Circular Permutation Research Articles

Synchronization of a genetic oscillator with the cell division cycle

Genetic circuits that control specific cellular functions are never fully insulated against influences of other parts of the cell. For example, they are subject to periodic modulation by the cell cycle through volume growth and gene doubling. To investigate possible effects of the cell cycle on oscillatory gene circuits dynamics, we modelled a simple synthetic genetic oscillator, the repressilator, and studied hallmarks of the resulting nonlinear dynamics. We found that the repressilator coupled to the cell cycle shows typical quasiperiodic motion with discrete Fourier spectra and windows in parameter space with synchronization of the two oscillators, with a devil’s stair case indicating the Arnold tongues of synchronization. In the case of identical parameters for the three genes of the repressilator and simultaneous gene duplication, we identify two classes of synchronization windows, symmetric and asymmetric, depending on whether the trajectories satisfy a discrete three-fold rotation symmetry, corresponding to cyclic permutation of the three genes. Unexpectedly changing the gene doubling time revealed that the width of the Arnold tongues is connected to that three-fold symmetry of the synchronization trajectories: non-simultaneous gene duplication increases the width of asymmetric synchronization regions, for some of them by an order of magnitude. By contrast, there is only a small or even a negative effect on the window size for symmetric synchronization. This observation points to a control mechanism of synchronization via the location of the genes on the chromosome.

Variance of the number of cycles of random A-permutation

Abstract We consider random permutations having uniform distribution on the set of all permutations of the n-element set with lengths of cycles belonging to a fixed set A (so-called A-permutations). For some class of sets A the asymptotic formula for the variance of the number of cycles of such permutations is obtained.

Engineering and exploiting synthetic allostery of NanoLuc luciferase

Allostery enables proteins to interconvert different biochemical signals and form complex metabolic and signaling networks. We hypothesize that circular permutation of proteins increases the probability of functional coupling of new N- and C- termini with the protein’s active center through increased local structural disorder. To test this we construct a synthetically allosteric version of circular permutated NanoLuc luciferase that can be activated through ligand-induced intramolecular non-covalent cyclisation. This switch module is tolerant of the structure of binding domains and their ligands, and can be used to create biosensors of proteins and small molecules. The developed biosensors covers a range of emission wavelengths and displays sensitivity as low as 50pM and dynamic range as high as 16-fold and could quantify their cognate ligand in human fluids. We apply hydrogen exchange kinetic mass spectroscopy to analyze time resolved structural changes in the developed biosensors and observe ligand-mediated folding of newly created termini.

Changing the vectorial synthesis of HaloTag through circular permutation modulates aggregation propensity in cells

Though in vitro studies have produced detailed 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, which builds polypeptide chains at a rate of around 10 residues per second in E. coli cells. This is much slower than the average folding rate of most small proteins, giving the nascent chain time to sample various conformations during translation.

Node Multiway Cut and Subset Feedback Vertex Set on Graphs of Bounded Mim-Width

The two weighted graph problems Node Multiway Cut (NMC) and Subset Feedback Vertex Set (SFVS) both ask for a vertex set of minimum total weight, that for NMC disconnects a given set of terminals, and for SFVS intersects all cycles containing a vertex of a given set. We design a meta-algorithm that allows to solve both problems in time 2^{O(rw^3)}cdot n^{4}, 2^{O(q^2log (q))}cdot n^{4}, and n^{O(k^2)} where rw is the rank-width, q the {mathbb {Q}}-rank-width, and k the mim-width of a given decomposition. This answers in the affirmative an open question raised by Jaffke et al. (Algorithmica 82(1):118–145, 2020) concerning an XP algorithm for SFVS parameterized by mim-width. By a unified algorithm, this solves both problems in polynomial-time on the following graph classes: Interval, Permutation, and Bi-Interval graphs, Circular Arc and Circular Permutation graphs, Convex graphs, k-Polygon, Dilworth-k and Co-k-Degenerate graphs for fixed k; and also on Leaf Power graphs if a leaf root is given as input, on H-Graphs for fixed H if an H-representation is given as input, and on arbitrary powers of graphs in all the above classes. Prior to our results, only SFVS was known to be tractable restricted only on Interval and Permutation graphs, whereas all other results are new.

Cyclic pattern containment and avoidance

The study of pattern containment and avoidance for linear permutations is a well-established area of enumerative combinatorics. A cyclic permutation is the set of all rotations of a linear permutation. Vella and Callan independently initiated the study of permutation avoidance in cyclic permutations and characterized the avoidance classes for all single permutations of length 4. We continue this work. In particular, we derive results about avoidance of multiple patterns of length 4, and we determine generating functions for the cyclic descent statistic on these classes. We also consider consecutive pattern containment, and relate the generating functions for the number of occurrences of certain linear and cyclic patterns. Finally, we end with various open questions and avenues for future research.

A general class of algebraic inequalities for generating new knowledge and optimising the design of systems and processes

A special class of general inequalities has been identified that provides the opportunity for generating new knowledge that can be used for optimising systems and processes in diverse areas of science and technology. It is demonstrated that inequalities belonging to this class can always be interpreted meaningfully if the variables and separate terms of the inequalities represent additive quantities. The meaningful interpretation of a new algebraic inequality based on the proposed general class of inequalities led to developing a light-weight design for a supporting structure based on cantilever beams, reducing the maximum force upon impact, generating new knowledge about the deflection of elastic elements connected in parallel and series and optimising the allocation of resources to maximise expected benefit. The interpretation of the new inequality yielded that the deflection of elastic elements connected in parallel is at least \(n^{2}\) times smaller than the deflection of the same elastic elements connected in series, irrespective of the individual stiffness values of the elastic elements. The interpretation of another algebraic inequality from the proposed general class led to a method for decreasing the stiffness of a mechanical assembly by cyclic permutation of the elastic elements building the assembly. The analysis showed that a decrease of stiffness exists only if asymmetry of the stiffness values in the connected elements is present.

Engineering a Fluorescent Protein Color Switch Using Entropy-Driven β-Strand Exchange.

Protein conformational switches are widely used in biosensing. They are often composed of an input domain (which binds a target ligand) fused to an output domain (which generates an optical readout). A central challenge in designing such switches is to develop mechanisms for coupling the input and output signals via conformational changes. Here, we create a biosensor in which binding-induced folding of the input domain drives a conformational shift in the output domain that results in a sixfold green-to-yellow ratiometric fluorescence change in vitro and a 35-fold intensiometric fluorescence increase in cultured cells. The input domain consists of circularly permuted FK506 binding protein (cpFKBP) that folds upon binding its target ligand (FK506 or rapamycin). cpFKBP folding induces the output domain, an engineered green fluorescent protein (GFP) variant, to replace one of its β-strands (containing T203 and specifying green fluorescence) with a duplicate β-strand (containing Y203 and specifying yellow fluorescence) in an intramolecular exchange reaction. This mechanism employs the loop-closure entropy principle, embodied by the folding of the partially disordered cpFKBP domain, to couple ligand binding to the GFP color shift. This study highlights the high-energy barriers present in GFP folding which cause β-strand exchange to be slow and are also likely responsible for the shift from the β-strand exchange mechanism in vitro to ligand-induced chromophore maturation in cells. The proof-of-concept design has the advantages of full genetic encodability and potential for modularity. The latter attribute is enabled by the natural coupling of binding and folding and circular permutation of the input domain, which theoretically allows different binding domains to be compatible for insertion into the GFP surface loop.

Application of Restriction Free (RF) Cloning in Circular Permutation.

The restriction free (RF) cloning has emerged as one of the highly efficient techniques in the area of genetic engineering. RF cloning has wide range of applications in plasmid DNA manipulation including cloning of a single gene, simultaneous assembly of multiple DNA fragments, and mutagenesis from single to multiple simultaneous alterations of a target DNA. Recently, we have developed a new technique of circular permutation using RF cloning. Circular permutation is widely used to investigate the mechanisms of protein folding and function. Previously, restriction enzyme based cloning was used to introduce circular permutation. Our RF cloning method made the protocol faster and more cost-effective. In this chapter, we describe a step-by-step protocol for generating circular permutants using RF methodology.

MixCycle: Unsupervised Speech Separation via Cyclic Mixture Permutation Invariant Training

We introduce two unsupervised source separation methods, which involve self-supervised training from single-channel two-source speech mixtures. Our first method, mixture permutation invariant training (MixPIT), enables learning a neural network model which separates the underlying sources via a challenging proxy task without supervision from the reference sources. Our second method, cyclic mixture permutation invariant training (MixCycle), uses MixPIT as a building block in a cyclic fashion for continuous learning. MixCycle gradually converts the problem from separating mixtures of mixtures into separating single mixtures. We compare our methods to common supervised and unsupervised baselines: permutation invariant training with dynamic mixing (PIT-DM) and mixture invariant training (MixIT). We show that MixCycle outperforms MixIT and reaches a performance level very close to the supervised baseline (PIT-DM) while circumventing the over-separation issue of MixIT. Also, we propose a self-evaluation technique inspired by MixCycle that estimates model performance without utilizing any reference sources. We show that it yields results consistent with an evaluation on reference sources (LibriMix) and also with an informal listening test conducted on a real-life mixtures dataset (REAL-M).

Obtaining protein foldability information from computational models of AlphaFold2 and RoseTTAFold

The recent breakthrough from AlphaFold2 and RoseTTAFold set a profound milestone for solving the protein folding problem, but they were not explicitly trained to predict protein foldability, i.e., if a protein can really fold into the predicted 3D structure. We wondered if the computational models from AlphaFold2 and RoseTTAFold might carry protein foldability information. Therefore, we predicted the structural models of 159 circular permutants and 158 alanine insertion mutants of the 159-residue dihydrofolate reductase. Our data showed that although AlphaFold2 and RoseTTAFold cannot directly identify unfoldable proteins, the RMSD values of computational models are correlated with protein foldability, with higher RMSD values indicating lower protein foldability. Furthermore, this correlation is independent of secondary structures, and the RMSD values of computational models are quantitatively correlated with protein foldability but not protein functions. Additionally, using a dataset of 129 de novo designed proteins, we showed that inter-model RMSD values between AlphaFold2 models and RoseTTAFold models are a good indicator of protein foldability. At last, we showed that inter-model RMSD values are also useful for evaluating protein solubility by modeling 1664 natural proteins. Our work could be of great value to the design of novel proteins and the prediction of protein foldability.

Computing fusion rules for spherical G-extensions of fusion categories

A G-extension of a fusion category \({\mathcal {C}}\) yields a categorical action of G on the center \(Z({{\mathcal {C}}})\). If the extension admits a spherical structure, we provide a method for recovering its fusion rules in terms of the action. We then apply this to find closed formulas for the fusion rules of extensions of some group theoretical categories and of cyclic permutation crossed extensions of modular categories.

Likelihood landscape and maximum likelihood estimation for the discrete orbit recovery model

AbstractWe study the nonconvex optimization landscape for maximum likelihood estimation in the discrete orbit recovery model with Gaussian noise. This is a statistical model motivated by applications in molecular microscopy and image processing, where each measurement of an unknown object is subject to an independent random rotation from a known rotational group. Equivalently, it is a Gaussian mixture model where the mixture centers belong to a group orbit.We show that fundamental properties of the likelihood landscape depend on the signal‐to‐noise ratio and the group structure. At low noise, this landscape is “benign” for any discrete group, possessing no spurious local optima and only strict saddle points. At high noise, this landscape may develop spurious local optima, depending on the specific group. We discuss several positive and negative examples, and provide a general condition that ensures a globally benign landscape at high noise. For cyclic permutations of coordinates on (multireference alignment), there may be spurious local optima when , and we establish a correspondence between these local optima and those of a surrogate function of the phase variables in the Fourier domain.We show that the Fisher information matrix transitions from resembling that of a single Gaussian distribution in low noise to having a graded eigenvalue structure in high noise, which is determined by the graded algebra of invariant polynomials under the group action. In a local neighborhood of the true object, where the neighborhood size is independent of the signal‐to‐noise ratio, the landscape is strongly convex in a reparametrized system of variables given by a transcendence basis of this polynomial algebra. We discuss implications for optimization algorithms, including slow convergence of expectation‐maximization, and possible advantages of momentum‐based acceleration and variable reparametrization for first‐ and second‐order descent methods. © 2021 Wiley Periodicals LLC.

Geometry of distributive multiparty entanglement in 4 − qubit hypergraph states

A detailed investigation of the multiparty entanglement present in the 4 − qubit quantum hypergraph states is presented, following a measurement-based geometrical approach. Considering a classification of the 4 − party quantum system represented by a mathematical hypergraph based on the connections between its vertices, the genuine 4 − party entanglement present in each bi-partition of the states have been measured. A strong correlation between the connectivity of the vertices of the hypergraphs and the genuine 4 − party entanglement has been found. The equivalence of the genuine 4 − party entanglement present in each bi-partition is shown considering similar connectivity of the vertices. This explicates the cyclic permutation symmetry of the multiparty entanglement present in the 4 − qubit hypergraph states. Physically, one may expect the quantum systems with superposition of many states to behave in this symmetric manner while mapped into a network-type picture, which the authors have quantified, as well as classified in this work.

Circularly Permuted Far-Red Fluorescent Proteins.

The color palette of genetically encoded fluorescent protein indicators (GEFPIs) has expanded rapidly in recent years. GEFPIs with excitation and emission within the “optical window” above 600 nm are expected to be superior in many aspects, such as enhanced tissue penetration, reduced autofluorescence and scattering, and lower phototoxicity. Circular permutation of fluorescent proteins (FPs) is often the first step in the process of developing single-FP-based GEFPIs. This study explored the tolerance of two far-red FPs, mMaroon1 and mCarmine, towards circular permutation. Several initial constructs were built according to previously reported circularly permuted topologies for other FP analogs. Mutagenesis was then performed on these constructs and screened for fluorescent variants. As a result, five circularly permuted far-red FPs (cpFrFPs) with excitation and emission maxima longer than 600 nm were identified. Some displayed appreciable brightness and efficient chromophore maturation. These cpFrFPs variants could be intriguing starting points to further engineer far-red GEFPIs for in vivo tissue imaging.

A Partially-Sorted Concentric Layout for Efficient Label Localization in Augmented Reality.

A common approach for Augmented Reality labeling is to display the label text on a flag planted into the real world element at a 3D anchor point. When there are more than just a few labels, the efficiency of the interface decreases as the user has to search for a given label sequentially. The search can be accelerated by sorting the labels alphabetically, but sorting all labels results in long and intersecting leader lines from the anchor points to the labels. This paper proposes a partially-sorted concentric label layout that leverages the search efficiency of sorting while avoiding the label display problems of long or intersecting leader lines. The labels are partitioned into a small number of sorted sequences displayed on circles of increasing radii. Since the labels on a circle are sorted, the user can quickly search each circle. A tight upper bound derived from circular permutation theory limits the number of circles and thereby the complexity of the label layout. For example, 12 labels require at most three circles. When the application allows it, the labels are presorted to further reduce the number of circles in the layout. The layout was tested in a user study where it significantly reduced the label searching time compared to a conventional single-circle layout.

TwinCons: Conservation score for uncovering deep sequence similarity and divergence.

We have developed the program TwinCons, to detect noisy signals of deep ancestry of proteins or nucleic acids. As input, the program uses a composite alignment containing pre-defined groups, and mathematically determines a 'cost' of transforming one group to the other at each position of the alignment. The output distinguishes conserved, variable and signature positions. A signature is conserved within groups but differs between groups. The method automatically detects continuous characteristic stretches (segments) within alignments. TwinCons provides a convenient representation of conserved, variable and signature positions as a single score, enabling the structural mapping and visualization of these characteristics. Structure is more conserved than sequence. TwinCons highlights alternative sequences of conserved structures. Using TwinCons, we detected highly similar segments between proteins from the translation and transcription systems. TwinCons detects conserved residues within regions of high functional importance for the ribosomal RNA (rRNA) and demonstrates that signatures are not confined to specific regions but are distributed across the rRNA structure. The ability to evaluate both nucleic acid and protein alignments allows TwinCons to be used in combined sequence and structural analysis of signatures and conservation in rRNA and in ribosomal proteins (rProteins). TwinCons detects a strong sequence conservation signal between bacterial and archaeal rProteins related by circular permutation. This conserved sequence is structurally colocalized with conserved rRNA, indicated by TwinCons scores of rRNA alignments of bacterial and archaeal groups. This combined analysis revealed deep co-evolution of rRNA and rProtein buried within the deepest branching points in the tree of life.

Structure and stability of the designer protein WRAP-T and its permutants

beta -Propeller proteins are common natural disc-like pseudo-symmetric proteins that contain multiple repeats (‘blades’) each consisting of a 4-stranded anti-parallel beta -sheet. So far, 4- to 12-bladed beta -propellers have been discovered in nature showing large functional and sequential variation. Using computational design approaches, we created perfectly symmetric beta -propellers out of natural pseudo-symmetric templates. These proteins are useful tools to study protein evolution of this very diverse fold. While the 7-bladed architecture is the most common, no symmetric 7-bladed monomer has been created and characterized so far. Here we describe such a engineered protein, based on a highly symmetric natural template, and test the effects of circular permutation on its stability. Geometrical analysis of this protein and other artificial symmetrical proteins reveals no systematic constraint that could be used to help in engineering of this fold, and suggests sequence constraints unique to each beta -propeller sub-family.

Design of a Marine Sediments Resistivity Measurement System Based on a Circular Permutation Electrode

Marine sediments are rich in mineral resources, organic resources, and microbial life. The study of marine sediments is of great significance for the development and utilization of marine resources and understanding the life process. Resistivity is the overall characteristic of the electrical conductivity of marine sediments. Measuring the resistivity of marine sediments is helpful to ascertain the marine geological structure, study the distribution of marine mineral resources, and evaluate the marine soil environment. Measuring the resistivity of marine sediments is of great significance to promote marine exploration. At present, the resistivity measurement device on the market can be directly used to measure soil and water on land, but if used to measure marine sediments, it will be disturbed by seawater temperature and pressure, resulting in large errors. In this paper, a high-precision pressure-maintaining transfer system of marine sediment resistivity measurement instrument based on circular permutation electrode is designed, which can measure the resistivity of marine sediment samples after pressure-maintaining transfer. At the same time, a new type of circular permutation electrode measurement method is proposed, which makes the resistivity value more accurate, reduces the length of the probe appropriately, and saves the cost. By measuring the resistivity of marine sediments, the type of sediments can be inverted, which provides a way of thinking about the promotion of the research and development and utilization of marine resources.

A Diagrammatic Approach to Information Transmission in Generalised Switches

The quantum switch is a higher-order operation that takes as an input two quantum processes and combines them in a coherent superposition of two alternative orders. Here we provide an approach to the quantum switch based on the methods of categorical quantum mechanics. Specifically, we represent the quantum switch as a sum of diagrams in the category of finite dimensional Hilbert spaces, or, equivalently, as a sum of diagrams built from Selinger's CPM construction. The sum-of-diagrams picture provides intuition for the activation of classical capacity of completely depolarising channels (CDPCs) and allows for generalisation to N-channel switches. We demonstrate the use of these partially diagrammatic methods by deriving a permutation condition for computing the output of any N-channel switch of CDPCs, we then use that condition to prove that amongst all possible terms, the interference terms associated to cyclic permutations of the N channels are the information-transmitting terms with maximum normalisation