Circular Permutation Research Articles

An Order on Circular Permutations

Motivated by the study of affine Weyl groups, a ranked poset structure is defined on the set of circular permutations in $S_n$ (that is, $n$-cycles). It is isomorphic to the poset of so-called admitted vectors, and to an interval in the affine symmetric group $\tilde S_n$ with the weak order. The poset is a semidistributive lattice, and the rank function, whose range is cubic in $n$, is computed by some special formula involving inversions. We prove also some links with Eulerian numbers, triangulations of an $n$-gon, and Young's lattice.

On the cyclic automorphism of the Cuntz algebra and its fixed-point algebra

We investigate the structure of the fixed-point algebra of On under the action of the cyclic permutation of the generating isometries. We prove that it is ⁎-isomorphic with On, thus generalizing a result of Choi and Latrémolière on O2. As an application of the technique employed, we also describe the fixed-point algebra of O2n under the exchange automorphism.

Generating Cyclic Permutations: Insights to the Traveling Salesman Problem

Some results for the traveling salesman problem (TSP) are known for a prime number of cities. In this paper we extend these results to an odd number of cities. For an odd integer n, we show that there is an algorithm that generates n – 1 cyclic permutations, called tours for the traveling salesman problem, which cover the distance matrix. The algorithm allows construction of a two-dimensional array of all tours for the TSP on an odd number of cities. The array has the following properties: (i) A tour on a vertical line in the array moves the salesman uniquely compared to all other tours on the vertical line. (ii) The sum of the lengths of all tours on a vertical line is equal to the sum of all non-diagonal elements in the distance matrix for the TSP.

Separate sensible and latent cooling: Carnot limits and system taxonomy

Energy efficiency and thermal comfort can be improved by independent control of temperature and humidity. In recent work, an emerging class of cooling systems known as separate sensible and latent cooling (SSLC) systems have demonstrated 14 to 47% energy savings, while simultaneously providing better comfort control than conventional air conditioning. In this work, SSLC is defined as using two or more cooling processes with unequal sensible heat ratio (SHR) to control temperature and humidity independently and dynamically. Fundamental efficiency limits of SSLC systems are developed for the first time, demonstrating that the Carnot efficiency for SSLC is higher than that of conventional systems. It is revealed that SSLC has particularly high potential for moderate outdoor temperatures, high sensible building loads, and drier indoor conditions. In addition, this work proposes a classification scheme for SSLC systems, revealing an enormous number of feasible cycle permutations. This provides a framework for the important research task of developing SSLC systems that will save energy, be practical to construct, and be controllable.

Circular permutation of concanavalin A: why the rarest protein modification in nature came to be.

Circular permutation of concanavalin A: why the rarest protein modification in nature came to be.

Classical communication with indefinite causal order for N completely depolarizing channels

If two identical copies of a completely depolarizing channel are put into a superposition of their possible causal orders, they can transmit non-zero classical information. Here, we study how well we can transmit classical information with $N$ depolarizing channels put in superposition of $M$ causal orders via quantum SWITCH. We calculate Holevo quantity if the superposition uses only cyclic permutations of channels and find that it increases with $M$ and it is independent of $N$. For a qubit it never reaches $1$ if we are increasing $M$. On the other hand, the classical capacity decreases with the dimension $d$ of the message system. Further, for $N=3$ and $N=4$ we studied superposition of all causal orders and uniformly superposed causal orders belonging to different cosets created by cyclic permutation subgroup.

Folding of Circularly Permuted and Split Outer Membrane Protein F via Electrostatic Interactions with Terminal Residues.

Membrane proteins are essential targets in drug design, biosensing, and catalysis. In this study, we explored the folding of engineered outer membrane protein F (OmpF), an abundant and functional β-barrel protein expressed in Gram-negative bacteria. We carried out circular permutation, splitting and self-complementation, and point mutation. The folding efficiency and kinetic analyses demonstrated that the N- and C-terminal residues of OmpF played critical roles in folding via electrostatic interactions with lipid headgroups. Our results indicate that native porins without charged terminal residues may be tightly downregulated to retain the integrity of the outer membrane, and this modification may facilitate the insertion and folding of modified membrane proteins under in vitro and in vivo conditions for various applications.

Operating principles of circular toggle polygons

Decoding the dynamics of cellular decision-making and cell differentiation is a central question in cell and developmental biology. A common network motif involved in many cell-fate decisions is a mutually inhibitory feedback loop between two self-activating ‘master regulators’ A and B, also called as toggle switch. Typically, it can allow for three stable states—(high A, low B), (low A, high B) and (medium A, medium B). A toggle triad—three mutually repressing regulators A, B and C, i.e. three toggle switches arranged circularly (between A and B, between B and C, and between A and C)—can allow for six stable states: three ‘single positive’ and three ‘double positive’ ones. However, the operating principles of larger toggle polygons, i.e. toggle switches arranged circularly to form a polygon, remain unclear. Here, we simulate using both discrete and continuous methods the dynamics of different sized toggle polygons. We observed a pattern in their steady state frequency depending on whether the polygon was an even or odd numbered one. The even-numbered toggle polygons result in two dominant states with consecutive components of the network expressing alternating high and low levels. The odd-numbered toggle polygons, on the other hand, enable more number of states, usually twice the number of components with the states that follow ‘circular permutation’ patterns in their composition. Incorporating self-activations preserved these trends while increasing the frequency of multistability in the corresponding network. Our results offer insights into design principles of circular arrangement of regulatory units involved in cell-fate decision making, and can offer design strategies for synthesizing genetic circuits.

Structural and biochemical analyses of concanavalin A circular permutation by jack bean asparaginyl endopeptidase.

Over 30 years ago, an intriguing posttranslational modification was found responsible for creating concanavalin A (conA), a carbohydrate-binding protein from jack bean (Canavalia ensiformis) seeds and a common carbohydrate chromatography reagent. ConA biosynthesis involves what was then an unprecedented rearrangement in amino-acid sequence, whereby the N-terminal half of the gene-encoded conA precursor (pro-conA) is swapped to become the C-terminal half of conA. Asparaginyl endopeptidase (AEP) was shown to be involved, but its mechanism was not fully elucidated. To understand the structural basis and consequences of circular permutation, we generated recombinant jack bean pro-conA plus jack bean AEP (CeAEP1) and solved crystal structures for each to 2.1 and 2.7 Å, respectively. By reconstituting conA biosynthesis in vitro, we prove CeAEP1 alone can perform both cleavage and cleavage-coupled transpeptidation to form conA. CeAEP1 structural analysis reveals how it is capable of carrying out both reactions. Biophysical assays illustrated that pro-conA is less stable than conA. This observation was explained by fewer intermolecular interactions between subunits in the pro-conA crystal structure and consistent with a difference in the prevalence for tetramerization in solution. These findings elucidate the consequences of circular permutation in the only posttranslation example known to occur in nature.

Circularly permuted LOV2 as a modular photoswitch for optogenetic engineering.

Plant-based photosensors, such as the light-oxygen-voltage sensing domain 2 (LOV2) from oat phototropin 1, can be modularly wired into cell signaling networks to remotely control protein activity and physiological processes. However, the applicability of LOV2 is hampered by the limited choice of available caging surfaces and its preference to accommodate the effector domains downstream of the C-terminal Jα helix. Here, we engineered a set of LOV2 circular permutants (cpLOV2) with additional caging capabilities, thereby expanding the repertoire of genetically encoded photoswitches to accelerate the design of optogenetic devices. We demonstrate the use of cpLOV2-based optogenetic tools to reversibly gate ion channels, antagonize CRISPR-Cas9-mediated genome engineering, control protein subcellular localization, reprogram transcriptional outputs, elicit cell suicide and generate photoactivatable chimeric antigen receptor T cells for inducible tumor cell killing. Our approach is widely applicable for engineering other photoreceptors to meet the growing need of optogenetic tools tailored for biomedical and biotechnological applications.

CirPred, the first structure modeling and linker design system for circularly permuted proteins

BackgroundThis work aims to help develop new protein engineering techniques based on a structural rearrangement phenomenon called circular permutation (CP), equivalent to connecting the native termini of a protein followed by creating new termini at another site. Although CP has been applied in many fields, its implementation is still costly because of inevitable trials and errors.ResultsHere we present CirPred, a structure modeling and termini linker design method for circularly permuted proteins. Compared with state-of-the-art protein structure modeling methods, CirPred is the only one fully capable of both circularly-permuted modeling and traditional co-linear modeling. CirPred performs well when the permutant shares low sequence identity with the native protein and even when the permutant adopts a different conformation from the native protein because of three-dimensional (3D) domain swapping. Linker redesign experiments demonstrated that the linker design algorithm of CirPred achieved subangstrom accuracy.ConclusionsThe CirPred system is capable of (1) predicting the structure of circular permutants, (2) designing termini linkers, (3) performing traditional co-linear protein structure modeling, and (4) identifying the CP-induced occurrence of 3D domain swapping. This method is supposed helpful for broadening the application of CP, and its web server is available at http://10.life.nctu.edu.tw/CirPred/ and http://lo.life.nctu.edu.tw/CirPred/.

Precise asymptotics of longest cycles in random permutations without macroscopic cycles

We consider Ewens random permutations of length n conditioned to have no cycle longer than nβ with 0<β<1 and study the asymptotic behaviour as n→∞. We obtain very precise information on the joint distribution of the lengths of the longest cycles; in particular we prove a functional limit theorem where the cumulative number of long cycles converges to a Poisson process in the suitable scaling. Furthermore, we prove convergence of the total variation distance between joint cycle counts and suitable independent Poisson random variables up to a significantly larger maximal cycle length than previously known. Finally, we remove a superfluous assumption from a central limit theorem for the total number of cycles proved in an earlier paper.

Stability Comparisons between Natural versus Engineered Archaeal Heat‐Shock Proteins

Group II chaperonins are the most abundant proteins in hyperthermophilic archaea of the order Sulfolobales. These “heat shock” proteins (HSPs) consist of three subtypes: HSPα, HSPβ, and HSPγ. Expression of HSPα and HSPβ is upregulated in response to thermal shock and these subunits form nonameric double-ring structures (18-mers) referred to as HSP complexes. HSPα and HSPβ subunits form complexes as homooligomers or heterooligomers at different combination ratios. The natural role of these chaperonin complexes is to protect other cellular proteins (e.g., enzymes) from damage during heat shock. In an effort to develop a thermotolerant enzyme binding platform for industrial and/or agricultural applications, an engineered fusion protein was developed by combining a circular permutant of HSPβ from Sulfolobus shibatae with Type 1 cohesin from the cellulolytic bacterium, Clostridium thermocellum with the idea of developing a thermotolerant “artificial” cellulosome. In this study, the relative thermal stability and acido-stability of the natural and engineered HSP subunits was assessed using biophysical methods. Intrinsic tryptophan fluorescence, differential scanning calorimetry (DSC), and far UV circular dichroism spectroscopy suggest that the native structure of HSPα and HSPβ is not significantly perturbed at neutral pH over a temperature range of 25-90⁰C. The engineered HSPβ-coh fusion protein is slightly less thermally stable and is prone to aggregation past 85⁰C when compared to the natural HSPs. All subunits show shifts in structural conformation under low pH conditions (pH 2 and 3). The results of limited trypsin digestion show that the backbone structure of HSPβ is the least flexible followed closely by HSPα. HSPβ-coh exhibited the highest backbone flexibility. All subunits appear to be more compact at neutral pH and higher temperatures (75⁰C, 85⁰C, and 90⁰C) as lower relative fluorescence intensity (RFI) upon ANS binding is observed under these conditions as compared to the RFI of the subunits at pH2 and 25⁰C. The observations regarding the increased flexibility and lowered thermal stability of HSPβ-coh is due to the added cohesin moiety. This is consistent with the conclusions drawn from intrinsic fluorescence and ANS binding data. Perturbations in secondary and tertiary structures are observed for all subunits at pH<3. However, the exact nature of these structural changes remains unknown. At pH<3, it appears that HSP structures are less compact while still avoiding aggregation. Overall, the HSP subunits exhibit high thermal stability but undergo structural changes in low pH conditions.

On the constructions of n-cycle permutations

Any permutation polynomial is an n-cycle permutation. When n is a specific small positive integer, one can obtain efficient permutations, such as involutions, triple-cycle permutations and quadruple-cycle permutations. These permutations have important applications in cryptography and coding theory. Inspired by the AGW Criterion, we propose criteria for n-cycle permutations, which mainly are of the form xrh(xs). We then propose unified constructing methods including recursive ways and a cyclotomic way for n-cycle permutations of such form. We demonstrate our approaches by constructing three classes of explicit triple-cycle permutations with high index and two classes of n-cycle permutations with low index, many of which are new both at levels of permutation property and cycle property.

PERIODIC AUTOMORPHISMS, COMPATIBLE POISSON BRACKETS, AND GAUDIN SUBALGEBRAS

Let \U0001d52e be a finite-dimensional Lie algebra. The symmetric algebra (\U0001d52e) is equipped with the standard Lie–Poisson bracket. In this paper, we elaborate on a surprising observation that one naturally associates the second compatible Poisson bracket on (\U0001d52e) to any finite order automorphism ϑ of \U0001d52e. We study related Poisson-commutative subalgebras (\U0001d52e; ϑ) of \U0001d4ae(\U0001d52e) and associated Lie algebra contractions of \U0001d52e. To obtain substantial results, we have to assume that \U0001d52e = \U0001d524 is semisimple. Then we can use Vinberg’s theory of ϑ-groups and the machinery of Invariant Theory.If \U0001d524 = \U0001d525⊕⋯⊕\U0001d525 (sum of k copies), where \U0001d525 is simple, and ϑ is the cyclic permutation, then we prove that the corresponding Poisson-commutative subalgebra (\U0001d52e; ϑ) is polynomial and maximal. Furthermore, we quantise this (\U0001d524; ϑ) using a Gaudin subalgebra in the enveloping algebra \U0001d4b0(\U0001d524).

CRISPR start-up Scribe raises $100 million

Scribe Therapeutics , a University of California, Berkeley, spin-off dedicated to designing new-and-improved CRISPR gene-editing systems, has raised $100 million in series B financing. Cofounder and CEO Benjamin Oakes says the money will help Scribe expand its 30-person team, build up its enzyme engineering platform, and accelerate development of gene-editing therapies for neurodegenerative diseases. As a PhD student in Jennifer Doudna’s and David Savage’s labs at Berkeley, Oakes designed new versions of Cas9, the DNA-cutting enzyme used in first-generation CRISPR gene editing. He created a version of Cas9 whose DNA cutting is allosterically controlled by small-molecule binding. And he used a protein-engineering trick called circular permutation to create a version of Cas9 whose DNA-cutting ability is kept inactive until protease enzymes cleave a chemical lock that lets Cas9 begin cutting. Doudna, Savage, Oakes, and Brett Stahl, a postdoc in Doudna’s lab, founded Scribe in 2018 to apply similar protein-engineering techniques

Error correction by Reed–Solomon codes using its automorphisms

The article explores the syndrome invariants of АГ-group of automorphisms of Reed–Solomon codes (RS-codes) that are a joint group of affine and cyclic permutations. The found real invariants are a set of norms of N Г-orbits that make up one or another АГ-orbit. The norms of Г-orbits are vectors with 2 1 Cδ− coordinates from the Galois field, that are determined by all kinds of pairs of components of the error syndromes. In this form, the invariants of the АГ-orbits were cumbersome and difficult to use. Therefore, their replacement by conditional partial invariants is proposed. These quasi-invariants are called norm-projections. Norm-projection uniquely identifies its АГ-orbit and therefore serves as an adequate way for formulating the error correction method by RS-codes based on АГ-orbits. The power of the АГ-orbits is estimated by the value of N2, equal to the square of the length of the RS-code. The search for error vectors in transmitted messages by a new method is reduced to parsing the АГ‑orbits, but actually their norm-projections, with the subsequent search for these errors within a particular АГ-orbit. Therefore, the proposed method works almost N2 times faster than traditional syndrome methods, operating on the basic of the “syndrome – error” principle, that boils down to parsing the entire set of error vectors until a specific vector is found.

DEVELOPMENT OF A CONTROL ALGORITHM DEVICE LOAD BALANCING DEVICE

The work is devoted to improving the control algorithm of the load balancing device for 0.4 kV electrical mains that are in pilot operation. The action of the device is aimed at balancing the load at the nodal point of the electrical network, where several 3-phase outgoing lines are connected, by changing the connection diagram of the phases of one of them. The device is capable of realizing 6 combinations obtained by circular permutation and changing the phase rotation. The classification of analogs of the device according to the type of connection to the network is proposed. The technique of collection and processing of experimental data on the operation of the device is presented. Analysis of primary data shows that among the consumers there are powerful energy receivers with automatic two-position control. As a criterion for the expediency of one or another method of connecting the phases of the outgoing line to the phases of the source, the coefficient of non-uniformity of conductivities is taken. Strict observance of the condition of the minimum coefficient of unevenness is impossible in practice with the existing nature of the load due to the need for too frequent operation of the device. This and other limitations given in the work are the reason for the systematization of requirements for the device control algorithm, one of which is the need to filter the initial data before executing the main algorithm. The paper provides a rationale for the use of a digital IIR low-pass filter. Based on the frequency analysis of the curves of the conductivity unevenness coefficients, the filter design was carried out and the digital transfer function coefficients were calculated. A difference equation is determined that is suitable for implementation in the program for the control microcontroller of the device. The developed filter is universal and suitable for use in the control algorithm of a load balancing device of any power and for any type of load. The results of modeling the proposed filter are presented using the example of experimental data. The prospect of the research is the collection of statistics on various types of consumers, analysis of practical situations, in order to further refine the algorithm.

Polynomial chaos expansion for permutation and cyclic permutation invariant systems: Application to mistuned bladed disks

This article deals with the stochastic modeling of industrial mistuned bladed disks. More specifically, a cost-efficient implementation of polynomial chaos expansion is proposed, it is dedicated to mathematical systems exhibiting permutation invariance or cyclic permutation invariance of their random variables. Significant gains are obtained in comparison to the classical implementation of polynomial chaos expansion since potentially costly evaluations of the investigated deterministic system are only required over a small subspace of the random space. The proposed methodology is detailed and validated on analytical test cases before it is applied to two mistuned bladed disks models. First, computations carried out with a simplified bladed disk model allow an in-depth comparison between Monte Carlo simulations, previously published results with a standard polynomial chaos expansion and the proposed methodology. The latter is then employed to assess the influence of mistuning on the eigenfrequencies and amplification magnification of an industrial compressor stage. It is evidenced that in comparison to the standard polynomial chaos expansion, the proposed methodology yields computational gains of the same order of magnitude as the ones obtained going from Monte Carlo simulations to polynomial chaos expansion. Alternately, the proposed methodology may be employed to significantly increase the accuracy of the standard polynomial chaos expansion while featuring an identical computational cost.

The symmetric designer protein Pizza as a scaffold for metal coordination.

Symmetric proteins are currently of interest as they allow creation of larger assemblies and facilitate the incorporation of metal ions in the larger complexes. Recently this was demonstrated by the biomineralization of the cadmium-chloride nanocrystal via the Pizza designer protein. However, the mechanism behind this formation remained unclear. Here, we set out to investigate the mechanism driving the formation of this nanocrystal via truncation, mutation, and circular permutations. In addition, the interaction of other biologically relevant metal ions with these symmetric proteins to form larger symmetric complexes was also studied. The formation of the initial nanocrystal is shown to originate from steric strain, where His 58 induces a different rotameric conformation on His 73, thereby distorting an otherwise perfect planar ring of alternating cadmium and chlorine ions, resulting in the smallest nanocrystal. Similar highly symmetric complexes were also observed for the other biological relevant metal ions. However, the flexibility of the coordinating histidine residues allows each metal ion to adopt its preferred geometry leading to either monomeric or dimeric β-propeller units, where the metal ions are located at the interface between both propeller units. These results demonstrate that symmetric proteins are not only interesting to generate larger assemblies, but are also the perfect scaffold to create more complex metal based assemblies. Such metal protein assemblies may then find applications in bionanotechnology or biocatalysis.