On the permutation behaviour of Dickson polynomials of the second kind

The known permutation behaviour of the Dickson polynomials of the second kind in characteristic 3 is expanded and simplified. (C) 2002 Elsevier Science (USA).

The compositional inverse of a class of permutation polynomials over a finite field

A new class of bilinear permutation polynomials was recently identified. In this note we determine the class of permutation polynomials which represents the functional inverse of the bilinear class.

Structure and folding of potato type II proteinase inhibitors: Circular permutation and intramolecular domain swapping

Potato type II serine proteinase inhibitors are proteins that consist of multiple sequence repeats, and exhibit a multidomain structure. The structural domains are circular permutations of the repeat sequence.. as a result or intramolecular domain swapping. Structural studies give indications for the origins of this folding behaviour, and the evolution of the inhibitor family.

Atomic Latin squares of order eleven

A Latin square is pan-Hamiltonian if the permutation which defines row i relative to row j consists of a single cycle for every i j. A Latin square is atomic if all of its conjugates are pan-Hamiltonian. We give a complete enumeration of atomic squares for order 11, the smallest order for which there are examples distinct from the cyclic group. We find that there are seven main classes, including the three that were previously known. A perfect 1-factorization of a graph is a decomposition of that graph into matchings such that the union of any two matchings is a Hamiltonian cycle. Each pan-Hamiltonian Latin square of order n describes a perfect 1-factorization of Kn,n, and vice versa. Perfect 1-factorizations of Kn,n can be constructed from a perfect 1-factorization of Kn+1. Six of the seven main classes of atomic squares of order 11 can be obtained in this way. For each atomic square of order 11, we find the largest set of Mutually Orthogonal Latin Squares (MOLS) involving that square. We discuss algorithms for counting orthogonal mates, and discover the number of orthogonal mates possessed by the cyclic squares of orders up to 11 and by Parker's famous turn-square. We find that the number of atomic orthogonal mates possessed by a Latin square is not a main class invariant. We also define a new sort of Latin square, called a pairing square, which is mapped to its transpose by an involution acting on the symbols. We show that pairing squares are often orthogonal mates for symmetric Latin squares. Finally, we discover connections between our atomic squares and Franklin's diagonally cyclic self-orthogonal squares, and we correct a theorem of Longyear which uses tactical representations to identify self-orthogonal Latin squares in the same main class as a given Latin square.

Acyclic permutants of naturally occurring cyclic proteins - Characterization of cystine knot and beta-sheet formation in the macrocyclic polypeptide kalata B1

Kalata B1 is a prototypic member of the unique cyclotide family of macrocyclic polypeptides in which the major structural features are a circular peptide backbone, a triple stranded beta-sheet, and a cystine knot arrangement of three disulfide bonds. The cyclotides are the only naturally occurring family of circular proteins and have prompted us to explore the concept of acyclic permutation, i.e. opening the backbone of a cross-linked circular protein in topologically permuted ways. We have synthesized the complete suite of acyclic permutants of kalata B1 and examined the effect of acyclic permutation on structure and activity. Only two of six topologically distinct backbone loops are critical for folding into the native conformation, and these involve disruption of the embedded ring in the cystine knot. Surprisingly, it is possible to disrupt regions of the p-sheet and still allow folding into native-like structure, provided the cystine knot is intact. Kalata B1 has mild hemolytic activity, but despite the overall structure of the native peptide being retained in all but two cases, none of the acyclic permutants displayed hemolytic activity. This loss of activity is not localized to one particular region and suggests that cyclization is critical for hemolytic activity.

Identifying rate-limiting nodes in large-scale cortical networks for visuospatial processing: an illustration using fMRI

With the advent of functional neuroimaging techniques, in particular functional magnetic resonance imaging (fMRI), we have gained greater insight into the neural correlates of visuospatial function. However, it may not always be easy to identify the cerebral regions most specifically associated with performance on a given task. One approach is to examine the quantitative relationships between regional activation and behavioral performance measures. In the present study, we investigated the functional neuroanatomy of two different visuospatial processing tasks, judgement of line orientation and mental rotation. Twenty-four normal participants were scanned with fMRI using blocked periodic designs for experimental task presentation. Accuracy and reaction time (RT) to each trial of both activation and baseline conditions in each experiment was recorded. Both experiments activated dorsal and ventral visual cortical areas as well as dorsolateral prefrontal cortex. More regionally specific associations with task performance were identified by estimating the association between (sinusoidal) power of functional response and mean RT to the activation condition; a permutation test based on spatial statistics was used for inference. There was significant behavioral-physiological association in right ventral extrastriate cortex for the line orientation task and in bilateral (predominantly right) superior parietal lobule for the mental rotation task. Comparable associations were not found between power of response and RT to the baseline conditions of the tasks. These data suggest that one region in a neurocognitive network may be most strongly associated with behavioral performance and this may be regarded as the computationally least efficient or rate-limiting node of the network.

A family of perfect factorisations of complete bipartite graphs

A 1-factorisation of a graph is perfect if the union of any two of its 1-factors is a Hamiltonian cycle. Let n = p(2) for an odd prime p. We construct a family of (p-1)/2 non-isomorphic perfect 1-factorisations of K-n,K-n. Equivalently, we construct pan-Hamiltonian Latin squares of order n. A Latin square is pan-Hamiltoilian if the permutation defined by any row relative to any other row is a single Cycle. (C) 2002 Elsevier Science (USA).

Overlarge sets of 2-(11, 5, 2) designs and related configurations

We consider the construction of several configurations, including: • overlarge sets of 2-(11,5,2) designs, that is, partitions of the set of all 5-subsets of a 12-set into 72 2-(11,5,2) designs; • an indecomposable doubly overlarge set of 2-(11,5,2) designs, that is, a partition of two copies of the set of all 5-subsets of a 12-set into 144 2-(11,5,2) designs, such that the 144 designs can be arranged into a 12 × 12 square with interesting row and column properties; • a partition of the Steiner system S(5,6,12) into 12 disjoint 2-(11,6,3) designs arising from the diagonal of the square; • bidistant permutation arrays and generalized Room squares arising from the doubly overlarge set, and their relation to some new strongly regular graphs.

Functional recycling of C2 domains throughout evolution: A comparative study of synaptotagmin, protein kinase C and phospholipase C by sequence, structural and modelling approaches

The C2 domain is one of the most frequent and widely distributed calcium-binding motifs. Its structure comprises an eight-stranded beta-sandwich with two structural types as if the result of a circular permutation. Combining sequence, structural and modelling information, we have explored, at different levels of granularity, the functional characteristics of several families of C2 domains. At the coarsest level,the similarity correlates with key structural determinants of the C2 domain fold and, at the finest level, with the domain architecture of the proteins containing them, highlighting the functional diversity between the various subfamilies. The functional diversity appears as different conserved surface patches throughout this common fold. In some cases, these patches are related to substrate-binding sites whereas in others they correspond to interfaces of presumably permanent interaction between other domains within the same polypeptide chain. For those related to substrate-binding sites, the predictions overlap with biochemical data in addition to providing some novel observations. For those acting as protein-protein interfaces' our modelling analysis suggests that slight variations between families are a result of not only complementary adaptations in the interfaces involved but also different domain architecture. In the light of the sequence and structural genomic projects, the work presented here shows that modelling approaches along with careful sub-typing of protein families will be a powerful combination for a broader coverage in proteomics. (C) 2003 Elsevier Ltd. All rights reserved.

Genome scan meta-analysis of schizophrenia and bipolar disorder, part II: Schizophrenia

Schizophrenia is a common disorder with high heritability and a 10-fold increase in risk to siblings of probands. Replication has been inconsistent for reports of significant genetic linkage. To assess evidence for linkage across studies, rank-based genome scan meta-analysis (GSMA) was applied to data from 20 schizophrenia genome scans. Each marker for each scan was assigned to 1 of 120 30-cM bins, with the bins ranked by linkage scores (1 = most significant) and the ranks averaged across studies (R-avg) and then weighted for sample size (rootN[affected cases]). A permutation test was used to compute the probability of observing, by chance, each bin's average rank (P-AvgRnk) or of observing it for a bin with the same place (first, second, etc.) in the order of average ranks in each permutation (P-ord). The GSMA produced significant genomewide evidence for linkage on chromosome 2q (P-AvgRnk

Extremal circular permutations with concave functions

Let {a(1), a(2), ..., a(n)} be a set of n distinct real numbers and let alpha(1), alpha(2), ..., alpha(n) an be a permutation of the numbers. We construct the permutation to maximise L-f = Sigma(i=1)(n) f(\alpha(i+1) - alpha(i)\), for any increasing concave function f, where we denote alpha(n+1) equivalent to alpha(1). The optimal permutation depends on the particular numbers {a(1), a(2), ..., a(n)} and the function f, contrary to a postulate by Chao and Liang (European J. Combin. 13 (1992) 325). (C) 2004 Elsevier Ltd. All rights reserved.

Mapping hippocampal and ventricular change in Alzheimer disease

We developed an anatomical mapping technique to detect hippocampal and ventricular changes in Alzheimer disease (AD). The resulting maps are sensitive to longitudinal changes in brain structure as the disease progresses. An anatomical surface modeling approach was combined with surface-based statistics to visualize the region and rate of atrophy in serial MRI scans and isolate where these changes link with cognitive decline. Fifty-two high-resolution MRI scans were acquired from 12 AD patients (age: 68.4 +/- 1.9 years) and 14 matched controls (age: 71.4 +/- 0.9 years), each scanned twice (2.1 +/- 0.4 years apart). 3D parametric mesh models of the hippocampus and temporal horns were created in sequential scans and averaged across subjects to identify systematic patterns of atrophy. As an index of radial atrophy, 3D distance fields were generated relating each anatomical surface point to a medial curve threading down the medial axis of each structure. Hippocampal atrophic rates and ventricular expansion were assessed statistically using surface-based permutation testing and were faster in AD than in controls. Using color-coded maps and video sequences, these changes were visualized as they progressed anatomically over time. Additional maps localized regions where atrophic changes linked with cognitive decline. Temporal horn expansion maps were more sensitive to AD progression than maps of hippocampal atrophy, but both maps correlated with clinical deterioration. These quantitative, dynamic visualizations of hippocampal atrophy and ventricular expansion rates in aging and AD may provide a promising measure to track AD progression in drug trials. (C) 2004 Elsevier Inc. All rights reserved.

Structure-function studies of the plant cyclotides: The role of a circular protein backbone

The traditional idea of proteins as linear chains of amino acids is being challenged with the discovery of miniproteins that contain a circular backbone. The cyclotide family is the largest group of circular proteins and is characterized by an amide-circularized protein backbone and six conserved cysteine residues. These conserved cysteines are paired to form a knotted network of disulfide bonds. The combination of the circular backbone and a cystine knot, known as the cyclic cystine knot (CCK) motif, confers exceptional stability upon the cyclotides. This review discusses the role of the circular backbone based on studies of both the oxidative folding of kalata B1, the prototypical cyclotide, and a comparison of the structure and activity of kalata B1 and its acyclic permutants.

The cross-entropy method for network reliability estimation

Consider a network of unreliable links, modelling for example a communication network. Estimating the reliability of the network-expressed as the probability that certain nodes in the network are connected-is a computationally difficult task. In this paper we study how the Cross-Entropy method can be used to obtain more efficient network reliability estimation procedures. Three techniques of estimation are considered: Crude Monte Carlo and the more sophisticated Permutation Monte Carlo and Merge Process. We show that the Cross-Entropy method yields a speed-up over all three techniques.