Circular Permutation Research Articles

Infinite cycles of interacting bosons

Abstract In the first-quantized description of bosonic systems permutation cycles formed by the particles play a fundamental role. In the ideal Bose gas Bose-Enstein condensation (BEC) is signaled by the appearance of infinite cycles. When the particles interact, the two phenomena may not be simultaneous, the existence of infinite cycles is necessary but not sufficient for BEC. We demonstrate that their appearance is always accompanied by a singularity
in the thermodynamic quantities
which in three and four dimensions can be as strong as a one-sided divergence of the isothermal compressibility. Arguments are presented that long-range interactions can give rise to unexpected results, such as the absence of infinite cycles in three dimensions for long-range repulsion or their presence in one and two dimensions if the pair potential has a long attractive tail.

A circularly permuted CasRx platform for efficient, site-specific RNA editing.

Inactive Cas13 orthologs have been fused to a mutant human ADAR2 deaminase domain at the C terminus to enable programmable adenosine-to-inosine (A-to-I) RNA editing in selected transcripts. Although promising, existing RNA-editing tools generally suffer from a trade-off between efficacy and specificity, and off-target editing remains an unsolved problem. Here we describe the development of an optimized RNA-editing platform by rational protein engineering, CasRx-based Programmable Editing of RNA Technology (xPERT). We demonstrate that the topological rearrangement of a CasRx K940L mutant by circular permutation results in a robust scaffold for the tethering of a deaminase domain. We benchmark our tool against the REPAIR system and show that xPERT exhibits strong on-target activity like REPAIRv1 but low off-target editing like REPAIRv2. Our xPERT platform can be used to alter RNA sequence information without risking genome damage, effect temporary cellular changes and customize protein function.

A noncoherent chaotic communication system with multidimensional index modulation based on cyclic permutation grouping

With the surge in technological advancements and application demands, the study of high-data-rate communication systems has become increasingly significant. To enhance the data transmission rate and spectral efficiency of traditional DCSK systems, we have designed a differential chaos shift keying (DCSK) system with multidimensional index modulation based on cyclic permutation grouping (CPG-MIM-DCSK). Multidimensional index resources are efficiently combined by the CPG-MIM-DCSK system using the proposed cyclic permutation grouping method. In the CPG-MIM-DCSK system, both selected and unselected subcarriers, as well as selected and unselected time slots, are distinguished through different groups of cyclic permutations. Permutation, carrier, and time slot resources are fully utilized by this modulation method, thereby enhancing the data rate and spectral efficiency of traditional DCSK systems. The theoretical Bit Error Rate (BER) expressions of the CPG-MIM-DCSK system under Additive White Gaussian Noise (AWGN) and multipath Rayleigh fading channels are derived. The data rate, complexity, and spectral efficiency of the CPG-MIM-DCSK system are covered in further analysis. The simulation results highlight the superiority of the CPG-MIM-DCSK system in terms of data rate and spectral efficiency. Additionally, the proposed CPG-MIM-DCSK system exhibits better BER performance in both Gaussian and multipath Rayleigh fading channels compared to its comparative systems.

Markov chains generating random permutations and set partitions

The Chinese Restaurant Process may be considered to be a Markov chain which generates permutations on n elements proportionally to absorption probabilities θ|π|, θ>0, where |π| is the number of cycles of permutation π. We prove a theorem which provides a way of finding Markov chains, restricted to directed graphs called arborescences, and with given absorption probabilities. We find transition probabilities for the Chinese Restaurant Process arborescence with variable absorption probabilities. The method is applied to an arborescence constructing set partitions, resulting in an analogue of the Chinese Restaurant Process for set partitions. We also apply our method to an arborescence for the Feller Coupling Process. We show how to modify the Chinese Restaurant Process, its set partition analogue, and the Feller Coupling Process to generate derangements and set partitions having no blocks of size one.

Structures Associated with the Borromean Rings’ Complement in the Poincaré Ball

Guided by physical needs, we deal with the rotationally isotropic Poincar´e ball, when considering the complement of Borromean rings embedded in it. We consistently describe the geometry of the complement and realize the fundamental group as isometry subgroup in three dimensions. Applying this realization, we reveal normal stochastization and multifractal behavior within the examined model of directed random walks on the rooted Cayley tree, whose sixbranch graphs are associated with dendritic polymers. According to Penner, we construct the Teichm¨uller space of the decorated ideal octahedral surface related to the quotient space of the fundamental group action. Using the conformality of decoration, we define six moduli and the mapping class group generated by cyclic permutations of the ideal vertices. Intending to quantize the geometric area, we state the connection between the induced geometry and the sine-Gordon model. Due to such a correspondence we obtain the differential two-form in the cotangent bundle of the moduli space.

Forbidden Subposets in the Cycle Poset

AbstractThe cycle poset consists of the intervals of the cyclic permutation of the elements 1, 2, ..., n, ordered by inclusion. Suppose that F is a set of such intervals, none of them is a less than s others. The maximum size of F is determined under this condition. It is also shown that if the largest size of a set in this poset without containing a small subposet P is known, it solves the same problem, up to an additive constant, in the grid poset consisting of the pairs $$(i,j) (1\le i,j\le n)$$ ( i , j ) ( 1 ≤ i , j ≤ n ) and ordered coordinate-wise.

The importance of the location of the N-terminus in successful protein folding in vivo and in vitro

Protein folding in the cell often begins during translation. Many proteins fold more efficiently cotranslationally than when refolding from a denatured state. Changing the vectorial synthesis of the polypeptide chain through circular permutation could impact functional, soluble protein expression and interactions with cellular proteostasis factors. Here, we measure the solubility and function of every possible circular permutant (CP) of HaloTag in Escherichia coli cell lysate using a gel-based assay, and in living E. coli cells via FACS-seq. We find that 78% of HaloTag CPs retain protein function, though a subset of these proteins are also highly aggregation-prone. We examine the function of each CP in E. coli cells lacking the cotranslational chaperone trigger factor and the intracellular protease Lon and find no significant changes in function as a result of modifying the cellular proteostasis network. Finally, we biophysically characterize two topologically interesting CPs in vitro via circular dichroism and hydrogen-deuterium exchange coupled with mass spectrometry to reveal changes in global stability and folding kinetics with circular permutation. For CP33, we identify a change in the refolding intermediate as compared to wild-type (WT) HaloTag. Finally, we show that the strongest predictor of aggregation-prone expression in cells is the introduction of termini within the refolding intermediate. These results, in addition to our finding that termini insertion within the conformationally restrained core is most disruptive to protein function, indicate that successful folding of circular permutants may depend more on changes in folding pathway and termini insertion in flexible regions than on the availability of proteostasis factors.

Study of noise in virtual distillation circuits for quantum error mitigation

Virtual distillation has been proposed as an error mitigation protocol for estimating the expectation values of observables in quantum algorithms. It proceeds by creating a cyclic permutation of M noisy copies of a quantum state using a sequence of controlled-swap gates. If the noise does not shift the dominant eigenvector of the density operator away from the ideal state, then the error in expectation-value estimation can be exponentially reduced with M. In practice, subsequent error mitigation techniques are required to suppress the effect of noise in the cyclic permutation circuit itself, leading to increased experimental complexity. Here, we perform a careful analysis of the effect of uncorrelated, identical noise in the cyclic permutation circuit and find that the estimation of expectation value of observables are robust against dephasing noise. We support the analytical result with numerical simulations and find that 67% of errors are reduced for M=2, with physical dephasing error probabilities as high as 10%. Our results imply that a broad class of quantum algorithms can be implemented with higher accuracy in the near-term with qubit platforms where non-dephasing errors are suppressed, such as superconducting bosonic qubits and Rydberg atoms.

Dynamic Injection and Permutation Coding for Enhanced Data Transmission.

In this paper, we propose a novel approach to enhance spectral efficiency in communication systems by dynamically adjusting the mapping between cyclic permutation coding (CPC) and its injected form. By monitoring channel conditions such as interference levels and impulsive noise strength, the system optimises the coding scheme to maximise data transmission reliability and efficiency. The CPC method employed in this work maps information bits onto non-binary symbols in a cyclic manner, aiming to improve the Hamming distance between mapped symbols. To address challenges such as low data rates inherent in permutation coding, injection techniques are introduced by removing δ column(s) from the CPC codebook. Comparative analyses demonstrate that the proposed dynamic adaptation scheme outperforms conventional permutation coding and injection schemes. Additionally, we present a generalised mathematical expression to describe the relationship between the spectral efficiencies of both coding schemes. This dynamic approach ensures efficient and reliable communication in environments with varying levels of interference and impulsive noise, highlighting its potential applicability to systems like power line communications.

The circular balancing problem

We propose a balancing problem with a minmax objective in a circular setting. This balancing problem involves the arrangement of an even number of items with different weights on a circle while minimizing the maximum total weight of items arranged on any half circle. Due to its generic structure, it may have applications in fair resource allocation schemes. We show the NP-hardness of the problem and develop polynomial-time algorithms when the number of distinct weights is a fixed constant. We propose for the general case a tight 7/6-approximation algorithm and show that it performs better than two existing algorithms designed for an equivalent problem in the literature. The worst-case performance ratio is derived through a linear combination of valid inequalities that are obtained from the problem definition, the properties of the proposed algorithm, and the optimal circular permutation structure. Furthermore, we formulate a more general problem of minimizing the maximum total weight of items on equally divided circular sectors and present its computational complexity and a tight approximation algorithm.

Assembly of the bacterial ribosome with circularly permuted rRNA.

Co-transcriptional assembly is an integral feature of the formation of RNA-protein complexes that mediate translation. For ribosome synthesis, prior studies have indicated that the strict order of transcription of rRNA domains may not be obligatory during bacterial ribosome biogenesis, since a series of circularly permuted rRNAs are viable. In this work, we report the structural insights into assembly of the bacterial ribosome large subunit (LSU) based on cryo-EM density maps of intermediates that accumulate during in vitro ribosome synthesis using a set of circularly permuted (CiPer) rRNAs. The observed ensemble of 23 resolved ribosome large subunit intermediates reveals conserved assembly routes with an underlying hierarchy among cooperative assembly blocks. There are intricate interdependencies for the formation of key structural rRNA helices revealed from the circular permutation of rRNA. While the order of domain synthesis is not obligatory, the order of domain association does appear to proceed with a particular order, likely due to the strong evolutionary pressure on efficient ribosome synthesis. This work reinforces the robustness of the known assembly hierarchy of the bacterial large ribosomal subunit and offers a coherent view of how efficient assembly of CiPer rRNAs can be understood in that context.

Novel mirror-flipped mode permutation technique for long-haul 6-mode transmission

Mode-division multiplexing based on few-mode fibers is expected to overcome capacity limits of current single-mode fiber communications. Mode permutation as an approach for extending transmission reach has attracted great attention. In this paper, we propose a mirror-flipped mode permutation (MFMP) scheme based on newly developed fibers with symmetric differential mode delay (DMD). We demonstrate a 28-GBaud transmission over a 1542-km 6-mode fiber link, which suppresses modal-dispersion impact by 36% and mitigates mode-dependent-loss effect with 3.1-dB Q-factor improvement, compared with the traditional cyclic mode permutation at 1233 km. Then, we experimentally verify that the proposed MFMP works well even using a fiber with a not exactly symmetric group delay distribution. Benefiting from a low DMD in this fiber, we demonstrate a 6-mode PM-QPSK transmission at 25 GBaud with 16-λ channels, extending the transmission distance to 3000 km.

Characterization and structural basis for the brightness of mCLIFY: a novel monomeric and circularly permuted bright yellow fluorescent protein.

We present mCLIFY: a monomeric, bright, yellow, and long-lived fluorescent protein (FP) created by circular permutation of YPet, the brightest yellow FP from Aequorea Victoria for use in cellular and in vitro single molecule studies. mCLIFY retains the enhanced photophysical properties of YPET as a monomer at concentrations ≤ 40 μM. In contrast, we determined that YPet has a dimerization dissociation constant (K D 1-2) of 3.4 μM. Dimerization of YPet can cause homo-FRET, which underlies quantitative errors due to dimerization and homo-FRET. We determined the atomic structure of mCLIFY at 1.57 Å resolution and used its similarity with Venus for guided chromophore-targeted substitution studies to provide insights into its enhanced photophysical properties. The mutation V58L within the chromophore pocket improved quantum yield and extinction coefficient, making mCLIFY ~30% brighter than Venus. The extensive characterization of the photophysical and structural properties of YPet and mCLIFY presented here allowed us to reveal the basis of their long lifetimes and enhanced brightness and the basis of YPet's dimerization.

An advance in the arithmetic of the Lie groups as an alternative to the forms of the Campbell-Baker-Hausdorff-Dynkin theorem

The exponential of an operator or matrix is widely used in quantum theory, but it sometimes can be a challenge to evaluate. For non-commutative operators X and Y, according to the Campbell-Baker-Hausdorff-Dynkin theorem, e X+Y is not equivalent to e X e Y , but is instead given by the well-known infinite series formula. For a Lie algebra of a basis of three operators {X, Y, Z}, such that [X, Y] = κ Z for scalar κ and cyclic permutations, here it is proven that e a X+b Y is equivalent to e p Z e q X e−p Z for scalar p and q. Extensions for e a X+b Y+c Z are also provided. This method is useful for the dynamics of atomic and molecular nuclear and electronic spins in constant and oscillatory transverse magnetic and electric fields.

On the order types of hammocks for domestic string algebras

In the representation-theoretic study of finite dimensional associative algebras over an algebraically closed field, Brenner introduced certain partially ordered sets known as hammocks to encode factorizations of maps between indecomposable finitely generated modules. In the context of domestic string algebras, Schröer introduced a simpler version of hammocks in his doctoral thesis that are bounded discrete linear orders. In this paper, we characterize the class of order types(=order isomorphism classes) of hammock linear orders for domestic string algebras as the bounded discrete ones amongst the class LOfp of finitely presented linear orders–the smallest class of linear orders containing finite linear orders as well as ω, and that is closed under isomorphisms, order reversal, finite order sums and antilexicographic products.In fact, we provide a multi-step algorithm to compute the order type of any closed interval in the hammock, and prove the correctness of this algorithm. A major step of this algorithm is the construction of a variation, which we call the arch bridge quiver, of a finite combinatorial gadget called the bridge quiver introduced by Schröer. He utilised the graph-theoretic properties of the bridge quiver for the computation of some representation-theoretic numerical invariants of domestic string algebras. The vertices of the bridge quiver are (representatives of cyclic permutations of) bands and its arrows are certain band-free strings. There is a natural but ill-behaved partial binary operation, ∘, on a superset of the set of bridges consisting of weak bridges such that bridges are precisely the ∘-irreducibles. We equip an even larger yet finite set of weak arch bridges with another partial binary operation, ∘H, to obtain a finite category. The binary operation ∘H uses isomorphisms between hammocks and explicitly relies on the description of the domestic string algebra as a bound quiver algebra. Each weak arch bridge admits a unique ∘H-factorization into arch bridges, i.e., the ∘H-irreducibles.

An n-dimensional polynomial modulo chaotic map with controllable range of Lyapunov exponents and its application in color image encryption

In chaos-based image encryption, better performance of the chaotic system represents higher security of the encryption algorithm, and the Lyapunov exponent (LE) is an important measure to evaluate the chaotic performance. In this paper, an n-dimensional(n-D) polynomial modulo chaotic map (nD-PMCM) is proposed as a general model, whose range of Lyapunov exponent is controlled by the system parameters and the robust chaotic behavior with the desired complexity can be obtained. And an instantiated 2D-PMCM is constructed by setting the corresponding parameters in nD-PMCM, which exhibits superior chaotic performance compared to other chaotic maps. Then a novel color image encryption scheme based on the 2D-PMCM is also constructed, which mainly uses cross-plane cyclic shift permutation and cross-plane XOR diffusion. Different from the traditional encryption scheme with separate processing of color planes, the cross-plane operation disrupts the correlation between color image planes and has higher encryption efficiency due to the ability to disrupt the pixel position while also changing the pixel value. The experimental simulation results show that this encryption scheme has better security and encryption effects.

Medium-reach six-mode multiplexed transmission with Kramers–Kronig receiver enabled by mode permutation technique

We investigate the transmission distance of a 20-Gbaud 6-mode dual-polarization QPSK transmission system using the Kramers–Kronig(KK) receiver based on the mode permutation (MP) scheme. Compared to the system without the use of MP scheme, the maximum transmission distance is increased from 460 km to 910 km and to over 1000 km via the cyclic mode permutation and the mirror-flipped mode permutation, respectively, at the KP4 forward error correction (FEC) threshold of 2.2×10−4. The required carrier signal power budget of KK receiver is also reduced. The KK receiver can detect the signal of each mode after the transmission of 1000 km with potential advantages of low cost and low power. It is shown as a competitive detection technique in medium-reach metropolitan networks.

Short cycles of random permutations with cycle weights: Point processes approach

We study the asymptotic behavior of short cycles of random permutations with cycle weights. More specifically, on a specially constructed metric space whose elements encode all possible cycles, we consider a point process containing all information on cycles of a given random permutation on {1,…,n}. The main result of the paper is the distributional convergence with respect to the vague topology of the above processes towards a Poisson point process as n→∞ for a wide range of cycle weights. As an application, we give several limit theorems for various statistics of cycles.

Central Limit Theorems via Analytic Combinatorics in Several Variables

The field of analytic combinatorics is dedicated to the creation of effective techniques to study the large-scale behaviour of combinatorial objects. Although classical results in analytic combinatorics are mainly concerned with univariate generating functions, over the last two decades a theory of analytic combinatorics in several variables (ACSV) has been developed to study the asymptotic behaviour of multivariate sequences. In this work we survey ACSV from a probabilistic perspective, illustrating how its most advanced methods provide efficient algorithms to derive limit theorems, and comparing the results to past work deriving limit theorems. Using the results of ACSV, we provide a SageMath package that can automatically compute (and rigorously verify) limit theorems for a large class of combinatorial generating functions. To illustrate the techniques involved, we also establish an explicit local central limit theorems for a family of combinatorial classes whose generating functions are linear in the variables tracking each parameter. Applications covered by this result include the distribution of cycles in certain restricted permutations (proving a limit theorem stated as a conjecture in recent work of Chung et al.), integer compositions, and n-colour compositions with varying restrictions and values tracked. Key to establishing these explicit results in an arbitrary dimension is an interesting symbolic determinant, which we compute by conjecturing and then proving an appropriate LU-factorization. It is our hope that this work provides readers a blueprint to apply the powerful tools of ACSV in their own work, making them more accessible to combinatorialists, probabilists, and those in adjacent fields.

Protein engineering using circular permutation - structure, function, stability, and applications.

Protein engineering is important for creating novel variants from natural proteins, enabling a wide range of applications. Approaches such as rational design and directed evolution are routinely used to make new protein variants. Computational tools like de novo design can introduce new protein folds. Expanding the amino acid repertoire to include unnatural amino acids with non-canonical side chains in vitro by native chemical ligation and in vivo via codon expansion methods broadens sequence and structural possibilities. Circular permutation (CP) is an invaluable approach to redesigning a protein by rearranging the amino acid sequence, where the connectivity of the secondary structural elements is altered without changing the overall structure of the protein. Artificial CP proteins (CPs) are employed in various applications such as biocatalysis, sensing of small molecules by fluorescence, genome editing, ligand-binding protein switches, and optogenetic engineering. Many studies have shown that CP can lead to either reduced or enhanced stability or catalytic efficiency. The effects of CP on a protein's energy landscape cannot be predicted a priori. Thus, it is important to understand how CP can affect the thermodynamic and kinetic stability of a protein. In this review, we discuss the discovery and advancement of techniques to create protein CP, and existing reviews on CP. We delve into the plethora of biological applications for designed CP proteins. We subsequently discuss the experimental and computational reports on the effects of CP on the thermodynamic and kinetic stabilities of proteins of various topologies. An understanding of the various aspects of CP will allow the reader to design robust CP proteins for their specific purposes.