Circular Permutation Research Articles

Scattering from a finite ribbed plate with attached oscillators.

Double layer structures consisting of stainless steel and polymer rods are designed to blur and attenuate Bragg and Bloch-Floquet scattering from a periodically ribbed plate in a given frequency bandwidth. These structures can be considered as ribbed plate-spring-mass systems, the resonance frequencies of which are obtained from random and circular permutations of five basic oscillators. Analytical and finite element methods are used to find their parameters and tank tests have been carried out to ensure the accuracy of the numerical results and validate the relevance of such a model. It has been found that the typical features associated with the periodicity of the ribs are replaced by a "Christmas tree" appearance when the far field pressure backscattered from the model is displayed in the (frequency, aspect angle) plane. An analysis of backscattering patterns is presented and comparisons with Naval Research Laboratory results are made.

Sorting signed circular permutations by super short operations

BackgroundOne way to estimate the evolutionary distance between two given genomes is to determine the minimum number of large-scale mutations, or genome rearrangements, that are necessary to transform one into the other. In this context, genomes can be represented as ordered sequences of genes, each gene being represented by a signed integer. If no gene is repeated, genomes are thus modeled as signed permutations of the form pi =(pi _1 pi _2 ldots pi _n), and in that case we can consider without loss of generality that one of them is the identity permutation iota _n =(1 2 ldots n), and that we just need to sort the other (i.e., transform it into iota _n). The most studied genome rearrangement events are reversals, where a segment of the genome is reversed and reincorporated at the same location; and transpositions, where two consecutive segments are exchanged. Many variants, e.g., combining different types of (possibly constrained) rearrangements, have been proposed in the literature. One of them considers that the number of genes involved, in a reversal or a transposition, is never greater than two, which is known as the problem of sorting by super short operations (or SSOs).Results and conclusionsAll problems considering SSOs in permutations have been shown to be in mathsf {P}, except for one, namely sorting signed circular permutations by super short reversals and super short transpositions. Here we fill this gap by introducing a new graph structure called cyclic permutation graph and providing a series of intermediate results, which allows us to design a polynomial algorithm for sorting signed circular permutations by super short reversals and super short transpositions.

Solution Strategy for One-to-One Pickup and Delivery Problem Using the Cyclic Transfer Approach

One-to-one Pickup and Delivery Problem (PDP) with loading constraints is studied. Mathematical model is presented using combinatorial configurations approach for pickup and delivery and the phi-function technique for loading constraints. Two-stage solution strategy is used. The first stage has two levels: a) clustering the set of pickup and delivery points regarding to the set of vehicles and b) solving the PDP problem with 3D loading constraints for each vehicle within the appropriate cluster. The second stage improves the solution found at the previous stage and is based on the cyclic transfer approach and cyclic permutations. Computational results are presented to support efficiency of the approach.

ОПТИМИЗАЦИЯ ЛИНЕЙНЫХ ФУНКЦИЙ НА МНОЖЕСТВЕ ЦИКЛИЧЕСКИХ ПЕРЕСТАНОВОК С ЛИНЕЙНЫМИ ОГРАНИЧЕНИЯМИ

Предметом статьи является процесс решения задач дискретной оптимизации на комбинаторных множествах различных классов. Целью является разработка методов решения задачи оптимизации линейной функции с линейными ограничениями на множестве циклических перестановок, погруженном в евклидово пространство. Задачи: найти точное или приближенное решение задачи оптимизации линейной функции с линейными ограничениями на множестве циклических перестановок, погруженном в евклидово пространство. Исследовать свойства задачи оптимизации, оценить приближенное решение. Основные результаты работы. Предложена стратегия решения с использованием алгоритма на основе случайного поиска. Для решения задачи оптимизации линейной функции на множестве циклических перестановок используется подход, основанный на идеологии случайного поиска и аналитическом решении систем линейных неравенств, описывающих ограничения задачи. В процессе решения исходной задачи необходимо многократное решение вспомогательной задачи оптимизации линейной функции на множестве циклических перестановок без ограничений. В работе приводится два подхода к решению вспомогательной задачи. Первый подход позволяет получить точное решение вспомогательной задачи методом ветвей и границ или приближенное решение при использовании дополнительных эвристик с оценкой полученного решения. Второй подход – эвристический метод на основе транспозиций специального вида. Для реализации подхода введен класс транспозиций, представители которого соответствуют критерию смежности в перестановочном многограннике. Предложенные стратегии реализованы программно и протестированы на задачах различной размерности с исходными данными, генерируемыми случайным образом. Проведены вычислительные эксперименты с целью сравнения точности и времени решения исходной задачи методом случайного поиска с использованием предложенных подходов к решению вспомогательной задачи. Выводы. Эксперименты показывают преимущество решения вспомогательной задачи методом ветвей и границ на малых размерностях. При этом на задачах больших размерностей метод на основе транспозиций существенно выигрывает в плане экономии вычислительных мощностей.

Affine equivalence classes of 2-rotation symmetric cubic Boolean functions

ABSTRACTA Boolean function in variables that is generated by applying all even powers of the cyclic permutation of the variables to a single cubic monomial is called a cubic monomial rotation symmetric 2-function (or cubic 2-MRS function for short). In 2015, Cusick and Johns gave a complete description of the affine equivalence classes for such functions with generating monomial with 1<r<s and r and s not both odd. In this paper we develop the theory further by determining the smallest sets that act on the set of all these cubic 2-MRS functions to give the affine equivalence classes. Also, suppose f and g are two of these functions such that there exists a permutation σ of the variables where We give a complete description of all such permutations σ and an exact count of their number.

Modulating the Nucleophile of a Glycoside Hydrolase through Site-Specific Incorporation of Fluoroglutamic Acids.

Understanding the detailed mechanisms of enzyme-catalyzed hydrolysis of the glycosidic bond is fundamentally important, not only to the design of tailored cost-efficient, stable and specific catalysts but also to the development of specific glycosidase inhibitors as therapeutics. Retaining glycosidases employ two key carboxylic acid residues, typically glutamic acids, in a double-displacement mechanism involving a covalent glycosyl-enzyme intermediate. One Glu functions as a nucleophile while the other acts as a general acid/base. A significant part of enzymatic proficiency is attributed to a "perfect match" of the electrostatics provided by these key residues, a hypothesis that has been remarkably difficult to prove in model systems or in enzymes themselves. We experimentally probe this synergy by preparing synthetic variants of a model glycosidase Bacillus circulans β-xylanase (Bcx) with the nucleophile Glu78 substituted by 4-fluoro or 4,4-difluoroglutamic acid to progressively reduce nucleophilicity. These Bcx variants were semisynthesized by preparation of optically pure fluoroglutamic acid building blocks, incorporation into synthetic peptides, and ligation onto a truncated circular permutant of Bcx. By measuring the effect of altered electrostatics in the active site on enzyme kinetic constants, we show that lowering the nucleophile p Ka by two units shits the pH-dependent activity by one pH unit. Linear free energy correlations using substrates of varying leaving group ability indicate that by reducing nucleophilic catalysis the concerted mechanism of the enzyme is disrupted and shifted toward a dissociative pathway. Our study represents the first example of site-specific introduction of fluorinated glutamic acids into any protein. Furthermore, it provides unique insights into the synergy of nucleophilic and acid/base catalysis within an enzyme active site.

Circular-Shift Linear Network Codes With Arbitrary Odd Block Lengths

Circular-shift linear network coding (LNC) is a class of vector LNC with low encoding and decoding complexities, and with local encoding kernels chosen from cyclic permutation matrices. When $L$ is a prime with primitive root $2$, it was recently shown that a scalar linear solution over GF($2^{L-1}$) induces an $L$-dimensional circular-shift linear solution at rate $(L-1)/L$. In this work, we prove that for arbitrary odd $L$, every scalar linear solution over GF($2^{m_L}$), where $m_L$ refers to the multiplicative order of $2$ modulo $L$, can induce an $L$-dimensional circular-shift linear solution at a certain rate. Based on the generalized connection, we further prove that for such $L$ with $m_L$ beyond a threshold, every multicast network has an $L$-dimensional circular-shift linear solution at rate $\phi(L)/L$, where $\phi(L)$ is the Euler's totient function of $L$. An efficient algorithm for constructing such a solution is designed. Finally, we prove that every multicast network is asymptotically circular-shift linearly solvable.

CiPerGenesis, A Mutagenesis Approach that Produces Small Libraries of Circularly Permuted Proteins Randomly Opened at a Focused Region: Testing on the Green Fluorescent Protein.

Circularly permuted proteins (cpPs) represent a novel type of mutant proteins with original termini that are covalently linked through a peptide connector and opened at any other place of the polypeptide backbone to create new ends. cpPs are finding wide applications in biotechnology because their properties may be quite different from those of the parental protein. However, the actual challenge for the creation of successful cpPs is to identify those peptide bonds that can be broken to create new termini and ensure functional and well-folded cpPs. Herein, we describe CiPerGenesis, a combinatorial mutagenesis approach that uses two oligonucleotide libraries to amplify a circularized gene by PCR, starting and ending from a focused target region. This approach creates small libraries of circularly permuted genes that are easily cloned in the correct direction and frame using two different restriction sites encoded in the oligonucleotides. Once expressed, the protein libraries exhibit a unique sequence diversity, comprising cpPs that exhibit ordinary breakpoints between adjacent amino acids localized at the target region as well as cpPs with new termini containing user-defined truncations and repeats of some amino acids. CiPerGenesis was tested at the lid region G134-H148 of green fluorescent protein (GFP), revealing that the most fluorescent variants were those starting at Leu141 and ending at amino acids Tyr145, Tyr143, Glu142, Leu141, Lys140, and H139. Purification and biochemical characterization of some variants suggested a differential expression, solubility and maturation extent of the mutant proteins as the likely cause for the variability in fluorescence intensity observed in colonies.

Kinetic Basis of the Bifunctionality of SsoII DNA Methyltransferase.

Type II restriction–modification (RM) systems are the most widespread bacterial antiviral defence mechanisms. DNA methyltransferase SsoII (M.SsoII) from a Type II RM system SsoII regulates transcription in its own RM system in addition to the methylation function. DNA with a so-called regulatory site inhibits the M.SsoII methylation activity. Using circular permutation assay, we show that M.SsoII monomer induces DNA bending of 31° at the methylation site and 46° at the regulatory site. In the M.SsoII dimer bound to the regulatory site, both protein subunits make equal contributions to the DNA bending, and both angles are in the same plane. Fluorescence of TAMRA, 2-aminopurine, and Trp was used to monitor conformational dynamics of DNA and M.SsoII under pre-steady-state conditions by stopped-flow technique. Kinetic data indicate that M.SsoII prefers the regulatory site to the methylation site at the step of initial protein–DNA complex formation. Nevertheless, in the presence of S-adenosyl-l-methionine, the induced fit is accelerated in the M.SsoII complex with the methylation site, ensuring efficient formation of the catalytically competent complex. The presence of S-adenosyl-l-methionine and large amount of the methylation sites promote efficient DNA methylation by M.SsoII despite the inhibitory effect of the regulatory site.

Circularly permuted variants of two CG-specific prokaryotic DNA methyltransferases.

The highly similar prokaryotic DNA (cytosine-5) methyltransferases (C5-MTases) M.MpeI and M.SssI share the specificity of eukaryotic C5-MTases (5’-CG), and can be useful research tools in the study of eukaryotic DNA methylation and epigenetic regulation. In an effort to improve the stability and solubility of complementing fragments of the two MTases, genes encoding circularly permuted (CP) variants of M.MpeI and M.SssI were created, and cloned in a plasmid vector downstream of an arabinose-inducible promoter. MTase activity of the CP variants was tested by digestion of the plasmids with methylation-sensitive restriction enzymes. Eleven of the fourteen M.MpeI permutants and six of the seven M.SssI permutants had detectable MTase activity as indicated by the full or partial protection of the plasmid carrying the cpMTase gene. Permutants cp62M.MpeI and cp58M.SssI, in which the new N-termini are located between conserved motifs II and III, had by far the highest activity. The activity of cp62M.MpeI was comparable to the activity of wild-type M.MpeI. Based on the location of the split sites, the permutants possessing MTase activity can be classified in ten types. Although most permutation sites were designed to fall outside of conserved motifs, and the MTase activity of the permutants measured in cell extracts was in most cases substantially lower than that of the wild-type enzyme, the high proportion of circular permutation topologies compatible with MTase activity is remarkable, and is a new evidence for the structural plasticity of C5-MTases. A computer search of the REBASE database identified putative C5-MTases with CP arrangement. Interestingly, all natural circularly permuted C5-MTases appear to represent only one of the ten types of permutation topology created in this work.

Front Cover: Construction of a Triangle‐Shaped Trimer and a Tetrahedron Using an α‐Helix‐Inserted Circular Permutant of Cytochrome c555 (Chem. Asian J. 10/2018)

Front Cover: Construction of a Triangle‐Shaped Trimer and a Tetrahedron Using an α‐Helix‐Inserted Circular Permutant of Cytochrome <i>c</i>₅₅₅ (Chem. Asian J. 10/2018)

Circular permutation profiling by deep sequencing libraries created using transposon mutagenesis.

Deep mutational scanning has been used to create high-resolution DNA sequence maps that illustrate the functional consequences of large numbers of point mutations. However, this approach has not yet been applied to libraries of genes created by random circular permutation, an engineering strategy that is used to create open reading frames that express proteins with altered contact order. We describe a new method, termed circular permutation profiling with DNA sequencing (CPP-seq), which combines a one-step transposon mutagenesis protocol for creating libraries with a functional selection, deep sequencing and computational analysis to obtain unbiased insight into a protein’s tolerance to circular permutation. Application of this method to an adenylate kinase revealed that CPP-seq creates two types of vectors encoding each circularly permuted gene, which differ in their ability to express proteins. Functional selection of this library revealed that >65% of the sampled vectors that express proteins are enriched relative to those that cannot translate proteins. Mapping enriched sequences onto structure revealed that the mobile AMP binding and rigid core domains display greater tolerance to backbone fragmentation than the mobile lid domain, illustrating how CPP-seq can be used to relate a protein's biophysical characteristics to the retention of activity upon permutation.

The calculus of multivectors on noncommutative jet spaces

The Leibniz rule for derivations is invariant under cyclic permutations of co-multiples within the arguments of derivations. We explore the implications of this principle: in effect, we construct a class of noncommutative bundles in which the sheaves of algebras of walks along a tesselated affine manifold form the base, whereas the fibres are free associative algebras or, at a later stage, such algebras quotients over the linear relation of equivalence under cyclic shifts. The calculus of variations is developed on the infinite jet spaces over such noncommutative bundles. In the frames of such field-theoretic extension of the Kontsevich formal noncommutative symplectic (super)geometry, we prove the main properties of the Batalin--Vilkovisky Laplacian and Schouten bracket. We show as by-product that the structures which arise in the classical variational Poisson geometry of infinite-dimensional integrable systems do actually not refer to the graded commutativity assumption.

Nonlinearity of k-cycle permutations on ℤn

In this paper, we explore the existence and nonlinearity of affine [Formula: see text]-cycle permutations on [Formula: see text] for different values of [Formula: see text] and [Formula: see text]. It is proved that a transposition [Formula: see text] on [Formula: see text] has nonlinearity [Formula: see text] for [Formula: see text], a [Formula: see text]-cycle permutation on [Formula: see text] has nonlinearity [Formula: see text] for [Formula: see text], [Formula: see text] a prime. The number of affine [Formula: see text]-cycle permutations on [Formula: see text] and a bound for their nonlinearity have been obtained. Some results on the cyclic decomposition of a permutation on [Formula: see text], [Formula: see text] a prime, have also been proved.

Construction of a Triangle-Shaped Trimer and a Tetrahedron Using an α-Helix-Inserted Circular Permutant of Cytochrome c555.

Highly-ordered protein structures have gained interest for future uses for biomaterials. Herein, we constructed a building block protein (BBP) by the circular permutation of the hyperthermostable Aquifex aeolicus cytochrome (cyt) c555 , and assembled BBP into a triangle-shaped trimer and a tetrahedron. The angle of the intermolecular interactions of BBP was controlled by cleaving the domain-swapping hinge loop of cyt c555 and connecting the original N- and C-terminal α-helices with an α-helical linker. We obtained BBP oligomers up to ≈40 mers, with a relatively large amount of trimers. According to the X-ray crystallographic analysis of the BBP trimer, the N-terminal region of one BBP molecule interacted intermolecularly with the C-terminal region of another BBP molecule, resulting in a triangle-shaped structure with an edge length of 68 Å. Additionally, four trimers assembled into a unique tetrahedron in the crystal. These results demonstrate that the circular permutation connecting the original N- and C-terminal α-helices with an α-helical linker may be useful for constructing organized protein structures.

On Conjectures of Andrews and Curtis

It is shown that the original Andrews--Curtis conjecture on balanced presentations of the trivial group is equivalent to its version in which, in place of arbitrary conjugations, one can use only cyclic permutations. This, in particular, proves a satellite conjecture of Andrews and Curtis made in 1966. We also consider a more restrictive cancellative version of the cyclic Andrews--Curtis conjecture with and without stabilizations and show that the restriction does not change the Andrews--Curtis conjecture when stabilizations are allowed. On the other hand, the restriction makes the conjecture false when stabilizations are not allowed.

On the cycle structure of Mallows permutations

We study the length of cycles of random permutations drawn from the Mallows distribution. Under this distribution, the probability of a permutation $\pi\in\mathbb{S}_{n}$ is proportional to $q^{\operatorname{inv}(\pi)}$ where $q>0$ and $\operatorname{inv}(\pi)$ is the number of inversions in $\pi$. We focus on the case that $q<1$ and show that the expected length of the cycle containing a given point is of order $\min\{(1-q)^{-2},n\}$. This marks the existence of two asymptotic regimes: with high probability, when $n$ tends to infinity with $(1-q)^{-2}\ll n$ then all cycles have size $o(n)$ whereas when $n$ tends to infinity with $(1-q)^{-2}\gg n$ then macroscopic cycles, of size proportional to $n$, emerge. In the second regime, we prove that the distribution of normalized cycle lengths follows the Poisson–Dirichlet law, as in a uniformly random permutation. The results bear formal similarity with a conjectured localization transition for random band matrices. Further results are presented for the variance of the cycle lengths, the expected diameter of cycles and the expected number of cycles. The proofs rely on the exact sampling algorithm for the Mallows distribution and make use of a special diagonal exposure process for the graph of the permutation.

Designing fluorescent biosensors using circular permutations of riboswitches.

RNA-based fluorescent (RBF) biosensors have been applied to detect a variety of metabolites in vitro and in live cells. They are designed by combining the ligand sensing domain of natural riboswitches with in vitro selected fluorogenic aptamers. Different biosensor topologies have been developed to accommodate the diversity of riboswitch structures. Here we show that circular permutation of the riboswitch ligand sensing domain also gives functional biosensors, using the SAM-I riboswitch as our model. We reveal that this design can enhance fluorescence turn-on and ligand binding affinity compared to the non-permuted topology.

Quantum Clique Gossiping

This paper establishes a framework of quantum clique gossiping by introducing local clique operations to networks of interconnected qubits. Cliques are local structures in complex networks being complete subgraphs, which can be used to accelerate classical gossip algorithms. Based on cyclic permutations, clique gossiping leads to collective multi-party qubit interactions. We show that at reduced states, these cliques have the same acceleration effects as their roles in accelerating classical gossip algorithms. For randomized selection of cliques, such improved rate of convergence is precisely characterized. On the other hand, the rate of convergence at the coherent states of the overall quantum network is proven to be decided by the spectrum of a mean-square error evolution matrix. Remarkably, the use of larger quantum cliques does not necessarily increase the speed of the network density aggregation, suggesting quantum network dynamics is not entirely decided by its classical topology.

Covering and Euler cycles on non-oriented graphs

A covering cycle is a closed path that traverses each edge of a graph at least once. Two cycles are equivalent if one is a cyclic permutation of the other. We compute the number of equivalence classes of non-periodic covering cycles of given length in a non-oriented connected graph. A special case is the number of Euler cycles (covering cycles that cover each edge of the graph exactly once) in the non-oriented graph. We obtain an identity relating the numbers of covering cycles of any length in a graph to a product of determinants.