathematics in assessing the phenomena of immediate pertinence to diagnosis, pathophysiology and therapeutics. The presentations are interposed with queries (by Coronary Artery Disease, abbreviated as CAD) simulating the nature of interactions that occurred during the course itself. Each article concludes with one or more examples illustrating application of the concepts covered to cardiovascular medicine and biology.

Cardiology↗

A graph theoretic approach to the analysis of DNA sequencing data.

The analysis of data from automated DNA sequencing instruments has been a limiting factor in the development of new sequencing technology. A new base-calling algorithm that is intended to be independent of any particular sequencing technology has been developed and shown to be effective with data from the Applied Biosystems 373 sequencing system. This algorithm makes use of a nonlinear deconvolution filter to detect likely oligomer events and a graph theoretic editing strategy to find the subset of those events that is most likely to correspond to the correct sequence. Metrics evaluating the quality and accuracy of the resulting sequence are also generated and have been shown to be predictive of measured error rates. Compared to the Applied Biosystems Analysis software, this algorithm generates 18% fewer insertion errors, 80% more deletion errors, and 4% fewer mismatches. The tradeoff between different types of errors can be controlled through a secondary editing step that inserts or deletes base calls depending on their associated confidence values.

Algorithms↗

Noise in hysteretic systems and stochastic processes on graphs

It is shown that the theory of stochastic diffusion processes on graphs is a natural tool for the analysis of noise in hysteretic systems. In particular, by using this theory, analytical expressions for stationary characteristics of random outputs of some hysteretic systems are derived.

Journal Article↗

Minimal vertex covers on finite-connectivity random graphs: a hard-sphere lattice-gas picture.

The minimal vertex-cover (or maximal independent-set) problem is studied on random graphs of finite connectivity. Analytical results are obtained by a mapping to a lattice gas of hard spheres of (chemical) radius 1, and they are found to be in excellent agreement with numerical simulations. We give a detailed description of the replica-symmetric phase, including the size and entropy of the minimal vertex covers, and the structure of the unfrozen component which is found to percolate at a connectivity c approximately 1.43. The replica-symmetric solution breaks down at c=e approximately 2.72. We give a simple one-step replica-symmetry-broken solution, and discuss the problems in the interpretation and generalization of this solution.

Journal Article↗

Eigenstate structure in graphs and disordered lattices.

We study wave function structure for quantum graphs in the chaotic and disordered regime, using measures such as the wave function intensity distribution and the inverse participation ratio. The result is much less ergodicity than expected from random matrix theory, even though the spectral statistics are in agreement with random matrix predictions. Instead, analytical calculations based on short-time semiclassical behavior correctly describe the eigenstate structure.

Journal Article↗

Topology of technology graphs: small world patterns in electronic circuits.

Recent theoretical studies and extensive data analyses have revealed a common feature displayed by biological, social, and technological networks: the presence of small world patterns. Here we analyze this problem by using several graphs obtained from one of the most common technological systems: electronic circuits. It is shown that both analogic and digital circuits exhibit small world behavior. We conjecture that the small world pattern arises from the compact design in which many elements share a small, close physical neighborhood plus the fact that the system must define a single connected component (which requires shortcuts connecting different integrated clusters). The degree distributions displayed are consistent with a conjecture concerning the sharp cutoffs associated to the presence of costly connections [Amaral et al., Proc. Natl. Acad. Sci. USA 97, 11 149 (2000)], thus providing a limit case for the classes of universality of small world patterns from real, artificial networks. The consequences for circuit design are outlined.

Journal Article↗

Complex-temperature phase diagrams for the q-state Potts model on self-dual families of graphs and the nature of the q-->infinity limit.

Exact calculations of the Potts model partition function Z(G,q,v) have been presented for arbitrary q and temperature-like variable v on self-dual strip graphs G of the square lattice with fixed width L(y) and arbitrarily great length Lx with two types of boundary conditions. Letting Lx-->infinity, the resultant free energy and complex-temperature phase diagram have been computed, including the locus B where the free energy is nonanalytic. Results are analyzed for widths L(y)=1,2,3. These results have been used to study the approach to the large-q limit of B.

Journal Article↗

Metastable configurations of spin models on random graphs.

One-flip stable configurations of an Ising model on a random graph with fluctuating connectivity are examined. In order to perform the quenched average of the number of stable configurations we introduce a global order-parameter function with two arguments. The analytical results are compared with numerical simulations.

Journal Article↗

Glassy dynamics in granular compaction: sand on random graphs.

We discuss the use of a ferromagnetic spin model on a random graph to model granular compaction. A multispin interaction is used to capture the competition between local and global satisfaction of constraints characteristic for geometric frustration. We define an athermal dynamics designed to model repeated taps of a given strength. Amplitude cycling and the effect of permanently constraining a subset of the spins at a given amplitude is discussed. Finally we check the validity of Edwards's hypothesis for the athermal tapping dynamics.

Journal Article↗

Exact, convergent periodic-orbit expansions of individual energy eigenvalues of regular quantum graphs.

We present exact, explicit, convergent periodic-orbit expansions for individual energy levels of regular quantum graphs in the paper. One simple application is the energy levels of a particle in a piecewise constant potential. Since the classical ray trajectories (including ray splitting) in such systems are strongly chaotic, this result provides an explicit quantization of a classically chaotic system.

Journal Article↗

Information geometry of the ising model on planar random graphs.

It has been suggested that an information geometric view of statistical mechanics in which a metric is introduced onto the space of parameters provides an interesting alternative characterization of the phase structure, particularly in the case where there are two such parameters, such as the Ising model with inverse temperature beta and external field h. In various two-parameter calculable models, the scalar curvature R of the information metric has been found to diverge at the phase transition point beta(c) and a plausible scaling relation postulated: R approximately |beta-beta(c)|(alpha-2). For spin models the necessity of calculating in nonzero field has limited analytic consideration to one-dimensional, mean-field and Bethe lattice Ising models. In this paper we use the solution in field of the Ising model on an ensemble of planar random graphs (where alpha=-1, beta=1/2, gamma=2) to evaluate the scaling behavior of the scalar curvature, and find R approximately |beta-beta(c)|(-2). The apparent discrepancy is traced back to the effect of a negative alpha.

Journal Article↗

Star-graph expansions for bond-diluted Potts models.

We derive high-temperature series expansions for the free energy and the susceptibility of random-bond q-state Potts models on hypercubic lattices using a star-graph expansion technique. This method enables the exact calculation of quenched disorder averages for arbitrary uncorrelated coupling distributions. Moreover, we can keep the disorder strength p as well as the dimension d as symbolic parameters. By applying several series analysis techniques to the new series expansions, one can scan large regions of the (p,d) parameter space for any value of q. For the bond-diluted four-state Potts model in three dimensions, which exhibits a rather strong first-order phase transition in the undiluted case, we present results for the transition temperature and the effective critical exponent gamma as a function of p as obtained from the analysis of susceptibility series up to order 18. A comparison with recent Monte Carlo data [Chatelain et al., Phys. Rev. E 64, 036120 (2001)] shows signals for the softening to a second-order transition at finite disorder strength.

Journal Article↗

Software systems as complex networks: structure, function, and evolvability of software collaboration graphs.

Software systems emerge from mere keystrokes to form intricate functional networks connecting many collaborating modules, objects, classes, methods, and subroutines. Building on recent advances in the study of complex networks, I have examined software collaboration graphs contained within several open-source software systems, and have found them to reveal scale-free, small-world networks similar to those identified in other technological, sociological, and biological systems. I present several measures of these network topologies, and discuss their relationship to software engineering practices. I also present a simple model of software system evolution based on refactoring processes which captures some of the salient features of the observed systems. Some implications of object-oriented design for questions about network robustness, evolvability, degeneracy, and organization are discussed in the wake of these findings.

Journal Article↗