Circular Permutation Research Articles

Two-point functions of composite twist fields in the Ising field theory

All standard measures of bipartite entanglement in one-dimensional quantum field theories can be expressed in terms of correlators of branch point twist fields, here denoted by and . These are symmetry fields associated to cyclic permutation symmetry in a replica theory and having the smallest conformal dimension at the critical point. Recently, other twist fields (composite twist fields), typically of higher dimension, have been shown to play a role in the study of a new measure of entanglement known as the symmetry resolved entanglement entropy. In this paper we give an exact expression for the two-point function of a composite twist field that arises in the Ising field theory. In doing so we extend the techniques originally developed for the standard branch point twist field in free theories as well as an existing computation due to Horváth and Calabrese of the same two-point function which focused on the leading large-distance contribution. We study the ground state two-point function of the composite twist field and its conjugate . At criticality, this field can be defined as the leading field in the operator product expansion of and the disorder field µ. We find a general formula for and for (the derivative of) its analytic continuation to positive real replica numbers greater than 1. We check our formula for consistency by showing that at short distances it exactly reproduces the expected conformal dimension.

An exact imaginary-time path-integral phase-space formulation of multi-time correlation functions.

An exact representation of quantum mechanics using the language of phase-space variables provides a natural starting point to introduce and develop semiclassical approximations for the calculation of time correlation functions. Here, we introduce an exact path-integral formalism for calculations of multi-time quantum correlation functions as canonical averages over ring-polymer dynamics in imaginary time. The formulation provides a general formalism that exploits the symmetry of path integrals with respect to permutations in imaginary time, expressing correlations as products of imaginary-time-translation-invariant phase-space functions coupled through Poisson bracket operators. The method naturally recovers the classical limit of multi-time correlation functions and provides an interpretation of quantum dynamics in terms of "interfering trajectories" of the ring-polymer in phase space. The introduced phase-space formulation provides a rigorous framework for the future development of quantum dynamics methods that exploit the invariance of imaginary time path integrals to cyclic permutations.

Big data processing with 1D-Crosspoint Arrays

Increased chip densities offer massive computation power to deal with fundamental big data operations such as searching and sorting. At the same time, the proliferation of processing elements (PEs) in such multicore chips together with the employment of more aggressive parallel algorithms cause the amount of space needed for interprocessor communications to dominate the overall chip space, potentially resulting in reduced computational efficiency. To overcome this issue, this paper introduces a new architecture that uses simple crosspoint switches to pair PEs instead of a complex interconnection network. This new architecture may be viewed as a ‘quadratic’ array of processors as it uses PEs rather than PEs as in linear array processor models. The switches between adjacent PEs are created using a cyclic permutation wiring idea with PEs and as many crosspoints. We demonstrate the versatility of this new parallel architecture by designing fast algorithms to sort and search a list of n elements with it. In particular, we show that finding a minimum, maximum, and searching a list of n elements can all be performed on this parallel architecture in time with additional elementary logic gates with fan-in and in time with fan-in. We further show that sorting a list of n elements can also be carried out in time using additional elementary logic gates with fan-in and threshold logic gates on the same parallel architecture. The sorting time increases to if only elementary logic gates with fan-in are used. In addition, we establish how similar queries can be handled within the same order of time complexities. We use this new parallel architecture to perform sorting and searching on big data on three different models. The first of these models provides an efficient implementation of enumeration sorting and searching for moderate size big data sets. The second model offers increased parallelism by replication of the new parallel architecture but its hardware complexity limits its use to moderate size big data sets as well. The third model removes this limitation by introducing a tradeoff parameter between the time and hardware complexity of the overall computation, thereby providing an optimal use of available resources within a given chip-set space.

Distribution of the number of cycles in directed and undirected random regular graphs of degree 2.

We present analytical results for the distribution of the number of cycles in directed and undirected random 2-regular graphs (2-RRGs) consisting of N nodes. In directed 2-RRGs each node has one inbound link and one outbound link, while in undirected 2-RRGs each node has two undirected links. Since all the nodes are of degree k=2, the resulting networks consist of cycles. These cycles exhibit a broad spectrum of lengths, where the average length of the shortest cycle in a random network instance scales with lnN, while the length of the longest cycle scales with N. The number of cycles varies between different network instances in the ensemble, where the mean number of cycles 〈S〉 scales with lnN. Here we present exact analytical results for the distribution P_{N}(S=s) of the number of cycles s in ensembles of directed and undirected 2-RRGs, expressed in terms of the Stirling numbers of the first kind. In both cases the distributions converge to a Poisson distribution in the large N limit. The moments and cumulants of P_{N}(S=s) are also calculated. The statistical properties of directed 2-RRGs are equivalent to the combinatorics of cycles in random permutations of N objects. In this context our results recover and extend known results. In contrast, the statistical properties of cycles in undirected 2-RRGs have not been studied before.

Examining the importance of termini location in successful protein folding with HaloTag circular permutants in vitro, in E. coli, and in mammalian cells.

Though in vitro studies have produced insight into mechanisms underlying protein folding, determining how cellular factors modulate folding remains challenging. Proteins have the opportunity to fold when synthesized from N- to C-terminus by the ribosome at a rate of around five to ten residues per second in cells. This is much slower than the average folding rate of most small proteins, giving the nascent chain time to sample many conformations during translation. When refolding in vitro, HaloTag populates an aggregation-prone intermediate that contains structure from both the N- and C-termini, but this intermediate does not form co-translationally. Could altering the chain connectivity of HaloTag decrease its folding efficiency? We have shown how changing the vectorial synthesis of HaloTag through circular permutation affects folding efficiency in vitro and in vivo. We assessed all 298 circular permutants of HaloTag in E. coli cell lysate and showed that circular permutation positions differentially affect protein solubility and function. FACS experiments in live mammalian HEK293T cells demonstrate that many circular permutants retain function in an alternative cellular environment. Combining these data with both standard in vitro protein folding assays and hydrogen-exchange MS studies to map the folding process yields insight into how circular permutation alters the folding landscape of HaloTag and the role of co-translational folding in this process.

Classification of full exceptional collections on smooth toric Fano varieties with Picard rank two

Abstract In this paper, we give a complete classification of full exceptional collections, up to cyclic permutations, normalizations and mutations, on smooth toric Fano threefolds and fourfolds with Picard rank two. For such varieties, we find all the exceptional collections of maximal length and show that they are in fact full. This gives a partial answer to a conjecture in [29] and [32]. Moreover, such full exceptional collections essentially arise from Orlov’s theorem on projective bundles.

Metal-binding and circular permutation-dependent thermodynamic and kinetic stability of azurin.

Native topology is known to determine the folding kinetics and the energy landscape of proteins. Furthermore, the circular permutation (CP) of proteins alters the order of the secondary structure connectivity while retaining the three-dimensional structure, making it an elegant and powerful approach to altering native topology. Previous studies elucidated the influence of CP in proteins with different folds such as Greek key β-barrel, β-sandwich, β-α-β, and all α-Greek key. CP mainly affects the protein stability and unfolding kinetics, while folding kinetics remains mostly unaltered. However, the effect of CP on metalloproteins is yet to be elaborately studied. The active site of metalloproteins poses an additional complexity in studying protein folding. Here, we investigate a CP variant (cpN42) of azurin-in both metal-free and metal-bound (holo) forms. As observed earlier in other proteins, apo-forms of wild-type (WT) and cpN42 fold with similar rates. In contrast, zinc-binding accelerates the folding of WT but decelerates the folding of cpN42. On zinc-binding, the spontaneous folding rate of WT increases by >250 times that of cpN42, which is unprecedented and the highest for any CP to date. On the other hand, zinc-binding reduces the spontaneous unfolding rate of cpN42 by ~100 times, making the WT and CP azurins unfold at similar rates. Our study demonstrates metal binding as a novel way to modulate the unfolding and folding rates of CPs compared to their WT counterparts. We hope our study increases the understanding of the effect of CP on the folding mechanism and energy landscape of metalloproteins.

Rank-one transformations, odometers, and finite factors

In this paper we give explicit characterizations, based on the cutting and spacer parameters, of (a) which rank-one transformations factor onto a given finite cyclic permutation, (b) which rank-one transformations factor onto a given odometer, and (c) which rank-one transformations are isomorphic to a given odometer. These naturally yield characterizations of (d) which rank-one transformations factor onto some (unspecified) finite cyclic permutation, (d′) which rank-one transformations are totally ergodic, (e) which rank-one transformations factor onto some (unspecified) odometer, and (f) which rank-one transformations are isomorphic to some (unspecified) odometer.

Ribonuclease T2 represents a distinct circularly permutated version of the BECR RNases.

Detection of homologous relationships among proteins and understanding their mechanisms of diversification are major topics in the fields of protein science, bioinformatics, and phylogenetics. Recent developments in sequence/profile-based and structural similarity-based methods have greatly facilitated the unification and classification of many protein families into superfamilies or folds, yet many proteins remain unclassified in current protein databases. As one of the three earliest identified RNases in biology, ribonuclease T2, also known as RNase I in Escherichia coli, RNase Rh in fungi, or S-RNase in plant, is thought to be an ancient RNase family due to its widespread distribution and distinct structure. In this study, we present evidence that RNase T2 represents a circularly permutated version of the BECR (Barnase-EndoU-Colicin E5/D-RelE) fold RNases. This subtle relationship cannot be detected by traditional methods such as sequence/profile-based comparisons, structure-similarity searches, and circular permutation detections. However, we were able to identify the structural similarity using rational reconstruction of a theoretical RNase T2 ancestor via a reverse circular permutation process, followed by structural modeling using AlphaFold2, and structural comparisons. This relationship is further supported by the fact that RNase T2 and other typical BECR RNases, namely Colicin D, RNase A, and BrnT, share similar catalytic site configurations, all involving an analogous set of conserved residues on the α0 helix and the β4 strand of the BECR fold. This study revealed a hidden root of RNase T2 in bacterial toxin systems and demonstrated that reconstruction and modeling of ancestral topology is an effective strategy to identify remote relationship between proteins.

Creative destruction: New protein folds from old

Mechanisms of emergence and divergence of protein folds pose central questions in biological sciences. Incremental mutation and stepwise adaptation explain relationships between topologically similar protein folds. However, the universe of folds is diverse and riotous, suggesting more potent and creative forces are at play. Sequence and structure similarity are observed between distinct folds, indicating that proteins with distinct folds may share common ancestry. We found evidence of common ancestry between three distinct β-barrel folds: Scr kinase family homology (SH3), oligonucleotide/oligosaccharide-binding (OB), and cradle loop barrel (CLB). The data suggest a mechanism of fold evolution that interconverts SH3, OB, and CLB. This mechanism, which we call creative destruction, can be generalized to explain many examples of fold evolution including circular permutation. In creative destruction, an open reading frame duplicates or otherwise merges with another to produce a fused polypeptide. A merger forces two ancestral domains into a new sequence and spatial context. The fused polypeptide can explore folding landscapes that are inaccessible to either of the independent ancestral domains. However, the folding landscapes of the fused polypeptide are not fully independent of those of the ancestral domains. Creative destruction is thus partially conservative; a daughter fold inherits some motifs from ancestral folds. After merger and refolding, adaptive processes such as mutation and loss of extraneous segments optimize the new daughter fold. This model has application in disease states characterized by genetic instability. Fused proteins observed in cancer cells are likely to experience remodeled folding landscapes and realize altered folds, conferring new or altered functions.

Cavity soliton in a cyclic polarization permutation fiber resonator.

Cavity solitons are shape-preserving waveforms infinitely revolving around a cavity, which have numerous applications from spectroscopy to telecommunications. Although the cavity solitons have been widely studied for their special time-frequency characteristics over the past decade, the spectral flatness will be a large limitation in some applications without any extra filtering process. In this paper, we report on the generation of a distinct cavity soliton in a cyclic polarization permutation fiber resonator. With the simultaneous excitation of two orthogonal polarization modes with equally opposite dispersion, vectorial cavity solitons possessing broader and flatter spectra can be generated. In order to verify the concept, a numerical model of the polarization-maintaining fiber is proposed and the soliton with a flattened spectrum can be formed. The results enrich the soliton dynamics in the vectorial dissipation system.

Preliminary Characterization of Phage-Like Particles from the Male-Killing Mollicute Spiroplasma poulsonii (an Endosymbiont of Drosophila).

Bacteriophages are vastly abundant, diverse, and influential, but with few exceptions (e.g. the Proteobacteria genera Wolbachia and Hamiltonella), the role of phages in heritable bacteria-arthropod interactions, which are ubiquitous and diverse, remains largely unexplored. Despite prior studies documenting phage-like particles in the mollicute Spiroplasma associated with Drosophila flies, genomic sequences of such phage are lacking, and their effects on the Spiroplasma-Drosophila interaction have not been comprehensively characterized. We used a density step gradient to isolate phage-like particles from the male-killing bacterium Spiroplasma poulsonii (strains NSRO and MSRO-Br) harbored by Drosophila melanogaster. Isolated particles were subjected to DNA sequencing, assembly, and annotation. Several lines of evidence suggest that we recovered phage-like particles of similar features (shape, size, DNA content) to those previously reported in Drosophila-associated Spiroplasma strains. We recovered three ~ 19kb phage-like contigs (two in NSRO and one in MSRO-Br) containing 21-24 open reading frames, a read-alignment pattern consistent with circular permutation, and terminal redundancy (at least in NSRO). Although our results do not allow us to distinguish whether these phage-like contigs represent infective phage-like particles capable of transmitting their DNA to new hosts, their encoding of several typical phage genes suggests that they are at least remnants of functional phage. We also recovered two smaller non-phage-like contigs encoding a known Spiroplasma toxin (Ribosome Inactivating Protein; RIP), and an insertion element, suggesting that they are packaged into particles. Substantial homology of our particle-derived contigs was found in the genome assemblies of members of the Spiroplasma poulsonii clade.

Crystal Structure of 4,6-α-Glucanotransferase GtfC-ΔC from Thermophilic Geobacillus 12AMOR1: Starch Transglycosylation in Non-Permuted GH70 Enzymes.

GtfC-type 4,6-α-glucanotransferase (α-GT) enzymes from Glycoside Hydrolase Family 70 (GH70) are of interest for the modification of starch into low-glycemic index food ingredients. Compared to the related GH70 GtfB-type α-GTs, found exclusively in lactic acid bacteria (LAB), GtfCs occur in non-LAB, share low sequence identity, lack circular permutation of the catalytic domain, and feature a single-segment auxiliary domain IV and auxiliary C-terminal domains. Despite these differences, the first crystal structure of a GtfC, GbGtfC-ΔC from Geobacillus 12AMOR1, and the first one representing a non-permuted GH70 enzyme, reveals high structural similarity in the core domains with most GtfBs, featuring a similar tunneled active site. We propose that GtfC (and related GtfD) enzymes evolved from starch-degrading α-amylases from GH13 by acquiring α-1,6 transglycosylation capabilities, before the events that resulted in circular permutation of the catalytic domain observed in other GH70 enzymes (glucansucrases, GtfB-type α-GTs). AlphaFold modeling and sequence alignments suggest that the GbGtfC structure represents the GtfC subfamily, although it has a so far unique alternating α-1,4/α-1,6 product specificity, likely determined by residues near acceptor binding subsites +1/+2.

Braid representatives minimizing the number of simple walks

Given a knot, we develop methods for finding the braid representative that minimizes the number of simple walks. Such braids lead to an efficient method for computing the colored Jones polynomial of $K$, following an approach developed by Armond and implemented by Hajij and Levitt. We use this method to compute the colored Jones polynomial in closed form for the knots $5_2, 6_1,$ and $7_2$. The set of simple walks can change under reflection, rotation, and cyclic permutation of the braid, and we prove an invariance property which relates the simple walks of a braid to those of its reflection under cyclic permutation. We study the growth rate of the number of simple walks for families of torus knots. Finally, we present a table of braid words that minimize the number of simple walks for knots up to 13 crossings.

SeqCP: A sequence-based algorithm for searching circularly permuted proteins

Circular permutation (CP) is a protein sequence rearrangement in which the amino- and carboxyl-termini of a protein can be created in different positions along the imaginary circularized sequence. Circularly permutated proteins usually exhibit conserved three-dimensional structures and functions. By comparing the structures of circular permutants (CPMs), protein research and bioengineering applications can be approached in ways that are difficult to achieve by traditional mutagenesis. Most current CP detection algorithms depend on structural information. Because there is a vast number of proteins with unknown structures, many CP pairs may remain unidentified. An efficient sequence-based CP detector will help identify more CP pairs and advance many protein studies. For instance, some hypothetical proteins may have CPMs with known functions and structures that are informative for functional annotation, but existing structure-based CP search methods cannot be applied when those hypothetical proteins lack structural information. Despite the considerable potential for applications, sequence-based CP search methods have not been well developed. We present a sequence-based method, SeqCP, which analyzes normal and duplicated sequence alignments to identify CPMs and determine candidate CP sites for proteins. SeqCP was trained by data obtained from the Circular Permutation Database and tested with nonredundant datasets from the Protein Data Bank. It shows high reliability in CP identification and achieves an AUC of 0.9. SeqCP has been implemented into a web server available at: http://pcnas.life.nthu.edu.tw/SeqCP/.

Fluorescence-Activating and Absorption-Shifting Tags for Advanced Imaging and Biosensing.

Fluorescent labels and biosensors play central roles in biological and medical research. Targeted to specific biomolecules or cells, they allow noninvasive imaging of the machinery that govern cells and organisms in real time. Recently, chemogenetic reporters made of organic dyes specifically anchored to genetic tags have challenged the paradigm of fully genetically encoded fluorescent proteins. Combining the advantage of synthetic fluorophores with the targeting selectivity of genetically encoded tags, these chemogenetic reporters open new exciting prospects for studying cell biochemistry and biology. In this Account, we present the growing toolbox of fluorescence-activating and absorption-shifting tags (FASTs), small monomeric proteins of 14 kDa (125 amino acids residues) that can be used as markers to monitor gene expression and protein localization in live cells and organisms. Engineered by directed protein evolution from the photoactive yellow protein (PYP) from the bacterium Halorhodospira halophila, prototypical FAST binds and stabilizes the fluorescent state of live-cell compatible hydroxybenzylidene rhodanine chromophores. This class of chromophores are normally dark when free in solution or in cells because they dissipate light energy through nonradiative processes. The protein cavity of FAST allows the stabilization of the deprotonated state of the chromophore and blocks the chromophore into a planar conformation, which leads to highly fluorescent protein-chromophore assemblies. The use of such fluorogenic dyes (also called fluorogens) enables the imaging of FAST fusion proteins in cells with high contrast without the need to remove unbound ligands through separate washing steps. Fluorogens with various spectral properties exist nowadays allowing investigators to adjust the spectral properties of FAST to their experimental conditions. Molecular engineering allowed furthermore to generate membrane-impermeant fluorogens for the selective labeling of cell-surface proteins. Over the years, we generated a collection of FAST variants with expanded spectral properties or fluorogen selectivity using a concerted strategy involving molecular engineering and directed protein evolution. Moreover, protein engineering allowed us to adapt FASTs for the design of fluorescent biosensors. Circular permutation enabled the generation of FAST variants with increased conformational flexibility for the design of biosensors in which fluorogen binding is conditioned to the recognition of a given analyte. Bisection of FASTs into two complementary fragments allowed us furthermore to create split variants with reversible complementation that allow the detection and imaging of dynamic protein-protein interactions. We provide, here, a general overview of the current state of development of these different systems and their applications for advanced live cell imaging and biosensing and discuss potential future directions.

An Efficient Chaos-Based Image Encryption Technique Using Bitplane Decay and Genetic Operators.

Social networks have greatly expanded in the last ten years the need for sharing multimedia data. However, on open networks such as the Internet, where security is frequently compromised, it is simple for eavesdroppers to approach the actual contents without much difficulty. Researchers have created a variety of encryption methods to strengthen the security of this transmission and make it difficult for eavesdroppers to get genuine data. However, these conventional approaches increase computing costs and communication overhead and do not offer protection against fresh threats. The problems with current algorithms encourage academics to further investigate the subject and suggest new algorithms that are more effective than current methods, that reduce overhead, and which are equipped with features needed by next-generation multimedia networks. In this paper, a genetic operator-based encryption method for multimedia security is proposed. It has been noted that the proposed algorithm produces improved key strength results. The investigations using attacks on data loss, differential assaults, statistical attacks, and brute force attacks show that the encryption technique suggested has improved security performance. It focuses on two techniques, bitplane slicing and followed by block segmentation and scrambling. The suggested method first divides the plaintext picture into several blocks, which is then followed by block swapping done by the genetic operator used to combine the genetic information of two different images to generate new offspring. The key stream is produced from an iterative chaotic map with infinite collapse (ICMIC). Based on a close-loop modulation coupling (CMC) approach, a three-dimensional hyperchaotic ICMIC modulation map is proposed. By using a hybrid model of multidirectional circular permutation with this map, a brand-new colour image encryption algorithm is created. In this approach, a multidirectional circular permutation is used to disrupt the image’s pixel placements, and genetic operations are used to replace the pixel values. According to simulation findings and security research, the technique can fend off brute-force, statistical, differential, known-plaintext, and chosen-plaintext assaults, and has a strong key sensitivity.

A Cyclic Analogue of Stanley's Shuffling Theorem

We introduce the cyclic major index of a cyclic permutation and give a bivariate analogue of the enumerative formula for the cyclic shuffles with a given cyclic descent number due to Adin, Gessel, Reiner and Roichman, which can be viewed as a cyclic analogue of Stanley's shuffling theorem. This gives an answer to a question of Adin, Gessel, Reiner and Roichman, which has been posed by Domagalski, Liang, Minnich, Sagan, Schmidt and Sietsema again.

Geometric phases in neutrino mixing

Neutrinos can acquire both dynamic and geometric phases due to the nontrivial mixing between mass and flavor eigenstates. In this paper, we derive the general expressions for all plausible gauge invariant diagonal and off-diagonal geometric phases in the three- flavor neutrino model using the kinematic approach. We find that diagonal and higher-order off-diagonal geometric phases are sensitive to the mass ordering and the Dirac CP violating phase [Formula: see text]. We show that, third-order off-diagonal geometric phase [Formula: see text] is invariant under any cyclic or non-cyclic permutations of flavor indices when the Dirac CP phase is zero. For nonzero [Formula: see text], we find that [Formula: see text]. We also prove that, only the third-order off-diagonal geometric phase is sensitive to the sign of [Formula: see text]. Further, we explore the effects of matter background using a two-flavor neutrino model and show that the diagonal geometric phase is either [Formula: see text] or [Formula: see text] in the MSW resonance region and takes nontrivial values elsewhere. The transition between zero and [Formula: see text] occurs at the point of complete oscillation inversion called the nodal point, where the diagonal geometric phase is not defined. Also, in two-flavor approximations, two distinct diagonal geometric phases are co-functions with respect to the mixing angle. Finally, in the two-flavor model, we show that the only second-order off-diagonal geometric phase is a topological invariant quantity and is always [Formula: see text].

Evaluation of steric entanglement in coiled-coil and domain-swapped protein interfaces using 3D printed models

Oligomeric protein interfaces involve non-covalent attractive forces plus potential steric entanglement. 70 years ago, Crick proposed a “Knobs in Holes” model for coiled-coil protein interfaces. Subsequently, modifications to this model have been proposed, describing either a “leucine zipper”, “jigsaw puzzle”, or a “peptide Velcro” interface. These principally describe forms of steric entanglement that may enhance oligomer stability. However, such entanglement has not been rigorously evaluated since it is not possible to experimentally eliminate intrinsic non-covalent attractive forces. 3D printing provides a novel means to evaluate steric entanglement of protein interfaces in the absence of attractive forces. Surprisingly, quantitation of the energy required to dissociate various coiled-coil protein interfaces of 3D printed protein models suggests minimal steric entanglement. Conversely, evaluation of domain-swapped interfaces of symmetric protein oligomers, differing by circular permutation, identifies extensive potential steric entanglement. Combined with available experimental data, the results suggest that steric entanglement of a protein interface can contribute to kinetic trapping of both folding and unfolding pathways. Steric entanglement of protein interfaces is therefore postulated to be an undesirable property for naturally evolved and designed protein oligomers.